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Ab initio theory for two-neutrino and neutrinoless double-beta decay Payne, Charlie G. 2018

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Ab Initio Theory For Two-Neutrinoand Neutrinoless Double-Beta DecaybyCharlie G. PayneB.Sc., The University of Waterloo, 2015A 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)January 2018c© Charlie G. Payne, 2018AbstractAlthough the complexity behind nuclear physics problems is often deemed too intensive evenfor modern supercomputers, non-perturbative ab initio (meaning “from first principles”) many-body technologies have made significant strides towards efficiently modelling low-energy QCDstructure. In particular, the similarity renormalization group, along with an “in-medium”reference state normal ordering (IM-SRG), has proven useful for making nuclear problemscomputationally manageable. This IM-SRG method translates chiral Hamiltonians, nuclearproperties, and decays into a numerically tractable framework, whilst capturing bulk effects offorces via normal ordering.A decay that has generated much interest in the nuclear community is double-beta decay.The two-neutrino mode has evaded a proper theoretical treatment, due to a disconcertingpuzzle known as “quenching.” The neutrinoless mode (though still hypothetical) could unveilfundamental properties of the neutrino, such as its absolute mass and potential Majorananature. In this dissertation, we will use IM-SRG, in a valence space shell model construction,to compute both the two-neutrino and neutrinoless double-beta decay of the doubly magicnucleus, Calcium-48.We conclude that the use of a fully ab initio method that models many-body effects, viaIM-SRG, have decreased the two-neutrino double-beta decay nuclear matrix element by a factorof 3 (without any quenching factor), and the neutrinoless counterpart by roughly 20%, com-pared to the standard phenomenology for Calcium-48. This result has concerning experimentalimplications, since the half-life of a decay is proportional to the square of the inverse of thenuclear matrix element.iiLay SummaryAt the centre of the atom lies a collection of protons and neutrons, bound together in what’sknown as the atomic “nucleus.” Although the nucleus represents one of the most fundamentalbuilding blocks of matter, and therefore of reality itself, its structure is still mysterious in thecontext of theoretical physics. In particular, using many-body quantum mechanics to predicthow the nucleus might decay and emit elementary particles is of primary interest to theoristsand experimentalists alike. One of these decays, which we focus on in this dissertation, iscalled “double-beta decay;” and it has generated much excitement in the community due toits potential to unveil the secrets of the most elusive elementary particle: the neutrino. Bystudying the nuclear structure of Calcium-48, and its ability to double-beta decay, we attemptto set bounds on the mass of the neutrino.iiiPrefaceThis dissertation is original, unpublished work by the author, C. G. Payne. The softwareimsrg++ and nutbar, developed by Dr. Ragnar Stroberg at TRIUMF [1, 2], and the shellmodel code NuShellX, developed by Alex Brown at Michigan State University [3] and BillRae from Garisgonton, Oxfordshire [4], were used to obtain the results of this dissertation, asdescribed in Chapter 6. Dr. Stroberg was the post-doctoral collaborator of the author’s researchsupervisor at TRIUMF, Dr. Jason D. Holt. Together, the author, Dr. Holt, and Dr. Strobergplan to publish refined results on Calcium-48 in collaboration with Dr. Gaute Hagen,∗ et al,whom use the coupled cluster many-body method. Additionally, work on Germanium-76 isexpected to be published in collaboration with Dr. Javier Mene´ndez.† Benchmarking of theneutrinoless double-beta decay matrix elements (see Chapter 4) was aided by Dr. JonathanEngel.‡ Appendices A, B, and C are based on the textbook “From Nucleons to Nucleus:Concepts of Microscopic Nuclear Theory” by J. Suhonen [5] for introductory purposes.∗Oak Ridge National Laboratory†University of Tokyo‡University of Northern Carolina, Chapel HillivTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList of Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.1 Many-Body Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . 22.1.1 Central Quantum Harmonic Oscillator . . . . . . . . . . . . . . . . . . . 22.1.2 The Talmi-Moshinsky Transformation . . . . . . . . . . . . . . . . . . . 42.1.3 Slater Determinants and the Pauli Exclusion Principle . . . . . . . . . . 72.1.4 The Hartree-Fock Method . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2 Nuclear Structure Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2.1 The Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2.2 Nuclear Shell Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2.3 Nuclear Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2.4 Nuclear Many-Body Methods . . . . . . . . . . . . . . . . . . . . . . . . 142.3 Beta Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.3.1 Parity Violation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.3.2 Allowed Fermi Transition . . . . . . . . . . . . . . . . . . . . . . . . . . 162.3.3 Allowed Gamow-Teller Transition . . . . . . . . . . . . . . . . . . . . . . 16vTable of Contents2.3.4 Quenching in Single-Beta Decay . . . . . . . . . . . . . . . . . . . . . . 183 Double-Beta Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.1 Two-Neutrino Double-Beta Decay . . . . . . . . . . . . . . . . . . . . . . . . . . 193.1.1 Experimental Confirmation of 2νββ . . . . . . . . . . . . . . . . . . . . 203.1.2 2νββ Half-Life Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.1.3 M2ν Matrix Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.1.4 Quenching in 2νββ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.2 From Two Neutrinos to None . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.3 Neutrinoless Double-Beta Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.3.1 Current Experimental Status for 0νββ . . . . . . . . . . . . . . . . . . . 313.3.2 0νββ Half-Life Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.3.3 Nuclear Matrix Elements . . . . . . . . . . . . . . . . . . . . . . . . . . 343.3.4 The Closure Approximation . . . . . . . . . . . . . . . . . . . . . . . . . 354 M0ν Two-Body Matrix Elements . . . . . . . . . . . . . . . . . . . . . . . . . . 364.1 Deconstructing M0ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.2 Tensor TBMEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.3 Gamow-Teller TBMEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.4 Fermi TBMEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.5 Relative Bessel’s Matrix Elements . . . . . . . . . . . . . . . . . . . . . . . . . . 454.5.1 RBMEs with Short-Range Correlations . . . . . . . . . . . . . . . . . . . 484.6 Reconstructing M0ν TBMEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495 Valence Space In-Medium Similarity Renormalization Group . . . . . . . . 535.1 Nuclear Core and Valence Space . . . . . . . . . . . . . . . . . . . . . . . . . . 535.1.1 Model Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545.2 Similarity Renormalization Group . . . . . . . . . . . . . . . . . . . . . . . . . . 555.2.1 Nuclear Potentials and Similarity Transformations . . . . . . . . . . . . 565.2.2 SRG Flow Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575.2.3 Consistent Operator Evolution . . . . . . . . . . . . . . . . . . . . . . . 595.3 Reference State Normal Ordering . . . . . . . . . . . . . . . . . . . . . . . . . . 605.3.1 Hartree-Fock Step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615.4 The Magnus Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 616 Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646.1 imsrg++ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646.2 NuShellX@MSU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 656.3 nutbar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 656.4 Two-Tiered Adaptive M0ν TBMEs . . . . . . . . . . . . . . . . . . . . . . . . . 656.4.1 Integration Techniques from GSL . . . . . . . . . . . . . . . . . . . . . . 67viTable of Contents7 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 687.1 2νββ NME for 48Ca . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 687.1.1 Phenomenological Benchmarking . . . . . . . . . . . . . . . . . . . . . . 697.1.2 VS-IM-SRG Using the EM 1.8/2.0 Interaction . . . . . . . . . . . . . . . 717.2 0νββ NME for 48Ca . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 727.2.1 Phenomenological Benchmarking . . . . . . . . . . . . . . . . . . . . . . 737.2.2 VS-IM-SRG Using the EM 1.8/2.0 Interaction . . . . . . . . . . . . . . . 757.2.3 VS-IM-SRG Using the 500/400 N3LO+3N Interaction . . . . . . . . . . . 777.2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 778 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82Appendix A Angular Momentum Coupling . . . . . . . . . . . . . . . . . . . . . 95A.1 Clebsch-Gordan Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95A.2 The Wigner 3j-Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99A.3 Coupling Three Angular Momenta . . . . . . . . . . . . . . . . . . . . . . . . . 100A.4 Coupling Four Angular Momenta . . . . . . . . . . . . . . . . . . . . . . . . . . 101Appendix B Spherical Tensor Operators . . . . . . . . . . . . . . . . . . . . . . . 103B.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103B.2 Reduced Matrix Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106B.3 The Wigner-Eckart Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106B.4 Decomposition Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108Appendix C Fock Space and Operators . . . . . . . . . . . . . . . . . . . . . . . . 114C.1 Normal Ordering and Contractions . . . . . . . . . . . . . . . . . . . . . . . . . 115C.1.1 Wick’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116C.2 One-Body Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117C.3 Two-Body Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119C.3.1 The Closure Approximation . . . . . . . . . . . . . . . . . . . . . . . . . 121Appendix D Summation Limits for the Talmi-Moshinsky Transformation . . 122Appendix E Derivation of Equation (D11) from PRC.88.064312(2013) . . . . 126Appendix F Miscellaneous Formulae . . . . . . . . . . . . . . . . . . . . . . . . . 129viiList of TablesTable 4.1: Common SRC parameters for the Jastrow-type function . . . . . . . . . 48Table 6.1: Two-tiered integration scheme for M0ν TBMEs . . . . . . . . . . . . . 66Table 7.1: Values of M0ν parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 73Table 7.2: Benchmarking for the neutrinoless double-beta decay NME of 48Causing a pf -shell valence space and the GXPF1A interaction . . . . . . . 74Table 7.3: Comparison of the neutrinoless double-beta decay NMEs, for bench-marking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75Table 7.4: Summary of the VS-IM-SRG evolved neutrinoless double-beta decayNMEs of 48Ca using a pf -shell valence space . . . . . . . . . . . . . . . 77viiiList of FiguresFigure 2.1: Oxygen-16, as represented by the nuclear shell model . . . . . . . . . . 12Figure 2.2: Hierarchy of nuclear forces in χEFT . . . . . . . . . . . . . . . . . . . . 14Figure 3.1: The nuclear Feynman diagram for double-beta decay . . . . . . . . . . 20Figure 3.2: Even and odd mass parabolas of A = 48 isobars . . . . . . . . . . . . . 21Figure 3.3: The nuclear Feynman diagram for neutrinoless double-beta decay . . . 26Figure 3.4: The nuclear Feynman diagram for 0νββ via a pion exchange . . . . . . 28Figure 3.5: Reconstructing the blackbox theorem of Schechter and Valle . . . . . . 30Figure 5.1: Core, valence space, and excluded region for the ground state of 48Cawithin the nuclear shell model . . . . . . . . . . . . . . . . . . . . . . . 54Figure 7.1: Benchmarking for the two-neutrino double-beta decay NME of 48Causing a pf -shell valence space and the GXPF1A interaction . . . . . . . 70Figure 7.2: VS-IM-SRG evolved two-neutrino double-beta decay NME of 48Causing a pf -shell valence space and the EM 1.8/2.0 interaction . . . . . . 71Figure 7.3: VS-IM-SRG evolved neutrinoless double-beta decay NME of 48Causing a pf -shell valence space and the EM 1.8/2.0 interaction . . . . . . 78Figure 7.4: VS-IM-SRG evolved neutrinoless double-beta decay NME of 48Causing a pf -shell valence space and the EM 1.8/2.0 interaction, includingAV18 short-range correlations . . . . . . . . . . . . . . . . . . . . . . . 79Figure 7.5: VS-IM-SRG evolved neutrinoless double-beta decay NME of 48Causing a pf -shell valence space and the 500/400 N3LO+3N interaction . 80ixList of AcronymsχEFT Chiral Effective Field Theory0νββ neutrinoless double-beta (decay)2νββ two-neutrino double-beta (decay)2N, 3N, ... Two-Nucleon, Three-Nucleon, ...amu (unified) atomic mass unitBCH Baker-Cambell-HausdorffBSM Beyond (the) Standard ModelCG Clebsch-GordanCKM Cabibbo-Kobayashi-MaskawaCoM Centre of MassCTML Chosen Talmi-Moshinsky LimitsF Fermig.s. ground stateGT Gamow-Tellerh.c. hermitian conjugateIM-SRG In-Medium Similarity Renormalization GroupLECs Low-Energy Coupling ConstantsLHS Left Hand SideLNV Lepton Number Violation (or Violating)MEC Meson Exchange CurrentNLO, N2LO, ... Next-to-Leading Order, Next-to-Next-to-Leading Order, ...NME Nuclear Matrix ElementOBMEs One-Body Matrix ElementsOBTDs One-Body Transition DensitiesODE Ordinary Differential Equationo.w. otherwisexList of AcronymsPDE Partial Differential EquationPMNS Pontecorvo-Maki-Nakagawa-SakataQCD Quantum ChromodynamicsQFT Quantum Field TheoryQHO Quantum Harmonic OscillatorRBMEs Relative Bessel’s Matrix ElementsRHS Right Hand SideShM (Nuclear) Shell ModelSRCs Short-Range CorrelationsSRG Similarity Renormalization GroupT TensorTBMEs Two-Body Matrix ElementsTBTDs Two-Body Transition DensitiesTISE Time-Independent Schro¨diner EquationVS Valence SpaceWLOG Without Loss of GeneralityxiAcknowledgementsOf foremost importance, I would like to thank my research supervisor, Dr. Jason D. Holt, fortaking me on as a M.Sc. student and guiding me through this challenging project. He gaveme the chance to tackle nuclear physics, a field which I had no prior experience in. Abundantthanks go to Dr. Ragnar Stroberg, our post-doctoral collaborator, for motivating this researchand providing me with steadfast assistance. During the course of my work, we had several co-opstudents in our small team, who provided input: David Livermore, Oliver Drozdowski, SamuelLeutheusser, and Chan Gwak.Many thanks are in order for TRIUMF, its facilities, and all its employees - in particularthose at the theory department. TRIUMF has been a truly wonderful experience. As a the-ory student, being so close to experiment and phenomenology helps to keep you grounded inreality, instead of becoming lost in abstraction. Lively discussions with all the students at theoffice, past and present, kept me sane. My sincere gratitude goes to Prof. Reiner Kruecken, thecurrent Deputy Director of TRIUMF, who acted as my academic supervisor.I would also like to thank Prof. Jon Engel, Dr. Javier Mene´ndez, Dr. Gaute Hagen, Dr. MirkoMiorelli, and Javier Hernandez, for their fruitful insight into components of this research.Special thanks go to my parents, Joe Payne and Pam Hudson, and my sister, MelissaHudson. My family have always cheered on my progress and supported me through the worstof times, and without them this thesis would not have been possible.xiiIn loving memory of Umpa, Rob, and GraceyxiiiChapter 1IntroductionDespite being extraordinarily unintuitive, quantum mechanics, in its many-body formulation,describes physics from solid state devices, all the way to the hadronization of jets in high-energysupercolliders. At lower energies, like those on the scale of the atomic nucleus, quantum many-body theory is currently undergoing a foundational refurbishment [6]. Theorists are migratingfrom phenomenological models to new technologies built from first principles (ab initio). Thisrequires taking realistic nuclear interactions, like those derived from chiral effective field the-ory [7], and combining them with novel many-body methods to make predictions of nuclearobservables. Such a daunting task has, historically, proven to be notoriously computationallyintensive [8]. Thus, nuclear structure theorists are focused on making the nuclear problem morenumerically tractable, whilst also capturing all the relevant higher-order phenomena.One missing piece that has been neglected by nuclear theorists is the inclusion of three-nucleon forces - until recently [9–11]. By using a redefined notion of normal ordering, thein-medium similarity renormalization group can capture three-nucleon interactions and con-sistently incorporate them into operators other than just the Hamiltonian [12]. Furthermore,by decoupling a nucleonic valence space with the SRG, diagonalization is made more efficient.With this method in place, an intriguing question is: how do these three-nucleon interactions,embedded into nuclear structure theory, affect nuclear properties and decays? A nuclear decayof primary interest is double-beta decay, with its neutrinoless mode as an exciting prospect dueto the discovery of neutrino oscillations [13, 14].∗ Before that can be approached, one hopesthat the problem of “quenching” (see Section 3.1.4) in two-neutrino double-beta decay can besolved using nuclear structure theory. We will attempt to model both of these decays using theso-called Magnus formulation of IM-SRG for the medium-mass nucleus, Calcium-48.†This dissertation is organized as follows. First, in Chapter 2, we will outline the preliminarycontent needed to develop nuclear many-body theory and the fundamentals behind double-betadecay, while making frequent reference to introductory material in Appendices A, B, and C.After overviewing the status of double-beta decay in Chapter 3, we’ll do a rigorous derivationof the neutrinoless double-beta decay two-body matrix elements in Chapter 4. To model thisdecay ab initio, we use the valence space formulation of IM-SRG, which we describe in detailin Chapter 5. Finally, after outlining our numerical methods in Chapter 6, we’ll conclude withthe results of computations for Calcium-48 in Chapter 7.∗Neutrino oscillations prove that (at least one) neutrino has a non-zero mass, which is a necessary (but notsufficient) requirement for the theory behind neutrinoless double-beta decay (see Section 3.2).†Calcium-48 is a doubly magic nucleus, making it an ideal benchmark for our theoretical framework.1Chapter 2PreliminariesBefore we can present the main work of this dissertation, it is essential to lay out the mathemat-ical framework for nuclear physics. This chapter will help give unfamiliar readers an overviewof the techniques we use, and also to establish the notation we’ll be using throughout the courseof this dissertation. Any readers who find themselves uncomfortable with the theoretical ma-chinery of this chapter should first read: Appendices A, B, and C. The prescriptions laid outin [5, 15, 16] will be closely followed. We’ll assume the reader has a background in: quantummechanics, statistical mechanics, Lagrangian and Hamiltonian dynamics, particle physics ofthe Standard Model, some quantum field theory (QFT), linear algebra over complex vectorspaces, multivariate calculus, vector calculus, ordinary differential equations (ODEs), partialdifferential equations (PDEs), complex analysis, numerical methods, and some group theory.2.1 Many-Body Quantum MechanicsLet’s begin by bridging the gap between the abstractions of Appendices A, B, and C and thecoveted nuclear landscape. Since the atomic nucleus is a collection of bound nucleons (protonsand neutrons) interacting through the strong nuclear force, many-body quantum mechanics willbe the path to nuclear structure theory.2.1.1 Central Quantum Harmonic OscillatorIn nuclear physics, the solutions to the Time-Independent Schro¨dinger Equation (TISE) undera spherically symmetric central potential are often used as a starting point, which is referredto as the “(harmonic) oscillator basis.” Specifically, under the central potentialV (r) =12mNω2r2 − V0where ω, or }ω [MeV], is the “(oscillator) frequency,” mN is the mass of a nucleon, and V0comes from an experimental fit - the radial TISE becomes− }22mNñ∇2r −l(l + 1)2ôRnl(r)− V0 + 12mNω2r2Rnl(r) = EnlRnl(r) (2.1)This equation can be thought of as describing a nucleon oscillating about a central effectivecore∗ with the total wave-function∗or simply another nucleon in relative coordinates22.1. Many-Body Quantum MechanicsΨnlm(⇀r , t) = ψnlm(⇀r )Tnl(t) = Rnl(r)Ylm(θ, φ)e− i}Enltwhere Rnl(r) is the (relative) radial wave-function, Ylm(θ, φ) are your standard spherical har-monics∗ as in Equation (F7), nlm are the three necessary quantum numbers as per usual, andthe quantized energy levels areEnl =(N +32)}ω − V0, N ≡ 2n+ l (2.2)As with common convention, we’ll label the principle quantum number, n = 0, 1, 2, ..., ascounting the number of nodes of the radial wave-function. It can be shown [17] that thesolution to Equation (2.1) isRnl(r) = Nnl(rb)le−r22b2Ll+ 12n (r2/b2) (2.3)where b is the “(harmonic) oscillator length,” given byb ≡ }mNω≈ 197.3269788»938.9187474× }ω [MeV]fm (2.4)Here we have taken our numerical values from CODATA-2014 [18], and set the nucleon massas the average of the proton mass and the neutron massmN ≡ mp +mn2= 938.9187474 MeVNnl in Equation (2.3) is the normalization constant, which can be found asNnl.=√2n!b3 Γ(n+ l + 32)(2.5)where Γ(x) is the Gamma function, as in Equation (F1). Finally, the Lβα(x) in Equation (2.3)are the generalized Laguerre polynomials with α, β ∈ R, which are defined as solutions to thefrequently reoccurring ODE in physics, Equation (F3).It’s important to note that the harmonic oscillator wave-functions form a complete basis,which will have much utility in the future∫ ∞0Rnl(r)Rn′l(r)r2dr = δnn′An optimal oscillator frequency can be found from the Blomqvist-Molinari formula of nuclearcharge radius predictions, from the well-known paper [19]}ω ≈Ä45A−13 − 25A− 23äMeV (2.6)where, of course, A = N + Z is the “mass number” of the nucleus of interest, i.e., the sum ofthe number of constituent neutrons, N , and protons, Z. By sneakily introducing some nuclearparlance, we’ve suggested that the quantum harmonic oscillator (QHO) described above has aninherent relation to nuclear structure. More on this will be explored soon in Section 2.2.2.∗via the angular TISE32.1. Many-Body Quantum Mechanics2.1.2 The Talmi-Moshinsky TransformationMarcos Moshinsky developed a method to transform between two particles in a harmonic os-cillator basis to their relative and centre of mass (CoM) coordinates, using work done by IgalTalmi in the early 1950’s [20, 21]. Let’s label the “lab frame” bases as {|ni li〉 | i = 1, 2}, therelative coordinate basis as {|nr lr〉}, and the CoM basis as {|NΛ〉}. To stay consistent withthe notation throughout this dissertation, the total angular momentum is labelled L, so that⇀l1 +⇀l2 =⇀L =⇀lr +⇀Λ (2.7)where, if ⇀r1 and⇀r2 represent the coordinates of each particle in the lab frame, then the relativecoordinates and CoM coordinates are given by⇀r ≡ 1√2Ä⇀r1 −⇀r2äand,⇀R ≡ 1√2Ä⇀r1 +⇀r2ä(2.8)Take close notice of the inverse factor of square root two in front of these definitions, since itwill cause some notational trouble later on.Just as the basis {|n1 l1, n2 l2 : LM〉} is complete, so is {|nr lr, NΛ: LM〉}, and so we canplay the well known trick of inserting an identity into an arbitrary state as so|ni li, nj lj :LM〉 = 1 · |ni li, nj lj :LM〉=(∑|nr lr, NΛ:LM〉〈nr lr, NΛ:LM |)|ni li, nj lj :LM〉=∑〈nr lr, NΛ:LM |ni li, nj lj :LM〉|nr lr, NΛ:LM〉=∑nrlrNΛDij |nr lr, NΛ:LM〉 (2.9)In Equation (2.9) we’ve slipped in a definition, so let’s make it official:Definition 2.1: “Talmi-Moshinsky Brackets”The coefficients from Equation (2.9) above are defined asDij ≡ 〈nr lr, NΛ:L|ni li, nj lj :L〉 (2.10)where we have the couplings ∆(li lj : L) and ∆(lr Λ : L) from Equation (2.7). Note that inmany papers authors will switch the Λ↔ L, including Moshinsky himself. Also, we’ve droppedthe M -dependence in this definition, since it is well known that the Talmi-Moshinsky bracketsare independent of the total orbital angular momentum projection [22]. Finally, the commonconvention is to take it thatDij = 〈nr lr, NΛ:L|ni li, nj lj :L〉 .= 〈ni li, nj lj :L|nr lr, NΛ:L〉In other words, like the Clebsch-Gordan (CG) coefficients, these brackets are real, so Dij = D∗ij .These coefficients can be calculated via algorithms given in [23, 24].42.1. Many-Body Quantum MechanicsA question that appears much less heinous than it actually is, is: in Equation (2.9), what limitsdoes the sum run over? Since this issue gets rather involved, we’ll leave it for Appendix D. Forinstance, see the chosen Talmi-Moshinsky limits (CTML) given in Equation (D13).The power of these Talmi-Moshinsky brackets is that they give us a means of doing physicsin the relative frame, which will often be more intuitive and convenient than the lab frame.Hence, when calculating two-body reduced matrix elements (see Section C.3), we’ll typicallytransform from lab frame into relative coordinates via these brackets. This step is importantenough that we label it as a theorem:Theorem 2.1: “Talmi-Moshinsky Transformation”Consider a two-body spherical tensor operator, “T , with rank R. We can transform from thelab frame representation to the relative/CoM coordinates using the formula〈l1, l2:L||“TR||l′1, l′2:L′〉 = ÛLÛL′ ∑nrlr,n′rl′rNΛD12D′12(−1)R+L′+lr+Λ×®L lr Λl′r L′ R´〈nr lr||“TR||n′r l′r〉 (2.11)where the CTML are given in Equation (D21), and the brackets areD12 = 〈nr lr, NΛ:L|n1 l1, n2 l2:L〉 and, D′12 ≡ 〈n′r l′r, NΛ:L′|n′1 l′1, n′2 l′2:L′〉 (2.12)The utility of this formula cannot be overstated. In essence, we now have a way of transformingtwo-body matrix elements in the lab frame representation into a sum of one-body matrixelements in the relative coordinate space! Remarkably, all the information involving the CoMframe has been encoded by the Talmi-Moshinsky brackets, the phase factor, and the 6j-symbol.Proof of Theorem (2.1).First, let’s transform the states of the matrix elements on the left hand side (LHS) of Equa-tion (2.11) using (2.9), in other words||l′1, l′2:L′〉 =∑n′rl′rN ′Λ′〈n′r l′r, N ′Λ′:L′|n′1 l′1, n′2 l′2:L′〉||n′r l′r, N ′Λ′:L′〉 and,〈l1, l2, L|| =∑nrlrNΛ〈n1 l1, n2 l2:L|nr lr, NΛ:L〉〈nr lr, NΛ:L||so take it thatD12 ≡ 〈n1 l1, n2 l2:L|nr lr, NΛ:L〉 and, D′′12 ≡ 〈n′r l′r, N ′Λ′:L′|n′1 l′1, n′2 l′2:L′〉 (2.13)52.1. Many-Body Quantum MechanicsThis gives the reduced matrix elements〈l1, l2:L||“TR||l′1, l′2:L′〉 = ∑nrlrNΛ∑n′rl′rN ′Λ′D12D′′12〈nr lr, NΛ:L||“TR||n′r l′r, N ′Λ′:L′〉 (2.14)Now, from Proposition B.1 and Equation (B4), we may take it that “TR = [“TR⊗10]R, andapply Theorem (B.2) - Equation (B15) to the reduced matrix elements on the right hand side(RHS) of (2.14)〈nr lr, NΛ:L||“TR||n′r l′r, N ′Λ′:L′〉 = ÛL ÙR ÛL′lr Λ Ll′r Λ′ L′R 0 R 〈nr lr||“TR||n′r l′r〉 〈NΛ||1||N ′Λ′〉(2.15)We know from the result in Equation (B12) that〈NΛ||1||N ′Λ′〉 = δNN ′δΛΛ′ ÛΛ (2.16)and also, from Equations (A38), (A39), and (A28), we can obtainlr Λ Ll′r Λ′ L′R 0 R = δΛΛ′(−1)lr+L′+Λ+R ÛΛ−1ÙR−1 ®L lr Λl′r L′ R´ (2.17)Plugging Equation (2.16) and (2.17) into (2.15) yields〈nr lr, NΛ:L||“TR||n′r l′r, N ′Λ′:L′〉 = δNN ′δΛΛ′(−1)R+L′+lr+Λ ÛLÛL′ ®L lr Λl′r L′ R´ 〈nr lr||“TR||n′r l′r〉(2.18)And now inserting Equation (2.18) into (2.14) gives〈l1, l2:L||“TR||l′1, l′2:L′〉 = ÛLÛL′∑nrlrNΛ∑n′rl′rN ′Λ′δNN ′δΛΛ′D12D′′12(−1)R+L′+lr+Λ×®L lr Λl′r L′ R´〈nr lr||“TR||n′r l′r〉 (2.19)Finally, we may perform the sum over the N ′Λ′ in Equation (2.19), which will retrieve (2.11),and likewise send D′′12 → D′12 from (2.13) to (2.12) respectively.Corollary 2.1.1Consider a two-body spherical tensor operator, “T , with rank 0. We can transform from the labframe representation to the relative/CoM coordinates using the formula〈l1, l2:L||“T ||l′1, l′2:L′〉 = δLL′ ÛL ∑nrlr,n′rNΛD12Dn′r1′2′Ûl−1r 〈nr lr||“T ||n′r lr〉 (2.20)where the CTML are given in Equation (D16), and the brackets are62.1. Many-Body Quantum MechanicsD12 = 〈nr lr, NΛ:L|n1 l1, n2 l2:L〉 and, Dn′r1′2′ ≡ 〈n′r lr, NΛ:L|n′1 l′1, n′2 l′2:L〉 (2.21)The proof of Corollary 2.1.1 above is a straightforward application of Theorem 2.1 and theidentity in Equation (A30), where one must keep track of the Kronecker-deltas accordingly,note that lr,Λ, L ∈ N0, and keep track of the change of notation in the brackets. Another easilyproven (but less physically applicable) corollary is given in Equation (F10).2.1.3 Slater Determinants and the Pauli Exclusion PrincipleIn single-body quantum mechanics, we describe a particle with a wave-function, which wenow label Φ(⇀r , t) = 〈⇀r |Φ(t)〉, which satisfies the Schro¨diner equation. By doing separationof variables, the corresponding time-independent wave-function will be written as φ(⇀r ). Thenext natural question is: for A particles, each tracked in their respective coordinates system by⇀r1,⇀r2, ...,⇀rA, how would we construct a many-body wave-function?Definition 2.2: “Slater Determinant”Let’s presuppose that each fermion in an A-body system satisfies their own TISE with thenormalized wave-function φi(⇀ri) for i = 1, 2, ..., A. Then an ansatz for the many-body wave-function isΨ0(⇀r1,⇀r2, ...,⇀rA) =1√A!∣∣∣∣∣∣∣∣∣∣φ1(⇀r1) φ1(⇀r2) . . . φ1(⇀rA)φ2(⇀r1) φ2(⇀r2) . . . φ2(⇀rA)....... . ....φA(⇀r1) φA(⇀r2) . . . φA(⇀rA)∣∣∣∣∣∣∣∣∣∣(2.22)where the prefactor is to ensure this “Slater determinant” wave-function is, itself, normalized.Notice that we’ve now inherently switched to describing a collection of fermions, which willbe convenient for us since the nucleon is a spin-1/2 particle. The naught subscript on Ψ0 isthere to distinguish this from the time-dependant many-body wave-function, Ψ(⇀r1,⇀r2, ...,⇀rA, t).Using the determinant in Definition 2.2 guarantees that this many-body wave-function will beproperly anti-symmetrized, in that it will obey the Pauli exclusion principleΨ0(⇀r1,⇀r2, ...,⇀rA) = (−1)Π(P ) Ψ0ÄP (⇀r1,⇀r2, ...,⇀rA)ä(2.23)where Π(P ) is the number of interchanges of two coordinates required to bring the permuta-tion P back to identity. For example, for P (⇀r1,⇀r2,⇀r3) = (⇀r3,⇀r1,⇀r2) then Π(P ) = 2 by firstinterchanging 3 and 1, and then 3 and 2. The canonical example is thatΨ0(⇀r1,⇀r2) = −Ψ0(⇀r2,⇀r1) =⇒∫dτÄΨ0(⇀r1,⇀r2) + Ψ0(⇀r2,⇀r1)ä= 0 (2.24)which, physically, states that the probability of having two fermions∗ in the same state, inte-∗of the same particle type, for instance proton or neutron72.1. Many-Body Quantum Mechanicsgrated over any region, is exactly zero. Equation (2.24) is indeed satisfied by the two-bodySlater determinantΨ0(⇀r1,⇀r2) =1√2Äφ1(⇀r1)φ2(⇀r2)− φ1(⇀r2)φ2(⇀r1)ä2.1.4 The Hartree-Fock MethodIt is often the case that we will take a Slater determinant as an ansatz for a ground state, butthere is no guarantee that a Slater determinant of single-particle ground state wave-functions,φi(⇀r ), will physically correspond to the collective ground state in the presence of particleinteractions. The Hartree-Fock method is an application of the variational principle to thisSlater determinant ansatz, |Ψ0〉, to make an optimal mean-field potential that minimizes anyresidual interactions between the single-particle states. To first order in perturbation theory,we know that the best approximation for the ground state energy of a Hamiltonian, “H, isEg.s. = 〈Ψ0|“H|Ψ0〉Hence, the natural variational condition in this case isδÇEg.s.〈Ψ0|Ψ0〉å= 0 (2.25)It was shown, by use of Lagrange multipliers, that Equation (2.25) gives the unconstrainedproblem known as the “Hartree-Fock equation”− }22mN∇2φa(⇀r ) + VHFÄ{φi(⇀r )}äφa(⇀r ) = εaφa(⇀r )i = 1, 2, . . . , A and, a = 1, 2, . . .