Two body (meson exchange) currents and Gamow-Teller quenching Javier Menéndez JSPS Fellow, The University of Tokyo International Symposium on "High-Resolution Spectroscopy and Tensor Interactions" Osaka, 17 th November 2015
Weak transitions in nuclei β and ββ decay processes, Weak interaction L W = G F � � j L µ J µ † √ + H . c . L 2 j L µ leptonic current: electron, neutrino J µ † hadronic current: quarks → nucleons L In nuclei (non-relativistic), β decay is � g V τ − + g A σ i τ − � F | i | I � i i Fermi and Gamow-Teller transitions corrections (forbidden transitions) expansion of the leptonic current 2 / 20
Matrix elements Nuclear matrix elements for weak transitions � dx j µ ( x ) J µ ( x ) | Initial � � Final |L leptons − nucleons | Initial � = � Final | • Nuclear structure calculation of the initial and final states: Ab initio, shell model, energy density functional... • Lepton-nucleus interaction: Evaluate (non-perturbative) hadronic currents inside nucleus: phenomenology, effective theory 3 / 20
Gamow-Teller transitions Single- β , Gamow-Teller (GT) transitions well described by theory... 1.0 ν 0.77 0.8 W 0.744 e T(GT) Exp. n p 0.6 0.4 � g eff A σ i τ − g eff � F | i | I � , A ≈ 0 . 7 g A 0.2 i 0.0 ...but need to “quench“ GT operator 0.0 0.2 0.4 0.6 0.8 1.0 T(GT) Th. ����� � ���� ����� � ���� ������� ��� ������� ��� ���������� ��� ���������� ���������� ��� ���������� ��� Martinez-Pinedo et al. PRC 53 2602 (1996) ��� ��� Iwata et al. ��� ����� ����� JPSCP 6 03057 (2015) ��� ��� � � �� �� �� �� �� � � �� �� �� �� �� 4 / 20
Double–Gamow-Teller transitions 2 νββ decays also well described with "quenched" GT operator 0 + � 1 + 1 + � 0 + � � n σ n τ − � � � � m σ m τ − � � � � � � β = n m M 2 νβ � f k k i E k − ( M i + M f ) / 2 k Table 2 The ISM predictions for the matrix element of several 2 ν double beta decays (in MeV − 1 ). See text for the definitions of the valence spaces and interactions. M 2 ν ( exp ) M 2 ν ( th ) INT q 48 Ca → 48 Ti 0 . 047 ± 0 . 003 0.74 0.047 kb3 48 Ca → 48 Ti 0 . 047 ± 0 . 003 0.74 0.048 kb3g 48 Ca → 48 Ti 0 . 047 ± 0 . 003 0.74 0.065 gxpf1 76 Ge → 76 Se 0 . 140 ± 0 . 005 0.60 0.116 gcn28:50 76 Ge → 76 Se 0 . 140 ± 0 . 005 0.60 0.120 jun45 82 Se → 82 Kr 0 . 098 ± 0 . 004 0.60 0.126 gcn28:50 82 Se → 82 Kr 0 . 098 ± 0 . 004 0.60 0.124 jun45 128 Te → 128 Xe 0 . 049 ± 0 . 006 0.57 0.059 gcn50:82 130 Te → 130 Xe 0 . 034 ± 0 . 003 0.57 0.043 gcn50:82 Caurier, Nowacki, Poves PLB 711 62 (2012) 136 Xe → 136 Ba 0 . 019 ± 0 . 002 0.45 0.025 gcn50:82 This puzzle has been the target of many theoretical efforts: Arima, Rho, Towner, Bertsch and Hamamoto, Wildenthal and Brown... Anything missing in the transition operator or in many-body approach? 5 / 20
