Science, Faculty of
Physics and Astronomy, Department of
DSpace
UBCV
Bruneault, Mathieu
Master of Science - MSc
University of British Columbia
Ab initio nuclear physics considers the problem of solving the many-body Schrödinger equation for protons and neutrons in a nucleus. Of the problems that arise in trying to solve this problem, the sheer size of the Hamiltonians that need to be diagonalized has been particularly limiting in trying to derive nuclear properties from the ground up. With recent advances in computational power, and by using the in-medium similarity renormalization group, a novel ab initio approach, computations of ab initio nuclear properties are now within reach. A problem of particular interest is the scattering of nuclei by weakly interacting massive particles. As one of the prime candidates for the origin of dark matter, weakly interacting massive particles have been the subjects of many direct detection experiments. Such experiments look for nuclear recoil events caused by the non-relativistic scattering of dark matter particles off nuclei, and require careful nuclear calculations of observables such as recoil rates. In this dissertation, we present the results of ab initio calculations in the form of nuclear response functions and observables for dark matter-nucleus scattering in 22 experimentally relevant nuclei. This analysis uses the Chiral effective field theory framework to derive nuclear interactions, and the non-relativistic effective field theory framework to derive observables from a general dark matter-nucleus interaction. This approach, coupled with the in-medium similarity renormalization group, allows for ab initio calculations in quite heavy nuclei, in particular in this work we consider isotopes of Xenon, among other elements. We find that while individual nuclear responses can vary quite a lot from phenomenological calculations, overall observables remain mostly unchanged. One improvement we report on is a quantification of theoretical uncertainty, which may not be obtained from phenomenological calculations.
Ab Initio Predictions for Dark MatterDirect DetectionbyMathieu BruneaultB.Sc., McGill University, 2020A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinThe Faculty of Graduate and Postdoctoral Studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)June 2023© Mathieu Bruneault, 2023The following individuals certify that they have read, and recommend to the Faculty of Graduate and Postdoctoral Studies for acceptance, the thesis entitled:Ab Initio Predictions for Dark Matter Direct Detectionsubmitted by Mathieu Bruneault in partial fulfilment of the requirements forthe degree of Master of Sciencein PhysicsExamining Committee:Jens Dilling, Adjunct Professor, Department of Physics and Astronomy, UBCSupervisor Jason D. Holt, Scientist, Theory Department, TRIUMFSupervisory Committee Member iiAbstractAb initio nuclear physics considers the problem of solving the many-bodySchro¨dinger equation for protons and neutrons in a nucleus. Of the problemsthat arise in trying to solve this problem, the sheer size of the Hamiltoni-ans that need to be diagonalized has been particularly limiting in tryingto derive nuclear properties from the ground up. With recent advances incomputational power, and by using the in-medium similarity renormaliza-tion group, a novel ab initio approach, computations of ab initio nuclearproperties are now within reach.A problem of particular interest is the scattering of nuclei by weaklyinteracting massive particles. As one of the prime candidates for the origin ofdark matter, weakly interacting massive particles have been the subjects ofmany direct detection experiments. Such experiments look for nuclear recoilevents caused by the non-relativistic scattering of dark matter particles offnuclei, and require careful nuclear calculations of observables such as recoilrates.In this thesis, we present the results of ab initio calculations in the formof nuclear response functions and observables for dark matter-nucleus scat-tering in 22 experimentally relevant nuclei. This analysis uses the Chiraleffective field theory framework to derive nuclear interactions, and the non-relativistic effective field theory framework to derive observables from a gen-eral dark matter-nucleus interaction. This approach, coupled with the in-medium similarity renormalization group, allows for ab initio calculations inquite heavy nuclei, in particular in this work we consider isotopes of Xenon,among other elements.We find that while individual nuclear responses can vary quite a lotfrom phenomenological calculations, overall observables remain mostly un-changed. One improvement we report on is a quantification of theoreticaluncertainty, which may not be obtained from phenomenological calculations.iiiLay SummaryOne of the most important unsolved mysteries of physics pertains to thenature of dark matter. Though proof of its existence is abundant, and de-spite the significant experimental effort that has been carried out, its natureremains unknown. One of the more accepted hypotheses for dark matteris that it is a weakly interacting massive particle. One way to look forsuch particles is via direct detection, in which detectors look for nuclearrecoil events caused by incoming dark matter particle scattering off of nu-clei. These detections would give insight on the interaction between darkmatter and regular matter, and potentially help unveil its nature. To helpguide experiments, robust understanding of the nuclear physics behind darkmatter scattering must be gained.ivPrefaceThis thesis is original, unpublished work by the author, M. Bruneault. Theimsrg [1] software, used to perform the IMSRG evolution described in Chap-ter 5, was developed by Dr. Ragnar Stroberg at TRIUMF. The KSHELL[2] software, used for shell model calculations and described in Section 5.3,was developed by Noritaka Shimizu and collaborators at the University ofTsukuba. Some of the calculations presented in Chapter 6 were performedby Dr. Baishan Hu, a post-doctoral collaborator of the author’s supervisorycommittee member, Dr. Jason D. Holt. Work on the response functions ofall 22 isotopes considered in this thesis is planned for publication.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . xAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Nuclear Physics Basics . . . . . . . . . . . . . . . . . . . . . . 42.1 A Qualitative Overview of the Nuclear Interaction . . . . . . 42.2 The Many-Body Schro¨dinger Equation and Basis Choice . . 82.2.1 The Mean-Field Approximation . . . . . . . . . . . . 82.2.2 The Hartree-Fock Method . . . . . . . . . . . . . . . 92.2.3 Harmonic Oscillator Basis . . . . . . . . . . . . . . . 102.3 Phenomenology . . . . . . . . . . . . . . . . . . . . . . . . . 112.3.1 Nuclear Shell Model and Valence Space . . . . . . . . 122.3.2 The Wood-Saxon Potential and the Standard Proce-dure Nuclear Physics . . . . . . . . . . . . . . . . . . 132.4 Matrix Elements . . . . . . . . . . . . . . . . . . . . . . . . . 142.4.1 Angular Momentum Coupling . . . . . . . . . . . . . 152.4.2 The Wigner-Eckart Theorem . . . . . . . . . . . . . . 15viTable of Contents3 Chiral Effective Field Theory . . . . . . . . . . . . . . . . . . 173.1 QCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.1.1 The QCD Lagrangian . . . . . . . . . . . . . . . . . . 183.1.2 Chirality . . . . . . . . . . . . . . . . . . . . . . . . . 193.1.3 Chiral symmetry of QCD . . . . . . . . . . . . . . . . 203.2 Building Chiral EFT . . . . . . . . . . . . . . . . . . . . . . 213.2.1 The Chiral EFT Lagrangian . . . . . . . . . . . . . . 213.2.2 Power counting . . . . . . . . . . . . . . . . . . . . . 223.2.3 Low-Energy Constants . . . . . . . . . . . . . . . . . 243.3 Chiral interactions . . . . . . . . . . . . . . . . . . . . . . . . 254 Dark Matter and Non-Relativistic Effective FieldTheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.1 Dark Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.1.1 Evidence for the Existence of Dark Matter . . . . . . 264.1.2 Dark Matter Models and Detection . . . . . . . . . . 274.2 Coupling Dark Matter to Nuclei . . . . . . . . . . . . . . . . 304.2.1 Formulation of the Effective Field Theory . . . . . . . 314.2.2 Multipole Expansion and Nuclear Responses . . . . . 324.2.3 Computing Matrix Elements . . . . . . . . . . . . . . 354.2.4 Connecting to Observables . . . . . . . . . . . . . . . 375 Valence Space In-Medium Similarity RenormalizationGroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395.1 Similarity Renormalization Group . . . . . . . . . . . . . . . 395.1.1 SRG Flow Equations . . . . . . . . . . . . . . . . . . 405.2 The IMSRG . . . . . . . . . . . . . . . . . . . . . . . . . . . 405.2.1 Nuclear Hamiltonian . . . . . . . . . . . . . . . . . . 415.2.2 Normal Ordering and Wick’s theorem . . . . . . . . . 435.2.3 ISMRG flow equations . . . . . . . . . . . . . . . . . 445.2.4 Magnus Formulation . . . . . . . . . . . . . . . . . . 475.3 A Tool for Valence Space Calculations: KSHELL . . . . . . . 485.4 Review of the Ab Initio Procedure . . . . . . . . . . . . . . . 495.4.1 Steps of the Ab Initio Calculations . . . . . . . . . . 495.4.2 Approximations and Truncations . . . . . . . . . . . 506 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536.1 Benchmarking and Result Verification . . . . . . . . . . . . . 546.2 Response functions and observables . . . . . . . . . . . . . . 566.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60viiTable of Contents7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63A NREFT Operators and Response Functions . . . . . . . . . 78A.1 WIMP-Nucleon Interaction Operators . . . . . . . . . . . . . 78A.2 Dark Matter Response Functions . . . . . . . . . . . . . . . . 79A.3 Nuclear Response Operators . . . . . . . . . . . . . . . . . . 80viiiList of Figures1.1 Progress of the reach of ab initio calculations . . . . . . . . . 32.1 Strength of the nuclear potential as a function of distance . . 52.2 Schematic view of the shell model of the nucleus . . . . . . . 133.1 The two cases for helicity . . . . . . . . . . . . . . . . . . . . 203.2 Allowed Feynman diagrams representing the first 4 orders ofChiral EFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.1 Galactic rotation curve, with expected and observed curves . 284.2 Sketch of the possible standard model dark matter interactionand their associated detection methods . . . . . . . . . . . . . 294.3 Exclusion plot for a (incomplete) variety of WIMP detectionexperiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305.1 Schematic view of the effect of the VS-IMSRG . . . . . . . . 425.2 Schematic representation of the evolution of the nuclear Hamil-tonian under the IMSRG evolution . . . . . . . . . . . . . . . 476.1 Convergence of the Φ′′ response function at zero momentumtransfer with respect to increasing emax . . . . . . . . . . . . 546.2 M and Φ′′ response function obtained from the IMSRG andNREFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566.3 Φ′′ response function with and without the IMSRG evolution 576.4 All 8 nuclear response functions for 27Al . . . . . . . . . . . . 586.5 All non-zero response functions for 40Ar . . . . . . . . . . . . 596.6 Transition amplitude for eight of the computed nuclei . . . . 60ixList of Abbreviations3N, 4N, ... Three Nucleon, Four Nucleon, ...BSM Beyond the Standard Model.ChPT Chiral Perturbations Theory.DM Dark Matter.EFT Effective Field Theory.HF Hartree-Fock.HO Harmonic Oscillator.IMSRG In-Medium Similarity Renormalization Group.LECs Low Energy Coupling Constants.LQCD Lattice Quantum Chromodynamics.MBPT Many-Bodu Perturbation Theory.NLO, N2LO, ... Next-to-Leading Order, Next-to-Next-to-Leading Order,...NN Nucleon-Nucleon.NREFT Non-Relativestic Effective Field Theory.xList of AbbreviationsOBDME One-Body Density Matrix Elements.ODE Ordinary Differential Equation.p-p, n-n, n-p proton-proton, neutron-neutron, neutron-proton.QCD Quantum Chromodynamics.QED Quantum Electrodynamics.ShM Shell Model.SM Standard Model.SNPA Standard Nuclear Physics Approach.SRG Similarity Renormalization Group.SU Special Unitary.VS Valence Space.WIMP Weakly Interactig Massive Particle.xiAcknowledgementsI am very thankful to my research supervisor, Dr. Jason D. Holt, for guid-ing me through my Master’s degree. His knowledge and advice have beeninvaluable in completing this work. Ample thanks also go to Dr. BaishanHu, for guiding me throughout this project. His vast knowledge of the fieldand guidance have been extremely valuable not only in helping me completethe project, but also in gaining a better understanding of nuclear physicsas a whole. Many thanks also to Dr. Takayuki Miyagi and Dr. RagnarStroberg for their crucial inputs on the the work presented here. I am alsovery grateful to Dr. Jens Dilling for acting as my academic supervisor atUBC.Abundant thanks to my parents, Me´lanie Lussier and Yannick Bruneault,my brother, Benjamin Bruneault, as well as to my friends. Their moralsupport and encouragements have helped me through a lot.Finally, I acknowledge the support of the Natural Sciences and Engineer-ing Research Council of Canada (NSERC), which has in part funded thisresearch through a graduate scholarship. Further acknowledgements go tothe support of the McDonald institute, which has also in part funded thisresearch.xiiTo my familyxiiiChapter 1IntroductionOne of the biggest unsolved mysteries in physics pertains to the nature ofdark matter (DM). In spite of the great experimental progress [3, 4] that hasbeen achieved, dark matter has yet to be detected in a laboratory, and assuch has not been identified yet. In light of this situation, many models havebeen proposed as one of the dark matter progenitors [5], based on variousrequirements that will be discussed in Chapter 4. One particular categoryof models that has been widely looked for experimentally is the weaklyinteracting massive particle (WIMP)[6]. A number of different experimentalmethods have been used to look for these particles, including indirect [7] anddirect [8, 9] detection. The latter technique consists of detecting nuclearrecoil events caused by the scattering of WIMPs off of a target nucleus.Indeed, a flux of dark matter particles is to be expected through Earthas it transits through the Milky Way [10]. To interpret results from theseexperiments, a precise calculation of observables, such as recoil rates andcross-section, must be carried out. The goal of this thesis is to computesuch observables in the ab initio framework of nuclear physics.The starting point of ab initio nuclear physics is to consider the many-body Schro¨dinger equation, including the protons and neutrons that makeup the nucleus of interest. This might initially seem like a relatively sim-ple calculation akin to the hydrogen atom, for which one ultimately usesquantum electrodynamics (QED) to derive corrections to the Coulomb in-teraction, then solves the system of interest with the Schro¨dinger equation.Naively, one would think of using a nuclear interaction derived from thestrong interaction equivalent of QED, quantum chromodynamics (QCD). Inreality, the non-perturbative nature of QCD [11] at low energies, that is, theinability to express the theory as a power series, makes it extremely chal-lenging to derive a nuclear interaction this way [12]. Hence, for the mostpart nuclear physics has used phenomenological interactions, such as theArgonne potentials [13] or meson models [14] for instance.Another issue lies in actually solving the many-body Schro¨dinger equa-tion once a potential has been obtained. Realistic nucleon-nucleon (NN)potentials are heavily coupled between high and low momentum modes,1Chapter 1. Introductionmeaning they are highly non-perturbative [15]. It quickly becomes