(2.26)where we’ve assumed we’re working with A nucleons, and the Lagrange multipliers turned outto be the single-particle energies, εa. This non-linear Schro¨dinger-type PDE can be solvediteratively with some guessed set of single-particle wave-functions [5].In the case that we can set some Fermi surface, εF , for our many-body state, Equation (2.26)can be rewritten in a simple way, in order to solve for the single-particle energies. That is, usinga particle-hole occupation number representation (see Appendix C), Equation (C12) of Wick’sTheorem can be used to transform (2.26) intotab +∑cεc≤εFvcacb = εaδab (2.27)where the kinetic energy has be written with one-body matrix elements (OBMEs) in secondquantization as in Equation (C13), and likewise for the anti-symmetrized version of the inputinteraction treated as a two-body operator as followsV =12∑abcdvabcd cˆ†acˆ†bcˆdcˆc82.2. Nuclear Structure Theoryvabcd ≡ vabcd − vabdcFurthermore, once these single-particle energies are computed, a Hartree-Fock mean field canbe constructed as “HHF = ∑aεacˆ†acˆagiving the final Hartree-Fock ground state energyEHF =∑aεa≤εFtaa +12∑a,bεa≤εFεb≤εFvababMore details on how one can recast Equation (2.27) as an eigenvalue problem can be foundin Section 4.6 of Suhonen [5]. Now that we have the mathematical tools to create a realisticmany-body eigenspace, let’s apply them to nuclear structure.2.2 Nuclear Structure TheoryThe atomic nucleus represents the quintessential object to be handled by quantum many-bodytheory, since it is composed of protons and neutrons, which both obey quantum physics at thisscale. Although the protons repeal each other by means of electric repulsion, the nucleus isheld together by the strong nuclear force which binds both protons and neutrons collectively.Yet, this is not the only mechanism at play, since nucleons are spin-1/2 objects that obey theexclusion principle. This sets up a nuclear shell model, in analogy to the atomic shell modelused to describe electron orbitals centred about the nucleus. Ultimately, nucleonic interactionsand how those nucleons are physically structured together within the nucleus represents afundamental problem in physics and natural philosophy; since without the stability of thenucleus, our reality would seize to exist.2.2.1 The BasicsIn the early 1930’s, Heisenberg noted that since the proton and neutron have such similarmasses, currently reported as [18]mp = 938.2720814 MeVmn = 939.5654134 MeV(2.28)=⇒ mn −mp ∼ 2methen perhaps they could be considered as the same particle, only manifested as different quan-tum states. This particle was called the “nucleon,” and (remarkably) it can be described bythe same group theoretic formalism as spin. The quantum number that flips a nucleon betweenits proton and neutron states is called “isospin,” and we’ll take the isospin convention that92.2. Nuclear Structure Theory|p〉 = |t = 12 , tz = +12〉, |n〉 = |t = 12 , tz = −12〉Of course, this is not an exact symmetry of quantum physics, since the proton and neutronmasses are slightly different∗ and the proton has a positive charge whereas the neutron iselectrically neutral. However, since the Coulomb force is orders of magnitudes weaker than thestrong force, it can be neglected, along with the small mass difference - thus making isospinformalism an important tool in basic nuclear physics.A nucleus is composed of Z protons and N neutrons,† which are both composite objectsmade of up and down quarks. Thus, upon zooming in, a nucleus is (locally) like a quark-gluonsoup with (global) structure. The mass number is defined as the number of nucleonsA ≡ Z +Nand nuclei may be labelled as (Z,A). Symbolically, atomic notation from chemistry is also usedin nuclear physics, however it is simplified toAZX# −→ AXsince the chemical symbol, X, is understood to label Z, and the ionic charge, #, is not oftenapplicable. Alternatively, a nucleus may be referred to by its name appended with the massnumber, for instance: Germanium-76.= 76Ge.= (32, 76). We’ll somewhat haphazardly switchbetween these notions. As usual, “isotopes” have the same proton number Z, “isotones” havethe same neutron number N , and “isobars” have the same mass number A. An interestingpoint about isobars is that, empirically, they are found to have the same nuclear radiusR ≈ r0A1/3, r0 = 1.2 fm (2.29)Hence, the volume of a nucleus is proportional to A.‡ The mass of a nucleus, however, is amuch more complicated beast.Semi-Empirical Mass FormulaWe can predict the mass of nucleus (Z,A) via the formulam(Z,A) = Zmp + (A− Z)mn −BE(Z,A) (2.30)where BE is the “binding energy.” The binding energy has a negative in Equation (2.30) sincefree nucleons which come together to make a stable nucleus will be sitting in an energy well viaattractive forces, and this binding together will be contained as mass via relativity.§ It is alsoexperimentally relevant to report the “mass excess” of a nucleus, which is given by∗The proton is made of two valence up quarks and one valence down quark, whereas the neutron is madeof one valence up quark and two valence down quarks. It turns out that the down quark’s constituent mass isheavier than the up’s, and the charge of the proton affects its mass via QED corrections.†Typically Z < N , since the positively charged protons repel each other electrostatically (long range force),but the neutrons hold the nucleus together via attractive strong interactions (short range force).‡see the volume term of Equation (2.33)§of course, here we have set c = 1102.2. Nuclear Structure Theory∆m ≡ m(Z,A)−A · u (2.31)where the “u” is the unified atomic mass unit (amu), defined asu ≡ 931.4940955 MeV (2.32)The binding energy is commonly modelled by the formula [16]BE(Z,A) = aVA− aSA2/3 − aC Z(Z − 1)A1/3− aA (A− 2Z)2A+ δ(Z,A) (2.33)The first term of Equation (2.33) is the “volume term” (V ), which accounts for the strong inter-actions of nucleons with their neighbours under a femtometre range; the second is the “surfaceterm” (S), which corrects the first term for nucleons near the surface and introduces the liquiddrop analog of surface tension; the third is the “Coulomb term” (C), which gives electrostaticrepulsions between protons; the fourth is the “asymmetry term” (A), which models the Pauliexclusion principle for protons and neutrons and sets up their respective Fermi surfaces; andthe fifth is the “pairing term” (P ), which accounts for pairwise spin coupling:δ(Z,A) =0, A = odd+aPAkP , A = even, Z = even−aPAkP , A = even, Z = odd(2.34)Though the terms can all be theoretically justified, the parameters are fit empirically. Forexample, [25] lists them asaV = 15.75 MeV, aS = 17.8 MeV, aC = 0.711 MeVaA = 23.7 MeV, aP = 11.18 MeV, kP = −1/2(2.35)When plotting for multiple isobars, the semi-empircal mass formula will give parabolic lookingshapes, which are sometimes referred to as “mass parabolas.” An example of this can be seenin Figure 3.2, which depicts the double-beta decay of 48Ca.2.2.2 Nuclear Shell ModelThe pioneers of nuclear structure theory noticed that nuclei would undergo discrete lookingjumps in properties like energy levels around certain Z and N . The numbers for which Z andN would display this phenomenon were called “magic numbers.” It was eventually realized thatthis pattern mimicked the electron orbital shell structure seen in atomic physics. Hence, itwas proposed that both protons and neutrons occupy their own respective shells. This ideais somewhat puzzling in nuclear physics, since atomic orbitals are set up via quantization ofthe electron revolving around a centrally attractive object.∗ But, what are the protons andneutrons inside a nucleus orbiting - a mean field set up by their neighbours? Nonetheless, thenuclear shell model (ShM) has been successful in predicting nuclear properties [16].∗namely, the nucleus112.2. Nuclear Structure TheoryFigure 2.1: Oxygen-16, as represented by the nuclear shell model. Both the protons (depictedby red balls) and the neutrons (depicted by blue balls) have been filled in their own respectiveshells. Here we’ve shown the ground state, which fills the single-particles in their orbits startingfrom 0s1/2 to 0p1/2, and portrays no excitations to higher orbits in the sd-shell or beyond. Thespectroscopic notation used to label the orbits is explained in Figure 5.5 of [27], except we countthe principle quantum number by shifting down by 1.Nuclear shells are inherently related to the Pauli exclusion principle, since the nucleon isa fermion. What makes the ShM distinct from atomic orbitals, is that since the proton andneutron are different particle types (they have different isospin), they each have their own shellsto fill, in tandem. For example, Figure 2.1 shows the ground state filling of 16O. To obtain thecorrect quantization of the shells, one begins with a QHO potential (see Section 2.1.1), whichwas found to be a good leading order approximation because it naturally describes self-boundsystems [26]. The inclusion of spin-orbit coupling (that is, the addition of an⇀S◦⇀L-dependentterm in the nuclear potential) will split the QHO shells and (magically) reproduce the observedmagic numbers. A good visual description of reproducing the magic numbers is given in Figure5.5 of Krane [27], which starts with a phenomenological Woods-Saxon potential∗ and thenadds in spin-orbit splitting. Overall, the ShM has acted as a paradigm for understanding thebehaviour of nuclei [26], and thus it is a common first step in nuclear structure theory. Adding innuclear interactions to capture dynamics within the shells, then, should give a good descriptionof nuclear physics.2.2.3 Nuclear InteractionsUltimately, the strong nuclear force is an emergent phenomenon, which arrises from the stronginteraction between quarks, as governed by quantum chromodynamics (QCD). Quarks, asfermions, interact via the exchange of gluons, the boson which mediates the strong interac-tion. The QFT which describes this physics is referred to as QCD, since the “charge” of thestrong interaction obeys SU(3) symmetry and is labelled by the three colours: red, green, and∗this creates similar bound states to the QHO, but it is tailored to nuclear phenomenology122.2. Nuclear Structure Theoryblue, and their corresponding anti-colours. Quarks will confine into colour neutral hadrons:either baryons (a bound set of three quarks)∗ or mesons (a bound pair of a quark and anti-quark).† Stepping back in energy scale, one can view the strong interaction less precisely as anexchange of mesons‡ between baryons, which is sometimes called “quantum hadrodynamics.”This emergence in nuclear physics can be seen schematically in Figure 2 of [28].PhenomenologicalThe baseline theoretical tool to build nuclear interactions is, obviously, nuclear phenomenol-ogy [29, 30]. That is, one begins with mean-field§ or effective¶ field theory perturbative methods,and then designs a particular potential in order to reproduce nuclear properties like: bindingenergies, energy levels, and so on. Using a large body of experimental data for a chosen nu-cleus (or several), phenomenologists tune the interaction parameters to make their theoreticalcalculations fit with the measured values. This is often done beginning with a shell model ap-proach, as was done for the GXPF1A interaction [31, 32], which is specifically designed to givegood nuclear predictions for nuclei which have their predominate nucleon excitations withinthe pf -shell. Historically, this has been a successful approach - but it is not ab initio. Theultimate goal in nuclear physics is to build nuclear structure from a framework rooted in QCD;so phenomenological interactions are simply a means to an end.Chiral Effective Field TheoryAnother more fundamental approach to nuclear interactions is to use an effective field theorythat reproduces low-energy QCD. A modern technique in this vain, which has proven to besuccessful for nuclear structure theory [7], is known as “chiral effective field theory” (χEFT).From a breakthrough paper by Weinberg [33], it was realized that chiral symmetry breakingin QCD Lagrangians can be used to build an EFT which reconstructs Yukawa’s original pionpropagator theory [34] and higher-order corrections to nuclear forces [35]. In practice, tech-niques pioneered by [36, 37] can be used to perturbatively expand QCD in terms of a chosenenergy cutoff scale, Λ, to capture low-energy QCD.‖ The advantage with this formalism isthat it gives theorists a way to distinguish between leading order (LO), next-to-leading order(NLO), next-to-nexto-to-leading order (N2LO), . . . contributions. This is typically done usinga “power counting” scheme in ν-powers of the expansion parameter, (Q/Λ)ν , which can be seenin Figure 2.2. Through this process, physics from neglected degrees of freedom is captured inshort-range contact interactions which contain a number of “low-energy constants” (LECs) to∗for example: a proton, neutron, or delta baryon†for example: a pion, rho meson, or omega meson‡mesons indeed are bosons, so they act as the force carrier of the strong nuclear force§treating a nucleon as affected by the potential of the collective interactions of its neighbouring nucleons¶treating mesons as the force carriers between baryons, built from a low-energy theory of QCD and hencetaking the nucleus as a “soup” of quark-gluon structures‖note that the chiral symmetry breaking scale is at Λ ≈ 1 GeV (the nucleon mass)132.2. Nuclear Structure TheoryFigure 2.2: “Hierarchy of nuclear forces in [χEFT]” taken from Figure 1 of [7]. Nucleons andpions are depicted as solid and dashed lines respectively in the perturbative nuclear Feynmandiagrams. The dots represent vertex type, see [7] for more details.be calculated perturbatively or fit to data (like nucleon-nucleon phase shits), such as with theso-called EM 1.8/2.0 interaction [38]. Once these Hamiltonians have been constructed, one canuse them as input for a nuclear many-body method to model nuclear structure.2.2.4 Nuclear Many-Body MethodsWe’ve already introduced one of the staple many-body methods, the nuclear ShM (see Sec-tion 2.2.2 above). But today, there exist plenty of competing nuclear many-body methods onthe market (IM-SRG, CC, NR-EDF, QRPA, IBM, etc) [13], each with their own advantagesand drawbacks. A glaring irony of these quantum many-body techniques is that they cannotgo beyond the nucleon-nucleon (2N) level, since higher body effects like three-nucleon (3N)configurations are still too mathematically complex. One method which remedies this issue isthe in-medium similarity renormalization group (see Chapter 5). By redefining normal orderingwith respect to the nuclear medium (see Section 5.3), this method can capture the bulk effectsof 3N physics using 2N technology! This will surely make appreciable differences in nuclearstructure calculations, such as those for double-beta decay.142.3. Beta Decay2.3 Beta DecayBefore we move on to double-beta decay, let’s first introduce one of its defining components:single-beta decay. There are two main types of beta decay - in which either the neutron decaysinto a proton and emits an electron (to conserve charge) with a electron anti-neutrino (toconserve lepton number), or the proton decays into a neutron, positron, and electron neutrino.n −→ p + e− + νe or, p −→ n + e+ + νeThe former is often called β− decay, whereas the latter is β+ decay. By experimentalconvention, we’ll focus on the former, and simply call this “beta decay”: β− → β. An interestingnote is that β+ cannot happen spontaneously in free space, since the final state has more restmass (see Equation (2.28), and note that mν ≈ 0). However, the proton can decay within anucleus, because it can be excited (thus deriving energy/mass) through strong interactions withneighbouring nucleons. Both β− and β+ nuclear beta decay led to the discovery of the neutrino,as posited by Pauli in 1930 to solve the problem of missing energy in laboratory measurementsof beta-radioactivity [39]. At this point, early particle physicists were confident that they wereon their way to completing a theory of fundamental physics, but even a decay as simple as betadecay kept more secrets lurking within.2.3.1 Parity ViolationParity is a spacial coordinate transformation whereby one axis is flipped. By classical intuition,such a simple inversion wouldn’t seem to change the laws of physics, so parity could be assumedto be a symmetry of the universe. Indeed, parity is conserved for electromagnetic and strongprocesses, but in 1956 it was realized by Lee and Yang that not enough evidence existed toestablish or refute parity conservation in the weak sector [40]. Subsequently, parity violation wasfound in the beta decay of 60Co [41], and a full V−A theory of the weak interaction was rapidlydeveloped [42, 43]. For the Standard Model, parity happens to be maximally violated in that:only left-handed particles and right-handed anti-particles participate in the weak interaction.A pedagogical introduction to the particle physics behind V −A theory can be found in [44].The constants which describe parity violation are the vector coupling constant, gV , and theaxial-vector coupling constant, gA. Modern experiments are still being designed to improve themeasurement of the V −A constants, such as [45]. Accordingly, we’ll take the values asgV = 1 and, gA = 1.27These constants will come up frequently in the equations describing beta decay.152.3. Beta Decay2.3.2 Allowed Fermi TransitionAmong many forbidden and super-allowed beta transitions [5], there are two main leading-order∗modes to beta decay: the Fermi (F) transition [46], and the Gamow-Teller (GT) transition [47].For the Fermi case, the spins of the electron and the neutrino are anti-aligned, hence theycouple to spin S = 0. Hence, the total spin of the nuclear state remains unchanged, giving theselection rule ∆J = 0. Such a simple selection rule is clearly modelled by using the identityoperator taken as a spherical tensor “OF ≡ gV 1τ+ (2.36)along with the isospin operator, τ+, which will flip the state of the nucleon from neutrontz = −1/2 to proton tz = +1/2 in accordance with the beta decay. To find the reduced OBMEsfor the Fermi transition (to be used in Equation (C26) for a nuclear matrix element calculation),we’ll omit the isospin operator† and apply Equation (B11) to (2.36), giving〈a||“OF||b〉 ∝ δnanbδlalbδjajb ÛaThis scenario is remarkably more simple than the Gamow-Teller case (see below), and moreso compared to neutrinoless double-beta decay two-body matrix elements (TBMEs), which aretediously derived in Chapter Allowed Gamow-Teller TransitionThe Gamow-Teller mode is characterized by the spin alignment of the emitted particles in thebeta decay. Therefore they couple to spin S = 1, yielding the selection rule ∆J = 0,±1. Thischange in angular momenta can be modelled by the use of a Pauli operator as follows“OGT ≡ gA“στ+ (2.37)To obtain the reduced matrix elements of the Gamow-Teller operator, 〈l 12 :j||“OGT||l′ 12 :j′〉,we’ll omit the analysis of the isospin operator,† and instead start by looking at the matrixelements of the spin operator (with } 6= 1)“S = 12}“σ (2.38)We’ll take our quantization axis as the z-axis, and update the z-component of the spin operatorto a vector (rank L = 1) spherical tensor, Ŝz → “S1. The non-reduced matrix elements of z-component of Equation (2.38) are built by acting on spin-1/2 states〈12 m|“S1|12 m′〉 = m′} δmm′ such that, m,m′ = ±12 (2.39)To reduced these matrix elements, we can simply use the Wigner-Eckart Theorem of Equa-tion (B9), labelling the projected spherical tensor rank as M , giving∗as taken in the non-relativistic limit of the weak interaction†since this will introduce a Kronecker-delta of some form, depending on the chosen isospin formalism162.3. Beta Decay〈12 m′|“S1M |12 m〉 = (−1) 12−m′ Ç 12 1 12−m′ M må 〈12 ||“S1||12〉 (2.40)Plugging Equation (2.39) into Equation (2.40) will yield 0 = 0 unless m′ = m on the LHS, and−m′ +M +m = 0 on the RHS by Equation (A24), =⇒M = 0. It can be easily found thatÇ12 112−12 0 12å=Ç12 11212 0 −12å=1√6so, overall, we obtainm′} =(−1) 12−m′√6〈12 ||“S1||12〉 =⇒ 〈12 ||“S ||12〉 = √62 } =  32} (2.41)where we’ve dropped the “rank 1” subscript for the “S ; which is to be understood as the sphericaltensor operator∗ of the z-component of the spin operator, as opposed to the (bolded) vectorspin operator. However, since the projected spin quantum number should be ±12 on any axis,Equation (2.41) will hold for any component of the spin treated as a vector spherical tensor,and so we could† explicitly write〈12 ||“S ||12〉 −→ 〈12 ||(“Sx, “Sy, “Sz)||12〉 =  32} (1, 1, 1)Finally, putting Equation (2.38) and (2.41) together gives us〈12 ||“σ||12〉 = 2}〈12 ||“S ||12〉 = 2} ·√62 } = √6 (2.42)Now we consider a spin-1/2 particle with total angular momentum, j, via the ls-coupling∆(l 12 :j). Since the Pauli operator is a vector spherical tensor operator, like the spin operator,we may apply Corollary B.2.4 to it in this ls-coupling. Using Equation (B27) gives〈l s:j||“σ||l′ s′:j′〉 = −δll′(−1)l+2( 12 )+j+1+ 12 Û Û ′ ® 12 12 1j′ j l´ 〈12 ||“σ||12〉 (2.43)and plugging Equation (2.42) into (2.43) yields the reduced matrix elements〈l 12 :j||“σ||l′ 12 :j′〉 = √6 δll′Û Û ′(−1)l+j+ 32 ® 12 12 1j′ j l′´ (2.44)Thus, all that one needs to do to calculate the one-body reduced matrix elements of the Gamow-Teller operator of Equation (2.37) is compute (2.44), account for the isospin formalism, andmultiply by the axial-vector coupling constant, gA. Note that we need not anti-symmetrizeEquation (2.44), since it doesn’t make physical sense to anti-symmetrize a one-body state.However, if we mathematically applied Equation (B31) and anti-symmetrized between l and s,we would nonetheless find that∗hence it is bolded†although we never will172.3. Beta Decay〈l 12 ; j||“σ||l′ 12 ; j′〉 = 〈l 12 :j||“σ||l′ 12 :j′〉since l cannot take on half-integer values.2.3.4 Quenching in Single-Beta DecayIt’s long been known, to the dismay of many theorists, that the predictions from Equation (2.44)in fact do not reproduce the experimental observations of single-beta decay [48]. To correctfor these discrepancies, the axial-vector coupling constant is often set to one, gA → 1, or a“quenching factor” is multiplied by the GT operator. That is,qf ∼ 1gA' 11.3' 77% (2.45)is multiplied by Equation (2.37). However, many different quenching factors (ranging from 0.6to 0.9) have been proposed [13, 49], and they seem to be nucleus dependent! This inconsistencyin the quenching factor begs a precarious question: where does quenching come from? Is it theaxial-vector coupling constant in Equation (2.37) that is effectively quenched, or does qf comefrom the “bare” GT matrix elements in (2.44) themselves, or a mixture of both?In the case that it is all coming from the nuclear matrix elements, this implies that thechosen nuclear structure model may be able to account for quenching. In other words, perhapsnot enough physics is being captured by the nuclear structure calculations of beta decay. Forinstance, those who work on the interacting ShM have stated that, given they could implementa full model space, then the problem of quenching will be solved within their framework [50,51].∗ Thus, quenching may simply be a byproduct of having a “truncated basis space” (seeSection 5.1). A somewhat related claim is that quenching indeed originates from the bare GToperator, due to an ill-considered analysis of the nuclear spectroscopic data [26]. However, allthese proposals are still argued to this day, and many alternative explanations exist [13]. Oneof the largest effects, which may solve the problem of quenching, comes from the inclusion ofmeson exchange currents [52]. Overall, this important question has not been settled, and thedebate is still very alive, in particular within the double-beta decay community [53]. For moreon this phenomenon, see Section 3.1.4.∗These citations are for the case of double-beta decay, but quenching is assumed to be a shared problembetween both single and double-beta decay (see Section 3.1.4).18Chapter 3Double-Beta DecayThe first theoretical suggestion of double-beta decay was made by Maria Goeppert-Mayer in1935 [54]. For her contributions to nuclear physics, she won the Nobel prize in 1963 along withWigner and Jensen, making her the second (and latest) female to win the prize, following MarieSk lodowska Curie. As discussed in Section 3.1.1 below, double-beta decay with the emissionof two neutrinos (2νββ) was subsequently measured. Furthermore, since the self-conjugatenature of the neutrino has not been experimentally established, it has been suggested that theemitted neutrinos in 2νββ could annihilate [55, 56], leading to a lepton-number violating processknown as “neutrinoless double-beta decay” (0νββ), covered in Section 3.3. The fundamentalmechanisms behind 0νββ are theoretically unclear, as pointed out in Section Two-Neutrino Double-Beta DecayThe dominate mode of double-beta decay happens via the emission of two electron anti-neutrinos, as opposed to the neutrinoless case. A common schematic used to visualize thisdecay is a Feynman diagram, as seen in Figure 3.1 below. On the LHS of the diagram, we beginwith two neutrons; and hence a charge of zero and a lepton number of zero. On the RHS of thediagram, we end with two protons, two electrons, and two electron anti-neutrinos; and hencea net charge of zero and a net lepton number of zero. The two electrons carry a charge of −2and a lepton number of +2, and so they act to cancel the charge of the two protons; whereasthe two electron anti-neutrinos carry a lepton number of −2 to cancel out the “electron-ness”of the final state. Thus, all the Standard Model quantum numbers are conserved in 2νββ.Figure 3.1 is misleading since it makes double-beta decay appear as though it is simplytwo single-beta decays happening concurrently. However, this is not the case, since it is notpossible for any arbitrary nucleus to undergo 2νββ sporadically. This is because the pairingterm in the semi-empirical mass formula of Equation (2.33) yields an asymmetry in nuclei. Forinstance, considering isobars of mass A = 48, we see in Figure 3.2 below that it is energeticallyfavourable for Potassium-48 to beta decay into Calcium-48, but it is not energetically favourablefor Calcium-48 to beta decay into Scandium-48. It is, however, energetically favourable forCalcium-48 to double-beta decay into Titanium-48. Hence, we should keep in mind thatββ 6= β + βand that only nuclei lying close to the bottom of the mass parabolas, in the same kind ofconfiguration as in Figure 3.2, represent double-beta decay candidates. This configuration is193.1. Two-Neutrino Double-Beta DecayFigure 3.1: The nuclear Feynman diagram for double-beta decay. In terms of a nucleus, thedecay is read as: (Z,A) −→ (Z+2, A) + 2 e− + 2 νefavoured experimentally since one can be confident that starting with a sample of (Z,A) andyielding products of (Z+2, A) has not occurred simply by two beta decays in a row.3.1.1 Experimental Confirmation of 2νββA list of important historical events involving double-beta decay can be seen in Table 4 of [58].The two-neutrino double-beta decay was theorized half a century before it was first directlymeasured in the laboratory, for the nucleus 82Se [59]. In fact, even indirect evidence for 2νββby means of geochemical analysis involving 130Te took 15 years following Goeppert-Mayer’sproposal [60]. Such experimental confirmations have, in turn, proven useful for theoreticalstudies of nuclear structure [61]. This is since one can benchmark nuclear many-body predictionsof a rare decay against actual experimental data, and give insight into mysteries surroundingthis decay, like quenching for instance (see Section 3.1.4 below).Since the first measurement of double-beta decay in 82Se, many more direct decays havebeen observed in numerous nuclei, such as: 48Ca, 76Ge, 96Zr, 100Mo, 116Cd, 130Te, 136Xe, and150Nd. These nuclei are even-even, of course, and have similar mass parabolic arrangements tothose in Figure 3.2. Measurements of 2νββ are still on-going, and experimental techniques inthis field are continually improving. A leading experiment in this search is known as NEMO-3.For instance, at the time of writing, the most recent measurement of double-beta decay in 48Cais given by the NEMO-3 Collaboration [62], which quotes the half-life asT 2ν1/2(48Ca) = [ 6.4+0.7−0.6(stat)+1.2−0.9(syst) ]× 1019 yr (3.1)Other half-lives have been compiled in [63].203.1. Two-Neutrino Double-Beta DecayFigure 3.2: Even (orange) and odd (green) mass parabolas of A = 48 isobars, showing thedouble-beta decay of 48Ca into 48Ti. To calculate the parabolas, we used Equations (2.30),(2.31), and (2.33), where we took the parameters from (2.35), but refit aC and aA to reproducethe precise mass excess of 48Ca and 48Sc given by [57]. The blue arrows represent β− decay, thepurple arrows represent β+ decay, and the red arrow is the double-beta decay. Notice that 48Arcould, hypothetically, double-beta decay into 48Ca, but this would be difficult to distinguishfrom two beta decays in a row, following the blue arrows. Also note that 48Ca could beta decayinto 48Sc, because of its minutely lower mass, but this is highly repressed relative to double-betadecay. This is not the case for many double-beta decay candidates, for example in Figure 1of [53], where clearly ∆m(76Ge) < ∆m(76As), in contrast to ∆m(48Ca) ≈ ∆m(48Sc).3.1.2 2νββ Half-Life FormulaAs presented by Tomoda [64] in their Equation (3.26), based on work by Doi et al. [65, 66], the2νββ half-life formula can be written as[T 2ν1/2 ]−1 ≈ G2ν |M2ν |2 (3.2)where G2ν is known as the “phase (space) factor,” which incorporates the interface between par-ticle and nuclear physics, and will be discussed more below; M2ν is the “nuclear matrix element”(NME), which incorporates nuclear structure theory; and Equation (3.3) indeed obeys (F11).The “approximately equals” sign mainly comes from neglecting the lepton energies relative tothe nuclear excitation energies, which are strategically replaced by averages [64]. From now on,such approximation signs will be dropped, as they are understood in this context.As made explicit in [67], it has become customary to rewrite Equation (3.2) as213.1. Two-Neutrino Double-Beta Decay[T 2ν1/2 ]−1 = G2νg4A|M2ν |2m2e (3.3)where gA is the axial-vector coupling constant from Section 2.3.1, and the units of me are takenas MeV. This is convenient since it puts the units of the phase factor into inverse years, andany hypothetical renormalization of gA can be handled a posteriori. However, it may becomecumbersome when looking for values of the phase factor in the literature, since the labelling forG2ν is not clearly distinguished between Equation (3.2) and (3.3).2νββ Phase FactorFrom the time leading up to its publication, [64] had reformulated the theory of double-betadecay into the construct still used to this day. They also presented numerical results for thephase factor, which they called the “lepton phase-space integral,” given in their Equation (3.28).However, a more modern and sophisticated analysis has been done in [67] and [68], which wewill refer to for our numerical values of G2ν . Ultimately, the calculation of these values involveshow the relativistic electron wave-functions of double-beta decay interact with the Coulombpotential, whilst taking screening effects and a realistic finite nuclear size into consideration.Additional nuclear physics is also ingrained into G2ν , but by a remarkable coincidence∗ thisdependence cancels out - thus leading to the succinct separation of the phase factor and theNME seen in Equation (3.3) above. We will not go into the details of this physics whatsoever,and instead leave it to the expert analysis as presented in [64, 67–69]. For instance, [67] (printedbelow) and [63, 68, 69] all give similar values for 2νββ in 48Ca, among othersG2ν(48Ca) = 1.555× 10−17 yr−1 (3.4)Example 3.1Let’s make a reverse prediction of the NME for the double-beta decay of 48Ca into 48Ti.That is, taking the theoretical value in Equation (3.4) above and the experimental valuein Equation (3.1), treating the latter as having indefinite precision, setting gA = 1.27 andme = 0.511 MeV, and plugging all these values into Equation (3.3), we should expect to see anNME in the range ofM2ν ' 0.03846 MeV−1 (3.5)However, considering the current status in nuclear structure theory, producing this number fromscratch will not necessarily happen. The reason for the discrepancies seen will be commentedon in Section 3.1.4 below, but first let’s outline how these matrix elements are calculated.