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The 2ν double beta decay in nuclear matter
ANNALS
OF PHYSICS
187, 79-96 (1988)
The 2v Double Beta Decay in Nuclear Matter W. M. ALBERICO,M. B. BARBARO,* A. BOTTINO, AND A. MOLINARI Dipartimento
di Fisica Teorica deN’Uniuersifri 10125 Torino, Italy; and Nazionale di Fisica Nucleare, Sezione Torino, Italy
di Torino,
Istituto
di Torino,
Received January 13, 1988
In the present paper we analyze the suppression mechanism of the (Zvfifl) amplitudes, in the framework of superfluid nuclear matter. We show that the interplay between spin-isospin modes and spin pairing vibrations in quenching the jj amplitude is particularly transparent in the context of the polarization propagator method. Our procedure is applied to finite nuclei of interest and our results are compared to other theoretical evaluations and to the experimental data. 0 1988 Academic Press. Inc.
I. INTRODUCTION
The hope of obtaining, from studies of double beta (BP) decays, through the neutrinoless channel (Ovp/?), reliable information about the neutrino properties (Majorana/Dirac nature, mass) and the strength of a possible V+ A term in the weak-interaction Lagrangian has been deceived to a large extent until very recently, because of serious discrepancies between experimental data of different groups and also on account of some disagreement between experimental results and theoretical lifetime evaluations, already at the level of the “standard” (2$/3) mode [ 11. In fact shell model calculations based on realistic nucleon-nucleon interactions appear to lead to an overestimate of the 2v/3b decay amplitude in medium-heavy nuclei. However, according to the most recent developments, some of the above-mentioned conflicts between experimental results seem now to be substantially resolved [2-S]. Furthermore, an interesting suggestion has been made [9, lo], which might lead to a significant suppression of the (2vfib) decay rate thus improving the agreement with the data. In this connection we recall that the nuclei of interest for the /?/? decay are, in general, suitably described in terms of quasi-particles; accordingly they are conveniently treated in the framework of the quasi-particle random phase approximation (QRPA). The latter, beyond the usual particle-hole (p-h) correla-
* Dottorato di Ricerca dell’Universit8 di Torino.
79 OOO3-4916/88 $7.50 5951187/l-6
Copyright 0 1988 by Academic Press, Inc. All rights of reproduction in any form reserved.
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ET AL.
tions, introduces particle-particle (p-p) correlations as well; it so happens that a destructive interference between these two forces arises in the evaluation of the j?fi decay rates [ 111. Although the inclusion of the p-p correlations provides a remarkable suppression mechanism for the (2v/?/I) decay rate, yet one should realize that: (i) The agreement between theory and experiment occurs through a drastic cancellation of large numbers and appears, as a consequence, to critically depend upon the (p-p) interaction strength, g,,, whose range, as determined from 6’ decay, appears to encompass the values at which the (2v/?p) decay rates vanish [12-13 J. Thus, for the (2v/?B) decay lifetimes, the theoretical calculations yield lower bounds, small enough to be compatible with the experimental data, rather than definite predictions. Accordingly, an alternative view could be to exploit the sensitivity of the (2vB/?) amplitudes to the p-p strength in order to fix g,, from the Dp decay experimental data and then to employ this value as an input for the calculation of the (Ov@) nuclear amplitudes [ 14-151. (ii) Since the phenomenon under investigation entails strong cancellations, other effects [such as meson exchange currents (MEC) and the delta-delta mechanism (Ad)], which were commonly believed to represent small contributions to the (2vjIj3) rates, could play a non-negligible role and should then be taken into proper account. These effects might turn out to be particularly relevant in connection with the above-mentioned determination of g,,. Our aim, in the present paper, is to shed further light on the suppression mechanism of the (2$/I) decay rates and to lay the groundwork for the inclusion of the contributions mentioned in (ii). For this purpose nuclear matter offers a convenient framework. In spite of its simple geometry this system, as we shall illustrate in the following, allows us to substantially reproduce the general trend of the results of accurate finite nuclei calculations [S-lo, 121, in both the RPA framework and in the QRPA framework. Therefore we believe nuclear matter to be a testing ground reliable enough for the inclusion of further contributions to the fi/? decay, namely the above-mentioned effects associated, e.g., with the MEC. The latter indeed are more easily dealt with in this system than in finite