com-putationally unfeasible to directly diagonalize nuclear Hamiltonians due totheir size [16], even on modern-day superclusters. As an example, it is esti-mated in [16] that, if assuming 50 possible proton and neutron states, writingdown and storing of the full 16O nuclear Hamiltonian would take ∼ 7× 1023terabytes of memory. For these reasons, until the last few decades, ab initiocalculations in nuclear physics have largely been considered unfeasible.Thanks to recent advances in effective field theories (EFTs) [17–19] andmany-body methods [20–24], however, ab initio calculations of nuclear prop-erties are now computationally feasible. In particular, there are two maintools we will use in this process. First, we will derive nuclear interactionsfrom first principles using Chiral EFT [17–19]. This framework begins bywriting the most general Lagrangian respecting the symmetries of QCD anduses Chiral perturbation theory (ChPT) to derive and compute Feynmanndiagrams. This approach describes the strong interaction at momenta ofless than 1 Gev, where quarks are confined to hadrons [25]. In our case itis used to derive nuclear interactions. Additionally, we will use the EFTframework to couple a WIMP current to a nucleus [26]. This will allow usto compute observables of WIMP-nucleus scattering for a general WIMPmodel, needing only to compute operators related to the nucleus.Secondly, we will use the valence space in-medium similarity renormal-ization group VS-IMSRG [20–24], a powerful many-body method, to solvenuclear Hamiltonians. In essence, this method evolves the Hamiltonian toa block-diagonal form, mapping the dynamics of the whole nucleus ontoits valence space, without affecting observables. The remaining part of theHamiltonian is then small enough to allow for direct diagonalization. Thisallows for ab initio calculations in fairly heavy nuclei, progress in the lastyears may be seen in Figure 1.1This thesis is organized as follows. In Chapter 2 we will go through someof the basics of nuclear physics that will be relevant to the key elements of thethesis. In Chapters 3 and 4 we will introduce the Chiral and non-relativisticEFTs, which will respectively be used to derive nuclear interactions and tocouple an external WIMP current to a nucleus. Additionally, in Chapter 4we will go through a short review of the current state of evidence for darkmatter, as well as the models and detection methods that are relevant to thiswork. In Chapter 5 we will introduce the VS-IMSRG, discuss how it may beused to evolve large valence space Hamiltonians, as well as its limitations.We will also discuss some of the other numerical methods and techniquesthat are used in this process. Finally, we present and discuss our results forWIMP-nucleus scattering in medium mass and heavy nuclei in Chapter 62Chapter 1. Introductionand conclude in Chapter 7. Through this thesis, the natural units ℏ = c = 1will be used.Figure 1.1: Progress of the reach of ab initio calculations from 2010 to 2022.Nuclei for which ab initio calculations had been performed at that pointare denoted by colored squares. Candidates for neutrinoless double betadecay (0νββ) and DM detection are also indicated. Calculations of heavy(in particular, up to 208Pb [27]) nuclei are now possible [22].3Chapter 2Nuclear Physics BasicsBefore going into the modern, ab initio approach to nuclear physics, let usreview some of the basics of nuclear physics which will be relevant throughthis thesis. This includes a qualitative review of the nuclear interaction,basis choices for the nuclear Hamiltonian, some phenomenological modelswhich have been used through the years and useful mathematical tools wewill need to evaluate matrix elements.2.1 A Qualitative Overview of the NuclearInteractionSince we will be spending significant effort determining a realistic expressionfor the nuclear interaction, let us start this review by describing some of itsqualitative properties known from experiments. When we discuss nuclearinteractions later, be it phenomenological or ab initio, we will see how someof these properties emerge or are taken into account by different models [28].Generally attractive interactionThe fact that nuclei are stable at all and not repulsing electromagneticallyshould be taken as indication that the nuclear interaction is generally attrac-tive. Additionally, its strength has to be much stronger than electromag-netic interactions at distance of a few femtometers, otherwise the repulsionof protons would split nuclei [28].Short range interactionUnlike the electromagnetic and gravitational interactions, the nuclear inter-action has a very limited range, its effective range being of about 3 fm [29].This range was first measured from nucleon scattering experiments, whereit was found to be (2.71± 0.13) fm [29].42.1. A Qualitative Overview of the Nuclear InteractionRepulsive interaction at very short rangeAlthough we have stated that the nuclear interaction is generally attractive,it is quite repulsive at ranges of up to about 0.8 fm [30]. This repulsion atshort distances is also referred to as the force having a ”hard core” [31]. Thiscore was first proposed in [30], to explain some of the qualitative featuresof n-p and p-p scattering data [32]. Hence, the overall behavior of thenuclear interaction goes as follows for increasing distance: repulsive up to adistance of about 0.8 fm, then attractive with a strength that starts to falloff exponentially at around 1 fm, until it is completely negligible beyond 3fm. This behavior is plotted in Figure 2.1 [29].Figure 2.1: Strength of the nuclear potential as a function of distance (infm). A positive potential indicates repulsion. The first three features wediscuss can be seen: greatly repulsive until 0.8 fm, then attractive andrapidly decaying, reaching its effective range at ∼2.5-3 fm [33].Spin dependent interactionAnother property of the nuclear interaction is its dependence on the par-ticles’ spin. This was first observed experimentally for the neutron-proton(n-p) force in scattering cross-sections of ortho- and parahydrogen [34]. Thedifference between these two molecules, both made up of two hydrogenatoms, is that the spins are aligned in orthohydrogen and form a tripletstate, whereas they are unaligned in parahydrogen, forming a singlet state[35]. The result is a total proton spin of 1 in the orthohydrogen (which isthus a higher energy state), and of 0 in the parahydrogen. Early neutron52.1. A Qualitative Overview of the Nuclear Interactionscattering experiments were able to measure n-p scattering cross-sectionsfor both molecules, and concluded from the differences observed that thenuclear interaction was spin dependent [34].Charge independent interactionThe strong force, unlike the electromagnetic force, is not directly dependenton electric charge. Indeed, QCD is a theory that acts on quarks and gluons,in which color, a three-valued quantum number that is a property of quarksand gluons, plays the role of charge [36]. Since the underlying theory ofthe nuclear force is not one of electric charge, nuclear interactions shouldalso be expected to not directly depend on charge, as no charge term shouldshow up in the Lagrangian [11]. The electric charge independence of nuclearinteractions had been known many years before the advent of QCD and thediscovery of quarks[37]. This could be verified experimentally by measuringthe cross-sections of p-p, n-n and p-n scattering for particles in the samestates. The absence of a difference between these measurements stronglyindicated that the strong interaction does not depend on electric charge[37, 38]. One important note is that electromagnetic interactions given bythe Coulomb terms have an important effect in nuclei [39], we have simplystated that the nuclear interaction itself does not depend on it directly.Non-central interactionOne property of the electromagnetic and gravitational interactions is thatthey are central, depending only on the distance between particles, andgreatly simplifying calculations. The nuclear interaction offers no such el-ements, and contains contributions from a force that is not invariant un-der rotations and hence depends on how it is oriented in space (a tensorforce)[14]. Explicitly, this implies that in a two-nucleon system, such as thedeuteron, a quadrupole moment will arise. This quadrupole moment wasfirst measured in the deuteron in 1939 [40–42], confirming the non-centralityof the nuclear interaction.Spin orbit force interactionOne important effect in nuclei is spin-orbit coupling, the same phenomenonthat is responsible for the fine structure of hydrogen [43]. Mathematically,this may be expressed as a term of the form Vso(r)l · s in the 2N potential.This effect was originally considered when potentials containing a centraland tensor term failed to reproduce p-p scattering polarization data [44].62.1. A Qualitative Overview of the Nuclear InteractionIt was noted that the inclusion of a spin-orbit term was necessary to con-sistently reproduce p-wave (total angular momentum l = 1) results in thetriplet (spin s = 1) state of the proton [44].3+ N contributions to the interactionThree (and more) nucleon (3N) forces are interactions that only appear inthe presence of three or more nucleons. These have been suspected to ex-ist in nuclear forces for quite some time [45], and it has historically been apoint of contention to determine whether these need to be considered whenbuilding nuclear potentials. While there are results which can be reproducedwithout considering these effects [46], 3N forces do turn out to be essentialin describing properties such as the binding energies of light (A < 10) nu-clei [47–50]. Hence, potentials with 3N contributions have been proposedsince the 1950’s [51] and have since been improved upon in many ways [18].For instance, considering 3N forces from two-pion exchange resulted in acorrection to the triton binding energy of about ten percent to two-bodycalculations [51]. In particular, when we consider Chiral EFT in Chapter3, we will see how 3N nuclear forces emerge from low-energy QCD and howtheir contributions can be systematically derived.Meson exchange in the long-range interactionAlthough we have mostly considered nucleons as the fundamental particlesof nuclear interactions, the particles that are traditionally seen as carryingthe nuclear force are the mesons (though we now know the gluons carrythe strong interaction). We will explicitly consider them as carrying the in-teraction in Chapter 3 when introducing Chiral EFT. Mesons are particlesmade of an equal number of quarks and anti-quarks [52]. First theorized in1935 [53] by Yukawa as the carriers of the nuclear force, they were subse-quently detected in 1947 [54]. Much experimental evidence was found that,in particular, the pions, and the one-pion exchange process were critical inexplaining the long-range component of the nuclear interaction [55]. Me-son theory has been widely studied phenomenologically before the adventof QCD and the discovery of gluons as the true carriers of the strong force,thorough review can be found in [14].72.2. The Many-Body Schro¨dinger Equation and Basis Choice2.2 The Many-Body Schro¨dinger Equation andBasis ChoiceThe main goal of theoretical nuclear physics is to solve the many-bodySchro¨dinger equation for a realistic nuclear interaction (exactly what ”real-istic” means will be made clearer in Chapter 3). This may be achieved bydiagonalizing the many-body Hamiltonian, which can be written as followsassuming only 2N forces:H = T + V =A∑i=1− ℏ22mN∇2i +A∑i,j<1,i<jv(ri, rj). (2.1)Here, mN =mp+mn2 ≈ 940MeV is the nucleon mass, where we makethe approximation that the masses of the proton and of the neutron are thesame. The ri are the positions of each nucleon and v is the NN potentialchosen. To diagonalize this, we will to choose a basis, which we will obtainvia the Hartree-Fock (HF) method [56].2.2.1 The Mean-Field ApproximationTo construct the basis, we will first need to make an assumption about theNN potential of interest. As previously stated, the nuclear interaction isnot central, and thus a nucleus cannot a priori be seen as resulting fromcentral forces. This is unlike atomic calculations, where one can assumethe Coulomb force from the nucleus to be central, as the distance betweenelectrons and the nucleus is quite large compared to the size of the nucleus[57]. Nonetheless, we make an approximation, known as the mean-field ap-proximation, that nucleons are moving in an external potential v(r), whichis generated by all of the other nucleons. Explicitly, the Hamiltonian willbe separated in two parts, an effective mean-field Hamiltonian HMF and aresidual interaction VRES, as follows:H =(T +A∑i=1v(ri))+(V −A∑i=1v(ri))= HMF + VRES. (2.2)Here we have only added and subtracted a (thus far undefined) potentialv(r). The residual interaction is82.2. The Many-Body Schro¨dinger Equation and Basis ChoiceVRES =A∑i,j<1,i<jv(ri, rj)−A∑i=1v(ri), (2.3)and becomes weaker than the actual interaction V as the mean fieldv(r) gets more realistic (closer to V). In the mean-field approximation, wewill solve the mean-field Hamiltonian, which is achieved by using the usualseparation ansatz, and considering the residual approximation to be smallenough to be ignored. Thus, this becomes a problem of minimizing the resid-ual interaction, which we will accomplish using the Hartree-Fock method.2.2.2 The Hartree-Fock MethodThe Hartree-Fock method [56] is a variational calculation applied to a Slaterdeterminant ansatz [39], used to minimize the residual interaction intro-duced in sect. 2.2.1. As a reminder, the Slater determinant is an expressionfor the wave function Ψ0(r1, r2, ..., rA) of a system of multiple fermions,which are half-integer spin particles obeying the Pauli exclusion principle[57]. The determinant is normalized and also satisfies the Pauli principlefor wave functions of fermions, in that it is anti-symmetric with respect topermutations of two particles. Explicitly, the wave function is given by adeterminant over the single-particles wave functions ϕi(rj), and is writtenas such:Ψ0(r1, r2, ..., rA) =1√A!∣∣∣∣∣∣∣∣∣ϕ1(r1) ϕ2(r1) ϕ3(r1) · · · ϕA(r1)ϕ1(r2) ϕ2(r2) ϕ3(r2) · · · ϕA(r2).......... . ....ϕ1(rA) ϕ2(rA) ϕ3(rA) · · · ϕA(rA)∣∣∣∣∣∣∣∣∣ .We now want to find a potential and a set of wave functions that minimizethe residual interactions. Equivalently, we want to minimize the ground-state energy, which is given by the expectation value of the Hamiltonian inthe ground-state:Egs = ⟨Ψ0|H|Ψ0⟩. (2.4)Here, we will use the Slater determinant as our ansatz for the ground-state |Ψ0⟩. Minimizing this is equivalent to the following functional problem:92.2. The Many-Body Schro¨dinger Equation and Basis Choiceδ(⟨Ψ0|H|Ψ0⟩⟨Ψ0|Ψ0⟩)= 0, (2.5)where δ denotes the variation of a functional [58]. This equation canbe solved using Lagrange multipliers methods [56]. Doing so will yield theHartree-Fock equation:−ℏ22mN∇2ϕα(r) + VHF({ϕi(r)})ϕα(r) = ϵαϕα(r), (2.6)where the ϵα are the eigenenergies. Since the potential VHF is a func-tional that depends on the wave functions, this is more complex than theSchro¨dinger equation despite its similar appearance. Since this equation isnonlinear it must be solved iteratively [39]. The outcome of this calculationis that we will have a set of wave functions and a mean potential that min-imize the residual interactions. While we will not use these directly in ourab initio calculations, we can use them to construct a basis in which we willcompute the matrix elements of the nuclear Hamiltonian and operators ofinterest.2.2.3 Harmonic Oscillator BasisA simpler basis to consider is the harmonic oscillator (HO) basis. In practice,we will initially compute our operator and Hamiltonian matrix elements inthis basis, then transform to the HF basis, as it is simpler to write downmatrix elements in the HO basis. The HO potential is given by:vHO = −V1 + 12mNω2r2, (2.7)where V1 is the potential depth, ω is the harmonic oscillator frequency,and both of these parameters are fit experimentally [59]. This potential issimple enough that it can be solved explicitly. It is central, therefore theangular parts of the wave function are given by the usual spherical harmonics[60], and only the radial Schro¨dinger equation needs to be considered. Thisresults in a set of Rnl(r) radial wave functions, explicitly given by:Rnl(r) =√2n!b3Γ(n+ l + 3/2)(rb)lexp(−r22b2)Ll+1/2n(r2b2). (2.8)The Ll+1/2n are the associated Laguerre polynomials and b is the oscillatorlength, given in terms of the frequency by b =√ℏmNω . These wave functions102.3. Phenomenologyfollow the orthonormality rules, and they can be used to construct the HObasis. Explicitly, when we write down matrix elements of a function oroperator hl′jlj′(r) in this basis (we later refer to such matrix elements assingle-particle HO matrix elements), we will be interested in the following:⟨n′l′j′|hl′jlj′(r)|nlj⟩ =∫ ∞0r2Rn′l′(r)hl′j′lj(r)Rnl(r)dr=2bl+l′+3√n!n′!