∗as first noted by [64] and explicitly shown in [67]223.1. Two-Neutrino Double-Beta Decay3.1.3 M2ν Matrix ElementsAs with single-beta decay, the modes of 2νββ are the Fermi (see Section 2.3.2) and Gamow-Teller transitions (see Section 2.3.3). Thus, the NME from Equation (3.3) can be split up asM2ν = M2νGT −ÇgVgAå2M2νF (3.6)where gV and gA are the vector and axial-vector coupling constants respectively, as introducedin Section 2.3.1. However, as shown in Section 7.1 of [66], we can make the simplification that|M2νGT|  |M2νF |since “[n]early all the Fermi stength goes into the isobar analog state in the daughter, so thatM2νF can be neglected.” [70] pp. 488From the derivation of Equation (3.3), it can be shown that the dominant GT componentfor a 2νββ decay from the 0+ ground state of a parent nucleus to the J+ state of the daughternucleus is given by [48, 64, 67, 69, 71]M2ν ≈M2νGT =1√J + 1∑k〈J+f ||“στ+||1+k 〉〈1+k ||“στ+||0+i 〉[Ek + Ed(J) ]J+1(3.7)where i represents the initial state, of the parent nucleus; k represents the intermediate state,of the virtual intermediate odd-odd nucleus; and f represents the final state, of the daughternucleus. The NMEs for the individual GT operators in the numerator of Equation (3.7) areevaluated simply by using (2.44) within one’s chosen nuclear structure model. Ek are theexcitation energies of the intermediate state, as measured relative to the intermediate nucleus’ground state. Ed(J) is simply an energy displacement defined byEd(J) ≡ 12Qββ(J+) + ∆M and, ∆M ≡Mk −Mi (3.8)where Qββ is the Q-value of the double-beta decay, and Mk,Mi are the masses of the interme-diate and parent nuclei respectively. As usual, the Q-value is defined as the sum of the massesof the initial configuration minus the sum of the masses of the final productsQββ = Mi −Mf − 2me (3.9)where here we’ve neglected the mass of the two neutrinos, mν¯e ≈ 0, and of course me is themass of the electron ejected from the corresponding beta decay.A common ambiguity within the literature [48, 71] is how all these energies and massesare defined. For instance, we assume that the parent nucleus is atomically neutral - that is, ithas Z atomic electrons in its electron cloud. The daughter nucleus will therefore also have Zatomic electrons, but it has Z+2 protons since two of the parent neutrons have simultaneouslybeta decayed. Thus, the daughter nucleus is actually atomically charged by +2 units of |e|,and the Mf in Equation (3.9) above should actually be labelled as M2+f instead. Since this iscumbersome, no one writes this; so to make the distinction clear, let’s at least use a superscripta to denote when masses are atomically neutral. That is, we may rewrite Equation (3.9) as233.1. Two-Neutrino Double-Beta DecayQββ = Mai − (Mf + 2me) .= Mai −Maf (3.10)Plugging Equation (3.10) into (3.8) and setting J+f = 0+f gives usEd(0) =12(Mai −Maf ) +Mk −Mai = Mak −me −12(Mai +Maf ) (3.11)where we note that we’ve used Mk = Mak −me, since the virtual intermediate nucleus wouldhave Z atomic electrons and Z+ 1 protons, and hence Mk is atomically charged by +1 units of|e|. We can’t take that electron mass from the beta decay, since they’ve already been absorbedby the Q-value via Equation (3.10). Putting Equation (3.11) into (3.7) with J+f = 0+f yieldsM2ν(0+ → 0+) =∑k〈0+f ||“στ+||1+k 〉〈1+k ||“στ+||0+i 〉[E′k − Eg.s.k ] +Mak −me − 12(Mai +Maf )(3.12)where we’ve redefined the excitation energies to make it computationally explicit that they arerelative to the intermediate nucleus’ ground state, which is valid as long as E′k and Eg.s.k aremeasured relative to the same scale. Also, notice that (although they are labelled by k) bothEg.s.k and Mak don’t actually change in value over the summation; only E′k and 1+k do.An important note is that the closure approximation (see Section C.3.1) is insufficient fortwo-neutrino double-beta decay [72]. This leads to an immediate problem, since obtaining acomplete set of intermediate states is not generally possible. This will induce discrepancies inpredicting values for equations like (3.7) and (3.12), since a truncation will have to be made inthe amount of states used in the summation. More on this will be discussed in Section 7.1.Finally, one may ask, “what are the dominate final state contributions to the 2νββ decayin general, using Equation (3.7)?” This kind of analysis has been conducted in [71] and [73]for 48Ca where they find that the decay 0+ → 2+ gives negligible contributions to the half-life.Hence, higher lying final states will be neglected, and most authors will focus on 0+ → 0+ first,since it generally dominates for double-beta decay.Example 3.2We’ll now set up the M2ν matrix elements for the case of 48Ca double-beta decaying into 48Tivia the 0+ → 0+ mode. The virtual intermediate nucleus will be 48Sc, which has a 6+ groundstate [74, 75]. To obtain the atomic masses required in the denominator of Equation (3.12),we’ll take precision values from [76] for 48Ca, [18] for the electron mass, and [77] otherwise. InEquation (3.11) we obtainme = 0.5109989461 MeVu = 931.4940955 MeVMai.= M(48Ca) = 47.95252276 uMak.= M(48Sc) = 47.952231 u243.2. From Two Neutrinos to NoneMaf.= M(48Ti) = 47.9479463 u=⇒ Mak −me −12(Mai +Maf ) = 1.348701071 MeV (3.13)Notice that had we used Equation (3.8) with a precision measurement of the Q-value, likeQββ = 4.26798 MeV from [78], then we would have arrived to roughly the same Ed as above.Putting Equation (3.13) into Equation (3.12) gives usM2ν(48Ca0+ → 48Ti0+) =∑k〈48Ti0+ ||“στ+||48Sc1+k 〉〈48Sc1+k ||“στ+||48Ca0+〉E′k − E(48Sc6+g.s) + 1.348701071 [MeV](3.14)where the superscripts on the nuclei represents J+, i.e., total angular momentum and parity,as opposed to atomic charge (as used in chemistry notation).3.1.4 Quenching in 2νββGiven the GT operator of single-beta decay requires quenching (see Section 2.3.4), then it isreasonable to presume that M2ν in Equation (3.7) is also quenched. Since the M2ν matrixelements are composed of two copies of the GT operator, all that is done to “quench” double-beta decay is multiply the matrix elements by q2f , as in Equation (2.45). Upon squaring theNME, the quenching will roughly cancel the four copies of gA in Equation (3.3). Thus, whatmany researchers do is model single-beta decay quenching factors within their nuclear structuremodel, and then use these qf for double-beta decay [26], and make half-life predictions viaM2ν −→ q2f M2νThough this procedure seems simple enough, it is important to point out, once again, thatthe community is still at odds with how to interpret the phenomenon of quenching in thecontext of double-beta decay [13]. In fact, the range in appropriate quenching factors for betadecay suggests that it’s the GT NMEs which are the source of quenching, as opposed to arenormalization of gA [49]. Therefore, in this research, we will only present unquenched results.3.2 From Two Neutrinos to NoneNeutrinoless double-beta decay was first proposed by Wendell H. Furry [55], based on therealization that a neutrino could be self-conjugate and the physics of single-beta decay wouldremain unchanged. Hence, the neutrino in 2νββ could annihilate, leaving only two protons andtwo beta particles in the final state, as in Figure 3.3. This process could also happen via theemission of a neutrino at the top weak vertex and an absorption at the bottom, and visa versa.Looking at Figure 3.3, it does not take long to realize that something with 0νββ is not quitekosher within the framework of the Standard Model. To highlight these discrepancies, we’llexplore the following questions:253.2. From Two Neutrinos to NoneFigure 3.3: The nuclear Feynman diagram for neutrinoless double-beta decay, where theneutrino has been given a dashed line to highlight that it is its own anti-particle. In terms ofa nucleus, the decay is read as: (Z,A) −→ (Z+2, A) + 2 e−Would 0νββ confirm that: Lepton number is violated?By definition, neutrinoless double-beta decay begins with two neutrons, and ends with twoprotons and two electrons. We can see that charge is conserved, however the net change inelectron lepton number of Figure 3.3 is non-zero:∆L0ν = Lf − Li = 2− 0 = 2 6= 0This is a significantly disturbing feature of 0νββ: it requires lepton number violation (LNV).Such a violation has never been measured before, and lepton number has been held as a fun-damental conserved quantity in the Standard Model. With this said, many studies [79–82] andsearches [83–85] for lepton number violation have already been conducted.A discovery of LNV would entail that the Standard Model of particle physics is funda-mentally incomplete, since much of it is built upon symmetries and the the conservation oftheir corresponding quantum numbers. In the case of experimentally confirmed LNV, physi-cists would need to develop new theories Beyond the Standard Model (BSM). An exceptionallyexciting aspect of LNV is that it can∗ account for baryogensis in such a way that explainsthe matter/anti-matter asymmetry in the universe [86, 87]. Thus, if found, 0νββ would haveprofound cosmological implications, since it can be thought of as a lepton generating decay.7−→ In conclusion, the answer to the title of this sub-section is: true.∗[86] shows specifically that this is possible without the need for grand unification263.2. From Two Neutrinos to NoneWould 0νββ confirm that: The neutrino is its own anti-particle?From Figure 3.1, it is clear that the only way that the emitted electron anti-neutrinos canannihilate is if the neutrino is its own anti-particle. Similarly, for the neutrino to be emittedfrom the top weak vertex (for instance) of Figure 3.3, and then absorbed at the bottom vertex,it must be able to flip chirality. That is, since the neutrino would be emitted from the topsingle-beta decay as a left-handed particle, to conserve chirality it must be right-handed to beabsorbed in the bottom single-beta decay. As discussed further in the next question below, theonly way this flip in chirality is possible is if the neutrino is its own anti-particle.However, we know from the question above that we must admit 0νββ is a LNV process. Oncea single LNV operator is incorporated into the Lagrangian, this induces many other potentialLNV mechanisms, as described in detail by [13]. Most of these BSM mechanisms involve theexchange of a heavy particle, for instance a meson exchange current (MEC) as in Figure 3.4 seenbelow. Since a MEC involves the exchange a heavy∗ particle, the nucleons involved in Figure 3.4must be “closer” (on average) than those in Figure 3.3, and hence any MEC contribution issuppressed relative to the traditional 0νββ. This dominance of the neutrino exchange diagramcan be shown definitively within the framework of χEFT (briefly introduced in Section 2.2.3).7−→ In conclusion, the answer to the title of this sub-section is: true.Does 0νββ require that: The neutrino has a non-zero mass?Mohapatra’s chapter in Current Aspects of Neutrino Physics gives an insightful overview intothe theory (or lack thereof) behind the neutrino mass [88]. He points out that the initial V −Amodel was developed in order to account for the parity violation† in the neutrino for single-beta decay. This was built on the γ5-invariance of the weak Lagrangian for all fermions, whichwas historically motivated by the presumed masslessness of the neutrino. In conjunction withB−L being an exact symmetry, this initial parity violating theory yielded a precisely masslessneutrino in the Standard Model (to all perturbative orders).Ironically, further investigation into neutrino physics has established that this symmetryin the Standard Model must be broken. As the Cabibbo-Kobayashi-Maskawa (CKM) matrixdictates the ability for quark flavour to change via the weak interaction, the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix predicted an oscillation of neutrino flavour [89, 90]. However,in order for the three mass eigenstates (m1,m2,m3) of the neutrino to mix between its flavoureigenstates (e, µ, τ) during propagation, the neutrino must have a non-zero mass so that itcan experience time in its rest frame.‡ In 2015, Canadian scientist Arthur B. McDonald andTakaaki Kajita of Japan accepted the Nobel prize in physics§ for their pioneering work showing∗relative to, say, the electron mass†specifically, a left-handed chirality preference‡It would take infinite energy to make a massive particle travel at the speed of light, therefore only masslessparticles can travel at c. But, said particles experience no time or distance in their rest frame.§on behalf of the SNO and Super-Kamiokande Collaborations respectively273.2. From Two Neutrinos to NoneFigure 3.4: The nuclear Feynman diagram for 0νββ via a pion exchange; therefore not requiringany information about neutrinos, but still LNV. This example of a MEC contribution turns outto be small relative to Figure 3.3, within the framework of χEFT. The filled circle representssome nuclear contact physics BSM.that neutrino flavours oscillate [91, 92], and hence that the neutrino has a non-zero mass.All these discoveries have solidified the fact that neutrino physics requires theoretical de-velopments BSM. Hence, if 0νββ requires a massive neutrino in order to happen, this aloneis enough to make the statement that 0νββ is BSM. It so happens that the neutrino indeedmust have mass for 0νββ to occur, for similar reasons that it requires mass to oscillate be-tween flavours. We know (from the previous question) that 0νββ requires the neutrino to beits own anti-particle. The only difference between such a neutrino and its anti-particle partneris that one is left-handed and the other is right-handed. This flip in chirality can be achievedby viewing this neutrino in non-trivially different reference frames, and hence it needs mass.∗This is why the discovery of neutrino oscillations has generated much excitement regarding thepossibility of seeing 0νββ.7−→ In conclusion, the answer to the title of this sub-section is: true.Would 0νββ confirm that: The neutrino is a Majorana particle?In 1937 [93], Ettore Majorana proposed a theory of a massive spin-1/2 particle which is its ownanti-particle, in an attempt to rid of the seemingly unsatisfactory interpretation of a Dirac seaof negative energy holes. His equation comes out to∗In relativity, a particle travelling at c in one reference frame travels at c in all reference frames. Thus, if aparticle is massless (and therefore travels at speed c) then it cannot change chirality.283.2. From Two Neutrinos to Nonei/∂ψ −mψc = 0 (3.15)where /∂ ≡ γµ∂µ (using Einstein summation convention) and γµ are the well-known gammamatrices. ψc is the “charge conjugate” of the four-component spinor, ψ, defined byψc ≡ Cψ¯T .= C(ψ†γ0)T = Cγ0ψ∗where C is the charge conjugation matrix. Upon the identification thatψ.= ψc (3.16)it is clear that the “Majorana equation” in (3.15) reproduces the Lorentz invariant physics ofthe Dirac equation for an electrically neutral field(i/∂ −m)ψ = 0Particles that satisfy Equation (3.15) and (3.16) are their own anti-particles, by construction,and are referred to as “Majorana particles.” In summary,Remark 3.1If a particle is Majorana =⇒ it is its own anti-particle.This implication has made the term “Majorana particle” synonymous with “a particle whichis its own anti-particle.” However, one may ask: is the converse of Remark 3.1 necessarily true?That is, could there exist an equation which is Lorentz covariant and reproduces the quantumphysics of a neutral particle being its own anti-particle, but that does not take on the form ofEquation (3.15) and (3.16)? In a related vain, could a particle exist which obeys Equation (3.15),but does not satisfy Condition (3.16), and would it be called “Majorana”? And furthermore, ifwe allow for LNV (as we must for 0νββ), then why should one believe that minimal alterationsto the Standard Model describe 0νββ, as opposed to more drastic revisions to QFT itself? Fromthe understanding of the author, there are currently no satisfactory answers to these questions,likely because the converse of Remark 3.1 has been put in place by convention.From the previous discussion, it is acceptable that we must admit the neutrino is its ownanti-particle in order for 0νββ to occur. And, disregarding the logical non-equivalence ofRemark 3.1, many arguments have been constructed showing that the exchange neutrino mustindeed be Majorana. The most noteworthy of these arguments is dubbed the “Schechter-ValleTheorem” [94]. Many would cite the blackbox developed by [94], reproduced in Figure 3.5 below,as definitively answering the title of this sub-section as: true. In essence, Schechter and Valleshowed that a Majorana mass term of the neutrino is generated, by assuming only that: crossingsymmetry holds in the weak interaction mediated by some “natural” local gauge theory, i.e., acancellation of Figure 3.5 (e) would require an unnatural fine tuning of the physical parameters;the electron and up/down quarks are massive, as they certainly are; and the standard left-handed weak interaction holds, as in Equation (3.17) below [95]. Additionally, [96] and [97]293.2. From Two Neutrinos to NoneFigure 3.5: Reconstructing the blackbox theorem of Schechter and Valle [94]. In order: (a)represents 0νββ, (b) connects the nucleon (or corresponding up and down quark) lines, (c)switches the emitted beta particles with the neutrino via crossing symmetry, (d) rotates thediagram to reestablish a conservation of all the electroweak quantum numbers involved exceptlepton number, and (e) is the neutrino Majorana mass generating diagram. The blackbox in (e)represents all the “natural” BSM physics that can drive 0νββ and ultimately gives the neutrinoits Majorana mass, up to tree level.nearly simultaneously came to the same conclusions as Schechter and Valle, whilst also provingthat 0νββ cannot happen via Dirac neutrinos.(νeLγµeL + uLγµdL)Wµ (3.17)However, recent reviews of the literature have compiled a list of many alternate 0νββmechanisms [48, 53, 70]. In fact, soon after the publication of the Schechter-Valle Theorem,proposals of supersymmetric theories have been made explaining 0νββ which “do not involvethe exchange of Majorana neutrinos.” [98] Furthermore, a general quantitative analysis hasbeen done by [95] which concludes that the blackbox analysis is insufficient, and leaves roomfor other contributions to the neutrino mass. These suggestions highlight the fact that oncewe admit that 0νββ requires physics BSM, we no longer have a conclusive grasp on whatmediates the neutrinoless part of this decay. Could it be, then, that 0νββ can still happenwithout the requirement that the neutrinos are Majorana? And what experiments analyzingneutrinos could definitively establish the Dirac/Majorana/other nature of the neutrino? Theseare questions that the author believes warrants more discussion and scrutiny, despite currentclaims within the community that a positive measurement of 0νββ confirms that neutrinos areindeed Majorana [99–101].With this said, theorists and experimentalists should not fret, since the discovery of 0νββ303.3. Neutrinoless Double-Beta Decaywould still underline the need for significant changes to the Standard Model. But, if we cannotdefinitively say that a positive measurement of 0νββ would prove that the neutrino is Majorana,then what new physics can be concluded from it exactly? As described in the previous questionsabove, 0νββ would require a massive neutrino and LNV, both of which make it a BSM decay.We stress that these discoveries alone would be worthy of a Nobel prize in physics.7−→ In conclusion, the question in the title of this sub-section is: (arguably) misleading.Would 0νββ require quenching?From the discussion in Section 3.1.4, a major concern for researchers studying 0νββ is: givenM2ν are quenched in the nuclear structure model used, should M0ν also be quenched in thatcontext? [53] The answer to this question is still very unclear. However, as we will see, sincethe closure approximation will be used for the M0ν two-body matrix elements (whereas inEquation (3.7) it was not) the operator structure within the matrix elements completely changes.Thusly, perhaps M0ν should not be quenched whatsoever. As stated previously, in this researchwe will only present unquenched results.7−→ Overall, the answer to the title of this sub-section is: (currently) inconclusive.3.3 Neutrinoless Double-Beta DecayNeutrinoless double-beta decay came into the limelight quickly after the proposal of its neutri-noful counterpart, since early theoretical predictions∗ suggested its lifetime was much shorterthan 2νββ [55]. Experimentally, it was expected that the neutrinoless version would be foundfirst [58]. Now we understand that this is not the case, but 0νββ is still an exciting prospect(see Section 3.2 directly above). In particular, due to Equation (3.18) and (3.19) mentioned be-low, 0νββ has far reaching implications for the neutrino mass hierarchy scenarios (degenerate,normal, or inverted) [99].† Overall, neutrinoless double-beta decay indeed represents a massiveundertaking for both the nuclear/particle experimental and theoretical communities.3.3.1 Current Experimental Status for 0νββThe hunt for the first direct measurement of 0νββ is fast-paced and constantly evolving [53, 58].The same nuclei which are candidates for 2νββ, as listed in Section 3.1.1, are also candidatesfor 0νββ. Hence, the main differences (detector type, location, sensitivity, etc) between double-beta decay experiments are decided by what nuclear source is chosen. To distinguish betweena confirmation of 2νββ and 0νββ, one uses the detected Q-value of the electron emission, seefor instance Figure 10 of [53]. To date, 0νββ has never been definitively confirmed.∗At this time, parity violation had not been discovered, which significantly altered the T1/2 estimates.†For 0νββ, experimentalists are currently on the edge of probing the inverted region [102].313.3. Neutrinoless Double-Beta DecayA curious point to note, though, is that in 2001, after extensive searches [103], 0νββwas claimed to have been found in 76Ge by a subset of the Heidelberg-Moscow Collabora-tion [104]. This claim generated excitement, but it was subsequently scrutinized to the pointthat the current community does not accept its results [53, 105, 106]. In fact, many membersof the collaboration came out to rebuke the prior publication [107], since they felt that thestatistical analysis was incomplete, despite the persistence of the original proponents in thegroup [108]. Furthermore, many experiments like [109] have not been able to reproduce thesupposed Heidelberg-Moscow measurement. This “Heidelberg-Moscow controversy” highlightsthe need for continuing experimental progress in the search for 0νββ; both so that detectortechnologies can be advanced, and theoretical methods can be benchmarked against data.Since the existence of the neutrinoless mode has not been established, many modern exper-iments are underway, such as [62, 100, 102, 110–112] among others [53]. But how do experi-mentalists know what sensitivities are required of their apparatus, in order to measure 0νββfor their nuclei of interest? The answer is that many of them take estimates from phenomeno-logical studies. However, it is well known that the spread in NME calculations from competingmethods is inconsistent, and theoretical uncertainties are currently out of reach [13]. So, inconjunction with the importance of experimental data in theoretical studies, in turn, theoreti-cians currently carry the responsibility to make accurate predictions of 0νββ NMEs for thesake of experimentalists. In this dissertation, we’ll calculate these NMEs using a fully ab initioapproach called IM-SRG (see Chapter 5).3.3.2 0νββ Half-Life FormulaTo move forward, we’ll now make the following assumption:Assumption 1The 0νββ mode occurs via the Majorana nature of neutrinos.The original analysis of this mechanism was explored in [65]. An in-depth review of the V −Aparticle physics and QFT involved can be found in [66], where they derive the half-life formula[T 0ν1/2 ]−1 = G0ν |M0ν |2Ç〈mββ〉meå2(3.18)where G0ν is known as the 0νββ “phase factor,” which incorporates the interface betweenparticle and nuclear physics, and will be discussed more below; M0ν is the NME, which incor-porates nuclear structure theory (see Section 3.3.3 below); Equation (3.18) indeed obeys (F11),as expected; me is the electron mass, and the “(effective) Majorana neutrino mass” is〈mββ〉 =∣∣∣∣∣ 3∑k= 1mkU2ek∣∣∣∣∣ (3.19)where Uek are the first (aka, electron neutrino) row elements of the PMNS matrix, and mk are323.3. Neutrinoless Double-Beta Decaythe corresponding mass eigenstate neutrino masses. Equation (3.19) ties in the BSM physicsinvolved with 0νββ, hence Equation (3.18) bridges: QFT, BSM physics, and nuclear theory.For us, this half-life formula is the prime motivator for the research conducted in this thesisproject. To illuminate this, it is more appropriate to rewrite Equation (3.18) as〈mββ〉 = me√G0ν T 0ν1/2 |M0ν |(3.20)Now we can see the rub! If the experimental physics community can measure the PMNS matrixprecisely,∗ and make a measurement of the neutrinoless double-beta decay half-life T 0ν1/2 (seeSection 3.3.1 above); and if the theoretical community can determine the value of the phasefactor, and the NME; then one can determine the absolute mass scale of the neutrino. ViewingEquation (3.18) in the form of (3.20) is indicative of the model that Figure 3.5 (e) above actsas a Majorana neutrino mass contribution, via 0νββ. Furthermore, since the decay is expectedto be very rare and so T 0ν1/2 is large, this Majorana theory of the neutrino mass could explainwhy the neutrino has such a small mass [88, 94].Making an accurate model of the NME is what we will focus on for the rest of this disser-tation. Before we move on, however, it is important to stress, once again, that all this relies onthe assumption that neutrinos are Majorana! As discussed in Section 3.2 above, this need notbe the case. It could very well be that 0νββ exists and is dominated by a BSM mechanism notinvolving Majorana neutrinos, and the determination of Equation (3.20) is simply a soft predic-tion. Thus, it is important that the experimental community does not come to the immediateconclusion: a positive measurement of 0νββ =⇒ the neutrino is Majorana. Importantly, ifit is the case that Equation (3.19) can be established numerically by alternative experiments,and combining this number into Equation (3.18) happens to match precisely with a confirmedmeasurement of the 0νββ half-life, only then it is likely that the neutrino is Majorana. Thisbold conclusion is reliant on having good 0νββ NMEs!0νββ Phase FactorWe’ll treat the neutrinoless phase factor in the same manner that we did for the two-neutrinocase (see Section 3.1.2) - as a black box. Cowell was one of the first to notice that the phase factorhad previously been treated inconsistently in the literature [114], which has now set a presidentfor their modern calculation [67, 68]. For the sake of comparison† with Equation (3.4), [67]reports the 0νββ phase factor for 48Ca asG0ν(48Ca) = 2.481× 10−14 yr−1which is generally agreed upon [69].∗which is being done continually by neutrino oscillation experiments [113]†Note that this comparison is contrived, since the structure of Equation (3.3) is different from that in (3.18),and the NMEs in either scenario are not necessarily comparable.333.3. Neutrinoless Double-Beta Decay3.3.3 Nuclear Matrix ElementsThe NME, M0ν , from Equation (3.18) may be decomposed into three dominate parts [115]M0ν = M0νGT −ÇgVgAå2M0νF +M0νT (3.21)This is very similar to M2ν in Equation (3.6), except with an additional “Tensor” (T) part. Thiscomponent has been induced specifically by the physics of neutrinoless double-beta decay. Thederivation of this part can be found in [66] up to their Equation (3.5.7). There they consideredtwo additional effects, being: alternate electron P -wave effects, and nucleon recoil. However,as is common with current trends in the literature, we will neglect these effects [69].Note that in the notational convention of Equation (3.21), the GT, F, and T components ofM0ν may each carry a sign (they have not been written as magnitudes). It will be found thatM0νGT comes out positive, M0νF comes out negative, andM0νT comes out negative (see Section 7.2).Hence, the negative in front of M0νF in Equation (3.21) will tend to cancel, whereas the inducednegative from M0νT will decrease the total sum. They are also written in order of largest tosmallest in magnitude from left to right. In other words, it is common to find that|M0νGT| > |M0νF | > |M0νT |therefore many authors will simply neglect the Tensor part [72, 116, 117]. In this research,however, all three components will be analyzed, for completeness.Now, how can we calculate each component of Equation (3.21)? First note that the NMEsare written in shorthand such thatM0να.= 〈f |M̂ 0να |i〉 (3.22)where, in the RHS, |i〉 is the state of the initial nuclei in the neutrinoless double-beta decay,likewise |f〉 is the state of the final nuclei, and M̂ 0να is the scalar spherical tensor operatorrepresenting 0νββ for α = GT, F, or T. The reason these spherical tensors are scalar is simple.Let’s consider the Fermi component; from Section 2.3.2 we know this happens for a single-betadecay when the neutrino and beta particle are anti-aligned, and hence spin couples to 0. Fortwo beta decays, then, total angular momentum must couple to 0 as well, and we know thatspherical tensors can only change angular momentum in steps of their rank.∗ Hence, due to theselection rules of 0νββ,† its spherical tensors are rank 0. To obtain values for Equation (3.22), wetherefore need to find the scalar TBMEs for 0νββ and the corresponding TBTDs, in accordancewith Equation (C33). The latter task will be dealt with using nutbar (see Section 6.3), andthe former will be discussed in full detail in Chapter 4.∗this can be seen clearly when considering Equation (B14), for instance†Also note that only spherical tensors of the same rank may be summed together, as in Equation (3.21).343.3. Neutrinoless Double-Beta Decay3.3.4 The Closure ApproximationNotice that, in the paragraph above, we referred to Equation (C33) as opposed to (C30) tocompute values for (3.22). This is because we will employee the so-called closure approximationfor 0νββ. [118] showed that the difference between using the closure energy (see Equation (4.9)and Assumption 2 below) for 0νββ, and using a non-closure method, is about 10% for thefinal NMEs. Furthermore, it is well known that varying the closure energy yields only smalldifferences in the final values. With this knowledge, it is justified to compare NMEs which usea different value for the closure energy - and many authors adhere to different conventions.∗Establishing which closure energy is optimal for the nuclei undergoing 0νββ decay will be leftfor future research.∗For example, 0.5 MeV or 7.72 MeV are both acceptable values for the 48Ca closure energy.35Chapter 4M0ν Two-Body Matrix ElementsIn this chapter, we will present the mathematical construction of the M0ν TBMEs, in full detail.First, recall from Equation (3.21) and (3.22) in Section 3.3.3 above thatM0ν = M0νGT −ÇgVgAå2M0νF +M0νT (3.21)where,M0να.= 〈f |M̂ 0να |i〉, α = GT, F, T (3.22)The individual NMEs of Equation (3.22) may be calculated using Equation (C33) and Equa-tion (B9). That is, with spherical tensor rank L = 0,∗ we have〈f ||M̂ 0να ||i〉 =∑abcd∑J,J ′〈a b :J ||M̂ 0να ||c d :J ′〉 〈ηfJf ||î[cˆ†a⊗cˆ†b]J ⊗ [c˜c⊗c˜d]J ′ó0||ηiJi〉 (4.1)To calculate the TBTDs (see Section C.3) in Equation (4.1), we will use a code called nutbar(see Section 6.3). Before we may compute these NMEs fully, an essential component are theTBMEs, which will be denoted by the shorthandMαabcd ≡ 〈a b :J ||M̂ 0να ||c d :J ′〉, α = GT, F, T (4.2)where a, b, c, d represent all the relevant quantum numbers to describe the two-body states inJ-scheme with jj-coupling (see Section A.1). That is, we have ∆(ja jb :J) and ∆(jc jd :J′).4.1 Deconstructing M0νWe begin our long dive into the TBMEs of 0νββ by specifying what each scalar sphericaltensor operator, M̂ 0να , in Equation (4.2) represents. From the involved derivations of [64, 66],we obtainM̂ 0νGT = HGT(r12, Ek) yGT(r̂12)“σ1◦“σ2 τ+1 τ+2 (4.3)M̂ 0νF = HF(r12, Ek) yF(r̂12) τ+1 τ+2 (4.4)M̂ 0νT = HT(r12, Ek) yT(r̂12)“S12 τ+1 τ+2 (4.5)where operators denoted by a subscript 1 (or 2) are meant to act on the first (or second) particle∗the spherical tensors for 0νββ must be scalar, as explained in Section 3.3.3364.1. Deconstructing M0νof a two-body state, and the y(r̂)’s carry the obviously required spherical harmonic dependance,as in Equation (F7), viayα(r̂12) =√4piY00(r̂12) = 1, α = GT, F»24pi5 Y2m(r̂12), α = T(4.6)The isospin operators, τ+, simply act to transform a nucleon from a neutron state to a protonstate, in accordance with the two beta decays. For the Tensor component, we have introducedthe “tensor spin operator,” which is defined by“S12 ≡ 3(“σ1◦ r̂12)(“σ2◦ r̂12)− “σ1◦“σ2 −→ [“σ1⊗ “σ2]2 (4.7)where,⇀r12 ≡ ⇀r1 −⇀r2, r12 .