nuclei. Also the A,, isobar, which specifically affects the spin-isospin channel considered here, is suitably inserted in the RPA framework of nuclear matter. We consider the response of the latter to the Gamow-Teller (GT) operator (the only one we shall take into account in the present work, since it dominates over the Fermi operator) in the scheme of the polarization propagator method. When one introduces the pairing interaction and nuclear matter becomes superfluid, the polarization propagator unfolds two components: one associated with the particle-hole (p-h) and the other with the particle-particle (pp) elementary excitations. The competition between these two terms, especially when the p-h and p-p interactions are switched on, is displayed by our formalism in a very trans-
2~ DOUBLEBETA
81
DECAY
parent way and accounts for the reduction of the fib decay amplitude through a cancellation occurring between the spin-isospin (Gamow-Teller) and what may be defined as a spin pairing vibration mode. Furthermore this approach appears particularly flexibile in order to follow the evolution of the process under investigation both as a function of the momentum associated with the first order transition leading to the intermediate nucleus and of the temperature. This may bear some relevance for double charge exchange reactions, particularly those sensitive to the spin-isospin degrees of freedom, as well as for heavy ion physics, where the temperature plays a decisive role. At zero temperature and zero momentum the total diamagnetism of the noninteracting nuclear matter entails the vanishing of the fib decay amplitude. The interaction shifts this diamagnetism at a finite q = 4 (when T= 0) or at a finite T= T (when q = 0). This can be viewed as a generalization of the Meissner effect to include the isospin degrees of freedom as well as the spin degrees of freedom. For temperatures different from T (at zero momentum) the amplitude for ,6p decay is no longer vanishing, but is still quite depressed, with respect to an independent particle model prediction, by pairing effects and by long range correlations, suitably described in QRPA. The same situation occurs for non-zero momentum (at T= 0), but now, as already mentioned, it is the interplay between the p-h and p-p channels (in other words, between the spinisospin mode and the spin pairing vibration), which yields large cancellations in the j3p amplitude. We illustrate these ideas in the present paper, which is organized as follows: in Section II we recall the general expression for the (2vbp) decay half-life and find its relationship to the spin-isospin polarization propagator for normal nuclear matter. In Section III we introduce spin pairing correlations and consider the polarization propagator for the superfluid system; the flp decay amplitudes are then obtained in both the free case and in the QRPA framework. Finally in Section IV we present the results of our calculations and discuss them in connection with previous work.
II. GENERAL FORMALISM Let us start from the general expression of the (2v@?) decay half-life for the o++o+ transitions; following the notations of Doi et al. [l] it reads (11.1) where
Came CO,+IICn~n+~rrI/N,(l+))(N,(l+)IlC,t,+a,II Eo - (lP)(Mi + M,) m, being the electron mass, E, the intermediate
0:)
*, tII2j
state energy, Mi (M,) the mass of
82
ALBERICO
ET AL.
the initial (final) nucleus, t + the isospin raising operator and the sum C, referring to the nucleons in the nucleus. Finally Go= is a phase-space factor defined and tabulated in Doi et al. [I]. For a free Fermi gas, Eq. (11.2) can be factorized into a spin part and a dynamical one as follows:
Before evaluating both factors we remark that symmetric nuclear matter may be thought of (in the phase space) as two Fermi spheres, one for neutrons and one for protons, with the same radius k,. Accordingly, we conceive our initial state as made up by two J” = I+ neutron-particle, proton-hole excitations coupled to J” = O+, lying very close to the Fermi surface. Once the first weak transition has taken place, the intermediate state, in a Fermi gas scheme, is an isovector J” = 1 + neutron-particle, proton-hole state (lp - lh). Then (see Appendix) one gets, for the spin factor,
Moreover, neglecting the energy difference between the initial and final systems, the following expression for the dynamical factor d(q) holds, W,lhl)~(lpl -k’)f(q2)s d(q) = c p,h tl=p2/2hf- ?w/2M 52 -2
=--
Re
n”(q,
0
=
_ q>P
b
O)f(q2),
0
where M is the nucleon mass, Q is the volume enclosing the system, andf(q2) axial-vector nucleon form factor,
is the
