Neutron ββ-decay Angular Correlations

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Neutron ββ-decay Angular Correlations Brad Plaster

Markisch, et al. Figure Credit: https://www.physi.

Wu et al., Phys. Rev. 105, 1413 (1957) H. Abele, Prog.

1 Neutron ββ-decay Angular Correlations Brad Plaster University of Kentucky Figure Credit: V. Cirigliano, B. Markisch, et al. Figure Credit: C.S. Wu et al., Phys. Rev. 105, 1413 (1957) H. Abele, Prog. Part. Nucl. Phys. 60, 1 (2008) Fundamental Neutron Physics Summer School

2 Outline From the Weak Interaction to Angular Correlations Experimental Approaches to Measurements of Angular Correlations Motivation for the Frontiers of Angular Correlations (i.e., What could you discover?) 2

Recap: Lee & Yang (1956) T.D. Lee and C.N. Yang, Phys. Rev.

ee 22 dd 33 pp νν 22EE νν δδ ΔΔ EE νν MM 22 pp ee nn CC νν ee

7777 kev MM = ii ψψ pp ΓΓ ii ψψ nn ψψ ee ΓΓ ii CC ii + CC ii γγ

Vector (A) Scalar (S) Tensor (T) Pseudo- Scalar (P) ψψ : Dirac

3 Recap: Lee & Yang (1956) T.D. Lee and C.N. Yang, Phys. Rev. 104, 254 (1956) nn pp + ee + νν ee dddd = dd 33 pp ee 22 dd 33 pp νν 22EE νν δδ ΔΔ EE νν MM 22 pp ee nn CC νν ee TT ee mmmmmm = kev TT pp mmmmmm = kev MM = ii ψψ pp ΓΓ ii ψψ nn ψψ ee ΓΓ ii CC ii + CC ii γγ 55 ψψ νν γγ μμ γγ μμ γγ 55 II σσ μμμμ γγ 55 Vector (V) Axial- Vector (A) Scalar (S) Tensor (T) Pseudo- Scalar (P) ψψ : Dirac Spinors (4-component) CC ii, CC ii : Effective Weak Couplings (could be complex numbers) 3

4 Vector and Axial-Vector: Parity MM VV,AA = ψψ pp γγ μμ ψψ nn ψψ ee γγ μμ CC VV + CC VV γγ 55 ψψ νν + ψψ pp γγ μμ γγ 55 ψψ nn ψψ ee γγ μμ γγ 55 CC AA + CC AA γγ 55 ψψ νν γγ 5 = II + γγ5 ψψ Left Handed 1 2 II γγ5 ψψ Right Handed Rewrite as: CC ii + CC ii γγ 55 = CC ii + CC ii 11 + γγ CC ii CC ii 11 γγ 55 Left Handed Right Handed See Appendix Parity Violation Requires CC VV,AA 00 and CC VV,AA 00 Maximal Parity Violation CC VV = CC VV and CC AA = CC AA MM VV,AA = CC VV ψψ pp γγ μμ ψψ nn ψψ ee γγ μμ 11 + γγ 55 ψψ νν + CC AA ψψ pp γγ μμ γγ 55 ψψ nn ψψ ee γγ μμ 11 + γγ 55 ψψ νν 4

5 Vector and Axial-Vector: Low-Energy Limit MM VV,AA = CC VV ψψ pp γγ μμ ψψ nn ψψ ee γγ μμ 11 + γγ 55 ψψ νν + CC AA ψψ pp γγ μμ γγ 55 ψψ nn ψψ ee γγ μμ 11 + γγ 55 ψψ νν Consider low-energy limit of nucleon matrix elements ψψ = EE + MM φφ σσ pp EE + MM φφ pp 00 22MM φφ 00 ψψγγ μμ ψψ ψψψψ 00 μμ = 00 μμ = 11, 22, 33 ψψγγ μμ γγ 55 ψψ 00 ψψσσ kk ψψ μμ = 00 kk = μμ = 11, 22, 33 σσ kk = 0 σσμμ σσ μμ 0 MM VV,AA = CC VV ψψ pp ψψ nn ψψ ee γγ γγ 55 ψψ νν + CC AA ψψ pp σσ kk ψψ nn ψψ ee γγ kk 11 + γγ 55 ψψ νν Fermi Matrix Element: ψψ pp ψψ nn 22 = 11 Gamow-Teller Matrix Element ψψ pp σσψψ nn 22 =