= ||⇀r12||, r̂12 ≡⇀r12r12(4.8)The relative coordinate, ⇀r12, tracks particle 1 and 2, before second quantization. That is, for thefinal TBMEs, r12 will be integrated over with respect to the appropriate probability amplitudesand spherical harmonics. It is in conjunction with this spacial integration that the mappingon the RHS of Equation (4.7) occurs; that is, the tensor spin operator, “S12, becomes a rank2 spherical tensor product between two Pauli operators. Likewise, the spherical harmonics ofEquation (4.6) will also be treated as spherical tensors, as we will see in Equation (4.21) below.The Hα’s in Equations (4.3) to (4.5) are known as “neutrino potentials,” which are integralsover the neutrino exchange momentum, q, and they depend on the energy of the intermediatestate k of the neutrinoless double-beta decay. In their most general form, they are written asHα(r12, Ek) =2Rpi∫ ∞0dqq2fα(q ·r12)hα(q2)ω[ω + Ek + (Ei + Ef )/2](4.9)where r12 is the same as in Equation (4.8), R is the nuclear radius given by (2.29), ω is theneutrino energy given by ω =»q2 +m2ν , and the functions, fα(q ·r12), are defined simply asfα(q ·r12) =j0(q ·r12), α = GT, Fj2(q ·r12), α = T (4.10)where the j(x)’s are spherical Bessel’s functions as in Equation (F5). The functions, hα(q2),are the finite-size“(nucleon) form factors,” which are listed belowhF(q2) ≡ g2V (q2)g2V,0(4.11)hGT(q2) ≡ 1g2A,0ñg2A(q2)− gA(q2)gP (q2) q23mp+g2P (q2) q412m2p+g2M (q2) q26m2pô.=g2A(q2)g2A,0ñ1− 23q2q2 +m2pi+13Çq2q2 +m2piå2ô+16g2M (q2)g2A,0q2m2p(4.12)374.1. Deconstructing M0νhT(q2) ≡ 1g2A,0ñgA(q2)gP (q2) q23mp− g2P (q2) q412m2p+g2M (q2) q212m2pô.=g2A(q2)g2A,0ñ23q2q2 +m2pi− 13Çq2q2 +m2piå2ô+112g2M (q2)g2A,0q2m2p(4.13)with the vector (V ), axial-vector (A), pion-exchange propagator (P ), and magnetic moment (M)“(regularized) g-factors” are as followsgV (q2) ≡ gV,0Ä1 + q2/Λ2Vä2 , gA(q2) ≡ gA,0Ä1 + q2/Λ2Aä2gP (q2) ≡ 2mp gA(q2)q2 +m2pi, gM (q2) ≡ǵp − µnµNågV (q2)(4.14)and the proton mass mp, pion mass mpi, “(finite-size) cutoffs” Λ, and magnetic momentµp, µn, µN values will be specified when the NMEs are calculated.There are two glaring points to make about the neutrino potentials before we move on.First, we will impose some approximations on the general form of Equation (4.9). It’s clearthat the neutrino mass is small [119], and hence: ω ≈ q, by taking q  mν . Next, to guaranteethat we may employ Equation (4.1), we must use the closure approximation (see Section C.3.1and 3.3.4) so that M̂ 0να is independent of k via:Assumption 2: “The Closure Energy”In order to make the 0νββ operators of Equations (4.3) to (4.5) independent of the intermediatenuclear state, we impose the closure approximation by introducing the “closure energy”Ek + (Ei + Ef )/2 ≈ E cl0 (4.15)For now, we’ll assume that this approximation holds, since [118] showed that varying the closureenergy only introduces small changes to the overall NME.Choosing the best closure energy seems to depend on the parent nucleus of the neutrino-less double-beta decay [118]; more on this should be explored in future research. PuttingEquation (4.15) and ω ≈ q into (4.9) gives us the more computationally manageable neutrinopotentialsHα(r12, Ek) −→ Hα(r12) = 2Rpi∫ ∞0dqq ·fα(q ·r12)hα(q2)q + E cl0, α = GT, F, T (4.16)It is evident that Equation (4.16) above behaves as a Yukawa potential in r12 [66].Secondly, the nuclear radius, R, has been included to make the neutrino potentials unitless,but this yields some confusion within the nomenclature. That is, the physics of neutrinolessdouble-beta decay itself should, intuitively, be independent of the nucleus of interest. But, withthe inclusion of R, this will make the 0νββ NMEs nuclei-dependent. This makes combining the384.2. Tensor TBMEsNME with phase factors∗ of Equation (3.18) more convenient. Thus, even though M̂ 0να shouldonly be 0νββ-dependant, and Mαabcd of Equation (4.2) are nuclear structure dependant; due tothe current literature conventions, both are nuclei-dependant.One final point to make in deconstructing M0ν is that we will deal with isospin in the “PN”formalism. That is, the projected isospin† of our nucleon states will be well-defined. Therefore,we’ll separate out the isospin component from the operators, M̂ 0να , before any coupling orreduction of the TBMEs in Equation (4.2). So each operator structure in Equations (4.3)to (4.5) will yield a prefactor of〈ta, tb|τ+1 τ+2 |tc, tb〉 = 〈ta|tc + 1〉〈tb|td + 1〉 = δta,tc+1 · δtb,td+1 ≡ δ isoac,bd (4.17)When the Kronecker-delta in Equation (4.17) appears, it should be understood that the isospinquantum numbers have been separated from a, b, c, d for (4.2). It can be easily seen that,physically, this Kronecker-delta represents the transformation of two neutrons to two protons,in accordance with the double-beta decay.4.2 Tensor TBMEsWe’ll begin with the most complicated structure; that of the Tensor component of the M0νTBMEs. Using Equations (4.2), (4.5), (4.17), and (4.16) we getMTabcd = δisoac,bd〈a b :J ||HT(r12)yT(r̂12)“S12||c d :J ′〉= δ isoac,bd2Rpi∫ ∞0dqq ·hT(q2)q + E cl0〈a b :J ||fT(q ·r12)yT(r̂12)“S12||c d :J ′〉 (4.18)Note that in Equation (4.18) we retained fT and yT within the reduced matrix elements, sincethey are coordinate-dependant and therefore must be integrated over via the product of two-body states. We pulled the integral with respect to the neutrino exchange momentum, q, outsideof the states since - due to the structure of the form factors in Equations (4.11) to (4.13) - wewill need to handle the integration numerically. This complicates matters, in that each reducedmatrix element will need to be integrated individually (see Section 6.4 for more details); butit balances out, since the integration with respect to the states can be done analytically (seeSection 4.5 below). To simplify the notation, let’s define the augmented operator“QT ≡ fT(q ·r12)yT(r̂12)“S12 (4.19)which we labelled as a “Q” to remind us that it has q-dependence.Now we must deal with the elements 〈a b :J ||“QT||c d :J ′〉, but how? The trick is to notice thatwe must transform from jj-coupling to ls-coupling, as to clarify the operator’s action on the spinquantum numbers. From that point on, the Talmi-Moshinsky transformation of Section 2.1.2can be used to resolve the relative radial coordinate dependence. Using Equation (A34) gives∗which are nuclei-dependant, by construction†proton with tz = +1/2 and neutron with tz = −1/2, see Section 2.2.1394.2. Tensor TBMEs||c d :J ′〉 .= ||(lc 12)jc, (ld 12)jd:J ′〉 =∑L′S′lc ld L′12 12 S′jc jd J′ ||(lcld)L′, (12 12)S′:J ′〉and likewise for 〈a b :J ||. Using the above re-coupling, and Equation (4.19), we may write〈a b :J ||“QT||c d :J ′〉 = ∑LL′∑SS′la lb L12 12 Sja jb Jlc ld L′12 12 S′jc jd J′ 〈LS:J ||“QT||L′ S′:J ′〉 (4.20)In this form it is clear that the spherical harmonics will act on the total orbital angular mo-mentum quantum numbers of our two-body states, whereas the “S12 will act on the total spin.We may now treat yT within “QT as carrying spherical tensor harmonic structure. Hence,through Equations (4.6), (4.10), and (4.7), we take the operator inside of (4.20) as a scalarproduct (see Definition B.5) between the two rank 2 spherical tensors, such that“QT = fT(q ·r12)yT(r̂12)“S12 −→  24pi5[j2(q ·r12)Ŷ 2(r̂12)]2 ◦ [“σ1⊗ “σ2]2 (4.21)Then using Equation (B16) of Corollary B.2.1 with (4.21) yields〈LS:J ||“QT||L′ S′:J ′〉 = δJJ ′(−1)S+J+L′ ÛJ ®L S JS′ L′ 2´× 24pi5〈L||j2(q ·r12)Ŷ 2(r̂12)||L′〉〈S||[“σ1⊗ “σ2]2||S′〉 (4.22)From Equation (B15) of Theorem B.2, we see that〈S||[“σ1⊗ “σ2]2||S′〉 .= 〈12 12 :S||[“σ1⊗ “σ2]2||12 12 :S′〉 = ÛS Û2 ÛS′12 12 S12 12 S′1 1 2 〈12 ||“σ1||12〉〈12 ||“σ2||12〉= 6√5 ÛS ÛS′12 12 S12 12 S′1 1 2 (4.23)where we used Equation (2.42) for the vector spherical tensor Pauli operators “σ1 and “σ2, eachof rank 1. It’s clear that S = 0 or 1 since ∆(1212 : S), and likewise for S′. Calculating the9j-symbol from Equation (4.23) over this range of values for S and S′ gives1212 S1212 S′1 1 2 ={1/9, S = S′ = 10, o.w.(4.24)and so, plugging Equation (4.24) into (4.23) with Û1 = √3, yields〈S||[“σ1⊗ “σ2]2||S′〉 = 2√5SS′ = {2√5, S = S′ = 10, o.w.(4.25)Equation (4.25) may seem innocuous, but it severely restricts the summation over S, S′ in404.2. Tensor TBMEsEquation (4.20). That is, since the 9j-symbol is zero unless S = S′ = 1, we only need toconsider these values and then sum over L,L′, which are coupled as ∆(la lb :L) and ∆(lc ld :L′)respectively. So, putting Equation (4.25) into (4.22), and the result into (4.20) gets〈a b :J ||“QT||c d :J ′〉 =δJJ ′ ÛJ ∑LL′2√5(−1)1+J+L′®L 1 J1 L′ 2´la lb L12 12 1ja jb Jlc ld L′12 12 1jc jd J 24pi5〈L||j2(q ·r12)Ŷ 2(r̂12)||L′〉(4.26)From here we note that the coordinate, ⇀r12 as defined in Equation (4.8), is measured radially,relative to particles 1 and 2. Thus, since L and L′ are set in the lab frame, we must transforminto the relative/CoM coordinates via the Talmi-Moshinsky transformation of Theorem 2.1 -in order to clarify the action of the rank 2 spherical tensor operator j2(q ·r12)Ŷ 2(r̂12) on thestates. Using Equation (2.11), we obtain〈la, lb:L||j2(q ·r12)Ŷ 2(r̂12)||lc, ld:L′〉 = ÛLÛL′ ∑nrlr,n′rl′rNΛDabD′cd(−1)L′+lr+Λ®L lr Λl′r L′ 2´× 〈nr lr||j2(q ·r12)Ŷ 2(r̂12)||n′r l′r〉(4.27)with the Talmi-Moshinsky bracketsDab = 〈nr lr, NΛ:L|na la, nb lb:L〉 and, D′cd ≡ 〈n′r l′r, NΛ:L′|nc lc, nd ld:L′〉 (4.28)and the corresponding CTML from Equation (D21). A tricky caveat to point out about Equa-tion (4.27) is that the relative/CoM coordinates used to construct the brackets are built as⇀r ≡ 1√2Ä⇀r1 −⇀r2äand,⇀R ≡ 1√2Ä⇀r1 +⇀r2ä(2.8)Comparing this with Equation (4.8) introduces a factor of√2, which must be accounted for⇀r12 =√2⇀r =⇒ r12 =√2r and, r̂12 = r̂ (4.29)Finally, we can separate out the action of the spherical Bessel’s function and the sphericalharmonics using Equation (B14) of Example B.3. This with Equation (4.29) gives us〈nr lr||j2(q ·r12)Ŷ 2(r̂12)||n′r l′r〉 .= 〈nr lr||j2(√2qr)Ŷ 2(r̂)||n′r l′r〉 = 54pi(lr 0 2 0 | l′r 0) Ûlr 〈nr lr|j2(√2qr)|n′r l′r〉 (4.30)Plugging Equation (4.30) into (4.27), and the result into Equation (4.26) yields414.3. Gamow-Teller TBMEs〈a b :J ||“QT||c d :J ′〉 = δJJ ′ ÛJ ∑LL′2√5 (−1)1+J+L′ ÛLÛL′®L 1 J1 L′ 2´×la lb L12 12 1ja jb Jlc ld L′12 12 1jc jd J×∑nrlr,n′rl′rNΛDabD′cd(−1)L′+lr+Λ®L lr Λl′r L′ 2´×√6 (lr 0 2 0 | l′r 0) Ûlr 〈nr lr|j2(√2qr)|n′r l′r〉(4.31)where “QT is defined in Equation (4.19), the sum over angular momentum runs as ∆(la lb :L)and ∆(lc ld :L′), the Talmi-Moshinsky brackets Dab and D′cd are defined in (4.28), and we mayadapt Equation (D21) for our CTML. We’ll call the integrals over dr “relative Bessel’s matrixelements” (RBMEs), which can be calculated analytically; see Section 4.5 below.4.3 Gamow-Teller TBMEsThe calculation of the Gamow-Teller component of the M0ν TBMEs is similar to that of theTensor component done in the section above, but it is somewhat simpler. Using Equations (4.2),(4.3), (4.17), and (4.16), we may writeMGTabcd = δisoac,bd2Rpi∫ ∞0dqq ·hGT(q2)q + E cl0〈a b :J ||“QGT||c d :J ′〉 (4.32)where, “QGT ≡ fGT(q ·r12) yGT(r̂12)“σ1◦“σ2 (4.33)and fGT, yGT, and hGT are defined in Equations (4.10), (4.6), and (4.12) respectively. As wedid with the Tensor component, we’ll transform from jj-coupling to ls-coupling via〈a b :J ||“QGT||c d :J ′〉 = ∑LL′∑SS′la lb L12 12 Sja jb Jlc ld L′12 12 S′jc jd J′ 〈LS:J ||“QGT||L′ S′:J ′〉 (4.34)Now we can promote the “QGT to spherical tensor operator status by taking the scalar productbetween two rank 0 operators as so“QGT = fGT(q ·r12) yGT(r̂12)“σ1◦“σ2 −→ √4pi[j0(q ·r12)Ŷ 0(r̂12)]0 ◦ [“σ1◦“σ2]0 (4.35)〈LS:J ||“QGT||L′ S′:J ′〉 = δLL′δSS′δJJ ′ ÛJÛS ÛL√4pi〈L||j0(q ·r12)Ŷ 0(r̂12)||L′〉 〈S||“σ1◦“σ2||S′〉 (4.36)To obtain (4.36), we plugged Equation (4.35) into (B21) of Corollary B.2.2. Notice that we’veomitted the triangular delta, δ(LS:J), since it is clear from the context of ls-coupling.We could construct the analogous equation to (4.23) to find 〈S||“σ1◦“σ2||S′〉, or we could makeuse of a commonly employed trick from many-body quantum mechanics. From Equation (2.38),424.3. Gamow-Teller TBMEsin units of } = 1, it’s clear that〈S|“σ1◦“σ2|S′〉 = 4〈S|“S1◦“S2|S′〉 (4.37)Additionally, we can make use of knowing the eigenvalues of “S2 via the following manipulation,“S2 = (“S1 + “S2)2 = “S21 + 2“S1◦“S2 + “S22 =⇒ “S1◦“S2 = 12î“S2 − “S21 + “S22 óand hence the eigenvalues are“S1◦“S2 −→ 12îs(s+ 1)− s1(s1 + 1)− s2(s2 + 1)ó(4.38)For two spin-1/2 particles∗ the total spin (labelled as s here) is coupled as∆(s1 =12 , s2 =12 :s) =⇒ s = 0, 1 (4.39)Plugging Equation (4.39) into (4.38), and the result into (4.37) yields〈S|“σ1◦“σ2|S′〉 = δSS′ ·σσ(S)where, σσ(S) ≡ 2S(S + 1)− 3 ={1, S = 1−3, S = 0(4.40)In analogy with Example B.2, using the Wigner-Eckhart Theorem (Theorem B.1), we can showthat the reduced version of Equation (4.40) is〈S||“σ1◦“σ2||S′〉 = δSS′ ÛS ·σσ(S) (4.41)Putting Equation (4.41) into (4.36) gives us〈LS:J ||“QGT||L′ S′:J ′〉 = δLL′δSS′δJJ ′ ÛJ σσ(S)√4piÛL 〈L||j0(q ·r12)Ŷ 0(r̂12)||L′〉 (4.42)As opposed to the case with the Tensor component, now we have a spherical tensor harmonic ofrank 0 within our operator, and thus we can use the much simpler form of the Talmi-Moshinskytransformation presented in Corollary 2.1.1. Using Equation (2.20) gives〈la, lb:L||j0(q ·r12)Ŷ 0(r̂12)||lc, ld:L′〉 = δLL′ ÛL ∑nrlr,n′rNΛDabDn′rcdÛl−1r 〈nr lr||j0(q ·r12)Ŷ 0(r̂12)||n′r lr〉(4.43)with the Talmi-Moshinsky bracketsDab = 〈nr lr, NΛ:L|na la, nb lb:L〉 and, Dn′rcd ≡ 〈n′r lr, NΛ:L|nc lc, nd ld:L〉 (4.44)and the corresponding CTML from Equation (D16). Once again, using Equation (B14)〈nr lr||j0(q ·r12)Ŷ 0(r̂12)||n′r lr〉 =Ûlr√4pi(lr 0 0 0 | lr 0) 〈nr lr|j0(√2qr)|n′r lr〉 (4.45)where the CG coefficient goes to unity via Equation (A14), and we’ve again corrected for thediscrepancy between the radial coordinate definitions for the Talmi-Moshinsky brackets using∗such as our nucleons434.4. Fermi TBMEsEquation (4.29). Inserting Equation (4.45) into (4.43), and the result into (4.42) and (4.34),whilst cancelling the appropriate hat factors and also summing over L′ and S′ produces〈a b :J ||“QGT||c d :J ′〉 = δJJ ′ ÛJ ∑S=0,1σσ(S)s∑Lla lb L12 12 Sja jb Jlc ld L12 12 Sjc jd J×∑nrlr,n′rNΛDabDn′rcd 〈nr lr|j0(√2qr)|n′r lr〉{ (4.46)where “QGT is defined in Equation (4.33), the spin eigenvalues labelled by σσ are given in (4.40),the sum over angular momentum runs as ∆(la lb :L) with the restriction that ∆(lc ld :L),∗ theTalmi-Moshinsky brackets Dab and Dn′rcd are defined in (4.44), and we may adapt Equation (D16)for our CTML. The RBMEs can be calculated analytically; see Section 4.5 below.4.4 Fermi TBMEsThe only part that differs between the operator structure of the Fermi component and theGamow-Teller component of M0ν is the presence of “σ1◦“σ2. Therefore, to get the Fermi com-ponent, we may adapt Equation (4.35) and Equation (4.36) by replacing “σ1◦“σ2 → 1,〈LS:J ||“QF||L′ S′:J ′〉 = δLL′δSS′δJJ ′ ÛJÛS ÛL√4pi〈L||j0(q ·r12)Ŷ 0(r̂12)||L′〉 〈S||1||S′〉 (4.47)whereMFabcd = δisoac,bd2Rpi∫ ∞0dqq ·hF(q2)q + E cl0〈a b :J ||“QF||c d :J ′〉 (4.48)where, “QF ≡ fF(q ·r12) yF(r̂12)1 (4.49)and fF, yF, and hF are defined in Equations (4.10), (4.6), and (4.12) respectively. From Equa-tion (B11), we see that〈S||1||S′〉 = δSS′ ÛSso plugging this into Equation (4.47), and repeating the identical math of Section 4.3 directlyabove yields the TBMEs of the Fermi component for M0ν∗for computational efficiency, where the 9j-symbols yield zero444.5. Relative Bessel’s Matrix Elements〈a b :J ||“QF||c d :J ′〉 = δJJ ′ ÛJ ∑L,Ssla lb L12 12 Sja jb Jlc ld L12 12 Sjc jd J×∑nrlr,n′rNΛDabDn′rcd 〈nr lr|j0(√2qr)|n′r lr〉{ (4.50)where the sum over spin runs as ∆(1212 :S), the sum over angular momentum runs as ∆(la lb :L)with the restriction that ∆(lc ld : L), the Talmi-Moshinsky brackets Dab and Dn′rcd are definedin (4.44), and we may adapt Equation (D16) for our CTML. Below, we’ll calculate the RBMEsusing an analytical integration.4.5 Relative Bessel’s Matrix ElementsIn this section we will follow the techniques laid out in [118] to find the analytic form of therelative Bessel’s matrix elements (RBMEs) appearing in Equations (4.31), (4.46), and (4.50)〈nr lr|jρ(√2qr)|n′r l′r〉 .=∫ ∞0dr r2Rnrlr(r)jρ(√2qr)Rn′rl′r(r) (4.51)where q is the momentum transfer of the Majorana neutrinos, jρ are the spherical Besselfunctions of order ρ as in Equation (F5), and Rnl(r) are the radial harmonic oscillator wave-functions given by Equation (2.3). Notice that we’ve used the r as in Equation (4.29), in orderto match the radial oscillator wave-functions with our construction of the Tamli-Moshinskybrackets. With 20/20-hindsight, we will evaluate the (more grotesque) integrals〈nr lr|W (q, r)|n′r l′r〉 .=∫ ∞0dr r2Rnrlr(r)rue−wr2jρ(sqr)Rn′rl′r(r)where, W (q, r) ≡ rue−wr2jρ(sqr) with, u,w, s ∈ R(4.52)and u= 0 =w, s=√2 retrieves our original integral. To solve Equation (4.52), let’s first makethe notational simplification nr, lr → n, l, and then plug the radial wave-functions〈n l|W (q, r)|n′l′〉 =∫ ∞0dr r2+ue−wr2jρ(sqr)NnlNn′l′(rb)l+l′e−r2b2 Ll+ 12n (r2/b2)Ll′+ 12n′ (r2/b2)= NnlNn′l′(1b)l+l′ ∫ ∞0dr rl+l′+2+ue−(1b2+w)r2× jρ(sqr)Ll+12n (r2/b2)Ll′+ 12n′ (r2/b2)(4.53)The L’s in Equation (4.53) are Laguerre polynomials, as in Equation (F3). Hence, we canuse the well known fact that, for some α ∈ N0 and β ∈ R, we can expand them as followsLβα(x) =α∑k= 0Çα+ βα− kå(−x)kk!(4.54)454.5. Relative Bessel’s Matrix Elementswhere the coefficient in the sum is a generalized binomial coefficient, as defined in Equation (F2).Using the expansion in Equation (4.54) for both the Laguerre polynomials in (4.53) gives〈n l|W (q, r)|n′l′〉 = NnlNn′l′(1b)l+l′ ∫ ∞0dr rl+l′+2+ue−(1b2+w)r2jρ(sqr)ñ×n∑k= 0Çn+ l + 12n− kå(−1)kk!(r2b2)k×n′∑k′= 0Çn′ + l′ + 12n′ − k′å(−1)k′k′!(r2b2)k′ô= NnlNn′l′∑kk′(−1)k+k′k! k′!b(−l−l′−2k−2k′)Çn+ l + 12n− kåÇn′ + l′ + 12n′ − k′å×∫ ∞0dr r(l+l′+2k+2k′+2+u)e−(1b2+w)r2jρ(sqr)(4.55)At this point, the utility of Equation (E16) derived in Appendix E becomes very clear! Wesimply make the notational identifications as followsξ ≡ l + l′ + 2k + 2k′ (4.56)d ≡ ξ + 2 + u =⇒ κ = ξ − ρ+ u2∈ N0ν ≡ 1b2+ w =⇒ z .= s2q24( 1b2+ w)(4.57)Rewriting Equation (4.55) with these identification yields〈n l|W (q, r)|n′l′〉 = NnlNn′l′∑kk′(−1)k+k′k! k′!b−ξÇn+ l + 12n− kåÇn′ + l′ + 12n′ − k′å ∫ ∞0dr rde−νr2jρ(sqr)Applying Equation (E16) to the equation above, and staying consistent with the identificationsin Equation (4.56) and (4.57), gives us〈n l|W (q, r)|n′l′〉 = NnlNn′l′∑kk′(−1)k+k′k! k′!b−ξÇn+ l + 12n− kåÇn′ + l′ + 12n′ − k′å×( 1b2+ w)−d−12√pi4κ!zρ2 e−zLρ+12κ (z)(4.58)We would like to make Equation (4.58) more explicit, but also simplify it down, so that itdoesn’t become too cumbersome. First, we notice that( 1b2+ w)−d−12 .= bξ+3+u(1 + wb2)−12(ξ+3+u) = bξ+3+u(1 + wb2)−κ−32− ρ2 (4.59)via Equation (4.56) and (4.57). Furthermore,464.5. Relative Bessel’s Matrix Elementszρ2.=Çs2q2b24(1 + wb2)åρ2=12ρ(sqb)ρ(1 + wb2)−ρ2 (4.60)and, by manipulating the normalization factors from Equation (2.5), we get√pi4NnlNn′l′ = b−3 · 12√pi n!n′!Γ(n+ l + 32) Γ(n′ + l′ + 32)(4.61)Overall, putting Equations (4.59) to (4.61) into (4.58) yields〈nr lr|rue−wr2jρ(sqr)|n′r l′r〉 =12ρ+1√pi nr!n′r!Γ(nr + lr +32) Γ(n′r + l′r +32)bu(sqb)ρe−z×nr∑k= 0n′r∑k′= 0s(−1)k+k′k! k′!Çnr + lr +12nr − kåÇn′r + l′r +12n′r − k′å× (1 + wb2)−κ−ρ− 32 κ!Lρ+12κ (z){where, κ =12(lr + l′r − ρ+ u) + k + k′ ∈ N0 and, z =(sqb)24(1 + wb2)(4.62)where we’ve restored the r for “relative,” and we remind the reader that the Gamma functionsare defined as in Equation (F1), the generalized binomial coefficients are defined as in (F2),and the b is the harmonic oscillator length as defined in Equation (2.4).In order to satisfy the condition that κ ∈ N0 in Equation (4.62), we’ll require thatlr + l′r − ρ ∈ 2N0 and, u ∈ 2N0 (4.63)where we’ve imposed the latter restriction, u ∈ 2N0, by choice. The LHS restriction in Equa-tion (4.63) is satisfied for us, since ρ represents the spherical harmonic rank of the two-bodyoperator. In general, an operator can only change angular momentum in steps of its rank,by construction.∗ Finally, let’s reduce Equation (4.62) down to find the solution for Equa-tion (4.51), by setting u = 0 = w and s =√2, giving〈nr lr|jρ(√2qr)|n′r l′r〉 =12ρ+1√pi nr!n′r!Γ(nr + lr +32) Γ(n′r + l′r +32)(√2qb)ρ exp(−q2b22)×nr∑k= 0n′r∑k′= 0s(−1)k+k′k! k′!Çnr + lr +12nr − kåÇn′r + l′r +12n′r − k′å×î12(lr + l′r − ρ) + k + k′ó!Lρ+ 1212(lr+l′r−ρ)+k+k′(q2b22){(4.64)∗this can be seen clearly when considering Equation (B14), for instance474.5. Relative Bessel’s Matrix Elements4.5.1 RBMEs with Short-Range CorrelationsIn the case of neutrinoless double-beta decay, one could speculate that the exchange of the neu-trino at high momenta (short range)∗ would dominate. This would require the two neutronsinvolved in the 0νββ decay to be probabilistically “closer” together, such that their relativewave-functions would overlap at the scale of roughly 3 fm [120]. This phenomenon is knownas “short-range correlations” (SRCs), and can be physically modelled using the original pre-scription of Miller and Spencer [121]. The importance of SRCs for 0νββ NMEs was recognizedby [116, 122], and further popularized by [123]. Now many modern calculations include severalparameterizations of SRCs for neutrinoless double-beta decay [71, 72, 115, 117, 118, 124–134].To implement SRCs, one simply updates the nucleon-nucleon radial wave-function asRnl(r) −→ [1 + J(r12)]Rnl(r) (4.65)where the function, J(r12), is fit as a “Jastrow-type” function, such thatJ(r12) = −ce−ar212(1− br212) (4.66)The numbers a, b, c ∈ R are called “SRC parameters,” and many physics-dependant fittingshave been made for them [121]. Notice that the correct variable to use for the Jastrow-typefunction in Equation (4.65) is r12, as in (4.29), since that is the coordinate space in which theSRC parameters have been fit. The most commonly used sets of SRC parameters for 0νββ aregiven in Table 4.1 below.Name Abbr. a [fm−2] b [fm−2] c [1]Argonne V18 Potential AV18 1.59 1.45 0.92Charge-Dependant Bonn Potential CD-Bonn 1.52 1.88 0.46Miller-Spencer Fitting MS 1.1 0.68 1Table 4.1: Common SRC parameters for the Jastrow-type function in Equation (4.66), to beused in (4.65) for neutrinoless double-beta decay.To implement SRCs in ourM0ν TBMEs, we apply the mapping in Equation (4.65) with (4.66)on the RBMEs as in Equation (4.51). That is〈nr lr|jρ(q ·r12)|n′r l′r〉 −→∫ ∞0dr r2[1− ce−ar212(1− br212)]2Rnrlr(r)jρ(q ·r12)Rn′rl′r(r) (4.67)It is easy to show that, for some s ∈ R,[1− ce−as2r2(1− bs2r2)]2 = 1− 2ce−as2r2 + 2bcs2r2e−as2r2+ c2e−2as2r2 − 2bc2s2r2e−2as2r2 + b2c2s4r4e−2as2r2(4.68)At this point, the utility of Equation (4.62) becomes clear. Let’s make the labellingI u,w;snrlrn′rl′r(q) ≡ 〈nr lr|rue−wr2jρ(sqr)|n′r l′r〉 (4.69)∗relative to inter-nucleonic momenta484.6. Reconstructing M0ν TBMEsso, plugging Equation (4.68) into (4.67), and using the notation in (4.69), gives〈nr lr|jρ(√2qr)|n′r l′r〉 −→ I 0,0;√2nrlrn′rl′r(q)− 2c I 0,2a;√2nrlrn′rl′r(q) + 4bc I 2,2a;√2nrlrn′rl′r(q)+ c2I 0,4a;√2nrlrn′rl′r(q)− 4bc2I 2,4a;√2nrlrn′rl′r(q) + 4b2c2I 4,4a;√2nrlrn′rl′r(q)(4.70)Using Equation (4.70) along with (4.62) will incorporate SRCs into our TBMEs from Equa-tions (4.31), (4.46), and (4.50). It should be noted that the b in Equation (4.70) above isan SRC parameter from (4.66), whereas the b in Equation (4.62) is the harmonic oscillatorlength from (2.4). Thus, it is necessary to distinguish b → bosc, or alike, when computingEquation (4.62), to avoid ambiguity.4.6 Reconstructing M0ν TBMEsSince we have introduced lots of new notation in this chapter, for the reader’s convenience wewill summarize the formulas derived above. To obtain the 0νββ NMEsM0ν = M0νGT −ÇgVgAå2M0νF +M0νT (3.21)where,M0να.= 〈f |M̂ 0να |i〉, α = GT, F, T (3.22)we use Equation (4.1) in combination with the TBMEsMαabcd ≡ 〈a b :J ||M̂ 0να ||c d :J ′〉, α = GT, F, T (4.2)In general, using the closure approximation via Assumption 2, we can writeMαabcd = δisoac,bd2Rpi∫ ∞0dqq ·hα(q2)q + E cl0〈a b :J ||“Qα||c d :J ′〉 (4.71)where the form factors, hα, arehF(q2) ≡ g2V (q2)g2V,0(4.11)hGT(q2) ≡ 1g2A,0ñg2A(q2)− gA(q2)gP (q2) q23mp+g2P (q2) q412m2p+g2M (q2) q26m2pô.=g2A(q2)g2A,0ñ1− 23q2q2 +m2pi+13Çq2q2 +m2piå2ô+16g2M (q2)g2A,0q2m2p(4.12)494.6. Reconstructing M0ν TBMEshT(q2) ≡ 1g2A,0ñgA(q2)gP (q2) q23mp− g2P (q2) q412m2p+g2M (q2) q212m2pô.=g2A(q2)g2A,0ñ23q2q2 +m2pi− 13Çq2q2 +m2piå2ô+112g2M (q2)g2A,0q2m2p(4.13)with the g-factorsgV (q2) ≡ gV,0Ä1 + q2/Λ2Vä2 , gA(q2) ≡ gA,0Ä1 + q2/Λ2Aä2gP (q2) ≡ 2mp gA(q2)q2 +m2pi, gM (q2) ≡ǵp − µnµNågV (q2)(4.14)The augmented operators, “Qα (which are q-dependent), have been defined respectively in Equa-tions (4.33), (4.49), and (4.19), which can be written collectively as“Qα ≡ √4pij0(q ·r12)Ŷ 0 “σ1◦“σ2, α = GT√4pij0(q ·r12)Ŷ 0 1, α = F»24pi5 j2(q ·r12)Ŷ 2 “S12, α = T (4.72)The TBMEs for the GT and F augmented operators are as follows:〈a b :J ||“QGT||c d :J ′〉 = δJJ ′ ÛJ ∑S=0,1σσ(S)s∑Lla lb L12 12 Sja jb Jlc ld L12 12 Sjc jd J×∑nrlr,n′rNΛDabDn′rcd 〈nr lr|j0(√2qr)|n′r lr〉{ (4.46)with the spin eigenvaluesσσ(S) ≡ 2S(S + 1)− 3 ={1, S = 1−3, S = 0 (4.40)and〈a b :J ||“QF||c d :J ′〉 = δJJ ′ ÛJ ∑L,Ssla lb L12 12 Sja jb Jlc ld L12 12 Sjc jd J×∑nrlr,n′rNΛDabDn′rcd 〈nr lr|j0(√2qr)|n′r lr〉{ (4.50)with the sums over S = 0, 1 and ∆(la lb :L), and the Talmi-Moshinsky brackets areDab = 〈nr lr, NΛ:L|na la, nb lb:L〉 and, Dn′rcd ≡ 〈n′r lr, NΛ:L|nc lc, nd ld:L〉 (4.44)with the CTML adapted from Equation (D16) as504.6. Reconstructing M0ν TBMEs∑nrlr,n′rNΛ=b(ab+L)/2c−(nr+N)∑(6)lr = d(ab−L)/2e−(nr+N)b(ab−L)/2c−nr∑(5)N = 0b(ab−L)/2c∑(3)nr = 0with the constraints: (2)∆(lcld:L), (1)∆(lalb:L),(7)Λ = ab−2(nr+N)−lr, (4)n′r = (cd−ab)/2+nr ∈N0ab≡ 2na+la+2nb+lb, cd≡ 2nc+lc+2nd+ld(4.73)where the bolded, parenthesized, left superscripts denote the order that each parameter in thelimits should be set. The TBMEs for the T augmented operator is as follows:〈a b :J ||“QT||c d :J ′〉 = δJJ ′ ÛJ ∑LL′2√5 (−1)1+J+L′ ÛLÛL′®L 1 J1 L′ 2´×la lb L12 12 1ja jb Jlc ld L′12 12 1jc jd J×∑nrlr,n′rl′rNΛDabD′cd(−1)L′+lr+Λ®L lr Λl′r L′ 2´×√6 (lr 0 2 0 | l′r 0) Ûlr 〈nr lr|j2(√2qr)|n′r l′r〉(4.31)with the sums over ∆(la lb :L) and ∆(lc ld :L′), and the Talmi-Moshinsky bracketsDab = 〈nr lr, NΛ:L|na la, nb lb:L〉 and, D′cd ≡ 〈n′r l′r, NΛ:L′|nc lc, nd ld:L′〉 (4.28)with the CTML adapted from Equation (D21) as∑nrlr,n′rl′rNΛ=Λ+L′∑(8)l′r = |Λ−L′|b(ab+L)/2c−(nr+N)∑(6)lr = d(ab−L)/2e−(nr+N)b(ab−L)/2c−nr∑(5)N = 0b(ab−L)/2c∑(4)nr = 0with the constraints: (9)∆(lrl′r:2), (3)∆(LL′:2), (2)∆(lcld:L′), (1)∆(lalb:L),(10)n′r = (cd−ab)/2+(lr−l′r)/2+nr ∈N0, (7)Λ = ab−2(nr+N)−lrab≡ 2na+la+2nb+lb, cd≡ 2nc+lc+2nd+ld(4.74)To analytically calculate the RBMEs, we may use〈nr lr|jρ(√2qr)|n′r l′r〉 =12ρ+1√pi nr!n′r!Γ(nr + lr +32) Γ(n′r + l′r +32)(√2qb)ρ exp(−q2b22)×nr∑k= 0n′r∑k′= 0s(−1)k+k′k! k′!Çnr + lr +12nr − kåÇn′r + l′r +12n′r − k′å×î12(lr + l′r − ρ) + k + k′ó!Lρ+ 1212(lr+l′r−ρ)+k+k′(q2b22){(4.64)or, to incorporate SRCs, use〈nr lr|jρ(√2qr)|n′r l′r〉 −→ I 0,0;√2nrlrn′rl′r(q)− 2c I 0,2a;√2nrlrn′rl′r(q) + 4bc I 2,2a;√2nrlrn′rl′r(q)+ c2I 0,4a;√2nrlrn′rl′r(q)− 4bc2I 2,4a;√2nrlrn′rl′r(q) + 4b2c2I 4,4a;√2nrlrn′rl′r(q)(4.70)514.6. Reconstructing M0ν TBMEswithI u,w;snrlrn′rl′r(q) =12ρ+1√pi nr!n′r!Γ(nr + lr +32) Γ(n′r + l′r +32)bu(sqb)ρe−z×nr∑k= 0n′r∑k′= 0s(−1)k+k′k! k′!Çnr + lr +12nr − kåÇn′r + l′r +12n′r − k′å× (1 + wb2)−κ−ρ− 32 κ!Lρ+12κ (z){where, κ =12(lr + l′r − ρ+ u) + k + k′ ∈ N0 and, z =(sqb)24(1 + wb2)(4.62)where the b in this formula represents the oscillator length,∗ andI u,w;snrlrn′rl′r(q) ≡ 〈nr lr|rue−wr2jρ(sqr)|n′r l′r〉The formulae above constitute the most mathematically laborious component of this re-search, and unfortunately that carries into their numerical treatment. Controlling the necessaryintegrations and computationally heavy summations involved, in both a precise and efficientmanner, is not a trivial matter. More on this will be discussed in Section 6.4. Once we’vedealt with the numerical issues, there is one last step that we should not forget! In order tocapture the fermionic statistics of our nucleons, we will need to anti-symmetrize the TBMEsusing Theorem B.4.〈a b :J ||M̂ 0να ||c d :J ′〉 −→ 〈a b ; J ||M̂ 0να ||c d ; J ′〉Now that the 0νββ operators are mathematically in order, we’d like to find a way to embedthis physics into nuclear structure. We’ll accomplish this task by using the nuclear many-bodymethod known as: the valence space in-medium similarity renormalization group.∗not to be confused with the b from the SRC parameters!52Chapter 5Valence Space In-Medium SimilarityRenormalization GroupAs introduced in Section 2.2, the ultimate goal of nuclear physics is to fully solve the nuclearSchro¨dinger equation. With this dynamics in place, all the properties of the nucleus, as a semi-classical quantum mechanical object, can be modelled completely. The two major issues are:how do we obtain an appropriate potential that describes low-energy QCD, and how can wenumerically handle such a large Hamiltonian? Making both of these problems computationallytractable is where χEFT and nuclear many-body methods come into play.The non-perturbative, ab initio approach that we will use is known as the “Valence Space In-Medium Similarity Renormalization Group” (VS-IM-SRG) method [135]. There are currentlyseveral versions of IM-SRG [136], but we’ll choose the Magnus Formulation, as outlined inSection 5.4. In terms of the acronym, the VS-IM-SRG will be summarized in the order:∗Valence Space (VS), Similarity Renormalization Group (SRG), and then In-Medium (IM).5.1 Nuclear Core and Valence SpaceAs with most nuclear many-body methods, we’ll start by modelling the nucleus using the ShM.However, as nuclei become large in mass number, the number of interactions and potentialexcitation configurations between protons and neutrons in their shells increases dramatically.How can we contain the nucleus in such a way that is computationally manageable? Referringback to Figure 2.1, one could imagine that there exists a Fermi energy that limits the dynamicsof nucleons lying in lower closed shells. We’ll take advantage of this, and define what is knownas a “nuclear core”† of nucleons which we force to be inert under renormalization.Assumption 3Nuclei with a large enough Z and N can be modelled statistically using a Fermi surface to definean inert nuclear core, within the context of the ShM. We’ll assume that this approximation isgood at reproducing reality.