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Nuclear structure corrections in muonic atoms with statistical uncertainty quantification Hernandez, Oscar Javier 2019

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Nuclear structure corrections in muonic atoms with statistical uncertainty quantificationbyOscar Javier HernandezMSc., The University of Manitoba, 2015BSc., University of Manitoba, 2013A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OFDoctor of PhilosophyinTHE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES(Physics)The University of British Columbia(Vancouver)December 2019©c Oscar Javier Hernandez, 2019The following individuals certify that they have read, and recommend to the Faculty of Graduate and Postdoctoral Studies for acceptance, the dissertation entitled: Nuclear structure corrections in muonic atoms with statistical uncertainty quantification submitted by Oscar Javier Hernandez in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics Examining Committee: Sonia Bacca, Physics Co-supervisor Reiner Kruecken, Physics Co-supervisor  Alireza Nojeh, ​Electrical & Computer Engineering Supervisory Committee Member Mark Thachuk, Chemistry University Examiner Mona Berciu, Physics University Examiner Additional Supervisory Committee Members: Jeremy Heyl, Physics Supervisory Committee Member Jens Dilling, Physics Supervisory Committee Member iiAbstractThe discovery of the proton and deuteron radius puzzles from Lamb shift measure-ments of muonic atoms has initiated experimental efforts to probe heavier muonicsystems and casts doubt on earlier analysis based on ordinary atoms. For muonicatoms, the large muon mass results in a Bohr radius about 200 times smaller withrespect to their electronic counterparts, making them sensitive to nuclear structureeffects. These effects dominate the uncertainty budget of the experimental analy-sis and diminish the attainable accuracy of charge radii determinations from Lambshift spectroscopy.This dissertation investigates the precision of nuclear structure corrections rel-evant to the Lamb and hyperfine splitting in muonic deuterium to support ongoingexperiments and shed light on the puzzles. Using state-of-the-art nuclear models,multivariate regression analysis and Bayesian techniques, we estimate the contri-bution of all relevant uncertainties for nuclear structure corrections in muonic deu-terium and demonstrate that nuclear theory errors are well constrained and do notaccount for the deuteron radius puzzle. This uncertainty analysis was carried outusing the “η-expansion” method that has also been applied to A ≥ 2 nuclei. Thismethod relies on the expansion of a dimensionless parameter η , with η < 1, upto second order. To estimate the truncation uncertainty of this method and to im-prove future calculations of nuclear structure effects in other nuclei, we introducean improved formalism based on a multipole expansion of the longitudinal andtransverse response functions that contains higher order terms in η , and general-ize the method to account for the cancellation of elastic terms such as the Friarmoment (or third Zemach moment). This method is then adapted to address thenuclear structure corrections to the hyperfine splitting.iiiThe hyperfine splitting is dominated by magnetic dipole transitions that aresensitive to the effects of two-body currents. Therefore, we develop the formal-ism of the next-to-leading-order two-body magnetic moment contributions to themagnetic dipole. These operators are applied to A = 2,3 and A = 6 systems inanticipation of the upcoming experiments in µ6,7Li2+ ions. We find that two-bodycontributions are important to reach agreement with experiment.ivLay SummaryMost of the visible matter in the Universe from the air that we breathe, to the starsthat light up galaxies, consist of atoms. Atoms are systems of orbiting electronsbound to a nucleus by light. In analogy with how a prism splits white light into arainbow, light passing through an atom decomposes it into a unique spectrum ofcolors revealing their internal nature. While the nucleus is 100,000 times smallerthan the atom, certain colors of this spectrum contain its faint imprints. Using state-of-the-art nuclear models we study these signatures from exotic atoms in whichthe nucleus is orbited by a muon instead of an electron and carry out uncertaintyquantification with modern statistical methods. This work is a step forward toilluminate our understanding of the interplay between the nucleus and light.vPrefaceThe work presented in this thesis is based on both published and unpublished ma-terial of the author Oscar Javier Hernandez under the directives of his researchsupervisor, Dr. Sonia Bacca. The published parts of this work have appeared in thefollowing references:• [1] N. Nevo Dinur, O. J. Hernandez, S. Bacca, N. Barnea, C. Ji, S. Pas-tore, M. Piarulli and R. B. Wiringa, “Zemach moments and radii of 2,3H and3,4He”, Phys. Rev. C, 99, 034004, 2019.This work was motivated by the authors, N.N.D., S.P. and S.B. In this workI contributed towards formalizing the theory of the qr-space method andderiving new formulas for the r-space method that were needed to connectthe two approaches. I was responsible for implementing all electromagneticmoment formulas for 2H and conducting the benchmark with calculationsby R.B.W. in Tables 6.2 and 6.3. The other authors were responsible for thetext. The derivations of select formulas that appear in this contribution areprovided in Sections 4.1.2 and 4.1.3.• [2] C. Ji, S. Bacca, N. Barnea, O. J. Hernandez and N. Nevo Dinur, “Ab ini-tio calculation of nuclear structure corrections in muonic atoms”, J. Phys. G:Nucl. Part. Phys. 45, 093002, 2018.This work is a review commissioned to authors S.B. and C.J. The text waswritten by C.J., S.B., and N.B. The authors C.J. and N.N.D. were responsiblefor calculations carried out in µ3H and µ3,4He+. I was responsible for allviresults for µ2H, proof-reading, and generating figures. Figs. 1.2 and 1.3 fromChapter 1 are taken from this work.• [3] O. J. Hernandez, A. Ekstro¨m, N. Nevo-Dinur, C. Ji, S. Bacca and N. Barnea,“The deuteron-radius puzzle is alive: A new analysis of nuclear structure un-certainties”, Phys. Lett. B 778, 377-383, 2018.I am the lead author of this work. I wrote the initial drafts of the manuscriptand developed codes required to conduct the statistical regression analysisfor µ2H. Author A.E provided the covariance matrices of the parametersof the NkLOsim and NkLOsep potentials, as well as the Fortran routines tocompute these potentials in momentum space. A.E. also provided the pythonscript to visualize the correlations in Fig. 6.1 based on the calculations ofO.J.H. The other authors motivated the work, aided in interpreting the resultsand helped to finalize the draft of the paper. The main findings of this workare adapted and presented in Section 6.2.• [4] O. J. Hernandez, S. Bacca and K. A. Wendt, “Recent developments in nu-clear structure theory: an outlook on the muonic atom program”, PoS. 041,BORMIO2017, 2017.This is a proceeding of author S.B. from an invited talk at “55th Interna-tional Winter Meeting on Nuclear Physics (BORMIO2017)”. I am the leadauthor of this proceeding since I was responsible for generating the figuresand carrying out all of the calculations on which this work is based. Thiscontribution is based on the theoretical formalism of the two-body, one-pion-exchange contributions that I developed based on the literature. Thetheoretical formalism of this work is adapted and presented in Section 3.3.2.The benchmarks for A = 2 from this contribution are presented in Tables6.17 and 6.18. The formulas derived in this work were applied to A = 3 andA = 6 systems in Section 6.5. Author S.B. wrote text of contribution usingin part my notes on the formalism. Author K.A.W. was responsible for in-dependent calculations with one-pion exchange meson exchange currents toviibenchmark the numerics.• [5] O. J. Hernandez, N. Nevo-Dinur, C. Ji, S. Bacca and N. Barnea “Updateon nuclear structure effects in light muonic atoms”, Hyper. Int. 237, 158,2016.I am the lead author of this contribution. This proceeding is based on mytalk given at the “6th International Symposium on Symmetries in SubatomicPhysics (SSP 2015)”. Author N.N.D wrote the initial draft of the contribu-tion. I was responsible for verifying all of the results and editing the formu-las and text before the publication. The other authors edited and providedfeedback prior to the final submission of the draft.The unpublished parts of this thesis are taken from the following works thathave either been submitted or will be submitted for publication in the near future:• [6] O. J. Hernandez, C. Ji., S. Bacca, and N. Barnea. “Probing uncertain-ties of nuclear structure corrections in light muonic atoms”. To appear inPhys. Rev. C., arXiv:1909.05717, 2019.I am the leading author of this publication. The idea stemmed from discus-sions with the other coauthors. I was responsible for developing the theoreti-cal formalism of the η-expansion and implementing the methodology for theµ2H calculation. I carried out the benchmarks, derived all formulas, gener-ated all of the results and wrote the manuscript. The other authors edited thetext and provided feedback about the results and structure of the paper. Thetheoretical formalism of this contribution is presented in Section 4.3. Theresults of this work are presented in Section 6.3.• [7] C. E. Carlson, F. Hagelstein, O. J. Hernandez, A. Pineda, O. Tomalak,and K. Pachucki. “Nuclear structure effects in light muonic atoms”. To besubmitted, 2019.viiiThis contribution has been carried out in collaboration with other theoriststhat attended the topical workshop “Precision Measurements and Fundamen-tal Physics: The Proton Radius Puzzle and Beyond” in Mainz, 2018, whereI was an invited speaker. I was responsible for the Section of the contribu-tion titled “Two-Photon exchange in µ2H, µ3H, µ3He+ and µ4He+ Lambshift from nuclear theory”. I wrote the section myself with the guidance ofS. Bacca and presented the recent results of µ2H. I also contributed text tothe Section “Three-photon Exchange”. Along with the other authors, weproof-read and reviewed all sections of this work. The updated results forthe deuteron from this review are provided in Section 6.3.3.• [8] O. J. Hernandez, S. Bacca, N. Barnea, N. Nevo Dinur, A. Ekstro¨m, andC. Ji. “A statistical analysis of the nuclear structure uncertainties in µD”. Toappear in FFB22 Proceedings, arXiv:1903.02451, 2019.Proceedings for the “XXII International Conference on Few-Body Problemsin Physics”, in Caen, France 2018. I am the lead author of the contributionwhich is based on my conference talk. I was responsible for all calculationsfor µ2H. The text of this contribution has been adapted for Section 5.4 andSection 6.2.3. Figure 6.5 is taken from this contribution and Table 6.5 hasbeen adapted for the thesis. The other authors provided feedback about thetext before the submission of the final draft.In addition to the above, the introductory Sections of Chapter 3 are based onthe lecture notes and papers of Arenhovel [9–11] and on the book “Theoreticaland subnuclear physics” from Ref. [12]. The procedure of the spherical tensordecomposition of electromagnetic currents in Section 3.3 is adapted from the workof Refs. [12–15].The derivation presented in Section 4.5.2 for the nuclear structure correctionsto the Hyperfine splitting has been extended from Ref. [16] to the case of inelasticnuclear structure corrections. The derivation of these corrections is the fruit of thework of the author of this thesis and his collaborators [17].Chapter 5 is based on the theory developed by Refs. [18–21]. Section 5.3adapts the work of Ref. [21] for the η-expansion formalism.ixFinally, the data for Fig. 6.9 has been obtained from collaborators Ref. [22]who performed the calculations of the SRG no-core-shell model using the two-body currents provided by author O.J.H.xTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviiiGlossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxivDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxv1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 The nuclear interaction . . . . . . . . . . . . . . . . . . . . . . . . . 92.1 Quantum chromodynamics . . . . . . . . . . . . . . . . . . . . . 92.1.1 Symmetries of low energy QCD . . . . . . . . . . . . . . 112.2 Chiral effective field theory . . . . . . . . . . . . . . . . . . . . . 132.3 The few-body nuclear physics problem . . . . . . . . . . . . . . . 172.4 Pionless effective field theory . . . . . . . . . . . . . . . . . . . . 20xi3 Electromagnetic interactions . . . . . . . . . . . . . . . . . . . . . . 223.1 Electromagnetic operators . . . . . . . . . . . . . . . . . . . . . 223.1.1 Gauge invariance of the electromagnetic hamiltonian density 233.2 Models of electromagnetic operators . . . . . . . . . . . . . . . . 253.3 Spherical tensor decomposition of electromagnetic operators . . . 303.3.1 Long wavelength approximations of electromagnetic tensors 333.3.2 Magnetic moment operators . . . . . . . . . . . . . . . . 353.4 The photoabsorption cross section . . . . . . . . . . . . . . . . . 374 Muonic Atom Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 424.1 The electromagnetic moments of nuclei . . . . . . . . . . . . . . 444.1.1 Momentum space formulation . . . . . . . . . . . . . . . 454.1.2 Coordinate space formulation . . . . . . . . . . . . . . . 464.1.3 Mixed momentum and coordinate-space formulation . . . 494.2 The Lamb shift . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.3 Nuclear structure corrections in the η-less expansion . . . . . . . 524.3.1 The non-relativistic limit . . . . . . . . . . . . . . . . . . 584.4 Nuclear structure corrections in the η-expansion . . . . . . . . . . 594.4.1 Subtraction of the elastic part . . . . . . . . . . . . . . . 624.4.2 Lamb shift nuclear structure at α6 order . . . . . . . . . . 664.5 The Hyperfine splitting . . . . . . . . . . . . . . . . . . . . . . . 694.5.1 The Hyperfine Fermi energy . . . . . . . . . . . . . . . . 704.5.2 Nuclear structure corrections . . . . . . . . . . . . . . . . 714.5.3 The multipole expansion of the hyperfine splitting contri-bution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744.5.4 The non-relativistic, long-wavelength reduction . . . . . . 775 Uncertainty quantification . . . . . . . . . . . . . . . . . . . . . . . 785.1 Uncertainty propagation . . . . . . . . . . . . . . . . . . . . . . 795.1.1 Monte Carlo sampling . . . . . . . . . . . . . . . . . . . 815.2 Chiral effective field theory uncertainties . . . . . . . . . . . . . . 815.3 η expansion truncation uncertainty . . . . . . . . . . . . . . . . . 855.4 Other uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . 88xii6 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 916.1 Zemach radius benchmark Calculations . . . . . . . . . . . . . . 926.2 Uncertainty of the two-photon exchange (TPE) in the µ2H LambShift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 946.2.1 Statistical uncertainty estimates . . . . . . . . . . . . . . 956.2.2 Systematic uncertainty estimates . . . . . . . . . . . . . . 996.2.3 Total uncertainty estimates . . . . . . . . . . . . . . . . . 1016.3 The η-less expansion . . . . . . . . . . . . . . . . . . . . . . . . 1076.3.1 Analytical pionless effective field theory . . . . . . . . . . 1086.3.2 Realistic case . . . . . . . . . . . . . . . . . . . . . . . . 1096.3.3 Vacuum polarization . . . . . . . . . . . . . . . . . . . . 1176.4 Bayesian analysis of the η-truncation uncertainty . . . . . . . . . 1206.5 Towards heavier muonic atoms . . . . . . . . . . . . . . . . . . . 1257 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136A The Lippmann-Schwinger equation . . . . . . . . . . . . . . . . . . 149A.1 Ground state equations . . . . . . . . . . . . . . . . . . . . . . . 149A.2 Excited states: Lippmann-Schwinger equation . . . . . . . . . . . 151B Notation, conventions and Feynman formulas . . . . . . . . . . . . . 156B.1 Gamma matrix identities . . . . . . . . . . . . . . . . . . . . . . 157B.2 The Coulomb gauge . . . . . . . . . . . . . . . . . . . . . . . . . 158B.3 Integral relations and principle valued integrals . . . . . . . . . . 158C Racah algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161C.1 The irreducible spherical tensor operators . . . . . . . . . . . . . 161C.2 The spherical tensor basis . . . . . . . . . . . . . . . . . . . . . . 162C.3 Wigner-Eckart theorem . . . . . . . . . . . . . . . . . . . . . . . 163C.4 Racah algebra for the hyperfine splitting . . . . . . . . . . . . . . 165C.5 Commutator notation for matrix elements . . . . . . . . . . . . . 168C.6 Reduced matrix elements . . . . . . . . . . . . . . . . . . . . . . 169xiiiD Spherical vector harmonics . . . . . . . . . . . . . . . . . . . . . . . 172E Pionless EFT matrix elements . . . . . . . . . . . . . . . . . . . . . . 176F Maximum entropy reconstruction of the response function . . . . . 178xivList of TablesTable 5.1 The prior probability distributions of coefficients cν and thescale parameter c¯. . . . . . . . . . . . . . . . . . . . . . . . . 84Table 5.2 The prior probability densities of η . . . . . . . . . . . . . . . . 87Table 6.1 The results of the r, q, qr -space procedures, respectively, forselected deuteron electromagnetic (EM) moments without formfactors using the AV18 potential. . . . . . . . . . . . . . . . . 92Table 6.2 The deuteron benchmark in leading-order (LO) for electric prop-erties: using the harmonic oscillator (HO) basis expansion orthe Numerov algorithm with the AV18 potential. The resultswithout form factors is denoted (w/o FF), and with form factorsusing the Kelly form factors (w FF). Results are compared toexperimental data. The Table is adapted from Ref. [1]. . . . . . 93Table 6.3 The deuteron benchmark in the LO for magnetic properties: us-ing the HO basis expansion or the Numerov algorithm with theAV18 potential. The results without form factors is denoted(w/o FF), and with form factors using the Kelly form factors (wFF). Results are compared to experimental data. The Table isadapted from Ref. [1]. . . . . . . . . . . . . . . . . . . . . . . 94Table 6.4 The results for δTPE at various orders with estimates for the nu-clear physics σNucl and the total σTotal uncertainties. The Tableis adapted from Ref. [3]. . . . . . . . . . . . . . . . . . . . . . 102xvTable 6.5 The uncertainty budget of the δTPE value from our analysis.We use two values for the single nucleon uncertainties σN , onewhere we adopted the strategy of Ref. [53] as well as using thelarger uncertainties from Ref. [123] for δNsub. The Table is takenfrom Ref. [8]. . . . . . . . . . . . . . . . . . . . . . . . . . . 104Table 6.6 The calculated values of the terms ∆L,κ that contribute to thenuclear TPE in meV as a function of the multipole κ forpi-EFTat next-to-next-to-leading-order for µ2H. The Table is adaptedfrom Ref. [6]. . . . . . . . . . . . . . . . . . . . . . . . . . . 108Table 6.7 The calculated values of the terms ∆L,κ that contribute to thetwo-photon exchange in meV as a function of the multipole κusing the realistic N3LOEM potential for µ2H. The values inthe brackets indicate the numerical uncertainties. The Table isadapted from Ref. [6]. . . . . . . . . . . . . . . . . . . . . . . 110Table 6.8 The calculated values of the terms that contribute to δRelpol inmeV as a function of the multipole κ for the N3LOEM potential.The brackets indicate the numerical uncertainties. The Table isadapted from Ref. [6]. . . . . . . . . . . . . . . . . . . . . . . 111Table 6.9 The calculated values of the terms that contribute to δApol in meVas a function of the multipole κ for the N3LOEM potential. Thevalues in the brackets indicate the numerical uncertainties. TheTable is adapted from Ref. [6]. . . . . . . . . . . . . . . . . . 112Table 6.10 A comparison of the results from the η-formalism to the fullη-less formalism in the non-relativistic point-proton, relativis-tic point-proton and relativistic with exact nucleon form factorcalculations. Units are in meV with δATPE = δApol + δAZem. TheCoulomb correction δ (0)C = 0.262 meV from [2], not treatedhere, has also been added to δNRpol ,δRelpol ,δApol, for comparison.The Table is adapted from Ref. [6]. . . . . . . . . . . . . . . . 116Table 6.11 The calculated values of the terms ∆vac, prop,κ that contribute tothe vacuum polarization of the TPE in meV as a function of themultipole κ using the realistic N3LOEM potential for µ2H. . . . 118xviTable 6.12 The results of the TPE including the vacuum polarization con-tributions and the three-photon exchange from Ref. [111]. . . . 119Table 6.13 The leading, sub-leading and sub-sub-leading order, non-relativistic,η-expansion values from Ref. [2] for A = 2,3 and 4 in meVunits. The calculations have been carried out with the chiraleffective field theory (χEFT) potential from Ref. [64]. . . . . . 120Table 6.14 The summary of the results of the Cη prior for different η-estimators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121Table 6.15 The maximum-likelihood estimate (MLE) or median-value esti-mate (MVE) estimates for η and corresponding truncation un-certainties for µ2,3H isotopes. See text for details. . . . . . . . 123Table 6.16 The MLE or MVE estimates for η and corresponding truncationuncertainties for µ3,4He+ isotopes. See text for details. . . . . 124Table 6.17 Sum rules of the magnetic response function for the deuteron,calculated with the AV18 potential [125], using the leading or-der magnetic moment operator µLO. The Table is from Ref. [4]. 126Table 6.18 Sum rules of the magnetic response function of the deuteron,calculated with the N3LOEM χEFT potential [64]. The cal-culation are carried out with the magnetic dipole operator atLO or including the two-body contributions at next-to-leading-order (NLO). The Table is from Ref. [4]. . . . . . . . . . . . . 127Table 6.19 The results of the M1 calculation in A = 3 systems with themagnetic moment operator from LO up to NLO in comparisonto experiment using the nucleon-nucleon / two-body (NN)-onlyN3LOEM potential. The calculations with µNLO[2] are carried outeither using only the intrinsic µ int[2] contribution, or with the fulloperator µNLO[2] that also includes the Sach operator µSachs[2] . Thevalues are in Bohr magenton units µN . . . . . . . . . . . . . . 128xviiList of FiguresFigure 1.1 The schematic representation of the hydrogen energy levels Efor select spectroscopic states nL j from different theories in-cluding the hyperfine states F for nuclei with non-zero totalangular momentum J. . . . . . . . . . . . . . . . . . . . . . 3Figure 1.2 The proton radius puzzle: the rp determinations in µH [37, 38]in comparison to the CODATA-2014 evaluation [46], the re-sults from Mainz [47], and JLab values [39]. The recent resultsfrom Beyer et al. [40], Fleurbaey et al. [41], Bezginov et al.[42] and the latest results from the PRad experiment [43] arealso included. The Figure is adapted from Ref. [2]. . . . . . . 5Figure 1.3 The current deuteron radius puzzle: Recent determinations ofrd from µ2H spectroscopy [50], ordinary deuterium [46] andelectron scattering [51]. The value obtained the isotope shift[52] is denoted with “µH+iso” is also shown. The Figure istaken from Ref. [2]. . . . . . . . . . . . . . . . . . . . . . . . 6Figure 1.4 The TPE contributions to the nuclear structure corrections. Thetop solid line represents the muon, while the bottom solid lineis the nucleus. The blob represents the inelastic excitations ofthe nucleus and the wiggly lines are the photons. . . . . . . . 7xviiiFigure 2.1 The hierarchy of nuclear forces in χEFT from [59]. The nu-cleons are represented by solid lines while the pions are thedotted lines. The solid symbols, squares, diamonds, big andsmall circles represent different possible pi-N couplings as inRef. [59]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Figure 3.1 Photon-matter coupling diagrams: (a) the one-photon currentoperator Jµ ; (b) the two-photon operator Bµν . . . . . . . . . . 23Figure 3.2 The interaction diagrams that make up the two-body currents atLO in χEFT: the pion in flight (left) and the seagull (right). Thewiggle represents the electromagnetic interaction (the photon). 28Figure 3.3 The photo absorption process. The incoming nucleus in state|ΨI〉 absorbs a photon and transitions to the final state |ΨF〉. . 37Figure 4.1 TPE diagrams: (a) elastic vs (b) inelastic. The empty box repre-sents any general electromagnetic processes involving the up-per half of the diagram. In the inelastic diagram (b) the excitedstates of the nucleus are represented by the grey blob. TheFigure is adapted from Ref. [8]. . . . . . . . . . . . . . . . . 62Figure 4.2 The α(Zα)6 vacuum polarization contribution. The circle inthe photon line represents the creation of an electron and positronpair. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66Figure 4.3 The radiative recoil corrections: (a) The spanning photon di-agram (b) the muon self energy correction and (c) the vertexcorrection. . . . . . . . . . . . . . . . . . . . . . . . . . . . 68Figure 4.4 The three photon exchange diagram. . . . . . . . . . . . . . . 68Figure 4.5 The hyperfine structure of µ2H for the 2S1/2 and 2P1/2 states. 69xixFigure 6.1 The correlation matrix of the deuteron ground state energyE0, rms radius Rp, quadrupole moment Qd , D-state probabilityPD, magnetic moment µd , electric polarizability αE , magneticsusceptibility βM, leading dipole polarizability correction δ(0)D1 ,magnetic polarization correction δ (0)M and δATPE for the N2LOsimpotential with Λ = 450 MeV and T MaxLab =125 MeV. The Figureis taken from Ref. [3]. . . . . . . . . . . . . . . . . . . . . . 97Figure 6.2 The calculated values of δATPE for different cutoffs Λ in MeVas a function of T MaxLab for the N2LOsim potentials. The Figureis taken from Ref. [3]. . . . . . . . . . . . . . . . . . . . . . 99Figure 6.3 The systematic uncertainties of δATPE as a function of the cutoffΛ for the N2LOsim potentials. The width of the blue (dark)band indicates the uncertainty from σT MaxLab . The (light) greenband also includes the σ∆ uncertainty. The difference betweenthe maximum and minimum width of these bands are the σΛuncertainty. The Figure is from Ref. [3]. . . . . . . . . . . . . 100Figure 6.4 δTPE as a function of the χEFT order with total uncertaintyσTotal (see text for details). The Figure is from Ref. [3]. . . . . 103Figure 6.5 The posterior distributions of the σ∆ uncertainties at differentorders in the χEFT expansion (A0∆(1)2 , A0∆(1)3 , A0∆(1)4 ) in meVunits for δTPE in the ∆(1)ν approximation for the NkLOEKM po-tentials with (R0,Λ)=(0.8, 600) [fm, MeV]. The expansion pa-rameter is Q= 0.23, wB=1 for prior B with c¯<=0.1 and c¯>=10.The Figure is taken from Ref. [8]. . . . . . . . . . . . . . . . 105Figure 6.6 The absolute value of the longitudinal multipole corrections inthe non-relativisitic (NR) point-nucleon limit, in the relativistic(Rel) point-nucleon limit, and with form factors (+FF) as afunction of the multipolarity κ using the N3LOEM potential.The Figure is adapted from Ref. [6]. . . . . . . . . . . . . . . 113xxFigure 6.7 The absolute value of the (a) transverse Siegert corrections andmagnetic transverse corrections (b) plotted as a function of in-creasing photon multipolarity κ for µ2H in the point protonlimit and with form factors (+FF) using the N3LOEM poten-tial. The Figures are adapted from Ref. [6]. . . . . . . . . . . 114Figure 6.8 The normalized posterior distributions P(η |D) for µ2H in sub-figures ( a-c), for µ3H in (d-f), for µ3He+ in (g-i) and µ4He+in (j-l). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122Figure 6.9 The B(M1,0+→ 1+) transition strength for 6Li calculated inthe SRG-evolved no-core shell model for different NN χEFTpotentials with consistent three-body forces using the µLO orµLO+µ int[2] operators. . . . . . . . . . . . . . . . . . . . . . . 130Figure 7.1 The final uncertainty budget of the TPE calculation from Ref. [3].133Figure F.1 The reconstructed test response functions using (a) 10 mo-ments, (b) 20 moments, (c) 30 moments, in the Legendre basis. 