6 Vector and Axial-Vector: Time-Reversal pp ee νν ee nn νν ee ee nn pp + ee + νν ee ee + pp nn + νν ee nn pp MM VV,AA = ψψ pp γγ μμ ψψ nn ψψ ee γγ μμ CC VV + CC VV γγ 55 ψψ νν + ψψ pp γγ μμ γγ 55 ψψ nn ψψ ee γγ μμ γγ 55 CC AA + CC AA γγ 55 ψψ νν + ψψ nn γγ μμ ψψ pp ψψ νν γγ μμ CC VV + CC VV γγ 55 ψψ ee + ψψ nn γγ μμ γγ 55 ψψ pp ψψ νν γγ μμ γγ 55 CC AA +CC AA γγ 55 ψψ νν h.c. Time-Reversal Invariance Requires: CC VV = CC VV and CC AA = CC AA CC VV,AA Real (i.e., not complex) See Appendix 6

7 Beyond SM : Scalar, Tensor, Pseudo-Scalar pp ee νν ee MM = CC ii ψψ pp ΓΓ ii ψψ nn ii ψψ ee ΓΓ ii 11 + γγ 55 ψψ νν γγ μμ γγ μμ γγ 55 II σσ μμμμ γγ 55 Vector (V) Axial- Vector (A) Scalar (S) Tensor (T) Pseudo- Scalar (P) nn Vector: ψψ ee γγ μμ 11 + γγ 55 ψψ νν = 11 ψψ 22 ee 11 γγ 55 γγ μμ 11 + γγ 55 ψψ νν Vector Preserves Chirality, ψψ ee 1 γγ 5 = 1 + γγ 5 ψψ ee γγ as does Axial-Vector 0 Scalar: ψψ ee II 11 + γγ 55 Cannot ψψ νν ψψ ee 11 γγ 55 II 11 + γγ 55 ψψ write as: νν Scalar Flips Chirality, as does Tensor 7

8 Why is the Pseudo-Scalar (Relatively) Small? ψψγγ 55 ψψ = ψψ γγ 00 γγ 55 ψψ = ψψ II II 00 II II 00 ψψ = ψψ 00 II II 00 ψψ In the low-energy limit for the nucleon spinors: ψψ = EE + MM φφ σσ pp EE + MM φφ pp 00 22MM φφ 00 ψψ 00 II II II ψψ 2222 φφ 00 II 00 φφ

9 Nucleon-Level Couplings Quark-Level Couplings uuuuuu pp nn uuuuuu WW ee GG FF VV uuuu 22 νν ee T. Bhattacharya et al., Phys. Rev. D 85, (2012) T. Bhattacharya et al., Phys. Rev. D 92, (2015) T. Bhattacharya et al., Phys. Rev. D 94, (2016) R. Gupta et al., arxiv: MM = CC ii ψψ pp ΓΓ ii ψψ nn ii MM = GG FFVV uuuu 22 ii Coupling relative to SM (i.e., 1) ψψ ee ΓΓ ii 11 + γγ 55 ψψ νν εε ii pp uuγγ ii dd nn uu ee ΓΓ ii 11 + γγ 55 uu νν = gg ii Form Factor γγ μμ (VV) γγ μμ γγ 55 (AA) II (SS) σσ μμμμ (T) εε ii 1 1 εε SS εε TT CC VV = GG FFVV uuuu 22 gg VV CC AA = GG FFVV uuuu 22 gg AA gg ii gg VV = 11 CVC Hypothesis gg AA ~ Exp t, Lattice QCD gg SS ~ Lattice QCD gg TT ~ Lattice QCD CC SS = GG FFVV uuuu 22 εε SSgg SS CC TT = GG FFVV uuuu 22 εε TTgg TT 9

