The effects of pion-exchange corrections on the 2vββ decay nuclear matrix elements

The effects of pion-exchange corrections on the 2vββ decay nuclear matrix elements

Nuclear Physics A495 (1989) 602-610 North-Holland. Amsterdam THE EFFECTS OF PION-EXCHANGE 2v p/3 DECAY NUCLEAR CORRECTIONS MATRIX ON THE ELEME...

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Nuclear Physics A495 (1989) 602-610 North-Holland. Amsterdam

THE EFFECTS

OF PION-EXCHANGE

2v p/3 DECAY

NUCLEAR

CORRECTIONS

MATRIX

ON THE

ELEMENTS

M. ERICSON CERN,

Geneva, and lnstitut de Physique Nucl@aire, 69622 Villeurbanne, France J.D. VERGADOS Physics Department, Uniuekty of loannina, GR45332, Greece Received 29 February (Revised 2 September

1988 1988)

We examine the rSle of pion-exchange diagrams in the nuclear matrix elements which enter the 2~ /3p decay. The corresponding effective operators are constructed using PCAC and soft-pion theorems. The matrix elements of these operators are evaluated in the case of the ‘%a + 4XTi decay. We find that these effects are small and cannot possibly account for the discrepancy between the shell-model results and the experimental values.

Abstract:

1. p/3 decay pp decay theoretically.

has been the subject of intense activity, both It can occur via second-order weak interactions, (A,Z)+(A,Z+2)+e-+e-+S,+C,,

In the presence

of lepton-violating

2vpp

interaction,

another

(A,Z)+(A,Z+2)+e-+e-, In geochemical

experiments

Ovpp

‘) which measure

experimentally e.g., decay.

process

(I)

is possible:

decay.

the relative

and

abundance

(2) of daughter

nuclei it is not possible to distinguish between processes 1 and 2. This is instead feasible in laboratory experiments with the measurement of the total energy of the two electrons 2-‘). The best limit on OV p/3 decay comes from the 76Ge + “Se decay which gives ‘) Tl12 > 5.0 x 1O23y. This limit has improved by an order of magnitude in a decade 3-5). The detection of the OV process quantum-number

conservation

would

signal

not only the violation

but also the fact that neutrinos

are massive

of lepton Majorana

particles. Interest has therefore focused on this process. However, 2v /3/3 decay is also interesting for the following reasons: (i) by the challenge raised by the measurement of lifetimes larger than 10’” years; from this point of view, the recent measurement “) of T,,*(~v) between 0.8 x 102’ and 1.9 x lo*’ y for the 82Se + KrF2 is a remarkable result; of the (ii) the width f,, of this process must be known for the interpretation geochemical experiments ‘) in order to extract the width r,, for the neutrinoless process from the total lifetime; 0375.9474/89/$03.50 @ Elsevier Science Publishers (North-Holland Physics Publishing Division)

B.V.

M. Ericson, J. D. Vergados / Pion-exchange corrections

(iii)

it can

operators

serve as a test of our ability

in systems

with a complex

to evaluate

structure,

matrix

603

elements

such as those

entering

of two-body p/S decay.

Thus, even though the more interesting operator entering Ou/3/3 decay is not identical to that associated with the 2u process, one can gain a certain amount of confidence in the prediction

of the Ou rate which is required

in order to extract

information

about the neutrino properties from the OV pp decay data. The value of the 2v nuclear matrix elements to the energetically allowed states is still under debate as evidenced by the large discrepancies between the results of the various calculations. The general consensus ‘-“) is that these large deviations are due to the fact that these matrix elements are small, i.e., in all cases they exhaust a small fraction of the total strength. Thus they can be affected by some detailed aspects of the wave function which are not fully under control. The smaflness of these nuclear matrix elements is also consistent with the avaifable experimental data. In fact the geochemical rates of Kirsten et al. ‘) of the 18’Te+ 13*Xe decay are about 130 times slower than the theoretical predictions “) based on large shell-model calculations. This discrepancy is not so striking if one considers the geochemical measurements of Munuel et al. ‘) in conjunction with a smaller value of gA, e.g. gA = 1.0. In any case the nuclear matrix element is tiny (it exhausts a small fraction of the sum rule). Thus it is not surprising that the evaluations of these tiny matrix elements come under periodic revision. For instance, it has recently been pointed mechanism required by the above data is present in out ‘*-14) that the suppression the quasiparticle random phase approximation (QRPA) when both the particleparticle and particle-hole interactions are incorporated. In view of the smallness of the matrix element for 2~ /3/3 decay, one may wonder about the rlile played by the pion-exchange effects which are ignored in present calculations: the ey pair is produced together with a pion which is reabsorbed by another neutron, leading to the emission of the second ev pair. The two-body operators for such processes are different from the usual one, since they involve the pion propagator. They are not automatically submitted to the same cancellations as the main term. In fact, meson-exchange corrections have a significant influence on many nuclear processes, in particular when cancellations occur in the impulse approximation matrix element. It is interesting to explore whether the same occurs in 22~p/3 decay, which is the aim of the present article. The specific case of 48Ca is considered. 2. The 21, &3 decay process A second-order weak interaction proceeds via the diagram shown in fig. la. In the standard evaluation, the virtual intermediate states are purely nuclear. The amplitude then takes the form ‘5-‘7)

