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Nuclear Matrix Elements for Fundamental Symmetries Javier Menéndez JSPS Fellow, The University of Tokyo 3 rd European Nuclear Physics Conference Groningen, 3 rd September 2015
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Nuclear Matrix Elements for Fundamental SymmetriesNuclear Matrix Elements for Fundamental Symmetries Javier Menéndez JSPS Fellow, The University of Tokyo 3rd European Nuclear Physics

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Page 1: Nuclear Matrix Elements for Fundamental SymmetriesNuclear Matrix Elements for Fundamental Symmetries Javier Menéndez JSPS Fellow, The University of Tokyo 3rd European Nuclear Physics

Nuclear Matrix Elementsfor

Fundamental Symmetries

Javier Menéndez

JSPS Fellow, The University of Tokyo

3rd European Nuclear Physics Conference

Groningen, 3rd September 2015

Page 2: Nuclear Matrix Elements for Fundamental SymmetriesNuclear Matrix Elements for Fundamental Symmetries Javier Menéndez JSPS Fellow, The University of Tokyo 3rd European Nuclear Physics

Neutrinoless ββ decay, Dark Matter detection

Dark matter scattering off nucleiWhat is Dark Matter made of?

Neutrinoless double-beta decay

Lepton number violationMajorana / Dirac nature of neutrinosNeutrino masses and hierarchy

2 / 30

Page 3: Nuclear Matrix Elements for Fundamental SymmetriesNuclear Matrix Elements for Fundamental Symmetries Javier Menéndez JSPS Fellow, The University of Tokyo 3rd European Nuclear Physics

Nuclear physics and fundamental symmetriesNeutrinos, Dark Matter can be studied with high-energy experiments

Nuclear physics offers an alternative:Nuclei are abundant in huge numbers NA = 6.02 1023 nuclei in A grams!

Lots of material over long times provides access to detectvery rare decays and very small cross-sections!

Isolate from other processes:very low background (underground)

Visible Energy (MeV)

1 2 3 4

Ev

ents

/0.0

5M

eV

-110

1

10

210

310

410

510 (a) DS-1 + DS-2 Bi

208

Y88

Ag110m

Th232U + 238

Kr85

Bi + 210 +

IB/External

Spallation

Data

Total

Total

(90% C.L. U.L.)

KamLAND-Zen3 / 30

Page 4: Nuclear Matrix Elements for Fundamental SymmetriesNuclear Matrix Elements for Fundamental Symmetries Javier Menéndez JSPS Fellow, The University of Tokyo 3rd European Nuclear Physics

Nuclear matrix elementsNuclear matrix elements are needed to study fundamental symmetries

〈Final |Lleptons−nucleons| Initial 〉 = 〈Final |∫

dx jµ(x)Jµ(x) | Initial 〉

• Nuclear structure calculationof 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

CDMS Collaboration4 / 30

Page 5: Nuclear Matrix Elements for Fundamental SymmetriesNuclear Matrix Elements for Fundamental Symmetries Javier Menéndez JSPS Fellow, The University of Tokyo 3rd European Nuclear Physics

Lepton-number conservationLepton number conservedin all processes observed to date

β decay, 2νββ decay...

Uncharged massive particleslike Majorana neutrinos (ν = ν)theoretically allow lepton number violation

Neutrinoless ββ (0νββ) decay5 / 30

Page 6: Nuclear Matrix Elements for Fundamental SymmetriesNuclear Matrix Elements for Fundamental Symmetries Javier Menéndez JSPS Fellow, The University of Tokyo 3rd European Nuclear Physics

Weak transitions in nucleiβ and ββ decay processes driven by Weak interaction

LW =GF√

2

(jLµJµ†L

)+ H.c.

jLµ leptonic current (electron, neutrino)

Jµ†L hadronic currentStandard Model: Jµ†L for quarks, need Jµ†L for nucleons

N

N

e ν

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 lepton current

6 / 30

Page 7: Nuclear Matrix Elements for Fundamental SymmetriesNuclear Matrix Elements for Fundamental Symmetries Javier Menéndez JSPS Fellow, The University of Tokyo 3rd European Nuclear Physics

Gamow-Teller transitionsSingle-β, 2νββ decays well described by nuclear structure: shell model...

