Two body (meson exchange) currents and Gamow-Teller quenching - - PowerPoint PPT Presentation

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, 17th November 2015

Weak transitions in nuclei

β and ββ decay processes, Weak interaction LW = GF √ 2

• jLµJµ†

L

• + H.c.

jLµ leptonic current: electron, neutrino Jµ†

L

hadronic current: quarks → nucleons In nuclei (non-relativistic), β decay is F|

• i

gV τ −

i

+ gA σiτ −

i |I

Fermi and Gamow-Teller transitions corrections (forbidden transitions) expansion of the leptonic current

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Matrix elements

Nuclear matrix elements for weak transitions Final |Lleptons−nucleons| Initial = Final |

• dx jµ(x)Jµ(x) | Initial
• Nuclear structure calculation
• f 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

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Gamow-Teller transitions

Single-β, Gamow-Teller (GT) transitions well described by theory...

p n W ν e

F|

• i

geff

A σiτ − i |I ,

geff

A ≈ 0.7gA

...but need to “quench“ GT operator

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 T(GT) Th. T(GT) Exp.

0.77 0.744

• Martinez-Pinedo et al.

PRC 53 2602 (1996) Iwata et al. JPSCP 6 03057 (2015)

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Double–Gamow-Teller transitions

2νββ decays also well described with "quenched" GT operator

Caurier, Nowacki, Poves PLB 711 62 (2012) M2νβ

β =

• k
• 0+

f

• n σnτ −

n

• 1+

k

1+

k

• m σmτ −

m

• 0+

i

• Ek − (Mi + Mf )/2

Table 2 The ISM predictions for the matrix element of several 2ν double beta decays (in MeV−1). See text for the definitions of the valence spaces and interactions. M2ν(exp) q M2ν(th) INT

48Ca→ 48Ti

0.047 ± 0.003 0.74 0.047 kb3

48Ca→ 48Ti

0.047 ± 0.003 0.74 0.048 kb3g

48Ca→ 48Ti

0.047 ± 0.003 0.74 0.065 gxpf1

76Ge→ 76Se

0.140 ± 0.005 0.60 0.116 gcn28:50

76Ge→ 76Se

0.140 ± 0.005 0.60 0.120 jun45

82Se→ 82Kr

0.098 ± 0.004 0.60 0.126 gcn28:50

82Se→ 82Kr

0.098 ± 0.004 0.60 0.124 jun45

128Te → 128Xe

0.049 ± 0.006 0.57 0.059 gcn50:82

130Te → 130Xe

0.034 ± 0.003 0.57 0.043 gcn50:82

136Xe → 136Ba

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?

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Shell model nuclear structure

Shell model in one-major-shell spaces, phenomenological interactions pf-shell, KB3G, GXPF1A // sd-pf space SDPFMU interaction: 48Ca p3/2, p1/2, f5/2, g9/2 space, GCN2850 int.: 76Ge, 82Se d5/2, s1/2, d3/2, g7/2, h11/2 space, GCN5082 int.: 124Sn, 130Te, 136Xe Experimental excitation spectra and occupancies well reproduced

Theory Exp 500 1000 1500 2000 2500 Excitation energy (keV)

4

+

6

+ +

3

+

4

+

6

+

2

+

5

+

2

+ +

2

+

4

+

2

+

6

+

3

+,4 +

6

+

2

+ 2 +

5 (4

+)

136Xe

2 4 6 8

76Ge 76Se

EXP ISM(GCN) ISM(RG) EXP ISM(GCN) ISM(RG)

Neutron Vacancies

1p 0f5/2 0g9/2

76Ge 76Se

EXP ISM(GCN) ISM(RG) EXP ISM(GCN) ISM(RG)

Proton Occupancies

1p 0f5/2 0g9/2

Exp: Schiffer et al. PRL100 112501(2009), Kay et al. PRC79 021301(2009) Th: JM, Caurier, Nowacki, Poves PRC80 048501 (2009)

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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

2

N LO N LO

3

NLO LO 3N force 4N force 2N force

N N e ν N N e N π N ν e ν N N N N

Weinberg, van Kolck, Kaplan, Savage, Epelbaum, Kaiser, Meißner... Park, Gazit, Klos, Baroni...

