PARTIAL RESTORATION OF SPIN-ISOSPIN SU(4) …PARTIAL RESTORATION OF SPIN-ISOSPIN SU(4) SYMMETRY AND DOUBLE BETA DECAY 06/29/2017 Seattle, WA, USA Arturo R. Samana Universidade Estadual

1

Neutrinoless Double-beta Decay (INT 17-2a)

PARTIAL RESTORATION OF SPIN-ISOSPIN SU(4) SYMMETRY AND DOUBLE BETA

DECAY

06/29/2017 Seattle, WA, USA

Arturo R. Samana Universidade Estadual de Santa Cruz – Ilhéus –Brazil Francisco Krmpotić & César Barbero – UNLP -Argentina Vitor dos S. Ferreira – UERJ- Brazil

2

A.R.Samana - PARTIAL RESTORATION OF SPIN-ISOSPIN SU(4) SYMMETRY AND -DECAY

Neutrinoless Double-beta Decay (INT 17-2a) Seattle, WA, USA

Outline

• Introduction

• Decay modes 2 and 0

• Main features

• Formalism

• Nuclear structure model - QRPA

• Numerical results

• Final remarks

• Acknowlegments

3


The MAJORANA Collaboration (USA )0 in 76Ge. MAJORANA plans to collaborate with GERDA for a future tone-scale 76Ge 0νββ search.

EXO-200: 0 in Xenon 136. y. NEMO3 (France) 0 in 100Mo

I. Introduction


4

)( ,2 ee

eeZNZN 22)2,2(),(

)( ,0 ee

eZNZN 2)2,2(),(


21

2/1 )(MFGT

• Kinematic factor (G) • Nuclear Matrix Elements (M)

0for ,

2for , 1

em

mF

I. Introduction


5


• Nuclear Matrix Elements (M) merges from a microscopic hamiltonian worked in the framework of mean-field theories and often violates the symmetries of hamiltionian. • BCS theory violates conservation of number of particle and the spin-isospin SU(4) symmetry: (i) SU(4) is to be restored by the residual interaction, (ii) This restoration must not be complete to inhibit -decay -> Partial SU(4)

Symmetry Restoration (PSU4SR) • Symmetries broken by BCS are restored by QRPA with a special adoption of parameters in the particle-particle (pp) and particle-hole (ph) channels, for example, in Simkovic et al., PRC 87, 045501 (2013); Fang et al., PRC 92, 044301 (2015); Hyvarinen et al. PRC 91, 024613(2015).

• We present a recipe to implement the PSU4SR based on energetic of F and GT resonances in (ph) channel, and on the minima of F and GT -strengths in the (pp) channel.

I. Introduction


6


• Our physical substratum is the same as in previous QRPA calculations in -

decay:

Hirsch & Krmpotic, PRC 41, 792 (1990); Hirsch & Krmpotic, PLB2 46, 5 (1990);

Hirsch, Bauer & Krmpotic, NPA 516, 304 (1990);

Krmpotic, Hirsch & Dias, NPA 542, 85 (1992); Krmpotic, PRC 48, 1452 (1993);

Krmpotic, Mariano, Kuo & Nakayama, PLB 319, 393 (1993);

Krmpotic & Sharma, NPA 572, 329 (1994); Krmpotic, RMF 40, 285 (1994);

Ferreira, Master thesis,UESC-BA, Brazil, (2016),

here we just bring up to date those studies, including the pseudoscalar (P) and

weak-magnetism (M) matrix elements M0P , and M0

M, as suggested by

Simkovic et al. PRC 60, 055502 (1999).

I. Introduction – Main features

)(2

)()( 55

† xqgM

qigggxxJ P

N

MAV

Simkovic et al., PRC 60, 055502 (1999); PRC 77, 045503 (2008).

Krmpotic et al., NPA 612, 223 (1997);

Barbero et al., NPA 628, 170 (1998); PLB 445, 249 (1999).


7


1. Residual interaction is d-force in units of MeV-fm3 :

spin-singlet parameter in pp chanel:

spin-triplet parameter in pp chanel:

.

,

,)()(4

0

1

T

pp

t

T

pp

s

t

t

s

s

gv

gv

rPvPv d

2. We solve the RPA equation only once for intermediate (N-1,Z+1) nucleus as

in J. Hirsch & F. Krmpotic, PLB 246, 5 (1990) .