181Figure F.2 The reconstructed test response functions using ten momentsand (a) the Legendre, (b) monomial or (c) the Chebychev basisfunctions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182xxiGlossaryχEFT chiral effective field theoryEFT effective field theoryEM electromagneticHO harmonic oscillatorHFS hyperfine splittingIID independent, identically distributedLO leading-orderLS Lippmann-SchwingerLECS low-energy constantsMLE maximum-likelihood estimateMVE median-value estimateMC Monte-CarloNLO next-to-leading-orderN2LO next-to-next-to-leading-orderN3LO next-to-next-to-next-to-leading-orderN4LO next-to-next-to-next-to-next-to-leading-orderxxiiNN nucleon-nucleon / two-bodypi -EFT pionless effective field theoryQCD quantum chromodynamicsQED quantum electrodynamicsRMS root-mean squareSM standard modelTPE two-photon exchangexxiiiAcknowledgmentsConducting the PhD has been a challenging experience and it would not have beenpossible without the many people in my life. I would first like to give my biggestthanks to my academic supervisor, Dr. Sonia Bacca for her guidance and contin-uous support during my PhD as well as during my Masters degree. Her patience,encouragement and instruction are what motivated me to keep going. I am alsograteful to my main collaborators, Nir Barnea, Chen Ji and Nir Nevo Dinur who Ilearned a great deal from through our discussions and working together. I wouldalso like to thank all of my PhD committee members, TRIUMF and to all the sup-port I received from UBC and the administrative staff.I would like to express my gratitude to my family, especially my mum, San-dra A. Melgar who has always been there to support and encourage my passionsall of my life. I would not have come this far without her. I would like to thankmy brother, I. Alfredo Hernandez and my sister S. Cristina Hernandez who werepatient and always there for me throughout this journey. To my niece, Naya Her-nandez, I dedicate this work.I would like to thank my friends from across continents and cultures who werealways there to talk to and offered me the strength that I needed to face difficultchallenges.Lastly, I would like to give infinite thanks to my loving partner, who was sup-portive and patient with my decision to leave Canada and to complete my researchin Germany. Thank you for always being there for me.xxivDedicationI dedicate this dissertation to my mother, Sandra A. Melgar and to the memory ofmy father, Israel Hernandez Marroquin 1963-1996.xxvChapter 1IntroductionAtoms, the basic building blocks of all matter, have intrigued intellectuals sincetheir existence was proposed on philosophical grounds by Democritus and histeacher Leucippus, in the fifth century BCE [23]. It was not until the early nine-teenth century that John Dalton provided firm evidence to support their existence.John Dalton proposed that matter must be made up of individual, indivisible, com-ponents called atoms [23] to explain observations that certain chemical reactionsalways occurred in fixed ratios. Furthermore, he postulated that different elementsconsisted of atoms with different properties. Later in the early 1900s Albert Ein-stein provided further evidence of this hypothesis through his atomic explanationof Brownian motion that firmly established the theory [24].Studies regarding the internal nature of atoms were carried out during the 1800s[25, 26] using primitive diffraction gratings and prisms. These early researchersobserved that light emitted from different elements were unique, but their under-lying physical laws where unexplained. The observed spectroscopy signature ofthe lightest atomic system, the hydrogen atom, was successfully explained by theBohr model in 1903 [27]. This model radically departed from the classical theoryof mechanics and electromagnetic (EM) established by Maxwell and Newton inthe previous century. As demonstrated by Bohr’s predecessors [28], the hydrogenatom consisted of a light and negatively charged particle, the electron, orbiting aheavy positively charged core, later identified as the proton. The radical idea thatBohr’s model proposed was that the electrons orbiting a nucleus could only assume1specific distances that corresponded to discretized energy levels. Furthermore, heproposed that atoms emitted or absorbed light only at frequencies equal to the en-ergy differences of the corresponding orbits. This contradicted the classical physicspicture that does not impose restrictions on the distances or energies of the electronorbits. The quantized energy levels of the hydrogen atom proposed by Bohr wasEn =−13.6Z2n2eV, (1.1)where En is the energy of the discrete-nth state of hydrogen, Z is the charge num-ber of the nucleus and -13.6 eV is the binding energy of hydrogen. These en-ergy levels reasonably explained the observed spectral lines of hydrogen. In thedecades that followed, the Bohr model was formalized through the discovery ofthe Schro¨dinger equation that placed Bohr’s model and related discoveries on solidtheoretical footing becoming known as quantum mechanics. This field was laterextended to incorporate relativistic effects through the Dirac equation [29].As the precision of the known atomic hydrogen spectrum increased, it wasobserved that spectral lines such as the Lamb shift [30] were unexplained by theDirac model. This energy shift would wait until the era when relativistic quantumphysics and EM were combined into a cohesive physical theory, quantum electro-dynamics (QED), that has pushed forward the limits of theoretical precision. Forexample, this new theory was able to improve the 0.1% disagreement betweenthe theoretical hyperfine splitting (HFS) in hydrogen and experiment [31, 32]. Aschematic comparison of the hydrogen spectral lines of hydrogen for the Bohr andDirac models including the effects of QED and the hyperfine splitting are illustratedin Fig. 1.1.The application of QED to atomic systems, made up of point-like particles, iswell understood [33–35] and makes precise predictions for light atomic systems.However, nuclear structure effects also play a crucial role when precision is needed.It is now understood that spectroscopic measurements of atomic energy shifts,like the Lamb shift, can be used to reveal the properties of the atoms’ core, such asthe charge radius of the nucleus. In particular, the measured spectroscopic energy2Bohr   Dirac QED  HyperfineLamb shiftFigure 1.1: The schematic representation of the hydrogen energy levels E forselect spectroscopic states nL j from different theories including the hyperfinestates F for nuclei with non-zero total angular momentum J.shifts (∆E) of the Lamb or HFS are related to the nuclear structure effects through∆E = δQED+δFS(Rx)+δnucl+ . . . , (1.2)where δQED are the known QED corrections and the finite size correction δFS(Rx)represents the term involving the ground state EM moment of the nucleus (thecharge radius RE for the Lamb shift, or the Zemach radius RZ for the HFS). Theterm δnucl encapsulates the contributions from nuclear structure and the ellipsis de-notes other effects such as recoil that may be separately treated. When the otherterms in Eq. (1.2) are known, this relation allows the extraction of ground statenuclear EM moments.In particular, measurements of the Lamb shift make it possible to probe theradii of the lightest nucleus, the proton, which is one of the most fundamentalstructures in physics. The accuracy to which the proton radius (rp) and the radiiof heavier nuclei can be determined serves as a critical probe to test our funda-mental understanding of nuclear and EM forces. Therefore, high accuracy and3precision determinations of nuclear radii are an important step towards expand-ing our knowledge of fundamental interactions. Using Eq. (1.2), prior to 2010,spectroscopic measurements of ordinary electron hydrogen, along with electron-proton scattering experiments [36], were used to determine the size of the proton.From the analysis of these experiments it was determined that the proton radiuswas rp = 0.8775(51) fm.To improve the precision of charge radius values, it was known that muonicatoms, where the orbiting electron is replaced with the more massive muon, offeredthe opportunity to vastly improve these radii determinations due to the enhancedspectroscopic signals of muonic atoms relative to their electronic analogues. Fur-thermore, due to the large mass of the muon, in comparison to the electron, itsCompton wavelength is comparable to the size of the nucleus making it very sen-sitive to the effects of nuclear structure δnucl in Eq. (1.2). Thus, it serves as anideal laboratory to study the interplay between nuclear and EM forces. For theLamb shift, the enhancement factor of the spectroscopic signal is approximately(mµ/me)3 ≈ 2003. However, advancements in the experimental setup would haveto be developed before these muonic spectroscopy experiments could be realized.The first experiments to determine nuclear charge radii from muonic Lambshift measurements were carried out in 2010 by the CREMA collaboration [37]for muonic hydrogen (µH) and verified in 2013 by Antognini et al. [38]. Theseexperiments improved the precision of rp to ten-times the accuracy of previousdeterminations. However, their extracted value of rp = 0.84184(67) fm disagreedwith the earlier work of Ref. [36] by 7 σ . Due to the large magnitude of thisdiscrepancy, it was coined the “proton radius puzzle” and brought into questionour comprehension of fundamental physics based on the standard model. The dis-covery of this puzzle has since attracted the attention of numerous experimentalgroups. For example, the high precision electron-proton scattering measurementsconducted at Jefferson Lab in 2011 in Ref. [39] and more recently, the hydrogenspectroscopy experiments by Beyer et al. [40] and Fleurbaey et al. [41] that haveadded to the puzzle. The smaller rp radius is confirmed by the recently publishedLamb shift measurements in ordinary hydrogen by Bezginov et al. [42] as wellas the latest results from the high precision electron-proton scattering experiment(PRad) conducted at the Jefferson Laboratory [43]. The present status of the proton4radius puzzle including all recent published developments is illustrated in Fig. 0.84 0.86 0.88 0.90 0.92proton charge radius [fm]H world data 2014JLab 2011Mainz 2010CODATA 2014H. Fleurbaey et al.   1S-3S, 2018N. Bezginov et al.    2S-2P, 2019 Mihovilovic et al.   2019  A. Beyer et al.   2S-4P, 2017 PRad 2019 H 2010H 2013Figure 1.2: The proton radius puzzle: the rp determinations in µH [37, 38]in comparison to the CODATA-2014 evaluation [46], the results from Mainz[47], and JLab values [39]. The recent results from Beyer et al. [40], Fleur-baey et al. [41], Bezginov et al. [42] and the latest results from the PRadexperiment [43] are also included. The Figure is adapted from Ref. [2].Following the discovery of the proton radius puzzle, many works have beenpublished seeking to resolve the puzzle through from a range of different methods,from studies proposing physics beyond the standard model reviewed in [48] to themore recent work of Ref. [49] that exploits the dispersively improved chiral effec-tive field theory to constrain the form factor data and reproduces the smaller protonradius. However, as-of-yet no agreed upon explanation exists. In 2016 the CREMAcollaboration published their analysis of the muonic deuterium (µ2H) Lamb shiftcollected at the same time as their original µH experiment [50]. The charge ra-dius of the deuteron (rd) extracted from this analysis was rd = 2.12562(78) fm.Similar to the case of the µH radius, the µ2H charge radius disagrees with previ-ous determinations by 6 σ with respect to the CODATA 2014 analysis [46]. Thediscrepancy of this radius with respect to spectroscopy measurements in electronicdeuterium is about 3.5 σ . However, unlike in the proton case, the large uncertainty1The alternate analysis of Refs. [44, 45] also demonstrates that it is possible to reproduce thesmaller µH radius within a dispersion theory framework based on electron-proton scattering data.5of the electron scattering experiments from Ref. [51] cannot distinguish betweenthe radii of ordinary electronic and muonic atoms. These new disparities consti-tute the “deuteron radius puzzle” that is distinct but analogous to the proton radiuspuzzle.2.12 2.13 2.14 2.15deuteron charge radius [fm]e-2H scatteringe2H spectroscopyCODATA-20142HH+isoFigure 1.3: The current deuteron radius puzzle: Recent determinations of rdfrom µ2H spectroscopy [50], ordinary deuterium [46] and electron scattering[51]. The value obtained the isotope shift [52] is denoted with “µH+iso” isalso shown. The Figure is taken from Ref. [2].In the case of the deuteron radius puzzle, independent of the Lamb shift spec-troscopy measurements, by assuming lepton universality it is possible to infer rddirectly from the electronic isotope shift. The isotope shift is the experimentallydetermined difference (∆(H,2 H)) between 1S−2S transitions in electronic hydro-gen and deuterium that is directly related to the squared differences of their chargeradii [52]∆(H,2 H) = r2d− r2p = 3.82007(65)fm 2. (1.3)This relation allows the µ2H charge radius to be calculated from the µH protonradius determination. The prediction of the deuteron charge radius from the isotopeshift results in a value of rd = 2.12771(22) fm, denoted as “µH+iso” in Fig. 1.3 anddiffers from the rd Lamb shift value by about 2.6 σ . From this radius and with the6known QED corrections, the experimental value of nuclear structure correctionsδnucl is determined through Eq. (1.2).Figure 1.4: The TPE contributions to the nuclear structure corrections. Thetop solid line represents the muon, while the bottom solid line is the nucleus.The blob represents the inelastic excitations of the nucleus and the wigglylines are the photons.The nuclear structure effects in muonic atoms are dominated by the TPE il-lustrated in Fig. 1.4. This diagram depicts the inelastic interaction between themuon and the nucleus via EM forces. The inferred experimental value is δTPE =−1.7638(68) meV from Ref. [50] that differs from the theoretical compilation ofKrauth et al. [53] by about 2.6 σ . This small discrepancy was argued to be theresult of underestimated nuclear physics uncertainties and missing contributions.Motivated by these puzzles, the CREMA collaboration has conducted soon-to-be-published Lamb shift experiments for µ3He [54] with plans to extend their mea-surements to the HFS in this and other light muonic atoms such as µ6,7Li+2 in thefuture. These upcoming experiments will give insight as to whether these discrep-ancies also exist in nuclei with larger mass numbers.The work presented in this dissertation is an attempt to link the realm of atomicsystems governed by EM forces with nuclear interactions, focusing on the effectsof nuclear structure. In this work we introduce and develop the framework requiredto study these nuclear structure corrections necessary to support the ongoing ex-perimental efforts of the CREMA collaboration.The layout of this dissertation is as follows: Chapter 2 introduces the theory ofthe strong force along with the modern tools of chiral effective field theory (χEFT)giving rise to the hierarchy of nuclear forces. The discussion of nuclear forcesis followed by a review of the EM force in Chapter 3 that discusses the one and7two photo-nuclear operators that couple nuclei with EM forces. In Chapter 4 weprovide a derivation of the nuclear structure effects relevant to the Lamb and HFSin muonic atoms. Because of the importance of uncertainty quantification, a thor-ough overview of the latest methods used to estimate the statistical and systematictheoretical model uncertainties is summarized in Chapter 5. In Chapter 6 the cal-culations of the nuclear structure effects relevant to the Lamb shift are providedalong with an analysis of the relevant systematic and statistical uncertainties thatarise in the calculation. Lastly in Chapter 7 the work carried out in this thesis issummarized and an outline of future work is discussed.8Chapter 2The nuclear interactionThe strong force is one of the four fundamental physical forces that governs theobservable Universe. The other fundamental forces are electromagnetism, gravityand the weak force. The strong force binds nucleons together and generates themultifaceted spectrum of nuclear properties. The underlying fundamental theoryof the strong interaction is quantum chromodynamics (QCD) that characterizes theinterplay of the matter fields (quarks) with the force-carrying bosons (gluons) ofthe theory. In this framework, in analogy to charge in electromagnetism, quarkscarry a property known as color charge. Color charge can take on six values incontrast to the two values, positive or negative, for electromagnetism.This Chapter introduces the formalism of QCD and describes its fundamentalproperties and symmetries. Using these properties, we motivate one of the pillarsof modern nuclear theory, χEFT and discuss its application to the nuclear few-bodyproblem. We conclude the Chapter by introducing another effective field theory ofthe strong interaction, pionless effective field theory (pi -EFT). This effective fieldtheory is a useful framework to approximate nuclear few-body interactions.2.1 Quantum chromodynamicsThe theory of the strong interaction within the standard model (SM) of particlephysics is a non-Abelian gauge theory with SU(3)color symmetry. The Lagrangiandensity of the strong interaction in the absence of explicit P and CP violating terms9is [55]LQCD =∑fq¯ f(i /D−m f)q f − 14GaµνGµνa , (2.1)where the summation f runs over the six quark flavours (up, down, charm, strange,top and bottom), q f are the quark fields and m f are the bare quark masses. Thecovariant derivative isDµ = ∂µ + igAµ , (2.2)whereAµ ≡ 12Aaµλa, andAaµ are the four-vector gauge potentials with the indicesa running over the dimensions of the SU(3)color Lie algebra. The matrices λa arethe Gell-Mann matrices that act in color space and satisfy the commutation relation[λa,λb] = 2i fabcλc, (2.3)where fabc are the totally anti-symmetric, real, structure constants. The constant gis the strong coupling constant and the gluon field strength tensor isGaµν = ∂µAaν −∂νAaµ −g fabcAbµAcν . (2.4)The strong-coupling constant at a finite momentum transfer q2 is given as in Ref. [29]byg(|q2|) = g(µ2)√1+ g2(µ2)48pi (11Nc−2N f )ln( |q2|µ2) , (2.5)where g(µ2) is the value of the strong coupling constant at a fixed momentum scaleµ2 with |q2|  µ2, Nc is the number of colors and N f is the number of flavors. Inthe SM these values are Nc = 3 and N f = 61. Unlike in QED , g(|q2|) increases atsmall momentum transfers and decreases at large momenta, approaching zero as|q2| → ∞. This property is known as the asymptotic freedom of QCD.Another important feature of QCD is color confinement. This is the observedproperty that all quark arrangements exist as color singlets and QCD systems witha net color charge do not manifest in Nature.1In the low energy, nuclear physics regime, the number of flavors is effectively N f = Symmetries of low energy QCDHere we discuss the important symmetries of the theory relevant for nuclear physicsthat paves the way for χEFT. It is observed that in QCD the six quark flavors areseparated into two distinct mass scales. The light quarks (up, down, strange) havemasses ∼ (2.5+0.49−0.26, 4.67+0.48−0.17, 93+11−5 ) MeV while the heavier quarks (charm, bot-tom, top) are approximately ∼ (1.27+0.02−0.02, 4.18+0.03−0,02, 172.9+0.4−0.4) GeV in mass fromdirect measurements [56]. This ≈ 1 GeV mass scale difference decouples the lightfrom the heavy quarks. The latter are not relevant for nuclear structure and will notbe treated in the formalism that follows.The matter fields of the light quarks are described by the light quark field vectorq asq =quqdqs , (2.6)where qu,q f , and qs are the quark fields corresponding to the up (u), down (d),strange (s) flavors, respectively. The mass matrix of these fields isM =mu 0 00 md 00 0 ms , (2.7)where the entries mu,md ,ms in the matrix are the quark masses for the u, d and squark flavors, respectively. The masses of the lighter quarks are small in compar-ison to typical baryon masses (∼ 1 GeV) and consequently we consider the lightquark Lagrangian density in the limit of vanishing quark masses. In this limitlim(mu,md ,ms)→0LQCD ≡L 0QCD =(q¯Ri /DqR+ q¯Li /DqL)− 14Ga,µνGµνa , (2.8)where qR = PRq, qL = PLq are the right and left handed projections of the light11quark fields and the projection operators PR,PL arePR =12(1+ γ5) , (2.9)PL =12(1− γ5) , (2.10)with γ5 = iγ0γ1γ2γ3. In this limit, the Lagrangian density is invariant with respectto arbitrary 3×3 unitary transformationsqL→UL(3)qL = e− i2θL,aλae−iθLqL, (2.11)qR→UR(3)qR = e− i2θR,aλae−iθRqR, (2.12)where UL(3) and UR(3) are the unitary operators for the left and right handedquark field projections, respectively. The UL(3)×UR(3) unitary operators havebeen decomposed on the right hand sides of Eq. (2.11) and Eq. (2.12) as SUL(3)×SUR(3)×UL(1)×UR(1). The terms θR,θR,a,θL, and θL,a are the parameters ofthe unitary transformation [55]. The space U(1)L×U(1)R can also be expressedas U(1)A×U(1)V , where A denotes axial vectors and V are vectors. In this de-composition the U(1)A symmetry is broken explicitly by the non-vanishing quarkmasses as well as through the axial anomaly [55]. The UV (1) symmetry results inthe conservation of baryon number.In general, the SUL(N)× SUR(N) invariance of L 0QCD, for N = 2 or 3, isknown as chiral symmetry. This symmetry is broken in two ways, explicitly andspontaneously. The explicit symmetry breaking is from the contribution of themass term LM = q¯M q, to the QCD Lagrangian which is not invariant underSUL(3)× SUR(3) transformations. The spontaneous symmetry breaking resultsfrom the fact that chiral symmetry is not realized in the ground state of the the-ory. Evidence of this spontaneous symmetry breaking is seen most directly fromthe baryon spectrum. If chiral symmetry were to hold in Nature, then the baryonspectrum would have parity octet doublets with similar masses. However, parityoctet doublets with similar mass scales do not manifest in Nature. For example,the nucleons form part of the baryon octet Jpi = 12+ which are almost degeneratein mass at ≈ 900 MeV and strangeness s = 0. However, baryons with the same12properties and negative parity are only realized as nucleon resonances at energiesabove 1500 MeV. If the quarks in the QCD Lagrangian were massless, then thisspontaneous symmetry breaking would imply the existence of Goldstone bosons.However, since the quarks have finite non-zero masses, the spontaneous symmetrybreaking of chiral symmetry results in pseudo-Goldstone bosons which are identi-fied as the pions. The pions have masses of about 140 MeV, which are smaller thantypical baryonic mass scales.2.2 Chiral effective field theoryChiral effective field theory, is a low-energy effective field theory of QCD pio-neered by the work of Steven Weinberg in Refs. [57, 58] and in subsequent papers[55, 59, 60] of various authors. The degrees of freedom of the theory are pions andnucleons. This effective field theory models nuclear forces as a systematic hierar-chy of pion-nucleon interactions from an effective χEFT Lagrangian. The χEFT-Lagrangian observes the same fundamental symmetries as the underlying theory ofQCD, such as chiral symmetry, and is constrained to reproduce low-energy nuclearphysics phenomena. The χEFT Lagrangian density is expanded as the sumLχEFT =Lpipi +LpiN +LNN + . . . , (2.13)where Lpipi describes only the pion-pion interactions, LpiN the pion-nucleus, andLNN is the nucleon-nucleon interaction Lagrangians. The ellipsis in the aboveexpression denotes terms with interactions of more than two-pion or two-nucleons.Each of these individual terms contains an infinite number of diagrams representingpossible interactions. To make calculations tractable, an expansion parameter anda power counting prescription are needed to order each diagram according to theexpected size of their contributions and render their number finite at each order.The appropriate expansion parameter of the theory is identified as the ratio Q/Λb,where Q is the soft scale of the theory, roughly the pion mass, and Λb is the chiralsymmetry breaking scale. The energy breaking scale has been determined to beapproximately 600 MeV based on Bayesian analyses of nucleon-nucleon / two-body (NN) data [21, 61]. Based on this expansion parameter, every interactionwithin the χEFT framework is ranked according to the associated power (ν) of the13expansion parameter Q/Λb. The conventional scheme determining the order ν istaken from the naive dimensional analysis of Ref. [57]. In this prescription, theorder of a diagram ν isν =−4+2A+2L+∑i∆i, (2.14)where A is the number of nucleons, L is the number of loops, the sum runs over allvertices i in the diagram, and ∆i is defined as∆i =(di+ni2−2). (2.15)Here, di are the number of pion mass insertions and ni are the number of nucleonfields involved in the vertex i. With this power counting prescription the effectiveNN χEFT Lagrangian is expanded in powers of the chiral order νLχEFT = ∑ν=0LNνLO, (2.16)where LNνLO is the effective Lagrangian density at order ν , representing a finiteset of interactions. At each order in the effective interaction the ratio (Q/Λb)νprovides a rough estimate of the expected size of the contribution [59] and theexpansion converges when Q/Λb 1. This systematic expansion allows nuclearforces to be modeled in a controlled way to a desired level of accuracy. At eachχEFT order, the effective Lagrangian densities involve a finite set of undeterminedlow-energy constants (LECS) {αi} that are fitted to few-body nuclear data. The de-termination of the set of LECS α at each order in χEFT is carried out by minimizingthe objective functionα 0 = argminα{χ2(α )}, (2.17)where χ2(α ) isχ2(α ) = ∑i∈M(Oth(α )i−Oexpiσi)2. (2.18)The set M denotes the collection of few body observables used in the fitting pro-cedure [62] such as NN phase shifts or the Nijmegen partial wave analysis. Theterms Oth and Oexp denote the theoretical and experimental values of the nuclear14observable O, respectively and σi is the weight of the residual. The power count-ing prescription in Eq. (2.14) gives rise to the hierarchy of the nucleon interactionsshown in Figure 2.1.Figure 2.1: The hierarchy of nuclear forces in χEFT from [59]. The nucleonsare represented by solid lines while the pions are the dotted lines. The solidsymbols, squares, diamonds, big and small circles represent different possiblepi-N couplings as in Ref. [59].At the leading-order (LO) depicted in Fig. 2.1, the NN potential is a sum of acontact term and the one-pion exchange potential given in momentum space asV1pi(p, p′) =− g2AF2pi(τ 1 · τ 2)(σ 1 · (p− p′))(σ 2 · (p− p′))(p− p′)2+m2pi, (2.19)where gA is the axial vector coupling, mpi is the mass of the pion, Fpi is the pion de-cay amplitude2 and τ 1,2 are the isospin operators of particles 1 and 2, respectively.2The pion-decay amplitude is Fpi ≈ 186 MeV [63].15The term V1pi denotes the one pion exchange potentialV LONN (p, p′) = αS+αTσ 1 ·σ 2+V1pi(p, p′), (2.20)where αS and αT are the unknown LECS at this order and σ 1/2 are the Pauli spinmatrices. The calculation of NN observables typically involves the use of the T -matrix. This T -matrix is calculated with the Lippmann-Schwinger (LS) equationin Appendix A asT (p′, p) =VNN(p′, p)+∫d3 p VNN(p′, p′′)mp2− p′′2+ iε T (p′′, p), (2.21)where m is the nucleon mass. The iteration of VNN in Eq. (2.21) requires integra-tion to high momenta. To avoid divergences a regulator is required to suppressthe high momentum contributions of the Feynman amplitudes. This regulator isimplemented by settingVNN(p′, p)→VNN(p′, p)F(p′, p), (2.22)in the LS equation, where F is the chosen momentum-space regulator. The regula-tor depends on the energy cutoff scale Λ used to enforce this constraint.In this work, we will study the order-by-order χEFT convergence of nuclearobservables related to the muonic atoms using NN forces from LO up to next-to-next-to-next-to-next-to-leading-order (N4LO) in χEFT. For benchmarking pur-poses, in Chapter 6 we use the N3LO χEFT potential of Entem and Machleidtfrom Ref. [64] denoted as N3LOEM. Two more recent groups of chiral interactionshave been constructed using different fitting procedures and slightly different oper-ational forms in the potentials, but with identical power counting. We use all ordersavailable from Ref. [65] and denote them as NkLOEKM and those from Ref. [66],as NkLOEMN . In addition to the order-by-order convergence of the χEFT we willexamine the statistical uncertainties from the LECS originating from the pool offitted data. This task is carried out using the chiral potentials available up to next-to-next-to-leading-order (N2LO) from Ref. [62] denoted as NkLOsim and NkLOsep.The NkLOsim potentials are the set of NN potentials that used a simultaneous fitting16procedure for all of the LECS at each chiral order, while the NkLOsep potentials fitthese LECS seperately [62]. These potentials also allow us to probe the systemat-ics of the fitting procedure by varying the maximum kinetic energy (T MaxLab ) of theNN data used to fit the potentials. The details about the precise cut-offs and T MaxLabenergies that were used are discussed in more detail in Chapter 6.2.3 The few-body nuclear physics problemThe aim of nuclear ab-initio theory is to solve the Shro¨dinger equation of an A-bodynucleusHnucl|ψ〉= E|ψ〉, (2.23)where |ψ〉 is the A-body wave function, E are the energies of the system and Hnuclis the nuclear Hamiltonian of the few body systemHnucl = T +A∑i jVi j +A∑i jkWijk+ ... . (2.24)Here T is the total kinetic energy of the nucleons and Vi j, Wi jk denote the two- andthree- body nuclear forces, respectively. The ellipses denote more than three-bodyinteractions. 3 Few-body nuclear physics has been traditionally concerned withsolving Eq. (2.23) for A = 2− 4 using different models for two and three bodyinteractions. However, the field has developed tremendously over the past fewyears encompassing many recent advances, such that even medium-mass nucleican be tackled these days (see, e.g, the review [67]). This fast evolving field isknown as ab-initio nuclear theory.The simplest bound few-body nucleus is the deuteron, consisting of a protonand neutron confined by the NN interaction. After the removal of the center ofmass coordinate different methods exist to solve Eq. (2.23) for the deuteron groundstate wave function and excitation spectrum. In this work we will focus on twomethods: the LS and truncated harmonic oscillator (HO) basis expansion. The3We note that the Hamiltonian in Eq. (2.24) only includes the terms pertaining to the nucleus doesnot include terms that describe the leptons of the atomic system. The energy separation of the atomicand nuclear systems allow the lepton and nuclear degrees of freedom to be treated independentlyfrom one another, a fact that will be exploited in Chapter 4 when the muonic system is treated.17former method solves the deuteron in a discretized momentum space grid and thesecond method discretizes the Hamiltonian as matrix elements of the HO basis. Forany method, the basis states describing the deuteron’s intrinsic wave function (thatdoes not depend on the center of mass coordinates between the two particles) withall relevant quantum mechanical degrees of freedom are|κ〉= |(`S)J,T ;M,MT 〉, (2.25)where ` is the relative orbital angular momentum, S is the total spin and J and Mare the total angular momentum and its projection, respectively. The total isospin isT with projection MT . The index κ is the collective index of the allowable quantumnumbers. The deuteron ground state consists of the states with `=0 and `=2 withJ=1 and T =0, MT = 0.In coordinate space, the deuteron wave function is〈r|ψ〉= u(r)Y M01(rˆ)χT=00 +w(r)Y M21(rˆ)χT=00 , (2.26)where u(r) and w(r) are the S and D state wave functions and χTMT is the isospincomponent of the wave function. The functions Y M01(rˆ) and YM21(rˆ) are the vectorspherical harmonics defined in Appendix D. The normalization of Eq. (2.26) isimposed by the condition that u(r) and w(r) satisfy1 =∞∫0dr(u2(r)+w2(r)). (2.27)The functions u(r) and w(r) are obtained by solving Eq. (2.23). In the truncatedHO basis expansion these deuteron ground state wave functions are determined byexpanding the general states |κ〉 of the deuteron as|κ〉=NMax∑λcλ |n(`S)J,T ;M〉, (2.28)where the terms cλ are the eigenvector components of the discretized nuclearHamiltonian. The parameter NMax truncates the dimensions of the model spacethrough the condition that NMax ≤ 2n+ `. In the coordinate representation, the HO18basis functions are〈r|n, `,m〉= 1√b3Nn`L`+ 12n(r2b2)e−12(rb)2 ( rb)`Y `m(rˆ), (2.29)Nn` =√2Γ(n+1)Γ(n+ `+ 32), (2.30)where the parameter b=√h¯mΩHO is the characteristic length of the HO that dependson the reduced mass m of the proton-neutron system with oscillator frequencyΩHO.The deuteron wave functions can also be projected into the partial wave momentumspace representation. In this basis, the states are〈p|κ〉= φκ(p)〈pˆ|p,(`S)JM;T MT 〉, (2.31)where the momentum functions φκ(p) correspond to the state κ . In this basis, theSchro¨dinger equation for the ground state is the coupled integral equation(p22mδκκ ′φκ(p)+∑κ ′∫d p′p′2VNN(κ, p, p′,κ ′)φκ ′(p′))= Eφκ(p), (2.32)where κ is either the 3S1 or the 3D1 channel4. This coupled integral equation canbe solved using the methods in Appendix A and is extended to solve the scatteringproblem. The advantage of using the LS method for the A = 2 problem is that thecontinuum wave functions are explicitly constructed to obey the scattering planewave boundary conditions as described in Appendix A. For this reason, the LSequations are the preferred method for the calculation of phaseshifts or responsefunctions at finite momentum transfers.The above methods are extended to solve A ≥ 3 nuclear systems where thechallenge lies in the antisymmetrization procedure. For the two body system thisis carried out trivially by the generalized Pauli exclusion principle that imposed(−1)T+`+S =−1 to the states |κ〉. For heavier systems, the Pauli exclusion princi-ple is not so easily imposed and other strategies have been devised. For example, in4Here we adopt the spectroscopic notation 2S+1L j , where S is the total spin, L is the orbitalangular momentum and j is the total angular momentum.19the truncated HO framework, the many-body antisymmetrizer is constructed recur-sively as in Refs. [68–70]. The generalization of LS equations to heavier systems,in either coordinate or momentum space, leads to the coupled integral equations ofthe Fadeev-Yakubovskii formalism [71, 72] that accounts for the antisymmetriza-tion of the wave function components in the integral equations. Another approachthat was shown to be very efficient at constructing antisymmetrized wave func-tions for A≥ 3 is the Hyperspherical Harmonics methods [73, 74]. This formalismhas been successfully applied to tackle few-body problems [75, 76] and is morecomputationally efficient than the HO basis expansion for nuclei in 3 ≤ A ≤ 6.Other approaches to solve the few-body systems are also possible, such as Quan-tum Monte Carlo techniques as in Refs. [77, 78]. A more complete description ofthese methods are found in the following reviews [67, 79] and references therein.2.4 Pionless effective field theoryAnother effective field theory used in this work was pionless effective field theory(pi -EFT). In nuclear systems at energies below the pion mass, the dynamics of thepion are unresolved and nuclear interactions can be described by short-range, inter-nucleon, contact interactions. It has been demonstrated in Ref. [80] thatpi -EFTtheory is equivalent to the effective range expansion as described in Ref. [81]. Theeffective range expansion relates the S-wave phase shift δ0(k), that depends onmomentum k, to the scattering length a and the effective range parameter ρt of theNN interaction ask cot(δ0(k)) =−1a +12ρtk2+ ... (2.33)where ρt = 1.765(4) fm. In the effective range expansion the deuteron S wavefunction is a decaying exponential function〈r|N0〉= Aνse−γdrrY M01(rˆ)χT=0MT , (2.34)with γ2d = 2mpnEB and where mpn is the reduced mass of the proton and neutron.The binding energy of the deuteron EB is −2.2245 MeV and Aνs is the asymptoticnormalization constant at order ν of thepi -EFT Lagrangian. The leading- and sub-20leading normalization constants of the deuteron areALOs =√2γd , (2.35)AN2LOs =√2γd1− γdρt . (2.36)Because the deuteron is a loosely bound system with no excited state the com-pleteness of states of the deuteron excited states, |N〉, can be represented as thecompleteness of plane waves∑N 6=N0|N〉〈N|=∫ d3k(2pi)3|k〉〈k|− |N0〉〈N0|, (2.37)where 〈r|k〉= eik·r and 〈r|N0〉 is given in Eq. (2.34). However, for deuteron S-wavetransitions a different parametrization is used that depends on the scattering length[82]〈r|ψ〉=√4pi(j0(kr)+aS1+ ikaSeikrr)Y M011(rˆ)χT=0MT . (2.38)Here, aS represents either the triplet or