Short-range couplings fitted to experiment once

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Oxygen dripline and 3N forces

O isotopes: ’anomaly’ in the dripline at 24O, doubly magic nucleus

2 8 20 28 Z N 2 8

stability line

O 1970 F 1999 N 1985 C 1986 B 1984 Be 1973 Li 1966 H 1934 He 1961 Ne 2002 Na 2002 Mg2007 Al 2007 Si 2007

unstable oxygen isotopes unstable fluorine isotopes stable isotopes unstable isotopes neutron halo nuclei

Calculations based on chiral NN+3N forces and MBPT correctly predict dripline at 24O

Otsuka et al. PRL 105 032501 (2010)

8 20 16 14 Neutron Number (N)

s 1/2

(c) G-matrix NN + 3N (∆) forces

d3/2 d5/2

NN NN + 3N (∆)

Single-Particle Energy (MeV)

4

• 4
• 8

Single-Particle Energy (MeV) 8 20 16 14

d3/2 d5/2 s 1/2

(a) Forces derived from NN theory V G-matrix

Neutron Number (N)

low k

4

• 4
• 8

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Calcium isotopes with NN+3N forces

Calculations with NN+3N forces predict shell closures at 52Ca, 54Ca

28 29 30 31 32 33 34 35 36

Neutron Number N

2 4 6 8 10 12 14 16 18 20 22

S2n (MeV)

MBPT CC SCGF MR-IM-SRG

42 44 46 48 50 52 54 56

Mass Number A

1 2 3 4 5

2

+ Energy (MeV)

MBPT CC 51,52,53,54Ca masses [TRIUMF/ISOLDE] 54Ca 2+ 1 state excitation energy [RIBF]

Hebeler et al. ARNPS 65 457 (2015)

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Two-body currents in light nuclei

Two-body (meson-exchange) currents tested in light nuclei, electromagnetic and weak interactions studied:

3H β decay

Gazit et al. PRL103 102502(2009)

A ≤ 9 magnetic moments

8Be EM transitions

Pastore et al. PRC87 035503(2013) =

⇒

Pastore et al. PRC90 024321(2014)

3H µ capture

Marcucci et al. PRC83 014002(2011)

• 3
• 2
• 1

1 2 3 4 µ (µN) EXPT GFMC(IA) GFMC(TOT) n p

2H 3H 3He 6Li 7Li 7Be 8Li 8B 9Li 9Be 9B 9C

2b current contributions ∼ few % in light nuclei (Q ∼ √ BEm) 2b currents order Q3 ⇒ larger effect in medium-mass nuclei (Q ∼ kF)

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Hadronic weak currents in chiral EFT

At lowest orders Q0, Q2 1b currents only

J0

i (p) = gV (p2)τ −,

Ji(p) =

• gA(p2)σ − gP(p2) (p · σi)p

2m + i (gM + gV ) σi × p 2m

• τ −,

N N e ν

At order Q3 chiral EFT 2b currents predicted Reflect interactions between nucleons in nuclei Long-range currents dominate

N N e N π N ν e ν N N N N

J3

12 = − gA

4F 2

π

1 m2

π + k2

• 2c4k × (σ× × k)τ 3

× + 4c3k ·

• σ1τ 3

1 + σ2τ 3 2

• k
• 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 N π N ν e ν N N N N

⇒ Jeff

n,2b normal-ordered 1b current

Corrections ∼ (nvalence/ncore) in Fermi systems The normal-ordered two-body currents modify GT operator Jeff

n,2b = −gAρ

f 2

π

τ −

n σn

• I(ρ, P)

2c4 − c3 3

• + 2

3 c3 p2 m2

π + p2

• ,

p independent p dependent

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2b currents and GT quenching

2b currents, p = 0: GT, 2νββ decays

Jeff

n,2b = − gAρ f 2

π τ −

n σn

• I(ρ, P)
• 2c4−c3

3

• 0.04

0.08 0.12

ρ [fm

• 3]

0.5 0.7 0.9 1.1

GT(gA+2b)/gA

p=0

1bc 2bc JM, Gazit, Schwenk PRL107 062501 (2011)

General density range ρ = 0.10 . . . 0.12 fm−3 Couplings c3, c4 from NN potentials

Entem et al. PRC68 041001(2003) Epelbaum et al. NPA747 362(2005) Rentmeester et al. PRC67 044001(2003) δc3 = −δc4 ≈ 1 GeV−1

2b currents predict στ quenching q = 0.85...0.66

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2b currents: Coupled-Cluster calculations

Coupled-cluster calculations for single-β decay (GT strengths) including chiral 1b+2b currents in light 14C, 22O and 24O

• 1
• 0.5

0.5 1

cD

0.8 0.9 1

q

2 = (S− − S+) [3(N − Z)]

