Science, Faculty of
Physics and Astronomy, Department of
DSpace
UBCV
Gysbers, Peter
Doctor of Philosophy - PhD
University of British Columbia
Atomic nuclei exhibit many phenomena not limited to excited states, decays, reactions, and clustering. Nuclear processes control the evolution of stars and explain the abundances of chemical elements in the universe. Nuclear physics can be used to answer fundamental questions about underlying particle physics and cosmology, such as the symmetry between matter and antimatter or the nature of neutrinos. The discrepancy between theoretical predictions and observations motivates improved theory and can provide evidence for new physics. A predictive model of nuclei is needed as input for experimental tests and for astrophysical models. Nuclei are complex strongly-interacting quantum many-body systems. Accurate the- oretical techniques are required to predict the rate of nuclear decay processes, the cross section of nuclear reactions and the distribution of the emitted particles. Ab initio nuclear theory takes advantage of the recent rapid increase in computing power to calculate nuclear structure and reactions solely from realistic interactions between the constituent nucleons. In this thesis, we first present beta-decay calculations using the ab initio no-core shell model. Our calculations provide an explanation for the quenching of Gamow-Teller beta- decays, provide nuclear structure corrections to the beta-spectrum necessary to interpret experiments seeking to find new physics and provide estimates for the hypothetical process of neutrinoless double-beta decay. Second, we present radiative capture calculations using the no-core shell model with continuum, an extension which places bound and scattering states on equal footing. The rate of radiative capture reactions in big bang nucleosynthesis is required to estimate the abundance of isotopes in the early universe. In addition, anomalies in recent radiative capture experiments claim the discovery of a new boson. Comparing to these experiments requires prediction of the distribution of electron-positron pairs produced by radiative capture.
Radiative Capture and Decays in Ab Initio Nuclear TheorybyPeter GysbersB.Sc., McMaster University, 2017a thesis submitted in partial fulfillmentof the requirements for the degree ofDoctor of Philosophyinthe faculty of graduate and postdoctoral studies(Physics)The University of British Columbia(Vancouver)March 2023© Peter Gysbers, 2023The following individuals certify that they have read, and recommend to the Faculty ofGraduate and Postdoctoral Studies for acceptance, the thesis entitled:Radiative Capture and Decays in Ab Initio Nuclear Theorysubmitted by Peter Gysbers in partial fulfillment of the requirements for the degree ofDoctor of Philosophy in Physics.Examining Committee:Petr Navra´til, Research Scientist, TRIUMFCo-supervisorReiner Kruecken, Adjunct Professor, Physics and Astronomy, UBCCo-supervisorMark Thachuk, Professor, Chemistry, UBCUniversity ExaminerJoanna Karczmarek, Professor, Physics and Astronomy, UBCUniversity ExaminerMona Berciu, Professor, Physics and Astronomy, UBCSupervisory Committee MemberAdditional Supervisory Committee Members:Rob Kiefl, Professor, Physics and Astronomy, UBCSupervisory Committee MemberFei Zhou, Professor, Physics and Astronomy, UBCSupervisory Commitee MemberiiAbstractAtomic nuclei exhibit many phenomena not limited to excited states, decays, reactions,and clustering. Nuclear processes control the evolution of stars and explain the abundancesof chemical elements in the universe. Nuclear physics can be used to answer fundamentalquestions about underlying particle physics and cosmology, such as the symmetry betweenmatter and antimatter or the nature of neutrinos. The discrepancy between theoreticalpredictions and observations motivates improved theory and can provide evidence for newphysics. A predictive model of nuclei is needed as input for experimental tests and forastrophysical models.Nuclei are complex strongly-interacting quantum many-body systems. Accurate the-oretical techniques are required to predict the rate of nuclear decay processes, the crosssection of nuclear reactions and the distribution of the emitted particles. Ab initio nucleartheory takes advantage of the recent rapid increase in computing power to calculate nuclearstructure and reactions solely from realistic interactions between the constituent nucleons.In this thesis, we first present beta-decay calculations using the ab initio no-core shellmodel. Our calculations provide an explanation for the quenching of Gamow-Teller beta-decays, provide nuclear structure corrections to the beta-spectrum necessary to interpretexperiments seeking to find new physics and provide estimates for the hypothetical processof neutrinoless double-beta decay. Second, we present radiative capture calculations usingthe no-core shell model with continuum, an extension which places bound and scatteringstates on equal footing. The rate of radiative capture reactions in big bang nucleosynthesis isrequired to estimate the abundance of isotopes in the early universe. In addition, anomaliesin recent radiative capture experiments claim the discovery of a new boson. Comparing tothese experiments requires prediction of the distribution of electron-positron pairs producedby radiative capture.iiiLay SummaryThe components of the atomic nucleus are protons and neutrons. The numbers of theseparticles determine the characteristics of the nucleus. For example, adding one more protonor neutron can destabilize the nucleus and cause it to decay. There are many types of decayswhich emit different particles. From measurements of decay rates and decay products, wecan refine our understanding of the known fundamental forces or find new forces.Accurate theoretical predictions are needed which can be compared to observations.In this thesis, we use sophisticated computational techniques to predict the properties ofmany different nuclei and the rates of their varied decay modes. We also predict the rateof nuclear reactions corresponding to recent measurements with unexplained anomalies.The discrepancies between theory and experiment point the way towards refinement oftheoretical models or may provide evidence for new physics.ivPrefaceThis work is based on published and unpublished material. Chapter 1 introduces the fieldof nuclear physics, defines ab initio nuclear theory and outlines the thesis. Chapter 2,Chapter 3 and Chapter 4 are based on common literature and describe the mathematicaland computational tools used in our calculations. Chapter 5 presents results which wereoriginally published in [1–3], with the exception of Section 5.2.2 which is original unpub-lished material. Chapter 6 provides the theoretical basis for the following chapter. The firsttwo sections are based on common literature, while the second two are original expositorymaterial. The first section of Chapter 7 presents results which were originally published in[4], while the second section presents original unpublished material.• The results of Section 5.1.1 were previously published in[1] P. Gysbers, G. Hagen, J.D. Holt, G.R. Jansen, T.D. Morris, P. Navra´til,T. Papenbrock, S. Quaglioni, A. Schwenk, S.R. Stroberg, K.A. Wendt,Nat. Phys. 15 (5) (2019) 428–431.G.H., T.D.M. and T.P. performed the coupled-cluster calculations. G.R.J. computedthree- nucleon forces for the coupled-cluster calculations. P.G., S.Q., P.N. and K.A.W.performed calculations for the two-body currents. P.N. developed the higher-precisionchiral three- nucleon interactions used in this work and performed no-core shell modelcalculations. G.H. and T.D.M. derived and implemented the new formalism to in-corporate higher- order excitations in coupled-cluster theory. S.R.S. and J.D.H. per-formed VS-IMSRG calculations. All authors discussed the results and contributed tothe manuscript at all stages.• The results of Section 5.1.2 were previously published in[2] A. Glick-Magid, C. Forsse´n, D. Gazda, D. Gazit, P. Gysbers, and P.Navra´til, Phys. Lett. B 832, 137259 (2022).A. G. was the principal author of the manuscript. A. G. and D. G.(2) derived theexpressions for the nuclear structure corrections. D. G.(1) and C. F. developed thevmomentum-dependent operators. P. N. developed the formalism for translationally-invariant operators and densities. P. G. implemented and tested the operators in theno-core shell model. All authors discussed the results and edited the manuscript.• The results of Section 5.2.1 were previously published in[3] S. Novario, P. Gysbers, J. Engel, G. Hagen, G. R. Jansen, T. D. Morris,P. Navra´til, T. Papenbrock, and S. Quaglioni, Phys. Rev. Lett. 126,182502 (2021).S. N. was the principle author of the manuscript. S. N., G. H., G. R. J., T. D. M. andT. P. performed the coupled cluster calculations and contributed to the developmentof improved coupled cluster methods and codes. J. E. provided the radial parts ofthe neutrino-less double beta decay matrix elements. P. G. implemented the doublebeta decay operators and performed the no-core shell model calculations with theassistance of P. N. and S. Q..• The results of Section 5.2.2 are unpublished. This work was performed in collaborationwith T. Miyagi, P. Navra´til, S. Quaglioni, J. Engel and S. R. Stroberg.J. E. provided the radial parts of the neutrino-less double beta decay matrix elements.P. G. implemented the double beta decay matrix operators and the SRG evolution of3-body operators in the no-core shell model with assistance from P.N. and S.Q.. T.M.developed SRG evolution technology within the IMSRG with assistance from S.R.S..Calculations were performed by P.G. and T.M..• The results of Section 7.1 were previously published in[4] C. Hebborn, G. Hupin, K. Kravvaris, S. Quaglioni, P. Navra´til and P.Gysbers, Phys. Rev. Lett. 129 042503 (2022).C. H. was the principal author of the manuscript, and further improved the capa-bilities of the NCSMC codes for two-nucleon projectiles. This was based on previ-ous development by G.H., P.N. and S.Q. and assisted by K.K.. P.G. performed theSRG evolution of the electromagnetic operators and benchmarked the contributionsto NCSMC transition matrix elements. All authors discussed the results and editedthe manuscript.viTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiList of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xixAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xx1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Nuclear Structure and Reactions . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Elementary Particles and Interactions . . . . . . . . . . . . . . . . . . . . . 31.3 Nucleons and Nuclear Forces . . . . . . . . . . . . . . . . . . . . . . . . . . 71.4 The Nuclear Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.5 Applications of Nuclear Theory . . . . . . . . . . . . . . . . . . . . . . . . . 111.6 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 Many-Body Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.1 The Quantum Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . 142.2 Spin and Angular Momentum Coupling . . . . . . . . . . . . . . . . . . . . 172.3 Spherical Tensor Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.4 Isospin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.5 Many-Body States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.6 Intrinsic Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.7 Solving the Schrodinger Equation . . . . . . . . . . . . . . . . . . . . . . . . 282.8 The No-Core Shell Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28vii2.9 Static Nuclear Properties and Many-Body Operators . . . . . . . . . . . . . 312.10 Charge and Current Densities of the Nucleus . . . . . . . . . . . . . . . . . 343 Electroweak Decays of Nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . 363.1 Electromagnetism: γ Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.1.1 Multipole Decomposition of Electromagnetic Current . . . . . . . . . 403.1.2 Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.1.3 Final Formula and Selection Rules . . . . . . . . . . . . . . . . . . . 453.2 Weak Interactions: β Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.2.1 Leading-Order Beta Decay Operators and Two-Body Currents . . . 503.2.2 The GT Beta Decay Spectrum . . . . . . . . . . . . . . . . . . . . . 513.2.3 Double-Beta Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524 Similarity Renormalization Group . . . . . . . . . . . . . . . . . . . . . . . 564.1 Softening Potentials with Flow Equations . . . . . . . . . . . . . . . . . . . 564.2 SRG Evolution of Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.2.1 Induced Many-Body Terms . . . . . . . . . . . . . . . . . . . . . . . 594.2.2 Calculations of SRG-evolved Operators . . . . . . . . . . . . . . . . 605 NCSM Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635.1 Beta Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635.1.1 SRG and 2BC in GT Decay . . . . . . . . . . . . . . . . . . . . . . . 635.1.2 Precision Beta Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . 685.2 Neutrino-less Double Beta Decay . . . . . . . . . . . . . . . . . . . . . . . . 715.2.1 Coupled-Cluster Benchmarks . . . . . . . . . . . . . . . . . . . . . . 725.2.2 VS-IMSRG Benchmarks . . . . . . . . . . . . . . . . . . . . . . . . . 746 Scattering and Reactions of Nuclei . . . . . . . . . . . . . . . . . . . . . . 776.1 Scattering Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 776.1.1 Partial Wave Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 786.1.2 Coulomb Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . 816.1.3 Multi-channel Scattering and Reactions . . . . . . . . . . . . . . . . 836.1.4 Cluster States from NCSM . . . . . . . . . . . . . . . . . . . . . . . 856.2 The No-Core Shell Model with Continuum . . . . . . . . . . . . . . . . . . . 876.2.1 A Unified Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 876.2.2 Solving the NCSMC . . . . . . . . . . . . . . . . . . . . . . . . . . . 886.2.3 Transition Matrix Elements . . . . . . . . . . . . . . . . . . . . . . . 896.3 Radiative Capture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 906.4 Pair Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 956.4.1 Pair Production in Bound-Bound Decay . . . . . . . . . . . . . . . . 966.4.2 Pair Production in Radiative Capture . . . . . . . . . . . . . . . . . 100viii6.4.3 Hypothetical Intermediate Particles . . . . . . . . . . . . . . . . . . 1077 NCSMC Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1087.1 4He(d, γ)6Li Radiative Capture . . . . . . . . . . . . . . . . . . . . . . . . . 1097.1.1 The 6Li Composite System and 4He(d, d)4He Elastic Scattering . . . 1097.1.2 The 4He(d, γ)6Li S-factor . . . . . . . . . . . . . . . . . . . . . . . . 1107.1.3 The Effect of SRG on M1 Contributions . . . . . . . . . . . . . . . . 1137.1.4 Thermonuclear Reaction Rate . . . . . . . . . . . . . . . . . . . . . . 1137.2 7Li(p, γ)8Be Radiative Capture . . . . . . . . . . . . . . . . . . . . . . . . . 1157.2.1 The 8Be Composite System . . . . . . . . . . . . . . . . . . . . . . . 1157.2.2 7Li, 7Be and 8Be NCSM Results . . . . . . . . . . . . . . . . . . . . 1157.2.3 Resonances and Phase Shifts . . . . . . . . . . . . . . . . . . . . . . 1177.2.4 p+ 7Li Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1217.2.5 The 7Li(p, γ)8Be Cross Section . . . . . . . . . . . . . . . . . . . . . 1217.2.6 Pair Production and the X17 . . . . . . . . . . . . . . . . . . . . . . 1228 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130A Preliminary Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144A.1 Pauli Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144A.2 Angular Momentum Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . 144A.2.1 Properties of Clebsch-Gordan Coefficients . . . . . . . . . . . . . . . 144A.2.2 Definitions and Properties of 3nj Symbols . . . . . . . . . . . . . . . 145A.3 Rotations and Wigner D-matrices . . . . . . . . . . . . . . . . . . . . . . . 146A.4 Useful Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148B The Connection between QED and Nucleon Currents . . . . . . . . . . . 149B.1 Relativistic Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149B.2 Non-relativistic Expansion of Dirac Spinors . . . . . . . . . . . . . . . . . . 150B.3 Nucleon Currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150B.4 Seven Multipole Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 152ixList of TablesTable 3.1 The selection rules for allowed γ-decay transitions. For a given ∆J and∆pi the furthest left multipole in the table will be dominant. . . . . . . . 46Table 6.1 Notation for Rj in terms of N . . . . . . . . . . . . . . . . . . . . . . . . . 105Table 6.2 Simplified formulation of kinematic factors vi corresponding to the ATOMKIexperimental setup. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106Table 6.3 Relevant values of the Wigner d-matrix factor [35]. . . . . . . . . . . . . 106Table 7.1 The bound states of 8Be are well reproduced with the NCSMC. One minorexception is that the 3rd 2+ is slightly above the 7Li + p threshold. . . . 120Table B.1 Equivalent matrix elements between Dirac spinors (left) and those takenin the nonrelativistic limit (right). . . . . . . . . . . . . . . . . . . . . . . 150xList of FiguresFigure 1.1 (a) The full chart of nuclides showing the observed isotopes arrangedaccording to the number of protons (Z) and the number of neutrons(N). The isotopes coloured black do not decay, i.e. they are stable.The other isotopes are coloured according to their most dominant decaymode. Several thousand isotopes are shown but several thousand moreare predicted to exist. “Existence” is generally defined as bound withrespect to the strong force, i.e. decays are via electromagnetic or weakforces which act at a much longer time scale compared to the strong force.(b) A subset of the chart of nuclides, including only up to Z = 9 andN = 18. Figures adapted from [5] (which sources data from [6]) . . . . . 2Figure 1.2 The Standard Model of elementary particles [17]: the 12 fundamentalfermions and 5 fundamental bosons. Brown loops indicate which gaugebosons (red) couple to which fermions (purple and green). The valuefor “charge” reflected in figure is the electromagnetic charge. The massvalues reflected in the figure are as of 2019 (periodic reevaluations arecarried out by the Particle Data Group, see [18] for the latest). . . . . . 4Figure 1.3 All vertices allowed in the Standard Model: q is any quark, g is a gluon,X is any charged particle, γ is a photon, f is any fermion, H is the Higgs,m is any particle with mass (with the possible exception of the neutri-nos), mB is any boson with mass. In diagrams with multiple particlelabels separated by /, one particle label is chosen. In diagrams with par-ticle labels separated by |, the labels must be chosen in the same order.For example, in the four boson electroweak case the valid diagrams areWWWW , WWZZ, WWγγ, WWZγ. Lines on the left correspond toincoming particles, while lines on the right are outgoing. The diagramscan be rotated such that an incoming particle becomes outgoing (or viceversa) resulting in conjugation of the corresponding field. The conjugateof each listed vertex (reversing the direction of arrows) is also allowed.Adapted from [20] and [21]. . . . . . . . . . . . . . . . . . . . . . . . . . 5xiFigure 1.4 Hierarchy of nuclear forces in χEFT from [29]. Solid (dashed) lines de-note nucleons (pions). The solid dots, filled circles, filled squares, filleddiamonds, and open squares refer to vertices of dimension ∆i = 0, 1, 2, 3and 4 respectively, as defined in (1.2). . . . . . . . . . . . . . . . . . . . 8Figure 2.1 Schematic diagram of the Jacobi coordinates with three particles. . . . . 26Figure 2.2 Truncation diagram for the NCSM. As an example, the solid circles indi-cate the lowest Pauli configuration of 6Li. Here “N” corresponds to theharmonic oscillator excitation level (N˜ in the text). . . . . . . . . . . . . 29Figure 3.1 The Feynman diagram and kinematics for photon emission. . . . . . . . 37Figure 3.2 The leading order Feynman diagrams for a β− decay at different levelsof EFT. Solid lines represent fermions (nucleons, quarks and leptons).Wavy lines represent bosons (the W ). (a) The diagram for a nucleardecay including the kinematics. The circle at the vertex represents aneffective operator. (b) The diagram for β decay including the W bosonat the level of quarks. (c) The diagram for β decay including the Wboson at the level of nucleons. (d) The diagram for β decay at the levelof nucleons, where the dot represents an effective operator. . . . . . . . 48Figure 3.3 Diagrammatic representations of terms that contribute to the axial-vectorcharge-changing current up to N2LO in the limit ~K → 0. Solid linesrepresent nucleons while wavy lines represent the W boson. The left-most diagram yields the standard (one-body) στ transition operator, theother diagrams form the leading 2BC. . . . . . . . . . . . . . . . . . . . 51Figure 3.4 Feynman diagrams for (a) two-neutrino double-beta decay (2νββ) and(b) neutrino-less double-beta decay (0νββ). . . . . . . . . . . . . . . . . 53Figure 4.1 A schematic representation of the effect of SRG evolution on the Hamil-tonian matrix. The matrix elements are arranged according to the totalnumber of harmonic oscillator excitations N (corresponding to N˜ = 2n+`in the text). At successively smaller stopping points, λ0 > λ1 > λ2, thematrix elements near the diagonal become larger compared to those fur-ther from the diagonal. Adapted from [72] . . . . . . . . . . . . . . . . . 58Figure 4.2 The process of preparing an operator for an A-particle calculation. Adiagrammatic summary of Section 4.2.2. Adapted from [72] . . . . . . . 61xiiFigure 5.1 Theory-to-experiment ratio for the Gamow-Teller matrix elements of sixstrong transitions in light nuclei for the NN-N4LO+3Nlnl interaction de-veloped in [1]. The subscripts in the legend denote the total angularmomenta of the parent and daughter states. All initial states are groundstates. In the case 3H → 3He,6He → 6Li and 7Be → 7Li32, the daughternucleus is in its ground state, while the 7Be → 7Li12,8He → 8Li1 and10C→ 10B1 are decays to the first excited state of the daughter nucleus,and the 14O→ 14N1 is a decay to the second excited state of 14N. Hollowsymbols correspond to results obtained with the standard Gamow-Tellerστ operator, and full symbols include 2BC. The results are convergedto within 3% with respect to the model-space size. This uncertainty isslightly larger than the marker size and is not shown for transparency.Figure reproduced from [1]. . . . . . . . . . . . . . . . . . . . . . . . . . 64Figure 5.2 Convergence of the 3H→ 3He (a) and 6He→ 6Li (b) Gamow-Teller ma-trix elements with respect to the NCSM basis size for three values of theSRG evolution parameter λ. Dashed and full lines show results obtainedwith one-body only and one- plus two-body operators respectively. TheNN-N4LO+3Nlnl interaction was used with both the interaction and tran-sition operators consistently SRG evolved. The dotted and dash-dottedlines show results obtained with the same SRG evolved interaction usingonly the one-body operator without any SRG evolution. The shadowbands represent the experimental values with their uncertainties. Figurereproduced from [1]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66Figure 5.3 Comparison of experimental [77] and theoretical Gamow-Teller matrixelements for medium-mass nuclei. Plots of the Gamow-Teller matrixelements: sd-shells (top) and pf -shells (bottom). Theoretical results wereobtained using phenomenological shell-model interactions [78, 79] with anunquenched standard Gamow-Teller στ operator (open orange squares)and using VS-IMSRG approach with the NN-N4LO+3Nlnl interactionand consistently evolved Gamow-Teller operator plus 2BCs (filled greendiamonds). The linear fit show the resulting quenching factor q given inthe panels, and shaded bands indicate one standard deviation from theaverage quenching factor. Experimental uncertainties, taken from [77],are shown as vertical error bars. Figure reproduced from [1]. . . . . . . 67xiiiFigure 5.4 Comparison of the Gamow-Teller strength |MGT |2 for the β-decay of100Sn calculated in [1] compared to data [80], systematics [81], and othermodels (extreme single-particle model (ESPM), shell-model Monte-Carlo(SMMC), large-space shell-model (LSSM), quasiparticle random-phaseapproximation (QRPA) and finite Fermi systems (FFS)) from [80]. Opensymbols represent results obtained with the standard Gamow-Teller oper-ator (στ), filled symbols also include two-body currents (2BCs) and par-tially filled symbols show values following from the multiplications of thecomputed Gamow-Teller strength by the square of a phenomenologicalquenching factor. Each of our 100Sn calculations carries a conservativelyestimated uncertainty of about 10% (not shown to avoid overcrowdingthe figure). Figure reproduced from [1]. . . . . . . . . . . . . . . . . . . 68Figure 5.5 Dependence of nuclear matrix elements on the NCSM model-space pa-rameters and nuclear Hamiltonians. The light to dark bands correspondto Nmax = 8, 10, 12 for the NNLOsat (blue) interaction and Nmax =10, 12, 14 for the NNLOopt (green) interaction. The width of the bandsshow the variation with HO frequency ~Ω = 16, 20, 24 MeV. The solid(dashed) line shows the result with the NNLOsat interaction at Nmax =12, ~Ω = 20 MeV computed with translationally-invariant (standard)one-body densities. Figure reproduced from [2]. . . . . . . . . . . . . . . 69Figure 5.6 Dependence of nuclear matrix elements on the NCSM model space param-eters and nuclear Hamiltonians. Light, medium, and heavy filled bandscorrespond to Nmax = 8, 10, 12 for the NNLOsat interaction including3NF (blue bands), and Nmax = 10, 12, 14 for the NNLOopt interactionwith only 2NF (green bands). The width of the bands show the variationwith HO frequency ~Ω = 16, 20, 24 MeV. The solid (dashed) line showsthe result with the NNLOsat interaction at Nmax = 12, ~Ω = 20 MeVcomputed with translationally-invariant (standard) one-body densities.Figure reproduced from [2]. . . . . . . . . . . . . . . . . . . . . . . . . . 70xivFigure 5.7 (a) Calculated energy dependence of the spectrum of 6He β-decay, inarbitrary units. Dashed line is the pure GT spectrum, while the filledbands include nuclear-structure dependent corrections. (b) The residualnuclear structure correction δ1+β−1 compared to the pure GT spectrum.(c) Energy dependence of the angular correlation aβν . Dashed line cor-responds to the SM value, aGTβν = −1/3. (d) Relative size of the δ˜1+β−acorrection. The width of the dark filled bands shows the variation withthe employed nuclear Hamiltonian and NCSM model space parametersfor HO frequency ~Ω = 16, 20, 24 MeV, Nmax = 8, 10, 12 (10, 12, 14) us-ing the NNLOsat (NNLOopt) interaction, using translationally-invariantone-body densities. The width of the light filled band shows the totalestimated theory error. Figure reproduced from [2]. . . . . . . . . . . . 71Figure 5.8 Comparison of the 0νββ nuclear matrix element in several light nucleicomputed with the coupled cluster method and the no-core shell model.The first two columns correspond to different choices for the coupled-cluster reference state, and results for the CCSD and CCSDT-1 approx-imations are shown in each. The error bars indicate the uncertaintiescoming from variations with the model-space size. Each case utilizes the1.8/2.0(EM) interaction [93] except for 22O→ 22Ne which disregards thethree-nucleon forces to more rapidly converge the NCSM results. Figurereproduced from [3]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73Figure 5.9 Comparison of the NME for the 0νββ decay of 48Ca, calculated withinvarious approaches. The coupled-cluster results use both the CCSD andCCSDT-1 approximations with both the spherical and deformed refer-ence states. For IMSRG+GCM [94], the double bars show the effects ofuncertainty in model-space size; otherwise they show those of uncertaintyin short-range correlation functions. The other results are the realisticshell model [95], the quasiparticle random phase approximation [96], theinteracting boson model [97], various energy-density functionals [98, 99]and phenomenological shell models [66, 71, 100, 101]. Figure reproducedfrom [3]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74Figure 5.10 Nuclear matrix elements of the neutrino-less double-beta decay transi-tions with ∆T = 0. From left to right, columns correspond to thetransitions: 6He(0+1 ) → 6Be(0+1 ), 10Be(0+1 ) → 10C(0+1 ) and 14C(0+1 ) →14O(0+1 ). The numerical calculations use the parameters ~Ω = 20 MeVand Nmax = 8 (emax = 12) in the NCSM (IMSRG). The NCSM calcula-tion at A = 14 uses an additional approximation known as “importancetruncation” [102] to obtain a result at Nmax = 8. . . . . . . . . . . . . . 75xvFigure 5.11 Nuclear matrix elements of the neutrino-less double-beta decay transi-tions with ∆T = 2. From left to right, columns correspond to the tran-sitions: 8He(0+1 ) → 8Be(0+1 ), 10He(0+1 ) → 10Be(0+1 ) and 48Ca(0+1 ) →48Ti(0+1 ). The numerical calculations use the parameters ~Ω = 20 MeV,except ~Ω = 16 MeV for A = 48, and Nmax = 8 (emax = 12) in theNCSM (IMSRG). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76Figure 6.1 A particle incident on area dσ scatters into the solid angle dΩ. b is theimpact parameter, the distance by which the incident particle would havemissed the scattering center, had it continued on its original trajectory.Figure reproduced from [11]. . . . . . . . . . . . . . . . . . . . . . . . . 79Figure 6.2 The Feynman diagram and kinematics for radiative capture. . . . . . . 91Figure 6.3 The Feynman diagram for the pair production process. . . . . . . . . . . 96Figure 6.4 The Feynman diagram and kinematics for radiative capture with pairproduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96Figure 7.1 Low-lying spectrum of 6Li obtained with the NCSMC at Nmax = 11 usingthe SRG-evolved N3NLO NN potential [28] (NN-only) with λ = 2 fm−1,the NN+3nloc [60, 140] without (NN+3Nloc) and with the phenomeno-logical energy adjustment (NN+3Nloc-pheno) compared to experiments[141]. The zero energy is taken as the predicted (resp. experimental)4He + d threshold for NCSMC (resp. Exp.) . . . . . . . . . . . . . . . . 110Figure 7.2 NCSMC predictions of 4He(d, d)4He elastic scattering. Both NCSMCpredictions obtained without (NN+3Nloc) and with the phenomenolog-ical adjustment (NN+3Nloc-pheno) reproduce the experimental elastic-scattering 4He(d, d)4He cross section at the deuteron backscattered tothe angle θd = 164◦ [142, 143]. . . . . . . . . . . . . . . . . . . . . . . . 111Figure 7.3 Top: Predicted S-factor for the 4He(d, γ)6Li compared with data takenfrom Refs [146] (red circles), [147] (blue squares), [148] (green up-triangles),[149] (black down-triangles). Calculations are obtained using the SRG-evolved N3LO NN potential [28] (NN-only) with λ = 2 fm−2, the NN+3Nloc[60] without (NN+3Nloc) and with the phenomenological energy adjust-ment (NN+3Nloc-pheno). Bottom: E2, E1 and M1 components to thepredicted S-factor for the 4He(d, γ)6Li obtained with the NN+3Nloc-pheno.112Figure 7.4 Influence of the SRG transformation on the M1 operator. The experi-mental data follow the same legend as in Fig. 7.3. . . . . . . . . . . . . 113xviFigure 7.5 Ratio of the predicted thermonuclear reaction rates (black line) for 4He(d, γ)6Liwith the NACRE-II evaluation (red line) [151] for different temperatureT9 in GK. Our results are also compared with the recent thermonuclearreaction rate derived from the measurements of the LUNA collaboration(blue line) [152]. The shaded areas correspond to the uncertainty of eachcalculation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114Figure 7.6 The experimental level scheme for 8Be. Adapted from [153]. . . . . . . . 116Figure 7.7 Excitation energies from the NCSM for (a)7Li, (b) 7Be and (c) 8Be usingthe NN+3Nlnl interaction [136]. The lowest states most relevant for inputto the NCSMC are in the correct order. Only the lowest 5 states areshown for 7Be. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117Figure 7.8 The convergence of the ground state energies of 7Li, 7Be and 8Be withthe model-space parameter Nmax using the NN+3Nlnl interaction [136]. 118Figure 7.9 7Li + p eigen-phase shifts at Nmax = 9, even (a) and odd (b) parity. . . 119Figure 7.10 Dependence of the 7Li + p eigen-phase shifts on the NCSMC basis sizecharacterized by Nmax for the 1+ (left) and 3+ (right) resonances. Ekinis the kinetic energy of the 7Li + p in the center of mass frame. . . . . 120Figure 7.11 Selected P-wave 7Li + p phase shifts. The 7Be + n threshold is at 1.64MeV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120Figure 7.12 The differential cross section for 7Li(p, p)7Li elastic scattering. Θ is theangle between the incoming proton and outgoing proton in the centerof mass frame. (a) A comparison between inverse scattering data at 5and 5.44 MeV/u and NCSMC calculations at the corresponding centerof mass energies: E = 4.38 and 4.76 MeV. Data from [155, 156]. (b) Theconvergence of NCSMC calculations at E = 4.76 MeV with the inputparameter Nmax. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121Figure 7.13 The cross-section for 7Li(p, γ)8Be capture. (a) The integrated cross sec-tion of 7Li(p, γ)8Be calculated at Nmax = 7 (and without7Be+n degreesof freedom). (b) The integrated cross section of 7Li(p, γ)8Be calculatedat Nmax = 9 with phenomenological adjustments. γ0 is the capture cross-section for proton capture resulting in the ground state of 8Be. γ1 is thecapture cross-section resulting in the first excited state (2+) of 8Be. Datapoints are taken from [157]. (c) The 7Li(p, γ)8Be astrophysical S-factor(proportional to the cross section in (b) via (7.3)) (d) The phenomeno-logically adjusted phase shifts at Nmax = 9. . . . . . . . . . . . . . . . . 123xviiFigure 7.14 The pair creation distribution calculated with NCSMC transition strengthsbut using the bound-bound formula and compared to the ATOMKI data[160]. The calculation are scaled to match data (a): at 65 degrees, (b):at 105 degrees. The dimensionless parameter y (6.111) is integrated from-0.5 to 0.5 as in [122, 123]. The E1 (1−) and M1 (1+) contributions aredisplayed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125Figure 7.15 The pair creation distribution calculating with the full continuum-boundformula and compared to the ATOMKI data [160]. The calculation arescaled to match data (a): at 65 degrees, (b): at 105 degrees. The dimen-sionless parameter y (6.111) is integrated from -0.5 to 0.5 as in [122, 123]. 125Figure 7.16 The comparison between approximate (Γ) and full calculations (σ). In(a,b) y is integrated from -0.5 to 0.5 and in (c,d) y is integrated from−(1− 2meω ) to 1− 2meω . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126Figure 7.17 The integrated cross section of various X17 candidates vs the standardradiative capture cross section. Interferences between γ and X are notincluded. The electron-positron integrated cross section is based on anumerical integral of the bound-bound formula. The coupling constantsare approximated based on [161], which scales the operators based on thestrength of the signal in the 2016/2019 data [160]. . . . . . . . . . . . . 127xviiiList of Abbreviations0νββ neutrino-less double beta decay2νββ two-neutrino double beta decay2BC two-body current3N three-nucleonχEFT chiral effective field theoryBSM beyond the Standard ModelCC Coupled-ClusterCG Clebsch-GordanEFT effective field theoryF Fermi: a type of β-decay with ∆J = 0GT Gamow-Teller: a type of β-decay with ∆J = 1IMSRG In-Medium Similarity Renormalization GroupLEC low-energy constantNCSM No-Core Shell ModelNCSMC No-Core Shell Model with ContinuumNME nuclear matrix elementNN nucleon-nucleonOBDME one-body transition density matrix elementRGM resonating-group methodSD Slater DeterminantSM Standard ModelSRG Similarity Renormalization GroupxixAcknowledgementsI am thankful for the many individuals who helped me through this degree.Of greatest influence was Petr Navra´til, who guided me through many projects, startingwith a co-op position during my undergraduate degree. I am very grateful for Petr’s patienceand mentorship, lending technical expertise when I ran into problems and granting manyopportunities to present my work and collaborate with other experts.Special thanks to Sofia Quaglioni for inviting me to spend two summers as an intern atLawrence Livermore National Lab (LLNL), that time was an excellent start to my degree.I really appreciate the continued support and encouragement.Thank you to Reiner for chairing my committee, as well as to Rob, Mona and Fei fortheir excellent feedback.This work would not be possible without collaboration with other scientists. I am verygrateful to Ayala, Chloe¨, Sam, Michele and many others. I’d like to acknowledge everyone- past and present - in the TRIUMF theory department, especially the nuclear theory post-docs, with whom I could discuss research problems and learned a lot from. Thanks toMatteo, Takayuki, Anna, Mack, Lotta, Mehdi, and especially Angelo for guiding my firstco-op project and explaining much of the nuclear theory basics.Thanks to Olivia, Richard, Eric and Wojtek for encouraging exciting side-projects inquantum computing and machine learning. Thanks to all the theory students I’ve interactedwith: Michael, Antoine, Riku, Callum, Kang, Mirko, David, Lindsay, Charlie and Javier.Shout-out to those I met at LLNL, esp. Evan, Matt, Ahsan, and Yijia and thank you tothe many other people I’ve met along the way, through TRIUMF, the UBC Physics andAstronomy Department, the UBC AMS Waterpolo Club and UBC AMS Triathlon Club.I acknowledge funding from National Science and Engineering Research Council ofCanada, the US Department of Energy and the University of British Columbia. TRIUMFreceives federal funding via a contribution agreement with the National Research Councilof Canada. Computing support came from an INCITE Award at the Oak Ridge LeadershipComputing Facility, from the LLNL Institutional Computing Grand Challenge program,from the Digital Research Alliance of Canada and the Oak cluster at TRIUMF.Thank you to all of my family and friends for their encouragement and support through-out my education. Finally, thank you to Laura, for joining me on this adventure and pushingme through the final stretch, I’ll value your support always.xxChapter 1Introduction1.1 Nuclear Structure and ReactionsSince its discovery by Rutherford in 1911, the atomic nucleus has been at the forefront ofmodern physics. Through increasingly sophisticated experiments, the internal structure ofnuclei has been revealed. The nucleus was discovered to be built of discrete components:the nucleons (protons and neutrons).The number of protons is Z, the atomic number, which controls the chemical propertiesof the atom (in a neutral atom there are also Z electrons). The total number of nucleons isA, the mass number, which differentiates “isotopes” (nuclei with the same Z but differentA). Different isotopes have a differing number of neutrons N (since A = Z + N). Forexample, the element helium (Z = 2) is found in nature as isotopes with N = 2 (4He) andN = 1 (3He).The nuclear chart (Figure 1.1) contains thousands of isotopes, each with its own spec-trum of bound-state energies, shapes and sizes. Most nuclear states are unstable and decayor break-up into smaller nuclei or free nucleons. Decays emit radiation, as the nucleons fallto a lower energy configuration. For example, γ-decays emit photons while β-decays changethe charge state by emitting electrons (or positrons). Emission of an entire 4He nucleus iscalled α-decay. The rates of decay range across many orders of magnitude.Nuclei can also transform through collisions with other nuclei. In the process of scat-tering they may undergo nuclear reactions: capturing nucleons, fusing together or breakingapart, releasing energy in the process. Nuclear reactions power the Sun and all other stars.All the chemical elements in the Earth were created in stars billions of years ago after manychains of reactions, building heavier and heavier nuclei.The use of rare-isotopes has applications in materials science and medicine. For manyyears, it was not possible to explore the vast nuclear landscape and all the possible nuclearreactions that can occur. From the 1980s until the present the acceleration of short-livednuclei has become possible allowing the study of exotic nuclei and nuclear reactions. Rare-1(a)(b)Figure 1.1: (a) The full chart of nuclides showing the observed isotopes arranged ac-cording to the number of protons (Z) and the number of neutrons (N). Theisotopes coloured black do not decay, i.e. they are stable. The other isotopesare coloured according to their most dominant decay mode. Several thousandisotopes are shown but several thousand more are predicted to exist. “Exis-tence” is generally defined as bound with respect to the strong force, i.e. decaysare via electromagnetic or weak forces which act at a much longer time scalecompared to the strong force. (b) A subset of the chart of nuclides, includingonly up to Z = 9 and N = 18. Figures adapted from [5] (which sources datafrom [6])2isotope production facilities have been constructed across the world including TRIUMF(Canada), FRIB (USA), RIKEN (Japan), GSI (Germany), GANIL (France) and ISOLDEat CERN (Switzerland).In experiments, which attempt to emulate the highly ionized and high-temperatureregime of stellar processes, it is extremely difficult to obtain the counts necessary to accu-rately measure nuclear reaction rates. Hence, predictive theories are needed to extrapolatebeyond data. These theories are also needed to explain the vast collection of nuclear struc-ture properties. In the past, simple models have been constructed to describe experimentalobservations. The liquid drop model was developed to account for trends in the bindingenergy of nuclei as a function of (A,Z). The deviations from this model motivated the moresophisticated shell model of Mayer and Jensen for which they won the Nobel prize.A major goal of theoretical nuclear physics is to describe all of these phenomena fromfirst principles. Recent advances in computing have enabled the study of nuclear propertiesin the low- and medium-mass regime, by the so-called “ab initio” methods: the no-coreshell model [7], quantum Monte Carlo [8], coupled-cluster [9], and in-medium similarityrenormalization group [10], to name a few. Their aim is to provide an accurate description ofnuclear properties with controlled approximations and uncertainties starting from nucleonsand the interactions between them.Although the nucleons have internal structure (unlike, for example, the electron whichappears to be an elementary particle), the nuclear interaction can be connected to a funda-mental theory.1.2 Elementary Particles and InteractionsThe Standard Model of particle physics is the quantum field theory that classifies all ofthe known elementary particles and describes three of the four fundamental forces: theelectromagnetic, weak and strong interactions1. This section is a summary, mostly basedon [11, 12], see [13–16] for more details.The particles of the Standard Model (shown in Figure 1.2) are categorized into twotypes, fermions and bosons, which obey different statistics. Fermions (quarks and leptons)make-up the bulk of the matter of the universe and follow Fermi-Dirac statistics. Twoidentical fermions cannot occupy the same quantum state (the Pauli exclusion principle)and hence collections of fermions take up volume. Bosons follow Bose-Einstein statisticsand multiple bosons can occupy the same quantum state. Hence, collections of bosons canadd coherently and act as force carriers between the fermions. Particles are identified asexcitations (or “quanta”) of their corresponding fields (e.g. an electron is an excitation ofthe electron field).1The fourth force, gravity, has not yet been integrated and its inclusion is a very active area of physicsresearch. However it is apparent that gravity acts in a “classical regime” and is not relevant at the energyscales discussed in this thesis.3Standard Model of Elementary Particlesthree generations of matter(fermions)I II IIIinteractions / force carriers(bosons)masschargespinQUARKSu≃2.2 MeV/c²⅔½upd≃4.7 MeV/c²−⅓½downc≃1.28 GeV/c²⅔½charms≃96 MeV/c²−⅓½stranget≃173.1 GeV/c²⅔½topb≃4.18 GeV/c²−⅓½bottomLEPTONSe≃0.511 MeV/c²−1½electronνe<1.0 eV/c²0½electronneutrinoμ≃105.66 MeV/c²−1½muonνμ<0.17 MeV/c²0½muonneutrinoτ≃1.7768 GeV/c²−1½tauντ<18.2 MeV/c²0½tauneutrino GAUGE BOSONSVECTOR BOSONSg001gluonγ001photonZ≃91.19 GeV/c²01Z bosonW≃80.433 GeV/c²±11W bosonSCALAR BOSONSH≃124.97 GeV/c²00higgsFigure 1.2: The Standard Model of elementary particles [17]: the 12 fundamentalfermions and 5 fundamental bosons. Brown loops indicate which gauge bosons(red) couple to which fermions (purple and green). The value for “charge”reflected in figure is the electromagnetic charge. The mass values reflected inthe figure are as of 2019 (periodic reevaluations are carried out by the ParticleData Group, see [18] for the latest).Interactions between different particles can be constructed by combining together thefundamental vertices shown in Figure 1.3. Each vertex involves a coupling constant, andincoming and outgoing particles (fields). The photon field mediates the electromagneticforce and has a non-zero coupling with any field that has electric charge. The W and Zfields mediate the weak force, coupling with all the fundamental fermion fields and eachother. The gluon fields mediate the strong force, coupling with quarks and the gluon fielditself. The Higgs field couples to the masses of the other fundamental fields and itself.In principle, the probability of any process is calculated by summing over all the rele-vant diagrams (with the required incoming and outgoing particles) using the Feynman rulesextracted from the corresponding Lagrangian. In practice, this is only possible when thecoupling constants at the vertices are small (e.g. in electromagnetism), where the contri-butions from complicated diagrams with large numbers of vertices and internal loops aresuppressed. The theory of electron-photon interactions, quantum electrodynamics (QED),4has made the most precise predictions in all of theoretical physics2. The vast fields ofchemistry and materials science depend on these interactions.Figure 1.3: All vertices allowed in the Standard Model: q is any quark, g is a gluon,X is any charged particle, γ is a photon, f is any fermion, H is the Higgs, mis any particle with mass (with the possible exception of the neutrinos), mBis any boson with mass. In diagrams with multiple particle labels separatedby /, one particle label is chosen. In diagrams with particle labels separatedby |, the labels must be chosen in the same order. For example, in the fourboson electroweak case the valid diagrams are WWWW , WWZZ, WWγγ,WWZγ. Lines on the left correspond to incoming particles, while lines on theright are outgoing. The diagrams can be rotated such that an incoming particlebecomes outgoing (or vice versa) resulting in conjugation of the correspondingfield. The conjugate of each listed vertex (reversing the direction of arrows) isalso allowed. Adapted from [20] and [21].2The original theory used only the electromagnetic vertex of Figure 1.3. Modern predictions (as collectedin [19]) include corrections from all the other vertices in the form of internal loops. At low energy, QED canbe considered an effective field theory of the Standard Model.5A fundamental concept in physics is Noether’s theorem, the idea that symmetries in theLagrangian of a theory result in conserved quantities and conversely, every conservation lawreflects an underlying symmetry. For example, energy and momentum conservation corre-spond to symmetries of translation in time and space respectively while angular momentumconservation corresponds to rotational symmetry. In special relativistic theories we alsohave invariance under boosts, i.e. Lorentz invariance. The combination of translations,rotations and boosts is called the Poincare´ group. Particles appear as different representa-tions of the Poincare´ group. The simplest representation are one-component scalar fields,while the next simplest is a four-component Dirac spinor. The four components correspondto two spin states plus the two spin states of a distinct antiparticle. For example, theelectron field is “spin-12” with four components: electrons with spin up and spin down, andanti-electrons (“positrons”) with spin up and down. The spin corresponds to an intrinsicamount of angular momentum. Bosons have integer spin while fermions have half-integerspin.The application of symmetries in particle physics invites the use of group theory whichwill be mentioned only in passing in this thesis. For example, the group of all rotationsin three dimensions is SO(3), it is closely related to the group SU(2) which describes thesymmetry of particle spin3. A spin-12 particle transforms under rotations according to thetwo-dimensional representation of SU(2) (a spin-0 particle would transform under the one-dimensional representation and spin-1 under the three-dimensional, etc).We associate with each particle several “charges”, corresponding to so-called “internalsymmetries” or “gauge symmetries”. For example, electric charge can be associated witha U(1) gauge symmetry, i.e. the dynamics of particles under the electromagnetic forcecan be described by a Lagrangian which is invariant to a U(1) gauge transformation (inaddition to Poincare´ transformations). Additional symmetries one might expect are timereversal, parity (~r → −~r) and charge conjugation (changing the sign of all internal charges,converting each particle into its antiparticle).The forces of the Standard Model are based on symmetries of the gauge group SU(3)C×SU(2)L × U(1)Y . The strong force is invariant under the SU(3)C gauge symmetry whichcorresponds to the so-called “colour-charge” of the quarks and gluons, hence the theory ofthis sector is termed quantum chromodynamics (QCD). The remaining SU(2)L ×U(1)Y istermed the electroweak sector (a unified theory of electromagnetism and weak interactions).The electroweak interaction is complicated, in fact weak processes violate parity andcharge-conjugation (and likely time-reversal) symmetries. The quark and lepton fields arechiral, i.e. they have left-handed and right-handed particles. In the massless limit, chiralityis equivalent to helicity and a right-(left-)handed particle is one whose spin and momen-tum are aligned (anti-aligned). Only left-handed fermions participate in weak interactions3The difference is that a spinor under a rotation of 360◦ differs from the original spinor by a minus sign.A spinor is symmetric under a rotation of 720◦.6and have charges (termed “weak isospin”) under SU(2)L. Both chiralities have a “weakhypercharge” under U(1)Y (which differs between left- and right-handed).The electroweak gauge symmetry should forbid a Dirac mass term in the Lagrangian,which would mix left-handed and right-handed fields, but we observe fermions with massin nature. A mechanism to generate mass for the charged fermions and weak gauge bosonsis provided by the Higgs field. The Higgs field is a complex scalar field and the Lagrangiancontains a peculiar potential. The minimum of the Higgs potential corresponds to a non-zero value of the field and so the local field about this minimum does not have the samesymmetry, one of the two degrees of freedom becomes fixed (this is known as spontaneoussymmetry breaking). The observed W±, Z and photon are linear combinations between theSU(2)L and U(1)Y gauge fields. Interactions with the Higgs field play the role of a Diracmass in the Lagrangian and so the measured masses of the fermions and weak gauge bosonsare a function of coupling strengths and the vacuum expectation value of the Higgs field. Thephoton remains massless as it is the gauge field of a U(1) subgroup of SU(2)L×U(1)Y , theelectric charge is a linear combination of the weak isospin and hypercharge. The discoveryof the Higgs boson in 2012 [22, 23] was a very important confirmation of the StandardModel.So far nuclei and nucleons have not yet been mentioned. They come about as anemergent phenomena of low-energy QCD. The QCD coupling strength between colour-charges does not decrease with distance resulting in the property of confinement, i.e. barequarks are not observable but rather are constituents of colour-neutral composite particles(hadrons). We see in nature combinations of quarks: three quarks (baryons), or a quarkand anti-quark pair (mesons). The physics of nuclei can be calculated at the level of theconstituent nucleons with effective interactions connected to the underlying QCD.1.3 Nucleons and Nuclear ForcesAt low energies relevant for nuclear physics (in the tens of MeV far below the baryon massscale Λχ ≈ 1 GeV), QCD is non-perturbative and thus very difficult to solve. While thetechnique of lattice QCD has been successful in describing hadronic spectra (possible statesand their masses), calculations of two or more baryons are at present basically impossibleusing quarks and gluons as the degrees of freedom [24].Consequently when calculating properties of atomic nuclei, we use nucleons and pions asthe degrees of freedom. Nucleons are the baryons which consist only of the lightest quarksu and d: protons p = uud and neutrons n = udd. Pions are the mesons: pi+ = ud¯, pi− = u¯d,pi0 = 1√2(uu¯− dd¯). The quantum field theory of nucleons and pions is an “effective fieldtheory” (EFT) of QCD.The connection to QCD is achieved via chiral effective field theory (χEFT) [25]. χEFThas the same symmetries as QCD, in particular chiral symmetry which is spontaneouslybroken. In the limit of massless quarks, QCD is a chiral theory in which left- and right-7handed quarks can transform independently under flavour. The non-zero expectation valueof the quark condensate due to the strong gluon interaction spontaneously breaks thissymmetry. The symmetries that remain correspond to conserved isospin (see Section 2.4)and baryon number. Pions are generated in the theory as Nambu-Goldstone bosons [26].χEFT describes interactions between nucleons by a hierarchy of terms consisting ofpion exchanges and contact interactions [27, 28]. The mass of the pion mpi ' 140 MeVgives a “soft” scale, above which there is a mass gap to the next lightest meson, the rho(mρ ≈ 770 MeV), giving a “hard” scale Λχ ≈ mρ. One can perturbatively expand theχEFT Lagrangian in powers of Q ∈ { pΛχ , mpiΛχ }, a simultaneous expansion in terms of theexternal three-momenta |~p| = p and the pion mass [29]. The result is a finite set of diagramsat each order (Q)ν (shown in Figure 1.4). The power of an irreducible diagram involving Anucleons with L loops isν = −4 + 2A+ 2L+∑i∆i , (1.1)where each vertex i has dimension∆i = di +ni2− 2 , (1.2)and di is the number of derivatives and/or pion-mass insertions, while ni denotes the numberof nucleon fields (nucleon legs) involved in vertex i [29, 30].Figure 1.4: Hierarchy of nuclear forces in χEFT from [29]. Solid (dashed) lines denotenucleons (pions). The solid dots, filled circles, filled squares, filled diamonds,and open squares refer to vertices of dimension ∆i = 0, 1, 2, 3 and 4 respectively,as defined in (1.2).8χEFT contains parameters (low-energy constants or LECs) at each vertex, that are inprinciple calculable from QCD. However, at present these parameters are fitted to proper-ties of few-nucleon systems (e.g. nucleon-nucleon scattering and properties (energies anddecay rates) of the deuteron (2H), triton (3H) and Helium-3 (3He). χEFT is systematicallyimprovable as including higher-order terms result in increasingly small corrections. Theimportance of spin-orbit terms and tensor forces arise naturally in nucleon-nucleon (NN)interactions in χEFT. Three-nucleon (3N) and higher-body forces also naturally appear andare required to reproduce experiment.Effective field theory techniques are also used to connect the nucleons to the otherStandard Model interactions. The electric charge of the protons and magnetic moments ofboth nucleons allow the nucleons to interact electromagnetically. Hence, the nucleons innuclei can rearrange to more energetically favourable configurations by emission of photons(γ-decay). The weak interaction allows the conversion of u↔ d quarks and therefore n↔ pnucleons by emission of W bosons (which quickly decay into a lepton and anti-neutrino, i.e.β-decay).The Standard Model (SM) is an extremely precise theory but it does not explain every-thing, for example: gravity, dark matter and the properties of neutrinos. Nuclear physicscan provide a test bed for beyond the Standard Model (BSM) theories as observed de-viations from SM predictions of nuclear properties and reactions would provide evidencefor new physics. However these predictions require reliable and accurate computationaltechniques such as ab initio methods.1.4 The Nuclear HamiltonianAb initio approaches start from a microscopic Hamiltonian of an A-nucleon system whichincludes the kinetic energy of each nucleon and a nuclear interaction. The interaction typ-ically includes nucleon-nucleon (NN) and three-nucleon (3N) terms which are realistic inthe sense that they reproduce few-nucleon data. In principle, higher many-nucleon contri-butions can be included but in general they are not needed.The Hamiltonian is written asH =1AA∑i<j=1(~pi − ~pj)22mN+A∑i<j=1V NNij +A∑i<j<k=1V 3Nijk + · · · , (1.3)where mN is the nucleon mass4 and ~pi is the momentum of the i-th nucleon. The sum overi < j prevents double-counting of identical pairs (i.e.∑i<j =12∑i 6=j). Since nuclei areself-bound, the kinetic energy term that appears is the intrinsic (relative) kinetic energyTrel, i.e. the difference between the total kinetic energy T and the kinetic energy of the4In calculations we take the mass of the proton mp and neutron mn to be equal, and use the averagenucleon mass: mN =12(mp + mn).9center of mass TCoM:Trel =T − TCoM ,1A∑i<j(~pi − ~pj)22mN=∑ip2i2mN− P22mNA, (1.4)where ~P =∑i ~pi is the total momentum. The electromagnetic interaction is included viathe Coulomb force within V NN , and applies only to pairs of protons. The nuclear interactionis more complicated.Since the work of Yukawa in 1935, there has been evidence of the exchange of heavyparticles. Since then, phenomenological forces have been fit to data e.g. meson exchangemodels such as CD-Bonn, Argonne V18, etc. However once QCD was discovered in the1980s, these pictures needed to be connected. The pioneering work of Weinberg in 1990[31] proposed to use χEFT to derive nucleon-nucleon forces from QCD as pion exchanges.Using the underlying theory of QCD would require interactions between the constituentquarks but if we zoom out to the level of nucleons we are able to use an effective fieldtheory. According to Weinberg [32], the steps of building an effective field theory are first,to identify the relevant degrees of freedom (in this case nucleons and pions) and second, towrite down all possible interactions which preserve the underlying symmetries.Nuclear interactions can be systematically derived in the framework of chiral effectivefield theory. The Lagrangian of QCD is expanded at low energy with protons, neutronsand pions as the degrees of freedom and arranged into a power series where the hierarchybetween two-, three- and higher-body forces emerges naturally. The high-energy physicsbeing left out is absorbed into the low-energy constants (LECs) which can be fitted toexperimental data (in principle they could be computed via lattice QCD but despite muchprogress this is currently infeasible).Effective field theories require choices in the regularization scheme and energy cutoff ofthe required integrals. Greater or fewer degrees of freedom can be used (e.g inclusion ofthe excited baryon state ∆ or pion-less schemes). In principle, baryons and mesons withheavier quarks (e.g. strange) can be used, creating “hyper-nucleon” interactions.The parametrization and fitting technique of nuclear forces are not unique. These choicescreate a zoo of theoretically equivalent forces but due to the change in the weighting ofdifferent LECs they may result in better predictions in different nuclei. This creates anobvious source of uncertainty in ab initio predictions but it is difficult to quantify withoutthe ability to perform a single prediction with every available force. In practice to makenew predictions we choose interactions with a good track record. The quantification ofuncertainty due to the choice of nuclear force is a highly active area of research [33]. Inprinciple χEFT will produce many-body interactions beyond three-body, but in this studywe use only up to three body. The contribution of higher-body terms is generally small. Ab10initio calculations of very heavy systems have shown that NN+3N is sufficient [34].The Hamiltonian is non-relativistic and hence we can use the framework of non-relativisticquantum mechanics. The properties of a system can be retrieved from the wavefunction,which is a solution to the Schrodinger equation. A nuclear Hamiltonian must conserve somesymmetries, in particular intrinsic angular momentum and parity. Isospin (Section 2.4) isapproximately conserved. The eigenstates of the Hamiltonian will be simultaneous eigen-states of the corresponding operators and hence the nuclear wavefunction will be labelledby J , pi and MT . The goal of ab initio nuclear theory is to find the nuclear wavefunctionand then expectation values of operators will correspond to different observables.1.5 Applications of Nuclear TheoryAb initio nuclear theory provides insight to explain experimental results and can be com-bined with experiment to make predictions.For example, theoretical calculations have recently been applied to solve the puzzle ofnuclear beta decay quenching. Historically beta decay rates were observed to be systemat-ically smaller than expected based on the rate of decay of free neutrons. This quenchingeffect can be explained by a better description of both the beta decay operator and thenuclear wavefunction. The operator contains a higher-order term in which the weak forceacts on two nucleons simultaneously. Ab initio methods also include stronger correlationsof nucleons compared to phenomenological shell models.An accurate description of the nucleus is needed to find new physics in experimentsinvolving nuclear systems. For example, nuclear structure can influence the distributionof momenta of outgoing electrons in beta decay. New experiments which measure thekinematics of the electrons must take into account these effects.Another example is the hypothetical process of neutrino-less double beta decay. Ifmeasured the neutrino mass can be extracted from the decay lifetime using the nuclearmatrix element from theory. The expected size of the nuclear matrix element influenceswhich nuclei would be good candidates for measuring this yet-unseen decay mode. Theobservation of this process would expand the Standard Model by providing more informationabout neutrinos and could provide part of an explanation for matter-antimatter asymmetryin the universe.Astrophysical quantities like cosmological isotopic abundances also require theoreticalinput. Reaction rates demand integrals over all energies but measurements are limitedto discrete points. Ab initio nuclear theory has the potential to resolve conflicts betweenprevious predictions and measurements of the cosmological lithium abundance.Another cosmological mystery is dark matter. Proposed theories require new particlesboth to form the dark matter and to connect between the Standard Model and dark sector.Recent experiments report seeing the potential signature of a new particle with mass 17MeV (the so-called X17 boson) which decays into an e+e− pair. This particle is emitted11during the decay of a 8Be resonance in proton scattering experiments on 7Li, i.e. theradiative capture reaction 7Li(p, e+e−)8Be. A predictive nuclear model is needed to ruleout a Standard Model effect and justify the claim.1.6 OrganizationThis thesis is organized as follows:• Chapter 2 introduces the mathematical tools necessary to build a nuclear wavefunc-tion. This background provides the basis for the no-core shell model (NCSM) tech-nique which can be used to compute the bound states of many light nuclei.