3. vspp is fixed in the same way as gT=1

pp .

We require that the strength S+F becomes minimal when s=vs

pp / vspair

become s syn =1. This is sign of spin restored symmetry leading to

and the concentration of S-F is in the IAS.

,0)0( ,0 ,0 02 JMMS v

V

v

FF



8


4. vtpp is fixed following a similar recipe that vs

pp , we require that GT

strength S+GT becomes minimal as it was shown in Fig. 2 and 3 of Krmpotic

& Sharma, NPA 572, 329 (1994)

then t=vtpp / vs

pair become tsym ≄1 with

and not all concentration S-GT in the GTR.

.0)1( ,0 ,0 02 JMMS v

A

v

GTGT



9


5.The difference with other studies is that the experimental 2 moments are

not used for gauging the isoscalar pp parameter t. In this way, the QRPA model turns out to be predictable regarding M2

GT.

6. The restoration of the isospin and SU(4) symmetries, broken in the mean

field, are also manifested ph channel. We have monitored the ph parameters

vphs and vph

t from the experimental energetic of the IAS and the GTR

[Nakayama etal, PLB 114, 217 (1982)]:

26A-1/3 -> from the SU(4) symmetry-breaking caused by the LS coupling,

18(N-Z)/A -> symmetry-restoration effect induced by the residual interaction,

which displaces the GT towards the IAS with increasing N-Z.

MeV 5.1826 31

A

ZNAEE IASGT



The 0 nuclear moment is

with

and

is the Fourier transformer of hadronic current and R=r0A1/3 , r0=1.2 fm and the neutrino potential

with

10

Arturo Rodolfo Samana - PARTIAL RESTORATION OF SPIN-ISOSPIN SU(4) SYMMETRY AND DOUBLE BETA DECAY


II. Formalism A. 0 Nuclear Moments

eZNZN 2)2,2(),(21;

0J ,

eeFf

Ii

N

NkNkvkdM );(M);(4

R 00

IkJNNkJFNk )()();(M ††0

,)()( †† xkiexJxdkJ

)(

12);(

NkkNkv

).(2

1FINN EEE


Within NRA (when velocity terms are omitted), the hadronic currents:

where

are one-body density current, with fM=gV+gM=4.7, f ’M=fM/(2MN),

g’P=gP/(2MN), and gV =1, gA =-1.27, gM=3.7 . Using Fourier-Bessel relationship for exponential eik.r , spherical vectors and

performing the angular integration on k, multiplying by Rk2v(k;N)/4 then

with X=V, A, P, M. 11



)(),()(NRA xjxxJ

n

nPnMnAnn

N

V

n

nnV

gfgrxM

gxj

rxgx

],)[(2

)(

),()(

'' d

d

X

X

00 MM


With:

12



),;''(]2/)1(2[ 1 1

' )''(

)( );''()|11( )|'( 'ˆ ˆ)1(M

),;''( 1 1

' )''(

)( );''()|11( )|'( 'ˆ ˆ)1(M

),;''()''()( );''()1(M

),;''()''()( );''()1(M

'1'

1

''

2/)'(0

'1'

1

''

2/)'(0

11

''

10

00

''

0

J

M

LLJL

JL

npnp

ph

J

LLJ

M

J

P

LLJL

JL

npnp

ph

J

LLJ

P

J

A

LLJLJL

npnp

ph

J

L

A

J

V

JJJJJJ

npnp

ph

J

J

V

npnpRllJ

lLLnpW

pnWJnpnpllLLLL

npnpRJ

lLLnpW

pnWJnpnpllLLLL

npnpRnpWpnWJnpnp

npnpRnpWpnWJnpnp


The angular parts are:

with the two-body radial integrals

and the neutrino potentials are

with and

13



,)|(ˆˆˆˆˆˆ2)(

21

21

nn

pp

pnpnnLSJ

jl

JSL

jl

lLljjlLJSpnW

),;''();();( R);''( '