10 Decay Rate (I) (Fixed Initial Spin, Sum Over Final Spins) MM = CC VV ψψ pp ψψ nn ψψ ee γγ γγ 55 ψψ νν + CC AA ψψ pp σσ kk ψψ nn ψψ ee γγ kk 11 + γγ 55 ψψ νν + CC SS ψψ pp ψψ nn ψψ ee 11 + γγ 55 ψψ νν + CC TT ψψ pp σσ kk ψψ nn ψψ ee σσ kk 11 + γγ 55 ψψ νν We can write: MM = M VV + M AA + M SS + M TT dddd = dd 33 pp ee 22 dd 33 pp νν 22EE νν δδ ΔΔ EE νν MM 22 MM 22 = (M VV + M AA + M SS + M TT ) (M VV + M AA + M SS + M TT ) Example #1: Calculate M VV M VV M VV = CC VV ( ψψ pp ψψ nn )( ψψ ee γγ 00 (11 + γγ 55 ) ψψ νν ) M VV = CC VV ψψ pp ψψ nn ( ψψ νν (11 γγ 55 )γγ 00 ψψ ee ) 10

11 Decay Rate (II) (Fixed Initial Spin, Sum Over Final Spins) M VV M VV = CC VV 22 ψψ pp ψψ nn 22 Tr (ppee + mm ee )γγ 00 (11 + γγ 55 )pp νν γγ 00 (11 + γγ 55 ) EE νν 11 + pp ee pp νν EE νν Example #2: Calculate M SS M VV M SS = CC SS ( ψψ pp ψψ nn )( ψψ ee (11 + γγ 55 ) ψψ νν ) M VV = CC VV ψψ pp ψψ nn ( ψψ νν (11 γγ 55 )γγ 00 ψψ ee ) M SS M VV = CC SS CC 22 VV ψψ pp ψψ nn Tr (ppee + mm ee ) (11 + γγ 55 )pp νν γγ 00 (11 + γγ 55 ) EE νν mm ee 11

12 Example #3: Calculate M AA M VV Decay Rate (III) (Fixed Initial Spin, Sum Over Final Spins) M AA = CC AA ( ψψ pp σσ kk ψψ nn )( ψψ ee γγ kk (11 + γγ 55 ) ψψ νν ) M VV = CC VV ψψ pp ψψ nn ( ψψ νν (11 γγ 55 )γγ 00 ψψ ee ) M AA M VV = CC AA CC VV ( ψψ pp ψψ nn )( ψψ pp σσ ii ψψ nn ) Tr (pp ee + mm ee )γγ ii (11 + γγ 55 )pp νν γγ 00 (11 + γγ 55 ) σσ ii σσ pp ee EE νν pp ee ii ii + pp νν EE νν σσ pp νν 12

13 And Finally Angular Correlations After evaluating all other products MM 22 = (M VV + M AA + M SS + M TT ) (M VV + M AA + M SS + M TT ) dddd 11 + bb mm ee + aa pp ee pp νν EE νν + + AA σσ pp ee + BB σσ pp νν EE νν + aa = 11 gg AA 22 gg VV bb = 22gg SSεε SS 2222 gg AA gg VV gg TT εε TT gg AA gg 22 VV gg 22 VV gg AA = 00 in Standard Model (VV AA) AA = 22 gg AA gg 22 VV gg AA gg VV gg AA gg 22 BB = 22 VV gg AA gg 22 VV + gg AA gg VV gg AA gg 22 VV 13

14 Full Expression(s) for Angular Correlations J.D. Jackson, S.B. Treiman, and H.W. Wyld, Jr., Phys. Rev. 106, 517 (1957) Polarized Neutrons, σσ Measurement of Electron Polarization, σσ ee Neutrino (i.e., Proton) Measurement dddd dd ddωω ee ddωω νν pp ee EE aa pp ee pp νν EE νν + bb mm ee Note: If unpolarized, σσ = σσ AA pp ee + BB pp νν EE νν + DD pp ee pp νν EE νν 14

15 Full Expression(s) for Angular Correlations J.D. Jackson, S.B. Treiman, and H.W. Wyld, Jr., Phys. Rev. 106, 517 (1957) Polarized Neutrons, σσ Measurement of Electron Polarization, σσ ee Neutrino (i.e., Proton) Measurement dddd dd ddωω ee pp ee EE bb mm ee + pp ee AA σσ + GG σσ ee + σσ ee NN σσ + QQ pp ee + mm ee σσ pp ee + RR σσ pp ee 15

16 Full Expression(s) for Angular Correlations J.D. Jackson, S.B. Treiman, and H.W. Wyld, Jr., Phys. Rev. 106, 517 (1957) Polarized Neutrons, σσ Measurement of Electron Polarization, σσ ee Neutrino (i.e., Proton) Measurement dddd dd ddωω ee ddωω νν pp ee EE aa pp ee pp νν EE νν + bb mm ee + σσ ee GG pp ee + HH pp νν EE νν Note: As far as I know, there have been no measurements of GG, HH, KK, LL. pp ee +KK + mm ee pp ee pp νν EE νν + LL pp ee pp νν EE νν 16