14C, Λχ = 500MeV 22O, Λχ = 500MeV 24O, Λχ = 500MeV 14C, Λχ = 450MeV 22O, Λχ = 450MeV 24O, Λχ = 450MeV 14C, Λχ = 550MeV 24O, Λχ = 550MeV 22O, Λχ = 550MeV

Ekström et al. PRL113 262504 (2014)

Calculation with chiral EFT NN+3N forces 1b+2b currents Normal-ordered 1b part with respect to Hartree-Fock state From 2b currents predict small στ quenching q = 0.96...0.92

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2b currents: transferred-momentum dependence

2b currents depend on transferred momentum p: − gAρ

f 2

π τ −

n σn

• 2

3 c3 p2 m2

π+p2

• 100

200 300 400

p [MeV]

0.6 0.7 0.8 0.9 1 1.1 0.5

GT(1b+2b)/gA

1bc 2bc JM, Gazit, Schwenk PRL107 062501 (2011)

N N e N π N ν

Quenching reduced at p > 0, relevant for 0νββ decay where p ∼ mπ and other weak processes e.g. muon capture

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Neutrinoless double-beta decay

Neutrinoless double-beta (0νββ) decay: Lepton-number violation, Majorana nature of neutrinos Nuclear matrix elements combined with 0νββ decay lifetimes will determine the mass hierarchy of neutrinos

X

M M

W p W p ν ν n e e n ν p e ν p e W W n n

In 2νββ decay, the momentum transfer limited by Qββ, while for 0νββ decay larger momentum transfers are permitted

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0νββ decay matrix elements with 1b+2b currents

48Ca 76Ge 82Se 124Sn 130Te 136Xe

1 2 3 4 5 6 7

M

0νββ

1b Q 1b Q

2

1b+2b Q

3 cD=0

SM (2009)

100 200 300 400 500 600

p [MeV]

CGT(p)

Q Q

2

Q

3

JM, Gazit, Schwenk PRL107 062501 (2011)

Order Q0+Q2 similar to phenomenological currents

JM, Poves, Caurier, Nowacki NPA818 139 (2009)

Order Q3 2b currents reduce NMEs ∼ 15% − 40% Smaller than −50% (q2 = 0.72) due to momentum-transfer p > 0 2b currents need to be included in all approaches calculating 0νββ decay

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Neutrinoless ββ decay matrix elements

Large difference in matrix element calculations, same transition operator

• ;

• 7

Figure 5: (Color online) (a) Decomposition of the total NMEs from the fi- nal GCM PNAMP (PC-PK1) calculation; (b) the total NMEs calculated with either only spherical configuration or full configurations, in comparison with those of GCM PNAMP (D1S) from Ref. [34]. The shaded area indicates the uncertainty of the SRC e ect within 10%. See text for more details.

the tensor terms were neglected. These two e ects can bring a di erence up to 15% in the NMEs. By taking into account this point, one can draw the conclusion from Fig. 5(b) that these two calculations give consistent results for the total NMEs for all the candidate nuclei with the exception of 150Nd. Moreover, we note that in the calculation with pure spher- ical configuration, PNP increases significantly the NMEs for the 0

• decay evolved with one (semi)magic nucleus, includ-

ing 48Ca (127%), 116Cd (49%), 124Sn (55%), and 136Xe (58%), where pairing collapse occurs in either protons or neutrons. The increase in the NMEs by the PNP is mainly through the su- perfluid partner nucleus. For 48Ca, pairing collapse is found in both neutrons and protons, leading to about twice enhanced normalized NME than the other three ones. It can be under- stood from Eq.(6) that the

F

0 ˆ0 ˆ PJ 0 ˆ PNI ˆ PZI

I

0 for

48Ca-Ti does not change by the PNP, while the normalization

factor

F for the daughter nucleus 48Ti is increased, resulting in

the enhanced normalized NME. The comparison of the results

• f “Sph PNP (PC-PK1)” and “Sph PNP (D1S)” in Fig. 5(b)

shows a large discrepancy in 100Mo-Ru and 150Nd-Sm. This discrepancy could be attributed to di erent pairing properties. However, after taking into account the static and dynamic de- formation e ects, which turn out to decrease the NME signif- icantly, the discrepancy in 100Mo-Ru is much reduced, while that in 150Nd-Sm remains and is mainly attributed to the di er- ence in the overlap between the initial and final collective wave functions, as already discussed in Ref. [37]. Figure 6 displays our final NMEs for the 0

• decay in

comparison with those by the ISM [23], renormalized QRPA (RQRPA) [30], PHFB [33], NREDF (D1S) [34], and the IBM2 [32]. There are also other calculations that are not taken

• <

Figure 6: (Color online) Comparison of the NME M0 for the 0

• decay from

di erent model calculations. The shaded area indicates the uncertainty of the SRC e ect within 10%. The adopted values are available on the web site [52]. Table 2: The upper limits of the e ective neutrino mass m (eV) based on the NMEs from the present GCM PNAMP (PC-PK1) calculation, the lower limits

• f the half-life T 0

1 2( 1024 yr) for the 0

• decay from most recent measure-

ments [56, 10, 57, 58, 8, 9, 59] and the phase-space factor G0 ( 10 15 yr 1) from Ref. [14].