singlet scattering lengths determined fromexperiment [83]aS=0 =−23.748(10) fm, (2.39)aS=1 = 5.4194(20) fm. (2.40)21Chapter 3Electromagnetic interactionsThe EM interaction is one of the crowning pillars of modern theoretical physics anddescribes to an unparalleled level of precision the interactions between light (i.e.photons) and matter. Vasts amounts of nuclear physics data have been gatheredfrom EM probes.In this Chapter we introduce the theoretical framework that describes EM pro-cesses in nuclei. The EM operators are introduced as functional derivatives of theEM-Hamiltonian density and their gauge conditions are described in Section 3.1.This is followed by the discussion of models of EM operators at LO and next-to-leading-order (NLO) order in Section 3.2. In Section 3.3 the formalism of the spher-ical tensor decomposition of EM operators is introduced and applied in Section3.3.2 to derive the LO and NLO contributions of two-body currents to the magneticmoment operator. Finally we apply this formalism to obtain the photoabsorptioncross section in Section Electromagnetic operatorsThe Hamiltonian density describing EM interactions can be expressed as in Ref. [9]byHem(Aµ) = H0+Hint(Aµ) (3.1)where Aµ is the EM field of the photon, H0 the free Hamiltonian density in theabsence of external interactions and Hint is the Hamiltonian density coupling the22nucleus to photons shown schematically in Fig. 3.1. The interaction Hamiltoniandensity is expanded to second order as a function of the photon field Aµ as1Hint(Aµ) =∫d3x Jµ(x)Aµ(x)+12∫d3x d3y Aµ(x)Bµν(x,y, t)Aν(y), (3.2)where Jµ is the current density of the nucleus and Bµν is the operator that couplesthe nucleus to two-photons. This two-photon operator gives rise to the seagull-term(see Chapter 4). The current density and two-photon operators are the functionalderivatives of the EM-Hamiltonian density with respect to the photon field Aµ(x)in the limit of a vanishing vector fieldlimAµ→0δHem(Aµ)δAµ(x)= Jµ(x), (3.3)limAµ→0δHem(Aµ)δAµ(x)δAν(y)= Bµν(x,y)δ (x0− y0). (3.4)Figure 3.1: Photon-matter coupling diagrams: (a) the one-photon currentoperator Jµ ; (b) the two-photon operator Bµν .3.1.1 Gauge invariance of the electromagnetic hamiltonian densityOne fundamental property of the EM-Hamiltonian density is that it must be gaugeinvariant. The gauge transformations of the photon fields and wave functions are1Here we introduce the four vector notation x = (t,x)23[9, 10]Aµ → Aµ −∂µλ , (3.5)ψ → eiχ(λ ,t)ψ, (3.6)where χ(λ , t) is the linear functional of λ definedχ(λ , t) =∫d3x λ (x, t)Ω(x, t), (3.7)and Ω(x, t) is an arbitrary function determined from gauge invariance. Under thetransformation in Eqs. (3.5) and (3.6), the EM-Hamiltonian density becomes2Hem(A)→ e−iχ(λ ,t) (Hem(A−∂ )− i∂t)eiχ(λ ,t), (3.8)and gauge invariance imposes the condition thatHem(A) = e−iχ(λ ,t)Hem(A−∂ )eiχ(λ ,t)+∂tχ(λ , t). (3.9)Expanding Eq. (3.9) to terms of linear order in λ , ∂tλ , and to terms that are inde-pendent of Aµ , results in the condition∫d3x(i [H0,Ω(x, t)]+∇tχ+∇ · J)λ (x, t)=∫d3x(Ω(x, t)−ρ(x, t))∂tλ (x, t). (3.10)This expression must hold for any arbitrary λ and ∂tλ that are treated indepen-dently. This only holds if the terms in the brackets of Eq. (3.10) identically vanish[9, 10], which results in the constraintsΩ(x, t) = ρ(x, t),∇ · J(x, t)+ i [H0,ρ(x, t)]+∂tρ(x, t) = 0. (3.11)To obtain the gauge conditions for the Bµν operator, the transformations in Eq. (3.9)are considered to linear order in λ , ∂tλ and the fields Aµ , resulting in the condition2Here we introduce the short hand notation ∂ = (∂t ,∇)24that [9, 10]∫d3x∫d3x′δ (x′− y) (i[Jµ(x′, t),ρ(x, t)]+∇ jB jµ(x,x′, t))λ (x, t)=∫d3x∫d3x′δ (x′− y)(B0µ(x,x′, t))∂tλ (x, t). (3.12)As before, the above expression must be satisfied for any arbitrary λ and ∂tλ . Thisonly holds if the individual terms in the brackets vanish, imposing the followingrestrictions on the two-photon operatorBµ0(x,x′, t) = 0,B0µ(x′,x, t) = 0,3∑k=1∇kBk j(x,x′, t) = i[ρ(x, t),J j(x′, t)], (3.13)where j and k are the spatial indices ( j,k) = 1,2,3.3.2 Models of electromagnetic operatorsFollowing the discussion of the gauge invariance conditions of the one and two-photon operators Jµ and Bµν , here we focus on specific models for the current Jµthat are relevant for our investigations of EM processes and that satisfy gauge invari-ance. To begin, the EM current is expanded into a sum of one, two and many-bodyoperators in direct analogy to the one, two and many-body hierarchy of nuclearforces asJν(x) =A∑iJν[1],i(x)+A∑i jJν[2],i j(x)+ . . . , (3.14)where the operators Jν[1],i and Jν[2],i j are the one and two -body current operators, re-spectively. Here the ellipsis denotes more than two-body operators. The one-bodyEM operators also known as the LO currents are derived from the non-relativisticexpressions of the EM interaction Hamiltonian of a A-body nucleus, from the min-imal substitution principle. The LO EM Hamiltonian up to terms linear in the field25Aµ is given byHem(Aµ) =A∑ieˆp,iA0+eˆp,i2m(pi ·A+A · pi)+µˆi2mσ i · (∇×A) , (3.15)where pi are the momenta of the nucleons of mass m, Aµ are the components ofthe electromagnetic 4-vector potential and σ i are the Pauli matrices of the nucleonspin operators. The operator eˆp,i is the charge projection operator of the protonand µˆi are the magnetic moment projectors of the nucleons. While the charge ofthe neutron is zero in the point nucleon limit, its charge projection operator is eˆn,iand will be defined here for completeness. These charge and magnetic projectionoperators areeˆp,i =(1+ τ3i2), (3.16) eˆn,i =(1− τ3i2), (3.17)µˆi = µp(1+ τ3i2)+µn(1− τ3i2), (3.18)with τ3i being the third component of the nucleon isospin (+1 for a proton and −1for a neutron) and µp = 2.793 and µn = −1.913 in nucleon magneton µN units.From Eq. (3.3) the charge and current density operators Jµ = (ρ,J) of Eq. (3.15)areρ[1],i(x) = eˆp,iδ (x− xi) (3.19)Jc[1],i(x) =eˆp,i2m{pi,δ (x− xi)}, (3.20)J s[1],i(x) = iµˆi2mσ i× [pi,δ (x− xi)] . (3.21)In the above expressions, the impulse approximation current J [1],i has been sepa-rated into the convection Jc[1],i and spin-magnetization Js[1],i contributions, with thespatial coordinates of the i-th nucleon xi. With these expressions it is straight-forward to show that the LO operators satisfy the continuity equationA∑i=1(∇ · J [1],i(x)+ i[H0,ρ[1],i (x)])= 0, (3.22)26where the non-interacting Hamiltonian density is given by just the kinetic energyH0 =A∑ip2i2m. (3.23)However, nuclear interactions VNN are included into this Hamiltonian density through,e.g., asH0→A∑ip2i2m+VNN , (3.24)if we have just two-body forces. In this case, the continuity equation in Eq. (3.22)is no longer satisfied because the LO charge density operator ρ[1](x) does not com-mute with the nuclear potential, i.e.,[VNN ,ρ[1](x)] 6= 0. Nevertheless, the nuclearcurrent can be made to satisfy the continuity equation by introducing the two-bodycurrent density J [2],i j (see Eq. (3.14))J(x) =A∑iJ [1](x)+A∑i jJ [2],i j(x), (3.25)that satisfies the conditionA∑i j(∇ · J [2],i j(x)+ i[VNN ,ρ[1],i (x)])= 0. (3.26)Within the χEFT framework, these two-body EM currents are derived as a hierar-chy of operators with a power-counting scheme analogous to the underlying χEFTtheoryJν[1],i(x) = JLO,ν[1],i (x)+ JNLO,ν[1],i (x)+ . . . , (3.27)andJν[2],i j(x) = JNLO,ν[2],i j (x)+ JN2LO,ν[2],i j (x)+ . . . . (3.28)The operator JLO,ν[1],i represents the leading order operators which in the power count-27ing of Refs. [84, 85] are the spin and convection currents3 in Eqs. (3.20) and (3.21).The ellipsis in the above equations denote higher order in χEFT contributions,such as N2LO and next-to-next-to-next-to-leading-order (N3LO) etc. The opera-tors JNLO,ν[1],i and JNLO,ν[2],i j , are the NLO contributions to the one and two -body EMoperators. In the χEFT power counting scheme there are no two-body contribu-tions to the NLO charge density term JNLO,ν=0[2],i j . The two-body current density JNLO[2],i jarises from the coupling of the photon to the pion and nucleon lines of the one-pionexchange potential from Fig. 2.1. This coupling results in two contributions [86],known as the seagull and pion-in-flight diagrams pictured in Fig. 3.2.Figure 3.2: The interaction diagrams that make up the two-body currentsat LO in χEFT: the pion in flight (left) and the seagull (right). The wigglerepresents the electromagnetic interaction (the photon).The contributions of the seagull and pion-in-flight current are denoted J s[2],i jand Jpi[2],i j, respectively. Their momentum space expressions are [87]J s[2],i j(k i,k j) =−ig2AF2pi(τ i× τ j)z(σ i(σ j · k jω2k j)−σ j(σ i · k iω2ki)), (3.29)Jpi[2],i j(k i,k j) =−ig2AF2pi(τ i× τ j)z (k j− k i)(σ i · k iω2ki)(σ j · k jω2k j), (3.30)3We note in passing that there exist other power counting conventions that place Jc and J s at ahigher order in the χEFT expansion.28and their sum is the total two-body NLO currentJNLO[2],i j(k i,k j) = Js[2],i j(k i,k j)+ Jpi[2],i j(k i,k j). (3.31)In these expressions, k i/ j denote the momentum transferred to the i-th or j-th nu-cleon and ω2ki/ j = k2i/ j +m2pi is the square of the total energy of the exchanged pion,with mass mpi , while τ i/ j are the nucleon isospin Pauli matrices.The coordinate space expressions of these currents are obtained by performingthe Fourier transform resulting inJ s[2],i j(q) =−m2pig2A4piF2pi(τ i× τ j)z eiq·R[e12 iq·rσ i (σ j · rˆ)+e−12 iq·rσ j (σ i · rˆ)](1+1mpir)e−mpi rmpir, (3.32)Jpi[2],i j(q) =g2A(2pi)3F2pi(τ i× τ j)z eiq·R(σ i ·(12q− i∇r))×(σ j ·(12q+ i∇r))∇rI(q,r), (3.33)where q is the momentum transfer, ∇r is the gradient with respect to the relativecoordinate r given byr = r i− r j, (3.34)and where the two-body center of mass coordinate isR =12(r i+ r j) . (3.35)In these expressions, the function I(q,r) arises from the Fourier transform of thepion-in-flight term and is defined asI(q,r) =∫d3 peip·r(m2pi +(p− 12 q)2)(m2pi +(p+ 12 q)2) . (3.36)293.3 Spherical tensor decomposition of electromagneticoperatorsBecause nuclear states are eigenstates of the total angular momentum of the nuclearHamiltonian, it will be computationally convenient to decompose the charge andcurrent density into spherical tensor operators of definite rank κ , projection ν , andparity Π, i.e. into operators that commute with the total angular momentum. Bydefinition, the momentum space expressions for the charge and current density arerelated to the coordinate space expressions via the Fourier transform throughρ˜(q) =∫d3x eiq·xρ(x), (3.37)J˜(q) =∫d3x eiq·xJ(x). (3.38)By expanding the exponential in Eq. (3.37) as plane waves, it can be shown thatthe current density is the sumρ˜(q) = (4pi) ∑κ≥0,νiκCκν(q)Y κ∗ν (qˆ), (3.39)where the spherical tensor operators Cκν(q) are the Coulomb operators/tensorsCκν(q) =14piiκ∫dqˆ ρ˜(q)Y κν (qˆ) =∫d3x ρ(x) jκ(qx)Y κν (xˆ). (3.40)Equation (3.40) gives the momentum and coordinate space definitions of the Coulombmultipole tensors, respectively4.The current density is a rank one tensor which means that it can be expandedas a sum of three spherical tensors of ranks `= κ−1,κ,κ+1J˜(q) = ∑κ≥1ν`Jνκ`(q)Yν∗κ`(qˆ),=∑κν(Jνκκ−1(q)Yν∗κκ−1(qˆ)+ Jνκκ(q)Yν∗κκ(qˆ)+ Jνκκ+1(q)Yν∗κκ+1(qˆ)), (3.41)4The integration of Eq. (3.40) and related momentum space expressions is carried out over theangles qˆ where the magnitude |q| of the momentum transfer is fixed.30where the operators Y ν∗κ` are the vector spherical harmonics defined in AppendixD. The current density multipole expansion operators Jνκ`(q) in momentum andcoordinate space areJνκ`(q) =∫dqˆ J˜(q) ·Y νκ`(qˆ) = 4pii`∫d3x j`(qx)J(x) ·Y νκ`(xˆ). (3.42)To simplify Eq. (3.41), the three operators, {Jνκκ+1(q), Jνκκ(q), Jνκκ+1(q)} will bere-written. Using the definitions of the vector spherical harmonics and other prop-erties from Appendix D the first of these operators isJνκκ+1(q) =∫dqˆ J˜(q) ·Y νκκ+1(qˆ)=∫dqˆ(−√κ+12κ+1(qˆ · J˜(q))Y κν (qˆ)+ i√κ2κ+1(qˆ× J˜(q)) ·Y νκκ(qˆ)), (3.43)while the operator Jµκκ−1(q) is similarly written asJνκκ−1(q) =∫dqˆ J˜(q) ·Y νκκ−1(qˆ)=∫dqˆ(√κ2κ+1(qˆ · J˜(q))Y κν (qˆ)+ i√κ+12κ+1(qˆ× J˜(q)) ·Y νκκ(qˆ)). (3.44)The sum of the operators in Eq. (3.43) and (3.44) using Appendix D simplifies tothe formJνκκ+1(q)Yν∗κκ+1(qˆ)+ Jνκκ−1(q)Yν∗κκ−1(qˆ) =(qˆY κ∗ν (qˆ))∫dqˆ(qˆ · J˜(q))Y κν (qˆ)− (qˆ×Y ν∗κκ(qˆ))∫ dqˆ(qˆ× J˜(q)) ·Y νκκ(qˆ). (3.45)The previous formula can be recast by introducing the multipoles of the electro-magnetic tensor operators. The multipoles of the longitudinal electric operator Lκνin momentum and coordinate space are definedLκν(q) =14pi∫dqˆ(qˆ · J˜(q))Y κν (qˆ),=iκ+1q∫d3x (∇ · J(x))Y κν (xˆ) jκ(qx). (3.46)31The above tensor is the longitudinal operator because according to Eq. (3.45) it isthe tensor component of the current J˜ parallel to the direction of q. The multipolesof the transverse electric tensor T Eκν areT Eκν(q) =14pii∫dqˆ (qˆ× J(q)) ·Y νκκ(qˆ),=iκq∫d3x (∇× jκ(qx)Y νκκ(xˆ)) · J(x). (3.47)Finally, the multipoles of the transverse magnetic tensor T Mκν areT Mκν(q) =14pi∫dqˆ J˜(q) ·Y νκκ(qˆ),= iκ∫d3x jκ(qx)J(x) ·Y νκκ(xˆ). (3.48)Using these definitions, the current J˜ isJ˜(q) = ∑κ≥1ν4pi [(qˆY κ∗ν (qˆ))Lκν(q)−(qˆ×Y ν∗κκ(qˆ)) iT Eκν(q)+Y ν∗κκ(qˆ)T Mκν(q)]. (3.49)In the special case where the momentum transfer q is parallel to the z-axis, thenY κ∗ν (qzˆ) =√2κ+14piδµ0, (3.50)Y ν∗κκ(qzˆ) =−ν√2√2κ+14pie∗ν , (3.51)qˆ×Y ν∗κκ(qzˆ) = iν√2√2κ+14pie∗ν , (3.52)where eν are the spherical basis unit vectors from Appendix D. The spherical tensorcomponents λ = {−1,0,1} of Eq. (3.49) are then [14, 15]Jλ (q) = (−1)λ√2pi(1+δλ0)×∑κ≥1√2κ+1(δλ0Lκλ (q)+δ|λ |,1(T Eκλ (q)+λTMκλ (q))). (3.53)323.3.1 Long wavelength approximations of electromagnetic tensorsIn the previous section we derived the exact expressions of the EM tensor operatorsL,T E, and T M that are involved in the infinite sum of Eq. (3.53). The numberof terms required to approximate this sum depends on the momentum transfer qconsidered. However, for many processes the momentum transfer involved is lowq→ 0 and only a few multipoles contribute. For these cases it is sufficient toconsider the behavior of these tensors in the long-wavelength approximation whereqr 1. In this regime the electric multipole tensor can be written as a sum of theSiegert term RE [88] and a Siegert correction term δRET Eκν(q) = REκν(q)+δREκν(q), (3.54)whereREκν(q) =−iκωq√κ+1κ∫d3x jκ(qx)Y κν (xˆ)ρ(x), (3.55)andδREκν(q) =−iκ+1√2κ+1κ∫d3x jκ+1(qx)Y νκκ+1(xˆ) · J(x). (3.56)Under the assumption of current conservation and by expanding the spherical Besselfunction in Eqs. (3.55) and (3.56) to leading order, the Siegert term and correctionoperators reduce to the form5REκν(q)→−iκqκ(2κ+1)!!√κ+1κOκE,ν , (3.57)δREκν(k)→ 0. (3.58)The general electric-multipole tensor operator OκE,ν isOκE,ν =∫d3x xκρ(x)Y κν (xˆ). (3.59)5The application of the current conservation, q · J˜(q) =ωρ˜(q), in Eq. (3.55) leading to Eq. (3.57)is the Siegert theorem which has many application in low energy nuclear physics.33The above tensor represents the κ-th spherical moment of the charge density oper-ator. In analogy to the transverse electric multipole tensor, the transverse magnetictensor operator may also be written as a sum of two tensors, the leading order RMand the correction term δRMT Mκν(q) = RMκν(q)+δRMκν(q), (3.60)with expressions given by6RMκν(q) =√κ+1κiκ+1∫d3x[x× J(x)κ+1+Ms(x)]·∇ ( jκ(qx)Y κν (xˆ)) , (3.61)δRMκν(q) =−iκ+1√2κ+1κq∫d3x jκ+1(qx)Y νκκ+1(xˆ) ·Ms(xˆ). (3.62)In the above equations the total spin magnetic moment operator Ms has been intro-duced, whereMs(x) =A∑iµˆi2mσ iδ (x− xi). (3.63)In the low momentum approximation, we haveRMκν(q)→ iκ+1qκ√(κ+1)(2κ+1)(2κ+1)!!OκM,ν , (3.64)δRMκν(q)→ 0, (3.65)where the operator OκM,ν is the general magnetic multipole tensor operatorOκM,ν =∫d3x xκ−1[Ms(x)+1κ+1x× J(x)]·Y νκκ−1(xˆ). (3.66)Unlike the case for the electric multipoles, there is no equivalent of the Siegerttheorem for the magnetic multipole operators.6The identity∫d3x jκ (qx)J(x) ·Y νκκ (xˆ) = 1√κ(κ+1)∫d3x (−ix×∇) [ jκ (qx)Y νκκ (xˆ)] · J(x) fromRef. [12] has been used.343.3.2 Magnetic moment operatorsIn the previous section we developed the formalism to describe the EM operators oflow momentum processes that are general for any nuclear current density J(x). Upto a simple factor, the transverse magnetic tensor operator in Eq. (3.66) for the caseof κ = 1 is the magnetic dipole operator, denoted µ . This is of particular interestto nuclear physics because it induces transitions that are directly measurable fromexperiment such as the HFS, the magnetic transition strength B(M1) and nuclearmagnetic moments. Due to its importance for low energy physics processes rele-vant to this work, we will derive formulas for the magnetic moment contributions atthe LO and also with the NLO contributions from two-body currents. In coordinatespace the magnetic dipole operator is defined7µ =12∫d3x x× J(x), (3.67)which in momentum space is [63]µ =− i2limq→0∇q× J˜(q), (3.68)where ∇q = (∂qx ,∂qy ,∂qz) is the gradient operator with respect to the componentsof q. In analogy to the current, this magnetic moment operator can be expanded inone and two -body terms asµ i =A∑iµLOi +A∑i jµNLO[2],i j. (3.69)The LO one-body magnetic moment operator is given byµLOi = µN (eˆp,i`i+ µˆiσ iδ (x− xi)) , (3.70)where `i is the angular momentum operator of the i-th nucleon and µN is the nuclearmagneton µN = 1/(2m). The NLO two-body currents are translationally invariantwith respect to the two-body center of mass coordinate R and are therefore of theform eiq·RJ [2],i j(q,r) as shown in Eqs. (3.32) and (3.33). In this form the applica-7If there is no convection current and only the spin current is present then µ (x) = Ms(x).35tion of Eq. (3.68) will break up the NLO magnetic moment operator into the sumof termsµNLO[2],i j(r,R) = µint[2],i j(r)+µSachs[2],i j (r,R), (3.71)where µ int[2],i j is the intrinsic magnetic moment operator definedµ int[2],i j(r) =−i2limq→0∇q× J˜ [2],i j(q,r), (3.72)that is only a function of the relative coordinate between the i-th and j-th nucleonsr, while the term µ Sachs[2],i j is the Sachs term that also depends on the two-body centerof mass coordinateµ Sachs[2],i j (r) =12R× J˜ [2],i j(q,0). (3.73)Using the current density expressions in Eqs. (3.32)-(3.33) the contributions fromthe seagull and pion-in-flight from Fig. 3.2 to the magnetic moment are, respec-tivelyµ s[2],i j(r) =−g2Ampi16piF2pi(τ i× τ j)z [(rˆ×σ i)(σ j · rˆ)− (rˆ×σ j)(σ i · rˆ)] f (r), (3.74)µ pi[2],i j(r) =−g2Ampi16piF2pi(τ i× τ j)z [(rˆ×σ i)(σ j · rˆ)− (rˆ×σ j)(σ i · rˆ)] f (r)− g2Ampi8piF2pi(τ i× τ j)z (σ i×σ j)Y (r), (3.75)where the constants in the above expressions are the same as in Eq. (2.19) and thefunctions f (r) and Y (r), with r = |r| aref (r) =(1+1mpir)e−mpi r, (3.76)Y (r) =e−mpi rmpir. (3.77)At NLO the total intrinsic two-body magnetic moment contribution is the sumµ int[2],i j = µs[2],i j + µpi[2],i j as in Ref. [86]. The intrinsic magnetic moment operator36is [89]µ int[2],i j(r) =−g2Ampi8piF2pi(τ i× τ j)z×[(1+1mr)((σ i×σ j) · rˆ) rˆ− (σ i×σ j)]e−mpi r. (3.78)The Sachs term at NLO is [63]µ Sachs[2],i j (r,R) =−12(τ i× τ j)zV1pi(r)(R× r) , (3.79)where V1pi(r) is the one-pion exchange potential from Eq. (2.19). In this work, weimplement these operators in several few-body A ≥ 2 systems. For the deuteron,only the intrinsic operator will contribute since R = 0. For larger systems, the Sachsterm also contributes. Eqs. (3.78)-(3.79) represents the full NLO-M1 operator thatwill be used in this work.3.4 The photoabsorption cross sectionOne of the most fundamental EM processes is the nuclear photo-absorption crosssection. This process is depicted schematically in Fig. 3.3 in which a real photon γis absorbed by an A-body nucleus in its ground state |ΨI〉 with its A-body center-of-mass 4-momentum PµI = (EI,PI). After absorbing a photon with 4-momentumqµ = (ω,q) the nucleus transitions into the final state |ΨF〉 with the new centerof mass momentum PµF = (EF ,PF). The translational invariance of the nuclearFigure 3.3: The photo absorption process. The incoming nucleus in state|ΨI〉 absorbs a photon and transitions to the final state |ΨF〉.Hamiltonian with respect to the A-body center of mass coordinate implies thatthe A-body nuclear wave function is a product state of center-of-mass plane-wave37motion |P〉 and an internal wave function given by|ΨI〉= 1√V|PI〉⊗ |N0J0;M0〉, (3.80)|ΨF〉= 1√V|PF〉⊗ |NJ;M〉, (3.81)where V is the normalization volume and |N0J0;M0〉 is the internal nuclear groundstate with total angular momentum J0 and projection M0. The final intrinsic nuclearstate is |NJ;M〉 with total angular momentum J and projection M. These internalstates have 3(A−1) degrees of freedom. The nuclear EM differential cross sectionof this process is [13, 15]dσ(ω) =12J0+1∑M012∑λ ∑J ∑MΓFI|Jin| , (3.82)where λ is the photon polarization and Jin is the initial photon flux. The term ΓFIis the photoabsorption transition density from the initial state I to the final state F .By Fermi’s golden rule [90] the transition density between initial and final states isΓFI = 2pi|〈F |Hint|I〉|2δ (EF −EI−ω)ρd , (3.83)where 〈F |Hint|I〉 is the matrix element of the EM interaction Hamiltonian Hint andρd is the density of final states of the nucleus8ρd =V(2pi)3d3PF . (3.84)The EM interaction Hamiltonian density isHint(A) =−∫d3x J(x) ·A(x), (3.85)where A is the external EM field and J is the nuclear current. To evaluate thematrix element 〈F |Hint|I〉 in Eq. (3.83), the external EM field in Eq. (3.85) mustbe quantized. This quantization is carried out over a volume V in the Coulomb8The momentum PF of the center-of-mass of the final state is integrated over all possible magni-tudes and directions.38gauge and by expanding A in terms of photon creation and anhilitation operatorsand plane waves summed over the two photon polarizations λA(x) = ∑λ=±1∫ d3q√2ωV[eq,λaq,λ eiq·x + e†q,λa†q,λ e−iq·x]. (3.86)Here eq,λ are the spherical basis unit vectors transverse to the direction of thephoton vector q. The operator aq,λ is the photon creation operator defined by itsaction on the photon vacuum state |0〉aq,λ |0〉= |q,λ 〉, (3.87)along with the commutation relation[aq,λ ,a†q′,λ ′]= δλλ ′δ (q′−q). (3.88)The state |q,λ 〉 is a photon with momentum q and polarization λ . Finally, todescribe the photoabsorption process we take the initial state |I〉 to be the productstate of |ΨI〉 with the photon vacuum state |0〉, while the final state |F〉 is theproduct state of |ΨF〉 with the external photon state |q,λ 〉|I〉= |ΨI〉⊗ |q,λ 〉, (3.89)|F〉= |ΨF〉⊗ |0〉. (3.90)With these states the EM transition matrix elements are〈F |Hint|I〉=−i∫d3x e−i(PF−PI)·x〈ΨF |J(0)|ΨI〉 · 〈0|A(x)|q,λ 〉, (3.91)where the translational invariance of the nuclear current has been usedJ(x) = eiP·xJ(0)e−iP·x. (3.92)39The photon transition matrix element in Eq. (3.91) can be evaluated directly fromthe relations in Eq. (3.87) and (3.88) to give〈0|A(x)|q,λ 〉=e†q,λ e−iq·x√2ωV. (3.93)The plane-wave center-of-mass wave functions in Eqs. (3.80) and (3.81) are usedin Eq. (3.91) to evaluate the nuclear matrix element as [15]〈ΨF |J(0)|ΨI〉= 1V 〈NJ;M|J˜(PF −PI)|N0J0;M0〉, (3.94)where J˜ is the Fourier transform of the current density. Combining these expres-sions together, the transition amplitude is|〈ΨF |Hint|ΨI〉|2 = (2pi)32ωVδ (PF −PI− k)×|e†q,λ · 〈NJ;M|J˜(PF −PI)|N0J0;M0〉|2. (3.95)Using the Wigner-Eckart theorem in Appendix C and summing over the Clebsch-Gordan coefficients [91, 92] we obtain∑J∑MJM0ΓI→F =(2pi)3αωV ∑κ≥1[|〈NJ||T Eκ (q)||N0J0〉|2+|〈NJ||T Mκ (q)||N0J0〉|2]δ (EN−E0−ω)δ (PF −PI−q), (3.96)where E0 is the energy of the ground state and EN is the excited state energy. Thislast expression can be integrated over the final momentum PF that eliminates themomentum delta function. The resulting expression is independent of the photonpolarization λ and consequently, the summation over λ introduces a factor of twointo the expression. The final result for the photoabsorption cross section isσκ(ω) =α(2pi)3ω(2J0+1) ∑N 6=N0∑κ≥1[|〈N||T Eκ (q)||N0〉|2+|〈N||T Mκ (q)||N0〉|2]δ (EN−E0−ω). (3.97)40In the long-wavelength approximation developed in Section 3.3.1, this cross sec-tion can be written as the sum of the electric and magnetic cross section contribu-tions σκ(ω) = σκ,E(ω)+σκ,M(ω) where the contribution from the electric tensorsisσκ,E(ω) =α(2pi)3(κ+1)κ [(2κ+1)!!]2ω2κ−1Sκ,E(ω), (3.98)and the magnetic contributions areσκ,M(ω) =α(2pi)3(κ+1)(2κ+1)[(2κ+1)!!]2ω2κ−1Sκ,M(ω), (3.99)where the magnetic/electric response function of the nucleus isSκ,E/M(ω) =12J0+1∑N 6=N0|〈N0||OκE/M||N〉|2δ (EN−E0−ω). (3.100)The construction of the above response functions can be carried out using maxi-mum entropy methods as described in Appendix F.41Chapter 4Muonic Atom TheoryThe accurate predictions of the energy levels of hydrogenic atomic systems1 aresignificant demonstrations of the validity of quantum mechanics. In the non-relativistic quantum mechanics framework, the Schro¨dinger equation for a leptonof mass m` interacting with a nucleus of mass MN via the Coulomb force (in unitswhere h¯ = c = 1) is given by(− ∇22mr− (Zα)r)ψn`m(r) = Enψn`m(r), (4.1)where n is the principal quantum number, ` is the angular momentum of the nucleus-lepton system and m is projection of the angular momentum. The term mr is thereduced mass of the lepton and nucleus and α is the fine structure constant. Thesolutions to this equation are the hydrogenic wave functionsψn`m(r) = φn(0)√4pi2`+1Rn`(r)Y `m(rˆ), (4.2)whereφn(0) =ν3/2√pi, (4.3)1These are ions with one negatively charged lepton orbiting any positively charged atomic nu-cleus.42and withν =mr(Zα)n. (4.4)Here Rn`(r) are the hydrogenic radial functions in Ref. [90] with bound state ener-giesEn =−mr(Zα)2n2 . (4.5)In this non-relativistic case the energies in Eq. (4.5) are degenerate with respect to`. In the generalized Dirac treatment of the hydrogenic atom, the energy levels aregiven by [93]En, j = mr1+ (Zα)2(√( j+ 12)− (Zα)2+n− j− 12)2− 12, (4.6)where j = `± 12 . This partially lifts the degeneracy of the atomic energy levelswith respect to the angular momentum. However, there is still a double degeneracybetween ` states of equal total angular momentum2 j = 12 ⊗ `. For example, the2S state where `= 0 and the 2P state where `= 1 can both have total angular mo-mentum j = 1/2 and consequently have the same energy level given by Eq. (4.6).In Nature the 2P and 2S states are however, not degenerate. The small splittingbetween the 2S and 2P energy levels is known as the Lamb shift [30], denoted∆ELS(2S− 2P), arises primarily from the effects of QED, and to a lesser extentfrom nuclear structure. For comparison, the QED effects contribute 228.7766(10)meV [50, 53] to the Lamb shift while the nuclear structure effects are about 100times smaller (see Sections 6.2 and 6.3.3).The atomic energy states n` j will be further separated according to the totalangular momentum of the lepton-nucleus system F = j⊗J, where j is the total an-gular momentum of the lepton and J is the total angular momentum of the nucleus.This energy separation arises from the interaction of the nuclear and leptonic spin2The operator ⊗ denotes the tensor product of two spherical tensors of different ranks. Forexample, for two spherical tensors of rank j1 and j2, respectively, their tensor product is j1⊗ j2denotes a new tensor of rank j12 where j12 = {| j1− j2|, | j1− j2|+1, . . . , | j1 + j2|−1, | j1 + j2|}.43currents. The resulting energy shift is the hyperfine splitting (HFS) and its effectsto the nS states is denoted ∆EHFS(nS).Entering into both the Lamb shift and the HFS are the nuclear structure correc-tions ∆Enucl(α), calculated in powers of the fine-structure constant α . The mainsubject of this work is to derive the formalism to calculate the nuclear structurecorrections to both the Lamb shift and the HFS. The nuclear structure correctionsto the Lamb and HFS are decomposed into elastic ∆Eelas and inelastic ∆Einel terms∆Enucl(α) = ∆Eelas(α)+∆Einel(α), (4.7)where in the latter contribution, the nucleus is excited to its intermediate states. Theelastic contribution to the Lamb shift is proportional to the third electric Zemachmoment, 〈R3E〉, while for the HFS, this term is proportional to the Zemach ra-dius 〈RZ〉. Due to the importance of the EM moments for the nuclear structurecorrections, we first outline the formalism to calculate the EM moments relevantfor ab-initio calculations in Section 4.1. Following this, we establish the general-ized covariant formalism to compute the Lamb shift in Section 4.2 and provide anoverview of the alternate η-expansion in Section 4.3. Finally, we extend this gen-eralized formalism to tackle the HFS and provide the non-relativistic expression inSection The electromagnetic moments of nucleiThe root-mean square (RMS) charge radius of a nucleus is one of the fundamentalnuclear properties and of interest to the broader physics community. As previouslydiscussed, precision muonic atom spectroscopy experiments of the Lamb shift andHFS probe the RMS charge radius as well as other EM moments such as the thirdZemach moment 〈R3E〉(2) and Zemach radius 〈RZ〉 that enter into the Lamb shiftand HFS formalism, respectively.In this Section we discuss general EM moments using the momentum space,coordinate space, and a newly developed hybrid formalism [1] focusing on thedeuteron that was used for benchmark calculations in its development. This hybridformalism has been found to be more precise and numerically stable in A≥ 3 nucleiand will be an important tool in the context of muonic atom spectroscopy.444.1.1 Momentum space formulationThe electric and magnetic form factors of the deuteron are definedFE(q2) =13 ∑M=±1,0〈N0J0;M0|ρ˜(qzˆ)|N0J0;M0〉, (4.8)FM(q2) =√2mdqIm{〈N0J0;M0|J˜y(qzˆ)|N0J0;M0〉}. (4.9)Here |N0J0;M0〉 introduced in Eq. (3.80), denotes the ground state wave function ofthe deuteron previously introduced. We have taken the momentum transfer q alongthe z-axis and md is the deuteron mass. These form factors have been normalized to1 and (mµ/m)µd , respectively, where 〈µLO〉≡ µd is the deuteron magnetic momentwith the LO magnetic dipole operator in Bohr magneton units µN . In the case ofthe deuteron, Eqs. (4.8) and (4.9) can be be evaluated in terms of the S and D wavefunctions in coordinate space asFE(q2) =(GpE(q2)+GnE(q2)) ∞∫0dr(u2(r)+w2(r))j0(12qr), (4.10)where GpE(q2) and GnE(q2) are the form factors of the proton and neutron, respec-tively. The magnetic form factor of the deuteron isFM(q2) =(GpM(q2)+GnM(q2)) ∞∫0dr[(u2(r)− 12w2(r))j0(12qr)+(√12u(r)w(r)+12w2(r))j2(12qr)]+34(GpE(q2)+GnE(q2)) ∞∫0dr w2(r)[j0(12qr)+ j2(12qr)],(4.11)where GpM(q2) and GnM(q2) are the proton and neutron magnetic form factors, re-spectively. From these form factors the even moments of the electric (magnetic)45nuclear charge densities can be extracted from Fx(q2) in the limit where q→ 0 as〈R2x〉=−3!