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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