48Ca 76Ge 82Se 100Mo 130Te 136Xe 150Nd

m 2.92 0.20 1.00 0.38 0.33 0.11 1.76 T 0

1 2

0.058 30 0.36 1.1 2.8 34 0.018 G0 24.81 2.363 10.16 15.92 14.22 14.58 60.03 for comparison. Here, only the calculations considering the SRC e ect with the UCOM (except for the IBM2 calculation with the coupled-cluster model (CCM)) and using the radius parameter R 1 2A1 3 fm are adopted for comparison. Our results are amongst the largest values of the existing calcula- tions in most cases, except for 100Mo-Ru, 124Sn-Te and 130Te-

• Xe. Moreover, the NME for 96Zr in both EDF-based calcu-

lations is significantly larger than the other results, which can be traced back to the overestimated collectivity. If the ground state of 96Zr was taken as the pure spherical configuration, the NME becomes 5.64 (PC-PK1) and 3.94 (D1S), respectively. We note that the consideration of higher-order deformation in nuclear wave functions, such as octupole deformation in 150Sm- Nd [53, 54], and triaxiality in 76Ge-Se [50, 51] and 100Mo- Ru [55], is expected to hinder the corresponding NMEs further in the DFT calculation. Table 2 lists the upper limits of the e ective neutrino mass m based on the present calculated NMEs for the nuclei whose lower limits of the half-life T 0

1 2 for the 0

• decay have

been recently measured [56, 10, 57, 58, 9, 59]. The smallest value ( 0 11 eV) for the upper limit m is found based on the combined results from KamLAND-Zen [9] and EXO-200 [8] collaborations for the0

• decay half-life (T 0

1 2

3 4 1025 yr at 90% confidence level) of 136Xe. This value is closest to but still larger than the estimated value (20 50 meV based on the inverted hierarchy for neutrino masses [19]) by a factor of 2 5. Summary and outlook. In summary, we have reported a 5

Yao et al. PRC91 024316 (2015)

Shell model small matrix elements: What is the effect of the valence space? EDF, IBM, QRPA large matrix elements: How well include nuclear structure correlations? Pairing, deformation, isoscalar pairing...

NME

QRPA IBM EDF SM SM SM (pf) (MBPT) (sdpf)

1 2 3

Ca

48

Iwata et al. (2015)

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Isoscalar pairing and 0νββ decay

0νββ decay very sensitive to isoscalar (proton-neutron) pairing Matrix elements too large if proton-neutron correlations are neglected

• 10
• 5

5 10 0.5 1 1.5 2 2.5 3

M0ν gT=0/ ¯ gT=1

GCM SkO′ QRPA SkO′ GCM SkM* QRPA SkM*

Hinohara, Engel PRC90 031301 (2014)

1 2 3 4 5 Ti → Cr 1 2 3 4 5 22 24 26 28 30 32 34 36 Cr → Fe KB3G Hcoll. Hcoll.(no P†P) MGT Nmother

JM et al. arXiv:1510.06824

Related to approximate SU(4) symmetry of the 0νββ decay operator

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Summary

Why nuclear structure calculations need to quench the στ operator to agree with experiment remains an open puzzle Corrections to 1b electromagnetic and weak operators shown to be needed in ab initio calculations of light nuclei Chiral EFT predicts 2b corrections for GT transitions: approximately calculated in medium-mass nuclei Long-range 2b currents contribute to GT quenching but actual size of this effect remains to be settled At larger momentum transfers p ∼ mπ the quenching due to 2b currents is reduced relevant for neutrinoless double-beta decay

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Collaborators

• G. Martínez-Pinedo
• A. Schwenk
• D. Gazit
• T. Otsuka
• T. Abe
• Y. Iwata
• N. Shimizu
• Y. Utsuno
• M. Honma
• N. Hinohara
• J. Engel
• A. Poves
• T. R. Rodríguez

E.Caurier F . Nowacki

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