∂Fx(q2)∂q2, (4.12)〈R4x〉=5!2∂ 2Fx(q2)∂ 2q2, (4.13)with x= E(M). Furthermore, the electric Zemach moments entering into the Lambshift, which will be explained in Section 4.2, can be calculated as〈RE〉(2) =−4pi∞∫0dqq2[F2E (q2)−1] , (4.14)〈R3E〉(2) =48pi∞∫0dqq4[F2E (q2)−1− q23〈R2E〉], (4.15)while the Zemach radius that enters into the HFS is the product of the electric andmagnetic form factors〈RZ〉=− 4pi∞∫0dqq2[FE(q2)FM(q2)FM(0)−1]. (4.16)4.1.2 Coordinate space formulationIn coordinate space the nuclear charge density is the sum of proton and neutroncharge densitiesρ(x) = ρ p(x)+ρn(x), (4.17)where ρ p and ρn are the normalized proton and neutron charge density operators,respectively, given as [82]ρ p(x) =1ZA∑iρp(x− xi)eˆp,i, (4.18)ρn(x) =1ZA∑iρn(x− xi)eˆn,i. (4.19)46These formulas are the sum of the charge densities of the individual nucleons thatmake up the nucleus. The vector xi are the coordinates of individual protons andneutrons, where∫d3xi ρp(xi) = 1 and∫d3xi ρn(xi) = 0 and Z is the charge num-ber (or number of protons) of the nucleus while N is the neutron number. Thecharge projection operators eˆp,i and eˆn,i are defined in Eq. (3.16) and Eq. (3.17),respectively. Using the total charge density the squared charge radius operator isR2E =∫d3x ρ(x)x2,=1ZA∑i∫d3x (eˆp,iρp(x− xi)+ eˆn,iρn(x− xi))x2,=1ZA∑i∫d3y (eˆp,iρp(y)+ eˆn,iρn(y))(y2+2y · xi+ x2i ),=1ZA∑ieˆp,ix2i +1ZA∑ieˆp,ir2p+1ZA∑ieˆp,ir2n,= Rˆ2p+ r2p+NZr2n. (4.20)In the intermediate steps of the above equations we used∫d3y y · xi = 0 , r2p =∫d3y y2ρp(y) and r2n =∫d3y y2ρn(y) where r2p and r2n are the squared charge radiiof the proton and neutron, respectively. The operator R2p is the squared chargeradius operator of the nucleus with the point-proton charge density operator3Rˆ2p =1ZA∑ieˆp,ix2i . (4.21)The squared-RMS radius is defined as the expectation value of R2E ≡ 〈R2E〉 operatoron the ground state nuclear nuclear wave function〈R2E〉= 〈N0J0;M0|R2E |N0J0;M0〉. (4.22)In analogy to the RMS charge radius derived in Eq. (4.20) other even-momentssuch as the fourth-moment of the nuclear charge density can be extracted from3We remark that the coordinate operators xˆi are denoted simply as xi for convenience since weare working in coordinate space.47point-nucleon calculations in coordinate space〈R4E〉=∫d3x ρ(x)x4,=1ZA∑i∫d3y(eˆp,iρp(y)+ eˆn,iρn(y))(y2+2y · xi+ x2i )2,=1ZA∑i∫d3y(eˆp,iρp(y)+ eˆn,iρn(y))×(y4+ x4i +4(y · xi)2+2y2x2i +4(x2i + y2)(xi · y)),=1ZA∑i∫d3y(eˆp,iρp(y)+ eˆn,iρn(y))(y4+ x4i +4(y · xi)2+2y2x2i ), (4.23)where in the last expression we have dropped the terms that integrate to zero. Afterexpanding the term (y · xi)2 in spherical harmonics and carrying out the resultingangular integral4, we arrive at〈R4E〉= 〈R4p〉+ r4p+NZr4n +103(r2p〈R2p〉+NZr2n〈R2n〉). (4.24)The definitions for the R4p is analogous to Eq. (4.21) and the squared neutron radiusoperator Rˆ2n isRˆ2n =1NA∑ieˆn,ix2i . (4.25)The squared electric moment for the deuteron is explicitly written as〈R2p〉=∞∫0dr( r2)2 (u2(r)+w2(r)). (4.26)4 The identity∫dqˆ(y · xi)2 =∫dqˆ( 23 ∑m Y2∗m (yˆ)Y2m(xˆi)+13Y0∗0 (yˆ)Y00 (xˆi))x2i y2 = 13 x2i y2 has beenused.48Similarly, the squared magnetic charge radius for the deuteron is〈R2M〉=12µd(gpr2p,M +gnr2n,M)1− 32∞∫0dr w2(r)+12µd(gp+gn)∞∫0dr r2[14u2(r)− 110√2u(r)w(r)− 710w2(r)]+34µd(r2p+ r2n) ∞∫0dr w2(r)+980µd∞∫0dr r2w2(r), (4.27)where r2p,M and r2n,M are the magnetic radii of the proton and neutron, respectively.The constants gp and gn are the magnetic g-factors of the proton and neutron. Thenth electric, or Zemach moments are defined as a convolution of charge or magneticdensities in coordinate space as〈Rn〉(2) =∫d3r∫d3r′ρx′(r′)ρx(r)|r− r ′|n, (4.28)where x,x′ = (E,M).4.1.3 Mixed momentum and coordinate-space formulationThe third formalism used to compute the EM moments is the hybrid qr-space for-malism developed in Ref. [1]. For a given form factor Fx(q2), the correspondingcoordinate space density function in the non-relativistic limit is the inverse Fouriertransform of the form factorρx(r) =∫ d3q(2pi)3Fx(q2)e−iq·r . (4.29)Substituting the above expression into Eq. (4.28), we have that the nth Zemachmoment is〈Rn〉(2) =∫d3r∫d3r′∫d3q∫d3q′Fx′(q′2)eiq′·r ′Fx(q2)eiq·r |r− r ′|n,=∫d3q Fx(q2)Fx′(q2)∫d3z zneiq·z, (4.30)49where we have integrated out the delta function in q and made the substitutionz = r−r ′. In Eq. (4.30) an evaluation of the integral over z would immediately leadto the momentum space expressions for the electric Zemach moments and Zemachradii in Eqs. (4.14)-(4.16). However, instead, we change the order of integration ofd3z and d3q which gives the alternate expression〈Rn〉(2) =∞∫0dz zn+1 2pi∞∫0dq q ·Fx(q2)Fx′(q2)sin(qz) . (4.31)The above relation is valid only for the “spherical” part of the density distributions,which is approximate for the deuteron, but exact for A = 3 and 4 nuclei. TheseZemach moments can also be related to the other moments through2〈R2E〉= 〈R2E〉(2), (4.32)2〈R4E〉= 〈R4E〉(2)−103〈R2E〉2. (4.33)These relations allow for the consistent calculation of virtually all moments andZemach moments with the hybrid qr-space method.4.2 The Lamb shiftThe Lamb shift energy splitting [30] is related to the charge radius 〈R2E〉 through∆ELS = δQED+AOPE〈R2E〉+δTPE, (4.34)where the term δQED denotes the QED corrections, dominated by the vacuum polar-ization and the self interaction of the muon. The next termsAOPE〈R2E〉 and δTPE arethe nuclear structure corrections stemming from the one-photon and two-photonexchange, respectively. For compound nuclei, as in Eq. (4.7), the nuclear structurecorrections δTPE are the sum of the elastic contribution ∆Eelas(α) and the inelasticnuclear polarizability corrections ∆Einel(α) that are calculated at order (Zα)5, i.e.,δTPE = ∆Eelas(α)+∆Einel(α). Each of these components can be further separatedinto terms originating from atomic-nucleus dynamics, denoted with A, from terms50that exclusively depend on the properties of the single-nucleon, denoted with N, as∆Eelas(α) = δAZem+δNZem, (4.35)∆Einel(α) = δApol+δNpol+δNsub. (4.36)The term δNsub is the nucleon subtraction term. This term arises from the calculationof the inelastic nucleon polarizabilites as integrals of nucleon structure functionswithin the dispersion theory framework [94, 95]. The subtraction term is requiredto regulate the low energy behavior of these integrands and is not well constrainedby data.The nuclear elastic component δAZem is related to the third Zemach moment byδAZem =−pi3|φ(0)|2(Zα)2mr〈R3E〉(2), (4.37)and is often called the Friar-term in the literature [53, 96, 97]. An analogous ex-pression holds for the nucleonic contribution δNZem. The evaluation of the nucleoniccontributions δNZem, δNsub, and δNpol requires the inclusion of sub-nuclear degrees offreedom, which is beyond the scope of our calculations. Therefore, these nucleoncontributions are scaled as in Ref. [2] from the results of dispersion analysis in µH[94, 95], for the proton and neutron. Both the nuclear and nucleonic TPE correc-tions are of order α5.A more convenient way to group these corrections is to collect all nuclear andnucleonic terms together. In this decomposition the TPE is the sumδTPE = δATPE+δNTPE, (4.38)where the TPE is now a sum of the nuclear and nucleonic TPEs, respectively, withδATPE = δAZem+δApol, (4.39)andδNTPE = δNZem+δNpol+δNsub. (4.40)514.3 Nuclear structure corrections in the η-less expansionThe nuclear structure contributions δApol to the Lamb shift are derived from sec-ond order perturbation theory. In the work of Refs. [5, 82, 98–101], these nu-clear structure corrections were calculated as a multipole expansion of the operatorη =√2mrω|X −X ′|, where mr is the reduced lepton-nuclear mass, ω = EN−E0 isthe nuclear excitation energy and |X −X ′| is the “virtual” distance traveled by thenucleons during the TPE process. The approximate scale of η can be understoodbased on the uncertainty principle. During the TPE process, the nuclear excitationenergy ω may be converted into the kinetic energy of a given proton; consequently,ω = p22mp, where mp is the mass of the proton and p is the momentum. Therefore,p=√2mpω and by Heisenbergs uncertainty principle, p|X −X ′| ∼ 1, which leadsto the estimate that|X −X ′| ∼ 1√2mpω. (4.41)Therefore, the above expression implies thatη ∼√mrmp≈√mµmp< 1, (4.42)and so an operator expansion in powers of η converges and is valid. This theo-retical formalism refered to as the “η-expansion”, will be reviewed in Section 4.4.The primary reasons that the η-expansion is convenient are:(i) That the η-expansion produces closed-form formulas that are readily amenablefor computation and;(ii) The η-expansion allows the analytical extraction of the elastic contributionsof δApol that cancel the analogous term from the elastic TPE contribution.Finally, we emphasize that for the η-expansion method to be valid, the parameter ηmust be smaller than 1. For the two and four body systems, the expansion rapidlyconverges as seen in Refs. [5, 99, 100]. However, in the A = 3 sector, the η-expansion showed a slow convergence [8, 102], which leads to a larger estimatedtruncation uncertainty from this method.Generalizing the η-expansion method beyond sub-subleading orders is ana-lytically challenging and propels us to adopt the computationally involved but52more systematic framework first introduced by Leidemann and Rosenfelder inRefs. [103, 104] for calculating the TPE. In this dissertation and in our workin Ref. [8] we refer to this method as the η-less expansion. This η-less expan-sion method has not recently been used for nuclear TPE calculations due to thehigher computational effort required for the procedure. Furthermore, the initial de-scription of the method did not demonstrate how to extract the elastic terms fromδApol and thus, analytical cancellations such as those between the Friar moments(δAZem) could not be used. In this Section, we revisit the original formalism ofRefs. [103, 104] and demonstrate in detail how the elastic contribution implicitlycontained in the generalized TPE process can be separated and extracted to exploitdirect cancellations.A charged lepton interactiong with a charged nucleus modifies the EM propa-gator of the photon (Dµν ) through the LS equation [35, 105]Dµν(x,x′) = (4piα)Dµν(x− x′)+(4piα)∫d4x1∫d4x2 Dµα(x− x1)Tαβ (x1,x2)Dβν(x2,x′),=D(1)µν (x,x′)+D (2)µν (x,x′)+ . . . , (4.43)The terms D (k)µν , denote the k-th contribution from perturbation theory to the ef-fective photon propagator, Dµν(x− x′) are the free photon propagators and thehadronic tensor Tαβ is defined as the expectation value of the time-ordered prod-uct of the nuclear current in its ground stateTαβ (x1,x2) = 〈N0|T{Jα(x1)Jβ (x2)}|N0〉, (4.44)The second order contribution to the effective photon propagator isD(2)µν (x,x′) = (4piα)2×∫d4x1∫d4x2 Dµα(x− x1)Tαβ (x1,x2)Dβν(x2− x′). (4.45)The contribution of this term to the Lamb shift defines the nuclear structure correc-53tions from the TPE given byδApol =−1mrIm∫d4Y∫d4Xφ¯(X)(−iγµ)SF(X−Y )(−iγν)D (2)µν (X ,Y )φ(Y ), (4.46)where φ(X) represents the atomic wave function in coordinate space and SF(X−Y )is the Feynman propagator of the lepton. For S-states the atomic wave function isapproximated by a delta function in momentum space [106] and the nuclear TPEcontributions, reduces to5δApol =−(4piZα)2mr|φ(0)|2Im∫ d4q(2pi)4×Dµρ(q)Dντ(−q)tµν(q,k)Tρτ(q,−q). (4.47)In the above expression and in the work that follows we write the φ(0) as a short-hand for φ2(0). The hadronic tensor Tµν(q,−q) is written explicitly as6Tµν(q,−q) = ∑N 6=N0〈N0|J˜µ(q)|N〉〈N|J˜ν(q)|N0〉q0−ω+ iε−〈N0|J˜ν(q)|N〉〈N|J˜µ(q)|N0〉q0+ω+ iε. (4.48)The lepton Compton tensor tµν(q,k) in Eq. (4.47) with the 4-momentum of themuon k = (mr,0) is defined astµν(q,k)s1s2 =u¯s1(k)(−iγµ)(γλ (k−q)λ +mr)(−iγν)us2(k)(k−q)2−m2r + iε, (4.49)where s1 and s2 are the spins of the lepton spinors us1/s2(k). For the Lamb shift weare only interested in the spin-independent components of this tensor. Therefore,5Note here q denotes the four momentum q = (q0,q)6In the limit where ε > 0 and ε → 0.54we take the spin-diagonal averaged elements of tµν14 ∑s1s2tµν(q,k)s1s2δs1s2 =14∑sTr[u¯s(k)(iγµ)(γλ (k−q)λ +mr)(iγν)us(k)](k−q)2−m2r + iε,=−((k−q)νkµ +(k−q)µkν −gµνk · (k−q))−m2r gµν(k−q)2−m2r + iε.(4.50)The last expression has been simplified by applying the trace formulas in AppendixB and symmetrizing the leptonic tensor. The evaluation of Eq. (4.47) is carried outin the Coulomb gauge defined in Appendix B. In this gauge, the non-vanishingelements of the tensor summations in the scattering amplitude areD00(q)D00(−q)t00(q,k)T00(q,−q) =− 2m2r(q2+ iε)2−4m2q201|q|2 TL(q,−q), (4.51)Din(q)D jm(−q)ti j(q,k)Tnm(q,−q) =−1(q2+ iε)22m2r q20(q2+ iε)2−4m2r q20TT (q,−q),(4.52)where TL and TT are the longitudinal and transverse tensors definedTL(q,−q) = T00(q,−q), (4.53)andTT (q,−q) =(δnm− qnqm|q|2)Tnm(q,−q), (4.54)respectively. These tensors are related to their respective response functions throughT00(q,−q) =∞∫ωthdω[1q0−ω+ iε −1q0−ω+ iε]SL(ω,q), (4.55)55and (δnm− qmqn|q|2)Tmn(q,−q) =∞∫ωthdω[1q0−ω+ iε −1q0−ω+ iε]ST (ω,q). (4.56)Here SL(q,ω) and ST (q,ω) are the longitudinal and transverse response functions,respectively. They are defined bySL(q,ω) = ∑N 6=N0|〈N|ρ˜(q)|N0〉|2δ (EN−E0−ω), (4.57)ST (q,ω) = ∑λ=±1∑N 6=N0|〈N|e†λ · J˜(q)|N0〉|2δ (EN−E0−ω), (4.58)where |N0〉 and |N〉 are the ground state and excited states of the nucleus, withenergies E0 and EN , respectively. The energy ωth is the threshold energy of the nu-clear response function and vector e†λ is the polarization vector which for λ =±1is purely transverse. The terms ρ˜ and J˜ are the Fourier transforms of the nuclearcharge and current densities, respectively. From these definitions the nuclear struc-ture corrections becomeδApol = (4piZα)2|φ(0)|2∫ d3q(2pi)3∞∫ωthdω×1|q|2 SL(|q|,ω)× Im∫ dq0(2pi)2mr(q2+ iε)2−4m2r q202ω(q0+ iε)2−ω2+ST (|q|,ω)× Im∫ dq0(2pi)2mr(q2+ iε)2−4m2r q20q20(q2+ iε)22ω(q0+ iε)2−ω2 .(4.59)The integrals over q0 are carried out over the upper-half of the complex planeusing contour integration techniques given in Appendix B. The evaluation of these56integrals yields the expressionδApol =−8(Zα)2|φ(0)|2∞∫0dq∞∫ωthdω [KL(q,ω)SL(q,ω)+KT (q,ω)ST (q,ω)+KS(q,ω)ST (0,ω)] , (4.60)where as in Refs. [103, 104] the kernels in the integrals areKL(q,ω) =12Eq(1(Eq−mr)(ω+Eq−mr) −1(Eq+mr)(ω+Eq+mr)), (4.61)KT (q,ω) =q24m2rKL(q,ω)− 14mrqω+2q(ω+q)2, (4.62)and the seagull kernel isKS(q,ω) =14mrω[1q− 1Eq]. (4.63)The kernel KS arises from the two-photon operator Bµν in Eq. (3.4) that preservesgauge invariance. Here, Eq =√m2r +q2 is the relativistic energy of the muon. Inthis work, the transverse response function is separated into the electric SET andmagnetic SMT response functions, so that the nuclear polarization from the TPE isdecomposed into the sumδApol = ∆L+∆T,E+∆T,M, (4.64)where the subscript L denotes the longitudinal and (T,E), (T,M) the transverseelectric and magnetic corrections, respectively. These corrections are given explic-57itly as∆L =−8(Zα)2|φ(0)|2∞∫0dq∞∫ωthdω KL(q,ω)SL(q,ω), (4.65)∆T,E =−8(Zα)2|φ(0)|2∞∫0dq∞∫ωthdω ×[KT (q,ω)SET (q,ω)+KS(q,ω)SET (0,ω)], (4.66)and∆T,M =−8(Zα)2|φ(0)|2∞∫0dq∞∫ωthdω KT (q,ω)SMT (q,ω). (4.67)The expressions in Eqs. (4.65)-(4.67) summarize the formalism of the nuclearstructure corrections in the η-less method. This method is more computationallyexpensive because the integration of the above expression requires an evaluationof two integrals, one in q and one in ω . The integral in ω is carried out by diag-onalizing the complete nuclear Hamiltonian which is computationally intractablefor A > 2. However, the Lanczos sum rule method [8, 107] can be easily adaptedto tackle the generalized sum rules in Eqs. (4.65)-(4.67) that would render theircalculations feasible. The integral in q is carried out on a Gaussian quadraturegrid.4.3.1 The non-relativistic limitIn the non-relativistic limit where q  mr, the non-relativistic energy is of theform Eq → mr +Tr, where Tr is the kinetic energy of the muon and Tr  mr. Inthis approximation the kernels in Eqs. (4.61)-(4.67) reduce toKL(q,ω)→ KNR(q,ω) =(1q2( q22mr+ω)), (4.68)and where the transverse and seagull kernels vanishKT (q,ω)→ 0, (4.69) KS(q,ω)→ 0. (4.70)58Using the above expressions the non-relativistic nuclear polarization contributionsfrom the TPE (denoted δNRpol ) reduces to the form in Ref. [106]δNRpol =−8(Zα)2|φ(0)|2∞∫0dq∞∫ωthdω KNR(q,ω)SL(q,ω). (4.71)This non-relativistic form is used to benchmark the η-less method with calculationsfrom the η-expansion. These benchmarks are carried out in Section Nuclear structure corrections in the η-expansionIn the earlier work of Refs. [2, 3, 82, 98, 99, 107], the nuclear structure correctionswere derived as a power expansion of the parameter η from Eq. (4.42). This “η-expansion” is briefly reviewed in this Section but for more details we refer thereader to Ref. [2]. In the non-relativistic limit the nuclear polarizability δNRpol iscalculated as a correction from the non-relativistic second order perturbation theoryas the matrix element7 [2, 108]δNRpol = 〈N0µ|∆HG∆H|N0µ〉, (4.72)where |µN0〉 is the tensor product of the nuclear ground state |N0〉 and the muonwave function |µ〉. The operator G is the inelastic Green’s function of the Hamilto-nian, H = Hnucl+Hµ , where Hµ is the muon Hamiltonian and Hnucl is the nuclearHamiltonian. The perturbation ∆H from nuclear structure is∆H =Z∑a∆V (x,X a), (4.73)with∆V (x,X a) =−α(1|x−X a| −1x), (4.74)7The detailed derivation of these expressions has been carried out in [108] and will not be rewrit-ten here.59being the difference from the point-Coulomb interaction of the lepton with thenucleus. The vector X a denotes the coordinates of the protons. The non-relativisticmatrix element in Eq. (4.72) is re-written in terms of the matrix element W asδNRpol = ∑N 6=N0∫d3X d3X ′ρ pN(X )W (X ,X′,ω)ρ pN(X′), (4.75)where ρ p(X ) is the proton charge density of the nucleus. The matrix element isthenW (X ,X ′,ω) =−Z2|φ(0)|2∫ d3q(2pi)3(4piαq2)2(1− eiq·X ) 1q22mr+ω(1− e−iq·X ′).(4.76)After carrying out the integral over the momentum |q| of the virtual photon, W is afunction of the dimension-less parameter η =√2mrω|X −X ′|W (X ,X ′,ω) =− pim2r(Zα)2φ 2(0)(2mrω)3/2 1η(e−η −1+η− 12η2). (4.77)As explained in Section 4.3 this dimensionless parameter has been qualitativelyargued to be of the order√mµmp 1 allowing W to be expanded in powers of η .The expansion of W in powers of η in Eq. (4.75) leads δNRpol to be a sum ofleading, subleading, etc., contributions with respect to their associated power of ηδNRpol = δ(0)NR +δ(1)NR +δ(2)NR + ..., (4.78)where δNRpol is dominated by the dipole correction δ(0)D1 . The leading dipole correc-tion is [2]δ (0)D1 =−16pi29|φ(0)|2(Zα)2∞∫ωthdω√2mrωSD1(ω), (4.79)60where SD1(ω) is the dipole response functionSD1(ω) =12J0+1∑N0 6=N,J|〈N0J0||O1E||NJ〉|2δ (EN−E0−ω). (4.80)The sub-leading term δ (1)NR is the sum of the elastic contributions δ(1)R3 and δ(1)Z3 [2]δ (1)R3 =−pi(Zα)23mr|φ(0)|2∫ ∫d3Xd3X ′|X −X ′|3ρ(pp)0 (X ,X ′), (4.81)δ (1)Z3 =pi(Zα)23mr|φ(0)|2〈R3E〉(2), (4.82)where ρ(pp) denotes the proton-proton correlation density. It is important to notethat δ (1)Z3 exactly cancels out the elastic contribution δAZem, ie. (δAZ3 = −δAZem inEq. (4.37))8. More detailed expressions for the other corrections in Eq. (4.78)above are given in Ref. [2]. In the work of Refs. [2, 82, 98–102], the η-expansion iscarried out up to sub-sub-leading order including relativistic and finite size effectsresulting in the sumδATPE = δ(0)+δ (1)+δ (2)+δAZem+δ(1)NS +δ(2)NS . (4.83)Here δ (0) is the leading, δ (1) the sub-leading and δ (2) the sub-sub-leading term thatalso include the non-relativistic terms in Eq. (4.78). The calculations of δ (0), δ (1)and δ (2) are carried out in the point nucleon limit. The effects of finite nucleon size(NS) are included through the contributions δ (1)NS and δ(2)NS . Their exact expressionsare given in Ref. [2]. The higher order terms in this expansion lead to difficult-to-evaluate, non-analytic contributions that that have so far only been estimated[2, 98].The sum in Eq. (4.83) summarizes the η-expansion approach for the TPE cal-culation. In the non-relativistic case, the expansion in η can be circumvented byintroducing the plane wave expansion into Eq. (4.76), integrating over the angles qˆand plugging the result back into Eq. (4.75). This will result in the non-relativistic8This is the kind of cancellation that was not possible in the original derivation from Leidemannand Rosenfelder [103, 104]61TPE correction in Eq. (4.71).While Eq. (4.71) provides a more systematic method to calculate the nuclearstructure corrections, it is not apparent that it contains the elastic term δ (1)Z3 thatcancels the corresponding term from the elastic TPE diagram. However, as we shalldemonstrate in the next Section, the cancellation of the elastic contributions fromany general elastic with the inelastic TPE diagrams can be easily accomplished bygeneralizing the formalism of Refs. [103, 104].4.4.1 Subtraction of the elastic partHere we first clarify the meaning of the “elastic” and “inelastic” TPE diagrams.The elastic TPE contribution corresponds to Fig. 4.1a where the nucleus remains inthe its ground state during the photon exchange. The inelastic TPE is the processillustrated in Fig. 4.1b where the nucleus absorbs sufficient energy from the photonto excite or break it apart. The calculation of the inelastic TPE term using theη-less expansion method implicitly incorporates elastic contributions, such as theZemach moments, that cancel the contributions from the elastic TPE diagram. Inthis Section we detail how the elastic terms can be extracted directly from the η-less method and provide the circumstances where they cancel with the inelasticcontributions of a general TPE at order α5 and higher.(a) (b)Figure 4.1: TPE diagrams: (a) elastic vs (b) inelastic. The empty box rep-resents any general electromagnetic processes involving the upper half of thediagram. In the inelastic diagram (b) the excited states of the nucleus arerepresented by the grey blob. The Figure is adapted from Ref. [8].Fig. 4.1 illustrates the general elastic and inelastic TPE diagrams, with the empty62square indicating processes that can be inserted into the photon or lepton propa-gators in the upper half of the diagram. The contribution of the elastic diagram inFig. 4.1a will be denoted as δAelas, while Fig. 4.1b is δApol.The contribution from the elastic diagrams is given byδAelas =−8(Zα)2|φ(0)|2∞∫0dq Kµν(q,ω = i0+)〈N0|J˜µ(q)|N0〉〈N0|J˜ν(−q)|N0〉,(4.84)where 0+ indicates that the limit with ε → 0+ is taken. The kernel Kµν(q, i0+) isdetermined by the processes inserted into the empty squares of the previous figures.The matrix elements 〈N0|J˜µ(q)|N0〉 represent the elastic electromagnetic verticesof the nucleus in the ground state. The general form of the contribution from theinelastic process isδApol =−8(Zα)2|φ(0)|2∞∫0dq∞∫ωthdω Kµν(q,ω)Sµν(q,ω). (4.85)Here, the kernel Kµν(q,ω) depends on the three-momentum transfer q and thenuclear excitation energy ω . The general nuclear response function Sµν(q,ω) isSµν(q,ω) = ∑N 6=N0〈N0|J˜µ(q)|N〉〈N|J˜ν(−q)|N0〉δ (EN−E0−ω). (4.86)The expression in Eq. (4.86) contains the electromagnetic matrix elements betweenthe ground state |N0〉 and the nuclear excited states |N〉. The analytical cancellationbetween the terms implicit in δApol and δAelas can be carried out by breaking-up δApolinto purely ω-dependent (∆inelω ) and ω-independent (∆inelω ) contributions asδApol = ∆inelω +∆inelω . (4.87)63These ω-dependent/independent corrections are∆inelω =−8(Zα)2|φ(0)|2∞∫0dq∞∫ωthdω[Kµν(q,ω)−Kµν(q, i0+)]Sµν(q,ω),(4.88)and∆inelω =−8(Zα)2|φ(0)|2∞∫0dq∞∫ωthdω Kµν(q, i0+)Sµν(q,ω), (4.89)respectively. In Eq. (4.89) the kernel Kµν(q, i0+) does not depend on the nuclearexcitation energy and is neglected in the ω integration. The remaining integral ofthe nuclear response function Sµν(q,ω) over ω is simplified using the relation∞∫ωthdω Sµν(q,ω) = 〈N0|J˜µ(q)J˜ν(−q)|N0〉−〈N0|J˜µ(q)|N0〉〈N0|J˜ν(−q)|N0〉,(4.90)where the matrix elements 〈N0|J˜µ(q)J˜ν(−q)|N0〉 are the electromagnetic correla-tion functions and 〈N0|J˜µ(q)|N0〉 are the matrix elements associated with the elasticelectromagnetic form factors of the nucleus. Therefore, one obtains∆inelω =−8(Zα)2|φ(0)|2∞∫0dq Kµν(q, i0+)×(〈N0|J˜µ(q)J˜ν(−q)|N0〉−〈N0|J˜µ(q)|N0〉〈N0|J˜ν(−q)|N0〉) ,(4.91)The first term in Eq. (4.91) has no corresponding elastic term in Eq. (4.84). Incontrast, the second term in Eq. (4.91) exactly cancels Eq. (4.84). Consequently,we arrive at the result that for any general TPE diagram the addition of δAelas and δApollead to the cancellation of the elastic TPE contributions and only the ω-dependent(∆inelω ) and correlation terms (∆corr) terms remain. Therefore, one obtainsδApol+δAelas = ∆corr+∆inelω , (4.92)64where the correlation term is∆corr =−8(Zα)2|φ(0)|2∞∫0dq Kµν(q, i0+)〈N0|J˜µ(q)J˜ν(−q)|N0〉. (4.93)The result derived in Eq. (4.92) is a general property of the TPE contributions. Toapply this result we demonstrate that in the point-nucleon limit, the η-less expres-sion in Eq. (4.71) can be expanded as δApol = ∆inelω + δ(1)R3 + δ(1)Z3 , where ∆inelω arethe ω-dependent nuclear polarizability corrections. In the limit where ω → i0+,the transverse Kernel KT (q,ω) vanishes and does not contribute to ∆inelω . However,KL(q,ω) does not vanish and provides the contribution given by∆L,ω =−8(Zα)2|φ(0)|2∞∫0dq∞∫ωthdω KNR(q, i0+) SL(q,ω). (4.94)After applying Eq. (4.90) into the above equation we obtain∆L,ω =−16mr(Zα)2|φ(0)|2P∞∫0dq1q4(〈N0|ρ˜ p(q)ρ˜ p,†(q)|N0〉− |〈N0|ρ˜ p(q)|N0〉|2) , (4.95)whereP denotes the principal value integral. The evaluation of the principal val-ued integrals giveP∞∫0dq|〈N0|ρ˜ p(q)|N0〉|2q4=pi48∫ ∫d3Xd3X ′|X −X ′|3ρ p0 (X )ρ p0 (X ′), (4.96)and similarly, the evaluation of the correlation term isP∞∫0dq〈N0|ρ˜ p(q)ρ˜ p,†(q)|N0〉q4=pi48∫ ∫d3Xd3X ′|X −X ′|3ρ(pp)0 (X ,X ′). (4.97)Using the results of Eq. (4.96) and Eq. (4.97) and substituting them into Eq. (4.95)shows that ∆L,ω = δ(1)R3 +δ(1)Z3 , and therefore δApol = ∆inelω +δ(1)R3 +δ(1)Z3 for the non-relativistic, point nucleon limit. This example establishes how the elastic contribu-65tions can be extracted from the η-less expansion formalism.4.4.2 Lamb shift nuclear structure at α6 orderThe nuclear structure corrections of the previous Sections are of the order (Zα)5.Therefore higher order α corrections should be suppressed relative to those contri-butions. The Coulomb distortion correction in the logarithmic approximation is ofthe order (Zα)6ln(Zα). This is given byδ (0)C =−16pi29(Zα)3φ 2(0)∞∫0dωmrωln(2(Zα)2mrω)SD1(ω). (4.98)The non-logarithmic α6 Coulomb distortion terms are also known, but found to benumerically negligible with respect to the logarithmically enhanced one[98, 100,102]. We include it in the calculation of the dipole term of Eq. (4.79). Morerecently, Kalinowski [109] found that the vacuum polarization contribution at α6,represented by the diagram in Fig. 4.2 was larger than expected. Its contributionshifted the δTPE value towards experiment.Figure 4.2: The α(Zα)6 vacuum polarization contribution. The circle in thephoton line represents the creation of an electron and positron pair.The effect of the vacuum polarization is to modify the wave function of the 2Sstates at the origin as well as the photon propagator. These effects are denoted δAwfand δAvac, pol, respectively [109]. In addition, the vacuum polarization modifies theelastic Zemach moment via a term denoted with δAvac, Zem. The total effect of thevacuum polarization is the sum of these effectsδAvac = δAwf+δAvac, pol+δAvac, Zem. (4.99)66The contribution to the total vacuum polarization from the modification of the 2Swave function at the origin can be calculated by the scaling the nuclear TPE cor-rections by [109]δAwf = 2×∣∣∣∣ φ˜(0)φ(0)∣∣∣∣δATPE, (4.100)where φ˜(0) is the perturbed 2S wave function of the muonic atom at the origin andthe factor of 2 is a symmetry factor. For the case of Z = 1, this scaling ratio is [109]∣∣∣∣ φ˜(0)φ(0)∣∣∣∣= 0.72615(αpi ) . (4.101)The vacuum polarization contribution from the modification of the photon propa-gator in the η-less formalism is (including a factor of 2 for symmetry purposes)isδAvac, pol =−2×8(Zα)2|φ(0)|2∞∫0dq∞∫ωthdω g(q)KNR(q,ω)SL(q,ω), (4.102)in the non-relativistic limit. The function g(q) is the parameterized vacuum polar-ization density functiong(q) =(2αpi) 1∫0dx x(1− x) · ln(1+q2m2ex(1− x)), (4.103)where me is the electron mass. As demonstrated in Section 4.4.1, Eq. (4.102)contains elastic terms denoted δAvac, Z3 and δAvac, R3. The term δAvac, Z3 cancels outthe contribution from the modified Zemach moment δAvac, Zem from the elastic TPEdiagram (ie. δAvac, Z3 = −δAvac, Zem). The term δAvac, R3 will be elastic contributionfrom the vacuum polarization that involves the charge correlation functions. UsingEq. (4.89) the elastic contribution δAvac, Z3 isδAvac, Z3 = 2×8|φ(0)|2(Zα)2∞∫0dq KNR(q,0+)g(q)|〈N0|ρ˜(q)|N0〉|2, (4.104)67and the charge correlation contribution δAvac, R3 isδAvac, R3 =−2×8|φ(0)|2(Zα)2∞∫0dq KNR(q,0+)g(q)〈N0|ρ˜†(q)ρ˜(q)|N0〉. (4.105)At the order α6, aside from the vacuum polarization there are three additionaldiagrams that contribute to the nuclear structure corrections, as pictured in Fig. 4.3from [110].Figure 4.3: The radiative recoil corrections: (a) The spanning photon diagram(b) the muon self energy correction and (c) the vertex correction.The elastic contributions of these diagrams have been calculated in Ref. [110]and demonstrated to be negligible to the Lamb shift, however, the inelastic con-tributions have not yet been calculated. These additional contributions have beenargued to be small relative to the vacuum polarization terms in Ref. [109], butfuture work will need verify these claims.Finally, the three-photon exchange shown in Fig. 4.4 has also been recentlyinvestigated for the deuteron [111].Figure 4.4: The three photon exchange diagram.68It has been shown that the elastic and inelastic contributions of this effect, in anal-ogy to the cancellation in the TPE, also cancel out to a large extent [111]. Theeffects of this contribution denoted as δA3PE are considered in Section The Hyperfine splittingThe HFS originates from the EM interaction of the nuclear and leptonic currents.This results in the splitting of the n` j states in hydrogenic systems into the states F ,where F is given by the coupling of the total angular momentum of the lepton j andthe total angular momentum of the nucleus J, so that F = j⊗J = | j−J|, ..., | j+J|.For the deuteron, J = 1 (`= 0,2; S= 1) and the orbiting lepton carries total angularmomentum j = `⊗1/2. The energy splittings for the 2S1/2 and 2P1/2 states in µ2Hare illustrated in Fig. 4.5.Figure 4.5: The hyperfine structure of µ2H for the 2S1/2 and 2P1/2 states.The energy difference of the S-states ∆EHF(nS) is the sum of terms∆EHF(nS) = EF (1+δQED+δR+δZem+δnucl) , (4.106)where EF is the Fermi-contact energy, δQED are the QED corrections, δR the recoilcorrections, δZem is proportional to the magnetic Zemach radius 〈RZ〉 and δnucl arethe nuclear structure corrections dominated by the TPE as in Ref. [93, 112]. Unlikein the case of the Lamb shift, non-relativistic quantum mechanics predicts that theFermi energy EF is the main contribution to the HFS.694.5.1 The Hyperfine Fermi energyThe Fermi energy arises from the EM interaction between the lepton and nuclearcurrents [12]EF = α∫d3x j`(x) ·A(x),= α∫d3x∫d3x′j`(x) · JN(x′)|x− x′| , (4.107)where A(x) is the EM vector potential, j` is the leptonic current and JN is thenuclear current. In the non-relativistic limit the leptonic current isj`(x) = iφ ∗(x)σ (σ ·∇φ(x))2m`− i(σ ·∇φ∗(x))σφ(x)2m`, (4.108)where m` is the lepton mass, φ(x) is the bound state wave function of the leptonwith spin operator σ . Using Eq. (B.9) in Appendix B the dot product of the leptonand nuclear currents are re-written asj`(x) · J(x′) =12m`σ · (∇|φ(x)|2× J(x′)) , (4.109)which gives the Fermi energy contribution asEF =α2m`∫d3x |φ(x)|2∫d3x′ J(x′) ·σ ×∇(1|x− x′|),=4piα3m`∫d3x|φ(x)|2 (Ms(x) ·σ ) ,=4piα3m`σ · JJ∫d3x|φ(x)|2ρM(x). (4.110)Here Ms(x) =µNJJ ρM(x) has been used and ρM(x) is the magnetic charge densityof the nucleus. In the case where the magnetic charge density of the nucleus ispoint-like, ρM(x)→ δ (x), the Fermi energy becomes [16]EF =4piαµN3m`|φ(0)|2σ · JJ. (4.111)704.5.2 Nuclear structure correctionsAs in the case of the nuclear structure corrections to the Lamb shift, the leading or-der nuclear structure effects are elastic and consequently the leading order inelasticcontributions to the HFS arise from the TPE. The interaction diagram is the sameas for the Lamb shift and illustrated in Fig. 4.1b. The covariant amplitude of thiscontribution is given by Eq. (4.47). Using Eq. (B.15) the leptonic tensor of the TPEamplitude in Eq. (4.113) is decomposed astµν =−u¯s(k)γν (k−q)µ + γµ (k−q)ν −gµν(/k−/q)+m`γµγν(k−q)2−m2` + iεus(k) (4.112)− u¯s(k) iεµνσλ γσ γ5 (q− k)λ(k−q)2−m2` + iεus(k). (4.113)Here Eq. 4.112 are the spin-independent contributions to the Lamb shift studiedin Section 4.2, while Eq. 4.113 are the spin dependent terms that contribute to theHFS. Through further simplifications this spin dependent contribution ist0i(k,q) =−iu†,s(k) (σ ×q)i(k−q)2−m2` + iεus(k), (4.114)and where ti0 =−t0i by symmetry. For brevity we write u†,s(k)σ us(k)≡ σ s. In theLorentz gauge and using Eq. (4.113) the TPE contribution to the HFS reduces toδATPE = (4piα)2|φ(0)|2∫ d4q(2pi)41(q2+ iε)2tµνT µν(q,−q), (4.115)= (4piα)2|φ(0)|2∫ d4q(2pi)4(σ s×q)m[T m0(q,−q)−T 0m(q,−q))](q2+ iε)2(q2−2m`q0+ iε) . (4.116)In the above expressions q2 = q20−|q|2 was used and we remind the reader that thelepton part is given by the small t while the hadronic contribution is the tensor de-noted with the capital T . The symmetry of Eq. (4.116) is exploited by introducingthe symmetrized tensorT˜ µν(q,−q) = T µν(q,−q)+T µν(−q,q), (4.117)71with components given byT˜ µν(q,−q) =∑N[J˜ν(q)|N〉〈N|J˜µ(−q)q0−ωq+ iε +J˜µ(−q)|N〉〈N|J˜ν(q)−q0−ωq+ iε]. (4.118)In addition, it is useful to define the vector U with componentsUm = T˜ m0(q,−q)− T˜ 0m(q,−q). (4.119)In Eq. 4.118 the nuclear excitation energy ωq contains the contribution from thenuclear recoil given byωq = ω+q22M. (4.120)From these definitions, the nuclear structure HFS energy shift isδATPE = (4piα)2|φ(0)|2σ s ·∫ d4q(2pi)4(q×U )(q2+ iε)2(q2−2m`q0+ iε) . (4.121)This TPE nuclear structure contribution is written in terms of the Fermi energy asδATPE = EFδHFS, (4.122)where the dimensionless TPE contribution isδHFS =12piαm`µN〈JJ|OˆTPE|JJ〉 (4.123)and the TPE operator to the HFS isOˆTPE =∫ d4q(2pi)4q×U(q2+ iε)2(q2−2m`q0+ iε) . (4.124)The above expression can be further manipulated to be more amenable to calcula-tions by exploiting symmetry properties. Under the parity transformation (q→−q)it can be shown thatq×U = 2ωq(q0−ωq+ iε)(q0+ωq− iε)q×{ρ˜(q), J˜(−q)}, (4.125)72where “{,}” denotes the anti-commutator. Furthermore, symmetrizing the denom-inator of the lepton tensor in Eq. (4.113) as1q2−2m`q0+ iε →q2(q2−2m`q0+ iε)(q2+2m`q0+ iε) . (4.126)with Eq. (4.126) and (4.125) the TPE operator in Eq. (4.124) isOˆTPE =∫ d4q(2pi)42ωqq×{J˜(−q), ρ˜(q)}(q20−|q|2+ iε)(q20−ω2+ iε)×1((q0−m`)2−E2q + iε)((q0+m`)2−E2q + iε). (4.127)where E2q = m2r + |q|2. The integral over q0 of Eq. (4.127) can be evaluated usingthe standard contour integration methods and the results are given in Appendix B.This evaluation will produce the HFS kernel function, denoted KHFS, in the expres-sionOˆµTPE =−i∫ d3q(2pi)3KHFS(q,ω)(q×{ρ˜(−q), J˜(q)})µ . (4.128)This last expression is the final form of the TPE nuclear structure operator that mustbe calculated.734.5.3 The multipole expansion of the hyperfine splitting contributionTo compute the operator in Eq. (4.128) on the partial-wave decomposed nuclearwave functions, the operator must be expanded in multipoles. Choosing the direc-tion of the spherical unit vector e0 to be parallel to q the charge and current densitycan be expanded using Eqs. (3.39) and (3.41) in terms of multipoles. Taking thecurl of Eq. (3.41), we haveq× J˜(q) =∑κµ4pi[−i(q× [qˆ×Y µ∗κκ(qˆ)])T Eκµ(q) +q×Y µ∗κκ(qˆ)T Mκµ(q)] , (4.129)where after applying the identities in Eq. (D.10), (D.11) and multiplying the resultby the multipole expansion of the charge density operator from Eq. (3.39), theresult is9ρ˜†(q)(q× J˜(q))=∑κµ∑κ ′µ ′(4pi)2(iq)(−i)κ ′C †κ ′µ ′(q)Y κ′µ ′ (qˆ)[Y ∗µκκ(qˆ)T Eκµ(q)−[√κ+12κ+1Y ∗µκ,κ−1(qˆ)+√κ2κ+1Y ∗µκ,κ+1(qˆ)]T Mκµ(q)].