Γ(n′ + l′ + 3/2)Γ(n+ l + 3/2)n′,n∑i,j=0(−1)i+ji!j!b2(i+j)(l′ + n′ + 1/2n′ − i)(l + n+ 1/2n− j)∫ ∞0drrρe−r2/b2hl′j′lj(r).(2.9)Here, ρ is given by ρ = l′ + l+ 2(i+ j) + 2, which is even, since the sumof the angular momenta is even. We will see in Section 2.4 how these matrixelements may be reduced in some of the quantum numbers. Note that thesematrix elements depend on j, j′ only through the function hl′j′lj(r), as theHO radial wave functions are independent on it. In fact, all operators wewill consider in this work are independent from j, j′, and thus so will be allmatrix elements.The basis generated by this is infinite, so we will need to introduce anexplicit cutoff. We will use a basis truncation emax, such that e = 2n+ l ≤emax, yielding an explicit upper limit on the n and l values of the matrixelements to be considered. This truncation is an approximation, and as suchwhen we compute operators, we will need to perform all computations atdifferent, increasing values of emax. This is done to ensure that the resultsare converged as a function of the truncation. We will introduce additionaltruncation parameters in Chapter 5 when discussing the IMSRG.2.3 PhenomenologyPhenomenology in nuclear physics refers the use of nuclear models in whichundetermined parameters are typically fit to reproduce particular experi-mental data in the regions of interest. These models are then used to makefurther calculations and predictions of nuclear properties (as an example, wewill consider how the Wood-Saxon potential may be used to make such pre-dictions). Phenomenology has been widely used in nuclear physics since the112.3. Phenomenology1960s [61], after unsuccessful attempts were made at creating a fundamentalnuclear theory. Such a theory would have been based on QFT, consideringthe nucleons and mesons as the fundamental degrees of freedom. In thissection we will discuss the basics of nuclear phenomenology, as well as itslimitations.2.3.1 Nuclear Shell Model and Valence SpaceOne of the early challenges of nuclear physics was to reproduce from a the-oretical model the ”magic numbers” of nuclear structure. These numbers(which go N,Z = 2, 8, 20, 28, 50, 82, as well as 126 for neutrons) referto a number of protons or neutrons of nuclei that are particularly stable(meaning having a high binding energy compared to nuclei with comparableN and Z). Nuclei experimentally show a shell structure much like the hydro-gen atom, and these magic numbers correspond to filled shells. This maybe observed as ”jumps” in the binding energies per nucleon of nuclei with amagic number of protons and/or neutrons, which can be seen in Figure 1.3of [62] for instance. The shell model (ShM) approach was first developed byEugene Paul Wigner, Maria Goeppert Mayer and J. Hans D. Jensen [63–66],work for which they shared the Nobel prize of Physics in 1963. A schematicview of the model for 23Na may be seen in Figure 2.2.In this model, the protons and neutrons each fill their separate shells.If one of these completes a shell, the nucleus is called semi-magic, and itis called magic if both particles completely fill a shell. In non-magic cases,the nucleus may be seen as being made up of two parts: an inert, stablecore made up of filled shells, and a valence (or model) space made of theremaining particles (there is also an excluded space, which we will talk a bitmore about in Chapter 5). In phenomenology, the nucleons in the core areconsidered to not contribute to the interactions, which greatly reduces thenumber of nuclear matrix elements to be computed. This has historicallybeen one of the most successful models in describing energies and transitionsin nuclei (see for instance [67, 68]). This is particularly true in the p, sd andpf [39] shells, which are the nuclei valence spaces (or groups of energy levels)shown in Figure 2.2. The names of these valence spaces refer to the angularmomentum numbers l that are present in that grouping of energy levels. Thisapproach is an approximation, and one that typically deteriorates the furthera nucleus is from a closed core (that is to say, the further N and Z are frombeing magic numbers), particularly in heavy systems (N + Z ⪆ 100) [39].Valence space calculations will nonetheless be important when discussingthe VS-IMSRG in Chapter 5.122.3. PhenomenologyFigure 2.2: Schematic view of the shell model of the nucleus for 23Na. Theorbitals are labeled in the same way as in the hydrogen atom, where thefirst number is the energy level n, the letter labels the angular momentumnumber l and the second number is the spin projection m. In this case, thecore of the nucleus (which is indicated by the red energy levels and nucleons)is made of the first 8 protons and neutrons forming a 16O core, which on itsown would be a magic nucleus. The valence space (blue energy levels andnucleons) is made of the remaining 3 protons and 4 neutrons, filling theirrespective shells. The splitting of the energy levels seen here (going fromleft to right) is an effect of considering spin-orbit coupling [39].2.3.2 The Wood-Saxon Potential and the StandardProcedure Nuclear PhysicsOne phenomenological potential which may be used with spin-orbit couplingand Coulomb effects to reproduce the shell model and magic numbers is theWood-Saxon potential [69] (another example is the optical model potential[70, 71]).vws(r) =−V01 + exp(r−Ra) . (2.10)132.4. Matrix ElementsHere, there are various parameters for which the numerical values are de-rived from fits to experimental observables, and a parameterization is givenin [39] by R = 1.27A1/3 fm, a = 0.67 fm and V0 = (51± 33(N −Z)/A)MeV(a more recent parameterization is given in [72]). These parameters arerespectively the average radius of a nucleus with A nucleons, the surfacediffuseness (corresponding to the surface ”thickness” of the nucleus) andpotential depth (roughly corresponding to the strength of the mean poten-tial at the center of the nucleus).This potential will not directly reproduce the shell structure and magicnumbers. As we mentioned when discussing the qualitative properties ofthe nuclear interaction, a spin-orbit term must be also included to split theenergy levels, as seen in Figure 2.2:vso(r) = Vso1rddr(11 + exp(r−Ra)) l · s. (2.11)The typically used value for the spin-orbit strength is Vso = 0.44V0 [39],and the values of R and a are the same as for eq. 2.11 [39]. Additionally, aCoulomb interaction term must be added to account for the electric repulsionbetween protons:vc(r) ={Ze2r if r ≥ RcZe2Rc(32 − r22R2c)if r < Rc(2.12)The total potential is given by summing up these three potentials. Wavefunctions can be obtained from numerically solving the Schro¨dinger equationwith the so-derived potential. Deriving energies from these will result in en-ergy levels reproducing the shell structure and magic numbers. We will referto this approach of obtaining wave functions from the Schro¨dinger equationwith phenomenological potential as the standard nuclear physics approach(SNPA). This approach with various nuclear potentials has historically beensuccessful in computing properties of few nucleon systems [73].2.4 Matrix ElementsIn this section, we will introduce some mathematical tools that will be usedin dealing with matrix elements of tensors/operators later on. In particular,we want to introduce the Wigner-Eckart theorem [39], which will reduce thenumber of matrix elements we need to consider. This section is based onparts of Chapters 1 and 2 of [39].142.4. Matrix Elements2.4.1 Angular Momentum CouplingWe begin with the coupling of two angular momenta. Starting from coupledbasis states |j1j2jm⟩ and uncoupled basis states |j1m1j2m2⟩, we define theClebsch-Gordan (CG) coefficients using:|j1j2jm⟩ ≡∑m1m2(j1m1j2m2|jm)|j1m1j2m2⟩, (2.13)where the (j1m1j2m2|jm) are the coefficients. These define a transforma-tion from one basis to the other and are used in angular momentum algebra[74]. An alternate way to carry the same information is by considering theWigner 3j symbols, which may be written as follows:(j1 j2 j3m1 m2 m3)= (−1)j1−j2−m3 1√2j3 + 1(j1m1j2m2|j3 −m3). (2.14)This is has nices symmetry properties than the CG coefficients, whichcan be seen by considering how both symbols change under changing of thecoupling order:(j2m2j1m1|jm) = (−1)j1+j2−j(j1m1j2m2|jm), (2.15)(j2 j1 j3m2 m1 m3)= (−1)j1+j2+j3(j1 j2 j3m1 m2 m3). (2.16)We can see the phase factor is more symmetric in angular momentumfor the 3j symbol. Additionally, the 3j symbol may be generalized to thecoupling of of three or four angular momenta, in the Wigner 6j and 9jsymbols [39]. For these reasons, we will make use of the Wigner symbolsrather than the CG coefficients in dealing with angular momentum coupling.2.4.2 The Wigner-Eckart TheoremIn the following sections, we will consider the matrix elements of not onlynuclear Hamiltonians, but also of tensor operators related to dark matterscattering. The details on these topics will be introduced in Chapters 3and 4, but for now let us give a theorem concerning the matrix elements ofa spherical tensor TLM . For general quantum numbers j, m and ξ (with ξbeing an umbrella symbol for any other good quantum number), the Wigner-Eckart theorem may be stated as:152.4. Matrix Elements⟨ξ′j′m′|TLM |ξjm⟩ = (−1)j′−m′(j′ L j−m′ M m)⟨ξ′j′||TL||ξj⟩. (2.17)Here, ⟨ξ′j′||TL||ξj⟩ is a reduced matrix element, called this way due tothe reduction in the quantum number m. These elements are very powerful,as they allow us to perform calculations of fewer matrix elements. Indeed,once the reduced matrix elements have been found, the matrix elementsincluding m may be recovered by applying eq. 2.17.16Chapter 3Chiral Effective Field TheorySo far, we have discussed some of the phenomenological models of nuclearphysics, and how these procedures may be used to predict properties of nu-clei. Although these models have worked well for many applications, such asdescribing the shell model and structure [63, 64, 66] or computing bindingenergies [47–50], they lack predictive power based on first principles thata theory derived from QCD would have. Additionally, due to the approxi-mation that only the valence space contributes to nuclear interactions, it isnot generally possible to estimate missing contributions to the theory. Thismeans is is not generally possible to derive a rigorous systematic theoreticaluncertainty, or error bars on observables that would estimate the missedcontributions from approximations. While it is possible to propagate ex-perimental uncertainties through fitting procedures, the biggest source ofuncertainty is the unclear connection to nuclear forces and truncations inmany-body physics. Since phenomenological models are not based on QCD,from which missing contributions could be estimated, this uncertainty can-not be quantified in a meaningful way. The more modern approach to ad-dress these issues is to turn to an EFT[19], in this case Chiral EFT [17, 18],which is a low-energy expansion of QCD. This approach was first developedat the beginning of the 1990s [75], and has since proven to be a powerfulway of deriving nuclear interactions [76–78].3.1 QCDQCD is the theory of strong interactions and is expressed at the quark andgluon level [36]. Originally developed in the 1970’s [79–81], it places quarksas the interacting particles and gluons as the strong force carriers (althoughcontrary to photons, gluons can also interact between themselves). QCD isa non-Abelian gauge theory with symmetry group SU(3), making it morecomplex than its electromagnetic analog, QED.Though perturbative QCD is possible at energies where quarks are notconfined to quarks [82, 83], it is extremely challenging to carry it out atmomenta smaller than about 1 GeV, the scale at which quarks are confined173.1. QCDto nucleons [11]. This may be seen by considering a property of QCD knownas asymptotic freedom [79, 80]. This is best seen by considering the strongcoupling constant of QCD, which can be expressed as a function of thetransfer momentum q [11]:αs(q2) =αs1 + (b0αs/2π) log (q/M)=2πb0log(q/Λ). (3.1)Here, the fixed strong coupling constant is αs =g24π , and Λ ≈ 200MeVis a momentum scale set by experiments [84]. To allow for perturbationtheory, the running coupling must generally be smaller than αs(q) ≈ 0.4[84], meaning the momentum transfer must be q ≥∼ 1 GeV [39]. This iswhere the crux of the issue lies, as this is the scale at which quarks are nolonger confined to hadrons [25]. One way to unveil the hadronic nature oflow-energy QCD is to turn to non-perturbative methods, but such methodsdo not currently exist and are still being developed [85]. We will thereforeturn to Chiral EFT, which we will review later in this chapter.3.1.1 The QCD LagrangianThe Lagrangian of QCD may be written as follows [86] (we will follow Ein-stein notation [87], and repeated indices are summed over):L = q (iγµDµ −M) q − 14GaµνGµνa . (3.2)Here, the q are the Dirac spinor quark fields. Since quarks carry bothflavor (with 6 possible flavors) and color charge (with 3 possible colors),there are associated implicit indices running respectively from 1 to 6 (ifconsidering all flavors) and from 1 to 3. For simplicity, in eq. 3.2 the quarkfields are written down as matrices. The γµ are the usual gamma matricesand M is the quark mass matrix. The covariant derivative Dµ is given by:Dµ = ∂µ − igλa2Aaµ, (3.3)and the gluon field strength tensor is:Gµν,a = ∂µAν,a − ∂νAµ,a + gfabcAµ,bAν,c. (3.4)183.1. QCDFinally, the Aµ,a are the gluon fields, the λa are the (3x3) Gell-Mannmatrices [88] that generate the SU(3) group, the fabc are the structure con-stants and g is the strong coupling constant. As mentioned earlier, this iswritten entirely in terms of quarks and gluons. This is important, as weknow the relevant degrees of freedom for nuclear physics to be the nucleonsand mesons [14].There are a few factors which make this a harder problem than QED.First, this is a non-Abelian theory, which is why the third term of eq. 3.4arises, even though there is no similar term in QED. This term allows forgluon-gluon interactions, and since the a, b, c indices run from 1 to 8 (whichwill be justified shortly), there are many terms that arise. Secondly, quarksboth carry a color charge which respects the special unitary group SU(3),the 3×3 unitary matrices with determinant 1 [60], as well as one of six flavors[36]. There are eight different gluons, which correspond to all the linearlyindependent, non-color neutral combinations of color/anti-color. Hence, thea,b,c indices in the above equations do indeed run from 1 to 8.This Lagrangian respects a number of symmetries (a review of thesesymmetries can be found in [89]), but to