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

T(GT) Th.

T(G

T)

Ex

p.

0.77

0.744

Martinez-Pinedo et al. PRC53 2602(1996)

〈F |∑

i

geffA σiτ

−i |I〉 , geff

A ≈ 0.7gA

For agreement theory needs to“quench“ Gamow-Teller operator

pn

W

ν

e

M2νββ =∑

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

Caurier, Nowacki, Poves PLB711 62(2012)7 / 30

Page 8: Nuclear Matrix Elements for Fundamental SymmetriesNuclear Matrix Elements for Fundamental Symmetries Javier Menéndez JSPS Fellow, The University of Tokyo 3rd European Nuclear Physics

Neutrinoless double-beta decayNeutrinoless double-beta decay (0νββ):Lepton-number violation, Majorana nature of neutrinos

Second order process only observable if single-β-decayis energetically forbidden or hindered by large ∆J

-75

-70

-65

30 31 32 33 34 35 36 37

Mas

s E

xces

s (M

eV)

Z

Se

GeAs

BrKr

Ga48Ca→ 48Ti76Ge→ 76Se82Se→ 82Kr96Zr→ 96Mo100Mo→ 100Ru110Pd→ 110Cd116Cd→ 116Sn124Sn→ 124Te130Te→ 130Xe136Xe→ 136Ba150Nd→ 150Sm

8 / 30

Page 9: Nuclear Matrix Elements for Fundamental SymmetriesNuclear Matrix Elements for Fundamental Symmetries Javier Menéndez JSPS Fellow, The University of Tokyo 3rd European Nuclear Physics

ββ decaysββ decays are quite different processes

(Ee−1+Ee−2

)/Qββ0.0 0.2 0.4 0.6 0.8 1.0

0.2

0.4

0.6

0.8

1.0

2ν2β spectrum 0ν2β signal

2νββ : Ee1 + Ee2 + Eν1 + Eν2 = Qββ0νββ : Ee1 + Ee2 = Qββ

X

M

M

W

p

Wp

ν

ν

n

e

e

n

ν

pe

ν

pe

W

W

n

n

In 2νββ decay, the momentum transferlimited by Qββ , while for 0νββ decaylarger momentum transfers are permitted

In 0νββ decay the Majorana neutrinos arepart of the transition operator,via the so-called neutrino potential

Lifetime limits: 76Ge (GERDA), 136Xe (EXO, KamLAND) T 0νββ1/2 >1025 y!

9 / 30

Page 10: Nuclear Matrix Elements for Fundamental SymmetriesNuclear Matrix Elements for Fundamental Symmetries Javier Menéndez JSPS Fellow, The University of Tokyo 3rd European Nuclear Physics

0νββ decay nuclear matrix elements

0νββ process needs massive Majorana neutrinos (ν = ν)⇒ detection would proof Majorana nature of neutrinos

(T 0νββ

1/2

(0+ → 0+

))−1= G01

∣∣M0νββ∣∣2(mββ

me

)2

M0νββ is the nuclear matrix element

G01 is the phase space factor: Qββ , electrons...

M0νββ is necessary to identify best candidates for experiment,and to obtain neutrino masses, with mββ = |∑U2

ek mk |

Compete with other determinations of neutrino masses:

single-β decay (√∑ |Uek |2m2

k ) and cosmology (∑

mk )10 / 30

Page 11: Nuclear Matrix Elements for Fundamental SymmetriesNuclear Matrix Elements for Fundamental Symmetries Javier Menéndez JSPS Fellow, The University of Tokyo 3rd European Nuclear Physics

Neutrino mass hierarchy

Nuclear matrix elements combined with 0νββ decay lifetimescan determine the mass hierarchy of neutrinos

10-4 0.001 0.010 0.100 110-4

0.001

0.010

0.100

1

m1,3 [eV]

|mee |[e

V]

NH

IH

QD

Pla

nck

1

KA

TRIN

Pla

nck

2

GERDA

Neutrino mass differences known from neutrino oscillation experiments11 / 30

Page 12: Nuclear Matrix Elements for Fundamental SymmetriesNuclear Matrix Elements for Fundamental Symmetries Javier Menéndez JSPS Fellow, The University of Tokyo 3rd European Nuclear Physics

Neutrinoless ββ decay matrix elementsLarge difference in matrix element calculations, same transition operator

!