∗this order somewhat reflects how VS-IM-SRG was developed chronologically†Such a core should not be confused with the antiquated idea of a “hard core” as defined by the nuclearpotential, which we discuss more in Section 5.2.1 below.535.1. Nuclear Core and Valence SpaceFigure 5.1: Core, valence space, and excluded region for the ground state of 48Ca withinthe nuclear shell model. The orange arrows represent undesirable, low-probability excitationspast the Fermi surface or into the excluded region, and the green arrows represent excitationsallowed within the defined valence space. The purple arrow schematically represents the use ofVS-IM-SRG to renormalize the dynamics to the chosen valence space, in this case the pf -shell(which rarely includes the 0g9/2 orbit, despite its small energy gap to 1p1/2 relative to the0f7/2 and 1p3/2 gap), thus leaving the core inert and the excluded region empty. Since48Ca isdoubly magic, this valence space accepts no protons, which makes 48Ca a favourable nucleus asa theoretical benchmark.Once a core has been established, this leaves a indefinite region of infinitely many shells thatnucleons may excite into. Yet, it’s clear that excitations from the ground state to higher lyingorbits are less probable than excitations into lower lying orbits. Thus, we find it reasonable toexcise higher lying shells from the nucleonic dynamics, in a space called the “excluded region.”What is left is known as the “valence space” (VS), since this is where the dynamics of thevalence nucleons∗ can occur. We see that an a priori choice of valence space automaticallydefines the core and excluded regions, as depicted for Calcium-48 in Figure 5.1.Assumption 4Coupled with Assumption 3 above, a suitable choice of valence space, tailored to a particular nu-cleus, should be able to capture enough dynamics in order to reproduce said nuclei’s properties.Of course, this inherently depends on the efficacy of the nuclear many-body method.5.1.1 Model SpaceTo build the nuclear ShM, we’ll start with the oscillator† basis (see Section 2.1.1) for the single-particle wave-functions. To construct the nuclear wave-function we’ll use a Slater determinant‡∗those which lie above the Fermi surface defining the inert core†including all the accepted corrections needed to reproduce the observed magic numbers‡our single-particles are nucleons, which are fermions, and hence a Slater determinant is justified545.2. Similarity Renormalization Group(see Section 2.1.3). However, as opposed to the standard ShM, we’ll now want to decouplethese wave-functions into the core and valence space respectively, as mentioned above. There-fore, within the valence space, we need to specify how many single-particle excitations will beacceptable for our calculations. This is done by setting a model space “size” known either as“Nmax” or “emax” for all single-particle states, in accordance with Equation (2.2), like2n+ l ≤ emaxUpon the inclusion of a 3N interaction, it would then also be important to set an “E3max” likee(1)max + e(2)max + e(3)max ≤ E3maxwhere the parenthesized superscripts label some set of three separate single-particles in thevalence space Slater determinant. We will exclusively set E3max = 14 for our calculations, butvary emax due to the following physically intuitive remark:Remark 5.1Upon applying the nuclear ShM for a valence space nuclear many-body method, as the sizeof the model space (emax, E3max, ...) increases, the results will become independent of theoscillator frequency (}ω) used for the basis states, resulting in “converge” [137].5.2 Similarity Renormalization GroupG lasek and Wilson took the renormalization group, the use of which in physics had beenpioneered in the early 1970’s by Wilson and alike [138], and modernized it in an ingeniousway [139, 140]. In essence, the renormalization group is a tool to systematically view physicalstructure at different scales (distance, energy, etc) [141]. The idea was to apply the renormaliza-tion group in a such a way that as an energy parameter decreases, the Hamiltonian diagonalizesin a self-consistent way. Exploiting this self-similarity of the Hamiltonian under the renormal-ization group is what gave this scheme the name “Similarity Renormalization Group” (SRG).The similarity transformation of the Hamiltonian can be written as‹H = U “HU−1 (5.1)where U is the unitary (similarity) matrix, hence U−1 = U †. The motivation behind Equa-tion (5.1) is that Wilson and G lasek wanted to find a way to remove the off-diagonal termsof the Hamiltonian, but in doing so, preserve the observables of the system of interest. Forinstance, we can calculate the spectrum normally or within the SRG spaceEn = 〈Ψn|“H|Ψn〉 = 〈Ψn|U †ÄU “HU †äU |Ψn〉 = 〈‹Ψn|‹H|‹Ψn〉The details behind how one drives the Hamiltonian to a band or block-diagonal form are outlinedin Section 5.2.2 below. Clearly, a block-diagonal Hamiltonian is not only advantageous in theframework of QFT, since all that would be needed to find the eigenstates of any physical555.2. Similarity Renormalization Groupsystem are a set of sub-block diagonalizations of the transformed Hamiltonian. The use of therenormalization group in nuclear physics, for instance, has been utilized in recent studies [6].5.2.1 Nuclear Potentials and Similarity TransformationsQCD has the favourable quality that as the energy of interest increases, the theory becomesperturbatively renormalizable [44], due to the running of the strong coupling constant for thegluon, αS , as a function of the transfer momentum∗αS(q2) =αS(µ2)1 +BαS(µ2) ln(q2/µ2)This property is known as “asymptotic freedom,” which was discovered independently in 1973by Gross and Wilczek [142] and Politzer [143]. The subsequent renormalizability and lack ofLandau poles make quantum chromodynamics UV complete. Unfortunately, for the low-energyregime,† such as those found in nuclear physics, the coupling constant is of order unity, yieldingan infrared divergence. Ergo, the standard perturbative techniques used in QFT commonly failwithin the context of nuclear theory.Since the Dirac equation governs physics at high energies, and the Schro¨dinger equationgoverns quantum mechanics at low energies, we can consider the nuclear problem as an exercisein solving the nuclear Schro¨dinger Hamiltonian. With this perspective, where do these pertur-bative divergences manifest? The off-diagonal terms of the Hamiltonian physically representthe correlation between high and low momentum states. Since the nuclear Hamiltonian is solarge, one may propose to use SRG technology, as in Equation (5.1), to “soften” the nuclearinteractions and drive the off-diagonal terms to zero [8]. A useful visualization of this can beseen in Figures 9 and 10 on page 102 of [8].‡This softening is an intuitive suggestion, since the renormalization group is what deter-mines: the running of LECs, power counting methodology, and regularization cutoff analysis inχEFT interactions (see Section 2.2.3). It therefore makes sense to adopt SRG technology fornuclear potentials in an ab initio model. However, with this SRG framework we must let go ofthe classical notion in nuclear physics that there exists a repulsive “hard core.”§ Why? ManySRG evolved potentials might not display this hard core, but they still describe the exact sameobservables as the bare potential. It is easy to argue that this is an acceptable theoretical com-promise, since the potential is ultimately not an observable itself, hence it cannot be measured,and it should not be interpreted physically [8]. Furthermore, these studies have motivated therecently discovered fact that many-body interactions, like 3N forces, are important for nuclear∗B > 0 for a QFT with three colours and six quarks, and q2 = µ2 is a mass scale where one can confidentlymeasure the colour charge of a quark.†at around roughly the mass of the proton, 1 GeV, and subsequent hadronization scales‡More diagrams, specifically in the context of IM-SRG, can been seen in Figure 26 of [28] or Figure 10.2of [144] for a numerical example, among others.§Note that the idea of this nucleon-nucleon repulsive “core” is very different from the nuclear core as depictedin Figure 5.1.565.2. Similarity Renormalization Groupphysics [145]. How one might capture these interactions will be touched on in Section 5.3. Butfirst, if we make the SRG a necessary step for our nuclear many-body method - how does onemathematically accomplish the desired decouplings?5.2.2 SRG Flow EquationsWe begin with the similarity transformation from Equation (5.1), but parameterize it by a “flowparameter,” s ∈ R, as [140] originally introduced.∗ As we increase this flow parameter, we’llevolve the Hamiltonian in infinitesimal regulations through the resolution scales of the renor-malization group, in order to drive “H towards a block-diagonal form. This can be accomplishedby first requiring that “Hs = Us “HU †s = “Trel + Vswhere, “H = “Trel + VThat is; the relative kinetic energy term remains unchanged under the similarity transformation.For a valence space formulation of the SRG, we decouple the valence space (VS) and excludedregion (excl), and write the similarity transformation as“Hs = Us “HU †s = “H VSs + “H excls (5.2)and then we obtain the eigenstates by a diagonalization within the valence space [137].In order to solve for “Hs, one can construct an ODE for it by taking the derivative ofEquation (5.2) with respect to s:d“Hsds=dUsds“HU †s + Us“HdU †sds=dUsds(U †s “HsUs)U †s + Us(U †s “HsUs)dU †sds=dUsdsU †s “Hs + “HsUsdU †sds (5.3)noting that d“H/ds = 0, since it is an initial condition; and that UsU †s = 1, since Us is unitary.By intuition, we’d like to isolate the derivative of U †s in terms of the derivative of Us, which wecan do by (once again) appealing to unitaritydds(U †sUs ) =d1ds= 0 =⇒ dU†sdsUs = −U †sdUsdsdU †sds= −U †sdUsdsU †s (5.4)Plugging Equation (5.4) into (5.3) yields∗They labelled the flow parameter λ, since it was thought of as a regularization cutoff, and they had alreadylabelled the unitary matrix performing the similarity transformation as S, instead of U .575.2. Similarity Renormalization Groupd“Hsds=dUsdsU †s “Hs − “HsdUsds U †s (5.5)And so, by making the definitionηs ≡ dUsdsU †s (5.6)we may rewrite Equation (5.5) in the compact formd“Hsds= [ηs, “Hs] (5.7)Equation (5.7) is referred to as the “flow equation,” and ηs is called the “generator” ofthis flow, of which many choices can be found [146, 147]. Notice that taking the dagger ofEquation (5.6) shows that ηs is anti-hermitianη†s =(dUsdsU †s)†= UsdU †sds.= −dUsdsU †s = −ηswhere the equivalence, denoted by.=, comes from Equation Us×(5.4). ηs being anti-hermitianjustifies referring to it as a generator, since we can mathematically identify Us by solvingdUsds= ηsUs =⇒ Us(Λ) = S expñ ∫ sΛds′ ηs′ô(5.8)where S is the s-ordering operator, such that it reorders operators from left to right in descend-ing s′; and Λ is some arbitrarily small cutoff, which yields self-similarity of the Hamiltonian,i.e., lims→Λ “Hs = “H. Physically, Equation (5.8) can be read as ηs infinitesimally generatingchanges to the resolution scale encoded by the similarity transformation in (5.2), which justifiescalling s a flow parameter. Notice that had we been interested in reproducing the high-energyphysics of QCD, one could imagine simply reversing the s-ordering operator, flip the integral,and set Λ to be arbitrarily large.For our purposes, we will take Λ = 0, and require that the unitary matrix, Us, be con-structed such that the Hamiltonian is block-diagonal as the flow increases. That is, splitting∗the similarity transformed Hamiltonian into “diagonal” (d) and “off-diagonal” (od) terms“Hs = “Hds + “Hods (5.9)we want a generator that guaranteeslims→∞“Hods = 0Not all generators yield the desired block diagonal form, and different choices of generators isstill an ongoing avenue of research [136].∗The decomposition of operators in this manner is somewhat roughly defined, in terms of what deems “band”or “block” diagonal, and depends on the intended purpose of the SRG.585.2. Similarity Renormalization GroupChoice of GeneratorSince the genesis of the SRG flow equations, several generators have been suggested which evolvethe Hamiltonian to a block diagonal form. One of the earliest choices of ηs with practical useis the Wegner generator [146], take asηs ≡ [“Hds , “Hods ] + h.c.It can be shown that the Wegner generator, under its use with Equation (5.7), will give thedesired asymptotics for the off-diagonal entries of the Hamiltonian“Hij(s) ∼ e−s∆2ij “Hij(0) where, ∆ij ≡ “Hdii(s)− “Hdjj(s)Shortly thereafter, in an application of quantum many-body theory to chemistry, White pro-posed a new generator of the form [147]ηij(s) =“Hodij (s)∆ij+ h.c. (5.10)which has the asymptotics “Hij(s) ∼ e−s“Hij(0)The White generator has a more controlled suppression of the off-diagonal terms than theWegner generator. Hence, despite potential numerical instabilities with the energy denominatorin Equation (5.10), unless otherwise stated we will opt to use the White generator.5.2.3 Consistent Operator EvolutionOf course, the energy levels are not the only property of a nucleus we seek to understand, andhence the Hamiltonian is not the only operator that we wish to SRG evolve. For instance, onemay wish to model: electromagnetic properties, radii, GT transitions, and double-beta decay.All these observables will be described by some Hermitian operator, which we’ll label genericallyas “O. One of the useful features about the SRG transformation is that the same evolution maybe applied to “O along with “H. That is, we can generalize Equation (5.2) to“Os = Us “OU †s (5.11)and use the same manipulations as above to arrive at the operator flow equationd“Osds= [ηs, “Os] (5.12)Applying Equation (5.12) along with (5.7), in tandem, can only be performed consistently aslong as the same generator is used. One should note that as we add more operators using thistechnique, we effectively must solve more ODEs - which highlights a computational limitation ofthis formulation. To over come this inefficiency, we’ll use the Magnus expansion with IM-SRG,595.3. Reference State Normal Orderingas described in Section Reference State Normal OrderingApplying the SRG flow equations to the nuclear Hamiltonian has typically been done in freespace, to make 2N and 3N nucleonic interactions more manageable [148]. For our purposes,however, we need a way to translate “Hs into the language of our desired valence space, asoutlined in Section 5.1. That is, we’d like to renormalize our χEFT interactions to matchFigure 5.1. Remarkably, this can be accomplished by redefining normal ordering “In-Medium”(IM) - with respect to the nuclear core. Along with this, we can induce 3N effects into 2Nterms, thus capturing many-body physics upon truncation for free.As introduced in Section C.1, the necessity to normal order creation and annihilation op-erators is ingrained in the machinery of second quantization. In QFT, this is often done withrespect to the vacuum, but can we generalize this notion? Instead of normal ordering againstthe vacuum, we can indeed normal order with respect to some reference state [12], call it |Φ0〉.All that we need to do is impose a new definition of normal ordering in Equation (C10) via〈0|cˆ†i · · · cˆj|0〉 = 0 −→ 〈Φ0|cˆ†i · · · cˆj|Φ0〉 = 0 (5.13)For a second quantized Hamiltonian,“H = ∑ab“Tab cˆ†acˆb + 12!2 ∑abcdV(2N)abcd cˆ†acˆ†bcˆccˆd+13!2∑abcdefV(3N)abcdef cˆ†acˆ†bcˆ†ccˆdcˆecˆf + . . .(5.14)we can then use Wick’s Theorem (see Equation (C12) of Section C.1.1) with the “referencestate normal ordering” from Equation (5.13) to rewrite (5.14) as [136, 149]“H = E + ∑abFab cˆ†acˆb +12!2∑abcdUabcd cˆ†acˆ†bcˆccˆd+13!2∑abcdefWabcdef cˆ†acˆ†bcˆ†ccˆdcˆecˆf(5.15)whereE =∑a“Taa na + 12∑abV(2N)abab nanb +16∑abcV(3N)abcabc nanbnc (5.16)Fab = “Tab + ∑cV(2N)acbc nc +12∑cdV(3N)acdbcd ncnd (5.17)Uabcd = V(2N)abcd +∑eV(3N)abecde ne (5.18)Wabcdef = V(3N)abcdef (5.19)605.4. The Magnus Formulationwith na and alike denoting the occupation numbers of nucleons in the reference state.This is a powerful result! We see that even if we truncate the reference state normalordered Hamiltonian in Equation (5.15) at the 2N level, we can still capture the bulk of the 3Ninteractions via Equations (5.16) to (5.18). Furthermore, by employing Equation (5.13), we canguarantee that we’ve set a nuclear core under the chosen valence space. Since the mathematicsat the 3N level is highly complex, we’ll use the two-body truncation with the above equationsin what’s known as the IM-SRG(2) formulation [10, 150]. However, we can still rest assuredthat 3N interactions have been included in our calculations.5.3.1 Hartree-Fock StepThere is an important criterion about the nuclear core that we neglected to mention in theabove discussion of normal ordering: Assumption 3. In particular, we want to make sure thatthe core is inert. That is, when taking all the single-particle oscillator basis states and using aSlater determinant to build a nuclear wave-function for the core, we want to make sure it is asclose the ground state as possible. In IM-SRG [135, 137], this is typically accomplish by usinga Hartree-Fock wave-function for the In-Medium reference state:|Ψ0〉 −→ |ΨHF〉Hence, we’ll use a Hartree-Fock variational step (see Section 2.1.4) before normal ordering canbe calculated in IM-SRG. This will also ensure that our oscillator basis states transform intosingle-particle wave-functions which realistically match our choice of interaction.5.4 The Magnus FormulationIn practice, implementing Equations (5.2), (5.7), and (5.12) is not always feasible because:the ODEs may become stiff, the advanced numerical integrations needed impose large memorycosts,∗ and solving for any additional operators requires another set of non-linear coupled ODEsto be solved. Furthermore, it is unclear how to systematically truncate the solution for theunitary matrix in Equation (5.8) under the s-ordering, and any such truncations will inevitablyinduce non-unitarity. These non-unitary resolution errors ultimately defeat the purpose ofpreserving observables under the SRG. To rectify these issues, we’ll use what is known as the“Magnus formulation”† of IM-SRG [149].The first step to this formulation is to ask: how can we construct the unitary matrixnecessary to similarity transform the Hamiltonian (and alike), in a computationally tacticalway? The idea is to take the form of the unitary matrix as followsUs = eΩ(s) such that, Ω†(s) = −Ω(s) and, Ω(0) = 0 (5.20)∗which is especially a problem for larger model spaces†One would expect that, upon the continuing success of this “Magnus” implementation, it will soon bereferred to as the “Morris” or “MPB formulation.”615.4. The Magnus FormulationIt can be proven that this form exists as a solution to the linear system∗ of ODEs in Equa-tion (5.8) [151]. The latter condition guarantees that Us=0 = 1 and hence “Hs=0 = “H. Butnow that we’ve introduced more information in terms of Ω(s), we must find a suitable way offinding it. By generalizing Equation (5.8), one can construct the following ODEdΩ(s)ds=∞∑k= 0Bkk!adkΩ(s)[ηs] = ηs +12[Ω, ηs] +112[Ω, [Ω, ηs]] + . . . (5.21)where Bk are the Bernoulli numbers,† and we’ve introduced the shorthand notation for therecursively defined nested commutators for the square matrices L (left) and R (right)adkL[R] = [L, adk−1L [R]] where, ad0L[R] = R (5.22)As discussed previously in Section 5.2.2, we’ll take ηs as the White generator.So how does the introduction of Equation (5.20) possibly help us with our original task ofusing the SRG on the Hamiltonian? We see that the condition Ω†(s) = −Ω(s) in combinationwith (5.2) gives us the innocuous modification“Hs = eΩ(s)“He−Ω(s) (5.23)From this point, one may use the Baker-Cambell-Hausdorff (BCH) formula on Equation (5.23)to obtain “Hs = ∞∑k= 01k!adkΩ(s)[“H] = “H + [Ω, “H] + 12[Ω, [Ω, “H]] + . . . (5.24)where we’ve used the same notation as defined in Equation (5.22). Now we can see the advantageof the Magnus formulation! We’ve effectively traded the cumbersome ODE in Equation (5.7)with a set of nested commutators for the Hamiltonian, in what is known as the “Magnus flowequation” (5.24), and the new ODE from (5.21).It turns out that all we will need to solve the ODE in Equation (5.21) is a first-order forwardEuler solver [149]. This will greatly cut memory costs when using IM-SRG, particularly formedium mass nuclei with large model spaces. Then, many computationally efficient diagram-matic methods for solving nested commutators can be exploited to SRG evolve the Hamiltonianvia the Magnus flow equation. Furthermore, we can systematically truncate Equation (5.21)and (5.24), thus minimizing any non-unitarity induced in Us. Thusly, the Magnus formulation:introduces significant memory savings, avoids problems with stiff ODEs, and gives a way tosystematically truncate the series sums whilst optimizing for unitarity loss.The next pressing question is, then, how can we accomplish appropriate truncations? Thecurrent technique is the so-called “Magnus(2) truncation,” in conjunction with IM-SRG(2)∗it’s considered a system in terms of the matrices (mathematical operators) involved†defined such that B1 = +1/2625.4. The Magnus Formulation(described in Section 5.3 above). That is, it has been found empirically “that the magnitude ofterms [in Equation (5.21) and (5.24)] decreases monotonically in k for all systems studied thusfar.” [149] So, for some tolerances, deriv > 0 and BCH > 0, we can truncate the sum of nestedcommutators at the k-th term where∣∣∣∣∣Bk ||adkΩ[η]||k! ||Ω||∣∣∣∣∣ < deriv (5.25)for Equation (5.21), and ∣∣∣∣∣ ||adkΩ[“H]||k! ||Ω|| ∣∣∣∣∣ < BCH (5.26)for Equation (5.24). The norms on the respective matrices are taken as the Frobenius norm||Ω|| ≡√∑ij|Ωij |2 .=»Tr(Ω·Ω†)and it is commonly found that a wide range in -tolerance values yields similar final results.Now, as in Section 5.2.3, we may SRG any other operator of interest by using the sameunitary matrix, Us = eΩ(s), as we did for the Hamiltonian. Thus, upon the application of theBCH formula, we obtain the analog to Equation (5.24) for an arbitrary operator“Os = ∞∑k= 01k!adkΩ(s)[“O] = “O + [Ω, “O] + 12[Ω, [Ω, “O]] + . . . (5.27)with a similar truncation as in Equation (5.26). In this sense, we can now handle the calculationof any nuclear observable without having to solve an additional set of ODEs, as was requiredwith Equation (5.12). Hence, we’ve translated the problem of solving multiple systems of(potentially stiff) ODEs for “Hs and any “Os into solving one (Euler solvable) ODE for Ω(s) anda BCH summation for Equation (5.24) and (5.27), whilst minimally violating unitarity. Overall,these non-perturbative equations for the nuclear many-body problem will require a numericaltreatment, which we present in the next chapter.63Chapter 6Numerical MethodsThis chapter will cover the numerical methods that we will use in order to obtain the results ofthis research (see Chapter 7). In particular: imsrg++ is used to evolve nuclear operators, viathe theory of VS-IM-SRG; NuShellX@MSU is used to create realistic wave-functions of nuclearstates, within the framework of the ShM; and nutbar bridges the latter two codes to make one-body transition densities (OBTDs) and two-body transition densities (TBTDs), to finally arriveat nuclear matrix element predictions. We’ll also detail how the M0ν TBMEs were calculatedcomputationally, within imsrg++.6.1 imsrg++To perform the valence space IM-SRG evolution of the Hamiltonian, and any additional desiredobservables, as described in Chapter 5, we’ll use a software developed by Dr. Ragnar Stroberg∗entitled imsrg++. It is written primarily in C++, using features from C++11 with python bind-ings, and the libraries: GSL,† BOOST,‡ Armadillo,§ OpenBLAS,¶ and OpenMP,¶ among others. Itis maintained by Dr. Stroberg under a GNU Public License,† and stored in the git repositorylocated at:‖ https://github.com/ragnarstroberg/imsrg.This software has been highly successful for valence space IM-SRG calculations [135, 137].Its source files are well organized and easy to compile in a Linux environment. Together, theyperform all the relevant mathematics described in this dissertation, such as: angular momentumcoupling (see Appendix A), the Hartree-Fock step (see Section 2.1.4), normal ordering (seeSection 5.3), TBME construction (as in Chapter 4), and of course IM-SRG evolution (seeChapter 5). Furthermore, it comes equipped with many operators (located in the imsrg_utilnamespace), from electromagnetic observables to nuclear radii to beta decay. However, NMEscannot be produced directly from it, since it needs to interface (see Section 6.1 further below)with a ShM code (see Section 6.2 directly below).∗Dr. Stroberg was Dr. Jason D. Holt’s post-doctoral collaborator at TRIUMF for the majority of this dis-sertation research on double-beta decay. He is missed.†GNU Public License: http://www.gnu.org/licenses/gpl.html‡BOOST Software License: http://www.boost.org/users/license.html§Mozilla Public License: https://www.mozilla.org/en-US/MPL/2.0/¶BSD License: https://opensource.org/licenses/BSD-3-Clause‖this is not necessarily a permanent link, contact Dr. Stroberg for maintenance issues646.2. NuShellX@MSU6.2 NuShellX@MSUTo construct a NME, we’ll need the appropriate wave-function information to build the OBTDsand TBTDs of Equation (C26) and (C33). That is, we’ll require the final nuclear state, |f〉,and the initial nuclear state, |i〉, to make〈ηfJf ||[cˆ†a⊗c˜b]L||ηiJi〉 and, 〈ηfJf ||î[cˆ†a⊗cˆ†b]Jab ⊗ [c˜c⊗c˜d]JcdóL||ηiJi〉for one-body operators (see Section C.2) and two body operators (see Section C.3), respectively.∗We’ll build these eigenvectors using the shell model code known as NuShellX@MSU [152]. To per-form the diagonalization of the Hamiltonian for low-lying states in the nuclear ShM, NuShellXuses the Lanczos algorithm [153]. It’s constructed in PN formalism and J-scheme, which corre-sponds with how we’ve treated our OBMEs and TBMEs throughout this text. NuShellX alsocomes with a library of interaction files suitable for many valence spaces, but one can interfacetheir own interactions, which we do from imsrg++ using SRG evolved χEFT input files.6.3 nutbarIn order to calculate the OBTDs and TBTDs via the wave-functions from NuShellX, andthen combine them with the IM-SRG evolved OBMEs and TBMEs to produce NMEs,† we’llemploy a code called nutbar. It is maintained by Dr. Stroberg at the git repository loca-tion:‡ https://github.com/ragnarstroberg/nutbar. nutbar is an acronym which standsfor: NuShellX Transitions from Binary Arrays.§ It’s a necessary interface between NuShellXand imsrg++, since the former does not provide the ability to calculate TBTDs for sphericaltensors. Furthermore, IM-SRG will push information between one-body and two-body matrixelements (see Section 5.3). Due to this fact, nutbar is also used to keep the OBTDs consistent.6.4 Two-Tiered Adaptive M0ν TBMEsFor the M0ν TBMEs (see Chapter 4), we first integrate with respect to r (see Equation (4.64)for instance) and then, after performing the summations for the relevant augmented operatorsin Equations (4.46), (4.50), and (4.31), we’ll integrate with respect to q for Equation (4.71).In practice, this turned out to be more awkward than desired. First, it is clear that Equa-tion (4.71) will require an adaptive integration technique over the interval [0,∞), so we turnedto the GNU Scientific Library for integration functions [154]. Since the accuracy of the inte-gral with respect to r, which is parameterized by the relative quantum numbers nr, lr, n′r, l′r,∗both have been written for a rank L spherical tensor operator, to get the amplitude 〈f ||T̂L||i〉†via Equation (C26) and (C33)‡this is not necessarily a permanent link, contact Dr. Stroberg for maintenance issues§NuShellX stores its wave-function data in binary656.4. Two-Tiered Adaptive M0ν TBMEsaffects the convergence of the q integration - for higher lying orbits we would often receive theGSL_EDIVERGE error:the integral is divergent, or too slowly convergent to be integratednumerically.To make matters more complicated, for higher sets of the relative quantum numbers, Equa-tion (4.64) and alike∗ would start to fail. This was because the Gamma functions in both thenormalization constant and the generalized binomial coefficients of Equation (4.64) would blowup, even when implementing clever and efficient uses of the tgamma_ratio function from BOOST.To rectify these issues, we developed a two-tiered adaptive integration scheme, as illustratedin Table 6.1. We called this “two-tiered” since we adaptively choose which adaptive integrationmethods to use based on the values of nr, lr, n′r, l′r. Ultimately, this scheme solved all the issuespresented above. However, to be properly implemented, we needed to expand the imsrg++framework with a package we named NDBD.hh and NDBD.cc,† which represents a modest 1,300lines of highly optimized object-oriented code written in C++. The NDBD class (which stands forNeutrinoless Double-Beta Decay) is exclusively used in imsrg_until.cc to build the TBMEs,and required slight modifications to imsrg++.cc and alike. It also takes advantage of symme-tries like I(nr, lr, n′r, l′r) = I(n′r, l′r, nr, lr) and more, and employs parallelization‡ and cacheingtechniques for memory cost reduction and significant speed increases.nr, lr, n′r, l′r dr dq[4, 8, 4, 8] analytic QAGIU[10, 10, 10, 10] analytic QAG(∞,∞,∞,∞) QAGIU QAGTable 6.1: Two-tiered integration scheme for M0ν TBMEs. The integration type is first usedwith respect to r (relative distance coordinate), and then q (neutrino momentum). This set ofintegrations for (dr, dq) are used for the relative quantum numbers nr, lr, n′r, l′r for values lessthan or equal to those found in the first column of the table. The analytic integration useseither Equation (4.64) or (4.70) with (4.62), depending on whether or not SRCs are included.QAG [155] and QAGIU [156] adaptive integration techniques are from GSL. This scheme wasfound to be acceptable for emax ≤ 10, and we expect it to work beyond this.Note that the values for the relative quantum numbers§ in the first column of Table 6.1were found by trial and error, and they (if not the whole structure of the table) may needto be changed with evolving versions of the code. We found that the first integration to failwas for dq, using the highly adaptive QAGIU algorithm (see QAGIU in Section 6.4.1 below).∗Equation (4.70) and (4.62) for the inclusion of SRCs†For details on this version of imsrg++, please contact me at: cgpayne@triumf.ca. At the time of writing, ithas not been pushed to the remote GitHub repository at [1].‡via OpenMP§nr and n′r will run from 0 to emax, and lr and l′r will run from 0 to 2·emax. To see this simply maximize thelimits from Equation (D21).666.4. Two-Tiered Adaptive M0ν TBMEsHence, for relative orbits greater than (nr, lr, n′r, l′r) = (4, 8, 4, 8) we switched from QAGIU tothe QAG algorithm (see QAG in Section 6.4.1 below). Since QAG requires a finite interval ofintegration, we found that taking a = 0 MeV to b = 2, 500 MeV was sufficient to approximatea semi-infinite interval, since all integrands would have decayed in this range. However, atroughly (nr, lr, n′r, l′r) = (10, 10, 10, 10) the analytic formula for the r integration would fail dueto reasons perviously mentioned. This is disappointing, since an analytic formula for an integralis more computationally efficient than an adaptive method. Nonetheless, QAGIU is optimized,so we switch to this to handle the RBMEs of Equation (4.51)∗ adaptively at this point.6.4.1 Integration Techniques from GSLBefore presenting the final results from these numerical methods for the neutrinoless double-beta decay of Calcium-48, we’ll briefly describe the adaptive integration methods that we usedfrom the GNU Scientific Library [154]. Note that, for both the gsl_integration_qagiu andgsl_integration_qag functions from GSL, we found (limit, epsabs, epsrel) = (1e 4, 1e –7, 1 e–5)to be appropriate for the integration with respect to r, and (limit, epsabs, epsrel) =(Nthreads×1e 4, 1e –7, 1 e–4) to be appropriate for the integration with respect to q. The epsabsand epsrel terminate the integration (and subsequently define “convergence”) when∣∣∣F (xi)− F (xi−1)∣∣∣ ≤ abs or, ∣∣∣1− F (xi)/F (xi−1)∣∣∣ ≤ relrespectively. F (xi) represents the sequence of summed integrals of some integrand, f(x), onthe set of adaptively chosen {xi s.t. i ∈ N0}.QAGIUTo integrate a function, f(x), over the interval [a,∞), the QAGIU algorithm [156] maps thedomain to (0, 1] using the coordinate transformation x = a+ (1− t)/t, giving∫ ∞adx f(x) =∫ 10dtf(a+ (1− t)/t)t2It is clear that the integrand on the RHS may have singularities. Hence, from this point,the QAGS algorithm [157] is implemented, which is an adaptive method that can handle in-tegrable singularities. QAGS uses adaptive bisection along with the Gauss-Kronrod 21-pointquadrature [158], and the Wynn’s -method [159] to speed up the convergence.QAGTo integrate a function, f(x), over the interval [a, b], the QAG algorithm [155] splits the domaininto subintervals and applies a Gauss-Kronrod quadrature [158]. We opted to use a 61-pointrule (key = 6). The algorithm focuses on subintervals where the integration is less well behaved,and then applies adaptive bisections.