with m, = 0.85 GeV. Formula (11.5) provides an interesting connection between the /3/i decay amplitude and the real part of the (static) free particle-hole polarization propagator at zero temperature, no. This is in line with the well-known relationship between the various susceptibilities of atomic nuclei and Re P '(4, 0) in the appropriate spin-isospin channels. Accordingly, in our model, the /$!l decay amplitude can be viewed as the paramagnetic susceptibility of nuclear matter (both in its normal and superconducting states). We notice that there is no explicit momentum dependence in the GT operator of formula (11.2); actually the momentum associated with the p-h excitation should be related to the first weak transition leading to the (virtual) intermediate isovector 1+
2V DOUBLE
BETA
DECAY
83
state; we shall see later that the interesting values of q range between 0.04 and 0.05 fm-i. However, the momentum dependence (at zero frequency) of the free polarization propagator 17’ is rather smooth for values of q up to 1 fm-‘. The expression (11.5) for the fl/? amplitude refers to the crude description of a non-interacting Fermi gas. In order to treat nuclear matter more realistically, we should dress with correlations the real part of the static particle-hole polarization propagator n(q, 0). To do this we remind the reader that RPA correlations, induced by the strong spin-isospin force, are essential in accounting, e.g., for the measured GT strength in nuclei. The corresponding RPA polarization propagator obeys the standard Dyson’s equation nRPA(q) = no(q) + no(q) J$,(q) ~RPAtqh
(11.6)
which, in nuclear matter and neglecting the exchange matrix elements of the p-h force, is algebraic and yields IIRPA( q) =
Accordingly,
no(q) 1 - V,,(q) no(q)’
(11.7)
one gets for the j?p amplitude, dRPA(q)
= -
T
Re URPA(q, O)f(q2)
0
(11.8) got% 0) f (q2) 1 - f$dq) flO(% 0)
[notice that for zero frequency and finite momentum no(q) is a real quantity]. For the p-h interaction in the spin-isospin channel, we include here only a short range repulsive component which, according to the Landau-Migdal parametrization, reads (11.9) where the factor 4 corresponds to the ph matrix elements of the (a, . (r2)(r1 . r2) operators, rz(q2) is the nNN form factor and the Landau-Migdal parameter g’ is given in pionic units. Indeed, in the cz channel, the rr- and p-meson exchange forces (V, and V,) should be included as well, but their direct matrix elements are negligible at small q. We also remind the reader that the parameter g’ approximately accounts (in the limit q -+ 0) for the exchange ph matrix elements of V, and I’,. Notice that, owing to the repulsive nature of the ph force, the denominator in (11.8) is larger than unity, thus reducing the free amplitude: the nuclear forces oppose the flipping of the nucleonic spins.
84
ALBERICOETAL.
Thus far we have only considered intermediate nucleon-particle nucleon-hole states, but the GT operator, responsible for b* decay, can also induce the transition from the initial state to a A,,-particle, nucleon-hole state. Obviously such a state would be far off-shell (by about 300 MeV) and for this reason it has been frequently neglected. Moreover, although the single B decay matrix elements leading to 1AN-‘) states are different from zero, the B/I matrix elements with a direct A,,hole contribution vanish [ 161. However, the RPA correlations in the spin-isospin channel are significantly altered when the A,,-hole polarization propagator, Z7A, is taken into account as well. This is easily done in nuclear matter, by replacing Z7’ in the denominator of (11.8) with the sum of the free NN-’ and AN-’ propagators: aiRPA(
-
t
Re
0
n,cs, 0) fW). i 1- ~,lmC~N(cl~ 0) + fl‘A% 011
(11.10)
Notice that ZZd does not appear in the numerator since it would imply a direct AN-’ contribution to the /@I amplitude. In concluding this section we remind the reader that the A,, can offer additional mechanisms for the O+ + O’pg decay, providing one takes into account two-body processes with intermediate A,, states [AA mechanism] [17]. This item, however, goes beyond the present RPA treatment. III.
PAIRING CORRELATIONS
It has been pointed out by several authors [9-121 that pairing correlations tend to reduce the BP decay amplitude, since it is more difficult for the single intermediate decay to occur, breaking a pair of correlated nucleons. This situation, where particles with opposite spins, namely (Nr, N 1 ), are paired by the attractive interaction Vpair, is described by replacing the (unstable) Fermi sea with the BCS ground state through the well-known canonical transformation to quasi-particle operators, which diagonalizes the pairing interaction. The latter modifies the singleparticle spectrum and occupation probability according to the substitutions WEk=&TZ
(111.1) (111.2)
where ,u~ is the chemical potential of nuclear matter and A the energy gap. Now, although real nuclei are at zero temperature in the ground state (recall, however, that considerably high temperatures may be reached in relativistic heavy ion reactions) it is convenient here to treat the superfluid nuclear matter within the temperature Green’s function formalism.