introduce Chiral EFT, we willneed to discuss Chiral symmetry, for which we first introduce the conceptof chirality.3.1.2 ChiralityBefore discussing chirality, let us discuss helicity. Helicity is the projectionof the spin of a particle onto the direction of its momentum. Thus, sinceangular momentum is quantized and takes a set of discreet values, so doeshelicity. We define a particle as right-handed when its spin and momentumare aligned and its helicity is positive, and as left-handed in the oppositescenario, these cases are shown in Figure 3.1.Chirality is a property of Dirac spinors [90] which happens to be thesame property as helicity for massless particles. As for helicity, particles canbe either right or left-handed, depending on how their wave function behaveswhen the particle is rotated. This property has particularly important im-plications for the weak interaction, which only affects left-handed particlesand right-handed anti-particles [91]. Mathematically, chirality may be con-sidered in terms of a projection operator, which projects upon the Chiralsubspaces:P± =1± γ52, (3.5)193.1. QCDFigure 3.1: The two cases for helicity. Only the direction of momentum isconsidered in computing helicity, not its magnitude.where ’+’ corresponds to the right-handed projection, and ’−’ the left-handed. γ5 is the product of the other gamma matrices, a reflection of thefact that chirality only applies to Dirac spinors. Since γ25 = 1, it can be seenthat Chiral states are eigenstates of the γ5 matrix.3.1.3 Chiral symmetry of QCDChiral symmetry in the QCD Lagrangian can be seen by considering theright and left-handed components of the quark fields, using the projectionoperator we have just introduced:qR =1 + γ52q, qL =1− γ52q. (3.6)In terms of these components, the Lagrangian can be rewritten as:L = qRiγµDµqR + qLiγµDµqL −mqRqL −mqLqR −14GaµνGµνa . (3.7)A Lagrangian respects Chiral symmetry if it is invariant under the fol-lowing transformation:qR → eiθqR, qL → eiϕqL, (3.8)with arbitrary phases θ and ϕ. While the first two and last terms of eq. 3.7are invariant under eq. 3.8, the mass terms are not. However, the massesof up and down quarks (mu = 2.15(15) MeV and md = 4.70(20) MeV[92]) are small in comparison to that of hadrons (mp = 938.27208816(29)MeV, mn = 939.56542052(54) MeV [93]). Therefore, we consider the Chiral203.2. Building Chiral EFTsymmetry for up and down quarks to be an approximate symmetry in thetheory.This symmetry is also spontaneously broken [94], meaning that the sym-metry is not realized in the ground state, despite being respected in theLagrangian [18]. In other words, it undergoes a process where the lowestenergy state does not respect Chiral symmetry, even though the physicallaws describing the system are invariant under that symmetry. A classicexample is the ”sombrero” potential, where the top of the potential is in-variant under rotations, while the trough of the potential (or in the lowestenergy state) observes no such symmetry.In the ground state, chiral symmetry is broken down to SU(2)V , theisospin symmetry [89]. According to Goldstone’s theorem [95], the sponta-neous breaking of a continuous symmetry implies the existence of a masslessboson with 0 spin [11]. Such bosons are called ’Goldstone bosons’ and theycarry the relevant interaction [11]. Since Chiral symmetry is approximate,the Goldstone bosons obtain a mass, a review of why that is may be foundin [96]. These massive particles with spin 0 which respect isospin symmetryand carry the strong interaction are identified to be pions [18]. This greatlyexplains the success of phenomenological models considering pion exchange,as we had discussed in the qualitative overview of Section 2.1.3.2 Building Chiral EFTIn this section, we will review how we can take advantage of the symmetrieswe have mentioned to build a theory that describes QCD at energies smallerthan 1 GeV, Chiral EFT. For more thorough reviews, see [17–19].3.2.1 The Chiral EFT LagrangianAs mentioned before, it is not possible to solve QCD at transfer momentalower than ∼1 GeV using usual perturbative methods. We have, however,successfully identified the degrees of freedom of the theory that is interest tous: hadrons and pions. We have also identified an important symmetry ofQCD, the approximate, spontaneously broken Chiral symmetry. The nextstep in building our EFT is to write down the most general Lagrangianthat is consistent with the symmetries of low-energy QCD, in terms of theeffective degrees of freedom at this scale.Schematically, we can divide the effective Lagrangian into sub-pieces,grouping the terms by the kind of interaction that they describe:213.2. Building Chiral EFTLeff = Lππ + LπN + LNN + ... (3.9)The terms shown explicitly in this equation respectively contain termsrelated to the pion-pion (ππ), pion-nucleon (πN) and nucleon-nucleon (NN)interactions. There are higher order terms than these, for instance involving3-nucleon interactions. These individual pieces of the Lagrangian will befurther divided by their order as follows:Lππ = L(2)ππ + L(4)ππ + ... (3.10)LπN = L(1)πN + L(2)πN + ... (3.11)LNN = L(0)NN + L(1)NN + ... (3.12)The superscripts correspond to the number of derivatives and pion massesin each term of that particular piece (in the next section we will motivatethis approach). The explicit expressions for these terms may be found interms of nucleon and pion fields in [18]. Here, we will view this processin terms of the Feynman diagrams that are brought about by the differentterms.3.2.2 Power countingUp to this point, we have taken a Lagrangian that can be written as ineq. 3.2 and made it more general. An infinite number of Feynman diagramscould be derived and computed from the general terms we have written downso far, as the Lagrangian contains infinitely many terms. Indeed, looking ateq. 3.9, we could always consider terms for interactions with more particles(NNN , ππN for instance).This EFT process would therefore not be very helpful without someway to characterize Feynman diagrams by how large their contribution is.Fortunately, by considering Chiral perturbation theory (ChPT)[86, 97], wewill have a scheme to organize the terms. Indeed, the benefit of this schemeis that all Feynman diagrams that can be derived from Chiral EFT obtaina leading factor of (Q/Λχ)ν [18]. Additionally, for any given value of theChiral order ν, ChPT ensures that there is a finite number of contributingdiagrams [18]. Here, Q is a momentum, in our case either the nucleon orpion momentum, and Λχ is a mass scale characteristic of Chiral symmetrybreaking, and is about 1 GeV [18]. This corresponds to the scale above which223.2. Building Chiral EFTquarks are no longer confined to hadrons [36], where our writing down of thetheory in terms of nucleons and pions is no longer valid. The Chiral orderν is given in terms of properties of the Feynman diagram of interest:ν =∑iVi(di − 12pi)−D + 3L. (3.13)Here, Vi is the number of vertices, di stands for the number of derivativesand pion mass terms, pi is the number of pion fields, D is the number ofintermediate states and L the number of loops. Using topological identities[75], this can be written in the following way:ν = 2−N + 2L+∑iVi(di +12ni − 2). (3.14)Here, we have introduced from the topological identities the incomingand outgoing nucleon number N (In Figure 3.2.2 this corresponds to thefields going in a coming out of the diagrams), as well as the number ofnucleon fields ni. Every Feynman diagram that can be derived from theChiral EFT Lagrangian may be characterized by this number, so diagramsare organized by order.In Figure 3.2.2, we can see some of the characteristics of the nuclearinteraction from the diagrams alone. We see at leading order the long rangeone-pion exchange (OPE) and the short range contact interaction being al-lowed. At next-to-next-to leading order (NNLO), 3N forces also contribute,and as we had suspected [47–50], they have a smaller effect . Additionally,forces involving more than three nucleons also start contributing at higherorders. Since each contribution is suppressed by a factor of (Q/Λχ)ν , wehave succeeded in creating a framework based on QCD that works a lowenergy. There are two main advantages to this framework, notably thatthe description of interactions based on it are systematically improvable byconsidering contributions of higher and higher orders. Another advantage isthat it is possible to quantify uncertainties by estimating the contributionfrom higher contributions. A priori, one could estimate that if contributionsup to order ν are considered, then missed contributions should be propor-tional to (Q/Λχ)ν+1. However, this is a much more subtle and complexproblem than this estimation, as at some orders contributions may cancelout or add in ways that stray from this estimate (for instance, 3N contri-butions are theoretically possible at NLO, but they happen to cancel outat that order). Hence, uncertainty quantification is still a work in progress[98].233.2. Building Chiral EFTFigure 3.2: Allowed Feynman diagrams representing the first 4 orders of Chi-ral EFT. The solid lines represent nucleons, while the dashed lines representpions. 3 and 4 body forces emerge naturally from this scheme, respectivelyat NNLO and N3LO. More detail on the vertex types may be found in [18].3.2.3 Low-Energy ConstantsWhen we presented the Lagrangian of Chiral EFT in eq. 3.9, the formula wassomewhat schematic, splitting the different contributions by their involvedparticles. In a more rigorous (but still general) way, the Lagrangian mayalso be written as:L =∑idiOi({Ψ}). (3.15)Here the Oi is the most general set of operators that respect the symme-tries of QCD, and the Ψ are the relevant fields, determined by the degreesof freedom. The di are called the low-energy constants (LECs), which are fitto experimental data [18]. Depending on which interactions the constantsare associated with, this experimental data may be n-n scattering data or243.3. Chiral interactionsbinding energies [78, 99] for instance. Therefore, part of the process of de-riving a nuclear interaction from Chiral EFT is to determine the LECs thatbest reproduce experiments. A step-by-step example of LEC optimizationin a Chiral interaction may be found in [100].3.3 Chiral interactionsIn principle there is an infinite number of Feynman diagrams that can be de-rived from Chiral EFT as there are infinitely many terms in the Lagrangian.Even after organizing the allowed interactions by Chiral order, a decisionmust still be made of where to truncate the contributions. Hence, there aremany different interactions that can be derived even to the first few orders.In the rest of this thesis, we will consider three different, realistic nuclearinteractions derived from Chiral EFT. We will carry out all calculations foreach interaction, and will consider the spread in results between these in-teractions to be an estimation of the main source of uncertainty. Hence, alluncertainty bands in result plots in this thesis will be given by the discrep-ancies between results of the nuclear interactions. A more thorough reviewof other approximations and sources of uncertainty will be given in Chapter5, with the goal of motivating this approach.The first interaction we will use is the 1.8/2.0 (EM)[76, 77], where 1.8and 2.0 respectively refer to the low-resolution scale λNN and the 3N cutoffΛ3N (these parameters are defined in [76]). The (EM) stands for Entemand Machleidt, who introduced this interaction [101]. This interaction re-produces the ground-state energies of all closed-shell nuclei studied in [77].The second interaction is the NNLOsat [99], where NNLO indicates that theLECs are optimized at NNLO, and ’sat’ indicates consistency with satura-tion properties of nuclear matter (a review of saturation and nuclear mattercan be found in [102]). This interaction is used to derived accurate bindingenergies and radii in some low to medium mass nuclei (up to 40Ca). Finally,we also use the ∆NNLOGO [78], where ∆ indicates the use of the ∆-isobaras an additional degree of freedom. Again, NNLO indicates optimization ofthe interaction at that order, and ’GO’ stands for ’Gothenburg-Oak Ridge’.Explicitly in our case, these interactions will be used to derive nuclear Hamil-tonians.25Chapter 4Dark Matter andNon-Relativistic EffectiveField Theory4.1 Dark MatterEarly evidence for the existence of dark matter dates back to the 1930’s[103]. Back then, it was found that the visible mass in clusters of galaxieswas too small to explain the formation of the cluster. It was then theorizedthat some other, non-visible matter must have been gravitationally holdingthe galaxies in the cluster, as the observed mass was too small to explainstability in the cluster. Since the almost 100 years of this discovery, a wealthof evidence has been found indicating the presence of dark matter in galaxiesand clusters [4]. Likewise, models [5, 6] and experimental effort [7–9, 104]have multiplied in the previous decades, and the origin of dark matter hassince been positioned as one of the most important unsolved mysteries ofmodern physics [3].4.1.1 Evidence for the Existence of Dark MatterThe very first piece of evidence for dark matter was obtained by observationsof galactic clusters. In 1933, it was estimated by Fritz Zwicky that the ratioof solar mass to solar luminosity in the Coma cluster was of 400 to 1 [103].While evaluating the light emitting mass of a cluster can be done directlybe measuring its luminosity, the total mass including dark matter has to befound in more roundabout ways, as dark matter does not emit light. Onesuch way is by considering gravitational lensing [105], which is one of themost important test of general relativity [106]. As we know from generalrelativity [87], light follows the curvature of space-time, and is hence affectedby gravity despite being massless. Since any mass will bend space-time,including dark matter, one can take advantage of this phenomenon, anddetermine how heavy a cluster is [105, 107]. By considering the degree264.1. Dark Matterto which light bends around a mass, the mass of a galaxy cluster may beinferred. This, in turn, is achieved by comparing the observed position ofa source, such as a distant quasar, to its expected position (assuming theobject we want to measure the mass of is between the source and us).This is not the only option, however, to compute the mass of a galaxycluster. The method initially used in 1933 makes use of the virial theorem[4], to obtain a relationship between the observed temperature of a galaxycluster and its mass.kT ≈ (1.3− 1.8)keV(Mr1014M⊙)(1Mpcr)(4.1)The large discrepancies derived between the measured temperature and thatpredicted by the cluster’s regular matter composition were the first convinc-ing evidence of the existence of dark matter [103].Evidence for the existence of dark matter is however not only present atthe scale of galactic cluster. Much evidence is also present at the galacticscale, namely by considering the rotation curves of galaxies [108]. Thesecorrespond to the velocity distribution of galactic matter as a function ofdistance r from the galactic center. From Newtonian dynamics, one wouldexpect this distribution to look as follows:v(r) =√GM(r)r, (4.2)where the mass distribution is given by a radial integral over ρ(r) ∝1/√r, hence the velocity should be proportional to r5/2. However, actualmeasurements [108] of this distribution show that the velocity actually be-comes constant with distance after a certain distance from the galactic cen-ter, as can be seen in Figure 4.1 and Figure 2 of [108].Such behavior in the rotation curve implies the existence of some halowith mass density proportional to 1/r2 (because the mass distribution hasto be proportional to r for the right hand side of eq. 4.2 