17 Beyond Leading-Order qq = 00: aa and AA MM = GG FFVV uuuu 22 ii εε ii pp uuγγ ii dd nn ψψ ee ΓΓ ii CC ii + CC ii γγ 55 ψψ νν S. Gardner and C. Zhang, Phys. Rev. Lett. 86, 5666 (2001) Vector Induced Tensor Weak Magnetism Induced Scalar ΓΓ VV = γγ μμ ΓΓ AA = γγ μμ γγ 55 Axial-Vector Induced Pseudo-Tensor Induced Pseudo-Scalar Weak Magnetism: OO qq MM ~ Correction to Vector-Axial-Vector Induced Scalar and Induced Pseudo-Tensor: Second-Class Currents (expected small) Induced Pseudo-Scalar: 0 per CVC Hypothesis 17

18 aa and AA Beyond Leading-Order qq = 00 S. Gardner and BP, Phys. Rev. C 87, (2013) λλ = gg AA /gg VV εε = mm ee 22 MM 22 RR = EE 00 MM xx = EE 00 RRRR = MM εε RRRR = mm ee MM mm ee 18

19 aa and AA Beyond Leading-Order qq = 00 S. Gardner and BP, Phys. Rev. C 87, (2013) λλ = gg AA /gg VV εε = mm ee 22 MM 22 RR = EE 00 MM xx = EE 00 RRRR = MM εε RRRR = mm ee MM mm ee 19

20 Angular Correlations Can Determine pp ee νν ee MM = CC ii ψψ pp ΓΓ ii ψψ nn ψψ ee ΓΓ ii 11 + γγ 55 ψψ νν nn CC ii γγ μμ γγ μμ γγ 55 II σσ μμμμ γγ 55 Vector (V) Axial-Vector (A) Scalar (S) Tensor (T) Pseudo-Scalar (P) gg VV = 11 CVC gg AA /gg VV aa, AA, BB, CC BSM bb, bb νν Small CC VV = GG FFVV uuuu 22 gg VV CC AA = GG FFVV uuuu 22 gg AA CC SS = GG FFVV uuuu 22 εε SSgg SS CC TT = GG FFVV uuuu 22 εε TTgg TT 20

21 Measurements of Angular Correlations 21

22 Common Features/Requirements of Experiments (I) dddd dd ddωω ee ddωω νν pp ee EE aa pp ee pp νν EE νν + bb mm ee + σσ AA pp ee + BB pp νν EE νν + DD pp ee pp νν EE νν Magnetic Fields (~Solenoidal): Acceptance Field Expansion High-Fidelity Electron Detection: Energy, Direction Problematic: Backscattering, Thresholds, etc. Figure Credit: J. Liu Need Neutrino Momentum: pp nn = pp ee + pp pp + pp νν = 00 pp νν = (pp ee + pp pp ) dddd ddωω Mott = αα 22 ZZ pp ee 22 ββ 22 sin 44 ( θθ 22) Requires Proton Detection: TT pp mmmmmm = kev Accelerating Electric Potential Figure Credit: J.D. Jackson, Classical Electrodynamics 11 ββ 22 sin 22 ( θθ 22) 22

Common Features/Requirements of Experiments (II)

Cold and Ultracold Polarization θ θ µ Cold:

23 Common Features/Requirements of Experiments (II) dddd dd ddωω ee ddωω νν pp ee EE aa pp ee pp νν EE νν + bb mm ee + σσ AA pp ee + BB pp νν EE νν + DD pp ee pp νν EE νν Neutrons: Cold and Ultracold Polarization θ θ µ Cold: supermirrors, 3 He spin filters, UCN: magnetic filters 23

24 Common Features/Requirements of Experiments (III) dddd dd ddωω ee ddωω νν pp ee EE aa pp ee pp νν EE νν + bb mm ee + σσ AA pp ee + BB pp νν EE νν + DD pp ee pp νν EE νν Many are measured via asymmetries (but not all!) Example: ddγγ 11 + σσ AA pp ee = 11 + AA ββ cosθθ ππ 22 ΓΓ ππ ΓΓ 22 ππ AAAAAAAAA sinθθ dddd = AAAA 11 + AAAAAAAAA sinθθ dddd = AAAA AA eeeeee = ΓΓ 11 ΓΓ 22 ΓΓ 11 + ΓΓ 22 = AAAA What if bb 00? 24