• Chapter 3 connects the Standard Model to expectation values of nuclear operatorsusing nuclear decay as a motivating example.• Chapter 4 describes the technique of Similarity Renormalization Group (SRG) whichis a method in which unitary transformations are used to improve the Hamiltonian.This pre-processing step results in faster convergence of calculations. The NCSMis a computational method which scales poorly with the number of nucleons A andquickly becomes very expensive. While the amount of available computing power iscontinually increasing, the development of techniques which can improve results forthe same amount of resources can be very useful.• Chapter 5 presents the results of NCSM calculations. The NCSM was used to bench-mark more approximate many-body methods, in this case “coupled-cluster” (CC) and“in-medium SRG” (IMSRG). All three ab initio methods were used to calculate thenuclear matrix elements for Gamow-Teller (GT) beta decays across a large range ofnuclear masses and demonstrate that quenching is not observed in ab initio methods.The NCSM calculations demonstrate the power and necessity of SRG. NCSM calcu-lations were also used to compute higher-order terms in the GT beta decay spectrumof 6He→ 6Li. These nuclear structure corrections will appear in future high-precisionexperiments and should not be mistaken for beyond the Standard Model physics. Fi-nally, the NCSM was used again to benchmark CC and IMSRG in the calculation ofneutrino-less double beta decay matrix elements.• Chapter 6 introduces nuclear wavefunctions of unbound (continuum) systems anddescribes the method of no-core shell model with continuum (NCSMC). The NCSM issolely a bound state method and does not have the capability to describe the unboundstates which occur in scattering and reactions. The NCSMC can describe both boundstates and scattering states by including explicit cluster states. The scattering stateamplitudes connect to cross sections of scattering processes and radiative capturereactions. The theory of electron-positron pair production in radiative capture ispresented and compared to pair production in bound-state decays.12• Chapter 7 presents the application of NCSMC to the radiative capture reactions4He(d, γ)6Li and 7Li(p, γ)8Be with extension to 7Li(p, e+e−)8Be. The first reactionhas applications to 6Li production in big bang nucleosynthesis.The second and thirdmay be used to benchmark experiments in the search for BSM physics, in particularthe claimed X17 boson.13Chapter 2Many-Body CalculationsNote: The mathematical expressions in this chapter and the following will use natural unitsi.e. ~ = c = 1. This means that masses are measured in MeV (rather than MeVc2). Distancesare sometimes quoted in fm= 10−15m (femtometre is often called “fermi” in nuclear physicsafter physicist Enrico Fermi). Momenta are quoted in either MeV or fm−1, the conversionfactor is ~c =197.327 MeV fm [19].2.1 The Quantum Harmonic OscillatorA very simple picture of the nucleus is of independent nucleons influenced by a centralpotential. (One can imagine this coming about as the averaged interaction with all the othernucleons). One model of the mean nuclear potential is the harmonic oscillator potential:V (r) ∝ r2 = ~r · ~r. This picture is clearly not the full story but introducing the process andsolutions with this potential is a useful step towards a more realistic model.We want to solve the time-independent Schrodinger equation:Hψ = Eψ , (2.1)where we have the Hamiltonian, with kinetic energy T and potential energy V :H =T + V ,H =A∑i=1[p2i2mN+12mNΩ2r2i], (2.2)with mN the nucleon mass and Ω the harmonic oscillator frequency. Each particle labeledby i is independent in this model and solves a single-particle Hamiltonian:H =p22mN+12mNΩ2r2 . (2.3)The momentum is calculated through the gradient operator in quantum mechanics (~p→14−i~∇) and so the momentum squared gives the Laplacian, i.e.p22mN→ − ∇22mN, (2.4)∇2 = 1r2∂∂r(r2∂∂r)+1r2(1sin θ∂∂θ(sin θ∂∂θ)+1sin2 θ∂2∂φ2),=1r2∂∂r(r2∂∂r)− L2r2, (2.5)where ~L is the angular momentum operator:~L = ~r × ~p = −i~r × ~∇ = −i(φˆ∂∂θ− θˆ 1sin θ∂∂φ). (2.6)The Hamiltonian is spherically symmetric and does not dependent on angles. Thisadmits the separable solutions:ψ(~r) = Rn`(r)Y`m(θ, φ) ≡ ϕn`m(~r) , (2.7)where the angular wavefunction Y`m are the standard spherical harmonics [35]:Y`m(θ, φ) =√(2`+ 1)4pi(`− |m|)!(`+ |m|)!eimφPm` (cos θ) , (2.8)and Pm` are the “associated Legendre polynomials”. The spherical harmonics are eigen-functions of the angular momentum operators:L2Y`m =`(`+ 1)Y`m , (2.9)LzY`m =mY`m . (2.10)The ` determines the parity of the state, by pi = (−)` . ` can be zero or any positive integerand for each value of ` there are 2`+ 1 possible values of m (−` ≤ m ≤ `).The remaining piece is the radial Schrodinger equation:{12mNr2[− ddr(r2ddr)+ `(`+ 1)]+mNΩ2r22}Rn`(r) =En`Rn`(r) , (2.11)which has another set of solutions related to generalized Laguerre polynomials (A.32) [36].n is any positive integer (or zero).The radial and angular wavefunctions are normalized such that they form orthonormalbases of functions of their respective coordinates, i.e.∫r2drRn′`(r)Rn`(r) = δnn′ (2.12)15and ∫d cos θdφY ∗`′m′(θ, φ)Y`m(θ, φ) = δ``′δmm′ . (2.13)The combined solutions are labeled as |n`m〉, where〈~r|n`m〉 = ϕn`m(~r) . (2.14)They are the eigenstates of the Hamiltonian operator:H |n`m〉 =En` |n`m〉 , (2.15)and are orthonormal, i.e.〈n′`′m′∣∣n`m〉 = 〈n′`′m′∣∣ (∫ d3r |~r〉 〈~r|) |n`m〉=∫d3~rϕ∗n′`′m′(~r)ϕn`m(~r)=δnn′δ``′δmm′ . (2.16)The energy depends only on the so-called “principal” quantum number N˜ = 2n + `,En` = EN˜ =(N˜ + 32)Ω. For reference, we can calculate the energy by evaluating theexpectation value of the Hamiltonian by (2.15) or via an integral i.e.〈n′`′m′∣∣H |n`m〉 =∫ d3rϕ∗n′`′m′(~r)(− ∇22mN + 12mNΩ2r2)ϕn`m(~r)=En`δnn′δ``′δmm′ . (2.17)The two components of this Hamiltonian are very similar:〈n′`′m′∣∣T |n`m〉 =δ``′δmm′([2n+ `+32]δnn′ +(δn,n′−1 + δn,n′+1)√(n+ 1)(n+ `+32))Ω2,(2.18)〈n′`′m′∣∣V |n`m〉 =δ``′δmm′([2n+ `+32]δnn′ −(δn,n′−1 + δn,n′+1)√(n+ 1)(n+ `+32))Ω2.(2.19)This potential creates a “ladder” of equally separated states (“rungs” or “shells”) whichthe nucleons can occupy. A model of the nucleus can be made by filling harmonic oscillatorstates, the most naive “shell model”.This filling must be limited by the fact that protons and neutrons are fermions with spin12 i.e. nucleons obey the Pauli exclusion principle (Fermi-Dirac statistics). For example,we can have 4 particles in the N˜ = 0 shell (two protons and two neutrons, one of each has16spin +12 and −12). When a shell is filled like this, it is referred to as “closed” and energy isrequired to add a nucleon as it must go to the next higher shell. 4He has this property andis more tightly bound than 5He or 5Li (both of which are in fact unbound).The particular combinations of Z, N that are strongly bound compared to their neigh-bours in the nuclear chart are known as the “magic numbers” (2, 8, 20, 28, etc.). Withrespect to Z these correspond to helium, oxygen, calcium, nickel, etc. These are analogous tothe numbers of electrons which fill the shells responsible for the periodic table in chemistry(2, 10, 18, 36, etc. correspond to the noble gases). These experimentally observed numberswere a motivation for the introduction of a “spin-orbit” term, described in Section 2.2.The exact energy differences between shells depends on the shape of the nuclear poten-tial. Historically phenomenological mean field models have been created, for example theWoods-Saxon potential [37], and were fitted to observed nuclear properties.To create a truly first-principles approach, motivated by the Standard Model, our goalis to build the nucleus from the NN+3N interactions. The harmonic oscillator states ϕn`mare still very useful as we can use a linear combination (of products) of these states as thesolution to the true Hamiltonian (as we’ll see in Section 2.7).2.2 Spin and Angular Momentum CouplingNucleons are spin-12 particles, so a nucleon wavefunction should include a spinor χ, i.e.ψ(~r, σ) =ϕn`m`(~r)χsms(σ) = 〈~r|n`m`〉 〈σ|sms〉 , (2.20)where the spinor is an eigenfunction of the spin operators:S2 |sms〉 =s(s+ 1) |sms〉 , (2.21)Sz |sms〉 =ms |sms〉 . (2.22)We expect that the Hamiltonian is symmetric under rotations and reflections and sothe total angular momentum and parity of a nuclear state, Jpi, should be conserved. Thetotal angular momentum operator is ~J = ~L + ~S. The orbital angular momentum ` addsto the spin s such that the state has a total angular momentum quantum number j (thiscorresponds to the notation (`s)j). We should therefore use states with good j, created bycombinations of (2.20) with the following expression (dropping n for brevity):|(`s)jm〉 =∑msm`〈`m`sms|(`s)jm〉 |`m`〉 |sms〉=∑msm`(`m`sms|jm) |`m`〉 |sms〉 . (2.23)The coefficients (`m`sms|jm) are known as the Clebsch-Gordan (CG) coefficients.17The observed magic numbers motivated the presence of an attractive, single-particlespin-orbit term in the nuclear potential [38], i.e.Vso = −v(r)~L · ~S . (2.24)The dot product between the spin and orbital momentum can be calculated via2~L · ~S |n`sjm〉 = (J2 − L2 − S2) |n`sjm〉= [j(j + 1)− `(`+ 1)− s(s+ 1)] |n`sjm〉= |n`sjm〉 ·` s = 12 , j = `+ 12(−(`+ 1)) s = 12 , j = `− 12 . (2.25)This shifts the energy of states with the highest ` and j in a shell down towards the nextlower shell. If the coefficients vn` =∫drr2Rn`(r)Rn`(r)v(r) are strong enough (En` =EN˜ − vn` < ~L · ~S >), the experimentally observed magic numbers can be reproduced. Forexample, the harmonic oscillator alone produces the magic numbers: 2, 8, 20, 40, 70, etc.The spin-orbit term pushes states from n = 0, ` = 3, j = 72 down, with sufficient energy gapthat these 8 states are labelled by a new magic number. States from higher energy levelsare also rearranged, with the result: 2, 8, 20, 28, 50, etc. [39].This addition to the nuclear potential was enough to make the single-particle shell modelpredictive, in the sense that the Jpi of the ground state of a nucleus can be accuratelypredicted simply by filling the shells. However there remain many exceptions and theexcited states of nuclei are not as easily predicted. To calculate these energies we need towork with a more realistic inter-nucleon interaction.Since the interaction should depend on the relative position of pairs of particles ~r1−~r2,we would like to use two-nucleon wavefunctions. We can again use the CG coefficients tocouple the individual angular momenta j1 and j2 to create states with good total angularmomentum j, i.e.|(j1j2)jm〉 =∑m1m2(j1m1j2m2|jm) |j1m1〉 |j2m2〉 . (2.26)For 3N interactions we expect to need three-particle wavefunctions. The total operatorwould be ~J = ~J1 + ~J2 + ~J3 which can be coupled by successive coupling of first ~J12 = ~J1 + ~J2and then ~J = ~J12 + ~J3, i.e.|((j1j2)j12j3)jm〉 =∑m3m12(j12m12j3m3|jm) |(j1j2)j12m12〉 |j3m3〉=∑m3m12(j12m12j3m3|jm)[ ∑m1m2(j1m1j2m2|j12m12) |j1m1〉 |j2m2〉]|j3m3〉 .(2.27)18But this is not unique, we could also couple ~J1 + ~J23 or ~J13 + ~J2. The values of the quantumnumber j do not depend on the coupling order but the states corresponding to differentcoupling schemes are not the same [40]. To convert between these different bases we needthe Wigner 6j symbol (A.11), the array with curly braces below:|(j1(j2j3)j23)jm〉 =∑j12(−)j1+j2+j3+j jˆ12jˆ23{j1 j2 j12j3 j j23}|((j1j2)j12j3)jm〉 . (2.28)2.3 Spherical Tensor OperatorsThe observable properties of a quantum state are extracted through the expectation valuesof operators i.e. the matrix element. For example, for an operator which depends only onthe spatial coordinates, we have〈n′`′m′∣∣O |n`m〉 =∫ d3rϕ∗n′`′m′(~r)O(~r)ϕn`m(~r) . (2.29)The Hamiltonian is the operator corresponding to the energy. For example, single-nucleon energies come from the matrix elements ((2.17) again)〈n′`′m′∣∣H |n`m〉 =En`δnn′δ``′δmm′ . (2.30)It useful to transform any operator with dependence on the nucleon coordinates into anexpansion over the spherical harmonics, i.e.O(~r) =∑λµOλµ(~r) =∑λµOλ(r)Yλµ(θ, φ) . (2.31)Each component Oλµ is a component of a spherical tensor of rank λ. The Hamiltonianis a scalar (rank 0), since it has no dependence on the angular coordinates.Another example is the Cartesian position vector ~r = (x, y, z) which can be convertedinto the so-called “covariant” components r± = ∓ 1√2 (x± iy) and r0 = z [40]. We haverµ =√4pi3rY1µ(θ, φ) (2.32)where µ = 0,±1 and r = |~r|. Therefore, the spherical components of the position vectorform a spherical tensor of rank 1.The Wigner-Eckart theorem [35] is a procedure which separates out the projection µdependence to simplify calculations. We obtain the relation:19〈j′m′∣∣Oλµ |jm〉 = (−)2λjˆ′−1 (jmλµ|j′m′) 〈j′∣∣ |Oλ| |j〉 , (2.33)where the “hat” notation means jˆ =√2j + 1 and the quantity 〈j′| |Oλ| |j〉 is the reducedmatrix element (which contains the physics of the matrix element and is rotationally in-variant). The CG coefficient immediately gives us selection rules, i.e. the matrix elementis zero unless ∆(j′λj) (the triangle condition: |j′ − j| ≤ λ ≤ j′ + j) and m+ µ = m′.Complicated operators such as the electromagnetic interaction will be decomposed asa series of spherical tensor operators, as will be shown in Section 3.1. However, someoperators are already a single irreducible tensor. For example, the identity 1 is a scalar,angular momentum ~J is a vector and Yλµ alone is a component of a rank λ tensor.It is useful to list the reduced matrix elements of these [40]:〈j′∣∣ |1| |j〉 =δjj′ jˆ , (2.34)〈j′∣∣ |J | |j〉 =δjj′ jˆ√j(j + 1) , (2.35)〈l′∣∣ |YL| |l〉 = (−)l′ lˆ′Lˆlˆ√4pi(l′ L l0 0 0), (2.36)〈12∣∣∣∣ |S| ∣∣∣∣12〉=√32, (2.37)where in (2.36) appears the 3j symbol (A.10).As an example, (2.37) can be determined by combining (2.22) and (2.33), i.e.〈s′m′∣∣Sz |sm〉 =mδss′δmm′ (2.38)=(sm10|s′m′)sˆ〈s′∣∣ |S| |s〉 . (2.39)Setting m = s = s′ then:s =(ss10|ss)sˆ〈s| |S| |s〉 , (2.40)and for s = 12 : 〈12∣∣∣∣ |S| ∣∣∣∣12〉= s · sˆ · 1(ss10|ss) =12·√2 · 1√13=√32. (2.41)A standard notation for s = 12 is to relate~S = ~σ2 , where σi are the standard Pauli matrices20(see Section A.1). The result is 〈12∣∣∣∣ |σ| ∣∣∣∣12〉=√6 . (2.42)When using a coupled basis, we may still need to compute expectation values of operatorswhich are also coupled, i.e.Oλµ = [Oλ1 ⊗Oλ2 ]λµ =∑µ1µ2(λ1µ1λ2µ2|λµ)Oλ1µ1Oλ2µ2 . (2.43)The reduced matrix elements, where Oλ1 acts on the j1, j′1 space and Oλ2 acts on the j2, j′2space, are calculated via [41]〈j′1j′2; j′∣∣ |Oλ| |j1j2; j〉 =jˆ′λˆjˆj′1 j′2 j′j1 j2 jλ1 λ2 λ〈j′1∣∣ |Oλ1 | |j1〉 〈j′2∣∣ |Oλ2 | |j2〉 . (2.44)The array in brackets is the Wigner 9j symbol (A.13). When λ = 0 then 9j reduces to a6j (see A.14), i.e.〈j′1j′2; j′∣∣ |[OLPL]0| |j1j2; j〉 =δj′j (−)j′2+j′+j1{j′1 j′2 j′j2 j1 L}〈j′1∣∣ |OL| |j1〉 〈j′2∣∣ |PL| |j2〉 .(2.45)An example is the spin-orbit term in the Hamiltonian, which is a scalar product betweenvectors:〈`′s′; j′∣∣ ∣∣∣[~L · ~S]0∣∣∣ |`s; j〉 =δj′j (−)s′+j′+`{`′ s′ j′s ` 1}〈`′∣∣ |L| |`〉 〈s′∣∣ |S| |s〉 . (2.46)We may also have operators which act only on a subspace. For example, we may need tocalculate just σ for state |(`s)j〉 by using the same formula (2.44), setting Oλ1 = 1 (λ1 = 0,λ = λ2 = 1) (again a zero in the 9j gives a 6j (A.14)). Therefore:〈`′12; j′∣∣∣∣ |σ| ∣∣∣∣`12; j〉=〈12∣∣∣∣ |σ| ∣∣∣∣12〉δ``′ jˆjˆ′ (−)`′+j′+ 32{1212 1j j′ `}=√6δ``′ jˆjˆ′ (−)`′+j′+ 32{1212 1j j′ `}. (2.47)One last (and defining) property of spherical tensors is how they transform under rota-tions. A rotation in three-dimensional Cartesian space induces a unitary transformation onangular momentum eigenstates |jm〉 in the Hilbert space of dimension 2j + 1 [41].21The matrix elements of such a transformation (for a rotation R about the Euler anglesα, β and γ) are the Wigner D-matrices, i.e.〈jm′∣∣R(α, β, γ) |jm〉 ≡Djm′m(α, β, γ) . (2.48)State vectors transform according toR|jm〉 =∑m′Djm′m∣∣jm′〉 , (2.49)and correspondingly the spherical tensor operators which act on those state vectors trans-form asROλµR† =∑m′Oλm′Dλm′µ . (2.50)See Section A.3 for more details.2.4 IsospinBecause of the very close masses of the proton and neutron, these particles can be modeledas the two projections of a two state system, i.e. a spinor χ. This approximate symmetryis known as isospin. The proton and neutrons are eigenstates of the isospin operator τwith quantum number t = 12 . Our convention is that protons have projection mt =12 andneutrons mt = −12 .Analogous to Section 2.3, operators can be separated into spherical tensor operatorsacting in isospin space. Rank-0 operators are known as “isoscalar” and have no isospindependence. Rank-1 operators that depend on τ (τz or τ±) are “isovector”.A single-nucleon basis therefore consists of the states〈~rστ |n`jmmt〉 =ϕn`jmmt(~r, σ, τ)=∑m`ms(`m`sms|jm)ϕn`m`(~r)χsms(σ)χtmt(τ) , (2.51)where s = 12 and t =12 is implicit in the first line. In the following sections we often use acollective index α = {n`jmmt}.Spin-12 particles transform under rotations in physical space according to the two-dimensional representation of the group SU(2). In the case of isospin-12 , the particlestransform under rotations in an abstract “isospin space” [11]. This is an internal symmetrycorresponding to the invariance of the nuclear force with respect to quark “flavour” i.e. thetype of quarks engaged in an interaction. Interactions between any pair of quarks or linearcombinations of quarks are identical. The full symmetry would be the group SU(6) (asthere are 6 quark flavours) but this symmetry is explicitly broken by the widely differing22masses of the quarks. The up and down quarks are sufficiently close in mass that they form avery good approximation of the subgroup SU(2). We can assign u(d) the isospin projection+12(−12), hence the combinations p = uud and n = udd have +12 and −12 respectively.Spin and isospin both combine according to the rules of angular momentum additionusing the CG coefficients (2.23). To be explicit, for spin-12 , let |↑〉 =∣∣s = 12 ,m = 12〉 and|↓〉 = ∣∣s = 12 ,m = −12〉. With two spinors (χ(1)χ(2)) we have the four combined states:|↑↑〉, |↑↓〉, |↓↑〉 and |↓↓〉. They are eigenstates of the combined spin projection operatorSz = S(1)z +S(2)z =12(σ(1)z + σ(2)z), i.e. they have the eigenvalues 1, 0, 0, and -1 respectively(as m = m1 +m2).The total spin corresponds to the operator:S2 =(~S(1) + ~S(2))·(~S(1) + ~S(2))=(S(1))2+(S(2))2+ 2~S(1) · ~S(2) . (2.52)The first two terms return eigenvalues of the total single-particle spin and will each bealways 34 e.g. (S(1))2χ(1)χ(2) =(34χ(1))χ(2) = 34χ(1)χ(2). The last term can be calculatedvia ~S(1) · ~S(2) = 14(σ(1)x σ(2)x + σ(1)y σ(2)y + σ(1)z σ(2)z), using the action of the Pauli matrices(Section A.1) which may flip the spins: σx(↑, ↓) = (↓, ↑), σy(↑, ↓) = (−i ↓, i ↑) and σz(↑, ↓) = (↑,− ↓). For example:~S(1) · ~S(2) |↑↓〉 = 14[(σ(1)x ↑)(σ(2)x ↓)+(σ(1)x ↑)(σ(2)x ↓)+(σ(1)x ↑)(σ(2)x ↓)]=14[(↓)(↑) + (i ↓)(−i ↑) + (↑)(− ↓)]=14[2 |↓↑〉 − |↑↓〉] . (2.53)It follows for the other states:~S(1) · ~S(2)|↑↑〉|↑↓〉|↓↑〉|↓↓〉 = 14|↑↑〉2 |↓↑〉 − |↑↓〉2 |↑↓〉 − |↓↑〉|↓↓〉 . (2.54)|↑↑〉 and |↓↓〉 are eigenstates of the total spin, e.g. S2 |↑↑〉 = 14 (3 + 3 + 2) |↑↑〉 = 2 |↑↑〉. Theyare therefore spin-1 states (|sm〉 = |11〉 and |1− 1〉 respectively) as s(s+1) = 2 =⇒ s = 1.The remaining two states must be mixed to create eigenstates, i.e. |10〉 = 1√2(|↑↓〉+ |↓↑〉)(spin-1) and |00〉 = 1√2(|↑↓〉 − |↓↑〉) (spin-0). The necessary coefficients in the linearcombinations, coupling two spin-12 to spin-1 or spin-0, are exactly the CG coefficients:(12m112m2|sm). A pair of spin-12 particles therefore can form four orthogonal states whichare eigenstates of the total spin, a symmetric triplet with total 1 and an antisymmetricsinglet with total 0.An identical argument follows for isospin, i.e. s → t, σ → τ (τi are the same Pauli23matrices as σi for spin but acting in isospin space). A pair of isospin-12 particles can bearranged into an isospin-1 symmetric triplet and an isospin-0 antisymmetric singlet. Com-bined with spin, two nucleons have sixteen possible states. However, due to the necessaryanti-symmetry of fermions, if we have a symmetric function of the spatial coordinates, theallowed states can only be in the combinations (symmetric spin, anti-symmetric isospin)or (anti-symmetric spin, symmetric isospin) e.g. (spin 1, isospin 0) or (spin 0, isospin 1).The explicit condition for anti-symmetry of two-nucleon states is (−)`+s+t = −1. Likewisethree-nucleon states have spin and isospin of 12 or32 and must be anti-symmetrized.2.5 Many-Body StatesThe naive way to model a system of A particles would be to assemble a many-bodystate as the direct product of single-particle wavefunctions ϕαi(~xi) (αi = ni`ijimimti,~xi = ~riσiτi) i.e. φα1···αA(~x1, · · · , ~xA) = 〈~x1 · · · ~xA|α1 · · ·αA〉 = ϕα1(~x1) · · ·ϕαA(~xA). How-ever this method cannot be correct as the states do not have the anti-symmetric ex-change property of fermions. To enforce the Pauli principle we must have φα1α2(~x1, ~x2) =−φα1α2( ~x2, ~x1). Two fermions should not be in the same state or at the same position, i.e.φαα(~x1, ~x2) = 0 = φα1α2(~x, ~x).We can enforce the Pauli principle by assembling states |φα〉, normalized such that〈φα|φα〉 = 1, through Slater determinants, i.e.〈~x1 · · · ~xA|φα〉 = 〈~x1 · · · ~xA|α1 · · ·αA〉SD =1√A!∣∣∣∣∣∣∣∣ϕα1(~x1) · · · ϕαA(~x1).... . ....ϕα1(~xA) · · · ϕαA(~xA)∣∣∣∣∣∣∣∣ . (2.55)Slater determinant (SD) states allow us to use the formalism of creation and annihilationoperators (“second quantization”). The algebraic relations of these operators preciselycapture the desired anti-symmetry.We start with a vacuum state |0〉 (no particles). The operator a†i creates a state whichhas a particle with index i (i.e. with the quantum numbers αi). However if that index isalready occupied we get zero. Similarly the operator ai destroys a particle with index i butreturns zero if that index is already empty. This is summarized bya†i |0〉 = |i〉 ,ai |i〉 = |0〉 ,a†i |i〉 =0 ,ai |0〉 =0 ,(2.56)with the algebraic properties:{ai, aj} = 0 ,{a†i , a†j}= 0 ,{ai, a†j}=δij , (2.57)where the curly braces denote the anti-commutator: {a, b} = ab+ba. The Slater-determinant24state is exactly recovered by successive application of creation operators, i.e.|α1 · · ·αA〉SD =a†α1 · · · a†αA |0〉 . (2.58)The calculation of many-body operators is greatly simplified by this formalism. If wehave a spherical tensor operator which acts on the coordinates of a single nucleon, we canevaluate the expectation value for the entire nucleus by summing over all of the single-nucleon expectation values. The rank and projection of an operator Oλµ are transferred toa pair of creation and annihilation operators, i.e.Oλµ =∑αβ〈α|Oλµ |β〉 a†αaβ , (2.59)Oλ =−∑ab〈a| |Oλ| |b〉 λˆ−1(a†aa˜b)λ, (2.60)where the labels a, b include n`j (in contrast to α, β: n`jm). A phase is added to theannihilation operator such that it obeys the correct tensor transformation properties: a˜jm =(−)j−m aj−m [42].If we have the SD wavefunctions |ψf 〉 and |ψi〉 then the reduced many-body transitionmatrix element is simply〈ψf | |Oλ| |ψi〉 = −∑ab〈a| |Oλ| |b〉 λˆ−1 〈ψf |∣∣∣(a†aa˜b)λ∣∣∣ |ψi〉 . (2.61)The 〈ψf |∣∣∣(a†aa˜b)λ∣∣∣ |ψi〉 term is referred to as the one-body transition density matrix element(OBDME). The only components of the OBDME which will be non-zero are those in which|ψi〉 and |ψf 〉 don’t have state b occupied and have state a empty respectively.Some operators are more complicated and intrinsically depend on the coordinates ofmore than one nucleon at a time. One clear example is double-beta decay (explained inSection 3.2.3) In this case there are two protons in the final state where there were twoneutrons in the initial state. Such operators are two-body operators and are computed via〈ψf |Oλµ |ψi〉 =∑αβγδ〈αβ|Oλµ |γδ〉 〈ψf | a†αa†βaδaγ |ψi〉 . (2.62)These can be computed with coupled pairs of nucleons via the two-body density matrixelements (TBDME) 〈ψf |∣∣∣∣[[a†aa†b]Jab [a˜ca˜d]Jcd]λ∣∣∣∣ |ψi〉, i.e.〈ψf | |Oλ| |ψi〉 = −14∑abcdDabDcdλˆ−1 〈ab; Jab|∣∣∣Oλ∣∣∣ |cd; Jcd〉 〈ψf | ∣∣∣∣[[a†aa†b]Jab [a˜ca˜d]Jcd]λ∣∣∣∣ |ψi〉 ,(2.63)where Dab =√1 + δab appears because |ab; Jab〉 are orthonormalized [43].25Similarly for three-body operators (in particular the 3N force), we have〈ψf |Oλµ |ψi〉 =∑αβγδκ〈αβγ|Oλµ |δκ〉 〈ψf | a†αa†βa†γaκaaδ |ψi〉 .It is far too memory intensive to store three-body densities and so the matrix elements aregenerally computed “on-the-fly”.2.6 Intrinsic CoordinatesThe nuclear Hamiltonian should describe the physics intrinsic to the nucleus. The harmonicoscillator Hamiltonian (2.2) has an external potential which cannot be physical. The trueHamiltonian (1.3) must use coordinates which are measured from the center of mass i.e.~ri → ~ri − ~R (where ~R = 1A∑i ~ri) and ~pi → ~pi − 1A ~P (where ~P =∑i ~pi). Alternativelythe Hamiltonian can use relative coordinates (i.e. T that depends on pi − pj and VNN thatdepends on ri−rj). The nuclear force should naturally depend only on the relative positionsof nucleons and cannot be easily calculated otherwise as the center of mass position is notknown a priori.However we can still use the harmonic oscillator states to build up the nuclear wavefunc-tion because they have very useful transformation properties. The center of mass depen-dence factors exactly [44]. We may use harmonic oscillator states which are functions of therelative coordinates. We choose to use the following set of so-called “Jacobi” coordinates:~ξ0 =√1A[~r1 + · · ·+ ~rA] =√A~R , (2.64)~ξ1 =√12[~r1 − ~r2] ,~ξ2 =√23[12(~r1 + ~r2)− ~r3],...~ξA−1 =√A− 1A[1A− 1 (~r1 + ~r2 + · · ·+ ~rA−1)− ~rA]. (2.65)Figure 2.1: Schematic diagram of the Jacobi coordinates with three particles.A many-particle state may be created simply by coupling harmonic oscillator states26that depend on each coordinate i.e. ϕn1`1m1(~ξ1)ϕn2`2m2(~ξ2) · · ·ϕnA−1`A−1mA−1(~ξA−1). Forexample, a two particle basis may use only the ξ1 coordinate and coupled spin and isospin,i.e.〈~ξ1σ1σ2τ1τ2∣∣∣n`sjmtmt〉 = ∑m`ms(`m`sms|jm)ϕn`m`(~ξ1) (χ(σ1)χ(σ2))sms (χ(τ1)χ(τ2))tmt .(2.66)Additional particles have a coordinate that depends on their position relative to the centerof mass of the other particles. ~ξ2 adds dependence on ~r3 but relative to12(~r1 +~r2) as shownin Figure 2.1. For three particles, we have to couple ϕn`sj(~ξ1) and ϕNLJ (~ξ2), i.e.〈~ξ1~ξ2σ1σ2σ3∣∣∣(n`sjNLJ )NJM〉 =∑mM(jmJM|JM)ϕn`sjm(~ξ1)ϕNLJM(~ξ2) (2.67)where N = 2n+ `+ 2N +L. Implicitly, we are adding along with ~ξ2 another spin-12 for σ3and we should also couple isospin:〈τ1τ2τ3∣∣(t12)T〉.In order to maintain anti-symmetry we need a specific combination of states, i.e.|NiJT 〉 =∑(n`sjt;NLJ ||NiJT ) |n`sjt;NLJ ; JT 〉 (2.68)where we have another set of numbers (n`sj;NLJ ||NiJT ), the “coefficients of fractionalparentage” (CFPs). There is a straightforward algorithm for constructing states of manyparticles with Jacobi coordinates and obtaining the CFPs [45]. An A-body state is a simpleextension onto an (A−1)-state. However, as A increases this becomes much more expensivethan a SD basis (and is not often done at A ≥ 7 [46]). The set of indices i increases withN .The connection between Slater determinant and Jacobi-coordinate bases comes from theTalmi-Moshinsky transformation [44]. With two nucleons the relation is∑m1m2(`1m1`2m2|LM)ϕn1`1m1(~r1)ϕn2`2m2(~r2) =∑n`mNLM(n`NLL|n1`1n2`2L)µ2µ1× (`mLM |LM)ϕn`m(~ξ1)ϕNLM (~ξ0) .(2.69)The quantities (n`NLL|n1`1n2`2L)µ2µ1are the so-called harmonic oscillator brackets for twoparticles of mass ratio µ2µ1 (in this case 1) [47].We have a choice of which basis to use, single-nucleon SD or Jacobi. In either case wehave to solve an equivalent equation.272.7 Solving the Schrodinger EquationWe may calculate the wavefunction of a nucleus by using an ansatz which is a linear com-bination of an (infinite) orthonormal basis, i.e.|ψ〉 =∑αcα |φα〉 , (2.70)where 〈φα|φβ〉 = δαβ.We may find an approximate solution by using a finite basis. This can be done bysolving the eigenvalue problem:H |ψλ〉 =Eλ |ψλ〉 , (2.71)where we must calculate all the matrix elements:Hβα = 〈φβ|H |φα〉 . (2.72)Through the variational principle, adding additional states to the basis will alwaysimprove the agreement between the lowest energy and the true ground state (if the trialwavefunction has any overlap with the ground state). The ground state |ψ0〉 correspondsto the set of coefficients c(0) which minimize the energy, i.e.E0 = 〈ψ0|H |ψ0〉=∑βc(0)∗β 〈φβ|H∑αc(0)α |φα〉=∑αβc(0)∗β c(0)α 〈φβ|H |φα〉 . (2.73)The next excited state has the minimal energy of states formed from the space orthogonalto the ground state. The next has minimal energy orthogonal to the first two. And so on...2.8 The No-Core Shell ModelThe no-core shell model (NCSM) is an ab initio technique which uses an expansion over amany-body harmonic oscillator basis to construct a many-nucleon wavefunction [7].The goal is to find the solutions of the Schrodinger equation (1.3) (i.e. the eigenvec-tors of H (2.71)). These solutions will be eigenstates with good JpiMT quantum numbers(approximate T ) and eigenvalues Eλ, i.e.∣∣∣ΨJpiTMTAλ 〉 = |AλJpiTMT 〉 , (2.74)28whereH |AλJpiTMT 〉 =Eλ |AλJpiTMT 〉 . (2.75)The strategy of the NCSM is to find the coefficients of an ansatz wavefunction. The basisis made up of SD states of single particle harmonic oscillator states or harmonic oscillatorstates with Jacobi coordinates. The eigenstates are a sum over configurations, i.e.|AλJpiTMT 〉 =Nmax∑Nex=0∑αcJpiTMTλNexα|ANexαJpiTMT 〉 . (2.76)The label Nex denotes the total number of excitations of nucleons above the lowest Pauli-allowed configuration. As the nucleons are in harmonic oscillator states, the excitation levelfor the i-th nucleon corresponds to its principal quantum number: N˜i = 2ni + `i. Theminimum configuration has Nex = Nmin (Nmax = 0) determined by the Pauli principle, e.g.in 4He all nucleons may be in N˜ = 0, therefore Nmin = 0, but for6Li, there must be at leastone proton and one neutron in N˜ = 1 therefore Nmin = 2 (see Figure 2.2). HenceNex =A∑iN˜i −Nmin . (2.77)The label α = {α1 · · ·αA} denotes the configuration of nucleons with that Nex. Forexample, Nex = 2 includes all possible combinations of two nucleons excited by +1 leveland one nucleon excited by +2. The sum over Nex is restricted by parity to either an evenor odd sequence. The number of configurations grows exponentially with Nex and A. Thisis the main limitation for computation.Figure 2.2: Truncation diagram for the NCSM. As an example, the solid circles in-dicate the lowest Pauli configuration of 6Li. Here “N” corresponds to theharmonic oscillator excitation level (N˜ in the text).The matrix elements of the kinetic energy can be easily computed analytically using(2.18). The matrix elements of the interaction from chiral effective field theory are computedvia numerical integrals of the radial harmonic oscillator wavefunctions e.g. 〈n′`′|V |n`〉 =29∫r2drR∗n`(r)V (r)Rn`(r). A useful property of harmonic oscillator states is that they areeigenfunctions of the Fourier transform [48], sometimes the matrix elements are easier tocalculate in momentum space i.e. 〈n′`′|V |n`〉 = ∫ p2dpR∗n`(p)V (p)Rn`(p).There is a compromise in the choice of basis between the scaling of the basis size withA versus the ease of removal of the center of mass. The Jacobi ξ are defined in an anti-symmetric way and therefore an A-nucleon eigenstate described by A − 1 intrinsic coor-dinates can be directly related to an eigenstate expanded in the Slater determinant basisusing (2.69), omitting spin and isospin, i.e.〈~r1 · · ·~rA|ψ〉SD =〈~ξ1 · · · ~ξA−1∣∣∣ψ〉ϕ000(~ξ0) . (2.78)When using the Jacobi coordinates, center of mass motion does not contribute at all tothe eigenenergies or eigenstates. However when using SD states, the excitation of the centerof mass state ϕNLM (ξ0) can create spurious eigenstates that can appear at similar energyto real states. To remove the ambiguity between physical and spurious states the methodof Lawson projection is used [49]. A term is added to the Hamiltonian controlled by theparameter β ∼ 1− 10 which pushes eigenstates with the center of mass in an excited stateto a higher energy above the physically relevant states, i.e.H =Trel + V + β(HCoM − 32Ω), (2.79)whereHCoM =P 22mNA+mNAΩR22, (2.80)Trel is defined in (1.4) and V consists of the remaining terms in (1.3).The main computational hurdle of the NCSM is the diagonalization of a very largesparse matrix. The Lanczos procedure is an effective tool as it finds the extremal eigen-values (and corresponding eigenstates) first (which are most physically relevant) [50]. TheHamiltonian is block-diagonal. For each JpiT block, we have a matrix with respect to the|Nα〉 (|NαJpiT 〉) states, schematically:H =〈00|H |00〉 〈00|H |10〉 〈00|H |11〉 · · · 〈00|H |Nmaxαmax〉.... . ....