2~ DOUBLE
BETA
85
DECAY
We have then solved the following gap equation 1 = g&J(O) j,“‘“” J&tanh(‘y),
(111.3)
where fi = l/k,T, T being the absolute temperature, ho,, = 0.49 pF [18], N(0) z A4k,/2n2h2 is the density of states at the Fermi surface, and g, = 120.4 MeV fm3, the strength of the pairing force Vpair. This value of g, corresponds to the parameter G = 28.84/A MeV frequently used in finite nuclei c191. The above equation allows the determination of the gap as a function of T; it stays almost constant for low temperatures and then drops to zero at the critical temperature T,, where the transition from the superfluid to the normal phase occurs. For nuclear matter (k, = 1.36 fm-‘) Tc turns out to be close to 0.8 MeV, if a nucleon effective mass larger than the free one (M,, = 1.15 M) is used. Indeed, at very low q values both the particles and the holes are close to the Fermi surface, where any realistic energy dependent mean field yields an increased nucleon mass, due to the coupling of the single particle motion with the nuclear surface vibrations. Let us then introduce the free quasi-particle-quasi-hole temperature polarization propagator [20] no,&% a) = @$q,
(111.4)
0) + ~~,“(q, 01,
where
X
Ly(q,w)=~
2m5 X
1
1
(III.Sa)
ho+E+-Ep+iq-fiw-E++E-+iq 4
--o+d2
(U+K
Cl-f(E+)-f(Ep)l
1 r%zo--E+-Ep+iv]-ko+E++E_+iq
1 >’
(IILSb)
In the above formulas the symbol + implies a wavenumber (p f q/2),
and f(E) is the modified Fermi function
S(E) = -1 :P
(111.7)
86
ALBERICO
For a better understanding
ET AL.
of (111.4) it is worth noticing that for temperatures
T> T,, where the gap vanishes, one has
24:=a B(k - kF)
(III.8a)
v; * 8(k, -k);
(III.8b)
this helps in recognizing particle-hole or particle-particle contributions. In particular, terms proportional to U: ~5, v: ~2, or UC v: stem from the convolution of two single particle temperature Green’s functions-
4a(x, x’) = - (0 I T C$&)
(111.9)
&AT(x’)llO>
whereas terms with the product (u, U_ u, v- ) stem from particle number nonconserving Green’s functions, 8 or St, which are specific for the superfluid phase and are defined as follows:
Ejdx> x’ ) = - (0 I T C$&) Thus the typical combination quasi-particle (or quasi-hole)
(111.10)
&qW)l lo>.
RF+ entering into (111.4) corresponds to a two Green’s function, which obviously vanishes for
T> Tc.
Let us now consider the behaviour of (111.4) at zero frequency (as in the previous section) both at 1q ) = 0 as a function of the temperature and at T= 0 as a function of momentum. For zero momentum the quantities indicated with suffixes + or coincide (e.g., U, = U- = u,), hence E$j’(q, 0) identically vanishes at any temperature. The function n!$(q, 0), instead, can be recast in the following form: l7~c”(q+O,o=O;
(111.11)
T)=,&%$
At zero temperature the expression (III.1 1) vanishes, thus entailing the total diamagnetism of the superfluid matter, and then it slowly increases with the temperature; above T,, it merges into the normal temperature-dependent particle-hole polarization propagator in the q -0 limit. Notice that the factor (u; + vi)*, not explicitly written in (III.1 1) since it is unity, still contains a p-h and a p-p contribution (the latter being associated with the double product). For the purposes of the present paper, the limit T + 0, q finite, is more interesting. In this case, as already pointed out, J7!$‘(q, o = 0; T= 0) = 0, whereas I7”,;(q,
which is conveniently
w=O;
T=O)=
-
-u-v+)’ sdp(u+v(2n)3
E,+E-
split into a p-h and a p-p contribution,
Zi’&Jh(q, w = 0; T= 0) = -
’
(111.12)
according to
dp cu:v2 +u?v$) s (2,rr)3 E++Ep
(111.13a)
87
2V DOUBLE BETA DECAY
and
Izos’cpp(q,o=o; T=0)=2@&,v-;~~+.