"

!

#

$

%

"

!

#

$

%

&"

&!

&#

&$

&%

!"

#$%&

''

((

))

**

+

!

')

!"#$%&"$'($!)$*+,

,"

+

+

+-./'($!)$*+,

+-./#$%$'($!)$*+,'

+ !"#$%&"$'($!)$*+,'

+-./#$%$(0+-,

+ !"#$%&"$+ 0+-,

+12

%3

+45

67

+42

87

+9:

-;

++5

!3

+22

"<

=5

>?

@9

-7

A5

7

:@

!B

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

the tensor terms were neglected. These two e ects can bring adi erence up to 15% in the NMEs. By taking into accountthis point, one can draw the conclusion from Fig. 5(b) that thesetwo calculations give consistent results for the total NMEs forall 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 forthe 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. Theincrease in the NMEs by the PNP is mainly through the su-perfluid partner nucleus. For 48Ca, pairing collapse is foundin both neutrons and protons, leading to about twice enhancednormalized NME than the other three ones. It can be under-stood from Eq.(6) that the F 0 ˆ0 PJ 0PNI PZI I 0 for48Ca-Ti does not change by the PNP, while the normalizationfactor F for the daughter nucleus 48Ti is increased, resulting inthe enhanced normalized NME. The comparison of the resultsof “Sph PNP (PC-PK1)” and “Sph PNP (D1S)” in Fig. 5(b)shows a large discrepancy in 100Mo-Ru and 150Nd-Sm. Thisdiscrepancy 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, whilethat in 150Nd-Sm remains and is mainly attributed to the di er-ence in the overlap between the initial and final collective wavefunctions, 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 theIBM2 [32]. There are also other calculations that are not taken

!

"

#

$

%

!"#

$ !"#

%&#'

( )* %+

,'-.

,/-

012

$3

045

67

042

87

0.9

/:

005

;3

022

-<

=5

>?

@.

/7

A5

B7

9@

;C

02D

Figure 6: (Color online) Comparison of the NME M0 for the 0 -decay fromdi erent model calculations. The shaded area indicates the uncertainty of theSRC 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 theNMEs from the present GCM PNAMP (PC-PK1) calculation, the lower limitsof the half-life T 01 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 150Ndm 2.92 0.20 1.00 0.38 0.33 0.11 1.76T 01 2 0.058 30 0.36 1.1 2.8 34 0.018G0 24.81 2.363 10.16 15.92 14.22 14.58 60.03

for comparison. Here, only the calculations considering theSRC e ect with the UCOM (except for the IBM2 calculationwith the coupled-cluster model (CCM)) and using the radiusparameter R 1 2A1 3 fm are adopted for comparison. Ourresults 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 canbe traced back to the overestimated collectivity. If the groundstate of 96Zr was taken as the pure spherical configuration, theNME becomes 5.64 (PC-PK1) and 3.94 (D1S), respectively.We note that the consideration of higher-order deformation innuclear 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 furtherin the DFT calculation.Table 2 lists the upper limits of the e ective neutrino mass

m based on the present calculated NMEs for the nucleiwhose lower limits of the half-life T 0

1 2 for the 0 -decay havebeen recently measured [56, 10, 57, 58, 9, 59]. The smallestvalue ( 0 11 eV) for the upper limit m is found based on thecombined results from KamLAND-Zen [9] and EXO-200 [8]collaborations for the0 -decay half-life (T 0

1 2 3 4 1025 yrat 90% confidence level) of 136Xe. This value is closest to butstill larger than the estimated value (20 50 meV based on theinverted hierarchy for neutrino masses [19]) by a factor of 2 5.Summary and outlook. In summary, we have reported a

5

Shell model small matrix elements:What is the effect of the small valence space?