〈Nmaxαmax|H |Nmaxαmax〉 . (2.81)The expansion has two input parameters which must be chosen to do a calculation,Nmax and the oscillator frequency Ω. The predictions of any NCSM calculation will havesome theoretical uncertainty based on the variation with respect to these parameters. Fora given A value there will be a value of Ω which gives the lowest ground state energy.30Generally the largest possible Nmax is used, but by computing at a succession of values wecan determine whether our calculation is converged. An uncertainty of the ground stateenergy can be given based on an ad hoc extrapolation to an infinite basis by fitting theparameters (E∞,a,b) of an exponential function, i.e.E(Nmax) = E∞ + ae−bNmax . (2.82)A fundamental limitation of the NCSM is that for finite Nmax the asymptotics of weaklybound states cannot be correctly reproduced and the properties of continuum states (E > 0)cannot be calculated. This is because the states of the harmonic oscillator states fall tozero very quickly past the oscillator length. This is addressed with the extension NCSMCdescribed in Section 6.2.2.9 Static Nuclear Properties and Many-Body OperatorsThe wavefunction of the nucleus contains all the information needed to predict observablequantities. Each observable property corresponds to the expectation value of an operator.One example is the size of the nucleus. The spatial extent of a single particle can bedirectly determined from the projection of the particle’s state on the spatial coordinates,i.e. 〈r|i〉 = ψi(r). The mass density distribution should be simply ψ∗(~r)ψ(~r), normalizedsuch that 〈ψ|ψ〉 = ∫ d3rψ∗(~r)ψ(~r) = 1. To determine, for example, the average radius weinsert the squared-radius operator:< r2 >= 〈ψ| r2 |ψ〉 =∫d3rψ∗(~r)r2ψ(~r) . (2.83)The average radius is r¯ =√< r2 >, as we require a scalar operator r2 = ~r · ~r to calculate ascalar quantity. < ~r > will give the dipole moment (for protons).For a many-body state |φα〉, the notation 〈φα|φα〉 implies that we integrate over all thenucleon coordinates (~xi = ~riσiτi) i.e.∫dξ =∫ ∏Ai d3ri∑ms,mtsuch that〈φα|φα〉 =∫dξφ∗α(~x1, · · · , ~xA)φα(~x1, · · · , ~xA) = 1 . (2.84)In the impulse approximation, we also attach to any operator a sum, the matter densityoperator ρm(~r) =∑Ai δ(~r − ~ri) and integrate over d3r. A many-nucleon operator is thenreduced to a sum of single-nucleon operators. The normalization factor is the mass number,i.e.〈φα| ρm(~r) |φα〉 =∫dξ∫d3rφ∗α(~x1, · · · , ~xA)A∑jδ(~r − ~rj)φα(~x1, · · · , ~xA) = A . (2.85)31The matter radius of a nucleus is:〈φα| r2m |φα〉 =1A∫dξ∫d3rφ∗α(~x1, · · · , ~xA)r2A∑jδ(~r − ~rj)φα(~x1, · · · , ~xA) , (2.86)using the operator:r2m =1A∫d3rA∑ir2δ(~r − ~ri) . (2.87)where ~r is taken relative to the center of mass. It is common to write this directly in theformr2m =1AA∑ir2i . (2.88)If we want to count only the protons (neutrons) we use the isospin projection operator1+τzi2 (1−τzi2 ). The point proton and neutron radius operators are thenr2p =1ZA∑i1 + τzi2r2i , (2.89)r2n =1NA∑i1− τzi2r2i . (2.90)In the above, for the operators we should have used coordinates with respect to thecenter of mass i.e. ~ri − ~R. Typically these are rewritten into a two-body form proportionalto ~ri − ~rj , which are then used with the two-body transition densities [51], e.g. the matterradius (2.88) would ber2m =1A∑i(~ri − ~R)·(~ri − ~R)=1A2∑i<j(~ri − ~rj)2 . (2.91)We would then integrate over only the internal coordinates, i.e.∫dξ =∫ ∏A−1i=1 d3ξi∑ms,mt.Some operators can be easily converted to two-nucleon relative coordinates. Differentoperators may have a different scaling factors depending whether they depend on the spatialcoordinates versus for example spin [51]. It depends whether we are summing A nucleons,A(A−1)2 pairs of nucleons orA(A−1)(A−2)6 triplets.The orbital angular momentum operator with coordinates defined with respect to the32center of mass can be converted as~L =∑i(~ri − ~R)× (~pi − 1A~P )=1A∑i<j(~ri − ~rj)× (~pi − ~pj) . (2.92)However, when the single-nucleon spin operator ~s is upgraded to a two-nucleon coupledspin, we need a different A-dependent scaling factor, i.e.~S =∑i~si=1A− 1∑i<j(~si + ~sj) . (2.93)General operators (i.e. spherical tensor operators) which depend on the single-nucleoncoordinates (and momenta) can be evaluated with Jacobi coordinate basis states by usingtranslationally invariant transition densities and replacing the coordinates in tandem withapplying appropriate scaling (when using a harmonic oscillator basis). The standard SDmatrix elements (2.61) contain center of mass contamination, i.e. the matrix elements:SD 〈AλfJf | |Oλ| |AλiJi〉SD =−1λˆ∑1,2〈n1l1j1| |Oλ(~r, σ)| |n2l2j2〉× SD 〈AλfJf |∣∣∣(a†n1l1j1 a˜n2l2j2)λ∣∣∣ |AλiJi〉SD . (2.94)However, eigenstates in the SD basis may still be used with the matrix elements of theeigenstates in the Jacobi basis and translationally-invariant densities, to compute the trans-lationally invariant matrix elements:〈AλfJf | |Oλ| |AλiJi〉 =− 1λˆ∑〈nlj|∣∣∣∣∣Oλ(−√A− 1A~ξ, σ)∣∣∣∣∣ ∣∣n′l′j′〉×(Mλ)−1nljn′l′j′,n1l1j1n2l2j2× SD 〈AλfJf |∣∣∣(a†n1l1j1 a˜n2l2j2)λ∣∣∣ |AλiJi〉SD , (2.95)where ~ξ is short-hand for ~ξA−1 (2.65) and the sum is over all quantum numbers which donot appear on the left. The transformation matrix Mλ is defined in [52] and combines withthe OBDME to form a translationally invariant quantity. The general strategy is to replace~r →√A−1A~ξ and ~∇r → −√A−1A~∇ξ.332.10 Charge and Current Densities of the NucleusTo calculate the charge density distribution ρ(~r) of the nucleons, we again use 1+τzi2 to selectonly the protons, i.e. with the operator:ρ(~r) =eA∑i=11 + τzi2δ(~r − ~ri) . (2.96)For generality we can evaluate the expectation of the operator acting between two differentnuclear eigenstates |i〉 and |f〉 (which are linear combinations of the basis states |φα〉), i.e.〈f | ρ(~r) |i〉 =eA∑j1 + τzj2∫dξψ∗f (~x1, · · · , ~xA)δ(~r − ~rj)ψi(~x1, · · · , ~xA) . (2.97)The total charge is normalized to Z, i.e.〈f | ρ |i〉 =∫d3r 〈f | ρ(~r) |i〉 = Zδfi . (2.98)The nucleons are moving within the nucleus. This creates a charged “convection current”density ~jc which can be summarized via the velocity distribution of the protons, using~v = ~pmN → −i~∇mN, i.e.~jc(~r) =e∑i1 + τzj2(~piδ(~r − ~ri) + δ(~r − ~ri)~pi2mN). (2.99)Because ~p and δ do not commute, we must use the symmetric combination [53, 54]. Theresult for many-nucleon states is〈f |~jc(~r) |i〉 =− ie∑j1 + τzj2mN∫dξψ∗f (~x1, · · · , ~xA)~∇jψi(~x1, · · · , ~xA) . (2.100)There are corrections to the current due to the presence of charged exchange particles(pions) in χEFT which will be discussed later (Section 3.1.2 and Section 3.2.1).The divergence free part of the current can be separated out and described as the curlof a magnetization, i.e.~jc(~r) =~j(~r) + ~∇× ~µ`(~r) , (2.101)defining ~j = ~jc− ~∇×~µ`, where ~∇·~j = ~∇·~jc. The magnetization comes from the componentof the protons motion contributing to their angular momentum i.e. it is proportional to the34orbital angular momentum operator ~L and can be written as~µ`(~r) =e2mNA∑i1 + τzi2(~Liδ(~r − ~ri) + δ(~r − ~ri)~Li2). (2.102)The nucleons also have intrinsic magnetic moments due to their internal structure.Part of the magnetization is simply a sum over the individual magnetic moments: protons~µp = gpµN~σ and neutrons ~µn = gnµN~σ, proportional to spin with the corresponding gfactors: gp = 2.793 and gn = −1.913, in units of the nuclear magneton µN = e2mN , i.e.~µs(~r) =A∑iµiδ(~r − ~ri)~σi , (2.103)µi =µN(1 + τzi2gp +1− τzi2gn). (2.104)The intrinsic magnetization is a clear example of the isoscalar/isovector breakdown of anoperator. It can be rearranged, i.e.µi =µIS + µIVi , (2.105)µIS =12(gp + gn) , (2.106)µIVi =τzi2(gp − gn) . (2.107)The total electromagnetic current operator is then~J (~r) =~jc(~r) + ~∇× ~µs(~r) = ~j(~r) + ~∇× (~µs(~r) + ~µ`(~r)) . (2.108)The charge and current determine the interaction between the nucleus and the elec-tromagnetic field. As will be discussed in Section 3.1, a nucleus may change state byinteraction with the electromagnetic field. The charge and current distribution can be de-composed into spherical tensor operators (multipoles) as in (2.31) and their interaction witha classical electromagnetic field defines the static electric (E) and magnetic (M) momentoperators [54]:MEλµ(~r) =rλYλµ(Ωr)ρ(~r) , (2.109)MMλµ(~r) =(~µs(~r) +1λ+ 1~r ×~jc(~r))· ~∇rλYλµ(Ωr) . (2.110)The multipole order is limited to λ ≥ 0 for E and λ ≥ 1 for M .35Chapter 3Electroweak Decays of NucleiAccording to Fermi’s Golden Rule [11], the rate of a decay is proportional to the productof the “amplitude”M for the process (corresponding to a nuclear matrix element) and theavailable phase space volume (all possible final states and combinations of momentum).The general formula for the decay rate Γ of an initial particle of mass MI to N particles is:Γ1→N =S2MI∑¯i∑f∫|M|2N∏F=1d3pF(2pi)32EF, (3.1)where i and f label available internal states of the initial particle I and final outgoingparticles F , respectively. S is the symmetry factor that corrects for double-counting ofidentical particles in the final state, it gets a factor 1/n! for each group of n identicalparticles [11].The bar on the first sum∑¯means an average, as an initial state may be in superpositionof quantum numbers. For example, in an unpolarized collection of spin-12 particles, theprojections +12 and −12 are equally likely and so we average by dividing by two. Forparticles with total angular momentum J the projection M may vary from −J to J and so∑¯M =12J+1∑M .The infinitesimal phase space volume for a particle of 4-momentum P = (E, ~p) is d4P(2pi)4.The 4-momentum phase-space is reduced to 3-momentum by the constraint that initial andfinal momenta be “on-shell” (P 2 = m2 = E2 − p2 and E ≥ 0), i.e.∫d4P(2pi)42piδ(E2 − p2 −m2)θ(E) =∫dEd3p(2pi)42pi2√p2 +m2δ(E −√p2 +m2)=∫d3p(2pi)32E. (3.2)The amplitude (or matrix element) will depend on the interaction of the initial and finalquantum fields of the process. For forces that can be described by a perturbative theory,the interactions can be understood as a hierarchy of diagrams as laid out by Feynman36(originally for QED [55]). The Feynman rules can be derived from the Standard Model (orEFT) Lagrangian. Each particle is represented by a line and the interactions are representedby a vertex (the connection between lines). A simplified version of the rules is as follows:1. For each incoming particle we insert a wavefunction |I〉. For each outgoing particlewe have a conjugate wavefunction 〈F |.• leptons (such as electrons and neutrinos) are described by Dirac spinors us(p)with spin s• photons are described by a polarization vector ~eλ(~q)2. At each vertex we insert the coupling constant and operator corresponding to theinteraction. In addition we have a δ over the 4-momenta going into and out of thevertex. Some examples:• For elementary particles the vertex is extremely simple, e.g. for an electroninteracting with a photon we have ieγµ where e is the charge coupling constant(e2 = 4piα, α ' 1137), γµ is a Dirac gamma matrix• For nuclei we have a more complicated operator. As we will see in Section 3.1,the interaction of a nucleus and a photon can be wrapped into the nuclear currentoperator Jµ(q). In principle, the current should have additive contributions fromeach nucleon but the vertices should take into account the internal structure [53].See Section B.3 for more details.3. Finally, if necessary, we add a propagator for particles corresponding to any internallines and integrate over the internal phase space.• For photons, insert − iηµνq2and integrate over∫ d4q(2pi)4The vertex factors arise from interaction terms in the Lagrangian of a theory while thepropagators come from the “free” terms [11].3.1 Electromagnetism: γ DecayψIψFγPIPFQFigure 3.1: The Feynman diagram and kinematics for photon emission.37Electromagnetic interactions are relevant to nuclear reactions (especially radiative cap-ture) but here we first present the formalism for bound-bound decays.In a gamma decay we have a one particle in the initial state (the initial nucleus) andtwo particles in the final state (a different nuclear state and a photon). The leading order(“tree level”) Feynman diagram corresponding to this process is shown in Figure 3.1. Aphysical (massless) photon with 4-momentum Q = (ω, ~q) is limited by the condition |~q| = ωand has two possible polarizations λ = ±1.The γ-decay rate for a nucleus at rest with mass MI (PI = (MI ,~0)) isΓ =12MI∑¯i∑f∑λ∫ ∣∣∣MλQFI ∣∣∣2 d3q(2pi)32ω d3pF(2pi)32EF . (3.3)The rate depends on an average over possible initial states and a sum over the possible finalstates (including an integral over the final phase space).We have the initial state with the nucleus in state |I〉 and no photons |0〉. The final stateis |F 〉 with a photon state |Qλ〉. The strength of the interaction between the photon fieldand the nucleus is the expectation value between these states of the overlap of quantumfields over all space, i.e.MλQFI = 〈F ;Qλ|∫d4xAµ(x)J µ(x) |I; 0〉 . (3.4)Due to gauge invariance we can set the scalar part of the radiation field to 0, i.e.choose A = (0, ~A). J = (ρ, ~J ) is the nuclear current operator which extracts the dis-tribution of electromagnetic charge and currents in the nucleus. Since the states can befactorized (〈F | 〈Qλ| and |I〉 |0〉), the operators act separately on the nuclear and photonspaces (〈Qλ|Aµ(x) |0〉 and 〈F | J µ(x) |I〉). The above assumes first-order perturbation the-ory because the coupling constant (the proton charge) e is small. Here e is absorbed intothe definition of the nuclear current. Because the quantum states are eigenstates of theirrespective Hamiltonians, the time dependent parts of the wavefunctions are exponential:ψa(~x, t) = 〈~x|a〉 e−iEat, and so the integral over time will give a δ for the energies i.e.∫dteiωt = 2piδ(ω). Therefore:MλQFI =∫dteiEF teiωte−iEI t∫d3x 〈F | 〈Qλ| ~A(~x) · ~J (~x) |I〉 |0〉=2piδ(EF + ω − EI)∫d3x 〈Qλ| ~A(~x) |0〉 · 〈F | ~J (~x) |I〉 . (3.5)The photon field has the form [13, 56]:~A(~x) =∫d3k(2pi)32ωk∑λ′(~eλ′(~k)ei~k·~xaλ′(~k) + ~e∗λ′(~k)e−i~k·~xa†λ′(~k)), (3.6)38where we have the creation and annihilation operators a†λ, aλ with commutation relationsfor bosons:[aλ(~q), a†λ′(~k)]= 2ωqδλλ′(2pi)3δ3(~q − ~k) with ωq = ω = |~q| and ωk = |~k|. ~eλ(~k)are polarization vectors with λ = ±1. They satisfy ~eλ · ~e∗λ′ = δλλ′ and ~e∗λ = (−1)λ~e−λ. Thefinal state is then 〈Qλ| = 〈~qλ| = 〈0| aλ(~q) and only the a† term of ~A survives, i.e.〈~qλ| ~A(~x) |0〉 = 〈0| aλ(~q)∫d3k(2pi)32ωk∑λ′~e∗λ′(~k)e−i~k·~xa†λ′(~k) |0〉=~e∗λ(~q)e−i~q·~x . (3.7)The nuclear states can be factorized into an intrinsic state and a center of mass wave[53], i.e.|I〉 = |PI , i〉 =√2EIei~pI ·~R |i〉 ,|F 〉 = |PF , f〉 =√2EF ei~pF ·~R |f〉 , (3.8)where the normalization treats the nucleus as a generalized Dirac spinor. The matrixelement of 〈F | ~J (~x) |I〉 is evaluated as an integral over the spatial center of mass coordinates~R and all of the nucleon intrinsic coordinates summarized by ξ. Inserting (3.7) and (3.8)the integral in (3.5) is∫d3x 〈~qλ| ~A(~x) |0〉 · 〈F | ~J (~x) |I〉=∫d3x∫dξ∫d3R〈F∣∣∣ξ, ~R〉 [e−i~q·~x~e∗λ · ~J (~x)] 〈ξ, ~R∣∣∣I〉=√2EI2EF∫dξ∫d3R 〈f |ξ〉 e−i~pF ·~R[∫d3xe−i~q·~x~e∗λ · ~J (~x)]ei~pI ·~R 〈ξ|i〉 . (3.9)We may change variables ~x′ = ~x − ~R to evaluate the matrix element with operators ofintrinsic coordinates (as in Section 2.9 and Section 2.10). The current operator should betranslationally invariant and the integration over ~R returns δ, i.e.∫dξ∫d3R 〈f |ξ〉 e−i~pF ·~R[∫d3x′e−i~q·(~x′+~R)~e∗λ(~q) · ~J (~x′ + ~R)]ei~pI ·~R 〈ξ|i〉=∫dξ 〈f |ξ〉[∫d3Re−i(~pF− ~pI+~q)·~R] [∫d3x′e−i~q·~x′~e∗λ(~q) · ~J (~x′)]〈ξ|i〉=(2pi)3δ(~pI − ~pF − ~q) 〈f |~e∗λ(~q) · ~J (~q) |i〉 . (3.10)The matrix element contains the standard nuclear current transition operator (the Fouriertransform of the spatial charged current density operator of Section 2.10):~J (~q) =∫d3re−i~q·~r ~J (~r) . (3.11)39Inserting (3.9) and (3.10) into (3.5) we have the amplitude for gamma-decay:MλQFI =(2pi)4δ4(PI − PF −Q)√2EI2EF 〈f |~e∗λ(~q) · ~J (~q) |i〉 . (3.12)We use the property [δ(x)]2 = δ(x)(2pi)4and EI = MI and the decay rate becomes:Γ =12MI∑¯i∑f∑λ∫(2pi)4δ4(PI − PF −Q)2MI2EF∣∣∣〈f |~e∗λ(~q) · ~J (~q) |i〉∣∣∣2 d3q(2pi)32ω d3pF(2pi)32EF=∑¯i∑f∑λ∫ ∣∣∣〈f |~e∗λ(~q) · ~J (~q) |i〉∣∣∣2 d3q(2pi)32ω2piδ(MI − EF − ω) . (3.13)3.1.1 Multipole Decomposition of Electromagnetic CurrentThe electromagnetic current operators J (~q) = (ρ, ~J ) (sometimes called “form factors” or“response functions”), needed to evaluate the rate, are the Fourier transforms of the spatialcharge and current distributions of the nucleus (ρ(~r) and ~J (~r) defined in Section 2.9). Thecharge density operator is defined byρ(~q) =∫d3re−i~q·~rρ(~r) . (3.14)The plane wave e−i~q·~r has a straight-forward decomposition into spherical tensor operators,i.e.e−i~q·~r =4pi∑JM(−i)JjJ(qr)YJM (Ωr)Y ∗JM (Ωq) (3.15)with J ≥ 0, −J ≤M ≤ J . To simplify later equations we combine the Bessel function andspherical harmonic into the useful quantity:MJM (q, ~r) = jJ(qr)YJM (Ωr). (3.16)The charge density operator is thenρ(~q) =4pi∑JM(−i)JY ∗JM (Ωq)∫d3rMJM (q, ~r)ρ(~r) (3.17)=4pi∑JM(−i)JY ∗JM (Ωq)CJM (q) , (3.18)which defines the “Coulomb multipole operator”CJM (q) =∫d3rMJM (q, ~r)ρ(~r) . (3.19)40This is simplified by setting ez = qˆ (then only M = 0 contributes as YLM (θq = 0) =Lˆ√4piδM0), i.e.ρ(q) =√4pi∑J(−i)J JˆCJ0(q); . (3.20)This also means that the operators don’t need to be written in terms of ~q, but depend onq = |~q|.A vector operator like the current can be spanned by the polarization vectors, includingthe longitudinal, ~e0 = ~ez, in addition to the transverse, ~e±1 = ∓ 1√2 (~ex ± i~ey). Using thisbasis, the current can be decomposed as~J (q) =∑λJ (q, λ)~e∗λ . (3.21)The components of the current density operator are thenJ (q, λ) =~eλ · ~J (q)=∫d3re−i~q·~r~eλ · ~J (~r)=∫d3r(4pi∑JM(−i)JjJ(qr)YJM (Ωr)Y ∗JM (Ωq))~eλ · ~J (~r) . (3.22)The spherical vector harmonics, which are defined as~YJLM (Ω) =∑mλ(Lm1λ|JM)YLm(Ω)~eλ , (3.23)can be combined with the Bessel functions to form~MJLM (q, ~r) =jL(qr)~YJLM (Ωr) , (3.24)and then with the vector current, to obtain the quantity:MJLM (q) =∫d3rjL(qr)~YJLM (Ωr) · ~J (r) . (3.25)Using the inverse relation for vector spherical harmonics:YJM (Ω)~eλ =∑Lm(JM1λ|Lm) ~YLJm(Ω) , (3.26)41the components can be simplified (preemptively swapping J ↔ L), i.e.J (q, λ) =4pi∑LM(−i)LY ∗LM (Ωq)∫d3rjL(qr)(∑Jm(LM1λ|Jm) ~YJLm(Ωr))· ~J (~r)=4pi∑LM(−i)L LˆδM0√4pi∑Jm(LM1λ|Jm)∫d3r ~MJLm(q, ~r) · ~J (r)=√4pi∑LJ(−i)LLˆ (L01λ|Jλ)MJLλ(q) . (3.27)We have two cases: λ = 0 and λ = ±1. For λ = 0 we haveJ (q, 0) =√4pi∑LJ(−i)LLˆ (L010|J0)MJL0(q) . (3.28)Alternatively, we can usee−iqz =√4pi∑J=0(−i)J JˆMJ0(q, ~r) , (3.29)thene−i~q·~r~e0 =e−iqz~e0=iq~∇e−iqz=√4pi∑J=0(−i)J Jˆ iq~∇MJ0(q, ~r) . (3.30)The result isJ (q, 0) =√4pi∑J(−i)J JˆLJ0(q) , (3.31)where we have the “longitudinal multipole operator”LJM (q) =∫d3riq(~∇MJM (q, ~r))· ~J (r) . (3.32)For λ = ±1 we do the sum over L, which is limited to {J−1, J, J+1} by the CG coefficients42(evaluated via (A.9)), i.e.J (q, λ) =√4pi∑J=1[(−i)J−1 ˆ(J − 1) (J − 101λ|Jλ)MJJ−1λ(q)+ (−i)J Jˆ (J01λ|Jλ)MJJλ(q) + (−i)J+1 ˆ(J + 1) (J + 101λ|Jλ)]=√2pi∑J=1(−i)J[i√J + 1MJJ−1λ(q)− λ√2J + 1MJJλ(q)− i√JMJJ+1λ(q)].(3.33)An equivalent relation ise−i~q·~r~eλ =√2pi∑J=1(−i)J(i√J + 1 ~MJJ−1λ(q, ~r)− λ√2J + 1 ~MJJλ(q, ~r)− i√J ~MJJ+1λ(q, ~r)).(3.34)Using the identity [56]:1q~∇× ~MJJM =− i√J2J + 1~MJJ+1M + i√J + 12J + 1~MJJ−1M , (3.35)the expression is further simplified, i.e.J (q, λ = ±1) =√2pi∑J=1(−i)J Jˆ (T EJλ − λT MJλ ) , (3.36)which defines the “transverse electric multipole operators”:T EJM (q) =∫d3r1q(~∇× ~MJJM (q, r))· ~J (r) (3.37)and the “transverse magnetic multipole operators”:T MJM (q) =∫d3r ~MJJM (q, r) · ~J (r) . (3.38)Real photons are limited to λ = ±1. The relevant operator for photon emission (as in(3.13)) is then〈f |~e∗λ · ~J (~q) |i〉 = 〈f | (−1)λ~e−λ · ~J (~q) |i〉=− 〈f |∑J(−i)J√2piJˆ[T EJ−λ(q) + λT MJ−λ(q)] |i〉 . (3.39)433.1.2 ApproximationsIn the long-wavelength approximation (qr 1), the Bessel functions can be expanded asj`(qr) =(qr)`(2`+ 1)!!{1−12(qr)22`+ 3+ · · ·}, (3.40)where (2`+ 1)!! denotes a “double factorial” (i.e. for odd n: n!! = n(n− 2) · · · 3 · 1).This leads to a decomposition in terms of the multipole operators we already defined inSection 2.9 (integrating 2.109 and 2.110):MEJM =A∑ie1 + τzi2rJi YJM (Ωi) , (3.41)MMJM =A∑i(2µi ~Si +1J + 1µN1 + τzi2~Li)· ~∇i(rJi YJM (Ωi)). (3.42)The Coulomb operators can then be approximated byCJM (q) ≈ qJ(2J + 1)!!MEJM . (3.43)Another approximation uses the conservation of vector current:~∇ · ~J =dρdt. (3.44)Substituting the time evolution operator, evaluated between energy eigenstates, we have<dρdt>= < −i [H, ρ] >= −i < ωρ > , (3.45)where ω = Ef − Ei and < · · · >= 〈f | · · · |i〉. This is known as Siegert’s theorem [57] andwith this relation the longitudinal multipoles can be replaced. First using integration byparts to move the gradient, i.e.LJM (q) = iq∫d3r~∇MJM (q, r) · ~J (r)=− iq∫d3rMJM (q, r)~∇ · ~J (r)=− ωqCJM (q) . (3.46)The transverse electric can first be directly related to the longitudinal, i.e.T EJM (q) =√J + 1JLJM (q)− i√2J + 1JMJJ+1M (q) . (3.47)44Then making use of Siegert’s theorem, and neglecting the second term of (3.35) (in whichthe integral (3.25) will have an extra factor of qr), we haveT EJM (q) ≈−ωq√J + 1JCJM (q) (3.48)=ωqJ−1(2J + 1)!!√J + 1JMEJM . (3.49)The transverse magnetic operator can be approximated asT MJM (q) ≈qJ(2J + 1)!!√J + 1JMMJM . (3.50)In all of the above we have focused on the tree-level diagram of a photo-nucleus inter-action. However there are higher order diagrams. The size of the electromagnetic couplinge makes any additional photon vertices small and therefore we neglect them in this study.But there are additional diagrams coming from the χEFT description of the nuclei. Thephoton is not simply interacting with the nucleons but also with the charged pions beingexchanged. Siegert’s theorem helps with this for the electric multipoles, as the pions willcontribute to the current J and the continuity equation (3.44) makes the matrix elementonly depend on the charge ρ. However for magnetic multipoles we do expect contributionfrom meson-exchange currents (MECs). There has been some effort in the literature toderive and compute the two-body operators corresponding to the MECs (e.g. [58]). Theoverall effect is on the order of 10% in some nuclei.In this study we include only E1, M1 and E2 and neglect two-body exchange currents.3.1.3 Final Formula and Selection RulesIf we insert (3.39) into (3.3), the rate of decay of an excited nuclear state isΓ =e22Ji + 1∑Mi∑Mf∑λ∫d3q(2pi)32ω2piδ(MI − EF − ω)×∣∣∣∣∣〈JfMf | √2pi∞∑J=1(−i)J Jˆ [T EJ−λ(q) + λT MJ−λ(q)] |JiMi〉∣∣∣∣∣2, (3.51)where the coupling constant is explicitly removed from the current operators. The integralmeasure is d3q = ω2dωdΩ and so we can use the δ to integrate dω, i.e.∫ω2dω2ωδ(MI − EF − ω) = ω2fr , (3.52)where on the right ω is fixed to the value MI−EF and fr is the recoil factor fr = (1− ωMF )−1(which takes into account the motion of the final nucleus EF 'MF + P2F2MF).45CG coefficients are inserted in order to use the reduced transition matrix elementsT σJ = 〈Jf | |T σJ | |Ji〉 where the σ and σ′ can take the values 0 or 1, corresponding to electricor magnetic respectively. The sum over the Ms in the CGs gives δJJ ′ , using (A.7) to reorderthe CGs, i.e.∑Mi∑Mf∑λ∣∣∣∣∣〈JfMf |∑J(−i)J Jˆ [T EJ−λ(q) + λT MJ−λ(q)] |JiMi〉∣∣∣∣∣2=∑Mi∑Mf∑λ(∑Jσ(−i)J Jˆ Jˆ−1f (JiMiJ − λ|JfMf )λσT σJ)(∑J ′σ′(−i)J ′ Jˆ ′Jˆ−1f(JiMiJ′ − λ|JfMf)λσ′T σ′J ′)∗=∑λ∑Jσ∑J ′σ′∑Mi∑Mf(−)Ji−M1 JˆfJˆ(JfMfJi −Mi|J − λ) (−)Ji−M1 JˆfJˆ ′(JfMfJi −Mi|J ′ − λ)× (−i)J Jˆ Jˆ−1f λσT σJ (−i)−J′Jˆ ′Jˆ−1f λσ′T σ′∗J ′=∑λ∑Jσσ′λσ+σ′T σJ Tσ′∗J ′=2∑Jσ|T σJ |2 (3.53)where the sum over λ fixes σ = σ’, i.e.∑λλσ+σ′= 2δσσ′ . (3.54)After integrating over the angles∫dΩ = 4pi and replacing e2 = 4piα, the final rate isΓfi =8piαω2Ji + 1fr∑Jσ|T σJ |2 . (3.55)The selection rules for a decay (displayed in Table 3.1) are enforced by the reducedmatrix elements. A decay will be dominated by the transition type furthest to the left inthe table. Higher multipoles are “forbidden” in the sense that their rates are suppressedby additional factors of ω2. For example, 1− → 0+ will be an “allowed” E1 transition andforbidden for M2 (still possible but suppressed). Similarly, for 1+ → 0+, M1 transitions areallowed and E2 are forbidden.σJE1 M1 E2 M2∆J 0, 1 (except 0→ 0) 0, 1, 2 (except 0→ 0, 1, 12 → 12)∆pi yes no yesTable 3.1: The selection rules for allowed γ-decay transitions. For a given ∆J and∆pi the furthest left multipole in the table will be dominant.463.2 Weak Interactions: β DecayA γ-decay changes only the internal configuration of the nucleus (observed by a change ofthe energy and possibly angular momentum and parity) but it remains the same isotope.A β-decay is a type of process which changes the nucleon make-up i.e. (A,Z)→ (A,Z±1),changing the isotope but remaining an “isobar”. A neutron within the nucleus decays toa proton by emission of an electron and anti-neutrino. The electron is referred to as a β−particle. In some nuclei it is possible to have β+ decay where a proton decays to a neutronby emission of a positron and neutrino. In this thesis we reduce the scope to β−decays.A β-decay is a 1 → 3 decay (corresponding to the Feynman diagram in Figure 3.2a),i.e. we have the rateΓ =12MI∑¯i∑f∑ss¯∫ ∣∣Mss¯F I(K)∣∣2 d3k(2pi)32E d3k¯(2pi)32E¯ d3PF(2pi)32EF , (3.56)where K = k + k¯ with k for the electron and k¯ for the anti-neutrino. The amplitude mustcontain δ(PF +K−PI) which can be used to do the integral over d3PF . To get the rate, wesum over the outgoing lepton spins and integrate over their momenta (usually neglecting theneutrino mass). This is a somewhat involved process which will not be described here, see[11] for details. The kinematics of an analogous process (electron-positron pair production)will be discussed in Section 6.4.The Standard Model describes β decay through an interaction between quarks andleptons via W boson exchange as shown in Figure 3.2b. The flavour of the quarks changesand the W carries the resulting difference in charge. The weak interaction is described by aso-called “V-A” theory meaning that the interactions have both vector (V) and axial-vector(A) contributions.We can successively apply EFT to simplify this diagram. First, we can consider the Was acting on the nucleon as a whole rather than at the quark level (Figure 3.2c). The vertexfactor for a W−lepton or W−quark interaction contains the “V-A” operator γµ(1−γ5). Fornucleons this must be upgraded with form factors FV (K2) and FA(K2): γµ (FV − FAγ5) toaccount for the internal structure, where FV (0) ≡ gV ' 1 and FA(0) ≡ gA ' 1.27.Second, the mass of the W boson is much heavier than the relevant scale (mW '80GeV). The original diagram depends on two vertices each with a −g2√2coupling as well asthe propagator of the massive particle ∼ 1K2+m2W. The large mass allows us to drop theK (as m2W K2) and replace the internal line in the diagram with a contact interac-tion (Figure 3.2d). The resulting lepton-nucleon coupling depends on the square of thequark/lepton-level coupling and the mass [54], i.e.G√2=g28m2W, (3.57)47ψIψFν¯ee−PIPFk¯k(a)dduudue−ν¯eW−(b)npν¯ee−W−(c)npν¯ee−(d)Figure 3.2: The leading order Feynman diagrams for a β− decay at different levelsof EFT. Solid lines represent fermions (nucleons, quarks and leptons). Wavylines represent bosons (the W ). (a) The diagram for a nuclear decay includingthe kinematics. The circle at the vertex represents an effective operator. (b)The diagram for β decay including the W boson at the level of quarks. (c)The diagram for β decay including the W boson at the level of nucleons. (d)The diagram for β decay at the level of nucleons, where the dot represents aneffective operator.where G is the “Fermi” constant which takes the value 1.1663787 × 10−5 GeV−2 [19]. Itwas measured first through nuclear β decay and the relation between g and mW was laterreverse-engineered through more sophisticated experiments.The final step is to sum over the single-nucleon operators and we get a current operatorevaluated between the initial and final nucleus wavefunctions just like in (3.12) (Figure 3.2a).Schematically, the amplitude of the diagram in the W picture (Figure 3.2c) isM∼∫d4K(2pi)4((2pi)4δ(k + k¯ −K) 〈eν¯| g2√2γµ(1− γ5) |0〉)×(ηµν − KµKνm2W)K2 +m2W((2pi)4δ(PF +K − PI) 〈p| γµ (gV − gAγ5) |n〉), (3.58)48which becomes (Figure 3.2a):M∼ G√2(2pi)4δ(PF + k + k¯ − PI)`µ 〈f | J µW |i〉 , (3.59)where ` is the lepton current defined through the Feynman rules by`µ = u¯s(k)γµ (1− γ5) νs¯(k¯) (3.60)and J µW is the nuclear current operator for the weak interaction.Just like in (3.4) this comes from the integral of interacting fields. We have the leptonfield:K(~x) =∫d3k(2pi)32Ek∑s[aksus(~k)ei~k·~x + b†ksvs(−~k)e−i~k·~x], (3.61)where a (b†) create leptons (anti-leptons) described by Dirac spinors u (v) with momentumk and helicity s. The amplitude is thenMss¯F I =〈F ; ksk¯s¯∣∣ G√2∫d4xKµ(x)J µW (x) |I; 0〉=G√22piδ(EF + E + E¯ − EI)∫d3x〈ksk¯s¯∣∣Kµ(~x) |0〉 (4EIEF 〈f | J µW (~x) |i〉) , (3.62)where〈ksk¯s¯∣∣Kµ(~x) |0〉 =`µe−i~q·~x . (3.63)Combining these, the rate isΓ ∝∑¯i∑f∑ss¯G22∣∣` · 〈f | (J V + J A) |i〉∣∣2 × (phase space factor) . (3.64)The weak-nucleon interaction is exactly analogous to the electromagnetic interactionwith two exceptions:1. In addition to a vector component, the current operator also has an axial-vectorcomponent.Exactly like electromagnetic theory, we can use a multipole expansion. Thistime every operator has two components CW → CV + CA, etc.2. It can be charge-changing1 (i.e. decays via W±).1The weak force also contains a neutral current (corresponding to the Z boson) but that will not bediscussed here.49A β decay requires the current to be charge-changing so we attach the isospinoperator:J ±W =τ±2(J V + J A) . (3.65)The summed squared matrix element is then (45.17 in [54]):12Ji + 1∑Mi∑Mf|M|2=G224pi2Ji + 1×[12∑λ=±1`λ`∗λ∑J≥1∣∣〈Jf | ∣∣λT MJ + T EJ ∣∣ |Ji〉∣∣2+∑J≥0`z`∗z |〈Jf | |LJ | |Ji〉|2 + `0`∗0 |〈Jf | |CJ | |Ji〉|2− (`z`∗0 〈Jf | |LJ | |Ji〉 〈Jf | |CJ | |Ji〉+ h.c.)], (3.66)where we use the covariant components of `µ = (`0, `1, `−1, `z).The vector part of the weak nuclear current is the same operator as the electromagneticnuclear current (conserved vector current (CVC) theory [54]). Therefore in the same way,the longitudinal multipoles can be removed by relating them to the Coulomb multipoles justlike in Section 3.1.2. However the axial-vector current is not conserved and must explicitlycontain longitudinal multipoles. The sum over lepton spins will create different kinematicdependence on each nuclear operator (as we will see in Section 6.4).3.2.1 Leading-Order Beta Decay Operators and Two-Body CurrentsThe long-wavelength approximation (| ~K| → 0) leads to a great reduction in the number ofnon-zero multipole operators [54]. The only remaining terms are CV and T A,E . The leadingorder terms of these operators correspond to two types of “allowed” decays.Decays which do not change the nuclear angular momentum are so-called Fermi (F)decays and correspond to the operator:CV±00 =1√4piA∑iτ± . (3.67)In these decays ∆J = 0, meaning the electron and anti-neutrino must couple to total spinS = 0.Decays which do change nuclear angular momentum are so-called Gamow-Teller (GT)decays. In this case ∆J = 1, meaning the electron and anti-neutrino must couple to total50spin S = 1. The corresponding operator isT A,E±1M =√2LA,±1M =i√6pigAA∑iτ±σ1M . (3.68)Experimentally certain nuclei (i.e. particular states of nuclei) may decay primarily byone operator or the other. As in Section 3.1 there are selection rules. Only first-forbiddenβ decays change parity. In this study, we focus on allowed GT decays where parity isconserved.Higher order diagrams emerge from χEFT in the same way as in electromagnetism. Twonucleons may simultaneously contribute to the matrix elements through a pion exchangeor a contact operator [57]. The additional operator corresponding to these terms is termeda two-body current (2BC). The 2BC is related to the nuclear force, as the pion exchangeand 3N contact term share LECs [59, 60].