to be constant). Nosuch halo of regular matter is seen in galaxies [4], hence the strong suggestionthat some massive dark matter makes up these halos [109]. A review of otherevidence for the existence of dark matter can be found in Section 2.1 of [3],but these two in particular are very convincing observations.4.1.2 Dark Matter Models and DetectionOver the past century, since the first evidence for dark matter, many differentmodels [5] have been proposed as its progenitor. Despite this, and despite a274.1. Dark MatterFigure 4.1: Galactic rotation curve, the different contributions are high-lighted, and the expected distribution is compared to the observed one.While the distribution would be expected to fall off as 1/√r as distance getslarge, it is in fact observed to be close to constant.significant, worldwide experimental effort to detect dark matter [7–9, 104],very little is know about its composition. In particular, it is not knownwhether dark matter is made up of a single or multiple particle types. Inthis thesis, we will focus on the WIMP hypothesis [6], which we will brieflyreview here. There are many reviews of other models, which can be foundin [5, 110–112].The name ’WIMP’ does not refer to any specific model of dark matter,but rather to a range of different models [6]. As the name would indicate,it refers to DM models that do not interact electromagnetically, but thatdo interact weakly (as well a gravitationally) with standard model (SM)particles such as quarks [6]. In this context, interacting weakly means eithervia the weak interaction, or via a different, beyond the standard model(BSM) interaction with a coupling constant smaller than that of the weakinteraction [6].For quite some time, the WIMP hypothesis has been one of the leadingtheories for DM. This is in no small part due to the ”WIMP miracle” [113–115], a famous calculation of DM density. This calculation is based onthe assumption that WIMP DM is produced by freeze-out, meaning that284.1. Dark Matteras the early universe expands, WIMPs become increasingly more unlikelyto annihilate with themselves and their density is effectively ”frozen-out”[116]. This density is dependent on the WIMP annihilation cross-section.For a particle interacting at the scale of the weak force, it can be foundthat the expected freeze-out DM density matches within a few orders ofmagnitude with the density that is observed today in the universe [117].This coincidence initially garnered interest in WIMP models [118].When considering the different ways that experiments could detectWIMPs, one must consider the different ways dark and regular matter couldinteract if they do indeed interact weakly. Experiments could look to createdark matter in colliders by having only regular matter interact [104]. Addi-tionally, experiments could look for SM matter as the result of dark matterannihilation, which is known as indirect detection [7]. Finally, experimentscould look for nuclear recoil events caused by the scattering of incoming darkmatter off of a nucleus standing in a lab [8, 9]. These three possibilities aresummed up in Figure 4.2, in the form of a simple diagram.Figure 4.2: Sketch of the possible Standard Model dark matter interactions.The arrows indicate the direction of time corresponding to each detectionmethod. The interaction at the center is a black box SM/DM interactionthat would depend on the DM model. Depending on the direction the dia-gram is looked at, it can either show DM creation from SM matter collision(collider), SM matter creation from DM annihilation (indirect detection) ornuclear scattering (direct detection). [119]In this work, we will focus on direct detection experiments [8, 9], andwe will be concerned with the nuclear scattering of WIMPs. We will be294.2. Coupling Dark Matter to Nucleifocused on ab initio calculations of observables related to scattering, suchas cross-sections, with the goal of guiding direct detection experiments. Avariety of direct detection experimental results are summarized up in Figure4.3Figure 4.3: Exclusion plot for a (incomplete) variety of WIMP detectionexperiments. The irreducible neutrino background is plotted in yellow atthe bottom, and the upper limits on WIMP-nucleon cross-sections fromdifferent experiments are indicated as a function of WIMP mass. The fulllines indicate exclusion limits set by experiments, while the dashed linesindicate projections of sensitivity for future and proposed experiments. Plotgenerated using the SuperCDMS dark matter limit plotter, created by TarekSaab, University of Florida and Enectali Figueroa, Northwestern University.References for the results may be found in: [120–136].4.2 Coupling Dark Matter to NucleiIn Chapter 3, we have gone through the process of creating and solving atheory based on QCD with Chiral EFT. This was appropriate in derivingnuclear interactions, which can in turn be used to derive properties of nucleithemselves. We are, however, interested in the problem of WIMP-nucleusscattering, which will additionally require coupling an external current, cor-responding to a WIMP interacting with a nucleus. While Chiral EFT can304.2. Coupling Dark Matter to Nucleibe used to achieve this [137–140], we will consider here non-relativistic EFT(NREFT) [26], as a complementary approach. Rather than based on thesymmetries of QCD, NREFT is based on the requirement of Galilean in-variance [141]. This is motivated by the fact that expected WIMP-nucleusrelative velocities in a laboratory are on the scale of a few hundred kilome-ters per second [142], which is about a thousandth of c = 299, 792.458 km/s.This is far too low for relativistic effects to be a considerable effect comparedto other sources of errors, which will be discussed in Section 5.4.2. The maindraw of the EFT approach to couple external currents is that it may be usedto couple a general WIMP to nuclei, rather than a specific WIMP. Addi-tionally, it will separate nucleon dynamics from the WIMP current, so thatwe will only need to compute operators related to the nucleon.4.2.1 Formulation of the Effective Field TheoryThe starting point of NREFT is to write down the Hamiltonian density,which must be integrated over space to recover the Hamiltonian. We assumeonly one-body interactions between dark matter and nucleons (this is knowas the impulse approximation, and is reflected in the sum over A):HˆχA =A∑i=1∑τ=0,1∑jcτj Oˆ(i)j tτ(i). (4.3)Here, the Oˆ(i)j are a set of the most general operators that obey Galileaninvariance [141], they are listed in Appendix A.1. These operators are writ-ten in terms of Galilean invariant, Hermitian operators, of which there are4 in total. These are the momentum transfer iq (the operator q itself isanti-Hermitian hence it must get a factor of i), the dark matter and nucleusspin operators, Sχ and SN , as well as a Hermitian velocity given by:v⊥ = v +q(2µN ). (4.4)Here, v is the WIMP-nucleon relative velocity and µN =mNmχmN+mχ is theirreduced mass. Going back to eq. 4.3, the sums are respectively over thenucleons (up to A), over the isoscalar (τ = 0) and isovector (τ = 1) compo-nents, as well as over the different allowed interaction operators. The tτ(i) arethe Pauli matrices in isospin space and the cτj are the WIMP-nucleus cou-pling constants, which are entirely dependent on the chosen WIMP model.The coupling constants will be kept general through the process of cou-pling the WIMP current, but we will need to assign them values when we314.2. Coupling Dark Matter to Nucleiplot observables in Chapter 6. This is because observables directly dependon the dark matter model, as we will see in the next section. In this work,the coupling constants will be assumed to be proportional to 1/m2v, theelectroweak scale. This ensures that we can generate a WIMP-nucleus in-teraction with a desired fraction of the weak interaction’s strength. Thischoice is arbitrary and does not correspond to any particular DM model.4.2.2 Multipole Expansion and Nuclear ResponsesAt the end of this NREFT process, we want to have a set of operators thatdepend only on the nucleus, which is what can be computed in an ab initioframework using the IMSRG. Thus, it is necessary to consider the differentcouplings of a general dark matter model to the nucleus. For this, we rewritethe Hamiltonian density of eq. 4.3 as follows:HˆχA =A∑i=1∑τ=0,1{lˆτ0δ(r − ri) + lˆτ0Ai2mN[←−∇ · σiδ(r − ri)− δ(r − ri)σi · ∇⃗]+ lˆτ5 · σiδ(r − ri) + lˆτM ·i2mN[←−∇δ(r − ri)− δ(r − ri)∇⃗]+ lˆτE ·12mN[←−∇ × σiδ(r − ri)− δ(r − ri)σi × ∇⃗]} tτ(i). (4.5)The only new part introduced in this equation is the Pauli matrix vectorfor nucleon i, σi, otherwise this is simply rewriting of the original Hamil-tonian. The lˆ operators represent the different way in which dark mattercouples to the nucleus. In order of apparition in eq. 4.5, these are the cou-pling to the nuclear vector, axial charge, nuclear spin current, convectioncurrent and nuclear spin-velocity current. Explicitly, these may be writtenas:lτ0 = cτ1 − icτ5 (q × Sχ) · v⊥T + cτ8Sχ · v⊥T + icτ11q · Sχ,lτ5 =12[icτ3q × v⊥T + cτ4Sχ + cτ6Sχ · qq + cτ7v⊥T + icτ9q × Sχ + icτ10q],lτM = icτ5q × Sχ − cτ8Sχ,lE =12cτ3q + icτ12 − cτ13q × Sχ − icτ15Sχ · qq,v⊥T = δ(x− y + r)(i∇xmT− i∇ymχ)+12µTq, (4.6)324.2. Coupling Dark Matter to Nucleiwhere we have introduced and made use of the velocity v⊥T . Ultimately,we are interested in connecting the operators to observables such as cross-sections. To do this we must consider the transition probability∑spin |M|2,which is obtained from the Hamiltonian:∑spin|M|2 = 12J + 1∑Mi,Mf∣∣∣⟨JMf |HˆχA|JMi⟩∣∣∣2 . (4.7)J is the angular momentum and Mi/Mf go over its possible projectionson the z axis (which is defined as being in the same direction as the trans-fer momentum q). Note that the Hamiltonian HˆχA is used, rather thanits density. This is obtained by integrating the Hamiltonian density overr. Now, making use of some spherical harmonics identities (the sphericalharmonics are introduced when the Hamiltonian density is integrated over,details can be found in [26]), the Hamiltonian can be expanded out in amultipole expansion. The main goal of this expansion is to separate out theDM dynamics from the purely nuclear ones. Once this expansion is carriedout, the transition probability may be written as follows:∑spins|M|2 =8π∑τ,τ ′=0,1 ∑k=M,Σ′Σ′′Rττ′k(v⊥2T ,q2m2N)W ττ′k(q2)+q2m2N∑k=Φ′′,Φ′′M,Φ˜′,∆,∆Σ′Rττ′k(v⊥2T ,q2m2N)W ττ′k(q2) .(4.8)The full expanded expression for this may be found in eq. (3.20) of[26]. Two sets of functions are introduced here. First are the dark matterresponse functions Rττ′k(v⊥2T ,q2m2N), which depend entirely on the cτj couplingconstants, v⊥2T = v2−q2/(4µ2χ) and q2m2N. These response functions are set bythe dark matter model that is being considered and are listed in AppendixA.2. Mathematically, they are purely given by the operators introduced ineq. 4.6 and the q operator.Second are the W ττ′k(q2)are the nuclear response functions. They areintroduced when the Hamiltonian is integrated a set of spherical harmonicswhich is introduced and separated out from DM dynamics in the multipoleexpansion. These responses depend on the target nucleus and are the interestof this thesis.334.2. Coupling Dark Matter to NucleiFor both of these response functions, the k index denotes one of the pos-sible combinations of nuclear response operators. These response operators,listed in Appendix A.3, are M , Σ′, Σ′′, ∆, Φ′′ and Φ˜′. M and Φ′′ constitutethe spin independent responses, the other operators are spin dependent. Thenuclear response functions are given in terms of reduced matrix elements ofthese operators:W ττ′AB (q2) =∑L⟨jN , T,MT ||AˆL;τ ||jN , T,MT ⟩⟨jN , T,MT ||BˆL;τ ′ ||jN , T,MT ⟩,(4.9)where A and B are the nuclear response operators, T is the total isospinandMT is the magnetic quantum number for isospin. There are eight possi-ble combinations of A and B, six for A = B, as well as two others: A =M ,B = Φ′′, and A = ∆, B = Σ′. Each of these operators may only take certainvalues of L. M and Φ′′ may take the values L = 0, 2, 4, ..., Σ′, while Σ′′, ∆may take the values L = 1, 3, 5, ... and Φ˜′ takes the values L = 2, 4, ..., all ofthese going up to 2jN . The operator M is a particular case, as it takes thefollowing value in the zero momentum transfer limit:M00(q → 0)→ 1√4πA∑i=11i. (4.10)Here, 1i is the identity operator for the i-th nucleon. This dependenceon the identity operator only implies that at zero momentum transfer, theM00 operator is entirely fixed by the nucleon content, and is not a predictionof the underlying nuclear physics. Additionally, a similar expression can befound for the isovector component of the operator. This operator providesthe biggest contribution to the transition probability. This can be seen inAppendix C of [143], whereW 00M (0) for nuclei we consider in this work can beseen to be at least an order of magnitude larger than response functions forother operators. Hence, observables are in large determined by the nucleoncontent, as will be seen in Chapter 6.This limit only applies to the M operator, other operators are a predic-tion and depend on the used nuclear model. For instance, the limit for theΦ′′ operator looks as follows:Φ′′00(q → 0)→−13√4π∑iσi · li. (4.11)344.2. Coupling Dark Matter to NucleiThis depends on the spin-orbit coupling in the nucleus, hence is notsomething that can be fixed by knowledge of the number of nucleons. Theother four operators are similar and their zero momentum transfer limit canbe seen in Table 1 of [26].4.2.3 Computing Matrix ElementsEvaluating the matrix elements in eq. 4.9 requires evaluating the operatorsin the ground-state wave function of the Hamiltonian. To simplify thisprocess, we will relate these matrix elements to the one-body density matrixelements (OBDME), as well as to the single-particle HO matrix elements forthe operators.The first step of this process is to apply the Wigner-Eckart theorem [39]of eq. 2.17 to the matrix elements:⟨jN , T,MT ||AL;τ ||jN , T,MT ⟩ =(−1)T−MT(T τ T−MT 0 MT)× ⟨jNT ||AˆL;τ ||jNT ⟩. (4.12)Note that the matrix elements were already reduced in the quantumnumber Mj , so they are doubly reduced, both in angular momentum andisospin. Special notation is sometimes used to denote elements reduced inthese two numbers (for example, in [143]), but we will take the absenceof both Mj and MT to be sufficient indication of this. We will introducesingle-particle HO states |α⟩. This is a simplification for a state with allthe single-particle quantum numbers, so |α⟩ = |nα(lα = 1/2)jα⟩, the samestates which were introduced when discussing the HO basis in Section 2.2.3.We may then use these single-particle states to expand operators in termsof the creation and annihilation operators:AˆL;τ =∑αβ⟨α||AˆL;τ ||β⟩ [a†α ⊗ a˜β]L;τ√(2L+ 1)(2τ + 1). (4.13)Since we want the reduced matrix elements of the operators, taking thematrix elements of eq. 4.13 we get:354.2. Coupling Dark Matter to Nuclei⟨jNT ||AˆL;τ ||jNT ⟩ =∑αβ⟨α||AˆL;τ ||β⟩⟨jNT ||[a†α ⊗ a˜β]L;τ ||jNT ⟩√(2L+ 1)(2τ + 1)≡∑αβ⟨α||AˆL;τ ||β⟩ΨL;ταβ , (4.14)where we have now defined the OBDMEs ΨL;ταβ mentioned earlier. Thiscan be multiplied by the appropriate factors from eq. 4.12, to yield anexpression for the matrix elements of DM scattering. This is still underthe assumption that interactions between DM and nucleons are only atthe one-body level. Similar expressions could also be derived for two-bodyinteractions, and those would result in consideration of the two-body densitymatrix elements (TBDME).Hence, the response functions can be computed as sums over the single-particle matrix elements of the operators, as well as the OBDMEs. Thesingle-particle matrix elements may be found analytically in the HO basis,and are not a prediction of the nuclear Hamiltonian, only the OBMEs are, asthey depend on the creation and annihilation