25 Measurements of aa dddd dd ddωω ee ddωω νν pp ee EE aa pp ee pp νν EE νν + bb mm ee pp νν ee pp nn = pp ee + pp pp + pp νν = 00 ee θθ eeee Particle Data Group (2018) 25

26 aacorn at NIST Slides: F. Wietfeldt 26

00, and the other to aa = 00. 11. Which is which?

27 Proton TT pp Spectrum aaspect at ILL Question: One of these (red or blue) corresponds to aa = 00, and the other to aa = Which is which? pp ee θθ eeee νν ee TT pp [kev] Slides: C. Schmidt 27

Naaaa at SNS: aa dddd dd ddωω ee ddωω νν pp ee EE 00 22 11 + aa pp ee pp νν EE νν + bb mm ee pp pp 22 = (pp ee + pp νν ) 22 = pp ee 22

28 Naaaa at SNS: aa dddd dd ddωω ee ddωω νν pp ee EE aa pp ee pp νν EE νν + bb mm ee pp pp 22 = (pp ee + pp νν ) 22 = pp ee 22 + pp νν pp ee pp νν cosθθ eeνν ddγγ = 11 + aa pp ee cosθθ eeee L.J. Broussard et al., J. Phys. Conf. Ser. 876, (2017) 28

Measuring bb is very challenging! Naaaa at SNS: bb 127 pixels 2 mm thick < 100 nm dead layer bb nn = 00. 11 bb nn = 11 K.P. Hickerson, Ph.D.

29 Measuring bb is very challenging! Naaaa at SNS: bb 127 pixels 2 mm thick < 100 nm dead layer bb nn = bb nn = 11 K.P. Hickerson, Ph.D. Thesis, California Institute of Technology (2013) bb nn = bb nn = 00 3 kev resolution at 30 kev Threshold < 10 kev Preamplifier electronics nonlinearities < 0.3% dddd pp ee EE bb mm ee Slides: J. Fry 29

30 Naaaa at SNS Slides: J. Fry 30

31 UCNA at LANL: First Extraction of bb < b n < Importance of Detector Non-Linearities and Absolute Energy Calibration 31

32 Measurements of AA dddd dd ddωω ee pp ee EE bb mm ee + σσ AA pp ee AA = 22 gg AA gg 22 VV gg AA gg VV gg AA gg 22 VV Pre-2002 Post-2002 δδδδ AA = δδgg AA gg AA = χ / ν = 2.4 Also includes results from aa, BB, AA BB 32

33 Concept for Measurements of AA θθ ddγγ 11 + σσ AA pp ee = 11 + PP nn AA ββ cosθθ ~ 4% hemisphere acceptance 33

34 PERKEO III at ILL 0.17% Result for AA Soon!! Slides: B. Markisch 34

35 UCNA at LANL 800 MeV proton accelerator UCN Facility 35

36 UCNA at LANL 36

37 UCNA at LANL 1.0 T 0.6 T 37

38 UCNA at LANL Field Dip 1.0 T 0.6 T 38

39 UCNA at LANL 0.67% Result for AA Calibration Residuals 39

40 PERKEO II: Measurements of BB ee pp M. Schumann et al., Phys. Rev. Lett. 99, (2007) M. Schumann et al., Phys. Rev. Lett. 100, (2008) 40

Measurements of DD: Time-Reversal Violation dddd dd ddωω ee ddωω νν pp ee EE 00 22 11 + aa pp ee pp νν EE νν + bb mm ee + σσ AA pp ee + BB pp νν EE νν +

41 Measurements of DD: Time-Reversal Violation dddd dd ddωω ee ddωω νν pp ee EE aa pp ee pp νν EE νν + bb mm ee + σσ AA pp ee + BB pp νν EE νν + DD pp ee pp νν EE νν DD (CC VV CC AA CC VV CC AA ) H.P. Mumm et al., Phys. Rev. Lett. 107, (2011) T.E. Chupp et al., Phys. Rev. C 86, (2012) 41