604

M. Ericson,

J.D.

Vergados / Pion-exchange

“e

corrections

je

e-

n

(4

eP

Ii>

If>

Fig. 1. (a) Diagrammatic representation of 2v pp decay which occurs as a second-order process in the ordinary weak interaction. (b) 2v pp decay induced by the direct decay of the A($, 5) resonance present in the nuclear medium. This process does not contribute to O+ --) O+ decays.

where H, is the weak-interaction hamiltonian. In principle one has to introduce labels for the leptons and ensure antisymmetrization for identical leptons. These subtle points, which are not made explicit here, are included in the actual calculations. In the energy denominator, which depends on the lepton energies, one cannot separate, in principle, the nuclear part from the phase space integral. To achieve this separation we replace E,+ EC
where

T, = M(Z)

- M(Z

mation of the standard in the form 15)

- 2) is the available V-A theory,

energy.

the amplitude

Then in the allowed for Ot + O+ decays

approxi-

can be cast

(4) where

1; are the relevant

leptonic

C=

currents

L2,

and Fermi matrix



elements

Y =C T+(i)a(i)

w-=(o;l~+~+IO+), The Fermi matrix element MF has been discussion since it vanishes for ground-state

by i =

u(Pei)YA(1-Y5b(Pvi),

and MGT and MF are the Gamow-Teller

Mm =

defined

T+=C

defined

by

,

7+(i).

included only to ground-state

for completeness of the transitions due to isospin

M. Ericson, 1. D. Vergados / Pion-exchange

conservation.

The quantity

p. is defined

M

G-r ---=x go

605

tarreelions

by

(O;(Y(l+n)(l+n(Y(Ot) n

(6a)

,

f%

where ~,=I+(E,(l+n)-Ei+;T,]/m,c2.

(6b)

At this point we should stress that if the quantity w is defined via eq. (6a), the amplitude of eq. (4) involves no approximation. In practice, however, most shellmodel calculations replace CL,,in eq. (6a) by an average value associated with some average excitation energy (E,( 1‘f} (closure approximatjon). The recent calculations ‘2--‘4) done in the context of QRPA have the advantage that the summation over the intermediate states is explicitly performed. Concerning the other nonstandard mechanisms for 2v emission, one possibility would be the A($,;) resonance contribution shown in fig. lb. The amplitude associated with the process of fig. lb is proportional to m, where p& is the probability of finding the A($, t) resonance in the nucleus I’). To this order, however, the A($, 2) resonance gives no contribution for OC+Ot decays due to angular momentum conservation selection rules. Thus one is left with meson-exchange contributions. The relevant meson exchange amplitudes which will be considered in this work are shown in figs. 2a-2d. For the vector part the vertices appearing in fig. 2 are obtained from CVC. In this way the two-body operators built from the graphs of fig. 2 are similar to those entering into the meson-exchange corrections to the Compton amplitude 19). For the vertex 7~+ rr+2e-+2fie entering in fig. 2d we have used a quark model of the pion which gives 21yl:.

%

e-

e;je ______-______ V

(4 Fig. 2. Pion-exchange

diagrams

for the vector part (a)-(d).