(4.130)The spherical tensor components of Eq. (4.130) are integrated over the angles qˆand simplified using the identities in Appendix D to giveeλ ′ ·∫dqˆ ρ˜†(q)(q× J˜(q))= (4pi)2(iq) ∑κµµ ′(−i)κCκµκµ ′1λ ′C †κµ ′(q)T Eκµ(q)+(−i)κ+1[√κ+12κ+1Cκµκ−1µ ′1λ ′C†κ−1µ ′(q)−√κ2κ+1Cκµκ+1µ ′1λ ′C†κ+1µ ′(q)]T Mκµ(q).(4.131)The above expression is now the spherically integrated and multipole decomposedoperator expression of current-charge density operator in Eq. (4.128). Using theprevious equation the multipole expanded TPE operator in Eq. (4.128) becomes9Here we have used ρ(−q) = ρ†(q).74OˆαTPE =−i∞∫0dq(2pi)3(4pi)2(iq3)KHFS(q,ω) ∑κµµ ′[(−i)κCκµκµ ′1α{C †κµ ′(q),T Eκµ(q)}+(−i)κ+1√κ+12κ+1Cκµκ−1µ ′1α{C †κ−1µ ′(q),T Mκµ(q)}−(−i)κ+1√κ2κ+1Cκµκ+1µ ′1α{C †κ+1µ ′(q),T Mκµ(q)}].(4.132)Next, we apply the Siegert theorem from Eq. (3.55) to simplify the transverseelectric tensor asT Eκ (q) =−(i)κωq√κ+1κCκ(q)+δT Eκ (q), (4.133)where the second term is the Siegert correction that we neglect. To simplify thetransverse electric tensor asOˆαTPE =−i∞∫0dq(2pi)3(4pi)2(iq3)KHFS(q,ω) ∑κµµ ′[−ωq√κ+1κCκµκµ ′1α{C †κµ ′(q),Cκµ ′(q)}+(−i)κ+1√κ+12κ+1Cκµκ−1µ ′1α{C †κ−1µ ′(q),T Mκµ(q)}−(−i)κ+1√κ2κ+1Cκµκ+1µ ′1α{C †κ+1µ ′(q),T Mκµ(q)}].(4.134)The expectation value of this operator on the ground state |J0;M0〉 is then taken. Ifwe insert the intermediate inelastic nuclear excited states, ∑NJ;MJ|NJ;MJ〉〈NJ;MJ|75in Eq. (4.132), this results in the expression〈J0M0|OαTPE|J0M0〉=−i ∑κ≥1,J∞∫0dq(2pi)3(4pi)2(iq3)KHFS(q,ω)×[−ωq√κ+1κ ∑µµ ′MCκµκµ ′1α{{C †κµ ′(q),Cκµ ′(q)}}MM0JJ0+(−i)κ+1√κ+12κ+1 ∑µµ ′MCκµκ−1µ ′1α{{C †κ−1µ ′(q),T Mκµ(q)}}MM0JJ0−(−i)κ+1√κ2κ+1 ∑µµ ′MCκµκ+1µ ′1α{{C †κ+1µ ′(q),T Mκµ(q)}}MM0JJ0]. (4.135)Where the double bracket notation from Appendix C has been introduced to sim-plify the formula. This expression can be further reduced by applying Eq. (C.37)to give〈J0M0|OαTPE|J0M0〉= ∑κ≥1,J2pi∞∫0dq q3KHFS(q,ω)(−1)1√2J0+1CJ0M0(1,α),(J0M0)×[(−1)κ+J0+J+1ωqκˆ{κ κ 1J0 J0 J}{{C †κ (q),Cκ(q)}}JJ01+(−i)κ+1√(κ+1)(2κ−1)2κ+1(−1)κ−1+J0+J{κ−1 κ 1J0 J0 J}{{C †κ−1(q),T Mκ (q)}}JJ01−(−i)κ+1√κ(2κ+3)2κ+1(−1)κ+1+J0+J{κ+1 κ 1J0 J0 J}{{C †κ+1(q),T Mκ (q)}}JJ01].(4.136)This final expression for the HFS contains the commutator matrix elements betweenthe Coulomb and the magnetic tensors. This is in contrast to the Lamb shift wherethe electric and magnetic multipoles were separated.764.5.4 The non-relativistic, long-wavelength reductionIn the non-relativistic limit the Kernel KHFS becomes KNR in Eq. (4.136) withKNR(q,ω) =12m`q4(ω+ q22mr) . (4.137)This last expression is different from the non-relativistic expression in Eq. 4.68 forthe Lamb shift. The first term in Eq. (4.136) is the dipole operator (κ = 1). Takingonly this leading term gives〈J0M0|OαTPE|J0M0〉=∑J2pi∞∫0dq q3KNR(q,ω)(−1)1√2J0+1CJ0M0(1,α),(J0M0)×[√23(−1)J0+J+1{0 1 1J0 J0 J}{{C †0 (q),T M1 (q)}}JJ01+√53(−1)J0+J{2 1 1J0 J0 J}{{C †2 (q),T M1 (q)}}JJ01].(4.138)In analogy for the Lamb shift in Section 4.4.1, the ω-independent contributions ofEq. (4.138) are extracted using the procedure in Eq. (4.89) with the result〈J0M0|OαTPE|J0MJ0〉= iCJ0M0(1,α)(J0M0)√2J0+1(−1)J0×[136√pim2rm`∑J(−1)J{0 1 1J0 J0 J}√2ωmr{{∑ipˆir2i Y0(rˆi),µ}}JJ01− 118√10pim2rm`∑J(−1)J{2 1 1J0 J0 J}√2ωmr{{∑ipˆir2i Y2(rˆi),µ}}JJ01].(4.139)Here µ is the magnetic moment operator. The operators in the double commuta-tors are mixed magnetic-electric sum rules. As for the Lamb shift, it is expectedthat these dipole terms will dominate the HFS and therefore, this demonstrates theimportant role of the dipole magnetic moment of the nucleus to the HFS.77Chapter 5Uncertainty quantificationThe comparison of nuclear properties calculated from ab-initio theories to exper-iment is essential to validate the theory. While experimentally measured valuescarry uncertainty estimates, the accompanying error estimates from theory calcu-lations are often not rigorously analyzed or calculated. A full calculation includingall relevant theoretical uncertainties is, at the moment, beyond the available com-putational limits for heavy mass nuclei. Nevertheless, the field of uncertainty quan-tification in ab-initio nuclear theory has seen substantial advances in recent years.For example, in the work of Ref. [62], the first consistent simultaneous, order-by-order fit of the NN-potentials that included the covariance matrix of the LECS wascarried out up to N2LO. The covariance matrices of these NN potentials introducedthe ab-initio nuclear physics community to the use of statistical regression analysisfor uncertainty propagation. Another advancement was developed in Refs. [18–21]with the Bayesian uncertainty analysis procedure for order-by-order χEFT calcu-lations. Finally, there have also been more recent attempts to incorporate machinelearning methods [113–116] into nuclear ab-initio calculations.In this Chapter we identify and discuss all relevant uncertainty sources that arecrucial for realistic estimates of nuclear structure uncertainties in muonic atoms aswell as for other EM properties. We begin with Section 5.1 where the methods ofstatistical regression analysis and uncertainty propagation are outlined, followedby a discussion of the methods used to estimate the systematic χEFT uncertain-ties. The methods used for this purpose were cut-off variation and the Bayesian78formalism. In Section 5.2 we adopt the Bayesian framework used for the χEFTtruncation uncertainty estimate to analyze the η-expansion truncation uncertainty.We conclude the Chapter with a discussion of remaining uncertainties that requireconsideration in Section Uncertainty propagationThe statistical uncertainties σA of the nuclear observable OA(β ) are induced fromthe variations of the parameters β used in the calculation of the observable (forχEFT calculations β represent the LECS). Assuming that the Hessian matrix of theobservable is small, OA(β ) can be expanded to leading order around the optimalset of parameters β 0, from the χ2 minimization, to first order asOA(β ) = OA(β 0)+(β −β 0) · JA+O{(β −β 0)2}, (5.1)where JA is the Jacobian of the operator at the points β 0 with vector componentsJA,k = limβ→β 0∂OA(β )∂βk, (5.2)and O{(β − β 0)2} denotes higher order terms involving the Hessian of OA. Inthe region around the optimal parameter set, β 0, the probability density of β is amultivariate Gaussian distribution centered on β 0 of the formP(β |β 0,Σ0) =√|Σ0|2piExp(−12(β −β 0)TΣ−10 (β −β 0)), (5.3)where Σ0 is the covariance matrix of the parameters at the optimal points withcomponentsΣ0,i j = Cov(βi,β j). (5.4)The expectation values (which we denote with the brackets “〈〉”) of Eq. (5.3) are〈β 〉= β 0, (5.5)〈(βi−β0,i)(β j−β0, j)〉= Cov(βi,β j). (5.6)79Using these results, the expectation value of Eq. (5.1) can be calculated as〈OA(β )〉= OA(β 0)+ 〈(β −β 0) · JA〉+ 〈O{(β −β 0)2}〉= OA(β 0). (5.7)The expectation value of the square of the operator O2A is,〈OA(β )2〉= OA(β 0)2+OA(β 0)〈(β −β 0)〉 · JA+OA(β 0)JTA · 〈(β −β 0)T 〉+JTA · 〈(β −β 0)T (β −β 0)〉 · JA+ . . .= OA(β 0)2+ JTAΣ0JA, (5.8)where the ellipses denote terms with more derivatives. Using Eq. (5.7) and (5.8),the variance of the observable OA becomesσ2A = 〈OA(β )2〉−〈OA(β )〉2, (5.9)= JTAΣ0JA. (5.10)The calculation of the Jacobian vector JA is carried out using 10 function evalua-tions1 within a small neighborhood of the optimal values β 0 and by determiningthe slope of the univariate linear spline fit. This formalism can be extended to studythe covariance between two observables, OA(β ) and OB(β ), that both depend onthe parameters β . The covariance between these two observables is definedCov(OA,OB) = JTAΣ0JB. (5.11)The covariance is used to measure the linear correlation coefficient between OAand OB defined asρ(OA,OB) =Cov(OA,OB)σAσB, (5.12)where σA and σB are the standard deviations of OA and OB, as defined in Eq. (5.10).The value ρ(OA,OB) = 1 (ρ(OA,OB) =−1) indicates fully (anti-) correlated quan-1It was found in Ref. [117] that this number of function evaluations was sufficient.80tities, while ρ(OA,OB) = 0 implies that OA and OB are uncorrelated.5.1.1 Monte Carlo samplingThe uncertainty propagation formalism outlined in the previous Section requiresthe Jacobian vector to be calculated at an optimal point and involves the first or-der derivative of OA. In some cases differentiation is not numerically stable andit is desirable to use a derivative-free method for uncertainty propagation. Monte-Carlo (MC) simulations provide a simple alternative method. To propagate theuncertainties of OA induced by the parameters of the model, repeated samples ofthe parameter vector β are drawn from the distribution in Eq. (5.3). Once N sam-ples, are collected the expected value of OA is the average value of the MC samples[118]〈OA(β )〉= 1NN∑i=1OA(β i), (5.13)and the unbiased estimate of σA is the standard deviation of the MC samples givenas in Ref. [118] asσ2A =1N−1N∑i=1(O2A(β i)−〈OA(β )〉2). (5.14)5.2 Chiral effective field theory uncertaintiesIn addition to the uncertainties arising from the scatter of the nuclear model param-eters, there are uncertainties that stem directly from the underlying χEFT frame-work that should also be accounted for. These χEFT uncertainties stem from twosources: (i) from the variations of the cut-off Λ in the NN potential and (ii) from thetruncation of the χEFT expansion at a finite order. We denote these uncertaintiesas σΛ and σ∆, respectively.As described in Chapter 2 the χEFT NN-potentials employ a regulator parame-ter Λ. Since the long range properties of the NN interaction have been tuned to thesame values during the fitting procedure, the variations of an observable OA withrespect to Λ implies a sensitivity to short-range physics. This sensitivity introduces81the uncertainty σΛ and is commonly estimated by performing several calculationsof OA over a small cutoff range and taking the span of these calculations as anestimate of this uncertainty. It has been argued in recent literature [119] that σΛgenerated from this practice do not reflect the full uncertainty of the χEFT cal-culations. In fact, these uncertainty estimates have been shown to underestimateor overestimate the total χEFT uncertainty over a chosen cutoff range [61, 65],which brings into question the statistical interpretation of these uncertainty esti-mates. Nevertheless, because this is still an unresolved issue in nuclear theory, andin order to be conservative in our uncertainty estimates, we will include the σΛuncertainty from the cut-off variation procedure into our uncertainty estimate.A complementary, more systematic and statistically rigorous approach to ac-count for uncertainties from χEFT models is to use the Bayesian methodology out-lined in Refs. [19–21] to compute the chiral truncation uncertainty, σ∆. This chiraltruncation uncertainties originates from the calculation of an observable OA(p) ata finite order ν within the χEFT power counting scheme, with associated momen-tum p. It is assumed that the order-by-order convergence of OA(p) obeys the sameexpansion as the underlying NN-force, so thatOA(p) = A0ν∑µ=0cµ(p)Qµ , (5.15)where A0 is the leading order result, Q is the effective field theory (EFT) expansionparameter, typically chosen asQ = max{pΛb,mpiΛb}, (5.16)with Λb being the χEFT break-down scale (of the order of 600 MeV). Here cµ(p)are the observable and interaction specific coefficients that are taken to be indepen-dent, identically distributed (IID) random variables of natural size. These coeffi-cients are extracted from the order-by-order χEFT calculations, ascν =|ONνLOA −ONν−1LOA |Qν, (5.17)82where ONνLOA denotes the calculation of OA at order NνLO. Based on this, thechiral truncation uncertainty is σ∆ = A0∆ν and ∆ν is the sum of all higher ordercorrections∆ν =∞∑µ=ν+1cµ(p)Qµ . (5.18)The calculation of σ∆ is made under the assumption that the leading order approx-imation of the uncertainty is σ∆ ≈ A0cµ+1(p)Qµ+1. This leading order estimate isdenoted as ∆(1)ν . Following the work of Ref. [19], we outline the method used toderive the posterior probability density P(∆(1)ν |c), where c is the vector of knowncoefficients c =(c0, . . . ,cν) up to order ν in χEFT. In this leading order approxima-tion, the posterior distribution of ∆(1)ν will depend only on the unknown coefficientcν+1 that is used as a marginalization parameter2P(∆(1)ν |c) =∫dcν+1P(∆(1)ν |cν+1)P(cν+1|c). (5.19)The independence of cν+1 and c has been used to simplify the above expression.Assuming that the coefficients c are IID random variables of natural size with anassociated scale parameter c¯, thenP(∆(1)ν |c) =c¯>∫c¯<dc¯∫dcν+1P(∆(1)ν |cν+1)P(cν+1|c, c¯)P(c¯|c),=c¯>∫c¯<dc¯1Qν+1P(cν+1 =∆(1)νQν+1|c, c¯)P(c¯|c), (5.20)where c¯< and c¯> is the minimum and maximum of the scale parameter c¯ and theconstraint from the power expansion used wasP(∆(1)ν |cν+1) = δ (∆(1)ν − cν+1Qν+1). (5.21)The posterior probability of the scale parameter c¯ conditioned on the calculated2The marginalization of a joint probability distribution P(x,y) over the variable y is carried out asP(x) =∫dy P(x,y) =∫dy P(x|y)P(y).83coefficients c isP(c¯|c) = P(c0|c¯)P(c2|c¯) . . .P(cν |c¯)P(c¯)∫dc¯ P(c0|c¯)P(c2|c¯) . . .P(cν |c¯)P(c¯) . (5.22)The combination of Eqs. (5.22) and (5.20) give the posterior distribution of thetruncation uncertaintyP(∆1ν) =∫dc¯ P(cν+1|c¯)P(c0|c¯)P(c2|c¯) . . .P(cν |c¯)P(c¯)Qν+1∫dc¯ P(c0|c¯)P(c2|c¯) . . .P(cν |c¯)P(c¯) . (5.23)The previous equation contains the conditional probability distributions P(cν |c¯)and the prior P(c¯) that has not been specified. Following Ref. [19] we use the threesets of conditional probability distributions and priors from Table 5.1.Priors P(cν |c¯) P(c¯)A 12c¯θ(c¯−|cν |) 1ln(c¯>/c¯<)c¯θ(c¯− c¯<)θ(c¯>− c¯)B 12c¯θ(c¯−|cν |) 1√2pi c¯wB Exp(− (ln(c¯))22w2B)C 1√2pi c¯ Exp(− c2ν2c¯2)1ln(c¯>/c¯<)c¯θ(c¯− c¯<)θ(c¯>− c¯)Table 5.1: The prior probability distributions of coefficients cν and the scaleparameter c¯.The priors listed in Table 5.1 represent different assumptions about the na-ture of the expansion coefficients cν . The first sets of priors, A and B, employa non-informative, normalized, prior distribution P(cν |c¯), with the restriction thatthe coefficients cannot exceed the size of c¯. However, they differ with respectto the choice of the prior P(c¯). In row A of Table 5.1, P(c¯) is chosen to bea non-informative distribution and in row B, it is a log-normal distribution ofwidth wB. The Set C, assumes that the coefficients are normally distributed withzero mean and variance c¯2. This corresponds to the maximum entropy distri-bution of an ensemble of IID random variables with 〈cν〉 = 0 and squared sum∑kν=1〈c2i 〉= (k+1)c¯2 as discussed in Ref. [120].From the posterior distributions, the uncertainty σ∆ is the 68% confidence in-84terval of the A0∆(1)ν distribution. Simplifications of this Bayesian analysis havebeen proposed by Ref. [65] where it was proposed that the 68% confidence regionof the chiral truncation uncertainties is given byσNkLO = A0Qk+2max{|c0|, . . . , |ck+1|}, (5.24)with the coefficients cν as in Eq. (5.17). This analytical formula is in semi-quantitativeagreement with the Bayesian approach with the prior assumption of a uniform andboundless distribution [19]. This simplified approach has been found to be tooconservative in many cases. The advantage of this method is that it is simpler toimplement. However, in Chapter 6 we will give results that include the uncer-tainty estimates from both of the Bayesian as well as the simplified calculations inEqs. (5.23) - (5.24) for the δATPE in µD.5.3 η expansion truncation uncertaintyThe calculation of δATPE from the η-expansion in Section 4.4 has been carried outto sub-sub leading order. Therefore the uncertainty ση from this truncation needsto be estimated. The general form of the δATPE calculation in the η-expansion canbe represented as a series expansion in powers of the dimensionless operator ηδApol = X0∞∑n=0cn(A,Z)ηn, (5.25)where X0 is the natural scale of the TPE, taken here to be the leading order dipolecorrection in Eq. (4.79). The dimensionless coefficients cn(A,Z) depend on themass number A and charge number Z of the nucleus. To analyze the convergencepattern of this expansion in the Bayesian formalism it is assumed that the coeffi-cients are IID numbers of natural size with an associated scale parameter c¯. Fur-thermore, at a fixed order k in the η expansion of the nuclear polarization δATPE, wedefine the vector D that contains the order-by-order corrections up to δ (k) asD =1δ (0)(δ (1),δ (2), . . . ,δ (k)), (5.26)85where all of the corrections have been divided by the leading term δ (0) that setsthe scale of the series. In order to compute ση we will first use D to calculate theposterior of η . By Bayes theorem the posterior distribution of η conditioned onthe data D isP(η |D) = P(D|η)P(η)P(D), (5.27)where P(η) is the prior distribution of η containing all previous knowledge of theη-parameter and P(D) is the probability of the data. Assuming that each termin the η-expansion is independent from one another the probability of the dataconditional on η is factored into the product of independent conditional probabilitydistributions asP(D|η) = P(D1|η)P(D2|η) . . .P(Dk|η). (5.28)Here we introduce and marginalize over the unknown IID coefficients in Eq. (5.25)collected into the vector c = (c1, . . . ,ck). These coefficients have an unknown scaleparameter c¯ that is also marginalized over. The result isP(D|η) =c¯>∫c¯<dc¯ P(c¯)∫dc P(D1|c1,η)P(c1|η , c¯) . . .P(Dk|η ,ck)P(ck|η , c¯),(5.29)where c¯< and c¯> are the minimum and maximum scales, respectively.As in the case of the χEFT truncation uncertainties the probability distributionsof Dk conditioned on ck and η satisfies the constraintP(Dk|ck,η) = δ (Dk− ckηk). (5.30)Substituting this last expression into Eq. (5.29) the marginalization integrals willsimplify toP(D|η) = 1ηk(k+1)/2c¯>∫c¯<dc¯ P(c¯)P(c1|c¯)P(c2|c¯) . . .P(ck|c¯), (5.31)where the condition ck = Dkηk has been imposed from Eq. (5.30) for all unknown co-efficients. The expression in Eq. (5.31) is analogous to the one derived in Ref. [21]86that inferred the χEFT breakdown scale from the NN data. Up to a normalizationconstant, the posterior distribution of η isP(η |D) ∝ P(η)ηk(k+1)/2c¯>∫c¯<dc¯ P(c¯)P(c1|c¯)P(c2|c¯) . . .P(ck|c¯). (5.32)The different η-prior distributions that were used in this work are listed below inTable 5.2.Priors P(η)Aη 1ln(η>/η<)η θ(η−η<)θ(η>−η)Bη 1√2piηwη Exp(− ln(η)22w2η)Cη θ(η−η<)θ(η>−η) 1η>−η<Table 5.2: The prior probability densities of η .In Table 5.2, the first η-parameter prior, Aη , represents a scale-invariant unin-formative prior where η lies between the maximum and minimum values η< andη>, respectively. This prior is appropriate under the assumption that we do nothave information about the possible magnitude of the parameter η . The next priorin the table, Bη , is the log-normal distribution of width wη . This prior captures ourknowledge that the η parameter is a positive number with a maximum probabilityat η0 = Exp(−w2η). The last prior, Cη , represents the uniform probability distri-bution of the parameter η in the region η< ≤ η ≤ η>. This is a reasonable prior ifwe are certain that the parameter η lies in a given range.The derivation of the posterior distribution of the truncation uncertainty ση forthe η-expansion truncation in the leading order approximation follows the sameprocedure as for the χEFT truncation uncertainty σ∆ presented in Section 5.2 andwe do not repeat the details here. The difference with respect to that derivation isthat for the η expansion c1 6= 0. This will result in the posterior distribution of the87truncation uncertainty ∆(1)ν in the first omitted term approximation asP(∆(1)ν |D,η) =c>∫c<dc¯ P(cν+1 =∆(1)νην+1 |c¯)P(c1|c¯) . . .P(cν |c¯)P(c¯)ην+1c>∫c<dc¯ P(c1|c¯) . . .P(cν |c¯)P(c¯), (5.33)where the conditional probability densities P(cν |c¯) for the coefficients cν and P(c¯)are the same as those given in Table 5.1. The truncation uncertainty is the 68%confidence interval of the distribution of δ 0D1∆(1)ν . The expression in Eq. (5.33) canbe directly evaluated using a fixed choice of η , based on the posterior distributionP(η |D). Two choices for selecting η are made in Chapter 6: the first is to choosethe median value and the second is the maximum likelyhood estimate of η . Theresults of these two approaches will be discussed in Section 6.4.Another approach to compute ση is to avoid choosing a fixed value of η andcompute the marginalized posterior distribution of the truncation uncertainty. Theexpression for the marginalized posterior distribution of the η-truncation uncer-tainty isP(∆(1)ν |D) =ηmax∫ηmindη P(∆(1)ν |D,η)P(η |D), (5.34)where ηmin and ηmax are chosen to be the regions that enclose the 68% or 95%confidence intervals of the P(η |D) distribution. This procedure is implemented inour δTPE calculations and results will be shown in Chapter 6.5.4 Other uncertaintiesIn addition to the uncertainties discussed above there are other systematic uncer-tainties that must be estimated in our nuclear physics calculations. These uncer-tainty arise from the following sources:• Uncertainties from the few-body algorithm, denoted with σFB;• Uncertainties from the numerical procedure used to calculate the elasticcomponent of the nuclear TPE, denoted with σEP;88• Uncertainties from the maximum lab energy T MaxLab used in the fits of the NNpotential, denoted with σT MaxLab ;• Uncertainties from systematic approximations in the electromagnetic opera-tors Jν(x), denoted with σJ;• Uncertainties due to single nucleon physics, denoted with σN and;• Uncertainties from neglected terms in the (Zα) expansion, denoted withσZα .The uncertainty σFB is the systematic uncertainty from choosing a specificfew-body algorithm. For the deuteron, this uncertainty is addressed in Chapter6 through benchmark calculations of EM moments using the truncated HO basisexpansion method or the Numerov algorithm [121].In Chapter 4, three different methods for computing the EM moments relevantto muonic atoms were developed. The associated numerical accuracies of theseprocedures are addressed in Section 6 and denoted as σEP.The uncertainty σT MaxLab represents the systematic uncertainty that arises fromchanging the maximum kinetic energy in the laboratory frame of the NN scatteringdata sets used to fit the nuclear potentials described in Chapter 2. This uncertaintywas probed in the calculation of δATPE with the NkLOsim potentials of Ref. [62].As discussed in Chapter 3, the EM currents obey a hierarchy that is analogousto the χEFT expansion and a consistent χEFT calculation of electromagnetic prop-erties should treat the NN potentials and currents Jµ at the same order leading tofurther systematic corrections. However, for the TPE in the Lamb shift the domi-nant term is the leading dipole correction. The electric dipole operator is protectedby the Siegert theorem in Eq. (3.57), so that two-body currents are implicitly in-cluded via the use of the continuity equation [11] and the systematic uncertaintyfrom two-body operators to this dipole operator are suppressed. These operatortruncation uncertainties of the EM currents are denoted σJ and estimated in Chap-ter 6 for the Lamb shift TPE.In the formalism laid out in Section 4, the nuclear TPE process is separated intothe nuclear and nucleonic contributions. The nucleonic contributions, are calcu-lated within data in the dispersion framework in Ref. [122] leading to uncertainties89σN in their determinations. The single nucleon uncertainties that enter the δTPEdetermination are taken from Refs. [53, 123, 124] and are used as inputs in ourcalculations.The final relevant uncertainty considered in this work arises from the (Zα)-expansion. The TPE calculations in this work are carried out at a specific orderin α and the estimates of the neglected higher order contributions need to be esti-mated. With the exception of the Coulomb correction and the vacuum polarizationcontribution discussed in Section 4, the TPE diagrams that we consider in this workare of the order (Zα)5. The (Zα)-truncation-estimates, denoted σZα , are discussedin Chapter 6.90Chapter 6ResultsIn this Chapter we use the χEFT potentials introduced in Section 2.2 with the for-malism of Chapter 4 to calculate EM observables and processes in muonic atomsand few-body systems. The calculations for the deuteron are based on the HO ex-pansion of the wave function outlined in Section 2.3. A discretized approach to thesum rules was used to compute the terms in Eq. (4.83), as in Refs. [5, 100]. Anappropriately large NMax was chosen to assure the convergence of the calculations.We have previously demonstrated that this approach is reliable and agrees well withother calculations, such as from Pachucki [98] and Arenho¨vel [11]. The first goalof this Chapter is to address the few-body method uncertainty (σFB) and numericalaccuracy (σEP) of the EM moment calculations for µ2H in Section 6.1. In Section6.2 we quantify all relevant statistical and systematic uncertainties entering intothe TPE calculation from the η-expansion framework outlined in Section 4.4 thatinclude estimates of the η-truncation uncertainty (ση ). In Section 6.3, using theη-less formalism from Section 4.3, we justify this uncertainty using realistic χEFTpotentials as well as thepi-EFT formalism introduced in Section 2.4. This η-lessexpansion method is extended in Section 6.4 for the vacuum polarization term fromSection 4.4.2. The ση uncertainty calculated explicitly in Section 6.3 has only beencarried out in the A = 2 system and estimated for A ≥ 3. In Section 6.4 we applythe Bayesian formalism from Section 5.3 to robustly estimate ση for A = 3 and 4with the results of A = 2 used to validate the procedure. Finally, in Section 6.5 webenchmark and apply the two-body currents developed in Section 3.3.2 to A = 291and 3 systems. In anticipation of the Lamb and HFS measurements that will becarried out in muonic lithium atoms we apply one- and two-body currents to 6Li tostudy its magnetic properties and compare the results to experimental data.6.1 Zemach radius benchmark CalculationsHere, we address the uncertainties σFB and σEP relevant to the calculations of theTPE in muonic deuterium. The procedural σEP uncertainties are estimated by com-paring the results from the r-space, q-space and the hybrid qr-space methods de-veloped in Section 4.1. The calculations of selected EM moments for the deuteronusing the different procedures are presented in Table 6.1. For the calculations weemploy the AV18 potential from Ref. [125] and do not use nucleon form factors.Moment r-space q-space qr-space〈1/RE〉(2) fm−1 0.5992268(5) 0.5992259(2) 0.599226129(1)〈RE〉(2) fm 2.4062611(4) 2.40628(1) 2.4062607(2)〈R3E〉(2) fm3 31.781135(8) 31.7812(3) 31.7812(2)Table 6.1: The results of the r, q, qr -space procedures, respectively, forselected deuteron EM moments without form factors using the AV18 potential.The values in the brackets above are the estimates of the numerical uncertain-ties from the Zemach moment algorithm. The different procedures are consistentwithin their uncertainties up to relative difference of about ≈ 10−4%, which issmaller than the largest numerical error σEP ≈ 1×10−3% from the procedures andindicates excellent agreement. Overall the mixed qr-space method is more reliablesince it achieves high precision without needing to regularize the low-q behaviorof the integrands in Eqs. (4.15) and (4.16). These findings are consistent with ourwork in Ref. [1] for A≥ 3 systems.Next, the σFB method uncertainty is assessed through benchmark calculationsof the EM moments of 2H with either the Numerov algorithm or HO basis expan-sion, respectively, using the AV18 potential. These calculation also include nucleonform factors from the Kelly parametrization of Ref. [126]. The results for both thepoint-proton limits are provided, i.e., without form factors (w/o FF) as in Table 6.1,92and with the Kelly parametrized form factors (w FF). The moments related to theelectric properties, RE , 〈RE〉(2), and 〈R4E〉 are given in Table 6.2, while those relatedto the magnetic properties, 〈RZ〉, RM and the magnetic moment of the deuteron µdin Table 6.3. For µd we take the expectation value of the LO operator from Section3.3.2 on the deuteron ground state µd = 〈N0|∑i µLOi |N0〉. Both tables include theavailable experimental values.RE 〈R3E〉(2) 〈R4E〉Method [fm] [fm3] [fm4]HO (w/o FF) 1.96734(1) 31.7812(3) 55.370(1)Numerov (w/o FF) 1.9674(1) 31.83(1) 55.376(1)HO (w FF) 2.1219(1) 38.2902(3) 64.809(1)Numerov (w FF) 2.1218(1) 38.33(1) 64.814(1)Exp. 2.1413(25) [46] n.a n.a2.1256(8) [50]Table 6.2: The deuteron benchmark in LO for electric properties: using theHO basis expansion or the Numerov algorithm with the AV18 potential. Theresults without form factors is denoted (w/o FF), and with form factors usingthe Kelly form factors (w FF). Results are compared to experimental data.The Table is adapted from Ref. [1].The electric moments calculated with the different few-body algorithms arein good agreement except for the third Zemach moment 〈R3E〉(2) and the fourthcharge moment 〈R4E〉. This demonstrates that these moments are more sensitiveto the numerical method. This is due to the dependency of these moments on theasymptotic region of the wave functions that are more sensitive to the algorithmused. However, their sensitivity is not significant (∼ 0.2%).Including nucleon formfactors in the RE calculation shifts it towards the experimental value.The results of the magnetic properties in Table 6.3 are in agreement except forthe Zemach radius, but the difference is again negligible (∼ 0.1%). The additionof form factors improves the agreement of 〈RZ〉 with experiment. The deuteron µdvalue in the point nucleon limit, with the leading order dipole magnetic operatorµLO and with finite nucleon size are identical, since the form factor effects areproportional to q and vanish in the limit where q → 0. The inclusion of form93factors in the calculation of the magnetic radius RM slightly worsens the agreementwith experiment but the values are still within the experimental error. This behaiourmay be the result of missing two body currents that are known to be important tomatch experiment [127].