Shell model nuclear structure Shell model in one-major-shell spaces, phenomenological interactions pf-shell, KB3G, GXPF1A // sd-pf space SDPFMU interaction: 48 Ca p 3 / 2 , p 1 / 2 , f 5 / 2 , g 9 / 2 space, GCN2850 int.: 76 Ge, 82 Se d 5 / 2 , s 1 / 2 , d 3 / 2 , g 7 / 2 , h 11 / 2 space, GCN5082 int.: 124 Sn, 130 Te, 136 Xe Experimental excitation spectra and occupancies well reproduced Neutron Vacancies Proton Occupancies + ) + (4 ISM(GCN) 2500 2 5 ISM(RG) + + 2 + 5 + 2 2 + + 6 EXP 4 Excitation energy (keV) + + ,4 + 6 3 + 3 ISM(GCN) 2000 8 ISM(GCN) ISM(RG) + + ISM(RG) 6 6 + 4 EXP + 4 EXP ISM(GCN) 1500 6 ISM(RG) + 2 + 2 0g 9/2 EXP 0g 9/2 1000 4 0f 5/2 500 2 0f 5/2 136 Xe 1p 1p + + 0 0 0 0 Exp Theory 76 Ge 76 Se 76 Ge 76 Se Exp: Schiffer et al. PRL100 112501(2009), Kay et al. PRC79 021301(2009) Th: JM, Caurier, Nowacki, Poves PRC80 048501 (2009) 6 / 20
Chiral Effective Field Theory Chiral EFT: low energy approach to QCD, nuclear structure energies Approximate chiral symmetry: pion exchanges, contact interactions Systematic expansion: nuclear forces and electroweak currents 2N force 3N force 4N force N ν e LO N NLO N N e ν N N e ν π 2 N LO N N N N Park, Gazit, Klos, Baroni... N LO 3 Short-range couplings fitted to experiment once Weinberg, van Kolck, Kaplan, Savage, Epelbaum, Kaiser, Meißner... 7 / 20
Oxygen dripline and 3N forces O isotopes: ’anomaly’ in the dripline at 24 O, doubly magic nucleus 4 Single-Particle Energy (MeV) (a) Forces derived from NN theory stability line d3/2 Z 0 Si 2007 Al 2007 -4 Mg � 2007 s 1/2 d5/2 Na 2002 Ne 2002 F 1999 -8 G-matrix 8 V O 1970 low k stable isotopes N 1985 8 14 16 20 C 1986 unstable isotopes Neutron Number ( N ) B 1984 unstable fluorine isotopes Be 1973 Single-Particle Energy (MeV) 4 (c) G-matrix NN + 3N ( ∆) forces Li 1966 unstable oxygen isotopes 2 He 1961 neutron halo nuclei H 1934 0 d3/2 N 2 8 20 28 -4 s 1/2 d5/2 Calculations based on chiral NN+3N forces -8 NN + 3N ( ∆) and MBPT correctly predict dripline at 24 O NN 8 14 16 20 Otsuka et al. PRL 105 032501 (2010) Neutron Number ( N ) 8 / 20
Calcium isotopes with NN+3N forces Calculations with NN+3N forces predict shell closures at 52 Ca, 54 Ca 5 22 20 MBPT 18 CC 4 + Energy (MeV) 16 S 2n (MeV) 14 3 12 10 2 8 MBPT 6 CC 2 1 SCGF 4 MR-IM-SRG 2 0 0 28 29 30 31 32 33 34 35 36 42 44 48 46 50 52 54 56 Neutron Number N Mass Number A 51 , 52 , 53 , 54 Ca masses [TRIUMF/ISOLDE] 54 Ca 2 + 1 state excitation energy [RIBF] Hebeler et al. ARNPS 65 457 (2015) 9 / 20
Two-body currents in light nuclei Two-body (meson-exchange) currents tested in light nuclei, electromagnetic and weak interactions studied: 4 3 H β decay 3 Gazit et al. PRL103 102502(2009) 9 B 7 Li p 9 Li 3 H 2 A ≤ 9 magnetic moments 8 Be EM transitions 1 8 Li 8 B µ ( µ N ) 2 H 6 Li Pastore et al. PRC87 035503(2013) = ⇒ 0 GFMC(IA) 9 C Pastore et al. PRC90 024321(2014) GFMC(TOT) 9 Be 7 Be EXPT -1 3 H µ capture 3 He n -2 Marcucci et al. PRC83 014002(2011) -3 √ 2b current contributions ∼ few % in light nuclei ( Q ∼ BEm ) 2b currents order Q 3 ⇒ larger effect in medium-mass nuclei ( Q ∼ k F ) 10 / 20