~KLO, στ~KHeavy mesonexchangecDpi~KPion exchangec3, c4Figure 3.3: Diagrammatic representations of terms that contribute to the axial-vectorcharge-changing current up to N2LO in the limit ~K → 0. Solid lines representnucleons while wavy lines represent the W boson. The left-most diagram yieldsthe standard (one-body) στ transition operator, the other diagrams form theleading 2BC.3.2.2 The GT Beta Decay SpectrumTo leading order, the GT decay rate depends on LA (3.68). Experiments are startingto reach the level of precision where corrections to this approximation are necessary intheoretical predictions.Assuming a neutrino with zero mass, k¯ = E¯, the phase space can be easily integrated51over the neutrino momentum. The rate is proportional to this phase space factor, i.e.Γ ∝∫d3k(2pi)3d3k¯(2pi)3δ(EF − EI − E − E¯)=∫k2dkdΩk(2pi)3E¯2dE¯dΩk¯(2pi)3δ(Emax − E − E¯)=∫kEdEdΩkdΩk¯(2pi)6(Emax − E)2 , (3.69)where E =√k2 +m2e, dE =Ek dk and Emax = EF − EI .The full expression for the differential distribution of electrons isdΓfidE dΩk4pidΩk¯4pi=4pi2(Emax − E)2kECcorrF(Z,E)∣∣〈f | ∣∣LA∣∣ |i〉∣∣2× 3(1 + δβ1)(1 + aβν ~β · νˆ + bFmeE). (3.70)The Fermi function F(Z,E) takes into account Coulomb distortion [61] of the electron’swavefunction and Ccorr contains non-nuclear corrections (e.g. finite size effects). Here~β =~kE , with~k and E the momentum and energy of the emitted electron and νˆ is the unitvector for the direction of the momentum of the emitted anti-neutrino. The V-A structureof the weak interaction in the Standard Model implies that for a Gamow-Teller transitionthe angular correlation coefficient aβν has the value of −13 while the “Fierz interference”term bF is expected to be 0. The deviations from this expectation are denoted δβa and δβbdefined by aβν = −13(1 + δβa)and bβF = δβb respectively.The presence of a beyond the Standard Model interaction could induce a non-zero valuein the δs. However this must be differentiated from Standard Model effects e.g. the finitemomentum transfer, higher-order transition operators and nuclear structure. The StandardModel corrections δβ1 , δβa and δβb are related to ratios of the leading LA1 operator withadditional multipole operators (CA1 and T V,M1 , see [2, 62] for more details).3.2.3 Double-Beta DecayWhen the energy of the adjacent (Z ± 1) nucleus is too high (such that single beta decay iskinematically forbidden) it may still be possible to double-beta decay, if the nucleus of theZ ± 2 isobar has low enough energy.This process has been observed [63] with a very low rate. Since two neutrinos areproduced, this is known as 2νββ. Like single β decay, it has two modes, Fermi and Gamow-Teller which are products of the same single-beta-decay operators. A difference is that thematrix element must also include a sum over intermediate virtual states of the nucleus anda corresponding energy dependent weighting.Given the unknown nature of neutrinos, the mass of the neutrino has yet to be incorpo-rated into the Standard Model, it may be possible that the neutrino is its own anti-particle.52npe−ν¯enpe−ν¯e(a)npe−npe−(b)Figure 3.4: Feynman diagrams for (a) two-neutrino double-beta decay (2νββ) and(b) neutrino-less double-beta decay (0νββ).If neutrinos are “Majorana” it would be possible to observe a different process, 0νββ, ifthe two neutrinos annihilate. The kinematics of the electrons would be used to differentiatethis mode. The ratio between the total energy of the emitted electrons in 2νββ and theavailable nuclear transition energy is a continuum between 0 and 1, peaked at 13 , while in0νββ the peak would be at 1 [64].Neutrino-less double-beta decay is a hypothetical process. If discovered it would provideevidence of lepton number violation and determine the Majorana character of neutrinos. Itis expected to be extremely rare. The rate of this decay depends on the neutrino mass andhence can be used to determine the mass; however, the nuclear matrix element is requiredto compute it. The halflife formula is[T 0ν1/2]−1 = G0ν |M0ν |2(〈mββ〉me)2, (3.71)where G0ν is a phase-space factor (an integral over the possible outgoing electron momenta),me the electron mass and 〈mββ〉 is the effective neutrino mass [65]. The quantity of interestthat will be computed here is the nuclear matrix element M0ν .In principle, the matrix element can be computed through multipole decomposition ofthe weak current and chiral effective theory. However its derivation is beyond the scopeof this thesis. The leading order operator consists of four contributions (the Gamow-Teller[GT], Fermi [F], Tensor [T] and contact [C]), and so the total operator isM0ν = M0νGT +M0νF +M0νT +M0νC . (3.72)Each component has different spin dependence and can be derived by connecting a neutrinopotential to χEFT [66]. The existence of the contact is a relatively recent theoretical53discovery [67]. For now we will neglect it, its inclusion is a very active area of research [68].The operator takes the following form:Mx =Hxyxτ+1 τ+2 , (3.73)where the spin dependence of the operator is contained withinyx =1 if x = F ,σ1 · σ2 if x = GT ,√24pi5 Y2(rˆ)S12 if x = T ,(3.74)where S12 = 3(~σ1 · rˆ)(~σ2 · rˆ)− ~σ1 · ~σ2.Hx depends on the relative coordinate r = ξ1 and is input through harmonic oscillatorradial matrix elements of a neutrino potential, i.e.Hx(r) =2Rpi∫dqq · fx(qr)hx(q2)q + Ecl0, (3.75)with the approximate nuclear radius R = A13 r0 (r0 = 1.2 fm). The fx are Bessel functions,i.e.fx =j0 if x = F,GT ,j2 if x = T , (3.76)and the hx are form factors to correct for the finite size of the nucleons. For example, hFdepends on the vector form factor gV and a “finite-size cutoff” parameter ΛV :hF (q2) =gV(1 + q2ΛV)2 . (3.77)The hGT and hT are more complicated (depending on gA, ΛA, mpi, mp,n and µp,n) and arenot shown for brevity. They are tabulated in [69], following the derivations of [65] and [70].Hx contains the radius R which could also be computed with ab initio techniques but inour application of this operator (benchmarks with other models) that would introduce moremodel dependence. This form of the operator makes use of the “closure” approximationto remove the sum over virtual nuclear states. The appropriate closure energy Ecl0 will bedifferent for different nuclei as we are averaging over the states of the particular intermediateisotope [71].It is important to note that by definition the 0νββ operator must depend on two-nucleoncoordinates. Therefore it is implemented using a coupled isospin basis. It must be a rank-254isotensor with projection ±2, i.e.〈t′1t′2; t′m′t∣∣ [τ1τ2]2−2 |t1t2; tmt〉 = (tmtTMT |t′mt′) tˆ′−1 〈t′1t′2; t′m′t∣∣ ∣∣∣[τ1τ2]2∣∣∣ |t1t2; tmt〉= (112− 2|1− 1) 1ˆ21212 11212 11 1 2〈12∣∣∣∣ |τ | ∣∣∣∣12〉2. (3.78)Of course〈12∣∣ |τ | ∣∣12〉 = √6, the same as 〈12 ∣∣ |σ| ∣∣12〉.55Chapter 4Similarity Renormalization GroupIn a basis expansion technique, the elements of the Hamiltonian matrix which inhibit con-vergence the most are those that connect high and low energy i.e. the off-diagonal elements.The method of Similarity Renormalization Group (SRG) applies a unitary transformationwhich is designed to suppress these elements. Effectively it reduces the resolution of theinteraction, so that short-distance (high-energy) features are removed but low-energy ob-servables are unchanged [72].A nuclear interaction with a strong short-distance component is often referred to as“hard”, hence applying SRG creates a “soft” interaction. The original Hamiltonian isreferred to as “bare” and after SRG the “evolved” Hamiltonian should have the sameexpectation values (energies) but they will converge more rapidly with respect to the model-space size (the parameter Nmax in NCSM calculations).In this section technique of SRG evolution is summarized. The implementation of generalSRG-evolved operators of three-nucleons in the NCSM contributed to the results presentedin Section 5.2.2.4.1 Softening Potentials with Flow EquationsThe nuclear Hamiltonian itself is not observable thus one can apply an arbitrary unitarytransformation without changing observable quantities. The Hamiltonian is taken to be afunction of a continuous parameter s, i.e.H(s) = U(s)HU †(s) , (4.1)where U is a unitary transformation parameterized by s (U(s)U †(s) = U †(s)U(s) = 1). Theparameter s sets the amount of renormalization. s = 0 is the original “bare” Hamiltonian:H(s = 0) = H, U(s = 0) = 1 . (4.2)56The evolution of the Hamiltonian is carried out by choosing a generator η(s) and integratingthe flow equation:dH(s)ds= [η(s), H(s)] , (4.3)from s = 0 up to a finite value of s.The choice of generator depends on the application, in this work we choose a commutatorwith the kinetic energy: η(s) = m2N [T,H(s)] [73]. The parameter s must have units thatare the inverse [-4] of those of the generator [+4] (in natural units). Typically the resultsare presented with respect to λ = s−1/4, as λ then has units of momentum (fm−1). λ =∞corresponds to the bare Hamiltonian. A typical stopping point for s is λ = 2fm−1 (' 400MeV), this corresponds to roughly twice the Fermi energy of the nucleons in the nucleusand is an arbitrary but reasonable separation between high- and low-energy.It is also possible to find a flow equation for U(s) as well, i.e.dUds= η(s)U(s) , (4.4)but in practice this is not necessary. Historically calculations only evolved H. We maycalculate U(s) from eigenvectors of H(0) and H(s) (|ψ(0)〉 and |ψ(s)〉 respectively), i.e.U(s) =∑i|ψi(s)〉 〈ψi(0)| . (4.5)An important point is that the Hamiltonian is defined in a two- (or three-) nucleonspace. It is upgraded to the form in (1.3) by summing over all pairs (triplets) in the A-body system. We perform SRG on the two- (three) body space and so the eigenvectorsmentioned in (4.5) will be two- (three-) nucleon states. After upgrading the A-body spaceand solving the A-body Hamiltonian we will have A-nucleon eigenstates so there exists anA-body version of U(s) but it is impractical to compute.Another note is that there is an ambiguity in the relative phase of the two eigen-states |ψ(s)〉 and |ψ(0)〉 that cannot be determined uniquely by the diagonalization of H(s)and H(0) respectively. Since we expect that SRG will only affect short-range terms andnot produce a drastic change to the wavefunction, the relative phase is chosen such that〈ψi(0)|ψi(s)〉 ≥ 0.The choice of generator determines how the structure of the Hamiltonian is modified.Since the kinetic energy is tri-diagonal in the harmonic oscillator basis, using it in the gen-erator forces the Hamiltonian to evolve towards being (tri-)diagonal. This is schematicallyshown in Figure 4.1.57Figure 4.1: A schematic representation of the effect of SRG evolution on the Hamilto-nian matrix. The matrix elements are arranged according to the total numberof harmonic oscillator excitations N (corresponding to N˜ = 2n+ ` in the text).At successively smaller stopping points, λ0 > λ1 > λ2, the matrix elementsnear the diagonal become larger compared to those further from the diagonal.Adapted from [72]4.2 SRG Evolution of OperatorsWhen the transformed Hamiltonian is used in a many-body calculation the resulting eigen-functions will be different but the eigenvalues should be the same (due to unitarity). Otherobservables have corresponding operators that should also be evolved and the result shouldbe the same between the evolved and unevolved wavefunction [74], i.e.〈ψ|O |ψ〉 = 〈ψ(s)|O(s) |ψ(s)〉 (4.6)There is an analogous flow equation to (4.3) for operators:dO(s)ds= [η(s), O(s)] , (4.7)but if we extract the transformation U(s) we can calculate O(s) directly in the 2- (3-)nucleon space, i.e.O(s) =U(s)OU †(s) . (4.8)584.2.1 Induced Many-Body TermsSRG transformations induce higher-body terms in operators (including the Hamiltonian)and these need to be included in order to maintain unitarity [75]. These terms will be missingif the evolution is carried out in a few-body space i.e. evolution in two-body coordinateswill be missing three-body terms. A two-body potential in the Hamiltonian will producea three-body term if evolved in three-body coordinates. This occurs as spectator nucleonswill be linked through the repeated product with the generator.One way to see this is to write the operator in terms of creation and annihilationoperators (a†,a respectively) [75]. Schematically (suppressing indices and sums):〈O〉 =〈O(1)〉a†a+〈O(2)〉a†a†aa+〈O(3)〉a†a†a†aaa (4.9)The product of two one-body pieces (i.e. a†a with a†a) will result in a two-body piecea†a†aa (after normal ordering).Another way to see this is to differentiate O = O12 + O13 + O23 + O123 explicitly withrespect to s and compare to the flow equation calculated in each subspace [76], i.e.dOds=dO12ds+dO13ds+dO23ds+dO123ds= [[T,H] , O] (4.10)= [[T12, V12] + [T13, V13] + [T23, O23] + [T, V123] , O] , (4.11)where O12 is a two-body operator operating on particles 1 and 2 and O123 is an irreducible3-body piece. The s-dependence of H, V and O operators is implied here. Each two-bodypart results from the sub-equation:dOijds= [[Tij , Vij ] , Oij ] , (4.12)where, even though we have three particles, the dependence on the third coordinate Tk com-mutes (T = Tij+Tk, where T =∑3 p2i2mN= q22mN+ 3k24mN, q = pi−pj , k = 23[pk − 12 (pi + pj)]).If we cancel the ij terms on both sides thendO123ds= [[T12, V12] , (O13 +O23 +O123)] + {123→ 132}+ {123→ 231}+ [[T, V123] , O123] . (4.13)Even if H123 = V123 = 0 and O123(s = 0) = 0, a non-zero O123 will be induced through theevolution as dO123ds will be non-zero.594.2.2 Calculations of SRG-evolved OperatorsThe induced terms need to be separated out in order to be used. The irreducible 2- and3-body components must be embedded into the many-body separately as they have differentscaling factors with the total nucleon number (due to combinatorics, see Section 2.9). Theevolution is carried out separately in each space and then the two-body total is subtractedfrom the three-body after embedding.To make the notation clear:O(1) =∑ij〈i|O |j〉 a†iaj , (4.14)O(2) =∑ijkl〈ij|O |kl〉 a†ia†jalak , (4.15)[O(1)](2)=∑ijkl〈i|O |k〉 〈j|l〉 a†ia†jalak , (4.16)O(3) =∑ijklmn〈ijk|O |lmn〉 a†ia†ja†kanamal , (4.17)[O(2)](3)=∑ijklmn〈ij|O |lm〉 〈k|n〉 a†ia†ja†kanamal . (4.18)Two-body evolution of a two-body operator is calculated from a two-body unitary, i.e.O(2)(s) = U (2)(s)O(2)U †(2)(s) . (4.19)The irreducible three-body induced operator is isolated by subtracting the two-body evolvedoperator from the operator evolved in three-body space:O(3)(s) = U (3)(s)[O(2)](3)U †(3)(s)−[U (2)(s)O(2)U †(2)(s)](3). (4.20)In principle evolution should be carried out in every subspace up to A-body as schemat-ically shown in Figure 4.2 but here we truncate at three-body. Generally a hierarchy ofimportance is expected in the induced terms, i.e. higher-body terms are smaller. The sizeof the missing terms can be determined by the variation of the parameter s. In addition,U (3) will depend on three-body interactions in the Hamiltonian if they are included.In the two- and three-body level the Hamiltonian is a block-diagonal operator in termsof the quantum numbers J , pi and T for total angular momentum, parity and isospinrespectively. The Hamiltonian and therefore the unitarity transformation U can be com-puted independently for each JpiT channel. Some operators connect blocks with differentspin and/or isospin. The operator must then be evolved by multiplying the corresponding60Figure 4.2: The process of preparing an operator for an A-particle calculation. Adiagrammatic summary of Section 4.2.2. Adapted from [72]blocks on either side, i.e.〈J ′pi′T ′∣∣∣O(s) |JpiT 〉 = 〈J ′pi′T ′∣∣∣UJ ′pi′T ′(s)OU †JpiT (s) |JpiT 〉 . (4.21)The additional expense of calculating the SRG evolution is not prohibitive as it iscarried out in the two-, or three-nucleon space in which the potential is formulated. Whilecomputationally expensive at the few-body level, the benefit is seen at the much moreexpensive many-body level. In addition, since 3N forces are included in the calculationanyway, the cost of including 3N induced terms is not insurmountable.SRG is performed with the Jacobi basis which may use a basis with much higher Nmaxcompared to the SD basis in the two- and three- body spaces. In this basis the numericalcalculations of evolution are more precise. In the two-body space we can include up toNmax = 300 but in the three- we use Nmax ∈ {30 − 50} with lower Nmax for higher Jblocks.In summary, the workflow of an NCSM calculation for a two-body operator with SRGis as follows:1. Choose an NN + 3N Hamiltonian (H(2) +H(3)).2. Solve (4.3) for H(2)s .3. Extract U(2)s and apply to O(2).4. Solve (4.3) for[H(2) +H(3)](3)s.615. Extract U(3)s and apply to[O(2)](3).6. Subtract O(2)s from[O(2)](3)sto obtain O(3)s .7. Embed[H(2)(+H(3))]sinto A-body and solve for A-body eigenstates.8. Calculate < Os >=< O(2)s > + < O(3)s >.In our terminology, using only H(2)s is the NN result,[H(2)](3)sis the NN+3N induced, and[H(2) +H(3)](3)sis the NN+3N full.62Chapter 5NCSM Results5.1 Beta DecayBeta decay is the dominant decay mode of many nuclei in the nuclear chart (blue andpink in Fig. 1.2), hence it is an important testing ground for comparing nuclear models toexperiment. The discrepancies between calculations and observations motivate ever moresophisticated theoretical techniques.One such discrepancy is the historical difference between theoretical predictions andexperimental observations. This is known as the quenching problem. A decay within anucleus is observed to be less likely than expected based on free neutron decay. For Gamow-Teller decays, phenomenological models can easily match the decay rate if the parametergA (in (3.68)) in calculations is modified (“quenched”) by a factor of about 0.75.In Section 5.1.1, we address quenching by using more accurate operators in ab initiomethods. Part of the quenching can be explained by inclusion of higher-order operators e.g.two-body currents (2BCs) where the presence of pions in the nucleus modifies the nucleartransition operator. In addition, the accuracy of the operator’s expectation values requiresthat the operator be SRG-evolved consistently with the input Hamiltonian.Future discrepancies may arise as experiments improve their precision beyond that oftheoretical predictions. In order to keep up, calculations must begin including correctionsto the leading order β decay operator.In Section 5.1.2, we address the search for beyond the Standard Model (BSM) physicsin measurements of β-decay spectra. In particular, we calculate the nuclear structure cor-rections that must be anticipated in order to identify BSM effects in measurements of the6He→ 6Li decay.5.1.1 SRG and 2BC in GT DecayIn our paper [1], we computed β-decay rates for nuclei across the nuclear chart. We chosestrong GT transitions for which quenching should dominate, rather than nuclei like 14C in63Figure 5.1: Theory-to-experiment ratio for the Gamow-Teller matrix elements of sixstrong transitions in light nuclei for the NN-N4LO+3Nlnl interaction developedin [1]. The subscripts in the legend denote the total angular momenta of theparent and daughter states. All initial states are ground states. In the case3H → 3He,6He → 6Li and 7Be → 7Li32, the daughter nucleus is in its groundstate, while the 7Be → 7Li12,8He → 8Li1 and 10C → 10B1 are decays to thefirst excited state of the daughter nucleus, and the 14O → 14N1 is a decay tothe second excited state of 14N. Hollow symbols correspond to results obtainedwith the standard Gamow-Teller στ operator, and full symbols include 2BC.The results are converged to within 3% with respect to the model-space size.This uncertainty is slightly larger than the marker size and is not shown fortransparency. Figure reproduced from [1].which fine-tuned cancellations occur.In light nuclei, we applied the NCSM to compute nuclear wavefunctions and the β-decaytransition matrix elements. These calculations contributed to an understanding of the effectof two-body currents (2BCs) in the decay. Figure 5.1 shows the result of inclusion of 2BCscompared to the leading order operator (3.68).The two-body currents which can be derived from χEFT [57, 60] share parameters withthe 3N terms in the Hamiltonian. In fact the low-energy constant cD was fit using the3H → 3He decay including 2BCs (which is precisely why that result agrees so well withexperiment) [60].The Gamow-Teller beta decay operator is a good example of SRG in action. Figure 5.2shows the effect of two-nucleon SRG on the GT operator in two different transitions. The64leading order GT operator (3.68) is denoted as “στ”. As the interactions used in calculatingthe wavefunctions were evolved, it is necessary to consistently evolve the operators. Notevolving the operator consistently results in the dotted lines which have a large spread withrespect to the evolution parameter of the input Hamiltonian. This spread is significantlyreduced when the operator is evolved (the dashed line).The inclusion of the higher order operator improves agreement with experiment, result-ing in the solid lines of Figure 5.2. The lack of spread with respect to λ indicates that theeffect of three-nucleon terms will be small enough to neglect. This is because the domi-nant term (GT) is a one-body operator and its two-body induced contribution is alreadysmall. The three-body induced term will be smaller and the two-body current (a two-bodyoperator) is already relativity small, and so will also have a small induced term.The decrease due to the 2BCs is not entirely consistent, for example in 8He the matrixelement increases, it is only when looking across the nuclear chart that this trend is seen.The NCSM was used to verify the effects of the 2BC in light nuclei and benchmark the op-erators for more approximate ab initio methods like coupled-cluster (CC) and in-mediumsimilariy renormalization group (IMSRG). The CC and IMSRG techniques were used tocalculate the matrix elements for much heavier nuclei. These more approximate methodswere used to calculate GT rates across a wide range of nuclear masses. Figure 5.3 showsIMSRG calculations for many-medium mass nuclei as compared to experiment and phe-nomenological shell model calculations. The overall theory vs experiment slope was greatlyimproved towards 1.The strongest GT transition is tin-100 (100Sn) which, while difficult to compute with abinitio techniques due to its large size, is a good candidate for coupled-cluster calculationsbecause it is doubly-magic (Z = N = 50). Figure 5.4 shows the GT matrix element of100Sn computed with CC for a number of nuclear interactions. The inclusion of the 2BCsreduces the size of the transition strength giving an equivalent result to phenomenologicalmodels where a quenching factor was applied.With ab initio methods the quenching factor becomes unnecessary. Two-body currentsin combination with an ab initio description of the nuclei explain the observed quenchingof the GT strength across the nuclear chart.65(a)(b)Figure 5.2: Convergence of the 3H→ 3He (a) and 6He→ 6Li (b) Gamow-Teller ma-trix elements with respect to the NCSM basis size for three values of the SRGevolution parameter λ. Dashed and full lines show results obtained with one-body only and one- plus two-body operators respectively. The NN-N4LO+3Nlnlinteraction was used with both the interaction and transition operators consis-tently SRG evolved. The dotted and dash-dotted lines show results obtainedwith the same SRG evolved interaction using only the one-body operator with-out any SRG evolution. The shadow bands represent the experimental valueswith their uncertainties. Figure reproduced from [1].660123|MGT|Experiment34P1 →34 S024Na4 →24 Mg433P1/2 →33 S3/234P1 →34 S024Ne0 →24 Na128Al3 →28 Si230Mg0 →30 Al126Na3 →26 Mg237K3/2 →37 Ar3/225Al5/2 →25 Mg5/237K3/2 →37 Ar5/219Ne1/2 →19 F1/2this workshell modelq = 1q = 0.96(6)q = 0.80(2)0 1 2 3|MGT| Theory (unquenched)0123|MGT|Experiment46Sc4 →46 Ti445Ti7/2 →45 Sc7/247Sc7/2 →47 Ti7/247V3/2 →47 Ti5/245V7/2 →45 Ti5/243Sc7/2 →43 Ca5/245Ti7/2 →45 Sc7/245V7/2 →45 Ti7/242Ti0 →42 Sc142Sc7 →42 Ca6this workshell modelq = 1q = 0.92(4)q = 0.75(3)Figure 5.3: Comparison of experimental [77] and theoretical Gamow-Teller matrixelements for medium-mass nuclei. Plots of the Gamow-Teller matrix ele-ments: sd-shells (top) and pf -shells (bottom). Theoretical results were ob-tained using phenomenological shell-model interactions [78, 79] with an un-quenched standard Gamow-Teller στ operator (open orange squares) and usingVS-IMSRG approach with the NN-N4LO+3Nlnl interaction and consistentlyevolved Gamow-Teller operator plus 2BCs (filled green diamonds). The linearfit show the resulting quenching factor q given in the panels, and shaded bandsindicate one standard deviation from the average quenching factor. Experi-mental uncertainties, taken from [77], are shown as vertical error bars. Figurereproduced from [1].675 7 9 11 13 15 17 19|MGT|2FFSQRPALSSMSMMCESPMBatist et al.Hinke et al.NNLOsatNN-N3LO+3NlnlNN-N4LO+3Nlnl2.8/2.0 (EM)2.0/2.0 (PWA)2.2/2.0 (EM)2.0/2.0 (EM)1.8/2.0 (EM)exptother modelsthis work2BCFigure 5.4: Comparison of the Gamow-Teller strength |MGT |2 for the β-decay of100Sn calculated in [1] compared to data [80], systematics [81], and other mod-els (extreme single-particle model (ESPM), shell-model Monte-Carlo (SMMC),large-space shell-model (LSSM), quasiparticle random-phase approximation(QRPA) and finite Fermi systems (FFS)) from [80]. Open symbols representresults obtained with the standard Gamow-Teller operator (στ), filled symbolsalso include two-body currents (2BCs) and partially filled symbols show valuesfollowing from the multiplications of the computed Gamow-Teller strength bythe square of a phenomenological quenching factor. Each of our 100Sn calcula-tions carries a conservatively estimated uncertainty of about 10% (not shownto avoid overcrowding the figure). Figure reproduced from [1].5.1.2 Precision Beta DecayIn the previous section, ab initio techniques were used to improve agreement between ex-perimental observations and theoretical predictions from the Standard Model (SM). Thesecalculations are approaching the precision needed to differentiate between the SM and newbeyond the Standard Model (BSM) physics.β-decay observables are sensitive to interference between currents of SM particles andhypothetical BSM particles. Such couplings are proportional to(vΛ)2where v ≈ 174 GeVis the SM vacuum expectation value and Λ is the new physics energy scale. A couplingon the order of 10−4 would suggest new physics at a scale beyond the reach of currentparticle accelerators. Such small deviations from the SM demand high-precision theoreticalcalculations of β-decay observables with quantified uncertainties.680.0 2.5 5.0q (MeV)0.2080.2120.2160.2200.224NNLOoptNNLOsat| f LA1 (q) i |20.0 2.5 5.0q (MeV)1.821.841.861.88×10 6| f MV1 (q)q i |20.0 2.5 5.0q (MeV)0.81.21.62.02.4×10 8| f CA1 (q)q i |2(MeV 2) (MeV 2)Figure 5.5: Dependence of nuclear matrix elements on the NCSM model-space pa-rameters and nuclear Hamiltonians. The light to dark bands correspond toNmax = 8, 10, 12 for the NNLOsat (blue) interaction and Nmax = 10, 12, 14for the NNLOopt (green) interaction. The width of the bands show the vari-ation with HO frequency ~Ω = 16, 20, 24 MeV. The solid (dashed) line showsthe result with the NNLOsat interaction at Nmax = 12, ~Ω = 20 MeV com-puted with translationally-invariant (standard) one-body densities. Figure re-produced from [2].In our paper [2], as a feasibility study, we computed β-decay observables of the pureGamow-Teller transition 6He(0+) → 6Li(1+). These are very light nuclei for which NCSMcalculations are very tractable, in addition there are several ongoing experimental campaignsto make very precise measurements with per-mil accuracy.We used technology developed in [82] to evaluate q dependent current operators in theNCSM. We calculated the wavefunctions of 6He and 6Li with the NCSM with two differentnucleon-nucleon forces (NNLOopt[83] and NNLOsat[84]). We computed the transition matrixelements for several multipole operators. For this transition the contribution of 2BCs issmall enough that they can be neglected (within the error budget of other quantities).Figure 5.5 shows the transition matrix elements for the axial Coulomb and axial longi-tudinal nuclear operators (CA1 and LA1 respectively (see Section B.3)) and the vector trans-verse magnetic nuclear operator T V,M1 (referred to as “MV1 ” in the figure). We estimateduncertainty bands for each nuclear force by varying the NCSM parameters: Ω and Nmax.In addition, we compared the results of computing the transition matrix element in thestandard way (2.94) and using translationally invariant densities (2.95). Figure 5.6 showssimilar uncertainty bands for the components of the nuclear operators (in terms of the“basic” multipole operators given in Section B.4).It is clear that there is greater dependence on the choice of nuclear interaction than onthe NCSM parameters. We also found that the transition matrix elements are sensitive to690.2550.2600.2650.2700.275|6 Li1+′ (q)6 He0+|2NNLOoptNNLOsat0.1280.1300.1320.1340.1360.138|6 Li1+′′ (q)6 He0+|2NNLOoptNNLOsat0 1 2 3 4 5q (MeV)0.00010.00020.00030.00040.00050.00060.00070.0008|6 Li1+′ (q)6 He0+|2NNLOoptNNLOsat0 1 2 3 4 5q (MeV)0.0030.0040.0050.0060.007|6 Li1+(q)6 He0+|2NNLOoptNNLOsatFigure 5.6: Dependence of nuclear matrix elements on the NCSM model space pa-rameters and nuclear Hamiltonians. Light, medium, and heavy filled bands cor-respond to Nmax = 8, 10, 12 for the NNLOsat interaction including 3NF (bluebands), and Nmax = 10, 12, 14 for the NNLOopt interaction with only 2NF(green bands). The width of the bands show the variation with HO frequency~Ω = 16, 20, 24 MeV. The solid (dashed) line shows the result with the NNLOsatinteraction at Nmax = 12, ~Ω = 20 MeV computed with translationally-invariant (standard) one-body densities. Figure reproduced from [2].contamination by the center of mass. The gradient operator induces spurious center of masseffects. The most sophisticated description of this transition is the Nmax = 12, ~Ω = 20MeV, NNLOsat calculation with translationally invariant densities.The total energy dependent corrections to the β spectrum are pictured in Figure 5.7.The total spectrum corresponds to the partially integrated rate (3.70) i.e.dΓfidE . Overallwe find that the Standard Model corrections do differ from zero and must be accounted forin experiments to differentiate from BSM effects. The corrections correspond to 1% of thespectrum and a 2% correction to the angular correlation.Our prediction for the Fierz term is q-independent at leading order and has the valuebF = −1.52(18) × 10−3 . The term bF meE should therefore make a non-zero appearance inthe spectrum and imitates BSM physics at the TeV scale. Our theoretical prediction carriesa cumulative uncertainty of less than 15% i.e. the Fierz term has a predicted value on theorder of 10−3 with an uncertainty of 10−4, so experiments at 10−4 precision must certainly70take into account these calculations.01020 spectrum (a.u.) (a)Gamow-TellerNucl. structure errorTotal theory error1011+ 1 (%)(b)0.3400.3350.3300.325a1+(c)0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5electron kinetic energy (MeV)2021+ a (%)(d)Figure 5.7: (a) Calculated energy dependence of the spectrum of 6He β-decay, inarbitrary units. Dashed line is the pure GT spectrum, while the filled bandsinclude nuclear-structure dependent corrections. (b) The residual nuclear struc-ture correction δ1+β−1 compared to the pure GT spectrum. (c) Energy depen-dence of the angular correlation aβν . Dashed line corresponds to the SM value,aGTβν = −1/3. (d) Relative size of the δ˜1+β−a correction. The width of thedark filled bands shows the variation with the employed nuclear Hamiltonianand NCSM model space parameters for HO frequency ~Ω = 16, 20, 24 MeV,Nmax = 8, 10, 12 (10, 12, 14) using the NNLOsat (NNLOopt) interaction, usingtranslationally-invariant one-body