∗alternatively, Equation (4.67) when SRCs are desired67Chapter 7ResultsIn Chapter 2, we gave a comprehensive review on all the basic elements needed to constructnuclear theory and the valence space IM-SRG framework from Chapter 5. Additionally, wereviewed the physics and current developments of double-beta decay in Chapter 3, which leadto a thorough derivation of the M0ν TBMEs in Chapter 4. With this footwork in place, we arenow finally ready to present the results of this research project, which were computed using thenumerical methods of Chapter 6.First we will put forth results for 2νββ and then for 0νββ, both for the nucleus Calcium-48, since its doubly magic shell closure makes it an ideal medium-mass nuclei to calibrate ourmany-body method with, and it is the lightest double-beta decay candidate. Accordingly, we’lltake the valence space to be the pf -shell (1p3/2, 1p1/2, 0f7/2, and 0f5/2 orbits). To demonstratethat imsrg++ is working as expected and that we have NMEs which are comparable with thedouble-beta decay community, we will benchmark based on the following principle:Remark 7.1Running VS-IM-SRG without any Magnus evolution, In-Medium normal ordering, or Hartree-Fock step (hence, using a pure harmonic oscillator basis) =⇒ we should be able to match withNMEs from standard ShM results using phenomenological interactions.With this remark in hand, we can check that we’ve properly solved for the M2ν OBMEs (seeSection 3.1.3) and M0ν TBMEs (see Section 4.6), which represents a extensive mathematicalundertaking. From there, we’ll evolve the NMEs using VS-IM-SRG with the Magnus(2) formu-lation (see Section 5.4) and χEFT interactions, and compare the results with our benchmarks.As discussed in Section 5.2.1 and 5.3, the main difference between the benchmarked NMEs andour evolved results will be the inclusion of ab initio interactions and 3N physics.7.1 2νββ NME for 48CaBefore we can test whether VS-IM-SRG is capable of producing neutrinoless double-beta de-cay NMEs which are more or less reliable than current values from other methods, we mustdemonstrate that 2νββ is under control within the theoretical framework. This is a convenientcourse of action using Calcium-48, since we know that it undergoes standard double-beta decay(see Section 3.1.1 and Example 3.1). Also, since no neutrinoless mode has ever been measured,687.1. 2νββ NME for 48Cacalibrating with respect to 2νββ is the first reasonable step. Furthermore, neither 2νββ nor0νββ have been modelled using a fully ab initio approach, so it is imperative that we benchmarkboth these decays against the current phenomenology.To calculate the 2νββ NMEs, we’ll use Equation (3.14), printed again below for convenience,with a summation over k = 1 to 250 intermediate excited states. To obtain the individualGamow-Teller NMEs in the numerator of Equation (7.1), we’ll use nutbar (see Section 6.3)to construct Equation (C26) using the OBMEs from (2.44). Note that the excitation energiesof 48Sc, and its 6+ ground state, will depend on the many-body method used, likewise withthe NMEs. This makes the choice of interaction somewhat tricky for the case of two-neutrinodouble-beta decay, since such an interaction must be tailored to produce both: good energylevels and good GT transitions. For now, we’ll leave this point for future considerations.M2ν(48Ca0+ → 48Ti0+) =250∑k= 1〈48Ti0+ ||“στ+||48Sc1+k 〉〈48Sc1+k ||“στ+||48Ca0+〉E′k − E(48Sc6+g.s) + δE1 +X [MeV](7.1)δE1 ={2.5173 MeV−îE′1 − E(48Sc6+g.s)ó, for phenomenology0, for ab intioX ={1.859700017, for phenomenology1.348701071, for ab intioOne last detail to mention before we proceed is that, for the phenomenological case ofmodelling 2νββ, it is common to shift Ek, for ∀ k, such that the lowest lying (k = 1) excited 1+state matches the experimental value. In Equation (7.1), we’ll include this correction by adding“δE1” into the denominator. The lowest lying excited 1+ state of 48Sc is at 2.5173 MeV,∗ whichwe obtained from [74]. The X term in the denominator is simply the mass contribution, ascalculated for Equation (3.14). The difference between the phenomenology and our ab initiocalculations will be by one me, which we believe to be an non-negligible error.7.1.1 Phenomenological Benchmarking (2νββ)To benchmark M2ν , we’ll apply Remark 7.1 and compare against [71, 73]. The results areplotted in Figure 7.1. As we can see, when we set the quenching factor (see Section 3.1.4), qf ,to 0.77,† the imsrg++ and nutbar codes precisely reproduce the running sum of the M2ν NMEfrom [73]. The match happens because no Magnus evolution was used and all other parametersare matched: 10.49 MeV basis frequency, pf -shell valence space, GXPF1A phenomenologicalinteraction [31, 32], etc. This should be no surprise, since the cited literature values were alsocalculated using NuShellX (see Section 6.2) and a valence space approach.Additionally, both panels show that convergence happens as long as the sum is taken to runover the Gamow-Teller resonances at ≈ 6 MeV. We report our phenomenological result as∗relative to its 6+ ground state†coloured as orange in the Figure 7.1697.1. 2νββ NME for 48CaFigure 7.1: Benchmarking for the two-neutrino double-beta decay NME of 48Ca using apf -shell valence space and the GXPF1A interaction. The oscillator basis frequency was takenas }ω = 10.49 MeV from Equation (2.6) to match with [126]. We’ve plotted the NME as therunning sum via Equation (7.1) as a function of the intermediate nuclei’s excitation energies(relative to the 6+ ground state). The left panel shows our results by running imsrg++ withoutany IM-SRG evolution by setting the maximum flow parameter to smax = 0, in accordancewith Remark 7.1. The right panel has been taken from [73] and adapted by highlighting theGXPF1A result in orange (qf = 0.77), for clarity. The red solid line represents the expectedNME from experiment, see Equation (3.5). To reproduce the phenomenology in the right panel,we adjusted 1.348701071 MeV −→ 1.859700017 MeV ≈ 1.9 MeV in Equation (7.1), which is aconvention used in the literature which we believe to be erroneous (hence this adjustment willnot be used for the ab initio results below).M2ν ={0.09080 MeV−1, qf = 10.05384 MeV−1, qf = 0.77(7.2)which matches the values from [71, 73]. However, without quenching (qf = 1), this result isclearly much larger than the experimentally expected value of 0.03846 MeV−1, and motivatesthe desire to quench the 2νββ in correspondence with single-beta decay (see Section 2.3.4).In conclusion, we’ve shown that imsrg++ can properly reproduce the current phenomenology,∗and so our code has been appropriately benchmarked for 2νββ.∗Note that the first excited states were corrected using δE1 in Equation (7.1), and without this correctionwe were not able to precisely reproduce the phenomenology.707.1. 2νββ NME for 48CaFigure 7.2: VS-IM-SRG evolved two-neutrino double-beta decay NME of 48Ca using a pf -shell valence space and the EM 1.8/2.0 interaction. We’ve plotted the NME as the runningsum via Equation (7.1) as a function of the intermediate nuclei’s excitation energies (relative tothe 6+ ground state). The left panel is identical to that of Figure 7.1, for comparison purposes(it has no Magnus evolution). The right panel is our ab initio result, using emax = 10 and}ω = 16 MeV. The teal curve includes MECs such as those in Figure 3.4 (of the 2νββ variant);the MEC input for the GT operator was provided courtesy of Petr Navra´til [161]. The red solidline represents the expected NME from experiment, see Equation (3.5).7.1.2 VS-IM-SRG Using the EM 1.8/2.0 Interaction (2νββ)We now use the Magnus(2) evolution with the so-called “EM 1.8/2.0” interaction [38],∗ and plotthe results in Figure 7.2. As with the phenomenological benchmarking, we see that convergencehas taken place, since we’ve summed over enough intermediate nuclear states to overcome theGT resonance at ≈ 6 MeV. However, in comparison with phenomenology (reprinted in theleft panel), the evolved M2ν is much lower! It even dips somewhat below the experimentallyexpected result (red solid line). This suggests that an ab initio calculation of nuclear double-beta decay includes significant contributions, such as higher-order many-body forces.Many interactions do not properly reproduce the energy spectra of nuclei from first princi-ples, but such an interaction is needed for Equation (7.1) because of the energy terms in thedenominator. Hence, it should be stressed that the choice of the EM 1.8/2.0 interaction wascrucial for this calculation. In fact, this interaction was so successful in energy calculationsthat we found no measurable difference between using a correction to the first excited states (asdiscussed beneath Equation (7.1) above) and running them fully ab initio. This gives credenceto fact that the EM 1.8/2.0 interaction can be trusted for 0νββ calculations.∗This interaction gets its name since it’s an SRG evolution of the χEFT from Entem and Machleidt (EM) [160]with the LECs labelled as 1.8/2.0. It has been so successful in ab initio calculations for medium mass nuclei thatit is sometimes referred to as the “magic” interaction.717.2. 0νββ NME for 48CaWe report our ab initio result asM2ν ={0.02977 MeV−1, qf = 1, without MECs0.01180 MeV−1, qf = 1, with MECs(7.3)Notice that when we include meson exchange currents (MECs), the NME is suppressed evenlower than desired. This is a conflicting result, since χEFT calculations including MECs mayaccount for quenching [52]. Hence, we had hoped to see that the MECs would correct forthe slight difference between the non-MEC result in Equation (7.3) and the experimentallyexpected value of 0.03846 MeV−1. It should be no surprise that this did not occur, however,since it is possible that important 1+ intermediate states were not captured using this valencespace approach. That is, we expect that there exist 1+ states of 48Sc that would correct forthe small difference between our quoted result and the experimental value, but such states lieoutside the pf -shell valence space.To test this idea in the future, we may: run a non-VS version of IM-SRG, or compare with adifferent many-body method which employs a large single-particle space. A collaborating group,under Dr. Gaute Hagen, uses the Coupled-Cluster (CC) theory, which can perform these typesof calculations. Hence, in upcoming studies, this discrepancy will be resolved. In any case,an argument can be made that both results (with and without MECs) are still significantlycloser to the experimentally expected value than previous calculations using phenomenologicalinteractions and no quenching. Thus, we conclude that 2νββ has been modelled well withinthe framework of VS-IM-SRG, and our following neutrinoless results are trustworthy. In fact,since 0νββ employs the closure approximation, we need not worry about capturing all theintermediate states in this case - and any problems with the above results are not expected totranslate to the following sections.7.2 0νββ NME for 48CaTo calculate the 0νββ NMEs, we’ll use the formulae from Section 4.6, and nutbar (see Sec-tion 6.3) to construct Equation (C33). In particular, to obtain the GT, F, and T TBMEs,we’ll use Equation (4.71) with Equations (4.46), (4.50), and (4.31). Note that these TBMEsare reliant on many parameters, which the double-beta decay community have been somewhatinconsistent with. For example, the µp − µn is often set to 3.7,∗ in order to compare withprevious results in the literature, such as [126]. We will take the parameters from Table 7.1below, and explicitly state whether the “outdated” or “updated” sets of parameters were used.To produce the total M0ν NME, we need to calculate the three decay transitions under the∗If the proton and neutron were elementary particles, then µp = 1 and µn = 0 and therefore µp − µn = 1.However, the measured value is ≈ 4.7, since nucleons are composed of quarks. Sometimes this experimentalparameter is thought of in terms of its deviation from the classical theoretical expectation, hence the confusionbetween 4.7 and 3.7. Surely, 3.7 is erroneous in the context of 0νββ.727.2. 0νββ NME for 48CaParameter UnitVersionOutdated UpdatedE cl0 (48Ca) [MeV] 7.72 7.72mp [MeV] 938.2720814 938.2720814mpi [MeV] 139.57018 139.57018µp − µn [µN ] 3.7 4.706gV,0 [unitless] 1 1gA,0 [unitless] 1.25 1.27ΛV [MeV] 850 850ΛA [MeV] 1086 1086r0 [fm] 1.2 1.2}·c [MeV ·fm] 197.3269718 197.3269718mN [MeV] 938.9187474 938.9187474Table 7.1: Values of M0ν parameters. Two sets of different parameters will be used: one forbenchmarking (third column), and one for the VS-IM-SRG evolution (fourth column). Thechoice of closure energy is nuclei-dependent, and we’ve taken it as 7.72 MeV for 48Ca, whichhas become a standard - although there is some debate as to what makes the optimal closureenergy [118], which should be researched more in the future. The only differences between theoutdated and updated parameters have been highlighted in red, and we’ll see that they indeedmake an effect, which makes benchmarking difficult.sum in Equation (3.21), which we print again below for the convenience of the reader.M0ν = M0νGT −ÇgVgAå2M0νF +M0νT (7.4)gV = 1, gA ={1.25, outdated1.27, updatedNotice that the factor in front of the Fermi term in the sum is there to make it independentof the value of the axial-vector coupling constant - which changes as experiments become moreprecise. First we will conduct the phenomenological benchmarking, as was done in Section 7.1.1above, and then we will present results using two different χEFT interactions, both with andwithout the inclusion of short-range correlations (SRCs).7.2.1 Phenomenological Benchmarking (0νββ)To benchmark M0ν , we’ll apply Remark 7.1 and compare against [71, 118, 126, 133]. To makea proper parallel between said references, we’ll use the parameters listed in the “outdated”column of Table 7.1; and then, to see what difference the corrected parameters induce, we’llalso run results for the “updated” column. We’ll see that the use of these different parametersmake a small difference, hence contrasting any results from Section 7.2.2 or 7.2.3 with thesebenchmarks is valid. The results are tabulated in Table 7.2 below.737.2. 0νββ NME for 48CaSRCs: none AV18Parameters: Outdated Updated Outdated UpdatedM0νGT 0.6767 0.7090 0.6487 0.6735M0νF –0.2074 –0.2074 –0.2041 –0.2041M0νT –0.06954 –0.07479 –0.07153 –0.07702M0ν 0.7400 0.7628 0.7078 0.7230Table 7.2: Benchmarking for the neutrinoless double-beta decay NME of 48Ca using a pf -shell valence space and the GXPF1A interaction. The oscillator basis frequency was takenas }ω = 10.49 MeV from Equation (2.6) to match with [126]. We ran imsrg++ without anyIM-SRG evolution by setting the maximum flow parameter to smax = 0, in accordance withRemark 7.1. Both sets of parameters from Table 7.1 were used, along with SRCs set to noneor the AV18 parameterization. Note that the concept of a model space size (emax) is irrelevanthere, since no decoupling will occur without running the IM-SRG (see Section 5.1.1). However,since an emax must be set to run the imsrg++ code, by construction, multiple emax’s were checkedand they all gave identical results, modulo random pivots used for the Lanczos algorithm inNuShellX (which changed the results negligibly to the 6th significant digit).Without SRCs (Benchmarking)First, we’ll discuss the results in Table 7.2 without any SRCs (see Section 4.5.1). M0ν isreported in the literature as: 0.711, from Table 3 of [126]; and 0.833, from Table 1 of [133].∗[126] used the outdated parameters from Table 7.1, hence why this value matches well with ourreported value of 0.7400 from Table 7.2. We propose that the slight difference in these values(0.711 and 0.7400) originates from [126] using a different ShM code. This builds our confidencethat we have properly coded 0νββ into the imsrg++ framework.We could not reproduce the value from [133], since the GXPF1B interaction is currentlynot available within NuShellX using the PN formalism. Nonetheless, their value is still closeto our reported result of 0.7628 from Table 7.2. Note that we’ve assumed they have used theupdated parameters listed in Table 7.1, since it is not made explicit in the text, but it is arecent publication. The authors have used a closure energy of 0.5 MeV, and they state thatusing 7.72 MeV would reduce their value by roughly 5%, giving 0.792. This is hopeful, since it isreasonable to presume that the difference between our NME and that from [133] originates fromthe difference between the GXPF1A versus GXPF1B interactions. More conclusive evidencethat we have properly benchmarked 0νββ will be discussed in the following section.The differences between using the outdated and updated parameters from Table 7.1 areapproximately: 5% for the Gamow-Teller, 0% for the Fermi, and 8% for the Tensor. This yieldsa 3% difference in the total NME, for the phenomenology without SRCs - which is negligiblecompared to the closure approximation (see Assumption 2 and Section 3.3.4). Note that the0% difference in the Fermi term is to be expected, since this transition has no axial-vector ormagnetic moment dependence.∗note that the GXPF1B interaction was used in the latter publication747.2. 0νββ NME for 48CaWith SRCs (Benchmarking)Now we’ll discuss the results from Table 7.2 including the SRCs; in particular, we choose theAV18 parameterization of the Jastrow-type function in Equation (4.66). That is, we’ll set a, b, cto the values found in row 1 of Table 4.1, and apply Equation (4.70) with (4.62) for the M0νTBMEs. M0ν is reported in the literature as: 0.779, from Table 3 of [126]; 0.82, from Table 2of [71]; 0.729, from Table 1 of [118]; and 0.801, from Table 1 of [133].∗ It is encouraging to seethat all these values are in the same ballpark, and the small variations can be account for bythe different methods and inputs used.The best match with our result of 0.7230 from Table 7.2 are those from [118]. To be explicitwith the comparison, see Table 7.3 below. The small difference between these values can beaccounted for by the use of gA = 1.254 in [118], whereas we used 1.27. Another source oferror comes from the use of different integration techniques (see Section 6.4). Otherwise, thisreplication is precise, which is to be expected since we’ve duplicated the methods used in [118];in contrast to [71, 126, 133]. From this, we conclude that the 0νββ TBMEs have been properlyincorporated into our framework, and we can now confidently proceed to evolving the NMEs.Our Values Values from [118]M0νGT 0.6735 0.676M0νF –0.2041 –0.204M0νT –0.07702 –0.077M0ν 0.7230 0.729Table 7.3: Comparison of the neutrinoless double-beta decay NMEs, for benchmarking.The first column of values has been taken from the fourth column of Table 7.2, and the secondcolumn of values are the “Closure” NMEs from Table 1 of [118]. This clearly shows that we havebenchmarked 0νββ properly, and that [118]’s values are indeed independently reproducible.It is also worth noting that the introduction of SRCs has made some effect. This totalchange of roughly 5% is arguably comparable to the use of the closure approximation, and soit can be said that SRCs may yield non-negligible contributions. However, this presents a newchallenge, since many different sets of SRCs exist (see Table 4.1), and they each give differentresults, see [118, 123]. Hence, future work must be conducted with the purpose of definitivelyestablishing which set of SRCs are optimal for 0νββ.7.2.2 VS-IM-SRG Using the EM 1.8/2.0 Interaction (0νββ)Now that we know the M0ν NMEs have been coded properly within the imsrg++ framework,we can proceed with evolving them using the Magnus(2) formulation (see Section 5.4). Thisevolution will capture more of the induced many-body physics happening with in the pf -shellvalence space (see Section 5.3), and hence it should give a more accurate NME than the standardphenomenological ShM calculations presented in Section 7.2.1. To ensure that we have a more∗note that the GXPF1B interaction was used in the latter publication757.2. 0νββ NME for 48Caprecise model of 0νββ, we’ll exclusively use the updated set of parameters from the fourthcolumn of Table 7.1. Coupled with an appropriate χEFT interaction, this is the first time that0νββ has been calculated using a fully ab initio, non-perturbative method.Since the EM 1.8/2.0 interaction [38] has been so successful, and it convincingly modelled2νββ in 48Ca within the pf -shell (see Section 7.1.2), we will start by using this as our interactioninput. MECs, on the other hand, will not be included since their contributions to 0νββ havenot been developed at this time, by any study. Since we use the closure approximation for0νββ, we need not worry about missing intermediate states as in 2νββ, which we’ve proposedare the source of the difference between the experimental M2ν and our evolved values.Without SRCs (EM 1.8/2.0)First, we’ll present results without any SRCs (see Section 4.5.1). They can be seen in Figure 7.3,along with an analysis of the convergence. These results clearly show that the inclusion of many-body physics, via a fully ab initio calculation, has decreased the total 0νββ NME by about20%, in comparison with the phenomenological results from Table 7.2. Via Equation (3.18), wepredict a half-life which is approximately 60% longer than the current phenomenology. Froman experimental perspective, this is clearly not favourable, since more source material will berequired. It is not clear if this extrapolates to heavier nuclei, and it should be stressed thatno theoretical arguments exist which forbid the induced many-body physics from giving theopposite trend. Furthermore, MECs will make an appreciable effect. VS-IM-SRG calculationsfor heavier nuclei, and the inclusion of MECs, will be studied in future work.With SRCs (EM 1.8/2.0)Now we’ll present results including SRCs; in particular, we choose the AV18 parameterizationof the Jastrow-type function in Equation (4.66). That is, we’ll set a, b, c to the values foundin row 1 of Table 4.1, and apply Equation (4.70) with (4.62) for the M0ν TBMEs. Theycan be seen in Figure 7.4, along with an analysis of the convergence. A significant differencehas been introduced by the inclusion of SRCs here. The evolved SRC NMEs are roughly30% lower than the phenomenological SRC values from Table 7.2. Via Equation (3.18), thiscorresponds to a half-life which is 2 times longer than the phenomenology. As mentioned atthe end of Section 7.2.1, however, more research is needed to develop a methodology for whichSRC parameters are optimal, so any interpretation of these results is likely premature.The evolved SRC values are approximately 20% lower those without any SRCs, in com-parison to the 5% drop seen for the phenomenology. This is well beyond any error introducedby the closure approximation, and therefore it could be proposed that SRCs are indeed non-negligible for a VS-IM-SRG evolution. It is interesting to note that the convergence patternseen in Figure 7.4 matches that seen in Figure 7.3, which provides evidence that any apparentconvergence irregularities are actually consistent within the calculations - as opposed to beingcomputational anomalies within the IM-SRG framework (see Equation (5.10), for instance).767.2. 0νββ NME for 48Ca7.2.3 VS-IM-SRG Using the 500/400 N3LO+3N Interaction (0νββ)Here, we’ll present results without any SRCs, and using a different interaction from Section 7.2.2above. The purpose is twofold: we have access to more basis frequencies for this interactionto assess convergence, and it is important to check how using a different interaction will affectthe NMEs overall. The interaction we choose here is referred to as “500/400 N3LO+3N.”∗ Theresults can be seen in Figure 7.5, along with an analysis of the convergence.Visually, it is much clearer that these calculations converge, as compared to those in Fig-ure 7.3, since more basis frequencies were available using this interaction. Of course, if wewould’ve had more basis frequencies for EM 1.8/2.0, the coloured lines would have approachedeach other in a similar manner, as emax increases. However, the speed of convergence is inter-action dependant. Although it appears that these values converge faster than those from theEM 1.8/2.0 interaction, we choose to trust the latter NMEs more, since this interaction was bet-ter suited for the 2νββ calculations. Thus, the much lower NME of approximately 0.27 shouldnot be interpreted physically - we only performed these calculations to check convergence.7.2.4 Summary (0νββ)Since we’ve developed the power to calculate 0νββ using imsrg++ in a general and efficientmanner, we were able to present many results. Hence, for succinctness, in Table 7.4 we sum-marize our final results for neutrinoless double-beta decay. We may extrapolate our NME for0νββ without SRCs as M0ν ≈ 0.6. Remarkably, this value matches with that reported in Table1 of [163], which used an interacting ShM approach with full calculations in the pf -shell thatimplemented: seniority truncations, nuclear currents, and took account for pairing affects.M0ν }ω [MeV]Interaction SRC 12 16 20 24EM 1.8/2.0 none - 0.5663 - 0.6300EM 1.8/2.0 AV18 - 0.4718 - 0.5219500/400 none 0.2846 0.2629 0.2674 0.2837Table 7.4: Summary of the VS-IM-SRG evolved neutrinoless double-beta decay NMEs of48Ca using a pf -shell valence space. All of the entries are the total sum, M0ν , using the updatedparameters from Table 7.1. Convergence is set at emax = 10 (see Figures 7.3 to 7.5). Resultsusing the EM 1.8/2.0 interaction are considered more trustworthy, as discussed in the text. The500/400 N3LO+3N interaction is only included to demonstrate the independence of NMEs as afunction of }ω at this emax. The latter interaction’s results vary over ≈ 8% with respect to }ω,and the former’s results vary over ≈ 11%; which is comparable to the closure approximation.The SRCs, induce a difference of ≈ 20%, and so they may be considered non-negligible.We now print the figures referenced to throughout the text of this section.∗This interaction first appeared in [135]. In accordance with [162], a χEFT interaction was built from N3LOfor NN and N2LO for 3N and then SRG evolved starting with ΛNN = 500 MeV and Λ3N = 400 MeV - hence thenaming of the interaction.777.2. 0νββ NME for 48CaFigure 7.3: VS-IM-SRG evolved neutrinoless double-beta decay NME of 48Ca using a pf -shellvalence space and the EM 1.8/2.0 interaction. For comparison, the black dashed line representsthe benchmarking values from Table 7.2, using the updated M0ν parameters and no SRCs. Inaccordance with Remark 5.1, we see that as emax increases, the NMEs indeed converge to aresult independent of the oscillator basis frequency }ω. Under the sum of Equation (7.4), we cansee that the Fermi and Tensor components roughly cancel, leaving the Gamow-Teller as a goodapproximation for the total NME, labelled as M (0ν) in the top-right panel. Note that emax = 4results are not expected to give a consistent calculation for the pf -shell, hence the apparentconvergence irregularities. Only two basis frequencies were available for this interaction.787.2. 0νββ NME for 48CaFigure 7.4: VS-IM-SRG evolved neutrinoless double-beta decay NME of 48Ca using a pf -shell valence space and the EM 1.8/2.0 interaction, including AV18 short-range correlations. Forcomparison, the black dashed lines represent the benchmarking values from Table 7.2, using theupdated M0ν parameters, both with SRCs set to none and AV18. We also plot the emax = 10values from Figure 7.3 (results without SRCs) in dashed lines, with the colours correspondingto their respective }ω. In accordance with Remark 5.1, we see that as emax increases, the NMEsindeed converge to a result independent of the oscillator basis frequency }ω. Under the sum ofEquation (7.4), we can see that the Fermi and Tensor components roughly cancel, leaving theGamow-Teller as a good approximation for the total NME, labelled as M (0ν) in the top-rightpanel. Note that emax = 4 results are not expected to give a consistent calculation for the pf -shell, hence the apparent convergence irregularities. Only two basis frequencies were availablefor this interaction.797.2. 0νββ NME for 48CaFigure 7.5: VS-IM-SRG evolved neutrinoless double-beta decay NME of 48Ca using a pf -shellvalence space and the 500/400 N3LO+3N interaction. For comparison, the black dashed linerepresents the benchmarking values from Table 7.2, using the updated M0ν parameters and noSRCs. In accordance with Remark 5.1, we see that as emax increases, the NMEs indeed convergeto a result independent of the oscillator basis frequency }ω. Under the sum of Equation (7.4),we can see that the Fermi and Tensor components roughly cancel, leaving the Gamow-Teller asa good approximation for the total NME, labelled as M (0ν) in the top-right panel. Note thatemax = 4 results are not expected to give a consistent calculation for the pf -shell, hence theapparent convergence irregularities. Four basis frequencies were available for this interaction,making the convergence clearer - however, the EM 1.8/2.0 results from Figure 7.3 are consideredmore trustworthy, as discussed in the text.80Chapter 8ConclusionsModern nuclear many-body methods, such as the VS-IM-SRG using χEFT interactions, havebeen developed beyond simply an “innovative” status - that is, they are now an indispensabletool for calculating nuclear properties and decays. In this dissertation, we have demonstratedthat VS-IM-SRG is capable of modelling rare decays, like two-neutrino double-beta decay, andexotic ones, like neutrinoless double-beta decay. What’s more is that this non-perturbative abinitio approach can capture three-body physics using two-body machinery, which is imperativefor making predictions in medium mass nuclei like Calcium-48.We report evolved NMEs for 2νββ which are roughly 3 times smaller than the phenomeno-logical NMEs, without quenching [71, 73]. This 2νββ NME for 48Ca from VS-IM-SRG is 23%lower than the experimentally expected value, which is the opposite direction to quenching(see Figure 7.2). The inclusion of meson exchange currents did not fix this issue, and insteadlowered the NME further. This is to be expected, since the 2νββ case does not use the clo-sure approximation, and important intermediate 1+ states are likely missing from the pf -shellvalence space. A comparison will be made against Coupled-Cluster theory in the near future.Our 0νββ NMEs were approximately 20% smaller compared to ShM calculations usingthe GXPF1A interaction [118]. By Equation (3.18), this represents a 60% increase in thehypothetical 0νββ half-life. The inclusion of the AV18 SRC parameterization dropped M0νanother 20%∗ - a larger difference than the 5% change seen in the phenomenological values. Theprimary interaction of choice in this dissertation was the EM 1.8/2.0 from χEFT [38], althoughmore options should be studied in the future, in order to understand which interactions aremost suitable for the double-beta decay operator structures (see Section 7.1 and Chapter 4).We propose that ab initio many-body contributions are the main source of our unique NMEs.However, more research is needed to establish that this is indeed the case, as opposed to spuriousCoM contamination [137] or generator cancellations (see Equation (5.10) in Section 5.2.2). Astudy directed at how perturbations to the closure energy alter the VS-IM-SRG evolved 0νββNMEs is recommended. Also, since SRCs come in many different types (see Table 4.1), it wouldbe ideal to find the optimal parameterization. A more pressing matter is running the IM-SRGevolution to higher emax = 12 and 14, which can be done now the the new “Oak” cluster† hasbecome available to the TRIUMF Theory Department. 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Academic Press, 1980.94Appendix AAngular Momentum CouplingWhen creating a state composed of many particles, we must ask: how are our good quantumnumbers coupled together? By “good” quantum numbers, we mean those which are eigenvaluesof a (time-independent) Hermitian operator, “T, which is compatible with the Hamiltonian, “H,so that ∂“T/∂t = 0 and [“H, “T ] = 0. This guarantees two things: i) the Hamiltonian and “T aresimultaneously diagonalizable, hence they share the same complete set of eigenfunctions; andii) the quantum numbers of “T will be conserved via the Heisenberg Equation of Motion [164, 165]ddt〈“T(t)〉 = i}〈[“H, “T(t)]〉+ 〈∂“T∂t〉For example, in a non-relativistic system with a spherically symmetric central potential,the orbital angular momentum, l, and the intrinsic angular momentum or “spin,” s, will bethe good quantum numbers, whereas in a relativistic system the good quantum number for theDirac equation is (total) angular momentum, also sometimes called “spin,” “J = “L + “S , whichgives j. So this begs the question, which coupling schemes are most useful for certain multipleparticle systems? As a brief reminder before we approach this question: the orbital angularmomentum, spin, and total angular momentum all follow the same algebraic structure[L̂i, L̂j ] = iijkL̂k and, [“L2, “L] = ⇀0where ijk is the Levi-Civita symbol, and the second commutation relation guarantees that thesquare of the vector angular momentum operator is compatible (simultaneously diagonalizable)with each component of the angular momentum, hence they share the same eigenstates. Usingladder operators, it is an undergraduate exercise to show that the eigenvalues areL̂z|l m〉 = m}|l m〉 for, − l ≤ m ≤ l“L2|l m〉 = l(l + 1)}2|l m〉 where, l ∈ N0A.1 Clebsch-Gordan CoefficientsWe start by defining the well known “Clebsch-Gordan (CG) coefficients.” We’ll denote thesecoefficients as (j1m1 j2m2 | j m), which couples the states |j1m1〉 and |j2m2〉 as|j1m1〉⊗|j2m2〉 .