2

' knpRkpnRqvkdknpnpR LLJXJ

X

LL

).;()](')(2)[(');(

),;()(');(

),;()();( ),;()();(

22222

222

2222

JPAPJP

JMJM

JAJAJVJV

kvkgkkgkgkkv

kvkfkkv

kvkgkvkvkgkv

,)()()();( 2

o

LlnlnL drrkrjrurukpnRnnpp

.2

1- ,

)(

12),(

0

QEEkk

kvIJJ

J

J


One-body state dependent ph density matrices

FNS effects are introduced by

SRC between two nucleon are

Then

14



1

0 fm93.3 ),(1)( CC

SRC krkjrf

.)(

)( ,)(

)(

2

22

2

2

22

2

kgkgg

kgkgg

A

AAAA

V

VVVV

.)2(),( 2

1),(

),,(),(),(

1

1

222

2

CJX

C

JX

JXJXJX

qxqkkqqvkdkdxq

qv

qvqvqv

d

,0)()(0ˆ);''( †

p

†

p

2 IJnJnF

ph aaJJaaJJnpnp


with

and

15


II. Formalism B. 2 Nuclear Moments

,222

GTF

,)''()(

)0;''(g

,00

0

000000

''

2

,0

002

npWpnWnpnp

D

OO

npnp

ph

V

f

if

F

0

†

0 1̂ np

pn

aanpO

.)''()(

)1;''(g

,00

1

011011

''

2

,1

112

npWpnWnpnp

D

OO

npnp

ph

A

f

if

GT

1

†

13

1np

pn

aanpO


• Most used model is charge-exchange QRPA - Halbleib and Sorensen-

Nucl. Phys. A 98, 524 (1967) where:

1) BCS equations for the initial even-even nucleus (N,Z) with (vn,vp) and

(un=(1-v2n)

1/2, up=(1-v2p)

1/2), q.p energies (en,ep) and chem. pot. (n,p), with

BCS vaccum

2) Transition densities Excitation energies in

neighboring odd-odd

pn-QRPA equation Ikeda sum-rule

16


III. Charge-exchange QRPA

.)12( ,)12(

),()(0

22

††††

np j

nn

j

pp

nnnnnpppppI

NvjZvj

aavuaavu

).()();(

),()();(

pnYvupnXvuJpn

pnYvupnXvuJpn

JpnJnp

JnpJpn

,0

1,1

pnJ

ZN

J I

EE

)1,1( ZN

,

J

J

JJ

J

JJ

JJ

Y

X

Y

X

AB

BA,ZNSSS


A. Method I

• Vogel and Zirnbauer (PRL57, 3148 (1986)) discovered GSC is suppresing

2 rates.

•QRPA for both initial and final nuclei and NME averaged. Step 1) and 2) are

repeated for (N, Z) and (N-2, Z+2 ) gs, and intermediary 1+ and 1’+ in

(N-1, Z+1) In the second case the BCS vaccum is

17



2.-)12( ,2)12(

),()(0

22

††††

np j

nn

j

pp

nnnnnpppppF

NvjZvj

aavuaavu

.

,2

1

),(2

11

101

00

QEE

EEQ

I

FIpn

)13.3.()1;''()1;''(

)''()(2

g

11

''

011011

22

npnpnpnp

npWpnW

phph

npnp

AGT


B. Method II

• Civitarese, Faessler and Tomoda [PLB 194, 11(1987)], repeating steps 1)

and 2) for g.s. of (N, Z) and (N-2, Z+2 ), and intermediary 1+ and 1+

’ in

(N-1, Z+1). Their 2 moment reads

where the overlap is

In recent applications of Method II, the denominator is replaced by

where 18



)14.3(

.

2

1

)1;''(11)1;''()''()(g

01

2

''

'''

011011

22

i

EEQcm

npnpnpnpnpWpnW

e

phph

npnp

AGT

.)]()()()([11'' 1111'

pn

pnYpnYpnXpnX

.)1;(11)1;''(

)''()(00g2

'11

''

'''

011011

22

pnnp

npWpnWnpnp

FIAGT

).()(00 nnnnnpppppFI vvuuvvuu


C. Method III

• Eqs. (3.13) and (3.14) are physically sound ansatz for HS eqs. Then in

Hirsch & Krmpotic PLB 246, 5 (1990) , is proposed to solve only one QRPA

equation

solved for the vacuum

and BCS eqs. were solved for even nuclei. The GT moment is

where and GT strengths

19



).()(0~ ††††

nnnnnppppp ccvuccvu

,~

~~

~

~

~~

~~

J

J

J

J

J

JJ

JJ

Y

X

Y

X

AB

BA

.~)1;''(~)1;(~

)''()(g

1''