42 PERC at FRM: aa, bb, AA, BB, Slides: B. Markisch 42

43 PERC at FRM: aa, bb, AA, BB, Slides: B. Markisch 43

44 PERC at FRM: aa, bb, AA, BB, Slides: B. Markisch 44

45 PERC at FRM: aa, bb, AA, BB, Slides: B. Markisch 45

46 PERC at FRM: aa, bb, AA, BB, Slides: B. Markisch 46

47 PERC at FRM: aa, bb, AA, BB, Slides: B. Markisch 47

48 Motivation for the Frontiers of Angular Correlations 48

49 Current Landscape of VV uuuu δv ud (10 4 ) experiment theory nucleus dependent EW gg AA τ EW nucleus dependent EW PDG τ : PDG gg AA : χ 2 / ν = 1.9 χ 2 / ν = 2.2 Superallowed Neutron T = ½ Mirror Pion Uncertainties from: Particle Data Group (2018) 49

Current Landscape of VV uuuu Lifetime: W. Marciano and A. Sirlin, PRL 96, 032002 (2006) W.

50 Current Landscape of VV uuuu Lifetime: W. Marciano and A. Sirlin, PRL 96, (2006) W. Marciano, Santa Fe Neutron Lifetime Workshop (2012) M. A.-P. Brown et al., Phys. Rev. C 97, (2018) = gg AA gg VV 50

51 Searching for BSM Scalar and Tensor MM = GG FFVV uuuu 22 ii εε ii pp uuγγ ii dd nn uu ee ΓΓ ii 11 + γγ 55 uu νν T. Bhattacharya et al., Phys. Rev. D 85, (2012) T. Bhattacharya et al., Phys. Rev. D 92, (2015) T. Bhattacharya et al., Phys. Rev. D 94, (2016) R. Gupta et al., arxiv:

52 Current Landscape of gg AA : Test of Lattice QCD C.C. Chang et al., Nature 558, 91 (2018) 52

53 Current Landscape of gg AA : Test of Lattice QCD Lattice QCD vs. Experiment gg AA Circa ~2012 T. Bhattacharya et al., Phys. Rev. D 85, (2012) Now in 2018! C.C. Chang et al., Nature 558, 91 (2018) 53

54 Summary Remarkable progress in neutron ββ-decay angular correlations over past several years. Exciting time for the field. Potential in next 5 years for: Value for VV uuuu competitive with superallowed ββ decay Limit (value) on Fierz interference term bb that will compete with the LHC in terms of limits on BSM Scalar and Tensor Robust test of Lattice QCD via gg AA Thanks to Organizers, Collaborators, and You for listening! This material is based upon work supported in part by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics under Award Number DE-SC

55 55

56 References E.D. Commins, Weak Interactions (McGraw-Hill, 1973) G. Kallen, Elementary Particle Physics (Addison-Wesley, 1964) M.E. Peskin and D.V. Schroeder, An Introduction to Quantum Field Theory (Perseus, 1995) P. Renton, Electroweak Interactions (Cambridge University Press, 1990) S.S.M. Wong, Introductory Nuclear Physics (John Wiley & Sons, 1998) C. Zhang, Ph.D. Thesis, University of Kentucky (2002) This is certainly not a comprehensive list of references. Instead, this is just meant to illustrate the list of books I read ~15 years ago, and consulted again in preparing these slides. 56

57 Appendix Pauli-Dirac Representation γγ 0 = II 0 0 II γγ μμ = 0 σσμμ σσ μμ 0 γγ 5 = 0 II II 0 Parity PP ψψγγ μμ ψψpp = ψψγγ 0 ψψ ψψγγ kk ψψ μμ = 0 μμ = kk = 1,2,3 PP ψψγγ 5 γγ μμ ψψpp = ψψγγ 5 γγ 0 ψψ ψψγγ 5 γγ kk ψψ μμ = 0 μμ = kk = 1,2,3 Time-Reversal TT ψψγγ μμ ψψtt = ψψγγ 0 ψψ ψψγγ kk ψψ μμ = 0 μμ = kk = 1,2,3 TT ψψγγ 5 γγ μμ ψψtt = ψψγγ 5 γγ 0 ψψ ψψγγ 5 γγ kk ψψ μμ = 0 μμ = kk = 1,2,3 57

PHL424: Nuclear Shell Model. Indian Institute of Technology Ropar

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