operators. The single-particlematrix elements are derived from eq. 2.9 and are given by a series of inte-grals over Bessel/vector spherical harmonics. Four of these operators, M ,∆, Σ′ and Σ′′ are the same as those derived from semi-leptonic electroweakinteractions, where a multipole expansion of the same Bessel functions andspherical harmonics is necessary to couple electroweak leptons to the nu-cleus [144, 145]. The other two are also related to semi-leptonic electroweakinteractions for the same reason, although they come from higher order cor-rections considered in [146]. Expressions for all matrix elements may alsobe found in Appendix B of [143]. We note that all single-particle HO matrixelements take the form e−u/2P (u), where u = q2b2/2, b =√ℏ/mNω andP (u) is a polynomial in u. This note will be of importance when we presentresults in Chapter 6.From this, one possible process would be to construct a nuclear Hamil-tonian, either from phenomenology or ideally one derived from Chiral EFT.The OBDMEs would be obtained by considering the valence space part ofthe Hamiltonian, in accordance with the SNPA. The VS-IMSRG will allowus to improve over this procedure, by considering the entire Hamiltonianand operators, not only the valence space.364.2. Coupling Dark Matter to Nuclei4.2.4 Connecting to ObservablesNow that we have an expression for the transition probability, we are ableto compute observables. The cross-section can be obtained using:dσ(v, q2)dq2=18πv212jN + 1∑spins|M|2 . (4.15)This is the most basic observable that can be obtained from this theory,but of particular interest to direct detection experiments is the differentialrecoil rate [8], written here in terms of the recoil energy ER and the targetmass is mT = AmN , where A is the nucleon number and mN =mp+mn2 ≈940 MeV is the average nucleon mass:dRdER(E) =ρ0mχmT∫vf (v)dσdER(ER, v)d3v=ρ0mχmT∫vf (v)[2mTdσ(v, q2)dq2]d3v. (4.16)Two new parameters of dark matter are introduced in this equation:the dark matter local density ρ0 and the WIMP velocity distribution f(v).We use the description ρ0 = 0.3 GeV/cm3 [142]. We further assume thevelocity distribution to be a Maxwell-Boltzmann distribution, motivatedby its standard use in the literature [142], as well its justification in self-interacting dark matter simulations [147]:f(v) =1√2πσexp(−|v|22σ2). (4.17)Here we use σ =√3/2vc =√3/2 · 220 km/s, assuming a standard valuefor the circular velocity vc = 220 [142]. The description of dark matter wehave chosen in ρ0, f(v) and vc is not reflected in the response functions andis only used to plot observables.One important note is that experiments will often interpret their resultsusing the Helm form factor [148, 149], which is derived from a Born ap-proximation analysis of elastic cross-sections. This is done to simplify theircalculations, as they only need to consider one response function, W 00M . In-deed, the Helm form factor can be formulated as follows:W 00,HelmM = A2(3j1(qR1)qR1)2e−q2s2 . (4.18)374.2. Coupling Dark Matter to NucleiHere, j1 is a spherical Bessel function, R1 =√c2 + 73π2a2 − 5s2 and s,c and a are parameters of the nucleus [150]. This equation assumes thatonly the isoscalar part of the WM response function is contributing to theWIMP-nucleus interaction. This not only misses contributions from theother response operators, but also from W 11M , W10M and W01M , which can leadto corrections as large as ∼ 20% in some nuclei (This can be seen from theresponse functions presented in Appendix C of [143], specifically in 40Ar).38Chapter 5Valence Space In-MediumSimilarity RenormalizationGroupThus far, we have discussed how to derive nuclear interactions and Hamilto-nians from the ground-up, as well as how to couple a nucleus to dark mattercandidates. In this chapter, we will discuss how to actually solve (diagonal-ize) these Hamiltonians using the VS-IMSRG, as well as how to evolve theoperators of interest to our problem. This method may be used to computeproperties of quite heavy nuclei (up to A ≈ 200), in particular calculationsin 208Pb were recently achieved [27]. It uses techniques very similar to thoseused in the the Similarity Renormalization Group (SRG) [151, 152], whichwill be shortly reviewed here. The key difference between both techniquesis that the IMSRG is performed directly in the system of interest, what werefer to as ’in-medium’. Explicitly, this entails that the evolution is appliedto the whole nuclear Hamiltonian, rather than only an NN or 3N nuclearinteraction.5.1 Similarity Renormalization GroupThe starting point of the SRG is to consider a Hamiltonian H for NN or 3Nnuclear interactions, which may be written as follows [152]:H = Trel + V. (5.1)Here, Trel is the relative kinetic energy and V is the nuclear potential. Re-alistic potentials such as those derived from Chiral EFT include correlationeffects between the high- and low-momentum physics, making them non-perturbative.395.2. The IMSRG5.1.1 SRG Flow EquationsThe main goal of the SRG is to evolve the Hamiltonian of interest with acontinuous unitary transformation:H(s) = U(s)HU †(s) = Trel + Vs. (5.2)Here, s is the flow parameter which represents the progression of the SRGevolution, Vs is the evolved potential and Trel is independent of s. The U(s)matrix is a unitary operator that is chosen to decouple high-momentum fromlow-momentum physics in the Hamiltonian, which is equivalent to evolvingthe Hamiltonian to a block-diagonal form. This operator must be unitary,so that applying the transformation will not change any of the observables.The Hamiltonian varies with the flow parameter as follows:dH(s)ds= [η(s), H(s)] . (5.3)η(s) is known as the generator of the transformation, and is related tothe unitary operator as follows:η(s) =[dU(s)ds, U †(s)]. (5.4)There are multiple choices for the generator that will evolve the chosenpotential to a block-diagonal form. Two choices are given by:η(s) = [Trel, H(s)] and η(s) =[Hd(s), Hod(s)], (5.5)where in the second choice (which was the choice when the SRG wasfirst proposed [151]) we have split the Hamiltonian into its diagonal andoff-diagonal elements. From eq. 5.3 and the choice of generator, the nuclearpotential can then be evolved. The evolved potentials are then uncorrelatedbetween high- and low-momentum, an illustration of the evolution may befound in Figure 1 of [152]. This process can be applied to any nuclearinteraction, including ones derived from Chiral EFT and potentials including3N interactions. The problem of considering 3N interactions does make theproblem more complex, a discussion of this can be found in [152].5.2 The IMSRGAs mentioned, the SRG flow equations can be used directly on a nuclearHamiltonian for A-bodies, resulting in the IMSRG 3-body (and more) con-tributions will be induced in this process even if they are not explicitly405.2. The IMSRGincluded in the nuclear Hamiltonian, an approximation which will simplifycalculations by reducing the number of terms to be considered. Before in-troducing the relevant equations for the IMSRG, we must first make a fewnotes about the way the nuclear Hamiltonian is found, as well as introducesome mathematical tools which will be necessary to use the IMSRG. Morethorough reviews of the IMSRG are found in these works: [20–24, 153]5.2.1 Nuclear HamiltonianWe begin this process with constructing the nuclear Hamiltonian in theHO basis, using a nuclear interaction derived from Chiral EFT. Operatormatrix elements will also be computed in the same way, using the elementsintroduced in Section 4.2.3.Shell Model RegionsWhen constructing the Hamiltonian, an assumption is made that the systemmay be looked at in the ShM framework [154] we discussed in Section 2.3.1,hence as being split into three spaces. The core space, which in phenomeno-logical shell model calculations is seen as being completely inert, is writtendown explicitly along with its effects. The valence space, in which nucleonsare essentially analogous to valence electrons in atoms, contains the bulk ofnuclear effects. While in phenomenology, only the valence space effects areconsidered, here, with our many-body method we will be able to considereffects from all regions. The excluded region represents the energy statesin the ShM framework that are higher than the valence space, and whichnucleons may excite to in higher energy states [153]. When a valence spaceis chosen it fixes both the core and excluded space, as we can see in Figure5.1.Our goal with the IMSRG is to make the nuclear Hamiltonian blockdiagonal, such that only the valence space remains to be solved. This is whythe use of unitary transforms is important, as it preserves the observableswe are interested in, while simplifying the problem. Given the wealth oftools that exist to solve nuclear valence spaces [2, 155], retrieval of theOBDMEs from the evolved nuclear Hamiltonians is very straightforward(we will discuss this more in Section 5.3). Additionally, we apply sometruncations on the considered matrix elements, this will be further discussedin Section 5.4.2.415.2. The IMSRGFigure 5.1: Schematic view of the effect of the VS-IMSRG. The differentregions of the shell model are presented for 48Ca, with the protons in redand neutrons in blue. The orange arrows represent the effects that wouldphenomenologically be ignored, while the green is for pure valence spaceeffects. By using the IMSRG evolution (going from the left to the rightnucleus), core effects are mapped onto the valence space. Hence the corespace is inert, the excluded space is isolated and only the valence space needsto be solved.Natural Orbitals BasisThough the Hamiltonian is initially written down in the HO basis for sim-plicity, we want to transform it, along with the operators, to a more efficientbasis. What we mean by more efficient is that the basis will result in fasterconvergence of nuclear calculations with respect to truncation parameters.This in turn is achieved by having basis states that are approximations ofthe nucleus states. In Section 2.2.2 we introduced a method for finding suchstates: the HF method [56]. One shortcoming of the states generated of thismethod is the use of the mean-field approximation, which assumes thereare no correlations between nucleons. These correlations are known to haveeffects in nuclei (a review of this can be found in [156]), hence in this workwe consider an improvement over the HF basis that does include correla-tion effects: the natural orbitals (Nat) basis [157–159]. The constructionof this basis is based on the application of many-body perturbation theory(MBPT) to HF states, with the goal of obtaining a many-body state |Ψ⟩which includes correlation effects:425.2. The IMSRG|Ψ⟩ = |ΦHF⟩+ |Ψ(1)⟩+ |Ψ(2)⟩. (5.6)The |Ψ(n)⟩ states are corrections to |ΦHF⟩ to n-th order in MBPT. Thedetails of the different order corrections and how these include correlationsmay be found in [160]. One may use the state |Ψ⟩ and creation/annihilationoperators to build a density matrix, with matrix elements ρpq:ρpq = ⟨Ψ|a†paq|Ψ⟩. (5.7)Once diagonalized, this matrix will yield the basis states |nljmt⟩NAT.These states are given as a superposition of HF states, which themselves area superposition of HO states:|nljmt⟩NAT =∑n′cljtnn′ |nljmt⟩HF=∑n′c˜ljtnn′ |nljmt⟩HO,(5.8)where we have introduced two sets of expansion coefficients, cljtnn′ andc˜ljtnn′ . Thus, we have a new basis to which we can transform our Hamiltonianand operators. The main advantage of using this basis is that convergencewith respect to our truncation parameters will be more rapid. In particularin [159], it was found that the faster convergence achieved in the naturalorbitals basis (compared to the HF basis) resulted in having to consider 2-3times fewer basis states.5.2.2 Normal Ordering and Wick’s theoremSome notation we must introduce is the normal ordering of creation andannihilation operators, a†i and aj . This starts by defining a basis for theA-body Hilbert space:|Φi⟩ =A∏k=1a†ik |0⟩, (5.9)where |0⟩ is the vacuum. When considering a nucleus, we want to replacethe |0⟩ in eq. 5.9 with a reference state |Φ⟩ which approximates the nucleus’sground state (a discussion on choice of reference state is found in Section6 of [22]). Normal ordering will be defined with respect to this referencestate. We start by considering the one-body density matrix elements, which435.2. The IMSRGwe simultaneously use to define the contraction of creation/annihilation op-erators:ρji = ⟨Φ|a†iaj |Φ⟩ = a†iaj . (5.10)We can now introduce normal ordering of the operators with referenceto these contraction:a†iaj := : a†iaj : +a†iaj , (5.11)where we use the colon notation to denote normal ordering of the op-erators. By construction it may be seen that taking the expectation valueof normal-ordered in the reference state will give 0. Let us now consider anarbitrary product of production and annihilation operators a1a2a3...aN . Wemay rewrite this as a sum over contractions and normal-ordered, in what isknow as Wick’s theorem [39]:a1a2a3... = : a1a2a3... : + : a1a2a3... : + : a1a2a3... : + all 1-contractions+ : a1a2a3a4a5... : + all 2-contractions+ ...+ all n-contractions+ ...+ all terms with all pairs contracted.(5.12)This theorem is valid for any reference state, given that the contractionsand normal-ordering are appropriately defined with respect to this state.5.2.3 ISMRG flow equationsUsing Wick’s theorem and the normal ordering notation, the nuclear Hamil-tonian of interest can generally be rewritten as follows, assuming it includescontributions up to 3-body (contributions from 4- and more body terms arenot included in any of the Chiral interactions we use):H =E0 +∑ijfij : a†iaj : +12!2∑ijklΓijkl : a†ia†jalak :+13!2∑ijklmnWijklmn : a†ia†ja†kanamal : . (5.13)445.2. The IMSRGHere, E0, f , Γ andW are the normal-ordered contributions, respectivelyfrom 0, 1, 2 and 3 body terms. If we let V (n) denote the initial (non normal-ordered) n-body interaction and T the initial kinetic energy, their normal-ordered counterparts may be obtained as:E0 =∑iTiini +12∑ijV(2)ijijninj +16∑ijkV(3)ijkijkninjnk,fij = Tij +∑kV(2)ikjknk +12∑klV(3)ikljklnknl,Γijkl = V(2)ijkl +14∑mV(3)ijmklmnm,Wijklmn = V(3)ijklmn. (5.14)Here we have used the occupation number in the reference state ni, whichis 1 if the state is filled and 0 otherwise. This ensures that the sums run overoccupied states [22]. We will make use of the IMSRG(2) [22] approximationto simplify the flow equations, meaning we truncate the explicit 3-bodycontributions given by Wijklmn. Since the initial 3-body contributions arestill included in the other normal-ordered terms, as can be seen in eq. 5.14,the 3-body contributions will still be implicitly induced when performingthe IMSRG evolution.The scale of the error induced by this approximation is discussed in[161], and approximations beyond this (namely, the IMSRG(3)) are stilla work in progress [161]. For now, this approximation will be used forall calculations, as it greatly reduces the complexity of flow equations (asan example, the IMSRG(3) flow equations may be found in Appendix Aof [161]). In addition to the complexity of the flow equations, there areconcerns with the computational cost of implementing the IMSRG(3). Thecost of IMSRG calculations using the IMSRG(2) truncation scales with thenumber of basis states N as O(N6), whereas IMSRG(3) implementationsscale at least as O(N7) [161]. In the natural orbitals basis we need N ≈ 1000to achieve convergence [159], meaning corrections to the IMSRG(2) wouldentail a factor of ∼ 103 increase in computational costs.The normal-ordered contributions can be evolved, and as in the SRG,there are multiple ways this can be done depending on the generator that ischosen. These evolutions are written in terms of the generator, which alsomust be truncated to the 2-body level (as part of the IMSRG(2) approx-imation). This results in a set of coupled flow equations for E0, f and Γ455.2. The IMSRGwith respect to s, explicitly splitting the generator in one- and two-bodycontributions η = η(1) + η(2):dE0ds=∑ijη(1)ij fij(ni − nj) +12∑ijklη(2)ijklΓklijninjn˜kn˜ldf12ds=∑i(η(1)1i f12 + 1←→ 2) +∑ij(ni − nj)(η(1)ij Γj1i2 − fijη(2)j1i2)+12∑ijk(η(2)klijΓijk2(ninjn˜k + n˜in˜jnk) + 1←→ 2)dΓ1234ds=12∑ij(η(2)12ijΓij34(n˜i − nj) + 1, 2←→ 3, 4)−∑ij(ni − nj)((η(2)j2i4Γi1j3 − η(2)ilj3Γj2i4)− 1←→ 2)+∑i[((η(1)1i Γi234 − f1iη(2)i234)− 1←→ 2)− ((η(1)i3 Γ12i4 − fi3η(2)12i4)− 3←→ 4)](5.15)Once the contributions are evolved, the nuclear Hamiltonian can be re-constructed from eq. 5.13, where it will have been evolved to a block diagonalform, as schematically shown in Figure 5.2.After the Hamiltonian has been properly evolved, properties of the sys-tem, such as the energy levels, can be derived from the OBDMEs. This ismade possible by ensuring that the transformation is unitary. Indeed, suchtransformations leave the observables unchanged, even though the Hamil-tonian itself is evolved. Of course, energy levels are not the only relevantobservable. As discussed in Chapter 4, there is a concrete need to derivecross-sections of WIMP-nucleus interactions, for which we will also want toevolve the six nuclear response operators, listed in Appendix A.3. Similarlyto the Hamiltonian itself, evolving operators will effectively map all the in-formation onto the valence space. A general nuclear operators O may beevolved in exactly the same way as the Hamiltonian:O(s) = U †(s)OU(s) ⇐⇒ ddsO(s) = [η(s),O(s)] . (5.16)This is the same unitary transformation, hence the observables relatedto these operators will also be left unchanged. An important note is that theoperators must also be normal-ordered using Wick’s theorem, like for the465.2. The IMSRGFigure 5.2: Schematic representation of the evolution of the nuclear Hamil-tonian towards a block-diagonal form under the IMSRG evolution. The flowfrom left to right represents the IMSRG evolution of the Hamiltonian undereq. 