= |j1m1, j2m2〉 into |j1 j2:jm〉, via|j1 j2:jm〉 =∑m1,m2(j1m1 j2m2 | j m)|j1m1, j2m2〉 (A1)or visa versa95A.1. Clebsch-Gordan Coefficients|j1m1, j2m2〉 =∑j,m(j1m1 j2m2 | j m)|j1 j2, jm〉 (A2)We’ll commonly refer to the coupling |j1 j2:jm〉 as “J-scheme,” and the coupling |j1m1, j2m2〉as “M -scheme.” The CG coefficients satisfy two conditions, by definition(j1m1 j2m2 | j m) = 0 unless,m1 +m2 = m and, (A3)∆(j1 j2 :j) (A4)where we define the “triangle condition” in Equation (A4), denoted as ∆(j1 j2 :j), via|j1 − j2| ≤ j ≤ j1 + j2 (A5)where j goes in increments of ±1.The triangle condition above encodes what is meant by “coupling” quantum numbers to-gether. You see that j is not simply the sum of the two quantum numbers j1 and j2, despitethe fact that we have taken “J = “J1 + “J2 in J-scheme, which we call “jj-coupling.” Comparethis with what we call “ls-coupling,” when we take “J = “L+ “S , which obeys ∆(l s :j). The CGcoefficients also obey both “orthogonality” and “completeness,” respectively written as∑m1m2(j1m1 j2m2 | j m)(j1m1 j2m2 | j′m′) = δjj′δmm′ (A6)∑jm(j1m1 j2m2 | j m)(j1m′1 j2m′2 | j m) = δm1m′1δm2m′2 (A7)Now we introduce an unfortunate notation commonly used in angular momentum coupling.It’s neither a quantum mechanical operator, nor a unit vector, nor just a fancy hat. It’s calledthe “hat factor,” and it’s simply a common prefactor written asÛ ≡ √2j + 1 (A8)As with standard convention, we define the CG coefficients such that they are real numbers.The most general form of the CG coefficients in this convention is:Definition A.1: “Clebsch-Gordan Coefficient”The coefficients, (j1m1 j2m2 | j m), that satisfy Equation (A1) and (A2), may be defined suchthat (j1m1 j2m2 | j m) ∈ R via the formula(j1m1 j2m2 | j3m3) =∑8k8(−1)kk!Û× 1(j1+j2−j3−k)!(j1−m1−k)!(j2+m2−k)!(j3−j2+m1+k)!(j3−j1−m2+k)!96A.1. Clebsch-Gordan Coefficients×Ã3∏k= 1[ 3∑n= 1(−1)δkn jn]!Ã3∏n= 1(jn +mn)!(jn −mn)!/Ã[(3∑n= 1jn) + 1]!where we have denoted 8k8 as running that particular sum component over ∀ k such that thearguments of any factorial are non-negative for the allowed values of the j’s and m’s.There are several identities that come from this formula that we shall make use of(j1m1 j2m2 | j m) = (−1)j1+j2−j (j2m2 j1m1 | j m) (A9)= (−1)j1+j2−j (j1, −m1, j2, −m2 | j, −m) (A10)= (−1)j1−m1 Û Û−12 (j1m1, j, −m | j2, −m2) (A11)= (−1)j1−m1 Û Û−12 (j m, j1, −m1 | j2m2) (A12)(j1m1 j2m2 | 0 0) = δj1j2δm1,−m2(−1)j1−m1 Û−11 (A13)(j1m1 0 0 | j2m2) .= (0 0 j1m1 | j2m2) = δj1j2δm1m2 (A14)(j1 0 j2 0 | j 0) = 0 unless, j1 + j2 + j = even (A15)Note that many of the equations above are derivative of each other; for instance, the identityin Equation (A14) comes from (A11) with (A13), and then the equivalence is from (A9).Example A.1Due to the fermionic nature of nucleons, we will commonly take our states and anti-symmetrizethem. Let’s start with our two particle states in M -scheme, with the shorthand, |12〉 ≡|j1m1, j2m2〉 and |34〉 ≡ |j3m3, j4m4〉. Then anti-symmetrization happens via|j1m1; j2m2〉 .= |1; 2〉 .= |12〉A ≡ 1√2Ä|12〉 − |21〉ä(A16)Notice the notation in the equation above, where we drop the subscript A, representing “anti-symmertization,” and simply use a semicolon within the ket to denote anti-symmetrization.To find the anti-symmetrized analog in J-scheme, all we need to do is transform between|j1m1; j2m2〉 −→ |j1 j2; jm〉 via the CG coefficients. Using Equation (A1) with Equation (A16),|j1m1; j2m2〉 −→ |j1 j2; jm〉 =∑m1m2(j1m1 j2m2 | j m)|j1m1; j2m2〉=∑m1m2(j1m1 j2m2 | j m) 1√2Ä|12〉 − |21〉ä=1√2[ ∑m1m2(j1m1 j2m2 | j m)|12〉− (−1)j1+j2−j∑m1m2(j2m2 j1m1 | j m)|21〉]97A.1. Clebsch-Gordan Coefficients|j1 j2; jm〉 = 1√2Ä|j1 j2:jm〉 − (−1)j1+j2−j |j2 j1:jm〉ä(A17)where in the third line we used the identity in Equation (A9).In this example, we want to compute the inner-product between 〈12| and |34〉 both forM -scheme and J-scheme. In M -scheme the anti-symmetrized inner-product is straightforward〈1; 2|3; 4〉 .= 1√2Ä〈12| − 〈21|ä· 1√2Ä|34〉 − |43〉ä=12Ä〈12|34〉 − 〈12|43〉 − 〈21|34〉+ 〈21|43〉ä=12(δ13δ24 − δ14δ23 − δ23δ14 + δ24δ13)〈1; 2|3; 4〉 = δ13δ24 − δ14δ23 (A18)Note that in Equation (A18) the short hand δab.= δnanbδlalbδsasbδjajbδmamb for a, b = 1, 2, 3, 4,although we will commonly omit the n’s because it is understood. Now we can use this result tofind the same inner-product in J-scheme. Let’s couple our anti-symmetric states as |1; 2〉 −→|j1 j2; jm〉 and |3; 4〉 −→ |j3 j4; j′m′〉, to find the inner-product〈j1 j2; jm|j3 j4; j′m′〉 =∑m1m2∑m3m4(j1m1 j2m2 | j m)(j3m3 j4m4 | j′m′)(δ13 · δ24 − δ14 · δ23)=∑m3m4(j1m3 j2m4 | j m)(j3m3 j4m4 | j′m′) δl1l3δs1s3δj1j3 · δl2l4δs2s4δj2j4−∑m3m4(j1m4 j2m3 | j m)(j3m3 j4m4 | j′m′) δl1l4δs1s4δj1j4 · δl2l3δs2s3δj2j3(A19)In the first line above we used Equation (A1) and (A18), and then we resolved the sums overthe δm1m3δm2m4 − δm1m4δm2m3 .At this point, it is convenient to introduce a new notation, where δab.= δnanbδlalbδsasbδjajb(notice the lack of m’s). On the leftmost sum of Equation (A19), the deltas give j1 = j3 andj2 = j4; on the rightmost sum, the deltas give j1 = j4 and j2 = j3, so we may write〈j1 j2; jm|j3 j4; j′m′〉 =∑m3m4(j3m3 j4m4 | j m)(j3m3 j4m4 | j′m′) δ13δ24−∑m3m4(−1)j3+j4−j′ (j3m3 j4m4 | j m)(j3m3 j4m4 | j′m′) δ14δ23= δjj′δmm′δ13δ24 − (−1)j3+j4−j′ δjj′δmm′δ14δ23=⇒ 〈j1 j2; jm|j3 j4; j′m′〉 = δjj′δmm′ [δ13δ24 − (−1)j3+j4−j′ δ14δ23].= δjj′δmm′ [δ13δ24 − (−1)j1+j2−j δ14δ23](A20)In the first line above we used Equation (A9), in the second (A6), and the equivalence in thelast line we leave as a simple exercise to the reader. We’ll eventually find it useful to leave thecoupling notation in Equation (A20) generalized, as in98A.2. The Wigner 3j-Symbols〈(x1y1)z1, (x2y2)z2; zm|(x3y3)z3, (x4y4)z4; z′m′〉 =δzz′δmm′ [δ13δ24 − (−1)z3+z4−z′ δ14δ23]δzz′δmm′ [δ13δ24 − (−1)z1+z2−z δ14δ23](A21)where here we used the notation δab.= δnanbδxaxbδyaybδzazb , for a, b = 1, 2, 3, 4.There is much physics encoded in the above math already. For instance, in the notationof Equation (A20), let’s consider the normalization where we set j1 = j2 = j3 = j4 ≡ j′′, andj = j′, m = m′, which will give〈j′′ j′′; jm|j′′ j′′; jm〉 .= 1 − (−1)2j′′−j ={2, j = even0, j = oddTo get the conditions above we used the fact that an individual angular momentum is ahalf-integer so 2j′′ = odd, and for the phase: odd − even = odd, and odd − odd = even. Thisequation reveals two important points about J-scheme:i) our anti-symmetric states only couple to even total j, which encodes the Pauli principleii) our anti-symmetric states are still not fully normalizedTo compensate for the second point above, it is common to redefine Equation (A17) as|j1 j2; jm〉 ≡ 1»2(1 + δ12)Ä|j1 j2:jm〉 − (−1)j1+j2−j |j2 j1:jm〉ä(A22)however, many do this normalization a posteriori. When we employ Equation (A22), it willbe made entirely explicit. Note that, in this scheme: |j1 j2; jm〉 .= |(l1s1)j1, (l2s2)j2; jm〉 and|j2 j1:jm〉 .= |(l2s2)j2, (l1s1)j1; jm〉. We leave it as a small exercise for the reader to check thatEquation (A22) indeed yeilds |j′′ j′′; jm〉 = |j′′ j′′:jm〉 for even total j.A.2 The Wigner 3j-SymbolsIn the equations of angular momentum coupling, Eugene Wigner, a pioneer of symmetries inquantum mechanics [166], must have noticed a peculiar pattern between phase factors, hatfactors, and alike. Hence, the following definition was made:Definition A.2: “(Wigner) 3j-Symbols”Çj1 j2 jm1 m2 må≡ (−1)j1−j2−m Û−1(j1m1 j2m2 | j, −m) (A23)99A.3. Coupling Three Angular MomentaPhysically, the 3j-symbol is the probability amplitude that three states are coupled to producezero angular momentum. That isÇj1 j2 j3m1 m2 m3å∝∑j′m′(j1m1 j2m2 | j′m′)(j′m′ j3m3 | 0 0)Clearly the 3j-symbol will inherit the relevant conditions in Equation (A3) and (A4) fromthe CG coefficient on the RHS of Equation (A23). So we’ll get the conditionsÇj1 j2 jm1 m2 må= 0 unless,m1 +m2 +m = 0 and, (A24)∆(j1 j2 :j) (A25)It is important to note the use of parenthesized matrix notation for the Wigner 3j-symbolitself, as the type of brackets used will distinguish it from the other Wigner symbols to come.The 3j-symbol has the symmetry that switching any two columns will yield a phase factor, forexample switching the second and third columns yieldsÇj1 j2 jm1 m2 må.= (−1)−j1−j2−jÇj1 j j2m1 m m2å.= ... (A26)A.3 Coupling Three Angular MomentaThe CG coefficients are useful for coupling two particles together between M -scheme and J-scheme. The next natural question is: how does one couple more than two particles? We start bycoupling three particles, each associated with compatible “J1, “J2, and “J3, into “J = “J1 + “J2 + “J3.There are three intermediate ways we can couple to produce “J : couple particle 1 to 2, and thenthe result to 3; couple 2 to 3, and then the result to 1; or couple 1 to 3, and the result to 2.To transform between these couplings, we define the analog to CG coefficients, known asthe “6j-symbols.” For instance:Definition A.3: “(Wigner) 6j-Symbols”To transform from the jj-couplings “J12 = “J1 + “J2, “J = “J12 + “J3 to “J23 = “J2 + “J3,“J = “J12 + “J3, we use the “(Wigner) 6j-symbol,” denote as®j1 j2 j12j3 j j23´in the J-scheme summation|j1, (j2 j3) j23:j m〉 =∑j12(−1)j1+j2+j3+j Û12 Û23 ®j1 j2 j12j3 j j23´ |(j1 j2) j12, j3:j m〉It can be shown that the 6j-symbols are related to the 3j-symbols, and in turn to the CG100A.4. Coupling Four Angular Momentacoefficients of two particle coupling, via the formula®j1 j2 j3j4 j5 j6´≡∑m1,...,m6(−1)∑6k=1(jk−mk)Çj1 j2 j3m1 m2 −m3åÇj1 j5 j6−m1 m5 m6å×Çj4 j5 j3m4 −m5 m3åÇj4 j2 j6−m4 −m2 −m6åand that they must obey the triangle conditions®J1 J2 J3j1 j2 j3´= 0 unless,∆(J1 J2 :J3), ∆(j1 j2 :J3)∆(j1 J2 :j3), ∆(J1 j2 :j3) (A27)Similar to Equation (A26) for 3j-symbols, the 6j-symbols will be invariant under a swap ofany two columns horizontally, or the vertical reflection of any two columns as so®J1 J2 J3j1 j2 j3´.=®J1 J3 J2j1 j3 j2´.=®J3 J2 J1j3 j2 j1´.= ... (A28)®J1 J2 J3j1 j2 j3´.=®J1 j2 j3j1 J2 J3´.=®j1 J2 j3J1 j2 J3´.= ... (A29)Finally, if the lower right entry of the 6j-symbol is zero, we can reduce the 6j-symbol as follows®J1 J2 J3j1 j2 0´.= δ(J1J2:J3)δJ1j2δJ2j1(−1)J1+J2+J3 ÛJ −11 ÛJ −12 (A30)where we’ve introduced the so-called “triangular delta,” defined byδ(j1j2:j) ≡{1, ∆(j1 j2 :j)0, o.w.(A31)A.4 Coupling Four Angular MomentaThe coupling coefficients for four particles are known as “normalized 9j-symbols.” Unlike the6j-symbols, these are denoted by square brackets:Definition A.4: “(Wigner) 9j-Symbols”To transform from the jj-couplings “J12 = “J1 + “J2, “J34 = “J3 + “J4, “J = “J12 + “J34 to“J13 = “J1 + “J3, “J24 = “J2 + “J4, “J = “J13 + “J24, we use the “normalized (Wigner) 6j-symbol” j1 j2 j12j3 j4 j34j13 j24 jin the J-scheme summation|(j1 j3) j13, (j2 j4) j24:j m〉 =∑j12j34 j1 j2 j12j3 j4 j34j13 j24 j |(j1 j2) j12, (j3 j4) j34:j m〉 (A32)101A.4. Coupling Four Angular MomentaThe standard “(Wigner) 9j-symbols” are denoted by curly brackets, and are related to thenormalized 9j-symbol simply by a product of hat factors j1 j2 j12j3 j4 j34j13 j24 j ≡ Û12 Û34 Û13 Û14 j1 j2 j12j3 j4 j34j13 j24 j (A33)By construction, the normalized Wigner 9j-symbols can also be used to transform fromls-coupling to jj-coupling, simply by relabelling Equation (A32) as|(l1s1)j1, (l2s2)j2:jm〉 =∑l,sl1 l2 ls1 s2 sj1 j2 j |(l1l2) l, (s1s2) s:jm〉 (A34)It can be shown that the 9j-symbols are related to the 3j-symbols, and in turn to the 6j-symbolsof three particle coupling, via the mnemonic “rap one byt, rob any pet” as followsr a po n eb y t ≡∑∀m′sÇr a pmr ma mpåÇo n emo mn meåÇb y tmb my mtå×Çr o bmr mo mbåÇa n yma mn myåÇp e tmp me mtå (A35).=∑M(−1)2M ıM ®r o by t M´®a n yo M e´®p e tM r a´(A36)and that they must obey the triangle conditions for each of their rows and columnsr a po n eb y t = 0 unless,∆(r o :b), ∆(an :y), ∆(p e : t)∆(r a :p), ∆(o n :e), ∆(b y : t) (A37)A particularly useful identity of the 9j-symbol is that by switching any two rows or columns,we pick up a phase factor of the sum of the components, Σ ≡ r + o+ b+ a+ n+ y + p+ e+ t.For example, we can switch the first and third rows, or the first and third columns, viar a po n eb y t .= (−1)Σb y to n er a p .= (−1)Σp a re n ot y b .= ... (A38)Finally, a 9j-symbol with a zero entry can be reduced down to a 6j-symbol, as followsj u Rl c rD d 0 .= δRr δDd (−1)u+R+l+D ÙR−1ÙD−1 ®j u Rc l D´ (A39)A nice here trick is: if any other arbitrary element of a 9j-symbol is zero, then by repeatedlyusing Equation (A38) and then (A39), such a 9j-symbol can be reduced to a 6j-symbol.102Appendix BSpherical Tensor OperatorsIn quantum mechanics, we need a meaningful definition of “tensors,” in conjunction with anyother field. Tensors in physics are defined by their ability to transform in a manner that encodesthe geometry of said physical system. In fluid mechanics, one uses tensors that transform in aCartesian space; they rotate in the three dimensions of Euclidean geometry. In General Rela-tivity, one uses tensors that transform in a spacetime; they hyper-rotate in a locally Lorentzianmanifold of four dimensional Riemannian geometry. The case is wildly different in quantummechanics, since our geometrical structure is encoded by the coupling of angular momenta.B.1 DefinitionsConsider a rotation, R ∈ SO(3), which may be parameterized by Euler angles or rotationsabout the x, y, z axes, etc. In quantum mechanics, the unitary form of R is generated by theangular momentum operator. That is, for the rotation about “n by the angle θ, thenU(R(θ)) = exp(−iθL̂/}) ∈ SU(2) (B1)We can encapsulate how these unitary rotations transform quantum states by making thefollowing definition:Definition B.1: “Wigner D-Matrix”For a unitary rotation, such as the one in Equation (B1) above, acting on the state |l m〉, the“Wigner D-Matrix” is defined byDlm′m ≡ 〈l m′|U(R)|l m〉 =⇒ U(R)|l m〉 =∑m′Dlm′m|l m′〉 (B2)Now that we have a way of rotating states, we can define a “spherical tensor (operator)” as anoperator which transforms under rotations as so:Definition B.2: “Spherical Tensor (Operator)”If an operator, labelled “TLM , transforms under a unitary rotation, such as the one in Equa-tion (B1), via the Wigner D-Matrix likeU(R)“TLMU †(R) = ∑M ′DLM ′M“TLM ′103B.1. Definitionsthen it is said to be a “spherical tensor (operator)” of rank L. It is called “spherical” since theindices will obey the same algebraic structure as spherical harmonicsM = −L,−L+ 1, ..., L− 1, LSince the range of M is determined by the value of L, it is common to make the shorthand“TLM → “TL. Spherical tensors with L = 0 or 1 will often be referred to as “scalar” or “vector”operators, respectively.You might be wondering about how to construct products of spherical tensor operators.Unfortunately, our typical matrix multiplication does not necessarily produce a spherical tensor!Symbolically, if ST represents the space of all spherical tensor operators, we want to formulatea tensor product such that [ · ⊗ · ] : ST × ST 7−→ ST . Since CG coefficients encode angularmomentum, the most intuitive move is to insert them within the matrix multiplication:Definition B.3: “(Spherical) Tensor Product”Consider two spherical tensor operators, denoted “AL1 and B̂L2 , then the “(spherical) tensorproduct” between these two spherical tensors is defined as“TLM = [ “AL1⊗B̂L2 ]LM ≡ ∑M1M2(L1M1 L2M2 |LM) “AL1M1B̂L2M2 (B3)Remark B.1The tensor product “TLM , as defined in Definition B.3, is indeed a spherical tensor of rank L,where L obeys the triangle condition ∆(L1 L2 :L).Let’s go over an important example with the definition of the tensor product - a sphericaltensor product with the identity. Our intuition says that this tensor product should return theoriginal tensor itself, especially if this product forms some kind of monoid or group structureon the set of all spherical tensors.Example B.1We note that the identity operator,1, is a scalar spherical tensor operator, hence its M valuesonly run over M = 0, ie)1.=10.=100. So, taking the tensor product of 100 with an arbitraryspherical tensor, “TLM ′ , will give a spherical tensor of rank R, where we have ∆(0L :R) =⇒R = L. And so Equation (B3) gives us[10⊗“TL]LM = ∑M ′(0 0LM ′ |LM)“TLM ′ = ∑M ′δM ′M “TLM = “TLM=⇒ “TLM = [10⊗“TL]LMwhere, in the second step of the first line, we used Equation (A14). Repeating similar math104B.1. Definitionswill give “TLM = [“TL⊗10]LM as well, hence“TLM = [10⊗“TL]LM = [“TL⊗10]LM (B4)In other words, the identity, as a spherical tensor operator of rank L = 0, gives us the identityelement under the spherical tensor product operation.A common scenario in many-body quantum mechanics is the need to take tensor productsof tensor products. Many of these types of formulae can be found in Section 3.3 of [15]. Beforewe present one of these, we must make another definition:Definition B.4: “Commutation Under the Tensor Product”Consider the “commutator of a tensor product” between the spherical tensors “AL1 and B̂L2J “AL1 , B̂L2KLM ≡ [ “AL1⊗B̂L2 ]LM − (−1)L1+L2−L [B̂L2⊗ “AL1 ]LM (B5)which is, itself, a spherical tensor operator of rank L. Then, when J “AL1 , B̂L2KLM = 0 thespherical tensors “AL1 and B̂L2 are said to “commute under the tensor product.” The phase infront of the second term on the RHS of Equation (B5) comes from the identity in Equation (A9)used within Definition B.3.For the spherical tensors “AL1 , B̂L2 , “CL3 , D̂L4 , which may not necessarily commute under thetensor product, Equation (20) in Section 3.3.3 of [15] gives the tensor recoupling formulaî[ “AL1⊗B̂L2 ]L12 ⊗ [“CL3⊗D̂L4 ]L34 óLM =∑L13L24L1 L2 L12L3 L4 L34L13 L24 L î [ “AL1⊗ “CL3 ]L13 ⊗ [B̂L2⊗D̂L4 ]L24 óLM+∑RS((−1)L1+L3+L4−L34+L−R ÛL12 ÛL34ÙR ÛS ®L1 L2 L12L34 L R ´®L3 L4 L34R L2 S ´×î “AL1 ⊗ [ JB̂L2 , “CL3KS⊗D̂L4 ]R óLM)(B6)where we’ve used Equation (B5) between B̂L2 and“CL3 in the bottom line. Finally, we maydefine the notion of a scalar product within the spherical tensor framework:Definition B.5: “(Spherical Tensor) Scalar Product”Consider two spherical tensors, “T and “S , both of rank L. Then we define their “(sphericaltensor) scalar product” as a rank 0 spherical tensor operator via“TL◦ “SL ≡ (−1)L ÛL [“TL⊗“SL]00 .= ∑M(−1)M “TLM “SL,−M (B7)The equivalence above can be shown easily by applying Equation (B3), then using Equa-105B.2. Reduced Matrix Elementstion (A13), and finally noticing that (−1)−M = ((−1)−1 )M = ( 1−1)M = (−1)M since M ∈ N0.B.2 Reduced Matrix ElementsIn nuclear physics, it will often be useful to (in some sense) “integrate out” the projectedquantum numbers of our nuclear states. Consider the two states |jm〉 and |j′m′〉, each with theirown corresponding wave-functions, ψjm(⇀r ) and ψj′m′(⇀r ), respectively. These wave-functionscould describe scalar particles, vector particles, or even tensor particles, and hence they maybe thought of as tensor operators, in the sense of quantum field theory. We may then make aseemingly random definition, which we take from [167]:Definition B.6: “Reduced Matrix Elements”For the states |jm〉 and |j′m′〉, the “reduced matrix elements” of a spherical tensor operator,“TL, are defined by〈j||“TL||j′〉 ≡ (−1)j−L+j′∫ dτ î ‹ϕj ⊗ [ “TL⊗ “ψj′ ]j ó00where the modified wave-function ϕ˜jm ≡ (−1)j+m ψ∗j,−m, and the wave-functions are treated asspherical tensor operators of rank j and j′ respectively.Definition B.6 gives us a way of writing matrix elements that do not explicitly depend onthe projected quantum numbers m and m′. The existence of these so-called reduced matrixelements is critical to developing nuclear structure theory within the context of many-bodyquantum mechanics. So the next natural question is: how do we relate them to what wealready know how to calculate?B.3 The Wigner-Eckart TheoremWe are now ready to introduce the first theorem relevant to the work presented in this disser-tation. We will state it without proof, but much deeper insight can be provided in the contextof nuclear theory by [167]. In the context of group theory, and the relationship between Liealgebras and angular momentum quantization representations via coupling coefficients, math-ematically rigorous proofs can be found in [168, 169]. Without further ado:Theorem B.1: “The Wigner-Eckart Theorem”For a spherical tensor operator “T , of rank L, with the set of quantum numbers η′ and ηadditional to those listed below, the matrix elements of “T will be related to the reduced matrixelements by a phase and a 3j-symbol, as follows106B.3. The Wigner-Eckart Theorem〈η j m|“TLM |η′ j′m′〉 = Û−1(j′m′ LM | j m)〈η j||“TL||η′ j′〉.= (−1)j−mÇj L j′−m M m′å〈η j||“TL||η′ j′〉 (B8)(B9)This is, quite simply, a stunning and highly useful theorem! In essence, what it says is thatthe coupling coefficients that we’ve developed can fully encode the decoupling of full matrixelements into their respective reduced matrix elements. Thus, through Theorem B.1, we havea means to convert between the matrix element scheme of our choice. If we can calculatethe reduced matrix elements, then we don’t have to carry projected quantum numbers untilthey are needed. Practically speaking, this will significantly reduce the amount of calculationsrequired for certain computations. All that is needed now, is a means of calculating the reducedelements themselves. First, we work through a couple simple examples:Example B.2We derive Equation (2.33) in [5], by noticing Equation (B8) with the rank L = 0 identityoperator, and dropping the additional quantum numbers η, η′, gives〈j m|10M |j′m′〉 .= 〈j|j′〉〈m|m′〉 = δjj′δmm′ = Û−1(j′m′ 0M | j m)〈j||10||j′〉 (B10)The CG coefficient on the RHS has to obey the condition in Equation (A3), or else Equa-tion (B10) above will be trivial. Hence, since it must be that m = m′ via the Kronecker-deltaon the LHS, we have: m′ +M = m = m′ =⇒M = 0. Equation (B10) becomesδjj′ Û = (j′m′ 0 0 | j m′)〈j||1||j′〉For j 6= j′ clearly we get 〈j||1||j′〉 = 0, and for j = j′ we may use Equation (A14) to obtain theuseful reduced matrix elements〈j||1||j′〉 = δjj′ Û or, 〈j||1||j′〉 = δjj′ Û ′ (B11)Notice that, had we used the quantum number l in replace of j, we can more explicitly write:〈n lm|1|n′ l′m′〉 = δnn′δll′δmm′ in Equation (B10). Hence, we’d need to carry the additionalquantum number η = n in the Wigner-Eckart Theorem, which would give〈n l||1||n′ l′〉 = δnn′δll′ Ûl .= δnn′δll′ Ûl′or, equivalently (B12)〈l||1||l′〉 = δll′ Ûl .= δll′ Ûl′The second line above is a common abuse of notation, where we will omit writing the n’s, eventhough they should certainly be included computationally.107B.4. Decomposition TheoremsExample B.3Another operator that often comes up within reduced matrix elements are the “spherical tensorharmonics.” These are simply the analog of spherical harmonics, YLM as in Equation (F7),treated as a spherical tensor operator, ŶL. The reduced matrix elements of a spherical tensorharmonic of rank L are presented in Equation (F8). In this example, we’ll calculate the reducedmatrix elements of ŶL augmented by some arbitrary, well-behaved function, f(r). We startwith the non-reduced matrix elements for one-body states〈n lm|f(r)YLM (r̂)|n′ l′m′〉 =∫dr r2∫dΩψ∗nlm(⇀r )f(r)YLM (r̂)ψn′l′m′(⇀r )=∫dr r2R∗nl(r)f(r)Rn′l′(r)∫dΩY ∗lm(θ, φ)YLM (r̂)Yl′m′(θ, φ).= 〈n l|f(r)|n′ l′〉〈l m|YLM (r̂)|l′m′〉=⇒ 〈n l||f(r)ŶL(r̂)||n′ l′〉 = 〈n l|f(r)|n′ l′〉〈l||ŶL||l′〉 (B13)where we separated the variables of the wave-functions as: ψnlm(⇀r ) = Rnl(r)Ylm(θ, φ), inaccordance with the TISE and the definition of spherical harmonics. Inserting Equation (F8)into (B13) with (A23) and l, L ∈ N0 gives〈n l||f(r)ŶL(r̂)||n′ l′〉 = (−1)L Ûl ÛL√4pi(l 0L 0 | l′ 0) 〈n l|f(r)|n′ l′〉 (B14)noting that Equation (B14) is 0 unless l + L+ l′ = even, by the identity in Equation (A15).B.4 Decomposition TheoremsWe’ll now put forth several theorems which expand the calculation tools for reduced matrixelements. The first of these “decomposition” theorems that we list is:Theorem B.2Consider the spherical tensor product made up of the spherical tensors “AL1 and B̂L2 by“TLM ≡ [ “AL1⊗B̂L2 ]LMthen, given that “J1 and “J2 commute and likewise for the primed versions, the symmetrizedJ-scheme reduced matrix elements of the tensor product and its component tensors obey〈j1 j2:j||“TL||j′1 j′2:j′〉 = Û ÛL Û ′ j1 j2 jj′1 j′2 j′L1 L2 L 〈j1|| “AL1 ||j′1〉 〈j2||B̂L2 ||j′2〉 (B15)One issue to point out is that since “TLM is, by construction, a two-body operator, the matrixelements can either be symmetrized or anti-symmetrized, depending on how the states are108B.4. Decomposition Theoremsset. For a fermionic system we would prefer the latter, so it’s important to take note thatEquation (B15) is the symmetrized version. To take care of this issue and anti-symmetrize, seeTheorem B.4 below. Theorem B.2 leads to a few useful corollaries:Corollary B.2.1Consider two spherical tensors, “T and “S , both of rank L. We can relate the symmetrizedJ-scheme reduced matrix elements of their scalar product by〈j1 j2:j||“TL◦ “SL||j′1 j′2:j′〉 = δjj′(−1)j2+j+j′1 Û®j1 j2 jj′2 j′1 L´ 〈j1||“TL||j′1〉 〈j2||“SL||j′2〉 (B16)Since it is a straightforward exercise in putting the technology we’ve developed to use, we’ll goahead and prove this corollary:Proof of Corollary (B.2.1).We begin by plugging Equation (B7) into the structure of Theorem B.2, hence〈j1 j2:j||“TL◦ “SL||j′1 j′2:j′〉 .= (−1)L ÛL 〈j1 j2:j||[“TL⊗“SL]0||j′1 j′2:j′〉Equation (B15) now gives〈j1 j2, j||“TL◦ “SL||j′1 j′2, j′〉 = (−1)L ÛL Û Û0 Û ′j1 j2 jj′1 j′2 j′L L 0 〈j1||“TL||j′1〉 〈j2||“SL||j′2〉 (B17)We can reduce Equation (B17) by noticing that Û0 = 1, and Equation (A39) yieldsj1 j2 jj′1 j′2 j′L L 0 = δjj′δLL(−1)j2+j+j′1+L Û−1 ÛL−1 ®j1 j2 jj′2 j′1 L´ (B18)Clearly δLL = 1 and subbing Equation (B18) into (B17) will cancel the hat factors ÛL and Û〈j1 j2:j||“TL◦ “SL||j′1 j′2:j′〉 = δjj′(−1)j2+j+j′1+2L Û ′ ®j1 j2 jj′2 j′1 L´ 〈j1||“TL||j′1〉 〈j2||“SL||j′2〉 (B19)By the δjj′ above, we have that j = j′ =⇒ Û ′ = Û. Also, since L ∈ N0 =⇒ 2L = even ∀L,therefore (−1)2L = +1. Putting all this into Equation (B19) retrieves (B16).Corollary B.2.2Consider two spherical tensors, “T and “S , both of rank 0. We note that their tensor product isequivalent to their scalar product, since by Equation (B7) we have“T0◦“S0 = (−1)0 Û0 [“T0⊗“S0]00 = [“T0⊗“S0]00 (B20)109B.4. Decomposition TheoremsWe can then relate the symmetrized matrix elements of their scalar tensor product by〈j1 j2:j||“T0◦“S0||j′1 j′2:j′〉 = δ(j1j2:j)δj1j′1δj2j′2δjj′ ÛÛ1Û2 〈j1||“T0||j′1〉 〈j2||“S0||j′2〉 (B21)Note that in the literature the Kronecker-deltas and the triangular delta might be under-stood, and so Equation (B21) can be written more compactly as〈j1 j2:j||“T0◦“S0||j1 j2:j〉 = ÛÛ1Û2 〈j1||“T0||j1〉 〈j2||“S0||j2〉However, for purposes of clarity, we will rarely use this convention, unless explicitly stated.Proof of Corollary (B.2.2).This is an extension of Corollary B.2.1 above, so we simply use Equation (B16) to get〈j1 j2:j||“T0◦“S0||j′1 j′2:j′〉 = δjj′(−1)j2+j+j′1 Û®j1 j2 jj′2 j′1 0´ 〈j1||“T0||j′1〉 〈j2||“S0||j′2〉 (B22)Within this equation, we see that we can apply the 6j-symbol identity in Equation (A30), so®j1 j2 jj′2 j′1 0´.= δ(j1j2:j)δj1j′1δj2j′2(−1)j1+j2+j Û−11 Û−12 (B23)Plugging Equation (B23) into (B22) gives us〈j1 j2:j||“T0◦“S0||j′1 j′2:j′〉 = δ(j1j2:j)δj1j′1δj2j′2δjj′(−1)j1+2j2+2j+j′1 ÛÛ1Û2 〈j1||“T0||j′1〉 〈j2||“S0||j′2〉(B24)The Kronecker-delta on j1, j′1 yields zero unless j1 = j′1, and hence we have the simplification(−1)j1+2j2+2j+j′1 = (−1)2(j1+j2) (−1)2jSince the j1 and j2 are half-integers, 2(j1 + j2) = even =⇒ (−1)2(j1+j2) = +1, and by thetriangular delta induced by Equation (B23), j ∈ N0 =⇒ (−1)2j = +1. Hence, the phase inEquation (B24) disappears, and restoring the triangle condition completes the proof.Corollary B.2.3Consider the spherical tensor, “T , of rank 0. We note that its tensor product with the identityis equivalent to their scalar product, since by Equation (B7) we have“T0◦10 = (−1)0 Û0 [“T0⊗10]00 = [“T0⊗10]00 .= “T0 (B25)The last equivalence above comes from Equation (B4) of Example B.1. However, we will usethe notation “T0◦10 to make it clear that “T is an operator acting on the space of particle 1 withj′1 → j1, WLOG. We can then write the reduced matrix elements of “T0 via〈j1 j2:j||“T0◦10||j′1 j′2:j′〉 = δ(j1j2:j)δj1j′1δj2j′2δjj′Û Û−11 〈j1||“T0||j′1〉 (B26)110B.4. Decomposition TheoremsBy extending Corollary B.2.2, all we have to do to prove Corollary B.2.3 is take “S0 = 10,and substitute Equation (B11) into (B21), et voila`. Corollary B.2.3 will come in handy in thefuture, but what if we wanted to take the tensor product of the identity with a spherical tensorof an arbitrary rank, for instance?Corollary B.2.4Consider the spherical tensor product made up of the spherical tensors “TL and the identityoperator10. By Equation (B4), this gives back the arbitrary spherical tensor, as expected“TLM .= [10⊗“TL]LMIn contrast with the corollaries above, let’s switch to ls-coupling, and assume that j′ is a half-integer and l, L ∈ N0. Then the symmetrized J-scheme reduced matrix elements of this tensorproduct comes to〈l s:j||“TL||l′ s′:j′〉 = −δll′(−1)l+2s+j+L+s′ Û Û ′ ®s s′ Lj′ j l´ 〈s||“TL||s′〉 (B27)Proof of Corollary (B.2.4).The first clear step is to input Equation (B4) into Equation (B15) of Theorem B.2, and nota-tionally switch from jj-coupling to ls-coupling via 〈j1 j2:j| → 〈l s:j| and|j′1 j′2:j′〉 → |l′ s′:j′〉. With the identity operator of rank 0 and Equation (B12), this gives〈l s:j||“TL||l′ s′:j′〉 = δll′Û ÛL Ûl′Û ′ l s jl′ s′ j′0 L L 〈s||“TL||s′〉 (B28)We notice that the 9j-symbol above has a zero entry, so now we can use Equation (A38) andthen (A39) to reduce the 9j-symbol down into a 6j-symbol, as follows (sparing the algebra)l s jl′ s′ j′0 L L = δll′(−1)1+l+2s+j+s′+L Ûl′ −1 ÛL−1 ®s j lj′ s′ L´ (B29)where we applied Equation (A28), used (−1)2j′ = −1 since j′ is a half-integer, that (−1)3L =(−1)2L (−1)L = (−1)L since L ∈ N0, and that the δll′ sets Ûl−1 = Ûl′ −1 and (−1)2l+l′ = (−1)3l =(−1)l since l ∈ N0. Plugging Equation (B29) into Equation (B28) will cancel the prefactors ofÛL and Ûl′, and by taking note of the redundancy of the Kronecker-delta on l, l′ and rearrangingthe phase we retrieve the desired result of Equation (B27).Theorem B.2 and its subsequent corollaries are useful for decomposing spherical tensoroperators when the states involved are already in J-scheme. But what about the case when thecoupling of the states is not defined a priori? For this case, we must use the following:111B.4. Decomposition TheoremsTheorem B.3Consider the spherical tensor product made up of the spherical tensors “AL1 and B̂L2 by“TLM ≡ [ “AL1⊗B̂L2 ]LMwhere all of these spherical tensors act on the same Hilbert space spanned by the basis states{|η J MJ〉}, and η are any additional quantum numbers needed to fully specify the states. Wemay “insert an identity” into the reduced matrix elements of the tensor product, via complete-ness, in the form of〈η J ||“TL||η′ J ′〉 = (−1)J+L+J ′ ÛL ∑η′′J ′′®L2 L1 LJ J ′ J ′′´〈η J || “AL1 ||η′′ J ′′〉〈η′′ J ′′||B̂L2 ||η′ J ′〉(B30)Now you may be thinking, “this is all great, but what if I want the anti-symmetrizedversions?” This is important in the case that the quantum mechanical system of interest obeysFermi-Dirac statistics, which it often will in nuclear physics. The following theorem comes inhandy, which can be applied to all the corollaries above:Theorem B.4Consider the spherical tensor operator, “TLM , with J-scheme reduced matrix elements in jj-coupling. We can anti-symmetrize its reduced (or standard) matrix elements using the (analo-gous) formula, as follows〈j1 j2; j||“TL||j′1 j′2; j′〉 = 〈j1 j2:j||“TL||j′1 j′2:j′〉 − (−1)j′1+j′2−j′ 〈j1 j2:j||“TL||j′2 j′1:j′〉 (B31)noting that the last term of the RHS has a swapping of j′1 ↔ j′2 compared to the first term, andthat the LHS has semicolons as opposed to the RHS which has colons (see the List of Symbols).To normalize Equation (B31), we simply send〈j1 j2; j||“TL||j′1 j′2; j′〉 −→ 1»(1 + δ12)(1 + δ1′2′)〈j1 j2; j||“TL||j′1 j′2; j′〉 (B32)The proof of this theorem, and the genesis of the normalization factor in Equation (B32),comes from applying Equation (A22). If you are unsure whether or not your matrix elementsare anti-symmetrized, due to notational inconsistencies, the following corollary is a good check:112B.4. Decomposition TheoremsCorollary B.4.1Consider the spherical tensor operator, “TLM , as in Theorem B.4. Its anti-symmetrized reduced(or standard) matrix elements obey the following (analogous) formula〈j1 j2; j||“TL||j′2 j′1; j′〉 = −(−1)j′1+j′2−j′ 〈j1 j2; j||“TL||j′1 j′2; j′〉 (B33)noting the difference in the positions of j′1 and j′2.Proof of Corollary (B.4.1).We simply plug Equation (B31) into the RHS of (B33) as so−(−1)j′1+j′2−j′ 〈j1 j2; j||“TL||j′1 j′2; j′〉 = −(−1)j′2+j′1−j′ 〈j1 j2:j||“TL||j′1 j′2:j′〉+ 〈j1 j2:j||“TL||j′2 j′1:j′〉= 〈j1 j2; j||“TL||j′2 j′1; j′〉where, in the first equality, we sneakily used the fact that(−1)2(j′1+j′2−j′) = (−1)2(j′1+j′2) (−1)−2j′ = +1× (−1)2j′ = +1and in the second line we used Equation (B31) on 〈j1 j2; j||“TL||j′2 j′1; j′〉, taking careful note∗ ofwhere the j′1 and j′2 are ordered throughout the arithmetic. Notice that this proof would stillhold if we had carried the normalization factors throughout the steps, since they would justcancel out from both sides.