Ignoring the small energy expressions, valid for O+-,O’ space. For the vector part,

For the axial part, only 2a and 2d enter.

of the exchanged pion, decays, for the two-body

we obtain operators

the following in momentum

(74

M. Ericson, J.D. Vergados / Pion-exchange corrections

606

C’b) (7c)

where g, = 1.24 and fnis the charged the standard decomposition 3(a1.4)(0,.9) where

T is the usual

tensor:

(f,= 130 MeV). We make

pion decay constant

= q2L%1*a2+ T(u,,

uu,,

u2, $) =3(ui.

u2,

(8)

$1,

~)(u2.ij)-u,

.u2. In the chiral

limit m, + 0, a cancellation occurs, similar to that of Compton scattering 19). Namely, the central pieces of V, , V2, V, , V, add to zero, and the tensor parts of V, , V,, V, also add to zero, so that V = V, + V2+ V, + V,-+ ( VJtenqor. Similarly, for the axial part, PCAC provides the vertices of fig. 2a which are thus derived from the TN scattering amplitude. The graphs of fig. 3 therefore have a connection to pion double-charge exchange. Thus the corresponding

amplitude

is

2

A,=$

(H -f

17

IyI;+(f,47Td,)21,.

I,(-f)(u,

x q)(uzx

+ + q) --!Ls!I q2+mt’

(9)

The first part corresponds to the rescattering of s-wave pions and the second part to that of p-waves. In expression (9) d, is the spin-dependent part of the charge exchange p-wave TN scattering volume, extrapolated for soft pions: d, = 0.07 mi3.

Fig. 3. Pion-exchange

diagrams

for the axial part.

In deriving the expression (9), we have, as usual in the meson exchange technique, retained only the non-Born parts (d, for instance is dominated by the intermediate virtual A excitation). It is now a straightforward matter to perform a Fourier transform of the amplitudes of eqs. (7a)-(7d) and (9) and get the corresponding effective operators in coordinate space. Thus eq. (4) becomes

M. Ericson,J.D. Vergados

/ Pion-exchange

607

c’orreclions

where

K ',2

f, 5, =f(47Td&JZ----0.19.

1 -....--.= m,m?r 2.2 x 1o-4, 3

.f”, 4-r

%r

(11) The operators

entering

the matrix

elements

V, A and A, are, respectively,

&?A= 1 T’(i)r+fj)(~uj~a;-~u~(x)-4as(x)~-gT(rrj, i tj

f& =

uj, $uf(x)},

(13)

c T+(i)T+wY”(X)>

(14)

i+j

where x = m,r,, , u;(x) = Ye(x) =

Yo( x) = x-’ e-’ ,

xl’ e-“,

U;(x) = z&x) = f(x -2) YJX) ,

u:(x)

= UT(X) =$(x+

1) YJX) , (15)

Note that the amplitude practice

(10) has been scaled by the electron

in 27.~J?p decay calculations.

The relative

importance

mass m,, a standard

of the exchange

current

contribution is set by the value of K, compared to PLO’,We note that K, is of order 10e4 while ~0’ is larger due to the smallness of the energy denominators. For the A = 132 system for instance the Los Alamos group 16) estimates pi’ = 4 x lo-“. Even if this must be decreased by an order of magnitude to bring the theoretical prediction into agreement with experiment, PC’ remains larger than K, . The meson-exchange terms could be significant only if M,, is greatly suppressed. We wiff now calculate

the relevant

nuclear

matrix elements

entering

the 48Ca+ 48Ti

(gs.) decay. This transition is particularly suited for our purposes since: (i) the nuclei involved have simple nuclear structure; in fact they can adequately be described in terms of the Of,/, shell, i.e., both protons and neutrons are restricted to occupy the Of,,, shell; (ii) the ground-state transition matrix element MGT is known to be suppressed 7~“*2”~2’); (iii) the quantity E,L~can be calculated exactly. In fact, within the Of,,, shell there is only one intermediate It state of 48Sc which is of the form [f;,2(n)f7,2(P)]I+. Thus, eq. (3) is reduced to eq. (4) without approximation with pO= ~(1~).

608

M. Ericson, J. D. Vergados / Pion-exchange corrections

In order to give a scale to the amount therefore Teller

compare

operator

the quantity

of cancellation

occurring

M GT to the sum rule involving

in MGT., we can

the double

Gamow-

22) Y+ = C i aiT+ S,,=~(O;~Y+Y+~O)=6(N-Z)(N-Z+l).

(16)

This result only holds for Pauli blocked nuclei, which 48Ca is not. But its order of magnitude still gives an element of comparison for the double GT strength to the ground state. We shall see below that the matrix element M,, for ground-state transition is 0.24. It thus exhausts only (O.24)2/So,= (0.24)2/(6 x 72) = 1.3 x 10e4 of the sum rule, evidentiating the smallness of the quantity MGT. The initial wave function in the above model is unique, The final-state wave function is of the form 7X2”) 48Ti(Ot) = 1 C,{Oc,,(

i.e., of the form of Off,,(n).

n)J; Of:,,( p)J; Of} ,

(17)

J

where J = 0,2,4,6. The coefficients C, depend, of course, on the two-body action. In this work we use the Kuo-Brown renormalized G-matrix elements. one finds ‘5-20) that the g.s. wave function corresponds to C, =0.908, respectively.

