stefanik, simkovic

Dedicated to Professor Apolodor Aristotel Radutas 70th Anniversary

DOUBLE FERMI MATRIX ELEMENT WITHIN PERTURBATION THEORY

DUSAN STEFANIK1, FEDOR SIMKOVIC1,2,3

1Comenius University, Mlynska dolina F1, SK842 48, Slovakia2BLTP, JINR, 141980 Dubna, Moscow region, Russia

3IEAP CTU, 12800 Prague, Czech Republic

Received May 7, 2013

The amplitude for double-beta decay with two-neutrino emission is related to and + transitions of Fermi and Gamow-Teller type connecting ground state ofinitial and final nuclei with virtual intermediate nuclear states. The suppression of dou-ble Fermi and Gamow-Teller matrix elements has origin in violation of the isospinSU(2) and spin-isospin SU(4) symmetries, respectively. We study double Fermi ma-trix elements within an exactly solvable model. By using perturbation theory up to thefirst order a dependence of the two-neutrino double beta decay matrix element on thelike-nucleon pairing, particle-particle and particle-hole proton-neutron interactions byassuming a weak violation of isospin symmetry of Hamiltonian expressed with gene-rators of the SO(5) group. It is found that there is a dominance of transition through asingle state of the intermediate nucleus.

Key words: two-neutrino double-beta decay, nuclear matrix element, SO(5)model.

PACS: 14.60.Pq, 23.40.Bw, 23.40.Hc, 95.30.Cq.

1. INTRODUCTION

Double beta decay has been a very interesting process as a probe examining thebreakdown of total lepton number conservation due to finite mass of the Majorananeutrinos [1]. Double beta decay occurs in the case that a nucleus cannot undergoordinary single beta decay because of energy conservation or angular momentummismatch.

Much theoretical work has been done to calculate matrix elements and decayrates for both lepton number conserving two-neutrino double decay (2-decay),

(A;Z)! (A;Z+2)+2e+2~e; (1)and total lepton number non-conserving neutrinoless double beta decay,

(A;Z)! (A;Z+2)+2e: (2)The 2-decay, which involves the emission of two electrons and two an-

tineutrinos, remains of major importance for nuclear physics. Till now, the 2-decay has been detected for 11 different nuclei for transition to the ground state andin two cases also to transition to 0+ excited state of the daughter nucleus. This rare

RJP 58(Nos. 9-10), 12511257 (2013) (c) 2013-2013Rom. Journ. Phys., Vol. 58, Nos. 9-10, P. 12511257, Bucharest, 2013

1252 Dusan Stefanik, Fedor Simkovic 2

process is one of the major sources of background in running and planned experi-ments looking for a signal of the more fundamental neutrinoless double-beta decay,which occurs if the neutrino is a massive Majorana particle.

The 2-decay can be expressed in terms of sequential single beta decayFermi and Gamow-Teller transitions through virtual intermediate states. The inversehalf-life of the 2-decay can be written as

T 21=2

1=G2g4A

M2GT gVgA

2M2F

2

; (3)

where G2 is the lepton phase-space factor, gA (gV ) is the axial-vector (vector) cou-pling constant. The double Gamow-Teller (GT) and double Fermi (F) matrix ele-ments are given by

M2F;GT =Xn

hf k OF;GT k 1+n ih1+n k OF;GT k iiEn (Ei+Ef )=2 (4)

with

OF =AXk=1

+k ; OGT =AXk=1

+k ~k: (5)

where ji > (jf >) are 0+ ground states of the initial (final) even-even nuclei withenergy Ei (Ef ), and j1+n > (j0+n >) are the 1+ (0+) states in the intermediate odd-odd nucleus with energies En.

The goal of this paper is to discuss the suppression mechanism of the doubleFermi matrix element close to the point of restoration of isospin symmetry of thenuclear Hamiltonian in the context of residual nucleon-nucleon interaction. For thesake of simplicity we consider a schematic Hamiltonian, describing the gross proper-ties of the beta-decay processes in the simplest case of monopole Fermi transitionswithin the SO(5) model [2, 3]. In order to find explicit dependence of M2F on dif-ferent parts of the nuclear Hamiltonian the perturbation theory up to the first order isexploited.

2. SCHEMATIC HAMILTONIANWITHIN THE SO(5) MODEL

In the model, protons and neutrons occupy only a single j-shell. The Hamilto-nian includes a single-particle term, proton-proton and neutron-neutron pairing, anda charge-dependent two-body interaction with both particle-hole and particle-particlechannels as follows:

H = epNp+enNnGpSypSpGnSynSn+2+2PP+; (6)RJP 58(Nos. 9-10), 12511257 (2013) (c) 2013-2013

3 Double Fermi matrix element within perturbation theory 1253

whereNi =

Xm

aym;tiam;ti ; =

Xm

aym; 1

2

am; 1

2

;

Syi =1

2

Xm

aym;ti~aym;ti

; P =Xm

aym; 1

2

~aym; 1

2

;(7)

with i= p;n and tn;p =1=2. aymt (amt) is creation (annihilation) operator of singleparticle state jjm;t > for protons and neutrons (t= tp; tn) and ~aymt = (1)jmaymt.