〈RZ〉 RM µdMethod [fm] [fm] [µN]HO (w/o FF) 2.3811(2) 1.9405(1) 0.84699(1)Numerov (w/o FF) 2.3795(1) 1.9405(1) 0.84699(1)HO (w FF) 2.5973(2) 2.0664(1) 0.84699(1)Numerov (w FF) 2.595(3) 2.0664(1) 0.84699(1)Exp. 2.593(16) [128] 1.90(14) [129] 0.8574382311(48) [46]Table 6.3: The deuteron benchmark in the LO for magnetic properties: usingthe HO basis expansion or the Numerov algorithm with the AV18 potential.The results without form factors is denoted (w/o FF), and with form factorsusing the Kelly form factors (w FF). Results are compared to experimentaldata. The Table is adapted from Ref. [1].Overall, we have demonstrated that for the deuteron the σEP uncertainty isnegligible, while the σFB uncertainty is at most about 0.2% for 〈R3E〉(2). For thedeuteron these uncertainties are negligible, but are more significant for larger sys-tems [1]. This uncertainty analysis is relevant for the ongoing experimental effortsin heavier muonic atoms.6.2 Uncertainty of the TPE in the µ2H Lamb ShiftIn this Section we apply the uncertainty quantification formalism developed inChapter 5 to the TPE calculation in the framework of the η-expansion for muonicdeuterium. As described in Section 4.4 the calculation of δTPE is separated intonuclear and nucleonic contributions, where the nuclear contributions δATPE are ourmain focus. In this Section we address the claim that the 2.6 σ disagreement be-tween the theory compilation of Krauth et al. [53] and the experimental value of[50] is the result of underestimated nuclear physics uncertainty. The elastic andinelastic nucleon TPE terms that constitute δNTPE in Eq. (4.40) are estimated to be94δNZem = −0.030(2) meV 1 and δNpol = −0.028(2) meV from data as described inRef. [122]. Unlike these previous terms the subtraction term δNsub is not well con-strained by experimental data. The contribution of this term in µH was calculatedby Birse and McGovern in Ref. [95] from χEFT and determined to be 0.0042(10)meV. By contrast, using the operator product expansion method, Hill and Pazreinvestigated the TPE calculation in µH. The central value of their result agreeswith Ref. [95] but carries a large uncertainty (see also Refs. [130, 131]). For µ2Hthe subtraction term δNsub is a sum of the proton and neutron subtraction terms. Inthis dissertation, we take the proton and neutron subtraction terms to be equal inmagnitude and designate a 100% uncertainty to the result as was carried out inRef. [53]. This results in δNsub = 0.0098(98) meV [53]. If the uncertainty of thisresult is increased to 200% then the value of δNsub = 0.0098(196) meV2 is of thesame magnitude as the value in Ref. [123].6.2.1 Statistical uncertainty estimatesThe uncertainties of δTPE due to statistical uncertainties in the LECS are probed withthe NkLOsim potentials of Ref. [62], described in Section 2.2, with k from 0 to 23.These sim potentials employ seven different cutoff values Λ = 450−600 MeV inincrements of 25 MeV. Furthermore, for each Λ the nuclear potential was fit to sixincreasing T MaxLab energy ranges of the SM99 world database. This database consistsof NN and piN scattering cross sections along with the ground state properties ofA =2,3 systems4. For each sim interaction the T MaxLab energies that were used wereT MaxLab = 125−290 MeV in increments of 33 MeV. The covariance matrix Σ0 of theLECS was determined numerically to machine precision. As described in Section5.1 the covariance matrices allow the forward propagation of the uncertainties ofthe LECS onto δTPE while the different combinations of the parameters Λ and T MaxLabprobe the systematics of the NN fitting method and cutoff sensitivity.1This value is estimated by rescaling the µH value adopted by [53] according to the scaling factordescribed in Refs. [5, 102].2Here the digits in the uncertainty have been kept to show that they are twice the value of δNsub =0.0098(98) meV from Ref. [53].3Note that k and ν are not exactly the same, even though there is a one-to-one correspondencebetween them; k = 0,1,2,3,4 corresponds to ν = 0,2,3,4,5 [3].4The radius Rp, binding energies E0, and also the quadrupole moment Qd for 2H.95The covariance matrix of nuclear structure corrections Cov(OA,OB) relevant tothe µ2H system is computed using Eq. (5.11) and the Σ0 of LECS is provided fromRef. [62]. The Jacobians generated with this procedure were tested by compar-ing our results for several deuteron ground state properties, such as the quadrupolemoment Qd , the RMS point nucleon radius Rp and the binding energy E0 withcalculations employing automatic differentiation. These benchmarks yielded ex-cellent agreement for all observables with relative differences at the 0.005% level,or better.The correlation analysis described in Section 5.1 provides a quantification ofthe correlations ρ(OA,OB) that exist in the χEFT model of the nuclear polariza-tion contributions. This determines the constraints between different observablespredicted within the NkLOsim models and is useful to validate the uncertainty prop-agation. Our correlation analysis focuses on δATPE and its components, such as theleading dipole term δ (0)D1 in Eq. (4.79) and the small magnetic term δ(0)M definedin Ref. [2]. Since these terms are related to the electric and magnetic dipole re-sponse functions we study the electric dipole polarizability αE and the magneticsusceptibility βM. The electric dipole polarizability is definedαE =2α3(2J0+1)(4pi3)∑N 6=N0|〈N0||OEκ=1||N〉|2EN−E0 , (6.1)while the magnetic susceptibility isβM =2α3(2J0+1)∑N 6=N0|〈N0||OMκ=1||N〉|2EN−E0 . (6.2)These expressions can be found in Refs. [5, 100]. In addition, we study the cor-relations of the ground state energy E0, the RMS point-proton distribution radiusRp of the deuteron, the electric quadrupole moment Qd and the magnetic dipolemoment µd , along with the D-wave probability PD. The correlation coefficientρ(OA,OB) in Eq. (5.12) between all pairs of observables OA and OB in the setO = {E0,Rp,Qd ,PD,µd ,αE ,βM,δ (0)D1 ,δ (0)M ,δATPE} are calculated and illustrated inFig. 6.1.96E0r QdPDµdαEβMδ(0)D1δ(0)M δA TPEE0rQdPDµdαEβMδ(0)D1δ(0)MδATPE1. 1.001.0 0.6 0.2 0.2 0.6 1.0CorrelationFigure 6.1: The correlation matrix of the deuteron ground state energy E0,rms radius Rp, quadrupole moment Qd , D-state probability PD, magnetic mo-ment µd , electric polarizability αE , magnetic susceptibility βM, leading dipolepolarizability correction δ (0)D1 , magnetic polarization correction δ(0)M and δATPEfor the N2LOsim potential with Λ = 450 MeV and T MaxLab =125 MeV. The Figureis taken from Ref. [3].The quadrupole moment Qd of the deuteron ground state is determined byits D-wave component and consequently the correlation coefficient ρ(Qd ,PD) isstrongly positive. The ground state magnetic moment of the deuteron induced bythe µLO operator from Eq. (3.70) is shown to be linearly related to PD through [81]µd =12(gp+gn)+34(1−gp−gn)PD, (6.3)where gp and gn are the magnetic g factors of the proton and neutron that areknown from experiment [46] to be gp=5.585694702(17) and gn=-3.82608545(90),respectively. As expected from this expression, the numerical analysis produces97ρ(PD,µd) =−1. Based on the observation that PD is strongly correlated to µd , onewould also expect that the magnetic polarization term δ (0)M is correlated with PD.This expected relation is confirmed in Fig. 6.1. Therefore, the magnetic propertiesof the deuteron are largely determined by its D-wave component.In contrast to the magnetic properties, the electric properties of the deuteronare found to be strongly related to Rp. In particular, we find that ρ(Rp,αE) isnearly +1. The correlation between these two observables is predicted by thepi-EFT model in Section 2.4 to be given by [82, 132]αE =α16|E0| 〈R2p〉. (6.4)This correlation has also been observed in heavier mass systems [133, 134]. In fact,we find that δATPE, which is dominated by the dipole term δ(0)D1 , is correlated to bothRp and αE . The production of expected correlations using the formalism of Section5.1 served as a useful way to validate the statistical analysis and these correlationsmay also serve to guide the use of constraints for alternate fitting procedures infuture investigations.The statistical uncertainties of the RMS point proton radius Rp, the electricdipole polarizability αE , and δATPE were determined to be 0.02%, 0.05%, and 0.05%respectively. This is negligible compared to the systematic uncertainties as demon-strated in Section 6.2.2. These statistical errors are significantly larger when theseparately (or sequentially) optimized NkLOsep potentials from Ref. [62] are used.The sep potentials are fit to the NN, piN and 3N sectors separately. The increaseof the uncertainties from this approach is due to the statistical covariances betweenthe NN, piN and 3N datasets that are neglected by this procedure and which alsolead to larger uncertainties in the LECS. The majority of NN χEFT potentials, suchas the 5th order χEFT forces employed in the next section, are optimized by thissequential methodology. The sub-leading piN LECS used in these χEFT poten-tials used a novel Roy-Steiner extrapolation of the piN scattering data [135]. TheLECS obtained from this approach have small statistical uncertainties [136], whichstrongly indicate that the forward propagation of these uncertainties to δATPE wouldproduce negligible errors.986.2.2 Systematic uncertainty estimatesHere we present an analysis of the contributions of the systematic uncertainties ofσ∆, σΛ and σT MaxLab , introduced in Chapter 5.125 158 191 224 257 290TmaxLab  [MeV]1.6821.6801.6781.6761.6741.6721.6701.6681.666δA TPE [ meV ]Λ= 450Λ= 500Λ= 550Λ= 600Figure 6.2: The calculated values of δATPE for different cutoffs Λ in MeVas a function of T MaxLab for the N2LOsim potentials. The Figure is taken fromRef. [3].In Fig. 6.2, the calculated values of δATPE are plotted as a function of the energy,T MaxLab , used in the optimization of the N2LOsim potentials for multiple values of thecutoff Λ. The error bars indicate the statistical uncertainties σstat, computed as de-tailed in Section 5.1. These uncertainties were on average found to be 0.001 meV,or 0.06% of the central value. It is clear from Fig. 6.2 that the statistical uncer-tainties are suppressed relative to the systematic uncertainties σΛ and σT MaxLab . Thespread of δATPE for different Λ values decrease at the largest T MaxLab energies, whichindicate that the nuclear dynamics contained in the LECS are better constrained athigher energies where more data is available.Next, we address the uncertainties σ∆ from the χEFT truncation at the orderν , first using the simplified formula of Eq. (5.24) to estimate the expected size ofthe next higher-order contribution in the χEFT expansion. The typical momentum99scale p that determines the χEFT expansion parameter Q of the nuclear structurecorrections is estimated by computing the average energy value of the δ (0)D1 term inδATPE5. The average value 〈ω〉D1 is calculated as the ratio〈ω〉D1 =∫dω ω√2mrωN SD1(ω)∫dω√2mrωN SD1(ω). (6.5)This procedure results in 〈ω〉D1 ≈ 7 MeV, corresponding to a momentum scale psmaller than mpi . Therefore, the χEFT convergence parameter Q in Eq. (5.16) forour σ∆ estimates is always taken to be mpi/Λb. The optimal value chosen for Λb is600 MeV as discussed in Section 2.2.450 475 500 525 550 575 600Λ [MeV]1.6601.6651.6701.6751.6801.685δA TPE [ meV ]Figure 6.3: The systematic uncertainties of δATPE as a function of the cutoffΛ for the N2LOsim potentials. The width of the blue (dark) band indicates theuncertainty from σT MaxLab . The (light) green band also includes the σ∆ uncer-tainty. The difference between the maximum and minimum width of thesebands are the σΛ uncertainty. The Figure is from Ref. [3].5We compute the typical momentum scale as the RMS momentum value√〈p2〉D1. The RMSmomentum value is related to the response function using 〈p2〉D1/2M ≈ 〈ω〉D1100In Fig. 6.3, the results of δATPE calculated with the N2LOsim potential are givenas a function of Λ. The solid black line indicates the central values and the coloredbands represent the uncertainty estimates. The blue band indicates the spread ofδATPE from the σT MaxLab uncertainty. With respect to the central values, this spreadamounts to a 0.004 meV (0.2%) contribution. The wider green band is the com-bination of the σT MaxLab and σ∆ uncertainty. Due to the fact that the σ∆ and σΛ un-certainties are not independent from one another, the σ∆ error is calculated fromthe simplified Bayesian formula in Eq. (5.24) and added to each point in T MaxLaband Λ. The green band encompasses the maximum and minimum values of δATPEobtained from this procedure which represent the combination of the σ∆, σΛ, andσT MaxLab uncertainties. The maximum σ∆ contribution is 0.007 meV for the N2LOsimpotentials. The total uncertainty of these potentials from the combination of σT MaxLab ,σ∆ and σΛ variations is 0.011 meV (0.65%).The convergence of δATPE with respect to higher order χEFT potentials is in-vestigated using the potentials described in Section 2.2 up to fifth order. For theNkLOEKM family [66] of potentials we use the cutoff combinations (R0,Λ) =(0.8,600),(1.0,600) and (1.2,400) [fm, MeV], where R0 is a coordinate-spaceregulator while for the NkLOEMN family of NN forces we use Λ = 450,500 and550 MeV. The calculation at higher order χEFT orders allow us to reliably estimateσ∆ and provide the most up-to-date results of the δATPE with respect to our previouswork in Refs. [5, 100].6.2.3 Total uncertainty estimatesAll of the systematic σ∆ and σΛ uncertainties from the χEFT potentials, statisticaluncertainties σstat, and σT MaxLab uncertainties for all interactions are combined intoσNucl, detailed in Table 6.4. The systematic errors σT MaxLab cannot presently be calcu-lated at the N3LO and N4LO χEFT orders. Therefore, the uncertainties from σTMaxLabfor the lowest order EKM and EMN potentials are estimated by using the corre-sponding values from the sim potentials. The sim potentials contain all of the abovesystematic and statistical uncertainties, determined consistently at each order. Thestatistical uncertainties are negligible in the sim potentials and are expected to besimilarly suppressed in the NkLOEKM and NkLOEMN families even though such an101evaluation in not currently feasible. Consequently, we use the statistical uncertain-ties obtained from the N2LOsim potential and apply them to the N2,3,4LOEKM/EMNforces. For the lower order LOEKM/EMN(NLOEKM/EMN) forces we use the corre-sponding uncertainties from the LOsim(NLOsim) potentials, respectively.Order Potential δTPE σNucl σTotal[meV] [meV] [meV]LOsimEKMEMN−1.616−1.767−1.599+0.11−0.11+0.18−0.17+0.095−0.097+0.11−0.11+0.18−0.17+0.097−0.099NLOsimEKMEMN−1.724−1.718−1.710+0.032−0.032+0.025−0.034+0.029−0.029+0.038−0.038+0.032−0.040+0.035−0.035N2LOsimEKMEMN−1.721−1.705−1.710+0.011−0.011+0.008−0.010+0.008−0.009+0.023−0.023+0.022−0.023+0.022−0.022N3LOEKMEMN−1.719−1.712+0.009−0.012+0.006−0.005+0.022−0.024+0.021−0.021N4LOEKMEMN−1.718−1.712+0.008−0.009+0.006−0.006+0.022−0.022+0.021−0.021Table 6.4: The results for δTPE at various orders with estimates for the nuclearphysics σNucl and the total σTotal uncertainties. The Table is adapted fromRef. [3].As discussed in Section 4.4, the current calculations are carried out withinthe η-expansion framework up to sub-subleading order. Thus, the η-truncationuncertainty ση needs to be estimated in the total error budget. Based on the resultsof the η-less formalism in Section 4.3 these ση contributions were estimated toprovide a 0.2− 0.3% effect. The uncertainties σJ from approximations in the EMcurrent operators Jν where estimated from a E1-response function provided byArenho¨vel [137] that included two-body currents and relativistic corrections as102described in Ref. [11] for the AV18 potential. The effects of these currents onthe leading dipole correction δ (0)D1 were both found to be negligible of the order of0.05%.The uncertainty estimates carried out thus far only probe δATPE, the term in δTPEthat depends only on the nuclear dynamics. A complete uncertainty estimate ofδTPE needs to consider the term δNTPE defined as the sum of terms in Eq. (4.40).The combination of the uncertainties of the individual components of this sumis denoted as σN introduced in Section 5.4. Furthermore, we must consider theZα truncation uncertainty discussed in Section 5.4. This σZα was estimated byRefs. [2, 98] to be of the order of 1%. We refer to the σZα uncertainty contributionsas the atomic physics uncertainty and add it to the other uncertainties in quadrature.The uncertainties σZα , σN , and ση of δTPE are combined with the nuclear physicsuncertainties into σTotal, given in Table 6.4.LO NLO N2 LO N3 LO N4 LO CREMA1.951.901.851.801.751.701.651.601.551.501.45δ TPE [meV]ExperimentTheoryNk LOEKMNk LOEMNNk LOsimFigure 6.4: δTPE as a function of the χEFT order with total uncertainty σTotal(see text for details). The Figure is from Ref. [3].In Fig. 6.4 and in Table 6.4, the convergence of δTPE is shown with its over-all uncertainty σTotal from LO up to N4LO in χEFT order where we use the δNsubvalue from Ref. [53]. The impact of the larger uncertainties of Ref. [123] are dis-cussed later in this Section. The uncertainty bands decrease as the order of the103χEFT potential increases as would be expected. Beside LO calculations, where thethree potential families are the most different, the results at higher orders are sta-ble around overlapping ranges irrespective of the potential. We observe that theN3LO and N4LO results are nearly identical indicating that the χEFT expansionhas converged for δTPE. The N3,4LO uncertainty estimates are larger due to theinclusion of σ∆ from Eq. (5.24) but remain consistent with our earlier estimatesin Refs. [5, 100]. The results given in Table 6.4 demonstrate that, while the nu-clear physics errors dominate the lower order χEFT calculations, at N4LO the mainuncertainty originates from other sources.The results obtained in this section are also compared to the experimentallyinferred δTPE = −1.7638(68) meV correction from Ref. [50] and the theoreticalcompilation of Krauth et. al from Ref. [53]. We find that the N4LO uncertaintyband is consistent with the theoretical compilation and encompasses the resultδTPE = −1.709 meV from our earlier work in Ref. [100] that was based on theAV18 potential [125], that was also included in the theory summary. Despite thisrefined uncertainty analysis our N4LO band is not yet compatible with the experi-mental δTPE determination.Source % Uncertainty Uncertainty in meVσsyst+0.5−0.6+0.008−0.011σstat 0.06 ±0.001ση 0.3 ±0.005σN 0.6 / 1.2 ±0.0102 [53] / ±0.0198 [123]σZα 1.0 ±0.0172σTotal 1.3 / 1.6-1.7+0.022 / +0.028−0.024 / −0.029Table 6.5: The uncertainty budget of the δTPE value from our analysis. We usetwo values for the single nucleon uncertainties σN , one where we adopted thestrategy of Ref. [53] as well as using the larger uncertainties from Ref. [123]for δNsub. The Table is taken from Ref. [8].From this analysis, we present the updated result of δTPE =−1.715 meV withthe uncertainty budget listed in Table 6.5 as in Refs. [3, 8]. The central value104is obtained from the average of the N4LO results of the EMN and EKM poten-tial families. These uncertainties are separated into systematic, statistic nuclearphysics, ση , σN and σZα uncertainties. The systematic nuclear physics uncertaintyσsyst is a combination of the σ∆, σΛ, σJ and σT MaxLab uncertainties. The contributionsof the uncertainties from σΛ and σ∆ are obtained from our N4LO studies by tak-ing the combined range of the EMN and EKM bands. The quadrature sum of alluncertainties is σTotal. As pointed out in Section 5.2 the truncation uncertainty cal-culated from Eq. (5.24) is a simplification of the Bayesian calculation in Eq. (5.20)that may be overly conservative. To address this and demonstrate the insensitivityof σTotal to the procedure used to calculate σ∆ we compute the 68% confidenceinterval of the Bayesian posteriors from Eq. (5.20). The posterior distributionsP(∆(1)ν |c) of the δTPE calculation are given Figs. 6.5 for the NkLOEKM potential atthree different χEFT orders. This more exact procedure for estimating σ∆ increasesthe lower bound of σTotal slightly from -0.024 meV in Table 6.5 to -0.023 meV us-ing the σN values of Ref. [53] since the values of σ∆ at N4LO for prior A in Table5.1 are smaller when calculated in this manner.1.0 0.5 0.0 0.5 1.010 2 A0 20100200P( (1)2 |c0, c2, c3)ABC2 1 0 1 210 3 A0 305001000P( (1)3 |c0, c2, c3, c4)ABC5.0 2.5 0.0 2.5 5.010 4 A0 40200040006000P( (1)4 |c0, c2, c3, c4, c5)ABCFigure 6.5: The posterior distributions of the σ∆ uncertainties at differentorders in the χEFT expansion (A0∆(1)2 , A0∆(1)3 , A0∆(1)4 ) in meV units for δTPEin the ∆(1)ν approximation for the NkLOEKM potentials with (R0,Λ)=(0.8, 600)[fm, MeV]. The expansion parameter is Q = 0.23, wB=1 for prior B withc¯<=0.1 and c¯>=10. The Figure is taken from Ref. [8].The TPE value determined in this work differs from the experimentally deter-mined value from Ref. [50] of δTPE = −1.7638(68) meV by less than 2 σ , whichis not significant. In Table 6.5 we find that the uncertainties arising from the nu-clear model dependence, σsyst and σstat, are small in comparison to the σZα or σN[123] uncertainties that dominate the final uncertainty budget. It is therefore un-105likely that the differences between the experimental and theoretical determinationsof δTPE stem from the NN-force models.Despite the controversial single-nucleon TPE uncertainty in Refs. [123, 130,131], it is evident from Table 6.5 that the 1% σZα error remains a major sourceof uncertainty. A more thorough estimate of α6 effects requires the calculation ofthird order photon exchange diagrams. In Section 6.3.3 we address the vacuumpolarization contributions [109] which are partial α6 contributions and that areenhanced relative to other contributions at the same order. Here we have demon-strated that the uncertainties from χEFT models of the NN interaction do not fullyaccount for the 2.6σ discrepancy between the theoretical and experimental δTPEextracted by Pohl et al. [50]. However, the significance has been reduced to below2σ from our uncertainty analysis. As we show in Section 6.3.3 the combinationof this error analysis with the aforementioned vacuum polarization resolve the dis-agreement to within 1σ .1066.3 The η-less expansionIn this Section we provide the results of the η-less formalism for the deuteron us-ing the analyticalpi-EFT case and with a realistic NN interaction. To calculate theterms in Eqs. (4.65)-(4.67) the longitudinal SL and transverse ST response func-tions defined in Eq. (4.57) and Eq. (4.58), respectively, are expanded as sums ofmultipole functions. These functions arise from the expansion of ρ˜ in Eq. (3.37)and J˜ according to Eq. (3.49). This results in the sums given bySL(q,ω) =∞∑κ=0SL,κ(q,ω), (6.6)ST (q,ω) =∞∑κ=1SE/MT,κ (q,ω), (6.7)where ST (q,ω) is further broken up into the sum of electric SET (q,ω) and magneticSMT (q,ω) responses. The response functions of each multipole is given bySL,κ(q,ω) =4pi2J0+1∑N 6=N0|〈N||Cκ(q)||N0〉|2δ (EN−E0−ω), (6.8)SE/MT,κ (q,ω) =4pi2J0+1∑N 6=N0|〈N||T E/Mκ (q)||N0〉|2δ (EN−E0−ω), (6.9)where Cκ(q) are the Coulomb multipoles defined in Eq. (3.40) and TE/Mκ (q) arethe electric or magnetic tensors in Eqs. (3.47) and (3.48), respectively. The Siegerttheorem is applied to relate SET (q,ω) to the longitudinal response functionSET,κ(q,ω) =ω2q2[κ+1κ]SL,κ(q,ω)+δST,κ(q,ω), (6.10)where δST,κ(q,ω) is the correction to the Siegert response. Following the workof Refs. [84, 138] this correction is neglected in the low q region that dominatesthe integrands in Eqs. (4.65)-(4.67). Through this multipole expansion, the nuclearstructure corrections in Eqs. (4.65)-(4.67) are the sum of terms∆x =∞∑κ=0∆κ,x, (6.11)107where the symbol x denotes (L), (T,E), or (T,M). In what follows, except for inthe Tables where the achieved numerical accuracy is interesting to compare, wequote the final values in the text to three significant digits because the numericaluncertainty of the calculations is negligible in comparison to the final error budgetwhen all relevant uncertainties from Section 6.2 are accounted for.6.3.1 Analytical pionless effective field theoryBefore tackling the deuteron calculation with a realistic potential, we adapt the for-malism ofpi-EFT at N2LO from Section 2.4 to compute the nuclear TPE correctionsin the non-relativistic, point-nucleon limit of Eq. (4.71).κ ∆L,κ ∑κmaxκ=0 ∆L,κ0 -0.05559 -0.055591 -1.451 -1.50662 -6.455×10−2 -1.57113 -1.182×10−2 -1.58304 -3.723×10−3 -1.58675 -1.545×10−3 -1.58826 -7.571×10−4 -1.58907 -4.145×10−4 -1.58948 -2.457×10−4 -1.58969 -1.546×10−4 -1.589810 -1.019×10−4 -1.589911 -6.972×10−5 -1.590012 -4.918×10−5 -1.590013 -3.560×10−5 -1.590114 -2.631×10−5 -1.590115 -1.981×10−5 -1.590116 -1.514×10−5 -1.590117 -1.174×10−5 -1.590118 -9.207×10−6 -1.590119 -7.281×10−6 -1.590120 -5.822×10−6 -1.5902Table 6.6: The calculated values of the terms ∆L,κ that contribute to the nu-clear TPE in meV as a function of the multipole κ forpi-EFT at next-to-next-to-leading-order for µ2H. The Table is adapted from Ref. [6].108This procedure is useful in order to quickly benchmark the numerical proce-dures of the η-less calculations. For example, thepi-EFT approach allows one tocompare the multipole contributions ∆L,κ in Table 6.6 and Table 6.7. The formulasto carry out this calculation are presented in Appendix E. In Table 6.6, the indi-vidual components ∆L,κ for κ = 0, ...,20 are given along with their running sum∑κmaxκ=0∆L,κ , that rapidly saturates. The equivalent result in the η-formalism is thesum of the terms δ (0)D1 +δ(1)Z3 +δ(2)R2 +δ(2)Q +δ(2)D1D3 = -1.590 meV from the realisticχEFT potential from [64]. These results differ by only about 0.01% indicating ex-cellent agreement between the two approaches. This demonstrates the validity ofthepi-EFT calculations for benchmarks.6.3.2 Realistic caseHere we consider the deuteron using the realistic χEFT nucleon-nucleon N3LOEMpotential from Ref. [64]. The calculations are performed as in Ref. [3]. The sourcesof numerical uncertainty that were considered are the HO frequency h¯Ω depen-dence, the model space size NMax, the maximum momentum value QMax used tointegrate the response functions and the number of quadrature points Nq in the mo-mentum integrals. For each κ , the total numerical uncertainty is the quadrature sumof these components. We begin with the non-relativistic formalism in Eq. (4.71) tocompute δNRpol .In Table 6.7, each contribution ∆L,κ is listed from κ = 0, ...,20 along with therunning sum. From this table we have ∆L = -1.588 meV. The equivalent resultin the η-formalism is again -1.590 meV. The difference between these values is0.002 meV highlighting the excellent agreement between both methods for µ2H.This shows that higher order corrections from the η-expansion for the deuteron arenegligible. In addition, we note that only the first four terms are required to reachagreement with δNRpol at the sub-percent level.We proceed to the evaluation of the relativistic terms in Eqs.(4.65-4.67) us-ing the point proton limit. The specific multipole contributions ∆x,κ to the TPEare listed in Table 6.8 and their sum is the final result δRelpol . In the relativis-tic, point-nucleon, η-expansion method the sum δ (0)D1 +δ(0)L +δ(0)T +δ(1)Z3 +δ(2)R2 +δ (2)Q + δ(2)D1D3 = -1.573 meV can be compared to the result ∆L + ∆T,E + ∆T,M= -109κ ∆L,κ ∑κmaxκ=0 ∆L,κ0 -0.0685620(1) -0.0685620(1)1 -1.436198(1) -1.504760(1)2 -6.442519(2)×10−2 -1.569185(1)3 -1.18696(1)×10−2 -1.581055(1)4 -3.7455(2)×10−3 -1.584800(1)5 -1.5570(2)×10−3 -1.586357(1)6 -7.649(2)×10−4 -1.587122(1)7 -4.201(3)×10−4 -1.587542(1)8 -2.502(3)×10−4 -1.587793(1)9 -1.583(4)×10−4 -1.587951(1)10 -1.051(4)×10−4 -1.588056(1)11 -7.25(4)×10−5 -1.588128(1)12 -5.16(4)×10−5 -1.588180(1)13 -3.77(4)×10−5 -1.588218(1)14 -2.82(4)×10−5 -1.588246(1)15 -2.15(4)×10−5 -1.588267(2)16 -1.66(4)×10−5 -1.588284(2)17 -1.31(4)×10−5 -1.588297(2)18 -1.04(3)×10−5 -1.588308(2)19 -8.4(3)×10−6 -1.588316(2)20 -6.8(3)×10−6 -1.588323(2)Table 6.7: The calculated values of the terms ∆L,κ that contribute to the two-photon exchange in meV as a function of the multipole κ using the realisticN3LOEM potential for µ2H. The values in the brackets indicate the numericaluncertainties. The Table is adapted from Ref. [6].1.571 meV, denoted δRelpol and amounts to a difference of 0.1%. The term ∆T,E,κ=1= -1.257×10−2 meV corresponds, in the long-wavelength approximation of thetransverse response function, to the term δ (0)T = -1.248×10−2 meV from the η-expansion shown in Section 4.4. They are in close agreement. The relativistic lon-gitudinal correction in the η-expansion, δ (0)L = 2.913×10−2 meV, is equivalent tothe difference between the results of ∆L,κ=1 in Tables 6.8 and 6.7 which amountsto 2.081×10−2 meV. The difference between the results arise from the distincttreatment of the response functions in the two approaches. In the η-expansion theresponse functions are treated in the long-wavelength approximation as in Section3.3.1, while in the η-less approach they are treated with the general q-dependence.110κ ∆L,κ ∆T,E,κ ∆T,M,κ ∑κmaxκ=0 ∆κ0 -0.0671358(1) 0.0 0.0 -0.0671358(1)1 -1.415390(1) -1.2570(7)×10−2 2.8243(2)×10−3 -1.492271(6)2 -6.186081(7)×10−2 1.01894(8)×10−4 5.197(3)×10−4 -1.553511(6)3 -1.113425(2)×10−2 9.00(1)×10−6 7.05(3)×10−5 -1.564565(6)4 -3.44382(3)×10−3 2.62(1)×10−6 1.007(6)×10−4 -1.567906(7)5 -1.40260(4)×10−3 1.34(1)×10−6 2.40(4)×10−5 -1.569283(7)6 -6.7570(4)×10−4 8.3(2)×10−7 4.9(1)×10−5 -1.569909(7)7 -3.6394(5)×10−4 5.7(2)×10−7 1.30(6)×10−5 -1.570259(7)8 -2.1261(6)×10−4 4.2(2)×10−7 3.0(1)×10−5 -1.570442(7)9 -1.3203(7)×10−4 3.3(2)×10−7 8.2(6)×10−6 -1.570565(7)10 -8.605(8)×10−5 2.6(2)×10−7 2.0(2)×10−5 -1.570631(7)11 -5.830(9)×10−5 2.1(2)×10−7 5.5(6)×10−6 -1.570684(7)12 -4.08(1)×10−5 1.7(2)×10−7 1.4(2)×10−5 -1.570710(7)13 -2.93(1)×10−5 1.4(2)×10−7 3.9(6)×10−6 -1.570736(7)14 -2.16(1)×10−5 1.2(2)×10−7 1.0(2)×10−5 -1.570747(7)15 -1.62(1)×10−5 1.0(2)×10−7 2.9(5)×10−6 -1.570760(7)16 -1.24(1)×10−5 9(2)×10−8 7(2)×10−6 -1.570765(8)17 -9.6(1)×10−6 7(2)×10−8 2.2(4)×10−6 -1.570773(8)18 -7.5(1)×10−6 6(2)×10−8 6(1)×10−6 -1.570775(8)19 -6.0(1)×10−6 5(2)×10−8 1.7(3)×10−6 -1.570779(8)20 -4.8(1)×10−6 5(2)×10−8 4(1)×10−6 -1.570779(8)Table 6.8: The calculated values of the terms that contribute to δRelpol in meV asa function of the multipole κ for the N3LOEM potential. The brackets indicatethe numerical uncertainties. The Table is adapted from Ref. [6].Next, we combine the relativistic kernels in Eqs. (4.61-4.63) with the nucleonform factors. The nucleon form factor parametrizations used here are [139]GpE(q2) =1(1+( qβ )2)2 , (6.12)GnE(q2) =λq2(1+( qβ )2)GpE(q2), (6.13)while for the magnetic form factors we choose [139]Gp/nM (q2) = µNgp/nGpE(q2). (6.14)111Here the parameters β and λ are related to the charge radii of the proton and neu-tron by β 2 = 12/r2p, and λ =− r2n6 , where r2n = -0.1161(22) fm2 and rp =0.84087(39)fm as in Refs. [2, 140] while µN is the nuclear Bohr magneton. This more generalcalculation is denoted δApol and the individual multipole contributions are given inTable 6.9.κ ∆L,κ ∆T,E,κ ∆T,M,κ ∑κmaxκ=0 ∆κ0 -0.0627430(1) 0.0 0.0 -0.0627430(1)1 -1.389312(1) -1.2729(8)×10−2 2.7400(2) ×10−3 -1.462044(7)2 -5.687366(7)×10−2 9.42909(5)×10−5 3.7810338(1)×10−4 -1.518446(7)3 -8.66690(1)×10−3 5.6589911(7)×10−6 3.4725465(8)×10−5 -1.527072(7)4 -2.499426(5)×10−3 1.189995(1)×10−6 3.046054(2)×10−5 -1.529540(7)5 -7.91031(2)×10−4 3.379354(4)×10−7 5.54470(1)×10−6 -1.530325(7)6 -3.73996(1)×10−4 1.83995(1)×10−7 7.40018(2)×10−6 -1.530692(7)7 -1.467700(5) ×10−4 7.10098(2)×10−8 1.67017(1)×10−6 -1.530837(7)8 -8.89794(3)×10−5 5.39528(9)×10−8 2.56595(3)×10−6 -1.530923(7)9 -3.81065(2)×10−5 2.25638(2)×10−8 6.4689(1)×10−7 -1.530960(7)10 -2.72129(1)×10−5 2.05594(9)×10−8 1.06086(4)×10−6 -1.530986(7)11 -1.211088(8)×10−5 8.7809(2)×10−9 2.8741(1)×10−7 -1.530998(7)12 -9.77783(7) ×10−6 9.0617(9)×10−9 4.8931(4)×10−7 -1.531008(7)13 -4.41778(4)×10−6 3.8693(2)×10−9 1.3972(1)×10−7 -1.531012(7)14 -3.93714(4)×10−6 4.3939(7)×10−9 2.4379(3)×10−7 -1.531016(7)15 -1.78341(2)×10−6 1.8585(2)×10−9 7.2510(8)×10−8 -1.531017(7)16 -1.72722(2)×10−6 2.2826(5)×10−9 1.2879(3)×10−7 -1.531019(7)17 -7.7872(1)×10−7 9.523(1)×10−10 3.9588(6)×10−8 -1.531020(7)18 -8.1058(1)×10−7 1.2503(4)×10−9 7.129(2)×10−8 -1.531020(7)19 -3.62163(8)×10−7 5.134(1)×10−10 2.2520(5)×10−8 -1.531021(7)20 -4.01814(8)×10−7 7.145(2)×10−10 4.101(2)×10−8 -1.531021(7)Table 6.9: The calculated values of the terms that contribute to δApol in meVas a function of the multipole κ for the N3LOEM potential. The values inthe brackets indicate the numerical uncertainties. The Table is adapted fromRef. [6].The inclusion of form factors in Table 6.9 reduces the absolute magnitude ofthe individual contributions to δApol with respect to Table 6.8 carried out in the point-nucleon limit. This is expected because the parametrizations of the form factors inEqs. (6.12)-(6.14) suppress the strength of the response functions at high q-values.The total sum of all contributions in Table 6.9 is δApol= -1.531 meV.1120 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20Photon multipolarity 10 610 510 410 310 210 1100|L,| [meV]NR point protonRel+point proton+FFFigure 6.6: The absolute value of the longitudinal multipole corrections inthe non-relativisitic (NR) point-nucleon limit, in the relativistic (Rel) point-nucleon limit, and with form factors (+FF) as a function of the multipolarityκ using the N3LOEM potential. The Figure is adapted from Ref. [6].The convergence of ∆L,∆T,el,∆T,mag with increasing photon multipoles is illus-trated in Figs. 6.6 and 6.7. In these figures the inclusion of nucleon form factorscreates a jagged convergence pattern. This behavior is understood from the fol-lowing arguments. When applied to the deuteron the Coulomb multipole tensoroperator isCκ(q) =(1+(−1)κ2)jκ(qr2)Y κ(rˆ)+(τ31 +(−1)κτ322)jκ(qr2)Y κ(rˆ),(6.15)where the isospin projection operators τ31 and τ32 act on particle 1 and 2 in thedeuteron, respectively.1131 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20Photon multipolarity 10 910 810 710 610 510 410 310 2|T,E,| [meV]Rel+point proton+FF(a)1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20Photon multipolarity 10 710 610 510 410 3|T,M,| [meV]Rel+point proton+FF(b)Figure 6.7: The absolute value of the (a) transverse Siegert corrections andmagnetic transverse corrections (b) plotted as a function of increasing photonmultipolarity κ for µ2H in the point proton limit and with form factors (+FF)using the N3LOEM potential. The Figures are adapted from Ref. [6].114When acting on the deuteron ground state, (T0 = 0,MT0 = 0,S0 = 1,J0 = 1,L0 =0, or 2), the reduced matrix elements for the even multipoles are isoscalar opera-tors, while odd multipoles of this operator are isovector transitions〈J,T ;MT ||C2κ(q)||J0,0;0〉=〈J|| j2κ(qr2)Y 2κ(rˆ)||J0〉δT,0δmT ,0, (6.16)〈J,T ;MT ||C2κ+1(q)||J0,0;0〉= 〈J|| j2κ+1(qr2)Y 2κ+1(rˆ)||J0〉δT,1δmT ,0. (6.17)These reduced matrix elements are almost numerically identical which producesthe smooth pattern observed in Fig. 6.6 in the point proton limit. The introductionof nucleon form factors to Eq. (6.16) gives〈J,T ;MT ||C2κ(q)||J0,0;0〉=(GpE(q2)+GnE(q2))〈J|| j2κ (qr2 )Y 2κ(rˆ)||J0〉δT,0δmT ,0(6.18)for the even Coulomb multipoles, while the odd Coulomb multipoles in Eq. (6.17)become〈J,T ;MT ||C2κ+1(q)||J0,0;0〉=(GpE(q2)−GnE(q2))〈J|| j2κ+1(qr2 )Y 2κ+1(rˆ)||J0〉δT,1δmT ,0.