(111.13b)
+
Figure 1 illustrates the behaviours of both Z70S,Cph and @,pp as well as their sum as functions of q [actually they are multiplied by - (Q/2)f(q2) in order to obtain the corresponding /?b amplitudes]. The amplitude for a normal fluid [Eq. (IIS)] is also reported, for comparison. One sees that the p-h and p-p contributions exactly cancel at q = 0, but then the p-h term prevails and approaches the free normal amplitude, although it is always smaller than the latter as a result of the pairing correlations. At this stage we should explore the most interesting effect of RPA correlations in the superfluid system. As suggested by several authors [9-121 this may be achieved in the quasi-particle random phase approximation. In principle, the corresponding equation for the polarization propagator (111.12) would mix the p-h and p-p channels; however, it has been shown [21] that if the gap d is much smaller than the Fermi energy and the density of the energy levels close to the Fermi surface is
-+%)/A 0.000
I“’ r I” n I I8 ’ I’1 _....._________....................................................................,.....~
0.004
_---
__--
__--
_----
--
_---
-_--
_----
--
_---
0.002
0.000
-0.002
_--0
_--QII tt 1 1 II II I 1 I I 1 I 1 1 I I -. -. 0.05
0.1
0.15
0.2
rm-’ FIG. 1. Behaviour of ZZ& (real part of) versus 4 (continuous line), together with its J75j$‘” and l7$‘r components (dashed lines). The curves correspond to A = 1.48 MeV, the value obtained by solving Eq. (111.3) with the strength of the force, g,, and the cutoff, tiw,, given in the text. The density of states at the Fermi surface is evaluated with an enhanced nucleon effective mass M,s= 1.15 h4. Note how f7& results from opposite contributions. Re ITo for a normal Fermi gas is also displayed (dotted line).
88
ALBERICO
ET AL.
constant (which is the case in our model), the QRPA equations for the p-h and the p-p polarization propagators decouple as follows:
In the above
ng;A’ph(q)
= n$;7Ph(q) + h?&“(q)
v,,(q) n,R,‘A’ph(q)
(111.14)
z7p.yq)
= @iy(q)
V,,(q)
(111.15)
eqUatiOnS
Vph
+ @pyq)
is given by (11.9) and VP, is written in the form
V,,(q) = -22 where matrix g,, in The QRPA
ITip-yq).
P?I
(111.16)
gppW2),
g,, is the strength of the particle-particle interaction and the factor - 2 is the element of the (a, .c2)(t, . r2) operators. We will comment on the value of the next section. solution of Eqs. (111.14) and (111.15) is straightforward and yields, for the j?fi amplitude, J&p(q)
= dig-yq)
(111.17)
+ dg-yq),
where dg-(q
)=-
0
5
Re
Ggh(q, 0) f (q2) i 1 - v,,(q) @,ph(q, O)
(III. 18a)
y
Re
@p(q, 0) 1 V,,(q) n$yq, i
(IIL18b)
and &;:A*
pp(q) = 0
0) “Oq2).
The behaviour of (111.17) as a function of q obviously depends on the precise values of g’ and g,, ; as a general remark we observe that, since both g’ and g,, are positive quantities, one has Vph > 0 and Vpp < 0 [see Eqs. (11.9) and (111.16)]. In turn, Zjzh ~0 whereas Z7‘&‘P > 0 so that the RPA correlations produce a quenching of the corresponding amplitudes in both channels. However, for values of g,, N g’, the Vpp is weaker than VP,, by a factor of 2; hence the RPA pp amplitude is less reduced than the corresponding ph amplitude and the total cancellation of the two components, which occurs at q = 0 in the free case, is displaced toward finite values of the momentum (which are the ones of interest for finite nuclei), as we shall see in the next section. For the sake of illustration, the QRPA amplitudes (111.18) are displayed, together with their sum, in Fig. 2, for g,, =g’ (the physical values for these parameters will be discussed later ).