011011

22

nppnnpWpnW

npnp

AGT

. ,

,)(~

)(~

);(~

,)(~

)(~

);(~

221-

n

221-

p nnpp

JnpJnpnp

JpnJnpnp

vuvu

pnYvupnXuvJpn

pnYvupnXvuJpn

.2~~~

,|)( )1 ,(~|~ 2

011

ZNSSS

pnWpnSpn

GT


D. Method IV Cha [PRC 27,2268 (1983)] in studies of single -decay: “because the intersection between two-qp's takes place in a residual nucleus, we should calculate e's, u's, and v's in the daughter nucleus.” Then, instead of dealing with two-

vacua QRPA, BCS eqs. are solved only for intermediary nucleus, with vacuum

and u's, and v's:

Sum rule:

The GT moment is

and the 0 NME are evaluated with the one-body densities

20



.2'

,1')12( ,1')12(

),''()''(0

22

††††

int

ZNS

NvjZvj

ccvuccvu

n

n

np

p

p

nnnnnppppp

.'

)1;''(' )1;(')''()(g

1''

011011

22

nppnnpWpnW

npnp

AGT

);''(' );(');''(

JnpJpnJnpnp


Method I and II ((N, Z) and (N-2, Z+2 ), and (N-1, Z+1)) involves also the

nuclei (N+1, Z-1) and (N-3, Z+3 ). This is because GSC for

correspond respectively, to

On the contrary, Method III and IV only involve nuclei within the isobaric triplet

(N, Z), (N-1, Z+1) and (N-2, Z+2 ) where -decay occurs.

21



)1,1()2,2()2,2()1,1( and ),1,1(),(

ZNZNZNZNZNZN

).3,3()2,2( and ),1,1(),(

ZNZNZNZN

N1, Z1

N2, Z2 N, Z

N1, Z1 N3, Z3


22


IV. Numerical results Code QRAP2DB * s.p.e in 76Ge

ttttttt vuvue )()(2 22

0,''0;ˆˆ 1

' JttVJttvujj tt

t

ttt

),1(),(2),1(2

1

)1,(),(2)1,(2

1

NZNZNZ

NZNZNZ

Z

N

BBB

BBB


23

* ph-channel •Systematic study of GT ressonancies Nakayama et al. PLB 114, 217 (1982).

-3

t

-3

s MeV.fm 92 ,MeV.fm 55 PHPH vv

• Fermi & Gamow-Teller theoretical

• For all nuclei (not 48Ca) were adopted:

MeV 5.1826 31

A

ZNAEE FIASGT

.10)1('

)1('

,)0 ('

)0 ('

2

1

2

2

0

2

MeVpn

pn

Epn

pnE

pn

pn

GT

pn

pn

F


IV. Numerical results

• Fermi experimental and Coulomb energies

.76.0.170.0),(

),,1(),1(

32

3/1

2

ZA

ZAZ

AZAZE

Coul

CoulCoulIAS

e

ee


24

* pp-channel parameters from P-SU4-SR




25

* pp-channel parameters from P-SU4-SR



,pair

s

PP

t

pair

s

PP

s

v

vt

v

vs

2

pairZ

s

pairN

spair

s

vvv

1s

The values exhibited in Table I are very close to those obtained previously in [NPA 572, 329 (2014), Table 4], where Method III was used to calculate the NM. The above similarity is the main reason for associating P-SU4-SR in -decay.


26

P-SU4-SR effects



Table II: 2-decay moments evaluated within the BCS (unperturbed) and QRPA (perturbed) approximations are compared with the experimental results recommended by Barabash [NPA 935, 52(2015)]. All the quantities are given in natural units.


27


IV. Numerical results TABLE III: 0-decay moments M0

X, as well the total moments M0=X M0X (normalized to g2

A, with

gA = 1,27), evaluated within the BCS (unperturbed) and QRPA (perturbed) approximations, are shown.