densities. The width of the light filled bandshows the total estimated theory error. Figure reproduced from [2].5.2 Neutrino-less Double Beta DecayNeutrino-less double-β decay is a hypothetical process in which a nucleus undergoes twosimultaneous β decays but emits no neutrinos, as discussed in Section 3.2.3. The observationof this decay would provide evidence of a lepton-number violating process and identify theneutrino as a Majorana particle, thus providing insights towards the origin of the matter-71antimatter symmetry in the universe and the neutrino mass. Current experiments haveestablished a lower limit on the lifetime of about 1026 years [85–87] and sensitivity will beimproved by several orders of magnitude further in the coming years.Essential for planning and interpreting these experiments are the nuclear matrix el-ements (NMEs) that relate the decay lifetime to the neutrino mass scale (3.71). Thesematrix elements are not known and cannot be measured. Theoretical computations basedon different models and techniques lead to factors of 3 to 5 in predicting these numbers.Further complication is the recent discovery that within χEFT, the standard decay oper-ator must be supplemented by a contact operator of uncertain strength [67]. Efforts areunderway to compute this contact from QCD and better understand its impact.The candidate nuclei for experimentally measuring 0νββ decay e.g. 48Ca [88, 89] areheavy and cannot be computed with the NCSM using current computing resources. There-fore numerical methods with additional levels of approximation are needed. Calculations offictitious decays of lighter nuclei are still of value, as they can be compared to the NCSMas a benchmark. In realistic scenarios these decays would not be detectable as they wouldbe swamped by single beta decays or would be energetically forbidden.5.2.1 Coupled-Cluster BenchmarksOur recent paper [3] presents calculations of the 48Ca → 48Ti decay via 0νββ. This wascalculated with the coupled-cluster method (CC). NCSM calculations were used to gaugethe quality of coupled-cluster in several light nuclei by calculation of fictitious decays. Usingseveral interactions, we calculated the NME for 0νββ using the closure energy of 5 MeV(defined in Section 3.2.3).In Figure 5.8, the NCSM NMEs are compared to the CC NMEs. The coupled-clustermethod approximates a nuclear state with an expansion over particle-hole excitations ofnucleons with respect to a particular reference state. Each calculation requires a choice ofreference state. The CC results are shown as two pairs. Each pair corresponds to a differentreference state. The first (second) point with each pair show CCSD (CCSDT-1) approx-imations, corresponding to the inclusion of singles+doubles [90] (singles+doubles+triples[91, 92]) excitations. The error bar on the NCSM results corresponds to the extrapolationto infinite model space. These results demonstrate that if the coupled-cluster calculationschoose the correct reference state and go to triples excitations, their calculations are com-parable to NCSM.The coupled-cluster method has much better scaling with A than the NCSM. Thisallows it to calculate the 0νββ matrix element for 48Ca which is a promising experimentalcandidate [88, 89]. A comparison of the 48Ca→ 48Ti NME with various phenomenologicalmethods (and the ab initio method IMSRG+GCM [94]) is shown in 5.9. While the groundstate of 48Ca is spherical, that of 48Ti is deformed. The error band corresponds to thechoice of deformed vs spherical CC reference state. Based on the benchmarks against the72Figure 5.8: Comparison of the 0νββ nuclear matrix element in several light nucleicomputed with the coupled cluster method and the no-core shell model. Thefirst two columns correspond to different choices for the coupled-cluster refer-ence state, and results for the CCSD and CCSDT-1 approximations are shownin each. The error bars indicate the uncertainties coming from variations withthe model-space size. Each case utilizes the 1.8/2.0(EM) interaction [93] ex-cept for 22O→ 22Ne which disregards the three-nucleon forces to more rapidlyconverge the NCSM results. Figure reproduced from [3].73NCSM, we would expect the CC prediction using a deformed reference state to be the mostaccurate.Figure 5.9: Comparison of the NME for the 0νββ decay of 48Ca, calculated withinvarious approaches. The coupled-cluster results use both the CCSD andCCSDT-1 approximations with both the spherical and deformed referencestates. For IMSRG+GCM [94], the double bars show the effects of uncer-tainty in model-space size; otherwise they show those of uncertainty in short-range correlation functions. The other results are the realistic shell model[95], the quasiparticle random phase approximation [96], the interacting bosonmodel [97], various energy-density functionals [98, 99] and phenomenologicalshell models [66, 71, 100, 101]. Figure reproduced from [3].5.2.2 VS-IMSRG BenchmarksThe same calculations with a different nuclear force was carried out for a similar suite oflight nuclei and the operators were consistently SRG-evolved. This was compared with thetechnique of VS-IMSRG (“valence space in-medium SRG”), which has an additional unitarytransformation on top of regular SRG [10]. VS-IMSRG, like coupled-cluster, is also capableof calculating the 48Ca decay.SRG calculations were carried out at both the two- and three-body level. As 0νββ isa two-body operator, it was expected that the three-body induced term would be bigger(compared to that of single beta decay). However our finding is that the three-body inducedeffect is quite small. The spread with λ is not always improved relative to two-body evolutionand the overall contribution is very small. Some examples are shown in Figure 5.10 andFigure 5.11.Figure 5.10 demonstrates the matrix element for light mirror nuclei where the isospinchanges by ∆T = 0. This is a hypothetical mode because single-β decays are possible andin the case of 6He → 6Be the change in charge is enough that the new nucleus would be74Figure 5.10: Nuclear matrix elements of the neutrino-less double-beta decay transi-tions with ∆T = 0. From left to right, columns correspond to the transi-tions: 6He(0+1 ) → 6Be(0+1 ), 10Be(0+1 ) → 10C(0+1 ) and 14C(0+1 ) → 14O(0+1 ).The numerical calculations use the parameters ~Ω = 20 MeV and Nmax = 8(emax = 12) in the NCSM (IMSRG). The NCSM calculation at A = 14 uses anadditional approximation known as “importance truncation” [102] to obtaina result at Nmax = 8.unbound due to Coulomb repulsion (see Figure 1.1). Figure 5.11 shows the more realisticcase of ∆T = 2 which corresponds to the situation in experimental candidates. The matrixelements here are significantly smaller.Overall, the spread in λ becomes smaller as the 2N-induced and 3N-induced contribu-tions are included, indicating the 4N induced term is less important. A few exceptions areobserved i.e. the M0νGT term for NCSM in6He and IMSRG in 14C. This benchmark indi-cates an difference of about 3, 8, 20% for A=6, 10, 14 respectively. According to a previousstudy, the A = 14 difference is especially large due to neglected triples in VS-IMSRG [94],inclusion of these in progress [103].These results indicate some deficiencies in the IMSRG for smaller nuclei. GenerallyNCSM agrees with the couple-cluster method (with the deformed basis) which has beenfound to give smaller 0νββ matrix elements than the IMSRG [94]. Overall, ab initio calcu-lations with different methods are relatively consistent and report smaller matrix elementsthan phenomenological models possibly reproducing a quenching-like effect.75Figure 5.11: Nuclear matrix elements of the neutrino-less double-beta decay transi-tions with ∆T = 2. From left to right, columns correspond to the transitions:8He(0+1 ) → 8Be(0+1 ), 10He(0+1 ) → 10Be(0+1 ) and 48Ca(0+1 ) → 48Ti(0+1 ). Thenumerical calculations use the parameters ~Ω = 20 MeV, except ~Ω = 16MeV for A = 48, and Nmax = 8 (emax = 12) in the NCSM (IMSRG).76Chapter 6Scattering and Reactions of NucleiCalculations of continuum phenomena like nuclear reactions, for example, radiative captureand photo-disintegration are important for the study of astrophysics. Nuclear reactionsdrive the life and death of stars but many of the reactions involved occur at energies wherecounting rates are too low to be reached by experiment. A predictive theory is needed to ex-trapolate to that domain from higher energy experiments (e.g. 12C(α, γ)16O measurementsat TRIUMF [104]) and therefore provide reliable input to astrophysical models.The basis expansion of the NCSM is made of square-integrable states which must goto zero at large distances. Weakly-bound states require the description of this far-distancebehaviour and so are difficult to construct, needing many more high-energy excitations.Unbound states such as resonances and scattering states are impossible to reproduce.The no-core shell model with continuum (NCSMC) [105] is a many-body method whichprovides a unified description of bound states and continuum dynamics. To introduce thistechnique, we will first discuss the general theory of scattering and reactions.6.1 Scattering TheoryIgnoring the internal structure of the particles, a two-particle collision (between a projectilewith mass MP and a target with mass MT ) can be simplified to a one variable problem asthe reaction should not depend on the overall center of mass motion. We can solve for thewavefunction of the system as a function of only the relative position.We expect that the initial state of scattering is a plane wave: eikz, and the final stateshould be a spherical wave: eikrr , where k =√2µE is the wavenumber, E is the availablekinetic energy and µ is the reduced mass of the system:µ =MPMTMP +MT. (6.1)77We look for solutions with the ansatz [36]ψ(r, θ) ≈ N(eikz + f(θ)eikrr), (6.2)where N is a normalization factor.The differential cross section is the observable which can be measured in an experiment,where a beam impinges on a target. It is defined as the proportionality constant D(θ)between the number of particles dN entering the infinitesimal area dσ and scattering intothe infinitesimal solid angle dΩ (see Figure 6.1). i.e.D(θ) =1LdNdΩ=dσdΩ, (6.3)where L is the luminosity (the number of incident particles per unit area per unit time).The probability dP is the probability density |ψ|2 times the infinitesimal volume dV .Conservation of probability equates the initial and final probabilities, for particles withspeed v, in time dt to relate the cross section to the wavefunction, i.e.dPI =dPF ,|ψI |2 dV = |ψF |2 dV ,|N |2 vdtdσ = |N |2 |f |2r2vdtr2dΩ ,dσdΩ= |f(θ)|2 . (6.4)Hence f(θ) is the scattering amplitude, the probability amplitude of elastic scattering inthe direction θ. The normalization N cancels out so in the following we will set it to 1.We must solve the Schrodinger equation for ψ(r, θ), giving f(θ) from which we candirectly extract the scattering cross sections. We can’t use Fermi’s golden rule because thenuclear interaction is not weak, but it will apply to electromagnetic transitions as will beshown in Section 6.3. If we already knew the wavefunction we could extract the scatteringamplitude via an integral [106], i.e.f(θ) =µ2pi∫eikrV (r)ψ(r)d3r , (6.5)but we will instead calculate it directly as a solution to the Schrodinger equation.6.1.1 Partial Wave AnalysisDue to conservation of angular momentum (and parity) different channels scatter indepen-dently. Correspondingly the Schrodinger equation with a spherically symmetric potentialadmits separable solutions ψ(~r) = R`(r)Y`m(θ, φ), u`(r) = rR`(r) satisfies the radial equa-78Figure 6.1: A particle incident on area dσ scatters into the solid angle dΩ. b isthe impact parameter, the distance by which the incident particle would havemissed the scattering center, had it continued on its original trajectory. Figurereproduced from [11].tion: [d2dr2− `(`+ 1)r2− 2µV (r) + k2]u` = 0 . (6.6)Each channel ` gives a different u`(r). The Y`m(θ, φ) are again the spherical harmonics(2.8).In the far field (r →∞), the interaction V (r) and the centrifugal term `(`+1)r2go to zeroand the spherical wave solves [d2dr2+ k2]u` = 0 . (6.7)In the intermediate region (assuming V (r) goes to zero more quickly than the centrifugal`(`+1)r2potential) we have the equation:[d2dr2− `(`+ 1)r2+ k2]u` = 0 , (6.8)which admits the Bessel function solutions R`(r) = j`(kr), n`(kr) which can behave likea outgoing spherical wave when combined. We therefore expect a solution to be a linearcombination:ψ(r, θ, φ) =∑`m(A`mj`(kr) +B`mn`(kr))Y`m(θ, φ) . (6.9)Only m = 0 can contribute due to the cylindrical symmetry of the system, i.e.ψ(r, θ) =∑`(A`j`(kr) +B`n`(kr))P`(cos θ) , (6.10)where P`(cos θ) are the Legendre polynomials (coming from Y0` (θ, φ) =ˆ`√4piP`(cos θ)).A plane wave is made from a combination of only j` (since n` is not finite at the origin),79i.e.eikz =∑`i`(2`+ 1)j`(kr)P`(cos θ) , (6.11)which can be expressed as a combination of an incoming spherical wave I` and an outgoingspherical wave O` via:eikz =∑`i`(2`+ 1)i21kr(I`(kr)−O`(kr))P`(cos θ) , (6.12)whereI`(kr) =kr (n`(kr)− ij`(kr)) , (6.13)O`(kr) =kr (n`(kr) + ij`(kr)) . (6.14)The wavefunction is often expressed in the form:ψ(r, θ) =∑`i`(2`+ 1)i21kr(I`(kr)− S`O`(kr))P`(cos θ) (6.15)where S` is the “S-matrix element” (not yet a true matrix but will be generalized into onein Section 6.1.3). The I`, O` asymptotically go to i`e−ikr, i−`eikr respectively. Thereforewe can match to our expected asymptotic solution:f(θ)eikrr=ψ(r, θ)− eikz=∑`i`(2`+ 1)i21kr(I`(kr)− S`O`(kr))P`(cos θ)−∑`i`(2`+ 1)i21kr(I`(kr)−O`(kr))P`(cos θ)=∑`i`(2`+ 1)i21kr([I` − I`] + (1− S`)O`)P`(cos θ) ,f(θ) =∑`(2`+ 1)i2k(1− S`)P`(cos θ) . (6.16)This can be again redefined in terms of the “partial-wave amplitudes”:a` =12ik(S` − 1) . (6.17)With the incoming plane wave explicit, our wavefunction solution looks like:ψ(r, θ) =eikz +∑`ˆ`2a`eikrrP`(cos θ) , (6.18)80and we can identify:f(θ) =∑`ˆ`2a`P`(cos θ) . (6.19)The problem of finding f(θ) has now been transferred to the problem of finding a`,which is generally done by finding the solutions to the interior wavefunction (where thepotential V (r) dominates) using a different method and then using matching conditionsat some r = a. This is the principle behind the R-matrix method described later in thissection and in Section 6.2.2.The scattering amplitude and S-matrix can be reparametrized using the phase shift δ`,i.e.a` =12ik(e2iδ` − 1)=1keiδ` sin δ` , (6.20)S` =e2iδ` . (6.21)The phase shift is the only effect of the central potential on a spherical wave, i.e. dueto conservation of probability the incoming and outgoing spherical wavefunctions can onlydiffer by a phase, the factor e2iδ` [36].6.1.2 Coulomb ScatteringIn nuclear scattering the incident particles have electric charge. This means the interactionincludes the Coulomb potential: VC(r) =q1q2r . Since this V (r) potential does not drop offmore quickly than the centrifugal term, we can no longer use the Bessel functions.In this case we have the equation:[d2dr2− `(`+ 1)r2− 2kηr+ k2]u =0 (6.22)where η = µq1q2k =q1q2v is the Sommerfeld parameter. The incoming and outgoing waves nowdepend on η i.e. I`(η, kr) and O`(η, kr) respectively, where I` = G`+iF` and O` = G`−iF`.F`(η, kr) and G`(η, kr) are the Coulomb functions [107, 108], which reduce to krj`(kr),krn`(kr) at η = 0.The Coulomb wave solution analogous to the plane-wave (6.12) is:ψC(r, θ) =∑`i`(2`+ 1)i21kr(I`(η, kr)−O`(η, kr))P`(cos θ) . (6.23)The asymptotic scattering states have the following form:ψC(r, θ) =ei[kz+η ln k(r−z)] + fC(θ)ei[kr−η ln 2kr]r, (6.24)81where fC =12ik∑`ˆ`2P`(cos θ)[e2iσ`(η) − 1] and σ`(η) = arg Γ(`+ 1 + iη). fC has the morecompact expression [109]fC =12k sin2 θ2e2i(σ0−η ln sinθ2) . (6.25)The presence of an additional short-range potential in addition to the Coulomb potentialadds a second term to the asymptotic wavefunction, i.e.ψ(r, θ) =ψC(r, θ) + f(θ)ei[kr−η ln 2kr]r. (6.26)The coefficient f(θ) is now a “Coulomb-distorted amplitude”[107, 109] which has an S-matrix element with an additional Coulomb phase σ`, i.e.f(θ) =12ik∑`ˆ`2e2iσ` (S` − 1)P`(cos θ) . (6.27)Analogous to (6.15) the asymptotic wavefunction is:ψ(r, θ) =∑`i`(2`+ 1)i21kreiσ` (I`(η, kr)− S`O`(η, kr))P`(cos θ) (6.28)The total elastic differential cross section will be the combination:dσdΩ= |fC + f |2 . (6.29)The matching condition used to find the wavefunction is the inverse logarithmic deriva-tive (at r = a). This defines R-matrix elementR` =1au(int)` (ka)u(int)′` (ka)=1au(ext)` (ka)u(ext)′` (ka), (6.30)where u(int)` denotes the radial part of the interior wavefunction, numerically integrated,and u(ext)` is the radial component of the exterior wavefunction (6.28), i.e.u(ext)` =i2[I` − S`O`] . (6.31)The matching conditions allows us to solve for S`:S` =I` − aR`I ′`O` − aR`O′`. (6.32)In practice the resolution of all radial equations is not computationally feasible and wemust truncate at some `. Thankfully, phase shifts are strongly suppressed by the centrifugal82term when `(`+1)a2 k2 [110]. Experimentally high ` states will also be suppressed byscreening of the Coulomb potential [106].6.1.3 Multi-channel Scattering and ReactionsThe nuclear reaction may change the composition of the colliding particles, i.e. the finalinternal states of the projectile and target may be different than their initial internal states.When the incoming particles are composite particles they may have many internal states.The different combinations of internal states as well as different mass partitions (i.e. adifferent number of nucleons in each nucleus) can be collected under the term “channels”.Each channel |ν〉 can be thought of as two internal wavefunctions φ coupled together [109],i.e.|ν〉 =[[φ(1)νS1⊗ φ(2)νS2](S) ⊗ YL(Ωrν )](JpiM) , (6.33)where ~rν is the relative coordinate between the center of mass of the two clusters. Thesystem with a total A nucleons is split between two clusters with A1 and A2 nucleonsrespectively (where of course A1 +A2 = A) and so the coordinate ~rν is defined by~rν = (rν , θrν , φrν ) =1A1A1∑i=1~ri − 1A2A∑j=A1+1~rj . (6.34)The total angular momentum J , projection M and parity pi must be conserved and sothere will be a different set of ν and corresponding equations for each JpiM . The JpiMlabels will be suppressed in much of the following. The stationary scattering wavefunctionwith entrance channel νi will look like:∣∣ΨJpiMνi 〉 = ∑νA |ν〉 uννi(rν)rν, (6.35)where A is the anti-symmetrization operator (with respect to exchanges of nucleons betweenthe clusters [111]). The radial part uννi must have the asymptotic form:uννi(rν) =i2√vν[δννiIν(ην , kνrν)− SννiOν(ην , kνrν)] , (6.36)and solves the Schrodinger equation:[d2dr2− `(`+ 1)r2− 2µνV + k2ν]uννi(r)− 2µν∑ν′∫dr′Wνν′(r, r′)uν′νi(r′) =0 . (6.37)Wνν′ is a non-local potential that occurs due to the necessary anti-symmetrization Wνν′ =〈ν| δ(rν − r)VAδ(rν′ − r′) |ν ′〉 − 〈ν|V |ν ′〉 δνν′δ(r − r′) [109]. The wavenumber is kν =83√2µν(E − Eν) where Eν is the threshold for the channel ν. The relative velocity is vν =kν/µν and the normalization v− 12ν keeps the S-matrix unitary.The S-matrix becomes Sννi which will have dimensions corresponding to the numberof initial channels and the number of available channels. This leads to the definition ofthe diagonal phase shifts δν : Sνν = e2iδν and the eigen-phase shifts δλ where e2iδλ are theeigenvalues of S. The S-matrix must be block-diagonal with respect to the total angularmomentum J and parity pi. The off-diagonal S-matrix elements where ν and νi have differentmass partitions would correspond to nuclear reactions. Off-diagonal matrix elements withthe same mass partitions but different internal states would be deemed inelastic scattering.Colliding systems in the entrance channel ν (now denoting just the mass partition andinternal states labeled by total spin S1, S2) can have different initial orientations M1 andM2. When those projections are selected, we have an asymptotic form analogous to (6.26),i.e.ψνM1M2(r, θ) =ψC(r, θ)φ(1)νM1φ(2)νM2+∑ν′M ′1M′2ei(kν′r−ην′ ln 2kν′r)rfνM1M2ν′M ′1M′2(θ)φ(1)ν′M ′1φ(2)ν′M2 .(6.38)This wavefunction can also be constructed by summing over coupled channels (6.33), i.e.ψνM1M2(r, θ) =√4pikν∑Jpi∑SLLˆeiσL (S1M1S2M2|SM) (L0SM |JM)ψνJpiM (r, θ) , (6.39)with M = M1 + M2. The projection of L must be zero due to the cylindrical symmetryabout the beam.Scattering and reaction rates will depend on the partial amplitudes fνM1M2ν′M ′1M′2which fur-ther depend on the multi-channel S-matrix (6.36), i.e.fνM1M2ν′M ′1M′2(θ) =√4pik∑Jpi∑SL∑S′L′Lˆei(σL+σL′ ) (S1M1S2M2|SM)× (L0SM |JM) (S′1M ′1S′2M ′2|S′M ′) (S′M ′L′M −M ′|JM)× (δνν′δSS′δLL′ − Sν′S′L′,νSL)YL′(M−M ′)(θ) . (6.40)A cross section is proportional to the square of the scattering amplitudes and we averageover initial orientations and sum over final orientations. For elastic scattering, the initialand final channels are the same and we have a pure Coulomb term:dσν→νdΩ=1Sˆ1Sˆ2∑M1M2∑M ′1M′2∣∣∣fC(θ)δM1M ′1δM2M ′2 + fνM1M2νM ′1M ′2 (θ)∣∣∣2 . (6.41)84For inelastic scattering, or a reaction, we havedσν→ν′dΩ=1Sˆ21 Sˆ22∑M1M2∑M ′1M′2∣∣∣fνM1M2ν′M ′1M ′2(θ)∣∣∣2 . (6.42)6.1.4 Cluster States from NCSMThe internal quantum numbers of the clusters can be included through the NCSM. A two-cluster state with total angular momentum J , parity pi and isospin T can be describedby:∣∣ΦJpiTνr 〉 = [(|A1λ1Spi11 T1〉 |A2λ2Spi22 T2〉)(sT ) Y`(θrν , φrν )](JpiT ) δ(r − rν)rrν . (6.43)The intrinsic (internally anti-symmetric) wavefunctions of the clusters are |A1λ1Spi11 T1〉 and|A2λ2Spi22 T2〉. They are individually eigenstates (i.e. NCSM solutions denoted by λ) of theA1- and A2-nucleon Hamiltonians respectively. The brackets denote implicitly the angularmomentum coupling of S1S2 to s and s` to J and the isospin coupling T1T2 to T . Theintrinsic cluster quantum numbers and the quantum numbers denoting different partialwaves are grouped together into the single index ν = {A1λ1Spi11 T1;A2λ2Spi22 T2; s`}.In the technique known as the resonating-group method (RGM), or “NCSM/RGM”[112], these states are used as a basis to construct an A-nucleon wavefunction for a givenentrance channel νi, i.e.∣∣ΨJpiTνi 〉 =∑ν∫drr2γJpiTννi (r)rAν∣∣ΦJpiTνr 〉 . (6.44)The γJpiTννi (r) are the variational amplitudes of the relative motion between the clusters.The operator Aν is the inter-cluster anti-symmetrizer. It depends on the ν because it willdepend on the mass partitioning.In this basis the A-nucleon Schrodinger equation can be mapped to a one-dimensionalcoupled-channel equation for the amplitudes:∑ν∫drr2[HJpiTν′ν (r′, r)− EN JpiTν′ν (r′, r)] γJpiTννi (r)r =0 , (6.45)where H and Nare the integration kernels of the Hamiltonian and norm respectively. Theyare simply the matrix elements connecting the basis states, i.e.Hν′ν(r′, r) = 〈Φν′r′ | Aν′HAν |Φνr〉 , (6.46)Nν′ν(r′, r) = 〈Φν′r′ | Aν′Aν |Φνr〉 . (6.47)85In practice, the kernels are calculated by discretizing the cluster states as an expansionover harmonic oscillator states, i.e.|Φνr〉 =Nrel∑n=0Rn`(r) |Φνn〉 , (6.48)where|Φνn〉 =[(|A1λ1Ipi11 T1〉 |A2λ2Ipi22 T2〉)(sT ) Y`(θrν , φrν )](JpiT )Rn`(rν) , (6.49)and the number of harmonic oscillator excitations Nrel used is comparable to the parameterNmax of the internal NCSM states [111].The Hamiltonian can be factorized into intrinsic parts H1, H2 acting on the A1, A2clusters separately plus part which depends only on the relative coordinate, i.e.H =Hrel(r) +H1 +H2 , (6.50)whereHrel(r) =Trel(r) + V¯C(r) + Vloc , (6.51)Trel(r) is the relative kinetic energy between the clusters, V¯C(r) =e2Z1νZ2νr is the averageCoulomb potential (Z1ν and Z2ν are the charge numbers of the clusters of channel ν). Vlocis the short-range part of inter-cluster potential, i.e. the nuclear interaction summing theover the nucleon pairs ij with i in 1 and j in 2 and triplets ijk with ij in 1 and k in 2 (andi in 1 and jk in 2). The long-range Coulomb term is subtracted i.e. Vloc = VNN+3N − V¯C .Despite the similarity, the γννi amplitudes do not represent a wavefunction. However,we can define the function χννi(r) which is [113], i.e.χννi(r)r=∑ν′∫dr′r′2N12νν′(r, r′)γν′νi(r′)r′. (6.52)It solves a Schrodinger equation analogous to (6.37):12µν{− 1r2ddr(r2ddr)+`(`+ 1)r2+ VC(r)}χννi(r)r+∑ν′∫dr′r′2Wνν′(r, r′)χν′νi(r′)r′= (E − Eν) χννi(r)r, (6.53)where µν is the reduced mass in channel ν and Eν = Eλ1 + Eλ2 is the threshold at whichchannels containing |λ1〉 |λ2〉 become open.When scattering is possible (at energies E > Eν), the asymptotic form of χ (in the86region where W = 0) should be the same as (6.36) i.e.χ(ext)ννi (r) =i2√vν[δννiI`(ην , kr) + SννiO`(ην , kr)] . (6.54)When E < Eν , the radial Schrodinger equation has the asymptotic solutions which are theWhittaker functions W`(ην , kr), i.e.χ(ext)ν (r) =CνW`(ην , kr) , (6.55)with Cν the asymptotic normalization constant. A discrete set of these solutions exist givingbound states of the A-nucleon system. Note the lack of second index νi, as it applies onlyto the scattering states.The solutions are found by an R-matrix method, analogous to that mentioned in Sec-tion 6.1.2. The internal wavefunction is discretized by expansion over a convenient basis,in particular a Lagrange mesh [114]. Then the matching conditions are implemented withthe introduction of a Bloch operator Lν into the Schrodinger equation:12µν{− 1r2ddr(r2ddr)+`(`+ 1)r2+ VC(r)− (E − Eν)}χ(int)ννi (r)r+∑ν′∫dr′r′2Wνν′(r, r′)χ(int)ν′νi (r′)r′+ Lν χ(int)ννi (r)r=Lν χ(ext)ννi (r)r, (6.56)where, with matching radius a,Lν = 12µνδ(r − a)(ddr− Bνr), (6.57)and Bν is an arbitrary parameter, tuned depending whether a scattering or bound solutionis sought. See [115] for details.6.2 The No-Core Shell Model with Continuum6.2.1 A Unified BasisA generalized cluster expansion can be used to describe both bound states and scatteringstates of an A-nucleon system on an equal footing.We write a new ansatz which is a superposition of discrete energy eigenstates of theA-nucleon system and continuous cluster states of A1-, A2-nucleon states, i.e.∣∣ΨJpiTAνi 〉 =∑λcJpiTλνi|AλJpiT 〉+∑ν∫drr2γJpiTννi (r)rAν∣∣ΦJpiTνr 〉 . (6.58)87The combined wavefunction contains the NCSM eigenstates of the full A-nucleon systemin the first term. The second term is the RGM wavefunction described in Section 6.1.4.Multiple NCSM calculations are needed to construct this basis and often fewer than thenumber of all possible NCSM eigenstates are included, but generally only the subset cor-responding to the energy region of interest are needed. Finding the unknown coefficientscJpiTλνiand amplitudes γJpiTννi (r) then allows us to construct realistic wavefunctions which canbe used to predict observable properties of the nuclear system.6.2.2 Solving the NCSMCSchematically, the Schrodinger equation in the NCSMC looks like(H hh H)(cγ)=E(I gg N)(cγ). (6.59)Making the conversion γ → χ via (6.52) (known as orthogonalization) this becomes:(E h¯h¯ H¯)(cχ)=E(δ g¯g¯ ∆)(cχ). (6.60)As the NCSM states are already eigenstates the upper left of the left-hand side is Eλλ′ =Eλδλλ′ while the upper left of the right hand side is δλλ′ . The diagonal norm kernel withthe orthogonalized cluster states is ∆νν′(r, r′) = δνν′δ(r−r′)rr′ .The Hamiltonian kernel in the orthogonal basis isH¯νν′(r, r′) =∑µµ′∫ ∫dydy′y2y′2N−12νµ (r, y)Hµµ′(y, y′)N−12µ′ν′(y′, r′) . (6.61)The remaining off-diagonal blocks are the coupling-kernels:g¯λν(r) =∑ν′∫dr′r′2 〈Aλ| A |φν′r′〉N−12ν′ν (r′, r) , (6.62)known as the cluster form factor, andh¯λν(r) =∑ν′∫dr′r′2 〈Aλ|HA |φν′r′〉N−12ν′ν (r′, r) , (6.63)the coupling form factor.Just as in the RGM we can use an R-matrix method. The addition of the A-nucleonNCSM states adds additional terms to the discretized RGM equations and upgrades theBloch operator L to a matrix of zeros with the RGM L in the lower right [111]. We are left88with a diagonalization problem:(H¯ + L)( c¯χ¯)= E(c¯χ¯), (6.64)with H¯ = N−12(E h¯h¯ H¯)N−12 ,(c¯χ¯)= N+12(cχ), N =(δ g¯g¯ ∆).6.2.3 Transition Matrix ElementsThe transition matrix elements of operators connecting NCSMC states will be a sum ofthe internal transitions (operators acting on each cluster) and the operator acting on therelative motion. Similarly to (6.50), operators can be separated into parts:O =O(1) +O(2) +O(rel)(rν) , (6.65)which act on the projectile, target and relative motion respectively.For electromagnetic operators of rank K, the component of the operator acting onthe relative motion O(rel)K will depend on the effective charge qσνK . This is because thecoordinates of the protons in the clusters differs a little from the center of mass of theclusters. In the first cluster rp∈Z1 =−A2A rν and the second rp∈Z2 =A1A rν [116]. Forexample:qσνK =Z1ν(A2A)K+ Z2ν(−A1A )K if σ = E(exact for K = 1) ,1A(Z1ν(A2A1)+ Z2ν(A1A2))if σ = M(K = 1) .(6.66)The electric and magnetic multipole operators are thenMσK =ME(1)K +ME(2)K + qEνKerKν YK(Ωrν ) if σ = E(exact for K = 1) ,MM(1)K +MM(2)K + qMνKµN√34piLrν if σ = M(K = 1) .(6.67)After we solve the NCSMC for the coefficients cλ and γν , we then assemble the transition89matrix elements by inserting the expansions on each side, i.e.〈ψJfpifTffA∣∣∣ |OK | ∣∣∣ψJipiiTiiA 〉 =∑λiλfc∗λf cλi 〈AλfJfpifTf | |OK | |AλiJipiiTi〉+∑λfνc∗λf∫drr2 〈AλfJfpifTf | |OKAν |∣∣φJipiiTiνr 〉 γννi(r)r+∑λiν′cλi∫drr2γν′νf (r)r〈φJfpifTfν′r∣∣∣ |Aν′OK | |AλiJipiiTi〉+∑νν′∫drr2∫dr′r′2γν′νf (r)r〈φJfpifTfν′r∣∣∣ |Aν′OKAν | ∣∣∣φJipiiTiνr′ 〉 γννi(r′)r′ .(6.68)The first term is simply the NCSM matrix elements of the full operator. For the otherterms we insert the operator components. Evaluations of O(i) will require insertions ofcompleteness over the internal states∑j |Aiλj〉 〈Aiλj |, resulting in a combination of theinternal matrix elements and the coupling kernels. The last term is expressed in terms ofthe norm kernels.6.3 Radiative CaptureRadiative capture is a type of fusion nuclear reaction in which two nucleons collide and forma larger bound nucleus and the extra energy is released through a photon. “Radiative” sinceradiation is emitted and “capture” because the projectile is captured by the target nucleus.Figure 6.2 shows this process diagrammatically.When interactions between particles are weak then again Fermi’s Golden Rule can beapplied. For example, when there are two particles in the initial state with 4-momentaP1 = (E1, ~p1) and P2 = (E2, ~p2) the cross section is [11]:σ =Sv12E12E2∑¯i∑f∫|M|2N∏F=1d3pF(2pi)32EF. (6.69)The amplitudeM contains the transition matrix element for the interaction that induces atransition between the initial and final states. An addition to (3.1), is the incoming relativevelocity v:v =√P 21P22 −m21m22E1E2. (6.70)In a radiative capture reaction T (P, γ)F , we calculate the cross section for a 2 → 290ψPψTψFγPPPTPFQFigure 6.2: The Feynman diagram and kinematics for radiative capture.process, i.e.dσ =1v14EPET∑¯i∑f∑λ∣∣∣MλQFI ∣∣∣2 d3q(2pi)32ω d3pF(2pi)32EF . (6.71)The amplitude is very similar to (3.3), but the initial state is two clusters: the target Tand projectile P . The average over initial states is an average over the angular momentaof both clusters (target |sTmT 〉 and projectile |sPmP 〉), i.e.∑¯i =1sˆ2P sˆ2T∑mPmT. The sumover final states |JfMf 〉 is∑f =∑Mf.Similarly to (3.12), the amplitude isMλQFI =(2pi)4δ(PP + PT − PF −Q) 〈f |~e∗λ · ~J (q) |i〉 8EFEPET . (6.72)Cancelling the 2E factors and doing the final state integral, the cross section is thendσ =e2vd3q(2pi)32ω2piδ(EF − EI − ω)∑¯i∑f∑λ∣∣∣Hfiλ ∣∣∣2 , (6.73)whereHfiλ = 〈f |~e∗λ · ~J (~q) |i〉 . (6.74)Implicit in the definition of the multipole operators is that the λ polarizations aretransverse to ~q, the direction of the photon. However the projection of the spherical tensoroperator should be defined in the same coordinates as the quantum states it acts on. In areaction, we define the angular momentum projection of the nuclear states with respect tothe relative velocity (i.e. the direction of the beam). This requires us to rotate the operators(or states) using the Wigner D-matrices (Section A.3).In principle, if the initial bound state in γ decay were polarized with respect to someaxis, the differential decay rate would require a rotation of the operator to change basis91from the axes set by measurement of the photon to the axes set by the polarization [54].The set of coordinates defined relative to zˆ along the beam is the “vˆ-frame”, the setof coordinates defined relative to zˆ along the photon is the “qˆ-frame”. In our case, thedirection of the photon is described by the polar coordinates (θq, φq) in the vˆ-frame. Thecorresponding rotation vˆ → qˆ is R(φq, θq,−φq) [117]. The state |i〉 in the qˆ-frame is |i〉q =R(φq, θq,−φq) |i〉v, related by a rotation operator to the state |i〉 in the vˆ-frame. The matrixelement is expanded over multipole operators via (3.39) in the qˆ-frame. In order to use statesin the vˆ-frame we apply the rotation operators. Using (A.19) and (A.29) the result containsthe D-matrices, i.e.