89
2~ DOUBLEBETADECAY
1~‘~~1~1”1~“” 0.002 -____.....________.__.......................,....,,,,...,,,,,,..,.,.,.,....,,~,,,....~,,.,. _----____-----___----0.001 7 g
0.000
-0.001 __--0.002 --
_---
_---
_---
__--
-
tl~~~l,,,~l,“‘l”‘~0 0.05
9 0.1
0.15
_ 0.2
fm-’ FIG. 2. The same as in Fig. 1, but with QRPA correlations present. The p-h and p-p interaction strengths are g’ = gpp = 0.7.
IV. RESULTS AND CONCLUSIONS In this section we present the results of our calculations, both for the normal and for the superfluid system. Although the present approach refers to infinite nuclear matter, we shall nevertheless apply it to a few nuclei, which are of particular interest for /?/I decays from the experimental point of view, namely Ge76, Se’*, Te13’, and Xe’36. Accordingly, our method is expected to yield the general trend of the amplitude versus the mass number A, but it is certainly not expected to account for specific features related to a given nucleus. In fact, in our model nuclear structure enters into the expression for the amplitude through the following three characteristic quantities: the Fermi momentum k,, the momentum q at which the amplitude is evaluated, and the mass number A, which is used to replace the nuclear volume Q through the customary relationship 52 = A/pF, OF being the average nuclear density. The latter is taken as the mean value of the Fermi distribution
PO(A) PF(r’ A)= 1 +exp((r-R(A))/a) with A po(A)= (47cR3/3)[1 + (uTc/R(A))‘]
(IV.2)
90
ALBERICO
ET AL.
In the above a=0.54fm is the surface thickness and R(A) the nuclear radius R, given by the usual expression [19] R=
(IV.3)
For the Fermi momentum (which in nuclear matter is 1.36 fm-‘), we utilize the average value, depending upon A, which can be deduced from PF itself: F,(A)
= [T
(IV.4)
p,~]“~.
We have then extracted the characteristic values of q for the different nuclei considered here from the corresponding energy differences of their typical single weak transitions; we have found q values ranging between 0.04 and 0.06 fm-‘. Let us first consider the normal (i.e., non-superfluid) system: we remind the reader that for q, say up to 1 fm-‘, the static polarization propagator (both in the free and in the RPA cases) has a very mild momentum dependence. Therefore the differences between the various nuclei essentially reflect, in this instance, the different sizes and kF values. In Table I we report the results of our calculation for 1M&)/p,/ for a noninteracting system (first row) and for the RPA correlated matter, both without (second row) and with (third row) the inclusion of the A,,-hole polarization propagator. For the Landau-Migdal parameter g’ we employ here the value 0.7, which in the RPA framework accounts fairly well for the giant GT resonance measured by (p, n) experiments. One clearly sees that the RPA correlations significantly reduce the free amplitudes, typically by a factor of 3; the inclusion of the d,, entails an additional 10% reduction. However the values of Table I do not match the available experimental data on Se** and Te13’, which require (see Table III, for reference) much smaller B/? decay amplitudes. Let us then consider the superfluid matter. By looking at Fig. 1, one realizes that TABLE Nucleus
I
Ge76
Se*?
Te”O
Xe’36
4 (fm-‘1
0.048
0.051
0.067
0.052
Free RPA RPA
0.568 0.203 0.179
0.609 0.217 0.191
0.927 0.325 0.285
0.966 0.339 0.297
(A331
Nore. Values of 1Mg;;Zp,,I. The RPA calculation is reported both without (second row) and with (third row) the inclusion of d-hole intermediate states; the q values utilized for the different nuclei are also reported here.