5.15. The remaining relevant block is small enough to be directly diago-nalized, using tools that will be discussed in section 5.3. Here, P denotes aprojection operator onto the valence space and Q is its complement, mean-ing P +Q = 1 [153].Hamiltonian in eq. 5.13 and eq. 5.14. Similarly to the Hamiltonian, theywill be split in normal-ordered 0-, 1-, 2- and 3-body contributions. Separateflow equations will be derived for these contributions, after having truncatedthe explicit 3-body contributions as per the IMSRG(2) approximation.5.2.4 Magnus FormulationThe flow equations presented in eq. 5.14 constitute a set of coupled ordinarydifferential equations (ODEs) which must be solved numerically. Attempt-ing to do so results in two major issues: the ODEs can become numericallyunstable [162] and IMSRG calculations become limited by the memory needsof the ODE solver [22]. Additionally, the inclusion of each operator intro-duces a new set of coupled ODEs from eq. 5.16, which also have the sameissues.For these reasons, we would like to be able to recast this problem as onethat does not involve ODEs. This is exactly what we will achieve by intro-ducing the Magnus formulation of the IMSRG [162]. This formulation is theaffirmation that the U operator may be written as an operator exponential:U(s) = eΩ(s). (5.17)Ω(s) is anti-Hermitian and known as the Magnus operator [153]. Plug-ging this in eq. 5.2 and expanding yields the following infinite series:475.3. A Tool for Valence Space Calculations: KSHELLH(s) = H(0) + [Ω, H(0)] +12[Ω(s), [Ω(s), H(0)]] + ... (5.18)Hence, once an expression has been determined for Ω(s), this is now onlya problem of evaluating a series of commutators with the original Hamilto-nian. To find an expression for Ω(s) in terms of the generator η(s), we mustalso evaluate sets of commutators, according to the following formula:Ω(s+ ds) = Ω(s) + η(s)ds+12[η(s),Ω(s)] ds+112[Ω(s), [Ω(s), η(s)]] + ...(5.19)Both eq. 5.18 and eq. 5.19 are infinite series, and hence need to betruncated. In practice, this is done by introducing two tolerances: ϵ1 andϵ2. Letting the k-th term of eq. 5.18 be denoted by Ak and the k-th term ofeq. 5.19 be denoted by Bk, we give the following conditions for truncationof the k-term of the corresponding series:∣∣∣∣∣ Bk√Tr(ΩΩ†)∣∣∣∣∣ < ϵ1 and |Ak| < ϵ2 (5.20)In [162] the prescription ϵ1 < 10−4 and ϵ2 < 10−4 MeV is given, and itis noted that stricter tolerances only result in relative differences of 10−7in observables. In Figures 3 and 4 of the same paper it is found that themagnitudes of eq. 5.20 decrease monotonically with k, hence it will always bepossible to perform calculations until the conditions of eq. 5.20 are satisfied.Overall, in introducing this formulation of the IMSRG, we have gonefrom having to solve sets of coupled ODEs to only needing to evaluate com-mutators of matrices. This has solved both the issues of the numericallyunstable ODEs and the limitation of IMSRG calculations by memory re-quirements of ODE solvers. Additionally, operators can be evolved usingthe same formula as the Hamiltonian [162, 163], eliminating the need tosolve ODEs for the operators.5.3 A Tool for Valence Space Calculations:KSHELLOnce the nuclear Hamiltonian and the operators have been appropriatelyevolved, we can obtain the OBDMEs using a numerical process. For this,we make use of the KSHELL code [2], a tool recently developed for nuclear485.4. Review of the Ab Initio Procedureshell model calculations. This tool solves the eigenvalue problem for largevalence space Hamiltonians and is capable of providing the desired densitymatrix and its elements. The code relies on input information on the nu-cleon of interest, on the different truncations of the nuclear Hamiltonian weare considering, as well as an input valence space Hamiltonian. Originallydesigned for phenomenological valence space calculations, KSHELL worksfor IMSRG calculations because of the diagonalization of the rest of theHamiltonian, and the effective mapping of ab initio effects onto the valencespace.5.4 Review of the Ab Initio ProcedureIn this section, we will review how the ab initio procedure works. It will bemade more apparent how the different topics we have covered in Chapters3 through 5 are brought together to make precise calculations of nuclearproperties based on QCD.5.4.1 Steps of the Ab Initio CalculationsWe first review the different steps taken in making ab initio calculationswith a focus on WIMP-nucleus scattering.1. Hamiltonian We begin the process by constructing the nuclear Hamil-tonian. We use interactions based on Chiral EFT and assume a valencespace, an excluded space and a core, and include 3-body interactions.Matrix elements are initially written in the HO basis, then we trans-form to the natural orbitals basis introduced in Section 5.2.1.2. Operators We then focus our attention to the NREFT operators. Theirmatrix elements are also constructed in the HO basis, using their ex-plicit form given in [143], then they are brought to the natural orbitalsbasis.3. IMSRG evolution The Hamiltonian and operators are then takenthrough the IMSRG evolution. The Hamiltonian is thus brought to ablock-diagonal form, with all effects and information mapped onto thevalence space.4. Matrix elements and KSHELL Since we can now consider only thevalence space of the Hamiltonian, we use standard shell model tools tocompute the density matrix elements. Namely, we use the KSHELL495.4. Review of the Ab Initio Proceduretool to derive those. The single particle matrix elements are retrievedfrom the evolved operators, and combined with the OBDMEs to givethe full matrix elements.5. Iterate These first 4 steps are then repeated for all the different inter-actions, as well as for a set of transfer momenta q. From these calcu-lations, we are then able to compute observables, as will be presentedin Chapter 6.5.4.2 Approximations and TruncationsIn this section we will go through a (non-exhaustive) list of approximationsand truncations used in the entire process of ab initio calculations. Someof these approximations have already been mentioned in the text, so theywill not be discussed as thoroughly. Some will be purely related to ChiralEFT, while others are introduced during the IMSRG step. One note is thatnumerical uncertainties are a contributing factor for all steps of this process,but they are very small compared to other sources we will consider in thissection. As an example it is taken to be a relative uncertainty of 0.01% in[98]. For numerically stable calculations this should be a reasonable upperbound, so we will take it to be of that order.Nuclear Interaction ApproximationsAs previously mentioned, an infinite number of Feynman diagrams can bederived from Chiral EFT. Even though power counting allows for organiza-tion of the contributions, a choice must be made to truncate contributionsup to a certain point. The truncation chosen will induce some amount of un-certainty. This truncation uncertainty was probed in [98] for computationsof n-p cross-sections, where it was found that uncertainties varied between4.8 mb and 6.1 mb for calculated cross-sections varying between 36.5 mband 42.7 mb. This implies relative truncation uncertainties in the ∼ 11-17%range. More detail may be found in Figure 10 of [98] and the surroundingdiscussion.Another source of uncertainty may be found in the LECs. Since theseare a best fit to experiments, there is some error propagation from theexperimental uncertainties, which is then reflected in the interaction andobservables. Such uncertainties were quantified for a two-body NNLO Chiralinteraction in [100]. It was found that uncertainties from the LECs resultedin relative uncertainties of about 0.5% for calculated observables.505.4. Review of the Ab Initio ProcedureIMSRG ApproximationsThere are many truncations we impose on the matrix elements of the Hamil-tonian when starting the IMSRG process. The goal of these truncations isto reduce computation times by reducing the amount of matrix elements tobe considered. One such truncation we impose is on the NAT basis itself,explicitly we cut any matrix element with quantum numbers not respect-ing e = 2n + l ≤ eNATmax . Additionally, the subspace we consider during theIMSRG evolution is further truncated, with a separate emax. Finally, wealso impose an explicit truncation on all the 3N matrix elements, explicitlye1+e2+e3 ≤ E3max. Motivated by the convergence achieved and illustratedin [164] with respect to these parameters, we employ the same values ofeNATmax = 14, emax = 8 and E3max = 22 through this work.As discussed in this chapter, the 3N forces are not explicitly included inthe nuclear Hamiltonian once its different contributions have been normal-ordered, truncation which we called the IMSRG(2). They are, however,induced by the 0-, 1- and 2-body contributions. The impact of this approx-imation, as well as corrections such as the IMSRG(3), are still a topic ofresearch [161].Seeing the nucleus as a core and a valence space is an approximation.The further a nucleus is from being magic, the further nuclear structuredeviates from the shell model. Additionally, when normal ordering, thechoice of a reference state is an approximation, albeit one that is mitigatedby choosing a more accurate reference state. However, in a perfect IMSRGtransformation (one without any truncation such as the IMSRG(2)), there isno dependence on the valence space or the reference state, so those would notbe a source of error. The dependence we see is purely caused by the use of theIMSRG(2) truncation, hence the valence space/reference state uncertaintiesare much more of a probe of the IMSRG(2), not an independent source ofuncertainty.Operator ApproximationsAs part of the IMSRG(2) approximation [22], operators also need to benormal-ordered. This is to ensure that 3N contributions are induced in theIMSRG evolution after the explicit 3-body contributions are cut [22].In building up NREFT, we made the assumption that the DM-nucleusinteraction was only at the one-body level, which is known as the impulseapproximation . Explicitly, this meant that in writing down the Hamiltonianof the theory we assumed the total interaction was equal to the sum of the515.4. Review of the Ab Initio Procedureindividual interactions with nucleons (reflected in the sum over A in eq. 4.3.Corrections beyond this may be derived, by considering two-body nucleoncurrents [165].52Chapter 6ResultsIn this chapter we present the converged ab initio nuclear response func-tions for 22 isotopes: 16O, 19F, 23Na, 27Al, 28−30Si, 40Ar, 70,72−74,76Ge, 127I,133Cs, 128−132,134,136Xe. These isotopes were chosen based on ongoing orplanned activities in dark matter direct detection experiments (a review ofexisting experiments may be found in Section 7 of [8]), except 16O, whichwas chosen for benchmark purpose due to its doubly magic nature and easeof computation. Calculations of the nuclear Hamiltonians and operators aredone for each isotope 3 times, once for every considered nuclear interaction.Additionally, calculations are performed on a fixed set of 15 transfer mo-menta q values ranging from q = 0 MeV to q = 500 MeV, which is thephysically allowed range of transfer momenta for our nuclei of interest. Thisis to ensure that after a fit (which we will discuss in the next paragraph),we can evaluate and plot the operators at any q, and not only at one of the15 chosen values. In non-relativistic WIMP-nucleus scattering, q may beobtained by considering non-relativistic 1D collisions [166]. It is given byq = 2vµχN , where v is the WIMP-nucleus relative velocity and µχN is theirreduced mass. Since there is a range in v (as we discussed in Section 4.2.4),there is also one in q. In the case of large WIMP mass and for expectedWIMP-nucleus velocities, this gives an estimation on the upper limit of thephysically allowed transfer momenta: qmax = 2A MeV. Plots in this sectionwill be presented over the physically allowed range of q.All results that will be presented here are different fits to the operators ofthe form e−u/2P (u), where as before u = q2b2/2, b =√ℏ/(mNω) and P (u)is a polynomial in u. This fit is motivated by the fact that single-particlewave functions in the HO basis can analytically be found to follow thatform, as was mentioned in Section 4.2.3. This form would not a priori beexpected to still apply, as those matrix elements are transformed to the Natbasis, evolved through the IMSRG and summed over OBDMEs. However,all of these steps only result in summing over HO configurations. Indeed, itcan be seen in eq. 5.8 that Nat basis states are a sum over HO states, ineq. 5.18 that evolved matrix elements are given as a sum over initial matrixelements, and in eq. 2.9 that the final matrix elements are given as a sum536.1. Benchmarking and Result Verificationover the single-particle HO matrix elements. Thus, the use of the e−u/2P (u)form is justified. Expressions will not be given here explicitly for all elementsor all response functions, but results are made available through a JupyterNotebook in an forthcoming publication [167]. These results represent afirst ab initio calculation of the response functions for isotopes heavier thanA = 4 [168] in the NREFT framework.6.1 Benchmarking and Result VerificationOne important part of ab initio calculations is making sure that they areproperly converged with respect to the truncation parameters listed in Sec-tion 5.4.2. By ”proper” convergence, we mean that the difference in opera-tors achieved by increasing the truncation parameters is smaller than othersources of errors. As an example we present a convergence plot with respectto emax in Figure 6.1.4 5 6 7 8 9 10emax0.10.20.30.40.50.6W′′ (0)eNATmax=141.8/2.0(EM)NNLOsatNNLOgoFigure 6.1: Convergence of the Φ′′ response function at zero momentumtransfer with respect to increasing emax for40Ar. As can be seen, thisoperator is quite well converged by emax = 8, justifying the use of this valueas a cutoff. This operator was chosen due to it not being normalized like theM operator is, hence being an actual prediction of the underlying nuclearphysics.In all calculations presented in this work, we use emax = 8. Convergenceis achieved by then, hence the use of this parameter value is justified. Otherparameters for which checking convergence would be relevant are ℏω, eNATmaxand E3max. Convergence plots with respect to these parameters may be546.1. Benchmarking and Result Verificationfound in Figures 1 and 8 of [164]. We use the same values that showedsufficient convergence in that work, as previously mentioned they are ℏω =16 MeV, eNATmax = 14 and E3max = 22.As mentioned, NREFT is not the only EFT path to deriving a WIMP-nucleus interaction. Chiral EFT may be used in much the same way, andafter a multipole expansion is carried out, a different set of nuclear operatorsis obtained. There are, however, two operators which are shared between thetwo theories, the M and Φ′′ operators. As a starting point to benchmarkingour results, we will compare these response functions between our resultsand results obtained from ShM calculations in the Chiral EFT framework[140]. This comparison is shown in Figure. 