Yao et al. PRC91 024316 (2015)

EDF, IBM, QRPAlarge matrix elements:How well they includenuclear structurecorrelations?

NM

E

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

0

1

2

3

Ca48

12 / 30

Page 13: Nuclear Matrix Elements for Fundamental SymmetriesNuclear Matrix Elements for Fundamental Symmetries Javier Menéndez JSPS Fellow, The University of Tokyo 3rd European Nuclear Physics

Pairing correlations and 0νββ decay

0νββ decay is favoured by pairing correlations

Maximum between superfluidnuclei, reduced with high-seniorities

0

2

4

6

8

10

12

0 4 8 12

M0ν

ββ

sm

A=48A=76A=82

A=124A=128A=130A=136

Caurier et al. PRL100 052503 (2008)

Related to two-nucleon transfers

0

2

4

6

8

0 2 4 6

0+

0

2

4

6

8

su

mm

ed 0νββ

NM

E

0+ + 2+

0

2

4

6

8

10

all Jm

76Se − 74Ge − 76Ge

ν all

0 2 4 6

Ex in 74Ge (MeV)

ν GT

0 2 76Ge74Ge

76Se

76As

Brown et al. PRL113 262501 (2014)13 / 30

Page 14: Nuclear Matrix Elements for Fundamental SymmetriesNuclear Matrix Elements for Fundamental Symmetries Javier Menéndez JSPS Fellow, The University of Tokyo 3rd European Nuclear Physics

Deformation and 0νββ decay

0νββ decay is disfavoured by quadrupole correlations0νββ decay very suppressed when nuclei have different structure

-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8

β 150Nd

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

β1

50S

m

0

1

2

3

4

5

6

4.5

2.5

0.5

0.5

0.5

0.5GT

(a)

Rodríguez, Martínez-PinedoPRL105 252503 (2010)

1.5

2

2.5

3

3.5

4

4.5

5

0 0.02 0.04 0.06

0.4

0.6

0.8

1

NM

E

overlap

∆β

A=66

overlap

NME

JM, Caurier, Nowacki, PovesJPCS267 012058 (2011)

Suppression also observed with QRPA Fang et al. PRC83 034320 (2011)14 / 30

Page 15: Nuclear Matrix Elements for Fundamental SymmetriesNuclear Matrix Elements for Fundamental Symmetries Javier Menéndez JSPS Fellow, The University of Tokyo 3rd European Nuclear Physics

Proton-neutron pairing and 0νββ decay

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

-10

-5

0

5

10

0 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)

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

22 24 26 28 30 32 34 36 38 40

MG

T

Nmother

Ca-->Ti Gamow-Teller Matrix Element for 0νββ decay

KB3G

M+P01+P10+QQ+στστ

M+P01+P10

M+P01+QQ+στστ

JM, Hinohara, Rodriguez, Engel, Martínez-Pinedo

Related to approximate SU(4) symmetry of the 0νββ decay operator15 / 30

Page 16: Nuclear Matrix Elements for Fundamental SymmetriesNuclear Matrix Elements for Fundamental Symmetries Javier Menéndez JSPS Fellow, The University of Tokyo 3rd European Nuclear Physics

0νββ decay without correlationsNon-realistic spherical (uncorrelated) mother and daughter nuclei:• Shell model (SM): zero seniority, neutron and proton J = 0 pairs• Energy density functional (EDF): only spherical contributions

0

1

2

3

4

5

22 24 26 28 30 32 34 36

MG

T

Nmother

Ca-->Ti (EDF Gogny)

Ca-->Ti (ISM KB3G)

Ca-->Ti (ISM GXPF1A)

In contrast to full(correlated) calculationSM and EDF NMEs agree!