Furthermore,

-0.418,

0.013, 0.004

the quantity

for J = 0,2,4,6.

interThen

(18)

p. is in this case /_Lo=ll.

The nuclear

matrix

element

is

MGT = -0.238

,

M

M,, = GT

= -0.0216

.

(19)

PO

The smallness of MGT can be traced back to the well-known the J = 0 and J = 2 components of the 48Ti wave function.

cancellation

between

The exchange-current matrix elements, associated with the operators of eqs. (12)-(14), including the factors to and 5, but not the scale K, , are given in table 1. The tensor contribution and, to a lesser extent, the scalar contribution also suffer from the above-mentioned decay is very small mainly

cancellation. The resulting contribution to the 2u BP due to the smallness of the parameter K, , i.e.,

K,( V+ &A) = 2.2 x lop4 x (0.016+0.19 -K, toA0 = -2.2 x lop4 x 2 x (-0.095) Thus the dominant time-component the usual matrix elements M2Y.

x 0.098) = 0.76 x lop5 , = 0.41 x lop4 .

term is three orders of magnitude

@@a) (20b)

smaller

than

M. Ericson, J. D. Vergados / Pion-exchange correctiom TABLE

609

1

The various exchange current matrix elements entering the 2v pp decay ‘xCa+4XTi in dimensionless units [here in units of K, (see eq. (1 I))]. The matrix element R,, , which does not interfere with the usual nuclear matrix element, is dominant, The matrix element V, is not given since V, = 2 V, Gaussian

No correlation scalar

tensor

scalar

tensor

0.018 -0.067 0.044 0.078 -0.095

-0.015 0.002 -0.020

0.079 -0.03 1 0.011 0.052 -0.086

-0.015 0.006 -0.014

V,+ V,+ V3+ V,+5,A=0.035 -&,A,, = 0.186

The above exchange-current the short-range two-nucleon nucleon

wave functions

qS,,, is the usual

of the type IS) = (1 - 4x,

shell-model

thus obtained

~s.m.(~, ,x2) ,

also appear

(21)

and

a=l.lfm-‘,

b=0.68fmp2.

in the table.

K,( V+ &A) = 0.62 x 10-5, i.e., a small modification. The above results indicate

- 4))

wave function

c(r)=emar’(l-br2),

The results

V,+ V,+ V,+ V,+f,A=0.028 -.&A0 = 0.168

matrix elements may be somewhat uncertain due to correlation. In order to test* this we employed two

~COLX,, 4 where

correlation

(22)

From these we obtain

- K,&,A, = 0.37 x lo-“,

that, even though there exist some cancellations

nuclear

in origin in the matrix elements of the exchange current operators, the main suppression comes from the smallness of K, compared to p(;‘. This is expected to be a general result since in the exchange mechanism there is no small energy (mass) scale to compete

with the small energy denominators

pn. A large pcLo can in principle

arise even if the energy denominators pu, are all small compared to WI,,, if it so happens that the various terms in the sum of eq. (6a) contribute destructively. But even then the value M,,/p,, must become much smaller than that dictated by experiment before the exchange current mechanism becomes effective. It thus appears that the smallness of the Mzy matrix element must be explained in terms of traditional nuclear physics. The QPRPA 12-14) calculations are in this direction. l

This is the reason

why no cut-off

factors

were employed

in deriving

eqs. (12)-( 19)

610

M. Ericson,

To summarize: exchange much

smaller

exhausts therefore, magnitude

we have

process

Vergados / Km-exchange

investigated

in the 2v /3p decay.

than those

associated

only a small fraction expected

J.D.

of the double

of the usual theoretical

into agreement

the possible

with those extracted

rble

played

by the meson

We have seen that such contributions

with the usual

to play a crucial

corrections

process

Gamow-Teller

are

the latter

sum rule. They are not,

r6le if only a reduction

nuclear

even though

matrix elements

of about is needed

an order of to bring them

from experiment.

One of us (J.D.V.) would like to thank the Theoretical and the Greek CERN Committee for providing support most of this work was performed.

Physics Division of CERN for a visit at CERN where

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