6.2 for 19F, for which Chiral EFTresults are readily available in the Jupyter notebook of [140]. In figures suchas that one, only the W 00k and W11k contributions will be shown explicitly.TheW 01k andW10k contributions will not be shown for simplicity and so as tonot clutter the figures, but they do contribute to calculations of observables.As would be expected, the WM response functions agree exactly overthis small range of momentum transfer, due to the normalization a q = 0.There are however discrepancies in the WΦ′′ response function of up to40%, indicating that some underlying nuclear physics are not considered byshell model calculations. Since the M response functions agree, however,we can rest assured that our benchmark is successful and we are able toappropriately reproduce shell model calculations.Another way to verify our calculations is to consider the effect of com-puting response functions without performing the IMSRG step, in differentbases. In Figure 6.3, we present theW 00Φ′′ response function, computed with-out the IMSRG step in the HO and NAT bases, and compared against theresults with using the IMSRG. The NAT and oscillator calculations pre-sented here are hence the same as the shell model calculations, the keydifference being the use of different nuclear interactions.There does not appear to be a significant difference in results in goingfrom the HO to the NAT basis if the IMSRG is not applied. Additionally,the shell model calculations lay within the uncertainty band of out unevolvedcalculations. All of these non-IMSRG calculations lay outside the band ofIMSRG calculations (even when doubling all the bands), explicitly showingoperator evolution in the IMSRG. This difference in non-IMSRG resultsthus indicates that the inclusion of core effects has a sizeable effect on thefinal response functions, not only in their values, but also on constrainingthem. It should be noted that the bands for unevolved results cannot reallybe trusted to give a good indication of uncertainty. They are effectivelyphenomenological calculations (the only differences from the ShM results556.2. Response functions and observablesq (MeV)10 1100101102WM(q2 )19FW00 IMSRGW11 IMSRGW00 SMW11 SM0 10 20 30 40 50q (MeV)10 2W′′ (q2 )Figure 6.2: M and Φ′′ response function obtained from the IMSRG in theNREFT framework, compared to the same response functions obtained fromShM calculations and Chiral EFT. ShM calculations are purely phenomeno-logical, hence there are no error bars on these results.being the interaction used and the frequency ℏω), so the uncertainty on thenuclear interaction really be taken to be the largest source of uncertainty (asevidenced by the disagreement of these results with IMSRG calculations).6.2 Response functions and observablesThe two response functions we have considered up to this point are not allthe possible ones, so to have an overview of all the discrepancies between theab initio approach and shell model calculation, we must consider the othersix response functions that contribute. We compare those to shell modelcalculations, this time in the NREFT framework. The full comparison (notexplicitly showing W 01k and W10k ) is shown in figure 6.4. The Helm form566.2. Response functions and observables0 20 40 60q (Mev)0.40.60.81.01.21.4M00 ′′29Si IMSRGNATOscillatorShell ModelFigure 6.3: Φ′′ response function with and without the IMSRG evolution.Both HO and Nat basis response functions are shown without the IMSRGevolution, and all is compared to shell model calculations. As previously,the bands are obtained by considering different interactionsfactor is also included in this figure to verify that its transfer momentumdependence agrees with our calculations. Since it only contributes to W 00Mand is hence normalized, it is not very visible on the plot.One important note is that we use the description that cpj = (c0j + c1j ),cnj = (c0j − c1j ), where cpj and cnj are the coupling constants at the pro-ton/neutron level. These definitions differ by a factor of 1/2 from the de-scription used in the ShM calculations of [26, 143, 169]. Hence, the nuclearresponse functions we will present will be a factor of 4 greater than thosewe compare to, so that the factor is not reflected in the observables. Wescale our response functions correspondingly when comparing to ShM cal-culations.We can once again see discrepancies in all the response functions that arenot normalized, further pointing to some underlying nuclear physics beingignored by the shell model. In particular the Σ′, Σ′′ and ∆Σ′′ ShM responsefunctions in 27Al lie well outside of the ± 2σ region of our calculations. Thisfurther points to some underlying nuclear physics effects being unaccountedfor in the ShM approach. We also present the response functions of 40Ar infigure 6.5, although only three of the response functions contribute in thiscase.576.2. Response functions and observables0255075100125WM(q2 ) 27AlW00 IMSRGW11 IMSRGHelm FFW00 SMW11 SM012345W′′ (q2 ) 27AlW00 IMSRGW11 IMSRGHelm FFW00 SMW11 SM0.0000.0040.0080.0120.016W′ (q2 )0.000.040.080.12W(q2 )0.0000.0150.0300.0450.060W′ (q2 )0.0000.0080.0160.024W′′ (q2 )0 20 40 60q (MeV)161284W′′ M(q2 )0 20 40 60q (MeV)0.080.060.040.02W′′ (q2 )Figure 6.4: All 8 nuclear response functions for 27Al, including the twointerference functions. This nuclei was chosen due to it having jN = 5/2and therefore having all response functions contributing. For most of thenon-normalized response functions, there are discrepancies between ShMand ab initio calculations.586.2. Response functions and observables10 1100101102WM40ArW00 IMSRGW11 IMSRGHelm form factorW00 SMW11 SM10 310 210 1W′′0 20 40 60 80 100q (MeV)3210WM′′Figure 6.5: All non-zero response functions for 40Ar. As mentioned in Chap-ter 4, the only contributing multipole operators for L = 0 are the spin-dependent responses, that is to say M and ΦFinally, we are ready to compute and present observables. We startwith the transition probability, which is given by summing over all the darkmatter and nuclear response functions. This is related to the differentialcross-section through a simple factor, as can be seen in eq. 4.15. Thetransition for eight of the computed isotopes are shown in Figure 6.6.Although we had seen differences in the response functions, IMSRG andShM calculation agree very well for observables. This is mostly a reflection ofthe fact that theW ττ′M response function is normalized. Since the Helm formfactor is missing the isovector contributions, it systematically overestimatesthe transition amplitude in all considered nuclei.596.3. Discussion0 25024||2×101519F a)0 2523Na b)0 4029Si c)0 5040Ar d)0 800.00.51.01.5||2×101373Ge e)0 80127I f)0 80129Xe g)0 80131Xe h)IMSRG Shell Model Helm Form Factorq(MeV)Figure 6.6: Transition amplitude for eight of the computed nuclei, from theIMSRG, ShM calculations and Helm form factor. The calculations fromthe Helm form factor are obtained by only considering the W 00M responsefunction.6.3 DiscussionWe have presented in this section response functions and scattering ampli-tudes (which are equivalent to differential cross-sections through eq. 4.15)for dark matter scattering based on ab initio calculations for eight different,experimentally relevant nuclei. Additionally, computations for a total 22nuclei have been carried out and are made available in a forthcoming pub-lication [167]. We have compared results to calculations made using botha simple Helm form factor and more refined phenomenological calculationsbased on the shell model. For the nuclei presented in Sections 6.1 and 6.2,we have found that individual response functions can show differences of upto 40% between methods (and in particular, all the response functions otherthan W 00M are taken to be 0 in the Helm form factor approach). However,transition probabilities from ShM response functions are all within the ± 1σregion of our calculations (except 40Ar, which is withing the ± 2σ region).This agreement in the transition probabilities is mostly a reflection ofthe normalization of the W ττ′M response functions. The overall agreementis not only due to how large the M response functions are with respectto the other operators’, but also a reflection of the fact that some of the606.3. Discussionresponse functions are suppressed by a factor of q2/m2N , as can be seen fromeq. 4.8. For many of the nuclei we have considered here, the maximumtransfer momentum is fairly low compared to mN , hence this factor is quitesignificant in reducing the contributions from the Φ′′, Φ′′M , Φ˜′, ∆ and ∆Σ′responses.One important note we have not touched on much pertains to the darkmatter coupling constants. In Chapter 4, we mentioned that those were setso that the interaction is a fraction of the weak interaction’s strength. Thisfraction was chosen to be the same for all responses. There is of course noguarantee that this is the case, and different DM models will have differentsets of coupling constants. In some models, the charge contribution M maybe smaller and the other responses may be more significant, resulting inmore significant discrepancies in the observables.61Chapter 7ConclusionIn summary, we have introduced in this work ab initio response functionsfor WIMP-nucleus scattering. We have made these calculations using theVS-IMSRG, for 22 different isotopes which are relevant in direct detectionexperiments: 16O, 19F, 23Na, 27Al, 28−30Si, 40Ar, 70,72−74,76Ge, 127I, 133Csand 128−132,134,136Xe. This constitutes a first ab initio calculation for theseisotopes in the NREFT framework, which is complementary to the ChiralEFT coupling of external WIMP currents. We have achieved our calcu-lations using nuclear interactions derived from Chiral EFT, yielding theo-retical error bands from the spread between different interactions. We willprovide these response functions in an upcoming publications, to allow formore careful analysis of results from DM direct detection experiments.While we observe differences in some of the nuclear response functionsof up to 40%, the observables are mostly unchanged, especially at low mo-mentum transfer. Despite this, these calculations represent an improvementover previous phenomenological calculations in two ways. The approach wehave taken is more fundamental, both by considering nuclear interactionsderived from low-energy QCD and by solving entire nuclear Hamiltonians,rather than only their valence space. Additionally, we provide theoreticaluncertainties, which may not be derived from phenomenology.There remain improvements over this, in particular the IMSRG(3), anext order approximation from IMSRG(2), remains to be implemented. Cal-culations with this are however not expected to make a large difference, asthe bulk of uncertainties and our main limitation in this work is coming fromChiral interactions and their truncations. Chiral uncertainties also remainto be quantified directly from the underlying EFT, which would yield morewell-motivated uncertainties than the interaction spread we are considering.Finally, while this work reports on the nuclear response operators for spin-dependent and spin-independent elastic scattering, calculations may also becarried out for inelastic scattering. 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Ab initionuclear response functions for dark matter searches. Physical ReviewD, 95(10):103011, 2017.[169] Riccardo Catena. Dark matter directional detection in non-relativisticeffective theories. Journal of Cosmology and Astroparticle Physics,2015(07):026, 2015.77Appendix ANREFT Operators andResponse FunctionsIn this section we will list the response functions and operators that wereintroduced and referred to when building NREFT.A.1 WIMP-Nucleon Interaction OperatorsWe start with the interaction operators, which correspond to the most gen-eral set of operators which could be built from the four Galilean invariant,Hermitian operators. Notice that the second operator is not given, as itturns out it does not contribute to the observables.Oˆ1 = 1χN (A.1)Oˆ3 = iSˆN ·(qˆmN× vˆ⊥)(A.2)Oˆ4 = Sˆχ · SˆN (A.3)Oˆ5 = iSˆχ ·(qˆmN× vˆ⊥)(A.4)Oˆ6 =(Sˆχ · qˆmN)(SˆN · qˆmN)(A.5)Oˆ7 = SˆN · vˆ⊥ (A.6)Oˆ8 = Sˆχ · vˆ⊥ (A.7)Oˆ9 = iSˆχ ·(SˆN × qˆmN)(A.8)Oˆ10 = iSˆN · qˆmN(A.9)Oˆ11 = iSˆχ · qˆmN(A.10)78A.2. Dark Matter Response FunctionsOˆ12 = Sˆχ ·(SˆN × vˆ⊥)(A.11)Oˆ13 = i(Sˆχ · vˆ⊥)(SˆN · qˆmN)(A.12)Oˆ14 = i(Sˆχ · qˆmN)(SˆN · vˆ⊥)(A.13)Oˆ15 = −(Sˆχ · qˆmN)[(SˆN × vˆ⊥)· qˆmN](A.14)A.2 Dark Matter Response FunctionsWe carry on with the dark matter response functions Rττ′AB. These are givenentirely in terms of the corresponding values of the Hermitian operatorsand the WIMP-nucleon coupling constants. They are not dependent onproperties of the chosen nucleus, only on the chosen dark matter model.Rττ′M(v⊥2T ,q2m2N)= cτ1cτ ′1 +14(q2m2Nv⊥2T cτ5cτ ′5 v⊥2T cτ8cτ ′8 +q2m2Ncτ11cτ ′11)(A.15)Rττ′Φ′′(v⊥2T ,q2m2N)=q24m2Ncτ3cτ ′3 +116(cτ12 −q2m2Ncτ15)×(cτ′12 −q2m2Ncτ′15)(A.16)Rττ′Φ′′M(v⊥2T ,q2m2N)=cτ3cτ ′1 +14(cτ12 −q2m2Ncτ15)cτ′11 (A.17)Rττ′Φ˜′(v⊥2T ,q2m2N)=116(cτ12cτ ′12 −q2m2Ncτ13cτ ′13)(A.18)Rττ′Σ′′(v⊥2T ,q24m2N)=q24m2Ncτ10cτ ′10 +14[cτ4cτ ′4 +q2m2N(cτ4cτ ′6 + cτ6cτ ′4 )+q4m4Ncτ6cτ ′6 + v⊥2T cτ12cτ ′12 +q2m2Nv⊥2T cτ13cτ ′13](A.19)79A.3. Nuclear Response OperatorsRττ′Σ′(v⊥2T ,q2m2N)=18(q2m2Nv⊥2T cτ3cτ ′3 + v⊥2T cτ7cτ ′7)+116[cτ4cτ ′4+q2m2Ncτ9cτ ′9 +v⊥2T2(cτ12 −q2m2Ncτ15)×(cτ′12 −q2m2Ncτ′15)+q22m2Nv⊥2T cτ14cτ ′14](A.20)Rττ′∆(v⊥2T ,q2m2N)=14(q2m2Ncτ5cτ ′5 + cτ8cτ ′8)(A.21)Rττ′∆Σ′(v⊥2T ,q2m2N)=14(q2m2Ncτ5cτ ′4 − cτ8cτ′9)(A.22)A.3 Nuclear Response OperatorsFinally, we present the nuclear response operators. These are dependent onlyon the chosen nucleus, and are what we run our calculations on. Once theare appropriately evolved, we evaluate their matrix elements appropriatelyand recover the nuclear response function W ττ′AB according to eq.4.9MLM ;τ (q) =A∑i=1MLM (qρi) ti(τ) (A.23)Σ′LM ;τ (q) = −iA∑i=1[1q∇⃗ρi ×MMLL(qρi)]· σ(i)ti(τ) (A.24)Σ′′LM ;τ (q) =A∑i=1[1q∇⃗ρiMLM (qρi)]· σ(i)ti(τ) (A.25)∆LM ;τ (q) =A∑i=1MMLL(qρi) ·1q∇⃗ρiti(τ) (A.26)Φ˜′LM ;τ (q) =A∑i=1[(1q∇⃗ρi)·(σ(i) ×1q∇⃗ρi)12MMLL(qρi) · σ(i)]ti(τ) (A.27)Φ′′LM ;τ (q) = iA∑i=1(1q∇⃗ρiMLM (qρi))·(σ(i) ×1q∇⃗ρi)ti(τ) (A.28)80
Thesis/Dissertation
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Physics
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
Graduate
Dilling, Jens
Ab initio predictions for dark matter direct detection
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