〈f |q ~e∗λ · ~J (q) |i〉q = 〈f |vR†(φq, θq,−φq)~e∗λ · ~J †(q)R(φ, θ,−φ) |i〉v= 〈f |vR†(φq, θq,−φq)∑jm(−i)j√2pijˆ[T Ej−λ(q) + λT Mj−λ(q)]R(φ, θ,−φ) |i〉v= 〈f |vR(−φq,−θq, φq)∑jm(−i)j√2pijˆ[T Ej−λ(q) + λT Mj−λ(q)]R†(−φ,−θ, φ) |i〉v= 〈f |v∑jm(−i)j√2pijˆ[T Ejm(q) + λT Mjm(q)]Djm−λ(−φq,−θq, φq) |i〉v .(6.75)In the following manipulations we use |i〉 = |i〉v and so we insert into (6.74) the operator:~e∗λ · ~J (q) =∑jm(−i)j√2pijˆ[T Ejm(q) + λT Mjm(q)]Djm−λ(−φq,−θq, φq) . (6.76)The initial state can be expanded over a coupled basis, i.e. we combine (6.39) and(6.43):|νimPmT 〉 =√4pikνi∑JLSLˆeiσL (sPmP sTmT |SM) (L0SM |JM) |νiLSJM〉 . (6.77)|νiLSJM〉 represent scattering states in a coupled basis (e.g. NCSMC states∣∣ψJpiMAνi 〉) wherehere νi describes the mass partition and internal states.The sum in (6.73) can be expanded∑¯i∑f∑λ∣∣∣Hfiλ ∣∣∣2 = 1sˆ2P sˆ2T∑mPmT∑Mf∑λ〈JfMf |∑σjmDjm−λ[√2pijˆλσ(−i)jT σjm]|sPmP sTmT 〉×〈JfMf | ∑σ′j′m′Dj′m′−λ[√2pijˆ′λσ(−i)j′T σ′j′m′]|sPmP sTmT 〉∗ ,(6.78)where σ = (0, 1) corresponding to (E,M) respectively and the νi labels and θq, φq are92suppressed.First, we substitute (6.77) (|sPmP sTmT 〉 = |νimPmT 〉), using the replacement: Xσjm =√4pik√2pijˆλσ(−i)jT σjm, i.e.sˆ2P sˆ2T∑¯i∑f∑λ∣∣∣Hfiλ ∣∣∣2 = ∑mPmT∑Mf∑λ{〈JfMf |∑σjmDjm−λXσjm×∑LSJLˆeiσL (sPmP sTmT |SM) (L0SM |JM) |LSJM〉}×{〈JfMf |∑σ′j′m′Dj′m′+λXσ′j′m′×∑L′S′J ′Lˆ′eiσL′(sPmP sTmT |S′M) (L′0S′M ′|J ′M ′) ∣∣L′S′J ′M ′〉}∗ .(6.79)We make use of the sum over mP ,mT (A.4):∑mPmT(sPmP sTmT |SM)(sPmP sTmT |S′M)=δSS′∑M, (6.80)and the Wigner-Eckart theorem (2.33):〈JfMf |Xσjm |LSJM〉 =Jˆ−1f (JMjm|JfMf ) 〈Jf |∣∣Xσj ∣∣ |LSJ〉 . (6.81)Defining XσjJfLSJ = 〈Jf |∣∣∣Xσj ∣∣∣ |LSJ〉 and making use of (A.31) with m = m′, we havesˆ2P sˆ2T∑¯i∑f∑λ∣∣∣Hfiλ ∣∣∣2 =Jˆ−2f ∑Mf∑λ∑σjm∑LSJM∑σ′j′∑L′J ′LˆLˆ′ei(σL−σL′ )×XσjJfLSJXσ′j′∗JfL′SJ ′∑K(−)m+λ (jmj′ −m|K0) (j − λj′λ|K0)DK00× (L0SM |JM) (L′0SM |J ′M) (JMjm|JfMf ) (J ′Mj′m|JfMf) .(6.82)We identify DK00(−φ,−θ, φ) as the Legendre polynomial PK(cos θq). The sums over mand M produce 6js by making use of (A.7) and (A.12) repeatedly, i.e.∑M∑m(−)m (jmj′ −m|K0) (L0SM |JM) (L′0SM |J ′M) (JMjm|JfMf ) (J ′Mj′m|JfMf)= (−)S−Jf Jˆ2f Jˆ Jˆ ′{J J ′ Kj′ j Jf}{J J ′ KL′ L S}(L0L′0|K0) . (6.83)The remaining sum over projections is∑Mf= Jˆ2f . And so converting X back to T and93using the shorthand T σjJfLSJ = 〈Jf |∣∣∣T σj ∣∣∣ |LSJ〉, we havesˆ2P sˆ2T∑¯i∑f∑λ∣∣∣Hfiλ ∣∣∣2 =Jˆ2f∑λ∑σj∑LSJ∑σ′j′∑L′J ′LˆLˆ′ei(σL−σL′ )XσjJfLSJXσ′j′∗JfL′SJ ′×∑KPK (−)S−Jf+λ Jˆ Jˆ ′{J J ′ Kj′ j Jf}{J J ′ KL′ L S}× (L0L′0|K0) (j − λj′λ|K0)=8pi2k2Jˆ2f∑σjσ′j′∑SLL′JJ ′J(∑λ(−)λ (j − λj′λ|K0)λσ+σ′)× (−i)j−j′LˆLˆ′ei(σL−σL′ )T σjJfLSJTσ′j′∗JfL′SJ ′∑KPK× (−)S−Jf jˆjˆ′Jˆ Jˆ ′{J J ′ Kj′ j Jf}{J J ′ KL′ L S}(L0L′0|K0) .(6.84)This can be simplified, similarly to (3.54), with:∑λλσ+σ′(−)λ (j − λj′λ|K0) = (−)1 (j − 1j′1|K0)+ (−)σ+σ′−1 (j1j′ − 1|K0)=−[1 + (−)σ+σ′+j+j′−K] (j − 1j′1|K0)=(j − 1j′1|K0)−2 if (−)σ+σ′+j+j′ = (−)K ,0 otherwise .(6.85)The result issˆ2P sˆ2T∑¯i∑f∑λ∣∣∣Hfiλ ∣∣∣2 =16pi2k2 Jˆ2f∑KPK′∑σσ′jj′∑SLL′JJ ′(−)S−Jf+1 jˆjˆ′Jˆ Jˆ ′LˆLˆ′ei(σL−σL′ )× (−i)j−j′T σjJfLSJTσ′j′∗JfL′SJ ′{J J ′ Kj′ j Jf}{J J ′ KL′ L S}× (L0L′0|K0) (j − 1j′1|K0) , (6.86)where∑′σσ′jj′ has the condition (−)σ+σ′+j+j′ = (−)K .The angular momentum algebra can be collected into a coefficient:aK =′∑σσ′jj∑SLL′JJ ′(−)S−Jf+1 jˆjˆ′Jˆ Jˆ ′LˆLˆ′ei(σL−σL′ )(−i)j−j′T σjJfLSJTσ′j′∗JfL′SJ ′×{J J ′ Kj′ j Jf}{J J ′ KL′ L S}(L0L′0|K0) (j1j′ − 1|K0) . (6.87)94Inserting everything into (6.73), where v has been absorbed into the normalization of thescattering wavefunctions, we first integrate over φq to get a factor of 2pi and the moved cos θq to the left-hand side to get the differential cross sectiondσd cos θq=4piα2piω2dω(2pi)32ω2piδ(EI − EF − ω)∑¯i∑f∑λ∣∣∣Hfiλ ∣∣∣2=αωfrJˆ2fsˆ2P sˆ2T16pi2k2∑KaKPK(cos θq) , (6.88)where fr = (1 +ω−pP cos θqMF)−1 is the recoil factor, with pP the projectile momentum andMF is the mass of the final state bound nucleus. All the PK(cos θq) with K 6= 0 integrateto 0 (neglecting recoil).The integrated cross section is thenσ =32pi2αωk2Jˆ2fsˆ2P sˆ2Ta0 , (6.89)wherea0 =′∑σσ′jj∑SLL′JJ ′(−)S−Jf+1 jˆjˆ′Jˆ Jˆ ′LˆLˆ′ei(σL−σL′ )T σjJfLSJTσ′j′∗JfL′SJ ′×{J J ′ 0j′ j Jf}{J J ′ 0L′ L S}(L0L′0|00) (j1j′ − 1|00)=∑σj∑SLJ∣∣∣T σjJfLSJ ∣∣∣2 . (6.90)This leaves us with the same result as the γ-decay rate (3.55) but with a sum over initialLSJ channels rather than just Ji and the additional normalization factor4pivk2.In the long-wavelength limit this becomesσ =32pi2αk2J2fsˆ2P sˆ2T∑σjω2j+1[(2j + 1)!!]2j + 1j∑LiSiJi∣∣∣〈ΨJfpifTf ∣∣ ∣∣Mσj ∣∣ ∣∣∣ΨJipiiTiSiLi 〉∣∣∣2 , (6.91)matching the literature up to normalization conventions [107, 108].6.4 Pair ProductionIf the photon emitted in a γ-decay or radiative capture has high enough energy (ω >2me) then an electron-positron pair may be produced. Diagrammatically this is shown inFigure 6.3 and Figure 6.4. The rate is much lower than photon decays (except in 0+ → 0+transitions where γ decays are forbidden [118, 119]) but recent experiments have been able tomeasure the rate, in particular in radiative capture reactions [120] where additional energy95is provided by the initial kinetic energy. The relevant nuclear operators and kinematics areanalogous to electron scattering [56, 121] but with the complication of a larger final phasespace.ψIψFe+e−PIPFQγP+P−Figure 6.3: The Feynman diagram for the pair production process.ψ(a)P ;PPψ(A−a)T ;PTψ(A)F ;PFe+, P+e−, P−γ,QFigure 6.4: The Feynman diagram and kinematics for radiative capture with pairproduction.6.4.1 Pair Production in Bound-Bound DecayThis process is simpler to calculate in a bound-bound transition, just as γ-decay (Section 3.1)is simpler than radiative capture (Section 6.3).We need to calculate the decay rate of a 1→ 3 process, i.e.dΓ =12MI∑¯i∑f∑s+s−∣∣Ms+s−FI ∣∣2 d3p+(2pi)32E+ d3p−(2pi)32E−d3pF(2pi)32EF, (6.92)where P+ = (E+, ~p+) and P− = (E−, ~p−) are respectfully the 4-momenta of the emittedpositron and electron and PF = (EF , ~pF ) is the 4-momentum of the final state nucleus. Theintermediate photon has 4-momentum Q = (ω, ~q).The amplitude of the diagram in Figure 6.3 is evaluated following the Feynman rules96listed in Section 3.1. We insert the Dirac spinors u¯s− for the outgoing electron and νs+ forthe outgoing positron, then insert the propagator−iηµνQ2for the exchanged photon betweenthe vertex factors ieγµ and ieJν where we use short-hand for the nuclear current Jν =〈f | Jν(q) |i〉. The amplitude isMs+s− = u¯s−(P−) (ieγµ)(−iηµνQ2)(ieJν) vs+(P+) . (6.93)We then insert the appropriate delta functions and normalization factors and integrate overthe intermediate 4-momentum Q, i.e.∑¯i∑s+s−∑f∫d4Q(2pi)4∣∣Ms+s−∣∣2 (2pi)4δ(PI − PF −Q)(2pi)4δ(Q− P+ − P−)4EIEF=∑¯i∑s+s−∑f∣∣Ms+s−∣∣2 (2pi)4δ(PI − P+ − P− − PF )4MIEF . (6.94)After this step, the symbol Q is defined by the external momenta:Q =P+ + P− , (6.95)Q2 =2(E−E+ − p−p+x+m2), (6.96)q2 =p2− + p2+ + 2p−p+x . (6.97)To simplify the expressions we define x = cos θee = pˆ+ · pˆ− and m = me.The lepton tensor is computed by via the lepton currents, summing over the final spins,i.e.`µν =∑ss¯(u¯s−(P−)γµvs+(P+)) (v¯s+(P+)γνus−(P−))=Tr[γµ(/P− +m)γν(/P+ −m)]=4[P−µP+ν + P−νP+µ − ηµν(P− · P+ +m2)]. (6.98)Folding this with the nuclear current, we get∑¯i∑s+s−∑f∣∣Ms+s−∣∣2 =∑¯i∑f`µνJ ∗µJν , (6.99)where the lepton tensor and nuclear current contractions result in14`µνJ ∗µJν = (P+ · J ) (P− · J ∗) + (P+ · J ∗) (P− · J )− (J · J ∗)(P+ · P− +m2)=Q4q4|ρ|2 (E+E− + ~p+ · ~p− −m2)+ ∣∣J T ∣∣2(E+E− +m2 − (~p+ · ~q) (~p− · ~q)q2),(6.100)97where ρ is the charge density and ~J T is the transverse current:~J T =∑λ=±1(~e∗λ · ~J)~eλ , (6.101)or equivalently~J T =(~ex · ~J)~ex +(~ey · ~J)~ey . (6.102)The result (6.100) is not obvious and comes about through several steps.• It can be shown that we may make the replacement J = (ρ, ~J ) →(−Q2q2ρ, ~J T)viasimilar arguments to Siegert’s theorem i.e. longitudinal components may be relatedto the time-like component [53]. This is shown later in the manipulations simplifying(6.118) into (6.122).• Defining % = −Q2q2ρ, we have the intermediate result:14`µνJ ∗µJν =2E+E− |%|2− E+%(~p− · ~J T∗)− E−%∗(~p+ · ~J T)− E+%∗(~p− · ~J T)− E−%(~p+ · ~J T∗)+(~p+ · ~J T)(~p− · ~J T∗)+(~p+ · ~J T∗)(~p− · ~J T)−(|%|2 − ∣∣J T ∣∣2) (E¯E − ~p+ · ~p− −m2) . (6.103)• The terms which mix % and ~J T must vanish as ρ does not change the M of the initialstate but ~J T does. This is because ~J T consists of the operators J±. The terms whichmix λ = 1 and λ = −1 will also be zero e.g. J+J ∗−.• ~p+ and ~p− lie in the same plane, so since ~q is along zˆ (pz zˆ = ~p·~qq2 ~q), we may extractthe transverse components of the momenta ~pT = ~p− ~p·~qq2 ~q, i.e.~p · ~J T = pxJx + py ,Jy (6.104)and so(~p+ · ~J T)(~p− · ~J T∗)+(~p+ · ~J T∗)(~p− · ~J T)=(~p+ · ~p− − (~p+ · ~q) (~p− · ~q)q2) ∣∣J T ∣∣2 .(6.105)The charge density relates directly to the Coulomb operators, i.e.∑¯i∑f∣∣ρ2∣∣ =4piJˆ2i∑J≥0|CJ |2 ≡∣∣MC∣∣2 , (6.106)while the transverse current relates to the transverse electric and magnetic operators98∑¯i∑f∣∣J T ∣∣2 =4piJˆ2i∑J≥1∣∣T EJ ∣∣2 + ∣∣T MJ ∣∣2 ≡ ∣∣MEM ∣∣2 . (6.107)Putting everything together, we first do the integral over the final nucleus momentum:dΓ =(4piαQ2)2 d3p+(2pi)32E+d3p−(2pi)32E−2piδ(MI − EF − E+ − E−)∑¯i∑f`µνJ ∗µJν . (6.108)Second, we do the angular integrals. There are four angles determining the directions ofthe positron and electron momenta. The amplitude only depends on the relative separationangle θee. Therefore by measuring the direction of ~p+ relative to ~p− we can replace theintegration d cos θ+ → d cos θee = dx. The remaining three angles (dφ+d cos θ−dφ−) thengive a constant factor (8pi2) in the total rate, i.e.dΓ =(4piαQ2)2( 1E+E−)2pidxp2+dp+(2pi)34pip2−dp−(2pi)32piδ(MI − EF − E+ − E−)∑¯i∑f`µνJ ∗µJν .(6.109)Third, we insert the lepton-nuclear tensor contraction and do the integral over dp−using the remaining δ. We get a factor of√p2−+m2p− =E−p− and a recoil factor fr =(1 + E−E+(1 + p+xp−))−1.We can change integration variables p+dp+ = E+dE+, resultingin the differential decay rate:dΓdxdE+=4α2piQ4p−p+fR[Q4q4∣∣MC∣∣2 (E+E− + p+p−x−m2)+∣∣MEM ∣∣2(E+E− +m2 − (~p+ · ~q) (~p− · ~q)q2)]. (6.110)This matches [122] where they make the further change of variables to the dimensionlesskinematic variable y (the “asymmetry parameter”) defined by:E+ − E− = yω . (6.111)The integral is then presented using dy = ω2 dE+. y can be integrated from −(1 − 2meω ) to1− 2meω , equivalent to integrating E+ from 0 to ω − 2me.996.4.2 Pair Production in Radiative CaptureWhen pair production occurs during a radiative capture reaction, i.e. T (P, e+e−)F thenwe must calculate the cross section for a 2→ 3 process. The cross section isdσ =1v14EPET∑¯i∑f∑s+s−∣∣∣Ms+s−fi ∣∣∣2 d3p+(2pi)32E+ d3p−(2pi)32E−d3pF(2pi)32EF. (6.112)Following Section 6.4.1, we havedσ =1v14EPETd3p+(2pi)32E+d3p−(2pi)32E−d3pF(2pi)32EF(2pi)4δ4(PP + PT − PF − P+ − P−)8EPETEF× e4Q4∑¯i∑f`µνJµJ ∗ν=1ve4Q4d3p+(2pi)32E+d3p−(2pi)32E−2piδ(EP + ET − EF − E+ − E−)∑¯i∑f`µνJµJ ∗ν . (6.113)Here again the lepton 4-momenta are P± = (E±, ~p±) and the final nucleus 4-momentum isPF = (EF , ~pF ). The lepton tensor (6.98) is again `µν = 4[P−µP+ν + P−νP+µ − ηµν(P− · P+ +m2)].The contraction between the lepton tensor and nuclear currents are much more compli-cated in this case. The kinematics mix in additional nuclear operators, and rather than thetwo terms in (6.110), the cross section will separate into six terms. Following the notationof [123, 124], we have14∑¯i∑f`µνJµJ ∗ν =6∑j=1vjRj . (6.114)The expansion of the contraction is done by collecting like terms, in this case we donot make the J →(−Q2q2ρ, ~J T)substitution, but rather make use of J = (ρ, Jx, Jy, Jz)explicitly where Jz =ω|~q|ρ. The lepton-nuclear tensor contraction is then14`µνJµJ ∗ν = (P+ · J ) (P− · J ∗) + (P+ · J ∗) (P− · J )− (J · J ∗)(P+ · P− +m2), (6.115)as in (6.100).Let’s define A = P+ and B = P− to avoid typing so many subscripts. Q = A + B andwe define C = A − B (such that A = 12(C + Q), B = −12 (C − Q)). This simplifies theequation, i.e.14`µνJµJ ∗ν =− |J |2(A ·B +m2)− 12|C · J |2 , (6.116)but the evaluation then expands it into something complicated. We write A = (α,~a) =(E+, ~p+) and B = (β,~b) = (E−, ~p−) so ω = α + β and ~q = ~a + ~b. Finally, C = (ς,~c) =100(ς, cx, cy, cz) where ς = α− β and ~c = ~a−~b.Expanding, we have14`µνJµJ ∗ν = = −(ρ2 − J2x − J2y − J2z) (αβ − ~a ·~b+m2)− 12[ς2ρ2 + c2xJ2x + c2yJ2y + c2zJ2z− ςcx (ρJ∗x + Jxρ∗)− ςcy(ρJ∗y + Jyρ∗)− ςcz (ρJ∗z + Jzρ∗)+ cxcy(JxJ∗y + JyJ∗x)+ cxcz (JxJ∗z + JzJ∗x) + cycz(JyJ∗z + JzJ∗y) ]. (6.117)We collect like terms (sometimes dropping the || for brevity). Rj here really means∑¯i∑f Rj .Term 1: ρ2, J2z , ρJ∗z , Jzρ∗v1R1 =−(ρ2 − J2z) (αβ − ~a ·~b+m2)− 12(ς2ρ2 − ςczρJ∗z − czςJzρ∗ + c2zJ2z)=− ρ2(1− ω2q2)(αβ − ~a ·~b+m2)− 12(ς − cz ω|~q|)2ρ2 . (6.118)Simplification requires some manipulation. We use cz =~c·~q|~q| , and some identities:~c · ~q =b2 − a2=β2 − α2=ςω , (6.119)Q2 =ω2 − q2 , (6.120)Q22=αβ − ~a ·~b+m2 . (6.121)We take R1 ≡ ρ2 and sov1 =(Q2q2)(αβ − ~a ·~b+m2)− 12(ς − ςω|~q|ω|~q|)2=(Q2q2)(Q22)− ς22(−Q2q2)2=(Q4q4)(q2 − ς22)=(Q4q4)(αβ + ~a ·~b−m2). (6.122)This is identical in form to the∣∣MC∣∣2 term in Section 6.4.1).101Term 2: ρJ∗x , Jxρ∗, JzJ∗x , JxJ∗zv2R2 =− 12[−ςcx (ρJ∗x + Jxρ∗) + czcx (JzJ∗x + JxJ∗z )] ,v2 ≡− 1√2cx(α− β − ω|~q|cz), (6.123)R2 ≡Re (ρ∗ (J+ − J−)) , (6.124)where we’ve used:Jxρ∗ + ρJ∗x =2Re(Jxρ∗)=2Re(−1√2(J+ − J−) ρ∗) . (6.125)Term 3: ρJ∗y , Jyρ∗, JzJ∗y , JyJ∗zv3R3 =− 12[−ςcy (ρJ∗y + Jyρ∗)+ czcy (JzJ∗y + JyJ∗z )] ,v3 ≡− 1√2cy(α− β − ω|~q|cz), (6.126)R3 ≡Im (ρ∗ (J+ + J−)) . (6.127)Since Jy has the additional factor of i:Jyρ∗ + ρJ∗y =2Re(Jyρ∗)=2Im(−1√2(J+ + J−) ρ∗) . (6.128)Jx, Jy Cross Terms :The remaining terms are(|Jx|2 + |Jy|2) (A ·B +m2)− 12(c2x |Jx|2 + c2y |Jy|2 + cxcy(JxJ∗y + JyJ∗x)). (6.129)We use|Jx|2 =12(|J+|2 − J+J∗− − J−J∗+ + |J−|2), (6.130)|Jy|2 =12(|J+|2 + J+J∗− + J−J∗+ + |J−|2), (6.131)JxJ∗y =−i2(|J+|2 + J+J∗− − J−J∗+ − |J−|2), (6.132)JyJ∗x =i2(|J+|2 − J+J∗− + J−J∗+ − |J−|2), (6.133)102and|Jx|2 + |Jy|2 = |J+|2 + |J−|2 , (6.134)JxJ∗y + JyJ∗x =− i(J+J∗− − J−J∗+). (6.135)Thenc2x |Jx|2 + c2y |Jy|2 + cxcy(JxJ∗y + JyJ∗x)=12(c2x + c2y) (|J+|2 + |J−|2)− 12(c2x − c2y) (J+J∗− + J−J∗+)− icxcy(J+J∗− − J−J∗+). (6.136)Term 4: |J+|2, |J−|2v4R4 =(|J+|2 + |J−|2)(αβ − ~a ·~b+m2)− 12· 12(c2x + c2y) (|J+|2 + |J−|2)=[−14(c2x + c2y)+ αβ − ~a ·~b+m2] [|J+|2 + |J−|2],v4 ≡− 14(c2x + c2y)+ αβ − ~a ·~b+m2 , (6.137)R4 ≡ |J+|2 + |J−|2 . (6.138)v4 exactly matches the coefficient in front of∣∣MEM ∣∣2 in Section 6.4.1, sincec2x + c2y =∣∣∣∣~c− ~c · ~qq2 ~q∣∣∣∣2 ,αβ − ~a ·~b+m2 − c2x + c2y4=αβ +m2 − (~a · ~q)(~b · ~q)q2. (6.139)Term 5: J+J∗− + J−J∗+v5R5 =12(c2x − c2y)Re(J∗+J−),v5 ≡12(c2x − c2y), (6.140)R5 ≡Re(J∗+J−). (6.141)103Term 6: J+J∗− − J−J∗+v6R6 =− 12[−icxcy (J+J∗− − J−J∗+)] ,=− cxcyIm(J∗+J−),v6 ≡− cxcy (6.142)R6 ≡Im(J∗+J−). (6.143)Differential Cross Section The six terms will be inserted in to the differential cross sectionafter shifting to spherical coordinates d3p = p2dpdΩ and using the δ to first do the integralover p−, then making the change of variables: p+dp+ = E+dE+, i.e.dσ =4vα2Q4frp−p+dE+dΩ+dΩ−(2pi)3∑jvjRj ,d5σdE+dΩ+dΩ−=4α2(2pi)3frvp+p−Q4∑jvjRj , (6.144)where appears the recoil factor fr =(1 +(E−p−)p−+p+x−pP cos θ−EF)−1.In radiative capture we have to compute on the matrix elements of the transverse elec-tromagnetic current (i.e. with λ = ±1). Those operators are Xσλjm =√4pik√2pijˆλσ)(−i)jT σjmwhere σ denotes electric or magnetic multipoles (σ = 0, 1 respectively). In pair productionwe have a virtual photon and hence the polarization can also be λ = 0. This is apparentdue to the presence of 〈f | ρ |i〉 terms in the cross section.To evaluate both ρ and J terms (and their combinations) we define the operator N µjmwhere µ = 0 corresponds to the charge operator ρ. µ = ±1 corresponds to the currentoperator with λ = ±1: N λjm =∑σXσλjm (Hλ in Section 6.3). We haveN 0jm =√4pik√4pijˆ(−i)jCjm , (6.145)N λjm =√4pik√2pijˆ∑σλσ(−i)jT σjm . (6.146)We will have to multiply by a more general D-matrix Djmµ where µ ∈ (0,±1). If wedefine the leptonic coordinate system to have an axis perpendicular to both the internalphoton and initial velocity, i.e. zˆ = qˆ and yˆ = vˆ × qˆ (xˆ = yˆ × zˆ), the necessary rotation isR(φq, θq, 0) and so Djmµ(−φq,−θq, 0) appears. We defineNµ =∑jmN µjmDjm−µ(−φq,−θq, 0) . (6.147)104j Rj NµN∗κ1 ρ2 N0N∗02 Re(ρ(J+ − J−)∗) Re(N0(N+ −N−)∗)3 Im(ρ(J+ + J−)) Im(N0(N+ +N−)∗)4 J2+ + J2− N+N∗+ +N−N∗−5 Re(J+J∗−) Re(N+N∗−)6 Im(J+J∗−) Im(N+N∗−)Table 6.1: Notation for Rj in terms of N .In the end we will evaluate in terms of the reduced matrix elements〈Jf | |N µj | |LSJ〉 = N µjJfLSJ , (6.148)analogous to XσjJfLSJ .The Rj terms correspond to the combinations in Table 6.1 whereNµN∗κ =∑¯i∑f〈f |∑jmDjm−µN µjm |i〉〈f |∑j′m′Dj′m′−κN κj′m′ |i〉∗ . (6.149)A general expression for NµN∗κ follows from very similar steps to Section 6.3, i.e.NµN∗κ =∑¯mPmT∑Mf〈JfMf |∑jmDjm−µN µjm |spmP sTmT 〉〈JfMf |∑j′m′Dj′m′−κN κj′m′ |sPmP sTmT 〉∗=(sˆP sˆT Jˆf )−2∑Mf∑jm∑LSJM∑j′∑L′J ′LˆLˆ′ei(σL−σL′ )Djm−µ(−)mκDj′−m−κN µjJfLSJNκj′∗JfL′SJ ′× (L0SM |JM) (L′0SM |J ′M) (JMjm|JfMf ) (J ′Mj′m|JfMf)=(sˆP sˆT Jˆf )−2∑Mf∑jm∑LSJM∑j′∑L′J ′LˆLˆ′ei(σL−σL′ )N µjJfLSJNκj′∗JfL′SJ ′×∑K(−)m+κ (jmj′ −m|K0) (j − µj′κ|K(κ− µ))DK0(κ−µ) (6.150)× (L0SM |JM) (L′0SM |J ′M) (JMjm|JfMf ) (J ′Mj′m|JfMf)=Jˆ2fsˆ2P sˆ2T∑jj′∑SLL′JJ ′(−)S−Jf−κ Jˆ Jˆ ′LˆLˆ′ei(σL−σL′ )N µjJfL′SJ ′Nκj′∗JfLSJ×∑K{J J ′ Kj′ j Jf}{J J ′ KL′ L S}(L0L′0|K0) (jµj′ − κ|K(µ− κ)) dK(µ−κ)0(θq) ,(6.151)where using (A.21), we have DK0M (−φq,−θq, 0) = dK0M (−θq) = dKM0(θq).The full differential cross section is difficult to compute explicitly for the 5-dimensional105phase space but this can be simplified and made into a form that may be compared toexperiment by considering only the case where the photon is perpendicular to the beamsuch that the electron and positron are in the same plane i.e. cz =~c·~qq ,cy = ~c − ~c·~qq2 ~q (ascx = 0 in this situation). In this case, v2 = v6 = 0, in addition R3 = 0, and so we actuallyonly need c2z and c2y. We havec2z =(~c · ~q)2q2=((~a−~b) · (−~a−~b))2q2=(b2 − a2)2q2, (6.152)andc2y =∣∣∣∣~c− ~c · ~qq2 ~q∣∣∣∣2=c2 − (~c · ~q)2q2=a2 − 2~a ·~b+ b2 − (b2 − a2)2q2. (6.153)i vi vi(θq =pi2 , φa =pi2 , φb = −pi2 )1 Q4q4(αβ + abx−m2) Q4q4(αβ + abx−m2)4 −14(c2x + c2y)+ αβ − abx+m2 −14(a2 + b2 − 2abx− (b2−a2)2q2)+ αβ − abx+m25 12(c2x − c2y) −12 (a2 + b2 − 2abx− (b2−a2)2q2 )Table 6.2: Simplified formulation of kinematic factors vi corresponding to theATOMKI experimental setup.K,M dKM0(θ) dK0M (pi2 )0, 0 1 11, 0 cos θ 01,±1 ∓ 1√2sin θ ∓ 1√22, 0 12(3 cos2 θ − 1) −122,±1 ∓√38 sin 2θ 02,±2√38 sin2 θ ±√38Table 6.3: Relevant values of the Wigner d-matrix factor [35].We then can convert the kinematics into a form (Table 6.2) that depends only on therelative angle through ~a ·~b = |~a||~b| cos θee = abx as in Section 6.4.1. Shown in Table 6.3 are106the simplified expressions that result for dK(µ−κ)0(θq) (which appears in (6.151)).6.4.3 Hypothetical Intermediate ParticlesThere have been claims that there are contributions to pair production in radiative cap-ture from BSM bosons. This hypothetical particle has an unknown spin and parity butthe presence of a signal could be enhanced by a resonance in the nuclear system, if thecorresponding operator has complementary Jpi. There have been a number of attempts tounderstand the observations of experimental anomalies from a particle physics perspective[125–127].Here we approximate the transition operators in the low-energy limit. In practice thecalculation is then to insert a new operator O(X) into the transition strength of the radiativecapture (6.91). The label σ would then be dropped and we look at the leading multipolej = 1. These operators may be sensitive to the momentum transfer carried by the newparticle i.e. kX =√ω2 −m2X . The total available energy ω is determined by the differencebetween the initial kinetic energy and final state bound state energy.For a vector particle, we set the operator proportional to the E1 [128], i.e.O(V ) =VME1kXω. (6.154)For a pseudo-scalar, the operator is Σ′′ [129] which reduces to S in the long-wavelengthlimit, i.e.O(P ) =PSkX . (6.155)For an axial vector we also get an operator proportional to S [130], i.e.O(A) =AS√2 +m2Xω2. (6.156)In principle, the couplings X would be fitted to match experiments.107Chapter 7NCSMC ResultsRadiative capture reactions are an important pathway by which heavier nuclei can be cre-ated in astrophysical environments. For example, d(p, γ)3He occurs as a primary step instellar burning. Using the NCSMC technique we may address radiative capture simulta-neously with the nuclear structure of the composite system, elastic scattering, inelasticscattering and reactions involving the same total mass number.In the A = 6 system, we computed 4He(d, γ)6Li. This reaction is an contributor to theprediction of the primordial abundance of 6Li. Accurate determination of the rate requiresprediction within the big bang energy window (30 keV - 400 keV), which experimentallyhindered by low counting rates, current data require extrapolation to low-energy. In our fullymicroscopic approach we include previously neglected M1 transitions which enhance thecapture probability below 100 keV. Overall, we reduce the uncertainty of the thermonuclearcapture rate by a factor of 7In the A = 8 system, we addressed reactions involving p + 7Li ↔ 8Be ↔ n + 7Be. Re-cent improvements of the NCSMC have made possible the investigation of reactions such as7Li(p, γ)8Be, 7Be(n, γ)8Be or 7Be(n, p)7Li. These reactions are important channels by which7Li is created or destroyed. They are the main contributors to the prediction of cosmolog-ical lithium abundance of which current estimates differ significantly from measurements[131]. A related reaction which is also accessible with NCSMC is 7Li(p, e+e−)8Be. A recentexperiment at ATOMKI reported an anomaly in the electron-positron spectrum from thisreaction [120]. The decay of the two 1+ excited states of the 8Be to the 0+ ground state pro-duces the emitted particles. These excited states are just above the 7Li + p threshold. Thespectrum of the particles from the second 1+ state suggests their production is delayed byan intermediate particle that is not the photon. One possible interpretation of this anomalyis a new boson with mass of 17 MeV [132], which could be a mediator between the visibleand dark sectors [127]. Our full ab initio calculation of this reaction has some discrepancywith the data, but is improved compared to previous theoretical calculations (e.g. [122]).1087.1 4He(d, γ)6Li Radiative CaptureThere are currently significant discrepancies between theory and experiment in the case ofthe primordial abundance of 6Li, which is under-predicted by three orders of magnitude[133]. This could be due to systematic discrepancies in extracting the abundances fromastronomical observations of metal-poor stars [134, 135]. Another possibility is an inaccurateprediction of the relevant reaction rates as nuclear reaction rates are the main input tomodels which predict the primordial abundances of elements. Constraining the uncertaintieson these reaction rates is an important goal.7.1.1 The 6Li Composite System and 4He(d, d)4He Elastic ScatteringThe NCSMC wavefunction needed for the system is given in terms of 6Li NCSM eigenstates∣∣6LiλJpiT〉 and continuous 4He-d cluster states ∣∣ΦJpiTνr 〉 built from 4He (α) and 2H (d) NCSMstates, i.e. ∣∣ΨJpiT 〉 = ∑λcλ∣∣6LiλJpiT〉+∑ν∫drr2γν(r)rAν∣∣ΦJpiTνr 〉 , (7.1)where∣∣ΦJpiTνr 〉 = [(∣∣4HeλαJpiαα Tα〉 ∣∣2HλdJpidd Td〉)(sT ) Y`(rˆα,d)](JpiT ) × δ(r − rα,d)rrα,d . (7.2)We use the first 15 NCSM eigenstates of 6Li, the single bound state of 4He (experimen-tally bound by 19.82 MeV below the 3He + p threshold) and the single bound state of 2Hin combination with 2H “pseudo-states”. The pseudo-states are NCSM eigenstates that donot correspond to real bound states, but their inclusion can partially account for the p+ ncontinuum as d is very loosely bound (experimentally the ground state is -2.22 MeV). Intotal 9 NCSM states are included (3 each from channels Jpi = {1+, 2+, 3+} correspondingto {3S1 −3 D1, 3D2, 3D3 −3 G3} (S = 1, L = {(0, 2), (2), (2, 4)}). The NCSM calculationswere up to Nmax = 12 and the relative motion between the clusters expanded to Nrel = 11culminating in an NCSMC calculation with Nmax = 11. NCSM calculations were repeatedwith two choices of 3N force (3Nloc [60] and 3Nlnl [136]). The first overbinds the6Li groundstate by 250 keV (compared to experiment) and the second underbinds by 590 keV. Thisdifference motivated the choice of 3Nloc in the NCSMC predictions.The ground state energy and the position of the low-lying resonances was predictedwith NCSMC (Figure 7.1) with different choices of the nuclear potential. The inclusionof a 3N force improved the 2+ − 3+ splitting, nearly exactly reproducing the 3+ energy.Despite inclusion of a 3N force, the ground state of 6Li was still overbound by about 350keV. By adjusting the ground state energy of 6Li in the NCSM (i.e. the values in the Eλλ′NCSMC kernel) we can accurately reproduce the ground state energy. The 1+ eigenstateswere shifted to reproduce the experimental 6Li ground state energy. This is denoted in theresults by NN+3Nloc-pheno. This phenomenological approach loosens the first-principles109paradigm to compensate for insufficiencies in the nuclear force as in several previous studies[137–139]. The exact energies and resonance positions are fitted to experiment but thewidths, structure and transition strengths are predictions.As a further benchmark we calculate the elastic scattering cross section 4He(d, d)4Heand compare to data. Figure 7.2. Without phenomenological adjustment the results alreadyagree well with experiment.−2−1012345Exp.NCSMCNN+3Nloc NN+3Nloc-phenoNN-only-1.471+-1.78-1.85-1.47g.s. g.s.g.s.g.s.Γ =0.02Γ =0.02Γ =0.08Γ =0.020.713+0.720.990.72Γ =1.30Γ =0.83Γ =0.63Γ =0.832.842+2.672.222.67Γ =1.50Γ =2.77Γ =2.22 Γ =2.774.171+3.743.70 3.74Figure 7.1: Low-lying spectrum of 6Li obtained with the NCSMC at Nmax = 11 usingthe SRG-evolved N3NLO NN potential [28] (NN-only) with λ = 2 fm−1, theNN+3nloc [60, 140] without (NN+3Nloc) and with the phenomenological energyadjustment (NN+3Nloc-pheno) compared to experiments [141]. The zero energyis taken as the predicted (resp. experimental) 4He + d threshold for NCSMC(resp. Exp.)7.1.2 The 4He(d, γ)6Li S-factorThe cross-section of the 4He(d, γ)6Li reaction is calculated via (6.91). We present the resultsin terms of the astrophysical S-factor. The S-factor is a rescaling of the cross section to110100 101Ed [MeV]10−1100(dσ/dΩ) c.m.[b/sr]NN+3NlocNN+3Nloc-phenoGalonsky et al. (168◦)Mani et al. (165◦)Mani et al. (163◦)Figure 7.2: NCSMC predictions of 4He(d, d)4He elastic scattering. Both NCSMCpredictions obtained without (NN+3Nloc) and with the phenomenologicaladjustment (NN+3Nloc-pheno) reproduce the experimental elastic-scattering4He(d, d)4He cross section at the deuteron backscattered to the angle θd = 164◦[142, 143].remove the Coulomb repulsion [107], i.e.S(E) =Ee2piησ(E) , (7.3)where η is the Sommerfeld parameter in this case η = Z1Z2αv =√3α√2Esince v =√2µE =√2(2·4mN6)E.The predicted S-factor agrees well with data as seen in Figure 7.3. The prediction agreeswell with LUNA at low energy already with only the NN potential but fails to reproducethe resonance peaks (3+ at 0.71 MeV and 2+ at 2.84 MeV). The resonances are reproducedwith the NN+3Nloc but the g.s. is overbound. The NN+3Nloc-pheno achieves both.The relative importance of the different electromagnetic multipoles (E1,M1,E2) varieswith energy. The 3+ resonance is predominately E2 but the low-energy is a mix of E2 andM1. Previous studies (e.g. [144, 145]) did not include the M1 transitions and we find theyhave a significant contribution at low-energy. The internal dipole magnetic moments ofthe 6Li and d contribute to this transition strength and therefore our microscopic model iswell-motivated. Phenomenological approaches often find sizable E1 contribution which werefute.1110.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0S-factor[MeVb]10−1 1001 −910−81 −710−610−5BBN rangeNN-onlyNN+3NlocNN+3Nloc-pheno10−1 100E [MeV]10−1210−1110−1010−910−810−710−6Total E2+E1+M1E2E1M1Figure 7.3: Top: Predicted S-factor for the 4He(d, γ)6Li compared with data takenfrom Refs [146] (red circles), [147] (blue squares), [148] (green up-triangles),[149] (black down-triangles). Calculations are obtained using the SRG-evolvedN3LO NN potential [28] (NN-only) with λ = 2 fm−2, the NN+3Nloc [60] with-out (NN+3Nloc) and with the phenomenological energy adjustment (NN+3Nloc-pheno). Bottom: E2, E1 and M1 components to the predicted S-factor for the4He(d, γ)6Li obtained with the NN+3Nloc-pheno.1127.1.3 The Effect of SRG on M1 ContributionsWe investigated the effect of consistent SRG evolution of the electromagnetic operators(since we use an SRG evolved potential). Because E1 and E2 operators are long-range,their matrix elements depend mainly on the wavefunctions are large distance and hence theSRG transformation has a negligible impact [74, 150].The electromagnetic transition strengths in the NCSMC are decomposed into fourcontributions (6.68). The dominant terms in this case are the〈6Liλ′∣∣ ∣∣T M1 ∣∣ ∣∣6Liλ〉 and〈2Hλ′d∣∣ ∣∣T M1 ∣∣ ∣∣2Hλd〉 transitions. In fact interference between these cause the minimumnear 0.5 MeV and enhancement at low-energy. This emphasizes the fact that a microscopicmodel is needed to describe this reaction.We compare in Figure 7.4 calculations obtained where the internal NCSM contributionsto the transitions are SRG-evolved (M1) and where they are not evolved (M1 bare). Theeffect is small, although visible and has a negligible effect on the total S-factor except atthe lowest energy.10−1 100E [MeV]10−1210−1110−1010−910−810−710−610−5S-factor[MeVb]Total E2+E1+M1Total E2+E1+M1 bareM1M1 bareFigure 7.4: Influence of the SRG transformation on the M1 operator. The experi-mental data follow the same legend as in Fig. 7.3.7.1.4 Thermonuclear Reaction RateFrom the S-factor (cross section), we obtain a thermonuclear reaction rate for 4He(d, γ)6Liand compared this to data (Figure 7.5). Our result falls within the bounds of the NACRE IIevaluation with reduced uncertainty by an average factor of 7. The uncertainty in our cal-culation is estimated based on the truncation error of the model space (Nmax convergence)113and the choice of 3N force.The reaction rate requires the folding of the energy-dependent cross section with theMaxwell-Boltzmann energy distribution as a function of temperature. Expressed as theMaxwellian-averaged rate < σv > times the Avogadro number NA [151]:NA < σv >=NA√8piµ(kBT )3∫ ∞0Eσ(E) exp[− EkBT]dE , (7.4)where kB is the Boltzmann constant, µ the reduced mass and T the temperature. A moretypical expression uses T9, the temperature in units of 109 K:NA < σv >=3.73× 109√µˆT 39∫ ∞0Eσ(E) exp[−11.605ET9]dE , (7.5)where E and σ(E) are in units of MeV and barn respectively, and µˆ is the reduced mass inatomic mass units. This results in a rate is in units of cm3 mol−1 s−1.We predict an even lower reaction rate than previous predictions, therefore the discrep-ancy of the predicted 6Li abundance is larger than previously reported.10−2 10−1 100T9 [GK]0.000.250.500.751.001.251.501.752.00RatiowithNACREIINACRE IILUNA 2017NN+3Nloc-phenoFigure 7.5: Ratio of the predicted thermonuclear reaction rates (black line) for4He(d, γ)6Li with the NACRE-II evaluation (red line) [151] for different temper-ature T9 in GK. Our results are also compared with the recent thermonuclearreaction rate derived from the measurements of the LUNA collaboration (blueline) [152]. The shaded areas correspond to the uncertainty of each calculation.1147.2 7Li(p, γ)8Be Radiative CaptureThe study of reactions involving 7Li, 7Be and 8Be has several motivations. 