NME scale set bypairing interactionJM, Rodríguez, Martínez-Pinedo,Poves PRC90 024311(2014)

NME follows generalizedseniority model:

M0νββGT 'απαν

√Nπ+1

√Ωπ−Nπ

√Nν√

Ων−Nν+1, Barea, Iachello PRC79 044301(2009)

16 / 30

Page 17: Nuclear Matrix Elements for Fundamental SymmetriesNuclear Matrix Elements for Fundamental Symmetries Javier Menéndez JSPS Fellow, The University of Tokyo 3rd European Nuclear Physics

Gamow-Teller transitions: quenchingSingle-β, 2νββ decays well described by nuclear structure: shell model...

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

T(GT) Th.

T(G

T)

Ex

p.

0.77

0.744

Martinez-Pinedo et al. PRC53 2602(1996)

〈F |∑

i

geffA σiτ

−i |I〉 , geff

A ≈ 0.7gA

For agreement theory needs to“quench“ Gamow-Teller operator

pn

W

ν

e

M2νββ =∑

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

Caurier, Nowacki, Poves PLB711 62(2012)7 / 30

Page 18: Nuclear Matrix Elements for Fundamental SymmetriesNuclear Matrix Elements for Fundamental Symmetries Javier Menéndez JSPS Fellow, The University of Tokyo 3rd European Nuclear Physics

Chiral Effective Field TheoryChiral EFT: low energy approach to QCD, nuclear structure energies

Approximate chiral symmetry: pion exchanges, contact interactions

Systematic expansion: nuclear forces and electroweak currents

2N LO

N LO3

NLO

LO

3N force 4N force2N force

N

N

e ν

N

N

e

N

π

N ν e ν

N

NN

N

Weinberg, van Kolck, Kaplan, Savage, Epelbaum, Kaiser, Meißner...

Park, Gazit, Klos...

Short-range couplingsfitted to experiment once

17 / 30

Page 19: Nuclear Matrix Elements for Fundamental SymmetriesNuclear Matrix Elements for Fundamental Symmetries Javier Menéndez JSPS Fellow, The University of Tokyo 3rd European Nuclear Physics

Calcium isotopes with chiral NN+3N forcesCalculations with NN+3N forces successful in medium-mass nuclei

28 29 30 31 32 33 34 35 36

Neutron Number N

0

2

4

6

8

10

12

14

16

18

20

22

S2

n (

MeV

)

MBPT

CC

SCGF

MR-IM-SRG

42 44 46 48 50 52 54 56

Mass Number A

0

1

2

3

4

5

2+ E

ner

gy (

MeV

)

MBPT

CC

Prediction of shell closures at 52Ca, 54Ca51,52,53,54Ca masses [TRIUMF/ISOLDE]54Ca 2+

1 state excitation energy [RIBF]18 / 30

Page 20: Nuclear Matrix Elements for Fundamental SymmetriesNuclear Matrix Elements for Fundamental Symmetries Javier Menéndez JSPS Fellow, The University of Tokyo 3rd European Nuclear Physics

2b currents in light nuclei2b currents (meson-exchange currents) tested in light nuclei:

3H β decayGazit et al. PRL103 102502(2009)

A ≤ 9 magnetic moments8Be EM transitionsPastore et al. PRC87 035503(2013)Pastore et al. PRC90 024321(2014)

3H µ captureMarcucci et al. PRC83 014002(2011)

-3

-2

-1

0

1

2

3

4

µ (µ

N)

EXPT

GFMC(IA)

GFMC(TOT)

n

p

2H

3H

3He

6Li

7Li

7Be

8Li 8B

9Li

9Be

9B

9C

In medium-mass nuclei, chiral EFT 1b + 2b currents (normal ordering)N

N

e ν

+

N

N

e

N

π

N ν e ν

N

N

N

N

19 / 30

Page 21: Nuclear Matrix Elements for Fundamental SymmetriesNuclear Matrix Elements for Fundamental Symmetries Javier Menéndez JSPS Fellow, The University of Tokyo 3rd European Nuclear Physics

2b currents in medium-mass nucleiNormal-ordered 2b currents modify GT operatorJM, Gazit, Schwenk PRL107 062501 (2011)