We rewrite Hamiltonian (6) with help of operators

Ay (Tz) =1p2

hayay

i1Tz; N =Np+Nn;

Tz =NnNp

2; T =

p2

Xm

aym; 1

2

am; 1

2

:(8)

Here, Ay (Tz) is the nucleon pair creation operator with angular momentumJ = 0, isospin T = 1 and its projection on z-axis Tz (Tz = 0;1). N , Tz and T arethe particle-number operator, the isospin projection and the isospin lowering opera-tors, respectively. It holds the identity T 2 = (TT++T+T)=2+T 2z . = j+1=2denotes the semi-degeneracy of the considered single level. The operators in theabove relations with their Hermitian conjugates represent ten generators of the SO(5)group [4]. We assume, the system is in seniority s=0. Then, [A

y ~A]00 expressed withthe SO(5) Casimir operator [4] is given by

[Ay ~A]00 =

1

2p3

[(2+3N=2)N=2T (T +1)] : (9)

For the Hamiltonian (6) we get

H =

en+ep 1

3

3+2 N

2

Gp+Gn

2+2

N

2

+ [enep2(Tz+1)]Tz+2+

1

3

Gp+Gn

2+2

T (T +1)

+

p2

GpGn

2

[A

y ~A]10+

r2

3

4 Gp+Gn

2

[A

y ~A]20:

(10)

As a consequence of the presence of the isovector and isoquadrupole terms inHamiltonian (10) the isospin is not conserved in general. It is due to differencesbetween proton and neutron pairing strengths and an arbitrary strength of the proton-neutron isovector pairing component. However, particle number and isospin projec-tion remains as good quantum numbers.

The kth eigenstates of the Hamiltonian (10) with quantum numbers N and Tzcan be expressed in terms of a basis labelled by a chain of irreducible representations

RJP 58(Nos. 9-10), 12511257 (2013) (c) 2013-2013


of the SO(5) group [4], namely

jk;NTzi=XT

c(k)NTTz

jNTTzi: (11)

A diagonalization ofH requires calculation of matrix elements hN;T;TzjH jN;T;Tziand hN;T 2;TzjH jN;T;Tzi. The corresponding reduced matrix elements aregiven in [2]. For Gp = Gn and (Gp+Gn)=2 = 4 the Hamiltonian (10) is diag-onal in the basis of states jN;T;Tzi.

3. DOUBLE FERMI MATRIX ELEMENTWITHIN PERTURBATION THEORY

We shall assume a small violation of the isospin symmetry due to isotensor termof nuclear Hamiltonian (10). For the numerical example we consider a large value ofj to simulate the realistic situation corresponding to medium- and heavy-mass nuclei.The parameters chosen are given by

= 10; N = 20; 1 Tz 5;ep = 0:3MeV; en = 0:1MeV; G= 0:165MeV;Gp =Gn =G; = 0:044MeV; 0:7 4=G 1:3:

(12)

For 4=G= 1 the isospin symmetry is restored. In Fig. 1 we present 0+ stateswith energy ETTz of different isotopes. This level scheme illustrates the situationfor the 2-decay of 48Ca. The isospin is known to be, to a very good approxima-tion, a valid quantum number in nuclei. The ground states of 48Ca and 48Ti can beidentified with T=4 Tz = 4 and T=2 Tz = 2, respectively, i.e. they are assigned intodifferent isospin multiplets. As the total isospin projection lowering operator T isnot changing the isospin the double Fermi matrix element M2F is non-zero only tothe extent that the Coulomb interaction mixes the high-lying T=4 Tz=2 analogue ofthe 48Ca ground state into the T=2 Tz = 2 ground state of 48Ti.

We shall study double Fermi matrix element in the perturbation theory withinthe discussed model close to a point of a restoration of the isospin symmetry (4=G=1). The isoscalar and isotensor terms of the Hamiltonian (10) represent the unper-turbated and perturbated terms, respectively. We denote perturbated states and theirenergies with a superscript prime symbol (jT 0Tzi, E0TTz ) unlike the states with adefinite isospin (jTTzi, ETTz ). Up to the first order of parameter (4G) we find

E044 = 14en+6ep110

3(G+2)

r2

3

(G4)h44j [Ay ~A]20 j44i ;

E043 = 13en+7ep+16110

3(G+2)

r2

3

(G4)h43j [Ay ~A]20 j43i ;

E022 = 12en+8ep124

3(G+2)

r2

3

(G4)h22j [Ay ~A]20 j22i ;

(13)

RJP 58(Nos. 9-10), 12511257 (2013) (c) 2013-2013


4 5 6 7 8 9 10Z

-10

-5

0

5

10

15E

[MeV

]

(10,5)

(8,5)

(6,5)

(10,4) (10,3)(10,2) (10,1)

(8,4) (8,3)(8,2) (8,1)

(6,4) (6,3)(6,2) (6,1)

(4,4) (2,2) (2,1)(4,3) (4,2)

(4,1)

Fig. 1 The energy of the 0+ states of different isotopes are shown for j = 19=2 (and the set ofparameters (12) with 4=G= 1) in MeV vs. Z plot. States are labeled by (T;Tz).