(6.19)The q-dependent factor in front of the reduced matrix elements in Eq. (6.18) andEq. (6.19) will enhance the value of ∆L,κ for the even-κs with respect to the odd κs,and the difference will be more pronounced at larger κs because the q-dependenceof the nucleon form factors is more pronounced at high momenta. This behavioris observed in Fig. 6.6 and Fig. 6.7a. The analogous pattern in Fig. 6.7b is not theresult of nucleon form factors, but due to the isovector spin-preserving transitionsthat dominate the even κ values which are enhanced relative to the odd κ isovectorspin-changing transitions.Finally, the results of δApol along with the non-relativistic point proton limit δNRpoland the relativistic point-proton δRelpol calculation are given in Table 6.10 in compar-ison to the η-expansion values. In this table we have that in the non-relativistic andrelativistic results, δAZem and δNR/Relpol are both in very good agreement between the115η-expansion η-lessδNRpol -1.328 -1.326δAZem -0.359 -0.359δATPE -1.687 -1.685δRelpol -1.308 -1.309δAZem -0.359 -0.359δATPE -1.667 -1.668δApol -1.248 -1.269δAZem -0.423 -0.406δATPE -1.671 -1.675Table 6.10: A comparison of the results from the η-formalism to the fullη-less formalism in the non-relativistic point-proton, relativistic point-protonand relativistic with exact nucleon form factor calculations. Units are in meVwith δATPE = δApol+δAZem. The Coulomb correction δ(0)C = 0.262 meV from [2],not treated here, has also been added to δNRpol ,δRelpol ,δApol, for comparison. TheTable is adapted from Ref. [6].η- and η-less methods. However, when nucleon form factors are included, thenδAZem and δApol are different from the η-expansion result by approximately 4% and2%, respectively.These differences suggest a larger-than-expected model dependence in the cal-culation of δAZem and δApol using the η-expansion that stem from the use of linearapproximations of the nucleon form factors in that method. However, we notethat these model dependencies are eliminated when the sum of these terms δATPE isconsidered and amounts to a final 0.2% difference with respect to the η-expansion.1166.3.3 Vacuum polarizationThe σZα uncertainty in Table 6.5 is one of the main bottle necks, the other be-ing σN , limiting the precision of the δTPE calculation. Following Section 5.4 weaddress here some of the α6 contributions from the vacuum polarization as inRef. [109]. Using the adapted η-less formalism of Section 4.4.2 we provide themost up-to-date δTPE value including the three-photon exchange contribution δA3PEfrom Ref. [111]. We then discuss the reduction of the uncertainty in the TPE deter-mination from the inclusion of these partial α6 contributions.In the work of Ref. [109], only the vacuum polarization contributions to theleading dipole operator were considered and higher multipoles were neglected.Using the expressions in Eq. (4.102) we compute δAvac within the η-less formalismas the sum of termsδAvac, prop =κmax∑κ=0∆vac, prop,κ , (6.20)from κ = 0, ...,20, and shown them in Table 6.11. The final sum in is δAvac, prop =−1.476×10−2 meV. The subtraction of the elastic term in this contribution is car-ried out by computing δAvac, Zem as the regularized integral of Eq. (4.104) (with anopposite sign) asδAvac, Zem =−32(Zα)2|φ(0)|2mr∞∫0dqg(q)q4×(|〈N0|ρ˜(q)|N0〉|2−1− q23〈RP〉2). (6.21)The result of this integral is δAvac, Zem =−0.00536 meV. The sum of these values isδAvac, Zem+δvac,prop =−0.0201 meV and exactly equal to δpotEpol =−0.0201 meVfrom the calculation of Ref. [109] that used the AV18 potential, but where our valueincludes corrections to all multipoles beyond the leading dipole operator. Thisdemonstrates that higher order multipole contributions to this term are negligibleat this precision for the µ2H system.To calculate the correction δAwf, we use the value δATPE = −1.675 meV in Ta-ble 6.10 from the η-less expansion using the χEFT potential. This leads to δAwf =117−0.00565(5) meV from Eq. (4.100) comparable to the term δwfEpol = −0.0064meV in Ref. [109], but which includes multipole corrections beyond the dipoleterm, finite size effects and relativistic contributions. In this case we find a rela-tive difference of about 12%. This difference demonstrates that the effects of finitenucleon size and relativity are significant. The final result of the vacuum polar-ization contribution is δAvac = δAwf + δAvac, prop + δAvac, Zem = −0.02581(5) meV. Theindividual contributions to δAvac, prop are itemized in Table 6.12.κ ∆vac, prop,κ ∑κmaxκ=0∆vac, prop,κ0 -8.9022×10−4 -8.9022×10−41 -1.2706×10−2 -1.3597×10−22 -8.5568×10−4 -1.4452×10−23 -1.7979×10−4 -1.4632×10−24 -6.0893×10−5 -1.4693×10−25 -2.6533×10−5 -1.4719×10−26 -1.3500×10−5 -1.4733×10−27 -7.6251×10−6 -1.4741×10−28 -4.6466×10−6 -1.4745×10−29 -2.9992×10−6 -1.4748×10−210 -2.0256×10−6 -1.4750×10−211 -1.4189×10−6 -1.4752×10−212 -1.0243×10−6 -1.4753×10−213 -7.5834×10−7 -1.4753×10−214 -5.7359×10−7 -1.4754×10−215 -4.4187×10−7 -1.4754×10−216 -3.4591×10−7 -1.4755×10−217 -2.7457×10−7 -1.4755×10−218 -2.2068×10−7 -1.4755×10−219 -1.7929×10−7 -1.4756×10−220 -1.4714×10−7 -1.4756×10−2Table 6.11: The calculated values of the terms ∆vac, prop,κ that contribute tothe vacuum polarization of the TPE in meV as a function of the multipole κusing the realistic N3LOEM potential for µ2H.Using the vacuum polarization contributions and the three-photon exchangecontributions from Ref. [111], Table 6.12 updates and supersedes the TPE valuefrom Table 6.5. To avoid double counting the σZα uncertainty, this 1% uncertainty118contribution [meV]δTPE -1.715+0.014−0.016δAwf -0.00565(5)δAvac,prop -0.01476δAvac,Zem -0.00536δAvac -0.02581(5)δA3PE -0.00875(92) [111]δTPE -1.750+0.014−0.016Table 6.12: The results of the TPE including the vacuum polarization contri-butions and the three-photon exchange from Ref. [111].is removed from the TPE error budget in the first row of Table 6.12. The uncer-tainties in δAvac, prop and δAvac, Zem are significantly smaller than the uncertainty inthe TPE and are therefore not listed in Table 6.12. The final result for TPE in-cluding δAvac and δA3PE is δTPE = −1.750+0.014−0.016 meV. This result demonstrates animproved agreement, within about 1σ , with respect to the experimental value ofδTPE =−1.7638(68) meV.In this Section the results of δATPE have been presented using the generalizedη-less method outlined in Section 4.3. The results of the non-relativistic, relativis-tic and the vacuum polarization were presented for µ2H. The ση uncertainty wasprobed by comparing the results of the η-less formalism to the results of the η-expansion outlined in Section 4.4. It was found that the inclusion of form factorsin µ2H calculations produces a difference of about 0.2% with respect to the η-expansion. This difference is the uncertainty ση of the η-formalism which has forthe first time been directly calculated for µ2H. In addition, we have demonstratedthat adapting this formalism for the vacuum polarization calculation is straight-forward and gives excellent agreement with the literature. In Table 6.12 we haveverified the results of Kalinowski [109] that demonstrate that the inclusion of δAvacis needed to reach agreement with experiment. In the previous sections we havedemonstrated that the η-less expansion method is a powerful and adaptable compu-tational framework to calculate nuclear structure corrections in muonic deuteriumto better precision and more systematically than the η-expansion. Furthermore,this framework can be extended to heavier mass systems as discussed in Ref. [6]119which indicates that the η-less scheme will be an important means for subsequentinvestigations of nuclear structure effects in muonic atoms.6.4 Bayesian analysis of the η-truncation uncertaintyThe extension of the η-less formalism of the last section provides a direct methodto address the ση uncertainty in A= 3 and 4 systems. However, as such calculationshave not yet been carried out, another method must be used to estimate the σηuncertainty. Here we present the results of the Bayesian formalism, introducedin Section 5.3, to estimate the ση uncertainty of δNRpol in heavier mass systems.The procedure requires the order-by-order results of the non-relativistic δNRpol fromRef. [2] for each light nuclei summarized in Table 6.13 for the N3LOEM potential.µ2H µ3H µ3He+ µ4He+δ (0) -1.912 -0.7848 -6.633 -4.701δ (1) 0.359 0.1844 -0.384 0.809δ (2) -0.037 -0.0247 0.83 0.101δNRpol -1.590 -0.6251 -6.187 -3.791Table 6.13: The leading, sub-leading and sub-sub-leading order, non-relativistic, η-expansion values from Ref. [2] for A= 2,3 and 4 in meV units.The calculations have been carried out with the χEFT potential from Ref. [64].The posterior distribution P(η |D) in Eq. (5.32) is calculated for each nucleuswith the data in Table 6.13. The resulting posterior distributions are shown inFig. 6.8 for the A = 2,3 and 4 systems that have been normalized over the plottedranges. The maximum and minimum values of the scale parameter c¯ was fixed tobe c¯< = 0.1 and c¯> = 100, respectively. The width parameter of wB = 1 has beenchosen for the ck-prior B. Each subfigure in the columns of Fig. 6.8 corresponds tothe Aη , Bη or the Cη priors for η in Table 5.2 where wη = 1 is chosen for the Bηprior. Furthermore, each figure provides the results of the different ck priors, A, Bor C from Table 5.1. The posterior distributions of P(η |D) allow the extraction ofη from different estimators. The two estimators that we choose are the maximum-likelihood estimate (MLE) and median-value estimate (MVE) of η .Once the value of η has been determined using one of the estimators, Eq. (5.33)120is used to compute the posterior distribution P(∆(1)ν |D,η) of the truncation uncer-tainty. The 68% confidence interval of this posterior is the truncation uncertaintyση . The results of ση for all combinations of priors and η-estimators are sum-marized in Table 6.15 for µ2H and µ3H and in Table 6.16 for µ3He+ and µ4He+which also includes the relative uncertainty. The estimates of η from the MVEincludes the 68% interval range. Due to the extended tails of the posterior distri-butions of η , the ση uncertainty from the MVE for all priors is larger than from theMLE.Estimator ση [meV] ση %µ2H MLE 0.004 0.3MVE 0.05 3.1µ3H MLE 0.004 0.6MVE 0.034 5.5µ3He+ MLE 1.7 27.5MVE 2.8 46.1µ4He+ MLE 0.014 0.4MVE 0.132 3.5Table 6.14: The summary of the results of the Cη prior for different η-estimators.The Cη prior in Table 5.1, corresponding to a uniform distribution for η overthe plotted ranges6, represents the most realistic distribution of η in this work. Byfixing this choice of prior, the results for the MVE or MLE in Tables 6.15 and 6.16are summarized in Table 6.14 by averaging ση for the ck-priors A, B and C. Therelative uncertainty from the MLE in µ2H is consistent with the value from the η-less expansion in Section 6.3. For µ3H and µ4He+, the relative ση uncertainty of0.4% and 0.6%, respectively, and consistent with the values in Ref. [2]. The MVEvalues of ση for these nuclei is approximately ten times larger than the MLE due tothe influence of the tails of the posterior distributions. In both cases the estimatesfor µ3He are significantly larger than the estimates in Ref. [2]. This indicates eitherthat the convergence pattern of δApol for µ3He+ is divergent or that the series doesnot converge monotonically. Future investigations will address these possibilities.6 ∫ η>η< dηP(η) = 1, where η> = 2 for µ2,3H, µ4He+ and η> = 8 for µ3He+.1210.0 0.5 1.0 1.5 2.00246P(|D)A2HABC(a)0.0 0.5 1.0 1.5 2.0012P(|D)B2HABC(b)0.0 0.5 1.0 1.5 2.00123P(|D)C2HABC(c)0.0 0.5 1.0 1.5 2.0024P(|D)A3HABC(d)0.0 0.5 1.0 1.5|D)B3HABC(e)0.0 0.5 1.0 1.5 2.0012P(|D)C3HABC(f)0 2 4 6|D)A3HeABC(g)0 2 4 6|D)B3HeABC(h)0 2 4 6|D)C3HeABC(i)0.0 0.5 1.0 1.5 2.00246P(|D)A4HeABC(j)0.0 0.5 1.0 1.5 2.0012P(|D)B4HeABC(k)0.0 0.5 1.0 1.5 2.00123P(|D)C4HeABC(l)Figure 6.8: The normalized posterior distributions P(η |D) for µ2H in sub-figures ( a-c), for µ3H in (d-f), for µ3He+ in (g-i) and µ4He+ in (j-l).122Nucleus η-Estimator ck-Prior η-Prior η ση meV ση (%)µ2H MLE A A 0.1 0.004 0.3MLE B A 0.1 0.004 0.2MLE C A 0.03 0.001 0.1MLE A B 0.13 0.006 0.4MLE B B 0.19 0.010 0.8MLE C B 0.2 0.020 1.0MLE A C 0.1 0.004 0.3MLE B C 0.11 0.004 0.3MLE C C 0.1 0.005 0.3MVE A A 0.1+0.3−0.05 0.005 0.3MVE B A 0.2+0.4−0.1 0.010 0.6MVE C A 0.1+0.4−0.04 0.006 0.4MVE A B 0.4+0.9−0.2 0.05 3.2MVE B B 0.3+0.7−0.2 0.04 2.6MVE C B 0.4+1.0−0.2 0.06 3.7MVE A C 0.4+1.2−0.1 0.05 2.9MVE B C 0.3+0.8−0.1 0.04 2.4MVE C C 0.4+1.2−0.1 0.06 4.0µ3H MLE A A 0.1 0.003 0.5MLE B A 0.1 0.003 0.5MLE C A 0.03 0.001 0.1MLE A B 0.1 0.004 0.6MLE B B 0.2 0.008 1.2MLE C B 0.2 0.011 1.7MLE A C 0.1 0.003 0.5MLE B C 0.1 0.003 0.5MLE C C 0.1 0.004 0.7MVE A A 0.1+0.4−0.1 0.004 0.6MVE B A 0.2+0.5−0.1 0.008 1.3MVE C A 0.1+0.5−0.05 0.005 0.7MVE A B 0.4+0.9−0.2 0.03 4.4MVE B B 0.4+0.8−0.2 0.03 4.3MVE C B 0.4+1.0−0.2 0.03 5.5MVE A C 0.4+1.2−0.2 0.03 5.0MVE B C 0.4+1.0−0.2 0.03 4.7MVE C C 0.5+1.3−0.2 0.04 6.9Table 6.15: The MLE or MVE estimates for η and corresponding truncationuncertainties for µ2,3H isotopes. See text for details.123Nucleus η-Estimator ck-Prior η-Prior η ση meV ση (%)3He MLE A A - - -MLE B A 0.4 0.3 5.2MLE C A 0.2 0.2 2.5MLE A B 1.0 0.8 13.2MLE B B 0.6 0.5 8.8MLE C B 0.8 0.7 10.6MLE A C 2.2 2.0 32.8MLE B C 0.6 0.7 10.6MLE C C 2.2 2.4 39.1MVE A A 1.9+3.8−0.6 1.5 24.9MVE B A 0.6+1.1−0.4 0.6 9.1MVE C A 1.5+3.3−0.5 1.3 20.6MVE A B 1.6+2.8−0.7 1.4 22.3MVE B B 0.7+1.2−0.4 0.7 10.7MVE C B 1.3+2.6−0.6 1.2 19.0MVE A C 3.1+5.6−1.6 4.0 64.0MVE B C 0.9+1.8−0.5 1.0 16.4MVE C C 2.8+5.0−1.3 3.6 57.74He MLE A A 0.01 0.01 0.3MLE B A 0.1 0.01 0.3MLE C A 0.03 0.003 0.1MLE A B 0.1 0.02 0.4MLE B B 0.2 0.03 0.7MLE C B 0.2 0.04 1.1MLE A C 0.1 0.01 0.3MLE B C 0.1 0.01 0.3MLE C C 0.1 0.02 0.4MVE A A 0.1+0.4−0.05 0.01 0.4MVE B A 0.2+0.4−0.1 0.03 0.7MVE C A 0.1+0.4−0.04 0.02 0.5MVE A B 0.4+0.9−0.2 0.1 3.1MVE B B 0.3+0.7−0.2 0.1 2.4MVE C B 0.4+1.0−0.2 0.1 3.9MVE A C 0.4+1.2−0.1 0.1 3.4MVE B C 0.3+0.8−0.2 0.1 2.3MVE C C 0.5+1.3−0.2 0.2 4.7Table 6.16: The MLE or MVE estimates for η and corresponding truncationuncertainties for µ3,4He+ isotopes. See text for details.1246.5 Towards heavier muonic atomsThe success of the muonic spectroscopy program for the µH and µ2H systems hasled to the measurements of the soon-to-be released Lamb shift results for µ3He+and µ4He+ ions [141, 142] and extensions of this experimental program towardsmeasurements of the Lamb and HFS in µ6,7Li2+ ions have been proposed [141].The extraction of the RE radii of these nuclei from the Lamb shift analysis requiresthe TPE calculations outlined in Section 4.2. It has been demonstrated that forA = 2− 4 systems the dominant EM operators contributing to the Lamb shift TPEis the electric dipole operator while the contributions from the magnetic operatorsare strongly suppressed.In contrast, as demonstrated in Eq. (4.139), the dominant operator of the nu-clear structure effects to the HFS is the magnetic dipole operator µ . Because theachievable precision of the extracted nuclear properties from the HFS measure-ments is determined by the accuracy of the nuclear structure effects, the success ofthe experimental analysis of the HFS will strongly depend on the accuracy that themagnetic moment operator in light nuclei can be modeled. It has been determinedfrom other calculations that two-body currents at NLO account for 70-80% of theeffects of two-body currents in nuclei [87]. Therefore, we focus on implement-ing and benchmarking the one and two body NLO contributions to the magneticmoment operator from Section 3.3.2 in light nuclei.For the deuteron, since the µNLOi j operator is an isovector operator, the con-tributions of the NLO currents to ground state observables, such as the magneticmoment, vanish. Consequently, we study the generalized sum rules for magneticdipole transitions which are break up observables. These sum rules are of the formmn =∫ ∞0dω R(ω)ωn, (6.22)where n = −1,0 and R(ω) is the response function defined in terms of reducedmatrix elementsR(ω) =12J0+1∑N 6=N0|〈N0||µ||N〉|2δ (E f −Ei−ω), (6.23)where µ is the magnetic dipole operator. As for the response functions in Section1254.3 the states |N〉 and |N0〉 denote the ground and excited states with total angularmomentum J and J0, respectively. The specific case where n =−1 is related to themagnetic susceptibility, βM, through the relationβM =2α3m−1, (6.24)where βM is the magnetic analogy of the electric dipole polarizability αE definedin Eq. 6.2. These sum rules have also been calculated in the past, see e.g. Ref. [11],which allows us to compare our results to similar calculations. As in the previoussections, these sum rules are calculated in the truncated HO basis. For benchmark-ing purposes, we performed the deuteron calculations using either the AV18 [125]or the N3LOEM chiral potential [64].The numerical implementation of our magnetic sum rule calculations werechecked by comparing our results using the LO magnetic moment operator µLO =∑i µLOi , against the work of Arenho¨vel [11, 137]. The original calculations inRef. [11] were obtained with the Bonn r-space potential, but here we compareagainst the modern AV18 potential. These comparisons are given in Table 6.17showing a sub percentage level of agreement. The source of these small differencesare traced to the limited precision of the magnetic dipole strength integration fromdata provided by Ref. [137]. In contrast, our sum rule calculations were directlycomputed as ground state expectation values. To further verify our calculations inTable 6.17 we compared against an independent implementation by K. Wendt [4]that achieved an agreement at the 0.1% level or higher.m−1 m0This work 13.9 fm3 0.245 fm2Ref. [11] 14.0 fm3 0.244 fm2Table 6.17: Sum rules of the magnetic response function for the deuteron,calculated with the AV18 potential [125], using the leading order magneticmoment operator µLO. The Table is from Ref. [4].Next, we introduce the two-body correction µNLO[2] = ∑i j µNLO[2],i j implementedin the HO basis to the magnetic dipole operator and compare these results to theLO calculation in Table 6.18. In the deuteron µNLO[2] = µint[2], since the Sachs term126contribution µ Sachs[2] in Eq. (3.79) vanishes in this coordinate system. For thesecalculations we use the χEFT potential at N3LO from Ref. [64]. We remark thatsuch a calculation is not fully consistent according to χEFT theory because thepotential and currents at the same χEFT order have not been used, however, themain purpose of these calculations is to develop the means for more sophisticatedcalculations in the future.µ m−1 m0LO 14.0 fm3 0.245 fm2LO+NLO 15.1 fm3 0.277 fm2Table 6.18: Sum rules of the magnetic response function of the deuteron,calculated with the N3LOEM χEFT potential [64]. The calculation are car-ried out with the magnetic dipole operator at LO or including the two-bodycontributions at NLO. The Table is from Ref. [4].To the best of our knowledge sum rule calculations with only the NLO correc-tion do not exist in the literature. Therefore, to test our implementation of the µ int[2]operator in the deuteron we have once again compared our results against the inde-pendent calculation of Ref. [143] and achieved a numerical agreement at the 0.1%level or better.In Table 6.18 we find that the effects of µ int[2] is about 7% and 11% for the m−1and m0 sum rules, respectively. The fact that the influence of the two-body currentsis greater for the m0 sum rule indicates that these currents have more effect athigher energies. This observation is consistent with the work of Ref. [11] that usedphenomenological currents and potentials. The 7%− 11% effect to the magneticsum rules highlights the importance of the µ int[2] contributions for the calculationsof the HFS in muonic atoms where the magnetic dipole operator dominates, see forexample Refs. [2, 16, 144].Following the implementation and successful benchmarks of the intrinsic µ int[2]M1 operator in the HO basis for A = 2, we extend our calculations to the HO Ja-cobi representation for A = 3 nuclei. In particular, we focus on the computation ofthe ground state M1 moments. The required three-body wave functions in the Ja-cobi coordinates were taken from Ref. [145] calculated from the momentum spaceFadeev equations in Refs. [71, 72]. To use the computational framework of the127µ int[2] matrix elements implemented and tested for the deuteron, it was necessary toproject the three-body wave functions into the HO basis. The implementation ofthis procedure was tested by comparing the calculated expectation values of the ki-netic energy and wave function normalization using either the Fadeev, A= 3, wavefunctions or the HO wave functions. The results of both methods were identicalup to machine precision demonstrating the validity of the numerical HO projectionprocedure.Next, the µLO operator was implemented in the A=3 systems using the HOmatrix elements in Appendix C to compute the M1 moment. The results of ourcalculations agreed exactly with Ref. [102] using the same potential. This con-firmed the implementation of the µLO operator in the A = 3 Jacobi frame. Finally,after testing the µLO operator the NLO two-body µNLO[2] contributions, they wereadded to the M1 operator. The addition of the µNLO[2] operator is carried out by sep-arating it into the sum of the intrinsic correction (µ int[2]) and the Sachs term (µSachs[2] )in Eqs. (3.78) and (3.79), respectively so that µNLO[2] = µint[2]+µSachs[2] . The results ofthese M1 calculations are given in Table 6.19.Nucleus 〈µLO〉 〈µLO+µ int[2] 〉 〈µLO+µNLO[2] 〉 〈µExp〉3H 2.6216(3) 2.8156(3) 2.832(4) 2.978962460(14)3He -1.7826(2) -1.9732(2) -1.989(4) -2.127625308(25)Table 6.19: The results of the M1 calculation in A = 3 systems with themagnetic moment operator from LO up to NLO in comparison to experimentusing the NN-only N3LOEM potential. The calculations with µNLO[2] are carriedout either using only the intrinsic µ int[2] contribution, or with the full operatorµNLO[2] that also includes the Sach operator µSachs[2] . The values are in Bohrmagenton units µN .The bracketed values in Table 6.19 for the theory represent the numerical un-certainties from the HO model space truncation. The effect of the µ int[2] operator hasa 7% and 11% effect on the ground state magnetic moment for 3H and 3He rela-tive to the LO result, respectively. The addition of the Sachs term µ Sachs[2] improvesthe agreement with the experiment, amounting to an additional 0.5% and 0.9% ef-fect with respect to the results with µLO+µ int[2]. These results demonstrate that theintrinsic magnetic moment is by far the dominant two-body contribution.128The 4He nucleus is a spin-less nucleus, it has no magnetic moment and con-sequently has no HFS, therefore, the next heavier mass nuclear system relevant forthe muonic spectroscopy program is 6Li. The 6Li nucleus is approximately a 4Hecore with a 2H halo. Therefore, the magnetic moment of the 6Li is almost the sameas the deuteron with an experimental M1 value of 〈µExp〉=0.8221(2) µN . This isbecause there is no contribution from the 4He core to the magnetic moment. Thisstrongly suggests that the Sachs term operator is a negligible contribution to mag-netic transitions in 6Li and that including only the µ int[2] operator to the magneticmoment operator accounts for the bulk of the two-body effects in this nuclei. As6Li is an isoscalar nucleus, the isovector two-body NLO contributions have no effecton the ground state magnetic moment. To study the effects of the NLO two-bodyoperator on this nucleus it is necessary to study excited state isovector EM transi-tions. One of these transitions is the reduced magnetic B(M1;Jpi0 → Jpif ) transitionstrength, where J0 and J f are the total angular momenta of the ground and excitedstates with parity pi respectively. This transition strength is the amplitudeB(M1;Jpi0 → Jpif ) =12J0+1|〈J0||µ||J f 〉|2, (6.25)where µ is the M1 operator [146]. This transition strength is related to the lifetimeof the state measured from γ-spectroscopy that allows for a direct experimentalcomparison.In Fig. 6.9 the results of the B(M1;1+→ 0+) magnetic transition strength withthe µLO operator and also including the two-body µ int[2] contribution are shown.The calculations were performed in the SRG-evolved no-core shell model for 6Liby the Darmstadt group [147] with the µ int[2] HO matrix elements developed andimplemented from this work. The calculations are carried out using four differentNN χEFT forces from the NkLOEMN potentials from N2LO up to N4LO or with theN3LOEM potential. Consistent three body forces have also been included into thecalculations.The results in Fig. 6.9 demonstrate that for all cases the magnetic transitionstrength shifts towards the experimental bands. The effect of the µ int[2] operator isapproximately 10% for all potentials. This result is approximately half of theeffect calculated in Ref. [77] using the two-body AV18 potential [125] with three12914.25 14.50 14.75 15.00 15.25 15.50 15.75 16.00B(M1, 0+ 1+) 2NExperimentN4LOEMNN3LOEMN3LOEMNN2LOEMNLOLO+ int[2]Figure 6.9: The B(M1,0+ → 1+) transition strength for 6Li calculated inthe SRG-evolved no-core shell model for different NN χEFT potentials withconsistent three-body forces using the µLO or µLO+µ int[2] operators.body Illinois-7 three-nucleon forces [148]. In that work they found that for 6Lithe B(M1;1+→ 0+) = 13.18(7) with the µLO operator, while B(M1;1+→ 0+) =16.08(8) after including two-body currents up to N3LO order in χEFT. We expectto write a common publication based on this work [22].130Chapter 7ConclusionsIn this work we have performed a detailed investigation of muonic atoms, a uniquelaboratory to study the interplay of EM and nuclear forces. This was carried outthrough a careful study of the nuclear structure effects arising from the two-photonexchange in these systems with an emphasis on the uncertainty quantification. Ourwork establishes the theoretical and computational framework needed to improvethe precision of present-day nuclear structure calculations for the Lamb and HFSas well as outlining a methodology to that account for all pertinent uncertainties.This work is highly relevant for the “proton” and “deuteron radius” puzzles, whichas-yet remain unsolved even though recent experimental work on ordinary atomsseem to indicate that there is a problem in the estimates of systematic experimentaluncertainties. Irrespective of these puzzles, precise nuclear structure effects in lightmuonic atoms for the Lamb and HFS are crucial to support the analysis of theongoing muonic atom spectroscopy experiments.With these applications in mind, we briefly summarize the work that has beenachieved in this dissertation. In Chapter 2, we introduced the formalism of thestrong interaction, along with a discussion of its underlying symmetries and lim-iting forms. This led to the study of chiral symmetry and motivated the modernframework of nuclear forces from χEFT. Employing this theory, the hierarchy ofnucleon-pion interactions arising from the χEFT power counting scheme was out-lined. We also discussed the fitting procedures of these forces to data and outlinedan alternate formalism to χEFT, thepi-EFT, that was used to benchmark the re-131sults of our work. In Chapter 3 we reviewed the theory of EM and establishedhow nucleons couple to the photon field with the photon-nucleon operators. It wasdemonstrated that the gauge invariance of the EM field imposes current conserva-tion. Subsequently, we examined specific models for the nuclear currents at LOand introduced two-body χEFT currents at NLO. The multipole formalism was in-troduced in Section 3.3 that allows the decomposition of any general current intoa sum of spherical EM tensors of a given rank. The multipole decomposition isa crucial tool for derivations in later sections. This Chapter concluded with thederivation of the general photo-absorption cross section.In Chapter 4, the theory of muonic atoms was outlined in detail. Due to the im-portance of nuclear EM moments to the Lamb and HFS, in Section 4.1, expressionswere derived for both the electric and the magnetic nuclear moments using the q-space, r-space and newly developed qr-space methods. Subsequently the nuclearstructure corrections relevant for the Lamb shift were derived in Section 4.3 usingthe adapted η-less expansion and compared against the η-full scheme in Section4.4. The general cancellation property of any nuclear TPE diagram at any orderin α was proven in Section 4.4.1, generalizing the result of Friar and Pachucki inRefs. [82, 98]. In Section 4.4.2 using this cancellation property, the η-less for-malism was adapted to tackle calculations of the nuclear structure corrections fromvacuum polarization first approached in the work Kalinowski [109]. The η-lessformalism was used to study the HFS and obtain non-relativistic expressions inSection 4.5. These derivations demonstrate the importance of the µ operator forthe HFS.Next, in Chapter 5, we accounted for all relevant uncertainty sources contribut-ing to nuclear structure calculations. This was carried out by introducing the sta-tistical regression analysis in Section 5.1 and the Bayesian uncertainty analysisfrom Sections 5.2 and 5.3. The Chapter concluded with a discussion of additionaluncertainties that were considered in our calculations.Finally in Chapter 6, we presented the findings of this work. In Section 6.1the results of benchmark calculations using the q-space, r-space and the hybridqr-space method were presented and established the equivalence of these methodsfor the deuteron, while also probing the systematic uncertainty. Next, to assesthe few-body method uncertainty, two different algorithms, the Numerov and HO132basis expansion method were compared in the deuteron and the differences werefound to be sub-percentage. These comparisons were useful to validate the newqr-space algorithm that was subsequently adapted for A ≥ 3 systems to improvethe accuracies of EM moment calculations in Ref. [1].In Section 6.2 the uncertainty estimates of the TPE calculation carried out withthe η-expansion were thoroughly estimated using the tools in Chapter 5. The re-sults of the statistical uncertainty analysis was presented in Section 6.2.1 while inSection 6.2.2 all additional systematic uncertainties were discussed. All uncertain-ties were combined in Section 6.2.3 into a final tabulated uncertainty budget. It wasdetermined that the (Zα) truncation uncertainty of about 1% dominates the errorbudget, followed by the uncertainty from nucleon contributions. While, we showedthat this new uncertainty analysis does not account for the 5.6 σ “deuteron radiuspuzzle”, they reduce the significance of the smaller 2.6 σ discrepancy between thetheoretical and experimental TPE to below 2 σ and prove that the uncertainty ofthe nuclear structure effect calculations are well under control. These results aresummarized in Fig. 7.1 that provides a visual representation of the final uncertaintybudget.meV+0.008+0.001+0.005+0.0102+0.022+0.0172Figure 7.1: The final uncertainty budget of the TPE calculation from Ref. [3].133In Section 6.3, the results of the TPE calculation carried out within the framework of the η-less expansion were given. To test the numerics of the η-less expan-sion procedure the results frompi-EFT were compared against numerical ones withrealistic potentials. The agreement between these methods allowed us to proceedwith more complicated calculations that included relativistic effects and nucleonform factors. These calculations allowed us to exactly quantify the η-truncationuncertainty to be ≈ 0.2%. In Section 6.3.3 the results of the nuclear structure cor-rections from the vacuum polarization were calculated in the η-less formalism.These results were in exact agreement with Ref. [109]. Following that work, wepresented our updated results of the TPE calculation that includes contributionsfrom the vacuum polarization and the three photon exchange from Ref. [111].While this does not represent a consistent calculation at the (Zα)6 order, we as-sume that the other diagrams at α6 are negligible as suggested by Refs. [109, 149].If these arguments are correct, then this removes the bulk of the (Zα) truncationuncertainties of the TPE calculation and further improves the agreement with theexperimental TPE. However future studies are needed to consistently carry out thiscalculation by including all other α6 corrections.Because the η-less method has not yet been applied to A ≥ 3 systems, inSection 6.4 we presented the η-truncation uncertainty estimates from the adaptedBayesian formalism of Section 5.3. We computed the posteriors of the parameterη that allowed the extraction of the most likely η parameter value used to estimatethe ση uncertainty. The results for the most realistic η prior were in agreementwith the estimates in Ref. [2], except for the case of µ3He+. For this system, theBayesian analysis suggests an issue with the convergence pattern of µ3He+ result-ing in a large uncertainty estimate. To verify these results, it will be necessary torevisit the TPE calculations of this nucleus using the η-less method.Lastly, in Section 6.5, having thoroughly studied and analyzed the Lamb shiftcalculations, we focused on the magnetic moment operator in light nuclei relevantfor the analysis of the HFS structure. This was carried out by performing bench-mark calculations of the magnetic dipole operator at LO and with the two-bodycorrections at NLO. In the A = 2 system this allowed us to test the implementationof the µ int[2] operator. Next, this µint[2] operator was implemented into the A = 3 sys-tems where it was shown to have about a 7−11% effect relative to the LO operator.134The Sachs µ Sachs[2] term was also investigated in these systems and shown to amountto less than a 1% effect. Due to the suppression of the Sachs term relative to the in-trinsic term, only the µ int[2] operator was implemented to study the B(M1;1+→ 0+)magnetic transition strength in 6Li that amounted to a 10% effect relative to theLO contribution and significantly improved the agreement with experiment. Thisdemonstrates the importance of the two-body magnetic contributions to achieveagreement with experiment.As a future outlook, we have demonstrated that the methods of this work arepromising ways to advance future nuclear structure calculations and achieve higherprecision with quantified uncertainty estimates. These calculations complement theanalysis of high precision measurements conducted by the CREMA collaborationin µ3He+ and 4He+ systems as well as future investigations planned for µ6Li2+and µ7Li2+ ions [141].The Bayesian approaches to uncertainty quantification adapted in this work areunder rapid development and offer powerful tools to augment traditional uncer-tainty analysis. 