Jeffn,2b ' −

gAρ

f 2π

τ−n σn

[I(ρ,P)

(2c4−c3)

3

]−

gAρ

f 2π

τ−n σn23

c3p2

4m2π + p2

,

N

N

e

N

π

N ν

0 0.04 0.08 0.12

ρ [fm-3

]

0.5

0.7

0.9

1.1

p=0

1bc

2bc

0 100 200 300 400p [MeV]

0.6

0.7

0.8

0.9

1

1.1

0.5

GT

(1b+

2b)/

gA 1bc

2bc

2b currents predict gA quenching q = 0.85...0.66Quenching reduced at p > 0, relevant for 0νββ decay where p ∼ mπ

20 / 30

Page 22: Nuclear Matrix Elements for Fundamental SymmetriesNuclear Matrix Elements for Fundamental Symmetries Javier Menéndez JSPS Fellow, The University of Tokyo 3rd European Nuclear Physics

Nuclear matrix elements with 1b+2b currents

48Ca

76Ge

82Se

124Sn

130Te

136Xe

0

1

2

3

4

5

6

7

M0ν

ββ

1b Q0

1b Q2

1b+2b Q3 c

D=0

SM (2009)

0 100 200 300 400 500 600

p [MeV]

CG

T(p

)

Q0

Q2

Q3

JM, Gazit, Schwenk PRL107 062501 (2011)

Order Q0+Q2 similar tophenomenological currentsJM, Poves, Caurier, NowackiNPA818 139 (2009)

N

N

e ν N

N

e

N

π

N ν

Order Q3 2b currentsreduce NMEs∼ 15%− 40%

Coupled-cluster β decay in C, O suggest smaller quenching q ∼ 0.9Ekström et al. PRL113 262504 (2014)

21 / 30

Page 23: Nuclear Matrix Elements for Fundamental SymmetriesNuclear Matrix Elements for Fundamental Symmetries Javier Menéndez JSPS Fellow, The University of Tokyo 3rd European Nuclear Physics

Dark Matter: evidence

Solid evidence of Dark Matterin very different observations:

Rotation curves, Lensing, CMB...Zwicky 1930’s, Rubin 1970’s..., Planck 2010’s

22 / 30

Page 24: Nuclear Matrix Elements for Fundamental SymmetriesNuclear Matrix Elements for Fundamental Symmetries Javier Menéndez JSPS Fellow, The University of Tokyo 3rd European Nuclear Physics

What is Dark Matter made of?The composition of Dark Matter is unknownHigh-energy physics: candidates proposed beyond Standard Model

• Weakly interacting massiveparticles (WIMPs)

• Sterile neutrinos

• Axions

• Gravitons

• . . .

Lightest supersymmetric particles(usually neutralinos)predicted in SUSY extensionsof the Standard Model

Expected WIMP-density agrees withobserved Dark Matter density

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Page 25: Nuclear Matrix Elements for Fundamental SymmetriesNuclear Matrix Elements for Fundamental Symmetries Javier Menéndez JSPS Fellow, The University of Tokyo 3rd European Nuclear Physics

WIMP scattering off nuclei

The challenge is direct Dark Matter detection

WIMPs interact with quarks⇒ nuclei

Direct detection experiments: XENON100, LUXnuclear recoil from WIMP scattering off nucleisensitive to Dark Matter masses & 1 GeV

WIMPs couple to the nuclear densityFor elastic scattering, coherent sumover nucleons and protons in the nucleus

WIMP spins couple to the nuclear spinPairing interaction: Two spins couple to S = 0Only relevant in stable odd-mass nuclei

N

N

χ

χ

N

NN

π

N χ

χ

CDMS Collaboration24 / 30

Page 26: Nuclear Matrix Elements for Fundamental SymmetriesNuclear Matrix Elements for Fundamental Symmetries Javier Menéndez JSPS Fellow, The University of Tokyo 3rd European Nuclear Physics

WIMP-nucleon interactionsThe WIMP-nucleus interaction is

Coupling to nuclear density: scalar-scalar, spin-independentCoupling to the spin: axial-axial, spin-dependent