The particular matrix elements of SO(5) operators connecting states with a definiteisospin and its projection are presented in [2]).

The double Fermi matrix element can be written as

M2F =10X

T=4;6;8;10

h202jT jT 03ihT 03jT j404iE0T3 (E044E022)=2

: (14)

It contains a sum over the states of the intermediate nucleus jT 03i. However, up tosecond order of perturbation theory there is only a single contribution through theintermediate state j403i. Thus, we have

M2F 'h202jT j403ih403jT j404i

E033 (E044E022)=2: (15)

If isospin symmetry is restored (4 = G) we end up with h202jT j403i =h22jT j43i = 0. For the energy denominator in (15) with help of Eqs. (13), weget

E043E044E022

=2 = 16+

7

3(G+2)

+

r1

6

(4G)

h2h43j [Ay ~A]20 j43ih44j [A

y ~A]20 j44ih22j [Ay ~A]20 j22i

i (16)We note that the energy denominatorE043(E044E022)=2 as well as the whole

RJP 58(Nos. 9-10), 12511257 (2013) (c) 2013-2013


double Fermi matrix elementM2F does not depend explicitly on mean field parame-ters ep and en. If we restrict our consideration to the first order perturbation theorywe find

M2F 'h42jT j43ih43jT j44i

16+ 73 (G+2)r

2

3

(G4) h42j [A

y ~A]20 j22i(28+14=3(G+2))

:

0.90 0.92 0.94 0.96 0.98 1.00 1.02 1.04 1.06 1.08 1.104/G

-1.00

-0.80

-0.60

-0.40

-0.20

0.00

0.20

0.40

0.60

0.80

1.00

MF

[MeV

-1 ]

exact solutionperturbation theory

Fig. 2 (Color online) Matrix element M2F for the double-Fermi two-neutrino double-beta decaymode as function of ratio 4=G for a set of parameters (12). Exact results are indicated with a solidline. The results obtained within the perturbation theory up to the first order in isotensor contribution toHamiltonian are shown with dashed line, respectively. The restoration of isospin symmetry is achievedfor 4=G= 1.

In Fig.2 M2F is plotted as function of ratio 4=G. We see that double Fermimatrix element calculated with perturbation theory up to the first order reproduceswell the exact solution close to the point of restoration of isospin symmetry (4=G=1).

4. CONCLUSIONS

The 2-decay processes of the Fermi type is analysed in the context of vi-olation of isospin SU(2) symmetry. Good isospin forbids the 2-decay as ground

RJP 58(Nos. 9-10), 12511257 (2013) (c) 2013-2013


states of initial and final nuclei belong to different isospin multiplets. One needs anisotensor force to mix T = 2. Naturally, the Coulomb interaction contains such aisotensor force. In our case we break isospin symmetry by hand. The only isospin vi-olation comes from the difference of the proton-proton (Gp) and the neutron-neutron(Gn) pairing force compared to the proton-neutron isospin = 1 pairing force ().

By taking the advantage of perturbation theory double Fermi matrix elements isanalysed in exactly solvable model, which Hamiltonian is expressed with generatorsof the SO(5) group. Up to the first order of perturbation theory in the isotensorcontribution to the Hamiltonian explicit dependence ofM2F on different componentsof residual interaction of nuclear Hamiltonian, namely like-nucleon pairing, particle-particle and particle hole proton-neutron interactions, is found. The mean-field partof Hamiltonian does not enter explicitly in this decomposition of double Fermi matrixelement and is related only to the calculation of unperturbated states of Hamiltonian.Further, a dominance of a contribution through a single state of the intermediatenucleus toM2F is established.

It goes without saying that more comprehensive calculation of double Fermimatrix element up to the second order in perturbation theory is desired. There isalso a possibility to study double Gamow-Teller matrix element in the context of theviolation of SU(4) symmetry [6] in the perturbation theory as well. First, this can beachieved within the SO(8) model. There is also a possibility to exploit perturbationtheory also for the case of realistic calculation of two-neutrino double beta decaymatrix elements.

Acknowledgements. This work is supported in part by the VEGA Grant agency of the SlovakRepublic under the contract No. 1/0876/12 and by the Ministry of Education, Youth and Sports of theCzech Republic under contract LM2011027.

REFERENCES

1. J.D. Vergados, H. Ejiri, and F. Simkovic, Rep. Prog. Phys. 75, 096301 (2012).2. J.G. Hirsch, P.O. Hess, and O. Civitarese, Phys. Rev. C 56, 199 (1997).3. F. Krmpotic, E.J.V. de Passos, D.S. Delion, J. Dukelsky, and P. Schuck, Nucl. Phys. A 637, 295

(1998).4. J.C. Parikh, Nucl. Phys. 63, 214 (1965).5. D.A. Varshalovich, A.N. Moskalev, and V.K. Khersonskii, Kvantovaja teorija uglovovo momenta,

(Izdatelstvo Nauka, Leningrad, 1975).6. J. Bernabeu, B. Desplanques, J. Navarro, and S. Noguera, Z. Phys. C 46, 323 (1990).

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