∗one should also be aware of the appearance of colons or semicolons113Appendix CFock Space and OperatorsIn second quantization, a Fock state is a way of representing a quantum many-body state interms of a well-defined particle number, as opposed to working with energy eigenstates as infirst quantization. That is, we label the many-body state by the occupation number of thesingle-particle states. In general, we can write|Ψ0{1, . . . , A}〉 = |n1, . . . , nA〉 (C1)where, for fermions, the state Ψ0 is an A-body Slater determinant (see Section 2.1.3) and nkare occupation numbers which are either equal to 0 or 1 for ∀ k, due to the Pauli exclusionprinciple. A simple example is where the single-particle states φ1 and φ2 are both occupied|11, 12〉 = φ1φ2 − φ2φ1√2This shows us the primary advantage of Fock states: we can model quantum many-body physicswithout any redundant reference to the particle labels.In Fock space (see Equation (C6) below), we generate the most general state by acting onthe vacuum state,∗ |0〉, with the fermionic creation operators, cˆ†. So, an equivalent way ofwriting the Slater determinant |Ψ0〉 in this occupation number representation would be|n1, . . . , nA〉 =Äcˆ†1än1 · · · Äcˆ†AänA |0〉The annihilation operator is defined by its action on the vacuum giving us nil. That is,cˆ|0〉 = 0 ⇐⇒ (cˆ|0〉)† = 0† ⇐⇒ 〈0|cˆ† = 0 (C2)We can capture the Dirac-Fermi statistics of fermions with the algebraic structure† of thefollowing anti-commutators of creation and annihilation operators{cˆi, cˆj} ≡ cˆicˆj + cˆj cˆi = 0 (C3){cˆ†i , cˆ†j} ≡ cˆ†i cˆ†j + cˆ†j cˆ†i = 0 (C4){cˆi, cˆ†j} ≡ cˆicˆ†j + cˆ†j cˆi = δij (C5)where we’ve used natural units, so } = 1. For bosons, we’d have the same algebra, except we’duse commutators instead of anti-commutators. However, in nuclear theory we won’t commonlymake reference to bosonic algebra, since protons and neutrons are both fermions.∗this represents the state devoid of any particles, i.e., it is empty†referred to as the “algebra” of the QFT by most physicists, which would likely drive a mathematician mad114C.1. Normal Ordering and ContractionsWe call the collection of all possible Fock states the “Fock space.” A more technical wayof describing the fermionic Fock space is that it is the direct sum of all the possible anti-symmetrized (AS) tensor products of single-particle Hilbert spaces. Symbolically, a Fock space,F , over the Hilbert spaces, H, can be represented asF(H) =⊕nAS(H⊗n) .= C⊕AS(H)⊕AS(H⊗H)⊕AS(H⊗H⊗H)⊕ . . . (C6)Note that separate Fock spaces can be defined for the protons and neutrons respectively, sincethey both constitute different states of the nucleon (see Section 2.2.1). Furthermore, excitationsof these nucleons can be modelled using Fock space in a “particle-hole” occupation numberrepresentation - see Figure 1 of [149], for instance. That is, whereby we can define a Fermisurface such that our nucleons are packed behind, particle operators cˆ†a and cˆa act above theFermi energy and hole operators hˆ†b = c˜b and hˆb = c˜†b act below the Fermi energy.∗C.1 Normal Ordering and ContractionsLet’s consider an arbitrary product of creation and/or annihilation operators, aˆbˆcˆdˆeˆ · · · . Be-cause of Equation (C2), it’s more convenient to deal with this product in a form such that allthe creation operators are put to the left and all the annihilation operators are put to the right.This is called the “normal ordered” product, which we will denote with our own notation†aˆbˆcˆdˆeˆ · · · To evaluate the normal ordered product, we may have to introduce a phase due to the commu-tation relations in Equations (C3) to (C5). For example, for a three fermion statecˆ1cˆ†2cˆ†3 = −cˆ†2cˆ1cˆ†3 = cˆ†2cˆ†3cˆ1 = −cˆ†3cˆ†2cˆ1Another tool in the context of second quantization is a “(Wick) contraction,” which isdefined as the expectation value of a product with respect to the vacuum (reference) stateaˆbˆ ≡ 〈0|aˆbˆ|0〉 (C7)An operator form which will equivalently give Equation (C7) is taken asaˆbˆ.= aˆbˆ− aˆbˆ (C8)where the normal ordering is defined with respect to the same vacuum, to stay consistent. Thisdefinition seems innocuous at first, but it quickly becomes rather involved. Before exposingsome of these complexities, notice that Equation (C8) corresponds with our intuitive notion ofnormal ordering. That is, when taking the expectation value of both sides of the equation∗see Equation (C17) below for the definition of this “tilde” operator†More common notations within the literature are N (· · · ) or : · · · :, which we find cumbersome and easyto miss respectively. We chose our new notation because the left bracket and right bracket form an “N” whenelongated and squished together.115C.1. Normal Ordering and Contractions〈0|aˆbˆ|0〉 = 〈0|aˆbˆ|0〉 − 〈0|aˆbˆ|0〉〈0|aˆbˆ|0〉·〈0|0〉 = 〈0|aˆbˆ|0〉 − 〈0|aˆbˆ|0〉 (C9)where the RHS remained unchanged, but we inserted Equation (C7) on the LHS, and thenpulled it out since the expectation value is a C-number. Generalizing Equation (C9) gives〈0|aˆbˆcˆdˆeˆ · · · |0〉 = 0 (C10)Why is this intuitive? Well, for aˆbˆcˆdˆeˆ · · · , we know that there exists at least one creationor annihilation operator. For the annihilation operator, normal ordering pushes it against thevacuum state, |0〉, giving zero by Equation (C2) - and similarly for a creation operator and thebra. Equation (C10) is important, since it can be taken as a defining feature when generalizingthe normal ordering to reference states other than the vacuum (see Section 5.3) [12]. Finally,some more involved arithmetic shows that the normal ordering of a contraction is as followsaˆbˆcˆdˆeˆfˆ gˆhˆ · · ·  .= (−1)P bˆeˆ dˆgˆ aˆcˆfˆ hˆ · · ·  (C11)where P counts the number of transpositions required to bring the contracted pairs to the leftof the final normal ordered product [5].C.1.1 Wick’s TheoremWe would like a way to rewrite large strings of creation and annihilation operators in a moremanageable form when doing derivations. Wick’s Theorem does just that. It states that anarbitrary product of creation and/or annihilation operators can be rewritten as its normalordering plus the sum of the normal orderings of all possible pair-wise contractions. Formally:Theorem C.1: “Wick’s Theorem”Consider the arbitrary product of creation/annihilation operators, aˆbˆcˆdˆeˆ · · · . We can expandthis product using normal ordering (as defined above) along with Equation (C11), as followsaˆbˆcˆdˆeˆ · · · = aˆbˆcˆdˆeˆ · · · + aˆbˆcˆdˆeˆ · · ·  + aˆbˆcˆdˆeˆ · · ·  + all other 1-contractions+ aˆbˆcˆdˆeˆ · · ·  + all other 2-contractions+ . . .+ all normal ordered terms of n-contractions+ . . .+ all normal ordered terms of all possible contractions(C12)This theorem remains valid for any reference state normal ordering, as mentioned in Section 5.3.116C.2. One-Body OperatorsAn in-depth proof of Equation (C12) can be found in QFT textbooks like [170].C.2 One-Body OperatorsIn the occupation number representation, we can construct a “one-body operator,” call it “T, as“T = ∑a,btab cˆ†acˆb (C13)where we’ve used the shorthand that a single nucleon a (or b, etc) has a.= na, la, sa, ja,ma.∗Why can we do such a thing? For a proof of concept, let’s go ahead and find the operatorcoefficients tab, using Equation (C2) to deduce the fact that〈a′|cˆ†a = δa′a〈0| and, cˆb|b′〉 = δbb′ |0〉so that sandwiching the operator “T with 〈a′| .= 〈ja′ma′ | and |b′〉 .= |jb′mb′〉 is〈a′|“T |b′〉 = ∑a,btab〈a′|cˆ†acˆb|b′〉 =∑a,btabδa′aδbb′〈0|0〉 = ta′b′So overall we can represent a spherical tensor operator, let’s make it rank L with projection-labelM , by its “one-body matrix elements” (OBMEs) and the creation/annihilation product“TLM = ∑a,b〈a|“TLM |b〉 cˆ†acˆb (C14)This can also be proven using Wick’s Theorem, see Example 4.5 of Suhonen [5].Example C.1The OBMEs of the number operator are〈a|“N|b〉 = 〈a|ηb|b〉 = ηb〈a|b〉 = ηbδabSo, using Equation (C14), the occupation number representation of the number operator is“N = ∑a,bηbδabcˆ†acˆb =∑aηacˆ†acˆa =∑acˆ†acˆa (C15)The last step in the line above occurs since, for fermions, the number eigenvalue is ηa = 0 or1 ∀ a, so we can simply reorder the summation index. In essence, Equation (C15) justifies whywe refer to this representation as occupation “number” representation, and it justifies why we’llcommonly think of cˆ†acˆb as a “density” operator.Using the Wigner-Eckart Theorem in Equation (B9), “T in (C14) can be rewritten as“TLM = ∑a,b(−1)ja−maÇja L jb−ma M mbå〈ja||“TL||jb〉 cˆ†acˆb (C16)∗i.e., all the relevant quantum numbers117C.2. One-Body OperatorsIt so happens that the creation operator is a spherical tensor of rank ja, however the annihilationoperator is not! So we’d like to find a form of the annihilation operator that is a spherical tensor,so that we can apply theorems developed in Section B.4 to this operator representation.Definition C.1: “Tilde (Annihilation) Operator”Consider the standard annihilation operator, cˆa.= cˆja,ma , which kills the particle (field excita-tion) a. Then, for future convenience, we make the definitionc˜a ≡ (−1)ja+ma cˆ−a where, cˆ−a ≡ cˆja,−ma (C17)Remark C.1The tilde operator, c˜a, and the creation operator, cˆ†a, are both spherical tensors of rank ja,whereas the annihilation operator is not spherical tensor operator. For proof of this statement,see the work leading up to Equation (A.81) and (A.82) of [171].Now we have a mechanism to turn the density in Equation (C16) into a spherical tensor,which we’ll do via a tensor product, as defined by Equation (B3). We may do this withconfidence considering Remark C.1, and so[cˆ†a⊗c˜b]LM =∑mamb(jama jbmb |LM) cˆ†ac˜b (C18)Let’s see if we can massage Equation (C18) out of (C16). By Equation (A26) and (A23), wecan rearrange the 3j-symbol from (C16) asÇja L jb−ma M mbå= (−1)−2jb−L−M ÛL−1(ja, −ma jbmb |L, −M) (C19)Equation (C19) turns (C16) into“TLM = ∑jajb∑mamb(−1)ja−ma−2jb−L−M ÛL−1(ja, −ma jbmb |L, −M)〈ja||“TL||jb〉 cˆ†acˆb (C20)Notice that since mb runs over −jb,−jb+1, ..., jb−1, jb, we can simply run mb backwards withinthe sum, and get the equivalent result in Equation (C20). Hence“TLM = ∑jajb∑mamb(−1)ja−ma−2jb−L−M ÛL−1(ja, −ma, jb, −mb |L, −M)〈ja||“TL||jb〉 cˆ†acˆ−b=∑jajb∑mamb(−1)2ja−ma−jb−2L−M ÛL−1〈ja||“TL||jb〉(jama jbmb |LM) cˆ†acˆ−bwhere we used Equation (A10) in the second line above. To recover the tilde operator, we useDefinition (C.1) with cˆ−b.= (−1)−jb−mb c˜b, which give us118C.3. Two-Body Operators“TLM = ∑jajb∑mamb(−1)2ja−ma−2jb−mb−2L−M ÛL−1〈ja||“TL||jb〉(jama jbmb |LM) cˆ†ac˜b (C21)In the sum over ma,mb, the CG coefficients will yield a zero unless they satisfy Condi-tion (A3), hence we must have that ma +mb = M , and so=⇒ (−1)−ma−mb−M .= (−1)2(−ma−mb) = (+1)−(ma+mb) = +1 (C22)Likewise, we notice that(−1)2ja−2jb−2L = (−1)2(ja−jb) (+1)−L = +1 (C23)Remarkably, by using Equation (C22) and (C23), the phase in Equation (C21) disappears“TLM = ÛL−1 ∑jajb〈ja||“TL||jb〉 ∑mamb(jama jbmb |LM) cˆ†ac˜b (C24)And now, comparing Equation (C24) with (C18), you can see that everything has been perfectlycooked, and we may write the compact version as“TLM = ÛL−1∑a,b〈a||“TL||b〉[cˆ†a⊗c˜b]LM (C25)Finally, consider sandwiching “TLM with some general final state 〈ηfJfMf | and initial state|ηiJiMi〉, where the labels ηf , ηi represent any additional quantum numbers we would need tofully characterize the quantum mechanical system of interest. Reducing both sides gives〈f ||“TL||i〉 = ÛL−1∑a,b〈a||“TL||b〉〈ηfJf ||[cˆ†a⊗c˜b]L||ηiJi〉 (C26)where we’ve introduced the notation 〈f | ≡ 〈ηfJf | and |i〉 ≡ |ηiJi〉, for simplicity. We call Equa-tion (C26) the “transition amplitude,” and the 〈ηfJf ||[cˆ†a ⊗ c˜b]LM ||ηiJi〉 pieces are referred toas the “one-body transition densities” (OBTDs). In the context of nuclear physics, a transitionamplitude is commonly called a “nuclear matrix element” (NME). For example, Equation (C26)could describe the decay of a nucleus from its initial state to a final state in terms of loopingover its constituent nucleons (as labelled by a, b).C.3 Two-Body OperatorsWe will now construct the analogous formula of Equation (C26) for the case of a “two-bodyoperator.” A two-body operator acts on two particles; let’s associate the sets of quantumnumbers for two particles in the ket as c and d, and then a and b for the corresponding bra. Forexample, in a hypothetical nuclear reaction, we might be turning particle 1 from the neutronstate c into the proton state a, and then particle 2 may undergo an electromagnetic transitionfrom d to b. We can express this two-body operator as a spherical tensor, “TLM , by119C.3. Two-Body Operators“TLM = ∑abcd〈a, b|“TLM |c, d〉 cˆ†acˆ†bcˆccˆd (C27)where the two-body matrix elements (TBMEs) of “T are naturally in M -scheme. Note that theorder of the creation/annihilation product has been determined in the same manner which wasused to obtain Equation (C14),∗ and that the index c is distinct from the label cˆ for cˆc.First by switching M -scheme to J-scheme in the TBMEs, then using similar algebra toSection C.2 above, whilst strategically applying Equations (A2), (B3), and (C17), one canshow that Equation (C27) may be rewritten as“TLM = ÛL−1 ∑abcd∑JabJcd〈a b:Jab||“TL||c d:Jcd〉î [cˆ†a⊗cˆ†b]Jab ⊗ [c˜c⊗c˜d]Jcd óLM (C28)As with Equation (C25), we can construct the NME of (C28) by sandwiching it between a finaland initial nuclear state and then reducing both sides, giving〈f ||“TL||i〉 = ÛL−1 ∑abcd∑JabJcd〈a b:Jab||“TL||c d:Jcd〉×〈ηfJf ||î[cˆ†a⊗cˆ†b]Jab ⊗ [c˜c⊗c˜d]JcdóL||ηiJi〉(C29)The piece after the TBMEs within the sum are called the “two-body transition densities”(TBTDs). To calculate the TBTDs, we will use a code called nutbar (see Section 6.3). Intu-itively, we’d like to be able to relate the TBTDs to the OBTDs of Equation (C26). We canaccomplish this by applying Equation (B30) of Theorem B.3 to Equation (C29), to obtain〈f ||“TL||i〉 = (−1)Jf+L+Ji ∑abcdk∑JabJcd®Jcd Jab LJf Ji Jk´〈a b:Jab||“TL(k)||c d:Jcd〉×〈ηfJf ||[cˆ†a⊗cˆ†b]Jab ||ηkJk〉 〈ηkJk||[c˜c⊗c˜d]Jcd ||ηiJi〉(C30)There are several important points to make about Equation (C30) before we move on.First, notice that inserting an identity has physically introduced an “intermediate” nuclearstate, denoted by k.† For instance, if we consider double-beta decay of Calcium-48 (for moreon this see Chapter 3), the initial, intermediate, and final nuclear states are|i〉 = |48Ca〉, |k〉 = |48Sc〉, and, |f〉 = |48Ti〉Another contrast between Equation (C29) and (C30) is that the latter has incorporated thefact that the spherical tensor operator, “T , could depend on the structure of the intermediatestate. Hence, we made the relabelling “TL → “TL(k) within the TBMEs, which will be discussedmore in Section C.3.1 below. Finally, we see that the OBTDs of Equation (C30) do not exactlymatch the form of the OBTDs of Equation (C26). To get an exact match of the OBTDs, onecould imagine using Equation (B6) on (C29) before inserting an identity via Equation (B30).∗this can be proven using Equation (C12) of Wick’s Theorem†The sum with respect to k has been appended to the leftmost summation symbol of Equation (C30), whichwe point out in contrast to Equation (C29).120C.3. Two-Body OperatorsExample C.2Let’s now push Example B.2 from its OBME version to its TBME version. Such TBMEs wouldbe useful in an Equation like (C30) above. First, we recall from Equation (A20) that〈j1 j2; jm|j3 j4; j′m′〉 = δjj′δmm′ [δ13δ24 − (−1)j3+j4−j′ δ14δ23]It’s easy to argue that the only thing to do in order to switch from 〈j1 j2; jm|j3 j4; j′m′〉 −→〈j1 j2; j||1||j3 j4; j′〉 is to include a factor of Û. This hat factor is induced in the same manneras it was for Equation (B11), with the small caveat that we must properly couple η = j1j2 toj and η′ = j3j4 to j′ in the Wigner-Eckart Theorem. Finally, since we are using the reducedmatrix elements, there is no need to include the δmm′ from Equation (A20). We obtain〈j1 j2; j||1||j3 j4; j′〉 = δjj′ [δ13δ24 − (−1)j3+j4−j′ δ14δ23] Û (C31)Note that we could prove this formally by using Equation (B15) with100.= [10⊗10]00 and theappropriate identities, and then anti-symmetrizing via Theorem B.4.C.3.1 The Closure ApproximationMathematically, we arrived at Equation (C30) by applying Theorem B.3 to Equation (C29),but this introduced the intermediate state k. At this point, we were forced to admit that thetwo-body operator itself could depend on the intermediate state, and hence we wrote “TL(k) inthe TBMEs. So, physically, in order to recover Equation (C29) when starting with the moregeneral result of (C30), we have to guarantee that the two-body operator is independent of k -so that the sum over k can be performed by reversing Equation (B30) of Theorem B.3.Some operators are naturally independent of k, but many are not.∗ In the unfortunate casethat an operator has a complicated dependence on k, theorists will often impose an approxi-mation that enforces an independence “TL(k) −→ “TL (C32)We will refer to the utilization of Equation (C32) as making a “closure approximation.” Thisnomenclature comes from its use in neutrinoless double-beta decay, where the operators involvedwill be “closed off” from their dependence on the intermediate nuclear state of the decay.Under such an approximation, or an exact guarantee of a spherical tensor’s independence ofthe intermediate state, we may sum over k and transform Equation (C30) into Equation (C29),which we copy here for the reader’s convenience〈f ||“TL||i〉 = ÛL−1 ∑abcd∑JabJcd〈a b:Jab||“TL||c d:Jcd〉×〈ηfJf ||î[cˆ†a⊗cˆ†b]Jab ⊗ [c˜c⊗c˜d]JcdóL||ηiJi〉(C33)∗see, for instance, the operator structure of neutrinoless double-beta decay in Section 4.1121Appendix DSummation Limits for theTalmi-Moshinsky TransformationTo get the summation limits for Equation (2.9), let’s consider the bracketD12 ≡ 〈nr lr, NΛ:L|n1 l1, n2 l2:L〉By Equation (2.7), we have ∆(lr Λ:L) and ∆(l1 l2 :L), hence|lr − Λ| ≤ L ≤ lr + Λ and, |l1 − l2| ≤ L ≤ l1 + l2 (D1)Now, the trick is to consider the energies from Equation (2.2), Enl[}ω] = 2n + l + 3/2, whichapply since the Talmi-Moshinsky brackets are constructed in the oscillator basis. Let’s definesomething akin to the total energies of the lab frame and the relative/CoM coordinates12 ≡ 2n1 + l1 + 2n1 + l2 (D2)and,rc ≡ 2nr + lr + 2N + Λ (D3)Why did we drop the 3/2’s? Well, by invoking conservation of energy we’d have E12 = Erc, orusing the definitions in Equation (D2) and Equation (D3), equivalently12 = rc =⇒ 2n1 + l1 + 2n1 + l2 = 2nr + lr + 2N + Λ (D4)Physically, it’s clear that the inherent energy content contained in particles 1 and 2 has to bethe same in the lab frame as it is in the relative/CoM coordinates. This implies that, given therestriction in Equation (D4) does not hold, then the bracket vanishes12 6= rc =⇒ D12 = 0 (D5)Since 12 will be set independent from the Talmi-Moshinsky transformation, as far as we’reconcerned it is just a constant. Hence, Equation (D5) is actually a constraint which will reducethe size of the summation from Equation (2.9). WLOG, let’s choose to isolate the indices inthe order: Λ, lr, N, nr. Isolating Λ from Equation (D4) gives the constraintΛ = 12 − 2(nr +N)− lr (D6)and we may now write122Appendix D. Summation Limits for the Talmi-Moshinsky Transformation∑nrlrNΛ=∑lr∑N,nrΛ = 12−2(nr+N)−lr(D7)since the brackets within the sum will obey Equation (D5).We started with four unknowns, and used one equation to reduced down to three unknowns;so we need to produce three more equations. Notice that the left triangle condition in Equa-tion (D1) actually gives us two equationsL ≤ lr + Λ (D8)and,|lr − Λ| ≤ L =⇒ − L ≤ lr − Λ ≤ L (D9)By adding Λ + lr to both sides of Equation (D9) and then inserting (D6) we get12 − 2(nr +N)− L ≤ 2lr ≤ L+ 12 − 2(nr +N)=⇒⌈12 − L2⌉− (nr +N) ≤ lr ≤⌊12 + L2⌋− (nr +N) (D10)Note that even though 12, L ∈ N0, 12±L need not be divisible by 2. Hence, to get the nearestinteger limits, we grabbed the maximal lower bound and the minimal upper bound; so we usethe ceiling and floor functions respectively in Equation (D10), for clarity within the summation.All that’s left to determine is the sum over nr and N , which we can get from puttingEquation (D6) into (D8) and manipulatingL ≤ lr + [12 − 2(nr +N)− lr] = 12 − 2(nr +N)L− 12 ≤ −2(nr +N) =⇒ nr +N ≤⌊12 − L2⌋(D11)noting that 12 ≥ L by Equation (D4). We want to separate out n and N into their own sums,which can be done by noticing that, for a, b, c ∈ N0 and some arbitrary f(a, b), we can writea+b≤ c∑a,b= 0f(a, b) = f(0, 0) + f(0, 1) + ...+ f(0, c)+ f(1, 0) + f(1, 1) + ...+ f(1, c− 1)+ ...+ f(c− i, 0) + f(c− i, 1) + ...+ f(c− i, i)+ ...+ f(c− 1, 0) + f(c− 1, 1)+ f(c, 0)=c∑b= 0f(0, b) +c−1∑b= 0f(1, b) + ...+i∑b= 0f(c− i, b) + ...+1∑b= 0f(c− 1, b) +0∑b= 0f(c, b)=c−a∑b= 0c∑a= 0f(a, b)123Appendix D. Summation Limits for the Talmi-Moshinsky TransformationOr, by switching notation, this shows that Equation (D11) gives usnr+N ≤b(12−L)/2c∑nr,N = 0.=b(12−L)/2c−nr∑N = 0b(12−L)/2c∑nr = 0(D12)Finally, putting Equation (D10) and (D12) into Equation (D7) yields∑nrlrNΛ=b(12+L)/2c−(nr+N)∑(4)lr = d(12−L)/2e−(nr+N)b(12−L)/2c−nr∑(3)N = 0b(12−L)/2c∑(2)nr = 0with the constraints: (5)Λ = 12−2(nr+N)−lr, (1)∆(l1l2:L)12≡ 2n1+l1+2n2+l2(D13)where we’ve appended the definition of 12 from Equation (D2), and included the coupling onthe RHS of (D2), for clarity. For further explicitness, the left superscripts, (i) for i = 1,2, ...,denote the order that each parameter in the limits is set, depending on one another. We willoften to refer to Equation (D13) and alike as our “chosen Talmi-Moshinsky limits” (CTML),since there are many different ways of writing them. A nice proof of concept is to check thatEquation (D13) indeed reproduces Table 1 of [172].Another necessary exercise is to extend our CTML to the summation limits in Equa-tion (2.11); we’ll start by analyzing (2.20). The derivation of Equation (D13) was performedusing conservation of energy between 12 and rc, so with the additional coordinates n′r, l′r (asdefined between particles with n′1, l′1 and n′2, l′2 respectively), we must introduce new information′12 ≡ 2n′1 + l′1 + 2n′1 + l′2 = ′rc ≡ 2n′r + l′r + 2N + Λ (D14)For Equation (2.20), we may take advantage of the δlrl′r employed in its proof, giving′12 = 2n′r + lr + 2N + Λ via, δlrl′r (D15)Quantization gives us the physical limitation that n′r ∈ N0, and additionally lr, N,Λ haveall been set for Equation (D13) using the conservation of energy in Equation (D4). So, isolatingfor n′r from Equation (D15) by subtracting it from (D4) gives the constraint′12 − 12 = 2(n′r − nr) =⇒ n′r =′12 − 122+ nr ∈ N0Since we have introduced particles 1′ and 2′, we must also ensure that their angular momentaare appropriately coupled to the total angular momenum, L, via ∆(l′1 l′2 : L). Thus, for theCTML of Equation (2.20), we combine the above constraints with Equation (D13) to obtain∑nrlr,n′rNΛ=b(12+L)/2c−(nr+N)∑(6)lr = d(12−L)/2e−(nr+N)b(12−L)/2c−nr∑(5)N = 0b(12−L)/2c∑(3)nr = 0with the constraints: (2)∆(l′1l′2:L),(1)∆(l1l2:L),(7)Λ = 12−2(nr+N)−lr, (4)n′r = (′12−12)/2+nr ∈N012≡ 2n1+l1+2n2+l2, ′12≡ 2n′1+l′1+2n′2+l′2(D16)124Appendix D. Summation Limits for the Talmi-Moshinsky TransformationFinally, to extend Equation (D16) further to encompass the limits in (2.11), we now mustaccount for: l′r 6= lr and L→ ∆(l1 l2 :L),∆(l′1 l′2 :L′). Taking the difference of the conservationof energy in Equation (D14) and (D4) yields′12 − 12 = 2(n′r − nr) + l′r − lr (D17)Since nr and lr have both been set, all we need to do is set l′r and then we may isolate for n′r as aconstraint. Now that we’ve introduced L′, which is coupled to 1′ and 2′, we must also guaranteethat it is coupled to the relative/CoM coordinates, via ∆(l′r Λ : L′). It is straightforward toconvince oneself, in this case, that|l′r − Λ| ≤ L′ ≤ l′r + Λ =⇒ |Λ− L′| ≤ l′r ≤ Λ + L′ (D18)Now that the summation limits over l′r have been set, and since Λ is known from Equation (D6),we isolate the constraint for n′r from (D17) to obtainn′r =′12 − 122+lr − l′r2+ nr ∈ N0 (D19)Equation (2.11) is unique in that we can also skip parts of the summation where the 6j-symbol vanishes. Hence, via Equation (A27), we obtain®L lr Λl′r L′ R´= 0 unless,∆(lr L :Λ), ∆(l′r L′ :Λ)∆(lr l′r :R), ∆(LL′ :R)(D20)where R is the rank of the spherical tensor operator. The top two triangle conditions ofEquation (D20) have already been implemented above, so only the bottom two need to beadded as constraints. Overall, putting Equations (D18), (D19), (D20), and (D16) togethergives the CTML for Equation (2.11) as follows∑nrlr,n′rl′rNΛ=Λ+L′∑(8)l′r = |Λ−L′|b(12+L)/2c−(nr+N)∑(6)lr = d(12−L)/2e−(nr+N)b(12−L)/2c−nr∑(5)N = 0b(12−L)/2c∑(4)nr = 0with the constraints: (9)∆(lrl′r:R), (3)∆(LL′:R), (2)∆(l′1l′2:L′), (1)∆(l1l2:L),(10)n′r = (′12−12)/2+(lr−l′r)/2+nr ∈N0, (7)Λ = 12−2(nr+N)−lr12≡ 2n1+l1+2n2+l2, ′12≡ 2n′1+l′1+2n′2+l′2(D21)We remind the reader that the primed quantum numbers are associated with the ket, theunprimed quantum numbers are associated with the bra, and the bolded, parenthesized, leftsuperscripts denote the order that each parameter in the limits should be set. An alternativeconstraint on n′r which yields equivalent results is: n′r = (′12 − l′r − Λ)/2−N ∈ N0.125Appendix EDerivation of Equation (D11) fromPRC.88.064312(2013)With 20/20-hindsight, we will want to deal with the integrals∫ ∞0dr rde−νr2jρ(qr) (E1)where d, ρ ∈ N0, ν ∈ R, and j is a spherical Bessel’s function as in Equation (F5). It is wellknown thatjn(x) ≡…pi2xJn+ 12(x) (E2)where J is a Bessel’s function as in Equation (F4). Hence, using Equation (E2), we may rewriteEquation (E1) as ∫ ∞0dr rde−νr2jρ(qr) = pi2q∫ ∞0dr rd−12 e−νr2Jρ+ 12(qr) (E3)As was done in [118], we can solve this from a standard integral table. For instance, in [173]from Equation (6.631) on page 716, it’s stated that∫ ∞0dxxµe−αx2Jν(βx) =Γĵ+ν+12äβ αµ2 Γ(ν + 1)e−z2Mµ2, ν2(z)where, z.=β24αand, Re(α) > 0, Re(µ+ ν) > −1(E4)where M is a Whittaker function as in Equation (F6). Using Equations (8.972.1) and (9.220.2),on pages 1027 and 1059 of [173] respectively, it can easily be shown thatMa,b(x) ≡ xb+12 e−x2Ça+ b− 12a− b− 12å−1L2ba−b− 12(x) (E5)where L are Laguerre polynomials as in Equation (F3), and the coefficient out front is thereciprocal of the generalized binomial coefficient as defined in Equation (F2). Hence, usingEquation (E5) and switching a→ µ2 , b→ ν2 , we can write thatMµ2, ν2(z) ≡ z ν+12 e− z2(µ+ν−12µ−ν−12)−1Lνµ−ν−12(z) (E6)Plugging Equation (E6) into Equation (E4), we see that126Appendix E. Derivation of Equation (D11) from PRC.88.064312(2013)∫ ∞0dxxµe−αx2Jν(βx) =Γĵ+ν+12äβ αµ2 Γ(ν + 1)zν+12(µ+ν−12µ−ν−12)−1e−zLνµ+ν−12(z) (E7)Now, let’s rewrite Equation (E7) in the notation relevant to Equation (E3), by switchingx→ r, µ→ d− 12, α→ ν, ν → ρ+ 12, β → qwhich gives∫ ∞0dr rd−12 e−νr2Jρ+ 12(qr) =ΓÄd+ρ+12äq νd2− 14Γ(ρ+ 32)zρ2+ 34(d+ρ−12d−ρ−22)−1e−zLρ+12d−ρ−22(z)where, z.=q24νand, Re(ν) > 0, Re(d+ ρ) > −1(E8)Here let’s introduce the convenient constantκ.=d− ρ− 22so that we can rewrite Equation (E8) as∫ ∞0dr rd−12 e−νr2Jρ+ 12(qr) =zρ2+ 34q νd2− 14· ΓÄκ+ ρ+ 32)Γ(ρ+ 32)Çκ+ ρ+ 12κå−1e−zLρ+12κ (z) (E9)Things are looking really messy, but there is hope! Recall that, for t ∈ R and any n ∈ Rsuch that t+ n > 0, the Gamma function obeys the recurrence formulaΓ(t+ n)Γ(t)= t(t+ 1) · · · (t+ n− 1) (E10)Also, with even more 20/20-hindsight, using Equation (F2), we find thatÇt+ n− 1nå=(t+ n− 1)(t+ n− 2) · · · ([t+ n− 1]− n+ 1)n!=(t+ n− 1)(t+ n− 2) · · · tn!=t(t+ 1) · · · (t+ n− 1)n!(E11)So putting together Equation (E10) and (E11) givesΓ(t+ n)Γ(t)·Çt+ n− 1nå−1= n! (E12)At this point we can see the trick; let’s take t = ρ+ 32 and n = κ in Equation (E12), so thatΓ(ρ+ 32 + κ)Γ(ρ+ 32)Çρ+ 32 + κ− 1κå−1=Γ(κ+ ρ+ 32)Γ(ρ+ 32)Çκ+ ρ+ 12κå−1= κ! (E13)Additionally, since z.= q2/4ν, it’s clear that127Appendix E. Derivation of Equation (D11) from PRC.88.064312(2013)z34q νd2− 14=q32434 ν34 · qν d2− 14=q12232 νd+12= ν−d−12…q8(E14)Now we can simplify Equation (E9) greatly, by plugging in Equation (E13) and E14 to obtain∫ ∞0dr rd−12 e−νr2Jρ+ 12(qr) = ν−d−12…q8κ!zρ2 e−zLρ+12κ (z) (E15)Finally, after much manipulation, we can evaluate the integral of interest, by putting Equa-tion (E15) into Equation (E3), giving∫ ∞0dr rde−νr2jρ(qr) = pi2q× ν −d−12…q8κ!zρ2 e−zLρ+12κ (z)Or, overall∫ ∞0dr rde−νr2jρ(qr) = ν−d−12√pi4κ!zρ2 e−zLρ+12κ (z)where, z.=q24νand, κ.=d− ρ− 22(E16)This formula indeed corresponds to Equation (D11) of [118], with the notational identificationsthat: m↔ d, ρ↔ l, and κ↔ k.128Appendix FMiscellaneous Formulae• We would like a function, call it Γ(x), over x ∈ R, such that it generalizes the notion of thefactorial with: Γ(n+1) = n! for n ∈ N0. Recall that the factorial is defined by the recurrencerelation: n! = n(n− 1)! with 0! = 1. Therefore, we want our “Gamma function” to obey itsown defining recurrence relation: Γ(x+ 1) = xΓ(x) with Γ(1) = 1. The following formulaΓ(x).=∫ ∞0sx−1e−sds (F1)solves the desired recurrence relation, which can easily be shown using integration by parts.• The “generalized binomial coefficient” is defined viaÇabå.=abb≡ a(a− 1)(a− 2) · · · (a− b+ 1)b(b− 1)(b− 2) · · · 1 or,Çabå=Γ(a+ 1)Γ(b+ 1) Γ(a− b+ 1) (F2)where a is in a commutative ring and b ∈ N0 for the left equation, or a, b ∈ R for the right.• The “(generalized) Laguerre polynomials,” Lβα(x), are solutions to the ODExd2ydx2+ (β + 1− x)dydx+ αy = 0 (F3)• The “Bessel’s functions (of the first kind),” Jn(x) are solutions to the ODEx2d2ydx2+ xdydx+ (x2 − n2) y = 0 (F4)• The “spherical Bessel’s functions (of the first kind),” jn(x), are solutions to the ODEx2d2ydx2+ 2xdydx+îx2 − n(n+ 1)óy = 0 (F5)• The “Whittaker functions (of the first kind),” Ma,b(x) are solutions to the ODEd2ydx2+Ç− 14+ax+14 − b2x2åy = 0 (F6)129Appendix F. Miscellaneous Formulae• The “spherical harmonics,” Ylm(θ, φ), are solutions to the PDEr2∇2Ylm(θ, φ) = −l(l + 1)Ylm(θ, φ)where we take the normalization such that: Y00 = 1/√4pi. Note that spherical harmonics aresimultaneous eigenfunctions of the angular momentum operators L̂z and “L2 viaL̂zYlm = mYlm and, “L2Ylm = l(l + 1)Ylm (F7)• The reduced matrix elements of the spherical tensor form of a spherical harmonic, YLM , are〈l||ŶL||l′〉 = (−1)lÛl ÛL Ûl′√4piÇl L l′0 0 0å(F8)We may extend Equation (F8), using ŶL = [ ŶL ⊗ 10 ]L and the analog of (B27) of Corol-lary B.2.4, to obtain the reduced matrix elements for a single spin-1/2 particle as [5]〈l 12 j||ŶL||l′ 12 j′〉 = (−1)L+j′− 121 + (−1)l+L+l′2Û ÛL Û ′√4piÇj j′ L12 −12 0å(F9)• Consider a two-body spherical tensor operator, “T of rank 0, such that, in relative coordinates〈nr lr||“T ||n′r l′r〉 ∝ δnrn′r=⇒ 〈l1, l2:L||“T ||l′1, l′2:L′〉 = δLL′ ÛL ∑nrlr,NΛD12D1′2′ Ûl−1r 〈nr lr||“T ||nr lr〉 (F10)where D1′2′.= 〈nr lr, NΛ:L|n′1 l′1, n′2 l′2 :L〉 and D1′2′ .= 〈nr lr, NΛ:L|n′1 l′1, n′2 l′2 :L〉, and theCTML can be found in Equation (D13).• For a substance with N0 number of constituents, whereby it decays with a half-life of T1/2,then the substance obeys the exponential decay formulaN(t) = N0 e−λt where, λ =ln(2)T1/2(F11)130


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