LSIχ + LSD

χ =GF√

2

∫d3r

[j(r)S(r) + jµ(r)JA

µ(r)]

j(r) = χχ = δsf si e−iqr is the leptonic (WIMP) scalar current

S(r) = c0∑A

i=1 δ3(r− ri ) is the hadronic scalar current

jµ(r) = χγγ5χe−iqr is the leptonic (WIMP) axial currentJAµ(r) =

∑Ai=1 JA

µ,i (r)δ3(r− ri ) is the hadronic axial current

Matrix element of the dark matter scattering: structure factor

SS(q) + SA(q) =1

4πG2F

∑sf ,si

∑Mf ,Mi

∣∣〈Jf Mf |LSIχ + LSD

χ |JiMi〉∣∣2

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Page 27: Nuclear Matrix Elements for Fundamental SymmetriesNuclear Matrix Elements for Fundamental Symmetries Javier Menéndez JSPS Fellow, The University of Tokyo 3rd European Nuclear Physics

Spin-independent structure factor for 130Xe

Coherent response at p = 0, lost at finite momentum transfers

SS(q) =∞∑

L=0

∣∣〈Jf‖c0

A∑i=1

jL(qri )YL(ri )‖ Ji〉∣∣2 →q→0

c20

4π(2J + 1)A2 ,

0 1 2 3 4 5 6u

10-4

0.001

0.01

0.1

1

10

100

1000

SS(u

)

This workHelm form factorFitzpatrick et al.

130Xe

Plot as function ofdimensionless u = p2b2/2b harmonic oscillator length

Only low-momentumtransfers up to u ∼ 2 relevantfor present experiments

Not very sensitive to nuclearstructure details: similarresults with model constantdensity + gaussian surface

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Page 28: Nuclear Matrix Elements for Fundamental SymmetriesNuclear Matrix Elements for Fundamental Symmetries Javier Menéndez JSPS Fellow, The University of Tokyo 3rd European Nuclear Physics

SD Structure Factors with 1b+2b currents

0 1 2 3 4 5 6 7 8 9 10

u

0.0001

0.001

0.01

0.1

Si(u

)

Sp(u) 1b currents

Sn(u) 1b currents

1 2 3 4 5 6 7 8 9

0.0001

0.001

0.01

0.1

Si(u

)

Sp(u) 1b currents

Sn(u) 1b currents

131Xe

129Xe In 129,131

54 Xe 〈Sn〉 〈Sp〉,Neutrons carry most nuclear spin

Couplings sensitive more toprotons (a0 = a1) or neutrons (a0 = −a1)

S(0) ∝∣∣∣ a0+a1

2 〈Sp〉+ a0−a12 〈Sn〉

∣∣∣22b currents involve neutrons + protons:

N

N

χ

χ

N

NN

π

N χ

χ

Neutrons always contribute with 2bcurrents, dramatic increase in Sp(u)

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Page 29: Nuclear Matrix Elements for Fundamental SymmetriesNuclear Matrix Elements for Fundamental Symmetries Javier Menéndez JSPS Fellow, The University of Tokyo 3rd European Nuclear Physics

SD Structure Factors with 1b+2b currents

0 1 2 3 4 5 6 7 8 9 10

u

0.0001

0.001

0.01

0.1

Si(u

)

Sp(u) 1b currents

Sn(u) 1b currents

Sp(u) 1b + 2b currents

Sn(u) 1b + 2b currents

1 2 3 4 5 6 7 8 9

0.0001

0.001

0.01

0.1

Si(u

)

Sp(u) 1b currents

Sn(u) 1b currents

Sp(u) 1b + 2b currents

Sn(u) 1b + 2b currents

131Xe

129Xe In 129,131

54 Xe 〈Sn〉 〈Sp〉,Neutrons carry most nuclear spin

Couplings sensitive more toprotons (a0 = a1) or neutrons (a0 = −a1)

S(0) ∝∣∣∣ a0+a1

2 〈Sp〉+ a0−a12 〈Sn〉

∣∣∣22b currents involve neutrons + protons:

N

N

χ

χ

N

NN

π

N χ

χ

Neutrons always contribute with 2bcurrents, dramatic increase in Sp(u)

27 / 30

Page 30: Nuclear Matrix Elements for Fundamental SymmetriesNuclear Matrix Elements for Fundamental Symmetries Javier Menéndez JSPS Fellow, The University of Tokyo 3rd European Nuclear Physics

Inelastic scattering?Can Dark Matter scatter exciting the nucleus to the first excited state?