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Advances in AppliedMathematics and Mechanics, 8(1):117127, 2016.doi:10.4208/aamm.2014.m504. → page 181148Appendix AThe Lippmann-SchwingerequationIn this Appendix we describe the LS algorithm used to solve the two-body problemintroduced in Chapter 2.A.1 Ground state equationsThe deuteron ground state wave function, denoted |N0〉, is expanded in the momen-tum space partial wave basis as〈p|N0〉=∑αφ`(p)〈pˆ|p,α〉 (A.1)where |p,α〉 are the partial wave basis states|p,α〉= |p,(`S)J,T ;MMT 〉, (A.2)with momentum p, total angular momentum J with projection M, orbital angularmomentum `, spin S, and total isospin T with projection MT . The functions φ`(p)are the momentum space wave functions for a fixed angular momentum channel.The sum over α for the deuteron ground state is over the channels (3S1 and 3D1).149The Schro¨dinger equation for the deuteron is(p22µ+Vˆ)|N0〉= E|N0〉, (A.3)where µ is the reduced mass of the proton and neutron system. Inserting the com-pleteness relation into the above equation1 =∑α∫d pp2|α, p〉〈α, p|, (A.4)results in the following coupled integral equation for the momentum space wavefunctions1E[φ0(p)φ2(p)]= p22µδ (p− p′)[1 00 1]+∞∫0d p′p′2[V0,0(p, p′) V0,2(p, p′)V2,0(p, p′) V2,2(p, p′)][φ0(p′)φ2(p′)].(A.5)This coupled integral equation is solved numerically by discretizing the momentumspace wave functions using a Gaussian quadrature grid. The discretization resultsin∞∫0d pp2→N∑i=1p2i wi, (A.6)where pi are the quadrature grid points and wi are the weights for N points. Tosolve Eq. (A.5) we introduce the vectorsv0 = [φ0(p0), . . . ,φ0(pk), . . . ,φ0(pN)] , (A.7)v2 = [φ2(p0), . . . ,φ2(pk), . . . ,φ2(pN)] , (A.8)1For brevity we will denote 〈pα|Vˆ |α ′p′〉 as V``′ .150containing the discretized momentum space wave functions. With this discretiza-tion Eq. (A.5) can be expressed as an eigenvalue problem[v0v2]= A[v0v2], (A.9)where A is a 2N×2N matrix. The components of A areAi j =p2i2µδi j +w j p2jV0,0(p j, pi) for i = 0, . . . ,N and j = 0, . . . ,N (A.10)Ai j =p2i2µδi j +w j p2jV0,2(p j, pi) for i = 0, . . . ,N and j = N+1, . . . ,2N (A.11)Ai j =p2i2µδi j +w j p2jV2,0(p j, pi) for i = N+1, . . . ,2N and j = 0, . . . ,N (A.12)Ai j =p2i2µδi j +w j p2jV2,2(p j, pi) for i = N+1, . . . ,2N and j = N+1, . . . ,2N.(A.13)Diagonalizing A in Eq. (A.9) and extracting the eigenvectors gives the deuteronbinding energy and ground state wave functions.A.2 Excited states: Lippmann-Schwinger equationFor scattering states, the equation that must be solved is|ψ±k 〉= |φk〉+1E− Hˆ0± iεVˆ |ψ±k 〉, (A.14)where |ψ±k 〉 denotes the incoming or outgoing scattered waves, Vˆ is the nuclearpotential and |φk〉 is the initial wave function. The operator Hˆ0 represents the free-particle Hamiltonian and E is the energy of the scattered wave E = k2/2µ . Thisexpression holds in the limit where the positive number ε approaches zero. Byconvention +iε represents outgoing waves and |φk〉 is taken to be a plane wave.The normalization condition of the plane waves is〈k|k ′〉= (2pi)3δ (k− k ′).151The solution to Eq. (A.14) is determined from the free reaction matrix R [150] suchthatR|φk〉= Vˆ |ψk〉. (A.15)The R matrix satisfies the integral equation given byRα ′,α(k′,k) =Vα ′,α(k′,k)−2µ∑α ′′P∞∫0dk′′ k′′2Vα ′,α ′′(k′,k′′)Rα ′′,α(k′′,k)k′′2− k20, (A.16)where E0 = k20/(2µ) and P denotes the principle valued integral. This R-matrixis related to the T matrix in Eq. (2.21) through T = R− ipiRδ (E −H0)T [151].The principle valued integral is carried out by subtracting the singular point of theintegrandP∞∫0dk′′ k′′2Vα ′,α ′′(k′,k′′)Rα ′′,α(k′′,k)k′′2− k20=∞∫0dk′′k′′2Vα ′,α ′′(k′,k′′)Rα ′′,α(k′′,k)− k20Vα ′,α ′′(k′,k0)Rα ′′,α(k0,k)k′′2− k20. (A.17)Rearranging Eq. (A.16), we have thatVα ′,α(k′,k) = Rα ′,α(k′,k)+2µ∑α ′′P∞∫0dk′′ k′′2Vα ′,α ′′(k′,k′′)Rα ′′,α(k′′,k)k′′2− k20=∑α ′′∫dk′′[δ (k′′− k′)δα ′′,α ′+2µk′′2Vα,α(k′,k′′)(k′′)2− k20]Rα ′,α(k′′,k)−2µ∑α ′′∞∫0dk′′ k20Vα ′,α ′′(k′,k0)Rα ′′,α(k0,k)k′′2− k20(A.18)For uncoupled channels, the solution to the above equation is carried out using theprocedure from Ref. [150]. The discretized momentum grid vector isk = [k1, . . . ,kN+1] , (A.19)152where kN+1 = k0. The (N+1)× (N+1) matrix F is constructed with entriesFi j = δi j +ω ′jVαα(ki,k j), (A.20)and where ω ′j are defined asω ′j = 2µk2j w jk2j − k20, for j = 1, . . . ,N (A.21)ω ′j =−2µk20N∑m=1wmk2m− k20, for j = N+1. (A.22)Once F is constructed the R matrix is obtained asRα,α(ki,kN+1) =N+1∑j=1(F−1)i jVαα(k j,kN+1). (A.23)This numerical method can be extended to the case of coupled channels by con-structing the 2(N+1)×2(N+1) matrices F,R and V partitioned into sub-matriceswhereR =[Rαα Rαα ′Rα ′α ′ Rα ′α ′], (A.24)V =[Vαα Vαα ′Vα ′α ′ Vα ′α ′], (A.25)andF =[Fαα Fαα ′Fα ′α ′ Fα ′α ′]. (A.26)153In the above expressions, α and α ′ are the coupled channels with ` = J+1,J−1respectively. The elements of V and R areVi j =Vα(i)α( j)(kn(i),kn( j)), (A.27)Ri j = Rα(i)α( j)(kn(i),kn( j)), (A.28)whereα(i) =α for i = 1, . . . ,N+1,α ′ for i = N+2, . . . ,2N+2, (A.29)andn(i) = i mod(N+1). (A.30)Using these definitions the elements of F areFi j = δi j +ω ′n( j)Vi j. (A.31)where kN+1 ≡ k0 as before. Using these definitions, we have thatRi j =2N+2∑j=1F−1i j Vi j. (A.32)Using the R matrix, the phase shifts for uncoupled channels are given by [151]Tan(δα(k0)) =−pi2 k0(2µ)Rαα(k0,k0). (A.33)154Once the R matrix is calculated, phase shifts can be computed. For coupled chan-nels the phase shifts in the Blatt and Biendenharn convention [59, 152] areTan(δα(k0)) =−pi4 k0(2µ)[Rα ′α ′(k0,k0)+Rαα(k0,k0)+Rα ′α ′(k0,k0)−Rαα(k0,k0)Cos(2ε)],(A.34)Tan(δα ′(k0)) =−pi4 k0(2µ)[Rα ′α ′(k0,k0)+Rαα(k0,k0)− Rα′α ′(k0,k0)−Rαα(k0,k0)Cos(2ε)],(A.35)Tan(2ε(k0)) =2Rαα ′(k0,k0)Rα ′α ′(k0,k0)−Rαα(k0,k0) . (A.36)These phase shifts are often converted to the Stapp et al. [59, 153] convention,denoted by δ¯ , whereδ¯α + δ¯α ′ = δα +δα ′ , (A.37)Sin(δ¯α ′− δ¯α) = Tan(2ε¯)Tan(2ε) , (A.38)Sin(δ¯α ′− δ¯α ′) = Sin(2ε¯)Sin(2ε) . (A.39)The scattering wave functions in coordinate space are obtained from the R matrixthrough [150][φα(r)φα ′(r)]=[jL(k0r)jL′(k0r)]−2µ∞∫0dkk2k2− k20[Rα,α(k,k0) Rα,α ′(k,k0)Rα ′,α(k,k0) Rα ′,α ′(k,k0)][jL(kr)jL′(kr)].(A.40)155Appendix BNotation, conventions andFeynman formulasHere we define all of the relativistic conventions and formulas used in the disserta-tion. The Minkowski metric, covariant and contravariant four-vectors are definedgµν = Diag{1,−1,−1,−1}, (B.1)V µ =(V 0,V), (B.2)Vµ =(V 0,−V ) . (B.3)The representation of the γ matrices used in this work areγ =[0 σ−σ 0], (B.4)andγ5 =[0 II 0]. (B.5)156The representation of the Pauli matrices areσx =[0 11 0], (B.6)σy =[0 −ii 0], (B.7)andσz =[1 00 −1]. (B.8)An important identity used in the derivations in Section 4.5 is [90](σ ·a)(σ ·b) = a ·b+ iσ · (a×b) . (B.9)B.1 Gamma matrix identitiesThe following γ-trace formulas were used to derive the results of this dissertation[35]Tr[γµ]= 0, (B.10)Tr[γµγν]= 4gµν , (B.11)Tr[γµaµγνbν]= 4a ·b, (B.12)Tr[γµaµγνbνγρcργτdτ]= 4((a ·b)(c ·d)− (a · c)(b ·d)+(a ·d)(b · c)) . (B.13)In addition to the above relations, it is useful to have expressions for the product ofγ-matricesγµγλ γν = gµλ γν +gλνγµ −gµνγλ − iεσµλνγσ γ5, (B.14)γµ/pγν = γν pµ + γµ pν −gµν/p+ iεµνσλ γσ γ5 pλ , (B.15)γ0γσ γ5 = δσ ,0γ5+δσ ,iσiI4. (B.16)157B.2 The Coulomb gaugeIn this work we choose to work in the Coulomb gauge that naturally separates outthe space and time components. The photon propagator in this gauge isD00(q) =i|q|2 , (B.17)D0i(q) = Di0(q) = 0, (B.18)Di j(q) =iq2+ iε(δi j− qiq j|q|2). (B.19)where i are the spacial components i = 1,2,3.B.3 Integral relations and principle valued integralsTo carry out many of the integrals in Chapter 4 it was very useful to symmetrizethe integrands using the formula∞∫−∞dx F(x) =12∞∫−∞dx [F(x)+F(−x)] . (B.20)In the results that follow, the contour integrals will be carried out over theupper-half of the complex plane denoted by γ+. The evaluation of the integral inEq. (4.59) for the longitudinal multipoles results inlimε→0∮γ+dq02pii1((q0−m)2−E2q + iε)((q0+m)2−E2q + iε)((q0+ iε)2−ω2)=− 18mωEq[1(Eq−m)(Eq−m+ω) −1(Eq+m)(Eq+m+ω)], (B.21)158while for the transverse multipoles, the evaluation of the integral giveslimε→0∮γ+dq02piiq20((q2+ iε)2−4m2q20)((q0+ iε)2−ω2)(q2+ iε)2=116m2qωω+2q(ω+q)2− q216m3ω(1Eq[1(Eq−m)(Eq−m+ω) −1(Eq+m)(Eq+m+ω)]). (B.22)For the derivations required in Section 4.5 for the HFS the following contour inte-gral was necessarylimε→0∮γ+dq02pii2ωq(q20−q2+ iε)((q0−m)2−E2q + iε)((q0+m)2−E2q + iε)(q20−ω2+ iε)=1(q2−ω2q )((q2−ω2q )2−4m2ω2q )+ωq4m2q3(q2−ω2q )+ωq4Eqm(Eq−m)((Eq−m)2−q2)((Eq−m)2−ω2q )+ωq4Eqm(Eq+m)((Eq+m)2−q2)((Eq+m)2−ω2q ).(B.23)where ωq = ω+ q22M . The last equation has useful limits that are relevant for elec-tronic or muonic atoms. For the case of an electronic atom we may use the limitwhere m→ 0 and Eq→ q that giveslimε→0∮γ+dq02pii2ωq(q20−q2+ iε)((q0−m)2−E2q + iε)((q0+m)2−E2q + iε)(q20−ω2+ iε)=−15q4ωq+10q2ω3q −3ω5q8q5(q2−ω2)3 +1(q2−ω2q )3.(B.24)159For muonic atoms, Eq→ m+ q22m ,qm  1 and ωm  1 which giveslimε→0∮γ+dq02pii2ωq(q20−q2+ iε)((q0−m)2−E2q + iε)((q0+m)2−E2q + iε)(q20−ω2+ iε)=18m2q4(4mq22m +ωq+2ω2( q22m +ωq)2−2− q−ωqq+ωq).(B.25)160Appendix CRacah algebraIn this appendix, we will define the algebra required to carry out the calculationsthroughout the dissertation.C.1 The irreducible spherical tensor operatorsAn irreducible spherical tensor operator of rank κ , with κ ∈N , and projectionsµ =−κ,−κ+1, ...,κ−1,κ , denoted as T κµ is defined as the set of operators whichunder the quantum mechanical rotation operator transform according to [154]U (R)T κν U†(R) =∑qDνν ′(α,β ,γ)T κν ′ , (C.1)where D(α,β ,γ) is the element of the rotation matrix with (α,β ,γ) being theEuler angles. Two spherical tensor operators T k1q1 and Tk2q2 can be coupled togetherto generate a new operator of rank κ by defining the tensor coupling operator [⊗]between the two tensors as[T k1⊗T k2]kµ= ∑q1,q2C(k,µ)(k1,q1),(k2,q2)Tk1q1 Tk2q2 , (C.2)161where C(k,µ)(k1,q1),(k2,q2) are the Clebsch-Gordan coefficients [91]. Furthermore, wenote that [T k1⊗T k2]k= (−1)k−k1−k2[T k2⊗T k1]k. (C.3)The tensor coupling in Eq. C.2 is inverted asT k1q1 Tk2q2 =∑kC(k,q)(k1,q1),(k2,q2)[T k1⊗T k2]kq. (C.4)For two Cartesian vectors A and B the following identities are usefulA ·B =−√3[A1⊗B1]00 =∑µ(−1)µAµB−µ , (C.5)A×B =−i√2∑λ(−1)λ [A1⊗B1]1λ e−λ , (C.6)(A×B)µ =−i√2[A1⊗B1]1µ . (C.7)C.2 The spherical tensor basisThe unit vectors of the spherical tensor basis are given by the expressionse+1 =− 1√2(xˆ+ iyˆ) , (C.8)e−1 =1√2(xˆ− iyˆ) , (C.9)e0 = zˆ, (C.10)where xˆ, yˆ and zˆ are the Cartesian unit vectors. The spherical tensor unit vectorssatisfy the following propertiese†λ = (−1)λ eλ , (C.11)e†λ ′ · eλ = δλλ ′ , (C.12)eλ ′× eλ = i√2C(1,λ+λ′)(1,λ ′),(1,λ )eλ+λ ′ . (C.13)162Using these properties a Cartesian vector V = (Vx,Vy,Vz) is decomposed into thespherical basis asV =∑qVqe†q, (C.14)where the component Vq is defined asVq = eq ·V . (C.15)As a special case, the directional vector rˆ can be written in the spherical tensorbasis asrˆ =√4pi3 ∑µY 1µ (rˆ)e†µ , (C.16)where Y 1µ (rˆ) is a spherical harmonic function.C.3 Wigner-Eckart theoremThe Wigner-Eckart theorem as defined by Edmonds [91] is given by〈N0J0M0|T kν |NJM〉= (−1)k−J+J01√2J0+1C(J0,M0)(k,ν),(J,M)〈N0J0||T k||NJ〉. (C.17)Using the symmetry properties of the Clebsch-Gordan coefficients this expressioncan also be written as〈N0J0M0|T kν |NJM〉= (−1)k+J0+M1√2k+1C(k,ν)(J,−M),(J0,M0)〈N0J0||Tk||NJ〉. (C.18)The Clebsch-Gordan coefficients satisfy the completeness relations given byj1+ j2∑J=| j1− j2|J∑M=−JC(J,M)( j1,m1),( j2,m2)C(J,M)( j1,m′1),( j2,m′2),= δm1,m′1δm2,m′2 (C.19)∑m1,m2C(J,M)( j1,m1),( j2,m2)C(J′,M′)( j1,m1),( j2,m2)= δJ,J′δM,M′ . (C.20)163Other Clebsch-Gordan coefficient symmetries that are very useful in deriving moregeneral results are [91, 92]C(J,M)( j1,m1),( j2,m2) = (−1)j1+ j2−JC(J,−M)( j1,−m1),( j2,−m2), (C.21)= (−1) j1+ j2−JC(J,M)( j2,m2),( j1,m1), (C.22)= (−1) j1−m1√2J+12 j2+1C( j2,−m2)( j1,m1),(J,−M), (C.23)= (−1) j2+m2√2J+12 j1+1C( j1,−m1)(J,−M),( j2,m2), (C.24)= (−1) j1−m1√2J+12 j2+1C( j2,m2)(J,M),( j1,−m1), (C.25)= (−1) j2+m2√2J+12 j1+1C( j1,m1)( j2,−m2),(J,M). (C.26)Using the completeness properties in Eq. (C.19),(C.20) and the Wigner-Eckart the-orem [91] the following identities are true [2, 91]〈J||T k||J0〉= (−1)J−J0〈J0||T k||J〉∗, (C.27)∑ν〈J0M0|Akν |JM〉〈J0M0|Bkν |JM〉=(−1)J0−J2J0+1〈J0||Ak||J〉〈J||Bk||J0〉, (C.28)∑ν〈J0M0|Akν |JM〉〈J0M0|Akν |JM〉=12J0+1|〈J0||Ak||J〉|2. (C.29)164C.4 Racah algebra for the hyperfine splittingHere we derive the relations that were necessary to obtain the relations in Section4.5.3. The first two useful properties are(T kµ )† = (−1)µT k−µ , (C.30)〈N0F0||T k||NF〉∗ = (−1)F0−F〈NF ||T k||N0F0〉. (C.31)Using the above formulas, product of matrix elements is written as〈J0M0|T k1†µ1 |JM〉〈JM|T k2µ2 |J0M0〉= (−1)µ1〈J0M0|T k1−µ1 |JM〉〈J0M0|(T k2µ2 )†|JM〉∗,= (−1)µ1+µ2〈J0M0|T k1−µ1 |JM〉〈J0M0|T k2−µ2 |JM〉∗,= (−1)µ1+µ2((−1)k1−J+J0√2J0+1C(J0,M0)(k1,−µ1),(J,M)〈J0||Tk1 ||J0M0〉)×((−1)k2−J+J0√2J0+1C(J0,M0)(k2,−µ2),(J,M)〈J0||Tk2 ||J〉)∗,=(−1)µ1+µ2+k1+k22J0+1(−1)2(J0−J)CJ0M0(k1−µ1),(JM)CJ0M0(k−µ2),(JM)×〈J0||T k1 ||J〉〈J0||T k2 ||J〉∗. (C.32)Using this result we can write〈N0J0M0|T k1†µ1 |NJM〉〈NJM|T k2µ2 |N0J0M0〉=(−1)µ1+µ2+k1+k22J0+1(−1)2(J0−J)CJ0M0(k1−µ1),(JM)CJ0M0(k2,−µ2),(J,M)×〈N0J0||T k1 ||NJ〉〈N0J0||T k2 ||NJ〉∗ (C.33)165The last two Clebsch-Gordan coefficients can be recast asCJ0M0(k1−µ1),(JM)CJ0M0(k2,−µ2),(JM)=((−1)k1+µ1 Jˆ0JˆCJ−M(k1,−µ1),(J0−M0))((−1)k2+µ2 Jˆ0JˆCJ−M(k2,−µ2),(J0−M0))=((−1)k1+µ1 Jˆ0Jˆ(−1)k1+J0−JCJM(k1,µ1),(J0M0))((−1)k2+µ2 Jˆ0Jˆ(−1)k2+J0−JCJM(k2,µ2),(J0M0))=2J0+12J+1(−1)2(k1+k2)+µ1+µ2+2(J0−J)CJM(k1,µ1),(J0M0)CJM(k2,µ2),(J0M0)(C.34)Inserting the above relation in Eq. (C.33) gives1〈N0J0|T k1†µ1 |NJ〉〈NJ|T k2µ2 |N0J0〉=(−1)k1+k22J+1CJM(k1µ1),(J0M0)CJM(k2,µ2),(J0M0)〈N0J0||T k1 ||NJ〉〈N0J0||T k2 ||NJ〉∗ (C.35)The sum over the following product of three Clebsch-Gordan coefficients is [92]∑µ1µ2MCk1µ1(k2µ2),(Kλ )CJM(k2µ2),(J0M0)CJM(k1µ1),(J0M0) =(−1)K√2J0+1CJ0,M0(K,λ ),(J0,M0)(−1)J+J0+k2 kˆ1Jˆ2{k1 k2 KJ0 J0 J}. (C.36)1Where we have used (−1)3(k1+k2)+2(µ1+µ2)+4(J0−J) = (−1)k1+k2 .166Using the above relation we have∑µ1µ2MCk1µ1(k2µ2),(Kλ )〈J0M0|Tk1†µ1 |JM〉〈JM|T k2µ2 |J0M0〉=(−1)K√2J0+1CJ0,M0(K,λ ),(J0,M0)(−1)J+J0+k2 kˆ1Jˆ2{k1 k2 KJ0 J0 J}(−1)k1+k22J+1×〈N0J0||T k1 ||NJ〉〈N0J0||T k2 ||NJ〉∗,=(−1)K√2J0+1CJ0,M0(K,λ ),(J0,M0) · (−1)k1+J0+J kˆ1{k1 k2 KJ0 J0 J}×〈N0J0||T k1 ||NJ〉〈N0J0||T k2 ||NJ〉∗. (C.37)Similarly, we have that∑µ1µ2MCk1µ1(k2µ2),(Kλ )〈J0M0|Tk2µ2 |JM〉〈JM|T k1†µ1 |J0M0〉(−1)K√2J0+1CJ0,M0(K,λ ),(J0,M0) · (−1)k2+K+J0+J kˆ1{k1 k2 KJ0 J0 J}〈N0J0||T k2 ||NJ〉〈N0J0||T k1 ||NJ〉∗.(C.38)Therefore,∑µ1µ2MCk1µ1(k2µ2),(Kλ )(〈J0M0|T k1†µ1 |JM〉〈JM|T k2µ2 |J0M0〉+ 〈J0M0|T k2µ2 |JM〉〈JM|T k1†µ1 |J0M0〉)(−1)K√2J0+1CJ0,M0(K,λ ),(J0,M0) · (−1)k1+J0+J kˆ1{k1 k2 KJ0 J0 J}×(〈N0J0||T k1 ||NJ〉〈N0J0||T k2 ||NJ〉∗+(−1)k1+k2+K〈N0J0||T k2 ||NJ〉〈N0J0||T k1 ||NJ〉∗).(C.39)167C.5 Commutator notation for matrix elementsThe following notation is used to simplify the derivations in Section 4.5. For twotensors Ak1 and Bk2 , the commutator of their matrix elements is denoted by thedouble commutator {{·, ·}}M1M2J1J2 which is defined{{Ak1µ1 ,Bk2µ2}}M1M2J1J2 = 〈J1M1|A†,k1µ1 |J2M2〉〈J2M2|Bk2µ2 |J1M1〉+ 〈J1M1|B†,k2µ2 |J2M2〉〈J2M2|Ak1µ1 |J1M1〉. (C.40)For the commutators of the reduced matrix elements we use the notation {{·, ·}}J1J2K ,i.e. without the projections M1 and M2, defined{{Ak1 ,Bk2}}J1J2K = 〈J1||Ak1 ||J2〉〈J1||Bk2 ||J2〉∗+(−1)k1+k2+K〈J1||Ak1 ||J2〉∗〈J1||Bk2 ||J2〉.(C.41)We also note that Eq. (C.41) can be cast in the following way that is amenable tocalculation when k1+ k2+K is even as{{Ak1 ,Bk2}}J1J2K = |〈J1||(Ak1 +Bk2)||J2〉|2−|〈J1||Ak1 ||J2〉|2−|〈J1||Bk2 ||J2〉|2.(C.42)168C.6 Reduced matrix elementsHere, useful reduced matrix elements are given that were used to derive manyresults in this work. By convention we choose the states to be |N(L,S)J〉. For twocoupled tensors T a(L) and T b(S), that act on the L and S subspaces, respectively,the reduced matrix elements satisfy〈N0(L0,S0)J0||[T a(L)⊗T b(S)]K ||N(L,S)J〉=∑N′JˆJˆ0KˆL0 L aS0 S bJ0 J K〈N0L0||T a(L)||N′L〉〈N′S0||T b(S)||NS〉. (C.43)If the tensors T a and T b act upon the same subspace then〈N0(L0,S0)J0||[T a⊗T b]K ||N(L,S)J〉=Kˆ(−1)K+J+J0 ∑N′′,S′′,L′′,J′′{a b KJ J0 J′′}〈J0||T a||J′′〉〈J′′||T b||J〉. (C.44)The following cases apply when the tensors in the reduced matrix elements actonly on a specific subspace [91]〈N0J0(L0,S0)||T k(L)||NJ(L,S)〉=(−1)L0+S0+J+kδSS0 JˆJˆ0{L0 J0 SJ L k}〈L0||T k(L)||L〉, (C.45)〈N0J0(L0,S0)||T k(S)||NJ(L,S)〉=(−1)L+S+J0+kδLL0 JˆJˆ0{S0 J0 LJ S k}〈S0||T k(S)||S〉. (C.46)169The following set of matrix elements where extensively used〈T0M0|[τ11 ⊗ τ12]1 |T M〉=√2(−1)T0+1×(δT01δT 0+δT00δT 1)δM00δM0, (C.47)〈L0||Iˆ||L〉=√2L+1δL0L, (C.48)〈N0J0(L0,S0)||Iˆ||NJ(L,S)〉=√2J+1δLL0δSS0δN0N , (C.49)〈L0||Lˆ||L〉= δL0L√L(L+1)(2L+1), (C.50)〈S0||S1||S〉= δSS0√S(S+1)(2S+1), (C.51)〈N0L0||Y `(rˆ) f (r)||NL〉= 〈N0L0|| f (r)||NL〉ˆ`Lˆ√4piC(L0,0)(L,0),(`,0),(C.52)〈N0J0(L0S0)||[Y k1(rˆ)⊗Σk212]K ||NJ(LS)〉= JˆJˆ0KˆL0 L k1S0 S k2J0 J K×〈L0||Y k1(rˆ)||L〉〈S0||Σk212||S〉, (C.53)〈N′1/2||τ1||N1/2〉=√6δN′N , (C.54)〈N′1/2||σ1||N1/2〉=√6δN′N , (C.55)〈N′S0||[σ11 ⊗σ12]k ||NS〉= 6Sˆ0Sˆkˆ1/2 1/2 11/2 1/2 1S0 S kδNN′(C.56)The matrix elements of the gradient operator are given as〈N0L0|| jk(qr)[Y k(rˆ)⊗∇]J||NL〉= (−1)L0+L+1+J Jˆ[{L L0 Jk 1 L−1}√L〈N0L0| jk(qr)(ddr+L+1r)|NL〉〈L0||Y k||L−1〉−{L L0 Jk 1 L+1}√L+1〈N0L0| jk(qr)(ddr− Lr)|NL〉〈L0||Y k||L+1〉].(C.57)170Using the recursion relations of the HO basis functions 2, we find3〈N0L0| jk(qr)(ddr− Lr)|NL〉=− 1b2〈N0L0| jk(qr)r|NL〉+2√Nb〈N0L0| jk(qr)|N−1 L+1〉 (C.58)〈N0L0| jk(qr)(ddr+L+1r)|NL〉= 〈N0L0| jk(qr)(2L+1r− rb2)|NL〉+2√Nb〈N0L0| jk(qr)|N−1 L+1〉 (C.59)2In this work we use the phase (−1)n in the HO basis functions3If the HO basis functions do not use the (−1)n phase in the definitions, the states with N−1 willhave an additional factor of −1.171Appendix DSpherical vector harmonicsIn this Appendix we provide the definitions of the vector spherical harmonics andprovide many of the properties that were used to derive expressions in the disser-tation. The spherical vector harmonics Y µκ`(qˆ) are the rank κ tensors formed bycoupling the a spherical harmonic of rank ` and the spherical tensor unit vector asY µκ`(qˆ) =[Y `(qˆ)⊗ e1λ]κµ=∑mλC(κ,µ)(`,m),(1,λ )Y`m(qˆ)eλ , (D.1)where the spherical components are given bye†λ ′ ·Yµκ`(qˆ) =∑mC(κ,µ)(`,m),(1,λ ′)Y`m(qˆ). (D.2)This allows the dot product of a vector v with a spherical vector harmonic to bewritten in tensor coupling notation asY µκ`(xˆ) · v =[Y `(xˆ)⊗ v1]κµ. (D.3)The spherical vector harmonics satisfy the following completeness relations∫dqˆ Y µ′∗κ ′`′(qˆ) ·Y µκ`(qˆ) = δκκ ′δµ ′µδ`′`, (D.4)∑κ`µY µ∗κ` (qˆ) ·Y µκ`(qˆ′) = δ (qˆ− qˆ′). (D.5)172These functions can also be integrated with respect to the spherical harmonic func-tions. These integrals appear in the derivations concerning the HFS and are givenby ∫dqˆ Y J′µ ′ (qˆ)Y∗µJ` (qˆ) = δJ′`∑λCJµJ′µ ′1λ e†λ . (D.6)The vector spherical harmonics are also eigenfunctions of the total angular mo-mentum operator κˆ and the orbital momentum operator ˆ`κˆzYµκ`(qˆ) = µYµκ`(qˆ), (D.7)κˆ2Y µκ`(qˆ) = κ(κ+1)Yµκ`(qˆ), (D.8)Lˆ2Y µκ`(qˆ) = `(`+1)Yµκ`(qˆ). (D.9)In momentum space, the following identities for the curl of the spherical vectorharmonic with the momentum vector q are needed in Section 4.5.3q×Y µ∗κκ(qˆ) =−iq[√κ+12κ+1Y ∗µκκ−1(qˆ)+√κ2κ+1Y ∗µκ,κ+1(qˆ)], (D.10)q× [qˆ×Y µ∗κκ(qˆ)]=−qY ∗µκκ(qˆ). (D.11)The following identities were used for the multipole expansion in Section 3.3Y µκκ+1(qˆ) =−√κ+12κ+1qˆY κµ (qˆ)− i√κ2κ+1qˆ×Y µκκ(qˆ), (D.12)Y µκκ−1(qˆ) =√κ2κ+1qˆY κµ (qˆ)− i√κ+12κ+1qˆ×Y µκκ(qˆ). (D.13)173Converting the momentum space expressions above into coordinate space resultsin the following identities from Ref. [91]−i(r×∇)Y `m(rˆ) =√`(`+1)Y m``1(rˆ), (D.14)rˆY `m(rˆ) =−√`+12`+1Y m`,`+1(rˆ)+√`2`+1Y m`,`−1(rˆ) (D.15)rˆ×Y m``(rˆ) = i√`2`+1Y m``+1(rˆ)+ i√`+12`+1Y m``−1(rˆ), (D.16)∇ f (r)Y `m(rˆ) =−√`+12`+1(ddr− `r)f (r)Y m`,`+1(rˆ),+√`2`+1(ddr+`+1r)f (r)Y m`,`−1(rˆ). (D.17)The curl identities for the vector spherical harmonics are∇× ( f (r)Y M`,`+1(rˆ))= i√ `2`+1[ddr+`+2r]( f (r)Y M`,`), (D.18)∇× ( f (r)Y M`,`(rˆ))= i√ `2`+1[ddr− `r]( f (r)Y M`,`+1)+ i√`+12`+1[ddr+`+1r]( f (r)Y M`,`−1), (D.19)∇× ( f (r)Y M`,`−1(rˆ))= i√ `+12`+1[ddr− `−1r]( f (r)Y M`,`). (D.20)The gradient identities of the vector spherical harmonics are∇ · ( f (r)Y M`,`+1(rˆ))=−√ `+12`+1[ddr+`+2r]( f (r)Y `M(rˆ)), (D.21)∇ · ( f (r)Y M`,`(rˆ))= 0, (D.22)∇ · ( f (r)Y M`,`−1(rˆ))=√ `2`+1[ddr− `−1r]( f (r)Y `M(rˆ)). (D.23)From these relations it follows that∇2 f (r)Y `m(rˆ) =(d2dr2+2rddr− `(`+1)r)f (r)Y `m(rˆ). (D.24)174Two last important identities areq jκ+1(qr)Yµκκ+1(rˆ) =−√κ+12κ+1∇(jκ(qr)Y κµ (rˆ))− i√ κ2κ+1∇× ( jκ(qr)Y µκκ(rˆ))(D.25)andq jκ−1(qr)Yµκκ−1(rˆ) =√κ2κ+1∇(jκ(qr)Y κµ (rˆ))− i√ κ+12κ+1∇× ( jκ(qr)Y µκκ(rˆ)) .(D.26)175Appendix EPionless EFT matrix elementsHere we present the matrix elements that are required to carry out thepi-EFT cal-culation of δApol from Section 2.4 at N2LO. The wave function of the deuteron andnormalization are given in Eq. (2.34) and Eq. (2.36), respectively. For these cal-culations E0= -2.224575(9) MeV is the deuteron binding energy and the numericalvalue of As=0.8845(8) fm−1/2 is taken for the asymptotic normalization constant.The nuclear excitation energy in this formalism isωk = Tk−E0 = k22mpn+κ22mpn. (E.1)The matrix elements required to evaluate δApol inpi-EFT are〈N0|1− ei 12 q·r |k〉=√4piAs[1κ2+ |k+ 12 q|2− 1κ2+ k2], (E.2)〈N0|1− ei 12 q·r |N0〉= A2s[12κ− 2qTan−1[ q4κ]], (E.3)〈N0|ei 12 q·r |N0〉= 2A2sqTan−1[ q4κ]. (E.4)These expressions are related to the η-less method in Section 4.3, through a mul-tipole decomposition of the matrix elements. This decomposition is carried out by176and introducing the gJ (k,q) functions1κ2+ |k+ 12 q|2− 1κ2+ k2=∞∑J=0gJ (k,q)PJ (x), (E.5)where these functions are defined asgJ (k,q) =2J +121∫−1dx PJ (x)[1κ2+ k2+ 14 q2+12 kqx− 1κ2+ k2]. (E.6)Integrating out the angular parts of k, we have∫dkˆ |〈N0|1− ei 12 q·k |k〉|2 =(4piA2s) ∞∑J=04pi2J +1g2J (k,q). (E.7)Combining all of these expressions, the leading order non-relativistic point nucleoncorrections are given byδNRpol =−|φ(0)|2(4pi)∞∫0dq(2pi)3(4piα)2q2× ∞∑J=0(4piA2s) 1(2J +1)∞∫0dk(2pi)3k2KNR(q,ωk)g2J (k,q)−KNR(q,0)A4s[12κ− 2qTan−1[ q4κ]]2δJ ,0].(E.8)The results of evaluating this expression are given in Table 6.6.177Appendix FMaximum entropy reconstructionof the response functionThe response function of Eq. (3.100) is one of the central objects in nuclear physics.This function is difficult to construct in practice because it requires the continuumstates |N〉. Bound state methods, such as the truncated HO basis representation,do not correctly represent these continuum states with the appropriate asymp-totic boundary conditions because the finite size of the basis imposes Dirichlettype boundary conditions [155]. Nevertheless, integrals of response functions con-structed from bound state methods will produce correct results as long as the Eigen-vectors of the finite basis form a complete basis.1It is computationally difficult to impose the appropriate continuum state bound-ary conditions and so alternate methods have been devised such as the Lorentzintegral transform reviewed in Ref. [156], to circumvent the need to calculate con-tinuum states by converting the original problem to that of an integral inversionand solving a bound state Schro¨dinger-like equation [156–159].Here we present an alternate method to compute the response function methodthat uses the calculated expectation values of the response. The method is adaptedfrom Refs. [160, 161] used to reconstruct a general probability density function1This is an application of the identity ∑N|〈N0|Oˆ|N〉|2 = 〈N0|Oˆ†Oˆ|N0〉 that holds for any operatorOˆ and a basis where ∑N|N〉〈N|= 1.178P(x) where x ∈ [0,1] and ∫ dx P(x) = 1. This reconstruction is carried out using Mcalculated moments φ = {1,〈φ1〉, ...,〈φM〉} of P(x). The components of φ are theexpectation values of the function φk(x)〈φk〉=1∫0dx P(x)φk(x), (F.1)and finding the density P(x) that minimizes the entropy and reproduces the mo-ments φ . The maximum entropy approximation of the underlying probability den-sity is denoted PME(x) and is obtained by maximizing the functional F given byF{PME(x)}=−1∫0dx PME(x)ln(PME(x))+M∑k=0λk〈φk〉− 1∫0dx PME(x)φk(x) , (F.2)where λ = (λ0, ..,λM) are the undetermined Lagrange multipliers. MaximizingEq. (F.2) with respect to the density PEM results in2PME(x|λ ) = e−∑Mk=1 φk(x)λkZ(λ ), (F.3)where Z(λ ) is the normalization factorZ(λ ) =1∫0dx e−∑Mk=1 φk(x)λk . (F.4)The remaining task is to determine the Lagrange multipliers λ . This can be ac-complished by finding the minima of the Legendre transform of Eq. (F.2) given by2We have changed the notation of the maximum entropy density estimate to PME(x|λ ) to make itmore explicit that this reconstructed probability density is conditioned on the Lagrangian multipliersλ .179[160]Γ(λ ,φ ) = ln(Z(λ ))+M∑k=1λk〈φk〉. (F.5)The critical points of Γ satisfy the Lagrange multiplier constraints1∫0dx PME(x|λ ) = 1, (F.6)and〈φk〉= 〈φk〉ME, (F.7)where the expectation value 〈φk〉ME denotes the expectation value of φk with respectto the maximum-entropy estimate of the density3. It is straight forward to show thatΓ is positive definite with respect to λ and therefore if the minima of the functionalexists, then it will be unique [160].This method is versatile and can be extended to the discrete case where it canreconstruct the probability density of an unknown distribution given a collectionof N samples D = {xi |i = 1, ..,N} from the distribution [162]. In this case theprobability of the set of parameters λ conditioned on the data D isP(λ |D) = P(D|λ )P(λ )P(D),∝M∏k=1PME(xi|λ )P(λ ),∝ e−NΓ(λ ,φˆ)P(λ ), (F.8)where Bayes formula has been used in the second line. The above equation con-tains the functional Γ(λ ,φ ) from Eq. (F.5) with the vector φˆ = {1, φˆ1, .., φˆM} inthe second argument. The vector φˆ consists of the expectation values of functions3i.e. 〈φk〉ME =1∫0dx PME(x|λ )φk(x)180φk(x) estimated from the observations D.4 The probability P(λ ) are the priors ofthe Lagrange multipliers λ . The posterior distribution of Eq. (F.8) is obtained bysampling the log-likelyhood using Markov chain Monte Carlo [120]. For practicalproblems the basis functions φk(x) used for the density reconstruction are [163]; themonomial basis functions xk, the Legendre basis functions Pk(x), and the Cheby-shev polynomials Tk(x).The nuclear response functions, SE/M(ω) can be reconstructed using the max-imum entropy method outlined above5 by computing the moments〈φk〉= 1ω f −ωthω f∫ωthdω SE/M(ω)φk(ω−ωthω f −ωth), (F.9)or〈φk〉= 1ω f −ωthω f∫ωthdω SE/M(ω)φk(ω f −ωω f −ωth), (F.10)where ωth is the threshold energy and ω f is the maximum energy of the calcula-tion. In practice, Eq. (F.10) was found to give better results for nuclear responsefunctions that are sharply peaked at the threshold energy.Next, we show the results of reconstructed test response functions with thismethod for a different number of moments using the Legendre basis in Fig. F.1.0.00 0.25 0.50 0.75 1.000123S O()TestReconstruction(a)0.00 0.25 0.50 0.75 1.000123S O()(b)0.00 0.25 0.50 0.75 1.000123S O()(c)Figure F.1: The reconstructed test response functions using (a) 10 moments,(b) 20 moments, (c) 30 moments, in the Legendre basis.4These expectation values are φˆk = 1N ∑Ni=1 φk(xi) where xi ∈ D.5Like a probability density function, nuclear response are positive definite and can be normalized.181We observe in the Fig. F.1 above that only a small number of moments areneeded to reconstruct a function that closely resembles the underlying responsefunction. Next, we test the sensitivity of the reconstruction for different basis func-tions and keeping the number of moments fixed. This is shown in Fig. F.2 were10 moments are used to reconstruct the test response function using three differentbasis functions.0.00 0.25 0.50 0.75 1.000123S O()TestReconstruction(a)0.00 0.25 0.50 0.75 1.000123S O()(b)0.00 0.25 0.50 0.75 1.000123S O()(c)Figure F.2: The reconstructed test response functions using ten moments and(a) the Legendre, (b) monomial or (c) the Chebychev basis functions.In Fig. F.2 the quality of the reconstruction depends on the basis used. How-ever, with enough moments, the reconstruction of the underlying response is achievedin all cases. This method has been tested for the deuteron in the HO basis however,further work is required to test if this method can be applied to other nuclear sys-tems where the moments may not converge as quickly.182


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