Hadronic weak currents in chiral EFT At lowest orders Q 0 , Q 2 1b currents only N e ν J 0 i ( p ) = g V ( p 2 ) τ − , � g A ( p 2 ) σ − g P ( p 2 ) ( p · σ i ) p + i ( g M + g V ) σ i × p � τ − , J i ( p ) = 2 m 2 m N At order Q 3 chiral EFT N ν N e N N ν e 2b currents predicted Reflect interactions π between nucleons in nuclei Long-range currents dominate N N N N 12 = − g A 1 � � J 3 2 c 4 k × ( σ × × k ) τ 3 σ 1 τ 3 1 + σ 2 τ 3 × + 4 c 3 k · � � k 2 4 F 2 m 2 π + k 2 π 11 / 20
2b currents in medium-mass nuclei Approximate in medium-mass nuclei: normal-ordered 1b part with respect to spin/isospin symmetric Fermi gas Sum over one nucleon, direct and the exchange terms N N e ν N N ν e ⇒ J eff n , 2 b normal-ordered 1b current Corrections ∼ ( n valence / n core ) π in Fermi systems N N N N The normal-ordered two-body currents modify GT operator p 2 n , 2 b = − g A ρ � � 2 c 4 − c 3 � + 2 � J eff τ − I ( ρ, P ) 3 c 3 , n σ n f 2 m 2 π + p 2 3 π p independent p dependent 12 / 20
2b currents and GT quenching � � �� 2 c 4 − c 3 n , 2 b = − g A ρ 2b currents, p = 0: GT, 2 νββ decays J eff π τ − I ( ρ, P ) n σ n f 2 3 1.1 General density range 1bc ρ = 0 . 10 . . . 0 . 12 fm − 3 GT(g A +2b)/g A 0.9 Couplings c 3 , c 4 from NN potentials 2bc Entem et al. 0.7 PRC68 041001(2003) p=0 Epelbaum et al. NPA747 362(2005) Rentmeester et al. 0.5 0 0.04 0.08 0.12 PRC67 044001(2003) -3 ] ρ [fm δ c 3 = − δ c 4 ≈ 1 GeV − 1 JM, Gazit, Schwenk PRL107 062501 (2011) 2b currents predict σ τ quenching q = 0 . 85 ... 0 . 66 13 / 20
2b currents: Coupled-Cluster calculations Coupled-cluster calculations for single- β decay (GT strengths) including chiral 1b+2b currents in light 14 C, 22 O and 24 O 1 14 C, Λ χ = 500MeV 22 O, Λ χ = 500MeV 24 O, Λ χ = 500MeV Calculation with chiral EFT 2 = (S − − S + ) [3( N − Z )] 14 C, Λ χ = 450MeV NN+3N forces 22 O, Λ χ = 450MeV 24 O, Λ χ = 450MeV 1b+2b currents 14 C, Λ χ = 550MeV 24 O, Λ χ = 550MeV Normal-ordered 1b part 0.9 22 O, Λ χ = 550MeV with respect to Hartree-Fock state q From 2b currents predict small σ τ quenching q = 0 . 96 ... 0 . 92 0.8 -1 -0.5 0 0.5 1 c D Ekström et al. PRL113 262504 (2014) 14 / 20
2b currents: transferred-momentum dependence p 2 � � 2b currents depend on transferred momentum p : − g A ρ 2 π τ − n σ n 3 c 3 f 2 m 2 π + p 2 1.1 1 GT(1b+2b)/g A 1bc 0.9 2bc N ν N e 0.8 0.7 π 0.6 N N 0.5 0 100 200 300 400 p [MeV] JM, Gazit, Schwenk PRL107 062501 (2011) Quenching reduced at p > 0, relevant for 0 νββ decay where p ∼ m π and other weak processes e.g. muon capture 15 / 20
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