In both cases the FNS and SRC effects are included. At the bottom of the table are shown the 76Ge

results: i) without SRC, in the row labeled as 76Ge*, ii) the bare values of moments, i.e., without the

FNS and SRC effects, in the row labeled as 76Ge** , and iii) the moments obtained in Ref. [10] and

derived from relations (4.5).

.MM

MMM

,MMM

,MM

,MM

0

0

0

0

(2015). 024613 91, PRC

Suhonen, &rinen aHyv Actual

AP

T

AP

GT

PP

T

PP

GTP

MM

T

MM

GTM

AA

GTA

VV

FV


28


IV. Numerical results TABLE IV: Fine structure of M0 moments

(normalized to g2A, with gA = 1.27) for 76Ge.

i) The residual interaction, through the

PSU4SR, is critical in reducing the nuclear

moments. The reduction for the

neutrinoless 0-decay NM is less

pronounced than in the case of 2-

decay.

ii) This quenching effect is smaller on

induced current moments M0P and M0

M

than on M0V and M0

A , which results from

the standard V-A weak current.

iii) Our M0M are, in principle, larger than in

other calculations by the factor (fM/gM)2 =

1.61, since we include the term gV =2MN in

the NRA of the weak Hamiltonian.

iv) Compared to the role played by the

residual interaction in the pp channel, the

FNS and SRC effects are relatively small.

FNS in bare are ~15-20% and SRC are

~3-5% similar results in Simkovic PRC

79,055501(2009)


29




30


IV. Numerical results FIG. 3: 0 nuclear moments

evaluated with several nuclear

structure model calculations:

i) QRPA by Tubingen (QRPA Tu)

[8] (gA = 1.27), Jyvaskyla (QRPA

Jy) [10] (gA = 1.26) groups,

and our results from Table III

(QRPA Ferr) (gA = 1.27),

ii) interacting shell model (ISM)

[52] (gA =1.25), Large-scale shell

model (SM (SDPFMU)) [60],

iii) interacting boson model

(IBM2) [61] (gA = 1.269),

vi) energy density functional

method (EDF) [62] (gA = 1.25),

and covariant density functional

theory (CDFT) [63] (gA = 1.254).

All results are normalized to gA2.

[8] Simkovic et.al, PRC 87, 045501 (2013). [10] Hyvarinen & Suhonen, PRC 91, 024613 (2015).

[52] Menendez et.al, NPA 818, 139 (2009). [60] Iwata et.al, PRL 116, 112502 (2016) .

[61] Barea, PRC 87, 014315 (2013). [62] Vaquero et.al, PRL 111, 142501 (2013).

[63] Song et.al, PRC 90, 054309 (2014).


31


V. Final remarks • The one-QRPA method is used for the first time to –decay. • Stress once again the strong bonding between the residual interaction, GSC, PSU4SR and quenching of the –decay NM. • To implement PSU4SR , we resort to energetic of GT resonances and minima of single . • The residual proton-neutron interaction plays a fundamental role in the PSU4SR, both ph and pp channels. • We find Method IV preferable over Method II, basically because it only involves the nuclei within the isobaric triplet (N, Z), (N-1, Z + 1), (N-2, Z + 2) where the decay occurs, while the last one involves also the nuclei (N + 1, Z-1) and (N-3, Z + 3). • Our results for 0 N are lower on average by 40%, attributing this difference to employ one-QRPA method instead usual two-QRPA-method. • It is hard to say which is the best way to the way of restoration of symmetry, since 0 NM are not experimentally measurable.


32

Acknowlegments



33


IV. Numerical results Table V: M2 and

M0 NM within

Method II and

Method IV :

(i) tsym from P-

U4SR

(ii ) t

(iii) t from

|M2exp|

2

Results for

gA=1.26 from

Simkovic PRC 91, 024613

(2016) should be

compared with

ours t of

Method II.

FIG. 2: Isoscalar parameters t in 76Ge within the

Method II for |M2exp|=0.113. The NM M2 is given

in natural units, while M0 is dimensionless. It

should be noted that M2n is negative at t = 0.


Documents

PARTIAL RESTORATION OF SPIN-ISOSPIN SU(4) …PARTIAL RESTORATION OF SPIN-ISOSPIN SU(4) SYMMETRY AND DOUBLE BETA DECAY 06/29/2017 Seattle, WA, USA Arturo R. Samana Universidade Estadual