Th Exp

0

100

200

300

400

500

600

700

800

900

En

erg

y (

keV

)

129Xe

3/2+

11/2-

1/2+

(9/2)-

3/2+5/2+

(5/2)+

1/2+

(5/2)+

7/2+

1/2+

1/2+

5/2+

5/2+

5/2+

7/2+

3/2+

3/2+

11/2-

9/2-

Th Exp

0

100

200

300

400

500

600

700

800

900

En

erg

y (

keV

)

131Xe

1/2+

11/2-

3/2+

9/2-

5/2+

3/2+

(1/2,3/2)+

7/2+

3/2+

7/2-

3/2+

3/2+

3/2+

1/2+

5/2+

11/2-

9/2-

7/2+

1/2+

7/2-

Very low-lying first-excited states ∼ 40,80 keV

If WIMPs have enough kinetic energyinelastic scattering possible

p± = µvi

(1±

√1− 2E∗

µv2i

)28 / 30

Page 31: Nuclear Matrix Elements for Fundamental SymmetriesNuclear Matrix Elements for Fundamental Symmetries Javier Menéndez JSPS Fellow, The University of Tokyo 3rd European Nuclear Physics

Spin-dependent inelastic WIMP scatteringInelastic structure factorscompete with elastic at p ∼ 150 MeV,in the kinematically allowed region

0 20 40 60 80 100

Evis

(keV)

10-7

10-6

10-5

10-4

R (

kg

-1d

-1)

129Xe inelastic

131Xe inelastic

Total inelastic129

Xe elastic131

Xe elastic

Total elastic

Total

Inelastic scattering⇒ spin couplingDensity coupling suppressed:coherence of all nucleons lost

0.5 1 1.5

10-4

0.001

0.01

0.1

Si(u

)

Sp(u) 1b + 2b inelastic

Sn(u) 1b + 2b inelastic

Sp(u) 1b + 2b elastic

Sn(u) 1b + 2b elastic

129Xe

Integrated spectrum for xenonshows expected signal frominelastic scattering including thegamma from excited state decay

One plateau per excited state

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Page 32: Nuclear Matrix Elements for Fundamental SymmetriesNuclear Matrix Elements for Fundamental Symmetries Javier Menéndez JSPS Fellow, The University of Tokyo 3rd European Nuclear Physics

Summary and OutlookNeutrinoless ββ decay detection will establishthe Majorana character of neutrinos andtogether with nuclear matrix elementsinform on neutrino masses

Theoretical nuclear matrix elements disagree:identify relevant correlations and configurationspaces for nuclear structure calculations

Chiral EFT predicts relevant 2b currentsthat modify the transition operators,need to be better evaluated

Nuclear physics helps Dark Matter searchesproblems because WIMPs interact with nuclei

Determination of nature and coupling ofWIMP-nucleon interaction, more generalcouplings to be considered

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Page 33: Nuclear Matrix Elements for Fundamental SymmetriesNuclear Matrix Elements for Fundamental Symmetries Javier Menéndez JSPS Fellow, The University of Tokyo 3rd European Nuclear Physics

CollaboratorsT. OtsukaT. Abe

Y. IwataN. Shimizu

Y. Utsuno M. Honma N. Hinohara

P. Klos, G. Martínez-Pinedo, A. Schwenk, L. Vietze

A. PovesT. R. Rodríguez

E.CaurierF. Nowacki

L. BaudisG. Kessler

J. D. Holt

D. Gazit

J. Engel

R. F. LangS. Reichard

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