7Li has acosmological abundance problem of smaller in magnitude to that of 6Li and in the oppositedirection (the ratio of predicted/measured is about 3). 7Li is stable in contrast to theBeryllium isotopes mentioned. 7Be has a 53 day half-life and therefore is most importantin its influence on the 7Li abundance in BBN.8Be is unstable and is a resonance of α + α at 91.8 keV (with a width of 6 eV). It isa bottleneck of nucleosynthesis and is part of the “triple-alpha” process by which 12C isproduced, leading to many of the most important elements for life (e.g.16O). The thresholdfor 7Li + p is at 17.2551 MeV below which there are 5 8Be states (α + α resonances).Reactions of 7Li + p are enhanced through 8Be resonances above this threshold. There aretwo 1+ resonances with only 7Li + p channels, above which is the 7Be + n threshold (since7Be is less bound than 7Li due to Coulomb repulsion). The spectrum becomes extremelycomplex above this point (see Fig. 7.6).8Be has recently been brought to attention in experiments. In particular, anomalieswere observed in the electron-positron pair production distribution from proton capture on7Li [120]. These were interpreted as the decay of a new boson. To assist in interpretationof the anomaly, the NCSMC can be applied to calculate radiative capture and the relatedpair production.7.2.1 The 8Be Composite SystemThe NCSMC wavefunction needed for this system is given in terms of 8Be NCSM eigenstatesand cluster states composed of 7Li NCSM eigenstates with a single proton and 7Be NCSMeigenstates with a single neutron, i.e.Ψ(8)NCSMC =∑λcλ∣∣8Be, λ〉+∑ν∫drγν(r)Aˆν∣∣7Li + p, ν〉+∑µ∫drγµ(r)Aˆµ∣∣7Be + n, µ〉 .(7.6)We use the lowest 5 NCSM eigenstates each from 7Li and 7Be. They are mirror nucleiand so the lowest states have the same Jpi: {32−,12−,72−,52−,52−}. We use the 30 lowest NCSMeigenstates of 8Be (15 positive parity and 15 negative parity), covering the energy range ofinterest.7.2.2 7Li, 7Be and 8Be NCSM ResultsOur NCSMC calculation requires as input NCSM eigenstates. We used the recently devel-oped nuclear force from chiral EFT NN+3Nlnl [136]. This Hamiltonian is fully determined inA = 2,3 and 4 systems (nucleon-nucleon scattering and properties of the deuteron, 3H, 3He(including the 3H half-life) and the 4He binding energy). It is SRG-evolved to λ = 2 fm−1.This interaction has already been tested in A = 7, 8 systems. The predicted excitation115Figure 7.6: The experimental level scheme for 8Be. Adapted from [153].116energies are shown in Figure 7.7. The convergence of the ground state is shown in Figure 7.8.(a) (b)(c)Figure 7.7: Excitation energies from the NCSM for (a)7Li, (b) 7Be and (c) 8Be usingthe NN+3Nlnl interaction [136]. The lowest states most relevant for input tothe NCSMC are in the correct order. Only the lowest 5 states are shown for7Be.7.2.3 Resonances and Phase ShiftsAs we do not include |α+ α〉 states in our basis we cannot expect to precisely predict theresonance positions below the 7Li + p threshold. In our calculations these are consideredas bound states. However inclusion of some cluster degrees of freedom still improves the1170 2 4 6 8 10 12Nmax-60-50-40-30-20-10E gs [MeV]7Be NCSM7Li NCSM8Be NCSM8Be NCSMC7Be Expt7Li Expt8Be ExptFigure 7.8: The convergence of the ground state energies of 7Li, 7Be and 8Be withthe model-space parameter Nmax using the NN+3Nlnl interaction [136].description of the asymptotics of the wavefunction. The resulting 8Be energies from NCSMCwith respect to the 7Li + p threshold are shown in 7.1. The improved ground state is alsoadded to Figure 7.8 to compare to the NCSM result.Figure 7.9 shows the eigen-phaseshifts for scattering with 7Li + p and 7Be +n channels.Calculations for Nmax = 9 are shown. The eigen-phaseshifts δ are calculated from theeigenvalues e2iδ of the S-matrix. The S-matrix is computed as a function of the center ofmass energy, shown here up to 8 MeV. The maxima in the derivative of the eigen-phaseshiftscorrespond to resonances. The eigen-phaseshift corresponding to the last 2+ state, which isbound in experiment (see Table 7.1), shows up as an extremely narrow resonance and canbe seen in the bottom-left corner of (a). The next two resonances are 1+ resonances whichcorrespond to the spin aligned and anti-aligned interactions between the proton and the32−ground state of 7Li. The 1+ resonances appear clearly in the level-scheme Figure 7.6.New phase shifts occur past the threshold of 7Be + n which is higher than 7Li + p. Wereproduce the 3+ pair in particular and find many resonances which have not yet been seenin experiment. The data are complex and the TUNL evaluation includes the results ofphenomenological R-matrix fits [154].The convergence of the eigen-phaseshift with Nmax is shown in Fig. 7.10 . The 1+do not change much. The 3+is not as well-converged, interestingly there is a crossing atNmax = 9. The 3+ resonances are mixed between the p+ 7Li and n+ 7Be mass partitionsand the amount of mixing depends on Nmax as seen in the diagonal phase-shifts (Fig 7.11).The same figure shows that the two 1+ have different interpretations, corresponding to totalchannel spin 1 and 2 respectively. We can also extract the isospin character, the first ispredominantly T = 1 and the second T = 0.1180 1 2 3 4 5 6 7 8Ekin [MeV]0306090120150180δ [deg]0+1+ 1+1+1+2+2+ 2+3+3+3+4+0+(a)0 1 2 3 4 5 6 7 8Ekin [MeV]0306090120150180δ [deg]0-2-2-1- 3-4-4-(b)Figure 7.9: 7Li + p eigen-phase shifts at Nmax = 9, even (a) and odd (b) parity.119Energy [MeV] Excitation Energy [MeV]NCSM NCSMC Expt. NCSM NCSMC Expt.0+ -15.96 -16.13 -17.25 0.00 0.00 0.002+ -12.51 -12.72 -14.23 3.45 3.41 3.034+ -3.97 -4.31 -5.91 11.99 11.82 11.352+ +0.76 -0.10 -0.63 16.72 16.03 16.632+ +1.09 +0.31 -0.33 17.05 16.44 16.92Table 7.1: The bound states of 8Be are well reproduced with the NCSMC. One minorexception is that the 3rd 2+ is slightly above the 7Li + p threshold.0 1 2 3 4 5 6 7 8Ekin [MeV]0306090120150180δ [deg]Nmax=9Nmax=7Nmax=51+ 1+1+(a)0 1 2 3 4 5 6 7 8Ekin [MeV]0306090120150180δ [deg]Nmax=9Nmax=7Nmax=53+ 3+ 3+(b)Figure 7.10: Dependence of the 7Li + p eigen-phase shifts on the NCSMC basis sizecharacterized by Nmax for the 1+ (left) and 3+ (right) resonances. Ekin is thekinetic energy of the 7Li + p in the center of mass frame.0 1 2 3 4 5 6 7 8Ekin [MeV]-60-300306090120150180δ [deg] 3P1(7Li3/2-)5P1(7Li3/2-)1P1(7Li1/2-)(a)0 1 2 3 4 5 6 7 8Ekin [MeV]-60-300306090120150180δ [deg]5P3(7Be3/2-)9P4(7Li7/2-)5P3(7Li3/2-)(b)Figure 7.11: Selected P-wave 7Li + p phase shifts. The 7Be + n threshold is at 1.64MeV.1207.2.4 p+ 7Li ScatteringThe differential and integrated cross sections can be calculated from the full multi-channelS-matrix. These quantities are observables that can be compared to experiments. The crosssection is enhanced by the presence of resonances.Figure 7.12 shows the differential cross section for 7Li(p, p)7Li elastic scattering (meaning7Li remains in the ground state and so kinetic energy is conserved). The integrated crosssection diverges due to the Coulomb repulsion but the differential cross section can be shownat each energy. The dependence of the differential cross-section on Nmax and the phenocorrection is shown. Data from inverse kinematics experiments i.e. p(7Li, 7Li)p is shown.The lab energy is 5 and 5.44 MeV per nucleon (i.e. 7Li with a beam energy of 35 and 38.1MeV) [155, 156]. To compare to our calculations dependent on the center of mass energy,we multiply by the ratio with the reduced mass 78 : 4.38 and 4.76 MeV. These energiesare beyond the point at which phenomenological shifts of the resonances are applied. Ourcalculations predict a huge number of resonances and their individual contributions to thecross sections has not yet been fully analyzed.The cross sections for inelastic scattering and transfer reactions are also obtained in ourformalism. Results will be compared to data in a future publication.0 25 50 75 100 125 150 175θ [deg]101102103dσ/dΩ[mb/sr]NCSMC: Nmax = 9, 4.76 MeVNCSMC: Nmax = 9, 4.38 MeVData: 5.43 MeV/uData: 5.00 MeV/u(a)0 25 50 75 100 125 150 175θ [deg]102103dσ/dΩ[mb/sr] (NCSMC: 4.76 MeV)Nmax = 7Nmax = 9Nmax = 9-pheno(b)Figure 7.12: The differential cross section for 7Li(p, p)7Li elastic scattering. Θ isthe angle between the incoming proton and outgoing proton in the centerof mass frame. (a) A comparison between inverse scattering data at 5 and5.44 MeV/u and NCSMC calculations at the corresponding center of massenergies: E = 4.38 and 4.76 MeV. Data from [155, 156]. (b) The convergenceof NCSMC calculations at E = 4.76 MeV with the input parameter Nmax.7.2.5 The 7Li(p, γ)8Be Cross SectionUsing (6.91) we calculate the radiative capture cross section between the initial clusterscattering state and the first 0+ and 2+ 8Be bound states, considering E1, M1 and E2121multipolarities.For much of the energy range the dominant term in the integrated cross-section is the E1transition strength, corresponding to transitions from the 1− cluster channels, in the caseof transitions to the ground state. However within the 1+ resonances, the M1 transitionstrength is greatly enhanced. The E2 transition strength is negligible at the order of 1%.In Figure 7.13, we compare the NCSMC predictions of the integrated cross section toan experiment [157]. After phenomenological adjustment the calculations match the dataextremely well. The adjustment shifts the position of the peaks but the size of the transitionstrength is already fairly well predicted, indicating that the ab initio wavefunctions arealready a reasonably described. We are missing some of the mixing between the T=1 andT=0 resonances, which could be another sign of the insufficiency of the nuclear force.For completeness, we include the S-factor and phase shifts with the resonances shiftedto emphasize the connection between the presence of a resonance and an enhancement inthe cross section.7.2.6 Pair Production and the X17Attention has been brought to the 8Be system due to recent claims of signals of the discoveryof a new BSM particle. Anomalies were observed in the electron-positron pair productionfrom proton capture on 7Li and 3H by the ATOMKI collaboration and interpreted as thedecay of a new boson with the mass 17 MeV (hence “X17”) [120, 158]. The signal of thisnew boson is the apparent excess that appears in the data at Θ ∼ 140◦ in Figure 7.14.The theory of Section 6.4 has been applied making use of the electromagnetic transitionmatrix elements calculated with the NCSMC.Combinations of the E1 and M1 transition strengths can be used to provide an ab initioprediction of the pair distribution. Using the Siegert’s theorem, Coulomb matrix elementsare replaced with electric multipoles. The E2 is suppressed by additional factors of q and anumerical coefficient < 1100 .Unlike [122] we do not fit the E1/M1 ratio but rather predict it. Our prediction issupported by the calculation of radiative capture (Fig. 7.13), the E1 and M1 componentsare plotted separately in Figure 7.17. In Figure 7.14 the pair production is predictedusing approximated transitions, i.e. the bound-bound theory is used as in [122] (but withNCMSC transition strengths inserted into (6.110) with appropriate normalization factors(such that we reproduce the radiative capture cross section through the scaled γ-decayformula)). In this method the 1+ and 1− scattering states do not interfere. In Figure 7.14,we compare scaling the predicted curve (as the data is in relative units) to match the dataat two different angles characterizing low and high angles. When matched at low angle, thepredicted background is much higher than data at high angles. In contrast, when matchedat high angle, the predicted background at low angles is lower than the data.Like in [123] we can include the interference from different partial waves in the scattering1220.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4E [MeV]10−710−610−510−410−310−2σ[b]Cross-section (7Li3/2−(p, γ)8Be)NCSMC: γ0NCSMC: γ0 + γ1Data: γ0Data: γ0 + γ1(a)0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4E [MeV]10−710−610−510−410−310−2σ[b]Cross-section (7Li3/2−(p, γ)8Be)NCSMC: γ0NCSMC: γ0 + γ1Data: γ0Data: γ0 + γ1(b)0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4E [MeV]100101102S[keVb]S-factor (7Li3/2−(p, γ)8Be)NCSMC: γ0NCSMC: γ0 + γ1Data: γ0Data: γ0 + γ1(c)0 0.2 0.4 0.6 0.8 1 1.2Ekin [MeV]-300306090120150180δ [deg]7Li+p phase shifts1+1+5P13P1(d)Figure 7.13: The cross-section for 7Li(p, γ)8Be capture. (a) The integrated crosssection of 7Li(p, γ)8Be calculated at Nmax = 7 (and without7Be + n degreesof freedom). (b) The integrated cross section of 7Li(p, γ)8Be calculated atNmax = 9 with phenomenological adjustments. γ0 is the capture cross-sectionfor proton capture resulting in the ground state of 8Be. γ1 is the capturecross-section resulting in the first excited state (2+) of 8Be. Data points aretaken from [157]. (c) The 7Li(p, γ)8Be astrophysical S-factor (proportional tothe cross section in (b) via (7.3)) (d) The phenomenologically adjusted phaseshifts at Nmax = 9.123state using the full theoretical prediction (6.144). The results are shown in Figure 7.15 andcompared against the more approximate method in Figure 7.16. The additional interferenceterms reduce the cross section at high angles and improve agreement with the data. Thedifference is less severe when the full range of possible electron-positron energies (in terms ofy (6.111)) are included but due to experimental limitations the limited range −0.5 < y < 0.5is better prediction of the observed background rate. This could give some support for theX17 boson claim.An estimate of the reaction 7Li(p,X)8Be was calculated with three X17 candidates: apseudoscalar [129], axial vector [130] and vector [126]. The cross section is computed bysubstituting a leading-order operator in the long-wavelength approximation into the formulafor radiative capture (6.91). The results are shown in Figure 7.17.We estimate the 7Li(p, e+e−)8Be cross section by integration of the bound-bound for-mula. It is essentially proportional to the γ-emitting proton-capture cross section with itsmagnitude scaled by a factor of ∼ α/2pi ∼ 10−3. It is understandable that no anomaly wouldbe observed at the first resonance which is swamped by the very high electromagnetic M1rate. The effect from the hypothetical X17 boson would be expected at second resonancein the case of a pseudoscalar or axial vector character. Any anomaly present between theresonances would be consistent with a vector particle and is the preferred candidate in themost recent ATOMKI publications [159].The claims of new BSM physics have spurred the development of new experimentalprograms to search for corroborating evidence. One example at TRIUMF is the DarkLightexperiment which will be installed in the new ARIEL e-linac beam hall. Another is theNewJEDI program at several facilities in Europe which has begun taking data.12440 60 80 100 120 140 160Θ [deg]10−2 10−210−1 10−1IPCC(relativeunits)NCSMC: Ekin = 0.9 MeVNCSMC: TotalNCSMC: E1NCSMC: M1ATOMKI 2019ATOMKI 2016(a)40 60 80 100 120 140 160Θ [deg]10−2 10−210−1 10−1IPCC(relativeunits)NCSMC: Ekin = 0.9 MeVNCSMC: TotalNCSMC: E1NCSMC: M1ATOMKI 2019ATOMKI 2016(b)Figure 7.14: The pair creation distribution calculated with NCSMC transitionstrengths but using the bound-bound formula and compared to the ATOMKIdata [160]. The calculation are scaled to match data (a): at 65 degrees, (b):at 105 degrees. The dimensionless parameter y (6.111) is integrated from -0.5to 0.5 as in [122, 123]. The E1 (1−) and M1 (1+) contributions are displayed.40 60 80 100 120 140 160Θ [deg]10−2 10−210−1 10−1IPCC(relativeunits)NCSMC: Ekin = 0.9 MeVNCSMC: d5σATOMKI 2019ATOMKI 2016(a)40 60 80 100 120 140 160Θ [deg]10−2 10−210−1 10−1IPCC(relativeunits)NCSMC: Ekin = 0.9 MeVNCSMC: d5σATOMKI 2019ATOMKI 2016(b)Figure 7.15: The pair creation distribution calculating with the full continuum-boundformula and compared to the ATOMKI data [160]. The calculation are scaledto match data (a): at 65 degrees, (b): at 105 degrees. The dimensionlessparameter y (6.111) is integrated from -0.5 to 0.5 as in [122, 123].12540 60 80 100 120 140 160Θ [deg]10−2 10−210−1 10−1IPCC(relativeunits)NCSMC: Ekin = 0.9 MeVNCSMC: dΓNCSMC: dσATOMKI 2019ATOMKI 2016(a)40 60 80 100 120 140 160Θ [deg]10−2 10−210−1 10−1IPCC(relativeunits)NCSMC: Ekin = 0.9 MeVNCSMC: dΓNCSMC: dσATOMKI 2019ATOMKI 2016(b)40 60 80 100 120 140 160Θ [deg]10−2 10−210−1 10−1IPCC(relativeunits)NCSMC: Ekin = 0.9 MeVNCSMC: dΓNCSMC: dσATOMKI 2019ATOMKI 2016(c)40 60 80 100 120 140 160Θ [deg]10−2 10−210−1 10−1IPCC(relativeunits)NCSMC: Ekin = 0.9 MeVNCSMC: dΓNCSMC: dσATOMKI 2019ATOMKI 2016(d)Figure 7.16: The comparison between approximate (Γ) and full calculations (σ). In(a,b) y is integrated from -0.5 to 0.5 and in (c,d) y is integrated from−(1− 2meω )to 1− 2meω .126Figure 7.17: The integrated cross section of various X17 candidates vs the standardradiative capture cross section. Interferences between γ and X are not in-cluded. The electron-positron integrated cross section is based on a numericalintegral of the bound-bound formula. The coupling constants are approxi-mated based on [161], which scales the operators based on the strength of thesignal in the 2016/2019 data [160].127Chapter 8ConclusionsAb initio techniques in nuclear theory allow accurate predictions of nuclear structure, decaysand reactions. The NCSM is one such technique and accurately calculates the spectrum ofbound states and β-decay rates in many light nuclei.Demonstrated in this thesis were predictions for the rates of GT β-decay transitions.Through collaboration with practitioners of more approximate ab initio methods, we couldcompute the nuclear transition rates for nuclei over a much wider range of masses. Theinclusion of two-body currents, more sophisticated many-body calculations and consistentSRG evolution resulted in nuclear transition matrix elements that were consistently moreagreeable with experiments than the phenomenological shell model. This provided a resolu-tion to the historical discrepancy between theory and experiment in strong GT transitions,removing the necessity of applying a quenching factor to the weak coupling constant gA.In the 6He → 6Li transition, we demonstrated that there are nuclear structure correc-tions to the leading-order β-decay operator that must be taken into account in experimentswhich measure the distribution of the emitted electrons. The level of precision in currentand future experiments is such that these higher-order effects will be detected and must notbe mistaken for new BSM physics.In addition, using the NCSM, we provided benchmarks of 0νββ nuclear transition matrixelements for several light nuclei, as a test for the ab initio methods of coupled-cluster andIMSRG. The NCSM generally supports coupled-cluster calculations which use deformedreference states, while IMSRG matrix elements are often larger. The coupled-cluster andIMSRG methods were used to predict the matrix element for the experimental candidate48Ca. These methods also provide predictions for 76Ge and in the future will also calculate136Xe, providing several avenues to study neutrino physics through this decay mode.Extensions to the NCSM, like NCSMC, provide access to an even greater range ofphenomena i.e. scattering and reactions, while simultaneously improving the description ofweakly bound states.In this thesis, we calculated properties of the 4He +d system, culminating in the predic-tion of the cross section for the radiative capture reaction 4He(d, γ)6Li. We revealed that128a fully microscopic approach is needed to achieve the important M1 contributions to thecross section. Our calculations provided state of the art predictions for the astrophysicalS-factor in the BBN energy range and reduced the uncertainty on predictions of the ther-monuclear reaction rate. This result indicates that the uncertainty in this reaction rate isnot the reason for the discrepancy between prediction and observation of the cosmological6Li abundance.We also calculated properties of the 7Li + p system, including both 8Be states and thepossibility of nucleon exchange to n + 7Be. The description of 8Be states was improvedcompared to existing NCSM calculations. In addition, the 1+ resonances above the 7Li + pthreshold were reproduced. This culminated in the prediction of the radiative capture crosssection 7Li(p, γ)8Be as well as novel extensions: 7Li(p, e+e−)8Be and 7Li(p,X)8Be. Thestandard radiative capture reaction could be calculated in a agreement with experiments.The radiative capture with pair production was first calculated in a simplified bound-to-bound-like formalism summing up separate electric and magnetic contributions in agreementwith other theoretical predictions. We demonstrated that a full-continuum calculation thattakes into account the interference between different multipoles improves agreement withthe observed pair-production distribution in the ATOMKI 7Li(p, e+e−)8Be experiments.The developments in this work leave much room for further study. For example, SRGevolution should be applied to q-dependent β-decay operators and the 0νββ contact opera-tor. The more accurate q-dependent multipole operators will be used in future calculationsof beta decays and electromagnetic observables. In particular nuclear structure correctionsto Fermi decays and GT transitions of heavier nuclei, for example supporting measurementsof 23Ne decay [162].In addition, the calculations of deuteron and proton capture presented here are thestarting point towards application of the NCSMC to many more reactions. The A = 8NCSMC calculations presented here contain the information needed to extract the rates ofthe transfer reactions 7Li(p, n)7Be and 7Be(n, p)7Li, which can be compared to experiment.The highly-complex spectrum of 8Be will be investigated through the effects of resonancesin scattering and reactions. Finally, the new capability to calculate electron-positron pairproduction and the emission of hypothetical particles will allow the investigation of similaranomalies in the 3H(p, e+e−)4He [163] and 11B(p, e+e−)12C [164] reactions. These calcula-tions of the Standard Model background may form the basis for comparison to the claimsof the discovery of new particles.129Bibliography[1] P. Gysbers, G. Hagen, J. D. Holt, G. R. Jansen, T. D. Morris, P. Navra´til,T. Papenbrock, S. Quaglioni, A. Schwenk, S. R. Stroberg, and K. A. Wendt.Discrepancy between experimental and theoretical β-decay rates resolved from firstprinciples. 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Further properties of CGs, 3njs andD-matrices are tabulated in [35, 41, 165, 166].A.1 Pauli MatricesWe use the convention:σx =(0 11 0)(A.1)σy =(0 −ii 0)(A.2)σz =(1 00 −1)(A.3)A.2 Angular Momentum CouplingA.2.1 Properties of Clebsch-Gordan CoefficientsThe orthogonality property:∑m1m2(j1m1j2m2|jm)(j1m1j2m2|j′m′)= δjj′δmm′ . (A.4)The completeness property:∑jm(j1m1j2m2|jm)(j1m′1j2m′2|jm)= δm1m′1δm2m′2 . (A.5)144The entries may be reordered by(j1m1j2m2|jm) = (−)j2+m2 jˆjˆ1(j2 −m2jm|j1m1) (A.6)= (−)j1−m1 jˆjˆ2(jmj1 −m1|j2m2) (A.7)= (−)j1+j2−j (j2m2j1m1|jm) . (A.8)Some particularly useful values (from [53]):((j − 1)01λ|jλ) =√j + 12(2j − 1) ,(j01λ|jλ) =− λ√2,((j + 1)01λ|jλ) =√j2(2j + 3). (A.9)A.2.2 Definitions and Properties of 3nj SymbolsThe 3j symbol is defined in terms of the CG coefficients:(j1 j2 j3m1 m2 m3)= (−)j1−j2−m3 jˆ−13 (j1m1j2m2|j3 −m3) . (A.10)Explicitly the 6j is defined by a sum over four CGs:{j1 j2 j12j3 j j23}=(−)j1+j2+j3+jˆj12 ˆj23∑m1m2(j1m1j2m2|j12m12) (j12m12j3m3|jm)× (j2m2j3m3|j23m23) (j1m1j23m23|jm) , (A.11)but it also appears in the appropriate sum over three CGs:(−)j1+j2+j3+j ˆj12 ˆj23{j1 j2 j12j3 j j23}(j1m1j23m23|jm) =∑m2(j1m1j2m2|j12m12) (j12m12j3m3|jm) (j2m2j3m3|j23m23) . (A.12)145The 9j symbol is defined by:j1 j2 j12j3 j4 j34j13 j24 j =∑m1m2∑m3m4∑m12m34∑m13m24m(j1 j2 j12m1 m2 m12)(j3 j4 j34m3 m4 m34)×(j13 j24 jm13 m24 m)(j1 j3 j13m1 m3 m13)(j2 j4 j24m2 m4 m24)(j12 j34 jm12 m34 m).(A.13)When one of the elements of a 9j is zero, we can reduce it to a 6j, i.e.j1 j2 j12j3 j4 j34j13 j24 0 = δj12j34δj13δ24 jˆ−112 jˆ−113 (−)j12+j13+j2+j3{j1 j2 j12j4 j3 j13}. (A.14)A 6j can be reduced , i.e.{j1 j2 j30 j′3 j′2}=(−)j1+j2+j3jˆ2jˆ3δj2j′2δj3j′3∆(j1j2j3) , (A.15)which introduces the triangular delta, defined as∆(j1j2j3) =1 if |j1 − j2| ≤ j3 ≤ j1 + j20 otherwise . (A.16)Sometimes angular momentum coupling is expressed in terms of the Racah symbols:W (j1j2l2l1; j3l3) = (−)j1+j2+l1+l2{j1 j2 j3l1 l2 l3}. (A.17)A.3 Rotations and Wigner D-matricesA rotation of the coordinate axes by the Euler angles α, β, γ induces a unitary transfor-mation on angular momentum eigenstates |jm〉. The rotation R(α, β, γ) corresponds to arotation about the z axis by the angle α, then about the new y′ axis by β, and then aboutthe new z′′ axis by γ (all right-hand rotations). This is equivalent to the following sequenceof rotations about the original axes: first γ about z, then β about y and then α about z.These steps correspond to the operator:R(α, β, γ) =e−iγJz′′e−iβJy′e−iαJz=e−iαJze−iβJye−iγJz (A.18)146where Jξ is the angular momentum projection operator for the axis ξ (following the con-ventions of [35]). The inverse of the rotation is:R−1(α, β, γ) = R†(α, β, γ) = R(−γ,−β,−α) . (A.19)The Wigner D-matrix is the matrix element of the rotation operator, i.e.〈j′m′∣∣R(α, β, γ) |jm〉 = δj′jDjm′m(α, β, γ) (A.20)whereDjm′m(α, β, γ) =e−im′αdjm′m(β)e−imγ (A.21)with the “small d-matrix” defined bydjm′m(β) =〈jm′∣∣ e−iβJy |jm〉 = Djm′m(0, β, 0) . (A.22)It follows from (A.19) thatDj∗m′m(α, β, γ) = Djm′m(−γ,−β,−α) . (A.23)The conjugate is also related by the identitiesDjmk(α, β, γ) = (−)m−kDj∗−m−k(α, β, γ); , (A.24)Dj∗mk(α, β, γ) = (−)m−kDj−m−k(α, β, γ) . (A.25)When both indices are set to zero, the D-matrix elements are given by ordinary Legendrepolynomials, i.e.Dj00(α, β, γ) = dj00(β) = Pj(cosβ) . (A.26)The D-matrices relate quantum states in the rotated frame to a linear combination of statesin the original frame, i.e.R(α, β, γ) |jm〉 =∑m′Djm′m(α, β, γ)∣∣jm′〉 . (A.27)The wavefunction in the rotated frame ψjm(r, θ′, φ′, σ′) is related to ψjm(r, θ, φ, σ) in theoriginal frame byψjm(r, θ′, φ′, σ′) =∑m′Djm′m(α, β, γ)ψjm′(r, θ, φ, σ) (A.28)Here θ, φ and θ′, φ′ are the polar coordinates in the initial and rotated frames (and σ andσ′ are the initial and rotated spin variables respectively). A spherical tensor operator147transforms asR(α, β, γ)OjmR†(α, β, γ) =∑m′Djm′m(α, β, γ)Ojm′ . (A.29)D-matrices (with the same angles) can be combined by the product rules:DjmkDj′m′k′ =j+j′∑J=|j−j′|(jmj′m′|J(m+m′)) (jkj′k′|J(k + k′))DJ(m+m′)(k+k′) , (A.30)DjmkDj′∗m′k′ =j+j′∑J=|j−j′|(−)m′−k (jmj′ −m′|J(m−m′)) (jkj′ − k′|J(k − k′))DJ(m−m′)(k−k′) .(A.31)A.4 Useful FunctionsExplicitly the radial harmonic oscillator wavefunctions are given byRnl(r, b) =√√√√ 2Γ(n+ 1)(b2)l+32 Γ(n+ l + 32)rl exp(− r22b2)Ll+12n(r2b2), (A.32)where b =√~mNΩis the oscillator length, Γ is the Euler gamma function and Ll+12n is ageneralized Laguerre polynomial.The first three associated Laguerre polynomials areLl+120 (x) =1 , (A.33)Ll+121 (x) =l − x+32, (A.34)Ll+122 (x) =12[(l +32)(l +52)− 2(l + 52)x+ x2]. (A.35)Further polynomials can be derived via the recursion relation [40]:Ll+12n (x) = Ll+32n (x)− Ll+32n−1(x) . (A.36)148Appendix BThe Connection between QED andNucleon CurrentsB.1 Relativistic FormalismWe use the metric:η =1−1−1−1 . (B.1)The 4-momentum of a particle is P = (E, ~p) where the energy is the first component E = P 0and the other components are the 3-momentum ~p. The square of the 4-momentum is thesquared mass for observable “on-shell” particles, i.e.P 2 = PµPµ = PνηµνPµ = E2 − p2 = m2 . (B.2)We use the following convention for Dirac gamma matrices (where each block is a 2 × 2matrix (1 = I2×2, σi are Pauli matrices)):γ0 =(1 00 −1), (B.3)~γ =γ0(0 ~σ~σ 0), (B.4)γ5 =(0 11 0). (B.5)149We can also form the anti-symmetric tensor:σµν =i2(γµγν − γνγµ) . (B.6)Any 4×4 matrix can be constructed as a linear combination of 1, γ5, γµ, γµγ5 and σµν . Wesandwich them between 4-component Dirac spinors to create scalars, pseudoscalars, vectors,axial-vectors, and anti-symmetric tensors respectively. For example, a lepton vector currentis: `µ = u¯γµu.B.2 Non-relativistic Expansion of Dirac SpinorsIn QED, fermions are 4-component spinors. However, in non-relativistic quantum mechanicswe have a 2-component spinor χ. A Dirac spinor of mass M takes the form:u(p, σ) =√E +M(χσ~σ·~pE+Mχσ). (B.7)We use the notation u¯ = u†γ0 and with this convention u†u = 2E (u¯u = 2M). We can takethe non-relativistic limits of matrix elements between u¯(p′, s′) and u(p, s) by expanding inpowers of inverse mass [57, 167] as shown in Table B.1. We neglect terms of order O( 1M2)and define q = p− p′.u¯(p′, s′) (· · · )u(p, s) 2Mχ†s′ (· · · )χs1 1γ5~σ·~q2Mγ0 1~γ ~p+~p′2M +i(~q×~σ)2Mγ0γ5~σ·(~p+~p′)2M~γγ5 ~σ1M σ0i 01M σij ijkσkM1M σ0iγ5iσiM1M σijγ5 0Table B.1: Equivalent matrix elements between Dirac spinors (left) and those takenin the nonrelativistic limit (right).B.3 Nucleon CurrentsFollowing [53], we can model the nucleon as a Dirac spinor but the vector interaction mustinclude two form factors F1(q2) and F2(q2). For protons: F1p(0) = F2p(0) = 1, while forneutrons: F1n(0) = 0, F2n(0) = 1. The single-nucleon vertex for a vector interaction (e.g.150with a photon) isγVµ =F1γµ + iκ2mNF2σµνqµ , (B.8)where κ is the anomalous magnetic moment. In principle κ is non-zero for electrons as welldue to QED corrections but is very small [106].The form factors are often rearranged into the linear combinations GE = F1 − κq24m2N F2and GM = F1 + κF2. To leading order inq2m2N, GEp = 1, GEn = 0, GMp = gp = 1 + κp,GMn = gn = κn. In the isospin picture, the form factors have isoscalar and isovectorcomponents Fi =12(FSi + τzFVi), and so FS1 = FV1 = 1, FS2 = gp + gn = 0.88, andF V2 = gp − gn = 4.706.We treat the nucleons as non-relativistic point particles with spin and a non-zero anoma-lous magnetic moment. Using the expansion above in Section B.2, we can attach the formfactors to the multipole operators. We have a clear correspondence between eγV0 and thecharge operator, i.e.12mNu¯(p′, s′)eγV0 u(p, s) 'eF1χ†s′χs (B.9)=e1 + τz2χ†s′χs , (B.10)whereas e~γV gives the convection current and magnetization (as in Section 2.10):12mNu¯(p′, s′)e~γV u(p, s) 'χ†s′[F1(~p+ ~p′2mN+i (~q × ~σ)2mN)+iκ2mNF2~q × ~σ]χs (B.11)=e1 + τz2χ†s′χs~p+ ~p′2mN+ χ†s′[µ~∇× ~σ]χs . (B.12)A free nucleon is a spinor times a plane wave. The nucleons in nuclei can be decomposedinto linear combinations of these states. The vector current density J Vµ (~r) is evaluated byinserting γVµ between those basis states. The spatial dependence can be organized intomultipole operators and we must attach the appropriate form factors.For µ = 0 we may writeρ(q) =∫d3r∑JMCVJM (q, r)CVJM (q, r) =MJM (q, r)F1(q2) . (B.13)For µ ∈ (x, y, z), we decompose into the longitudinal and transverse multipoles (as in151Section 3.1.1). Schematically (dropping q, r and JM):LV =− QqF1M , (B.14)T V,E = qmN[F1∆′ +GM2Σ], (B.15)T V,M =− i qmN[F1∆− GM2Σ′], (B.16)where we use the short-hand multipole operators listed in Section B.4.The nucleon axial-vector current is:γAµ =FAγµγ5 − iFP qµγ5 , (B.17)where FA(0) = gA and the pseudo-scalar form factor is determined via FP =2mNFAq2+m2pidue tothe Goldberger-Treiman relation [54]. And so we have the operators:CA =− iqM[FAΩ− 12(FA − Q2mNFP)Σ′′], (B.18)LA =i[FA +q22mNFP]Σ′′ , (B.19)T A,E =iFAΣ′ , (B.20)T A,M =FAΣ . (B.21)B.4 Seven Multipole OperatorsSingle-particle multipole operators can be expressed as product of form factors and thefollowing seven operators [168, 169]:1.MJM (q, r) , (B.22)2.∆JM (q, r) = ~MJJM (q, r) ·~∇q, (B.23)3.∆′JM (q, r) = −i(~∇q× ~MJJM (q, r))·~∇q, (B.24)4.ΣJM (q, r) = ~MJJM (q, r) · ~σ , (B.25)1525.Σ′JM (q, r) = −i(~∇q× ~MJJM (q, r))· ~σ , (B.26)6.Σ′′JM (q, r) =(~∇qMJM)· ~σ , (B.27)7.ΩJM (q, r) = MJM (q, r)~σ ·~∇q, (B.28)where MJM (q, r) = jJ(qr)YJM (Ωr) and ~MJLM (q, r) = jL(qr)~YJLM (Ωr) and the gradient ~∇is with respect to ~r.In some situations it is useful to form the combination:Ω′JM =MJM~σ · ~∇+12Σ′′JM . (B.29)153
Thesis/Dissertation
eng
Physics
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
Graduate
Navrátil, Petr
Kruecken, Reiner
Radiative capture and decays in ab initio nuclear theory
Text