2V DOUBLE
BETA
DECAY
91
in the range of the q values pertinent to the /?/I decays considered in the present paper (see Table I) the pairing correlations sizably reduce the p/I amplitude, already in the absence of RPA correlations. However, as was pointed out in Section III, an even more pronounced quenching can be obtained within the QRPA framework. In order to evaluate (111.17) one would need, as an input, the value of the p-p interaction strength, g,, . In principle, the latter should stem from a realistic NN force and indeed a microscopic G-matrix calculation of g,,, to be utilized in a shell model framework, has been carried out (see Haxton and Stephenson in [ 1 ] ). Here, however, we are utilizing effective residual interactions, which need to be fixed on the basis of some experimental constraints. Indeed the value of the p-h strength g’ has been determined from the giant GT resonance in a fairly unambiguous way; similarly the p-p strength can be extracted from the available data on fl’ decays in several nuclei [9, 121, but the resulting uncertainties on g,, are too large to allow precise predictions on the 2vb/? amplitudes. In view of this fact, we adopt here the heuristic attitude of considering g,, as a free parameter. Our 2vpfl amplitudes, evaluated according to (111.17), for the four nuclei under consideration, are displayed in Fig. 3 as a function of g,, (we remind the reader that an additional parameter characterizes, here, the different nuclei, namely the energy gap, A; instead of solving Eq. (111.3) we use, for finite systems, the value 12/fi MeV [19]). Also shown in the figure are the ranges of the experimental values, corresponding to the recent measurements of /?/I decay in Ses2
FIG. 3. The 2$/3 amplitudes versus the strength of the particle-particle interaction g,, are shown for Ge76 (dashed line), Ses2 (dotted line), TelW (dotdashed line), and Xe’36 (continuous line). They have been obtained by adapting the nuclear matter amplitude to suit the various nuclei according to the simple prescription given in the text. Also displayed (by vertical bars) are the experimental results for Seg2 and Tei3’ reported in Table III.
92
ALBERICO ET AL.
and Te13’ [6-81; f or both nuclei there exists a range of g,, values (0.42 6 g,, < 0.51) which render our theory compatible with the experiments. It is remarkable that the general trend of our p/3 amplitude as a function of g, agrees with the calculations of Refs. [9, 10, 123; the different slope has to be essentially ascribed to the differences in the Fermi surface of a nucleus with respect to the Fermi surface of nuclear matter. Moreover the interplay between p-p and p-h components of the amplitude, which is most apparent in the framework of the polarization propagator, seems to act somewhat differently within the QRPA equations for the nuclear wavefunctions. In order to compare our results with previous ones [9, 10, 121 without reference to this cancellation mechanism induced by the p-p interaction, we report in Table II the results of Refs. [9, 10, 121 at g,,,=O and our evaluation of the /3/I RPA amplitude in the superfluid system omitting the influence of the p-p channel. To achieve this goal, we either set g,, = 0 or wash out completely the p-p contribution (namely we set 17$2” = 0). Indeed it is not clear whether the g,, = 0 calculations of Refs. [9, 10, 121 include the p-p contribution at the free level. Indeed this second instance seems to provide results which are in closer agreement with the ones of Refs. [9, 10, 121. It is thus justified to utilize our superfluid matter approach for an evaluation of the /I/l decay amplitudes and half-lives in Ge76, Ses2, and Te13’. Both spin-isospin and pairing correlations are included, by utilizing the full RPA p-h and p-p polarization propagators [ Eqs. (III. 17) and (III. 18)]. One should recall that we express both g’ and g,, in pionic units, at variance TABLE Nucleus H-S C-F-T (g,,=O) M-K (g,,=O) Normal RPA (g’=O.7) Superfluid RPA (g,,=O) Superfluid RPA Ep = 0
II
Ge76
Se*?
Te”O
Xe’36
0.139
0.095
0.114
0.330
0.270
0.205
0.345
0.292
0.173
0.203
0.217
0.325
0.339
0.098
0.098
0.082
0.123
0.156
0.169
0.268
0.272
Note. Values of 1M,$)/&(, The first three rows are extracted from Haxton and Stephenson (H-S) [l], Civitarese, Faessler, and Tomoda (C-F-T) [lo], and Muto and Klapdor (M-K) [12], respectively. In the fourth row our RPA value for normal matter is shown, whereas in the last two rows the RPA evaluations in the superfluid matter are given, once with g,,=O only and once without the whole p-p contribution.
2V
93
DOUBLE BETA DECAY
with previous works [9, 10, 123; in the latter the commonly utilized value of g’ is close to unity. Thus, to facilitate the comparison, we point out that, at fixed g’ = 0.7, we use values of the ratio g,,/g’ in the range 0.6 < g,,/g’ < 0.73, which are rather close to the ones recommended by other authors [ 121. The results of our calculation are presented in Table III, where the pfl decay amplitudes (first column) and half-lives (second column) in Ge76, SeE2, and Te13’ are compared with the experiments. The theoretical values correspond to the abovediscussed range of gpp. One sees that the present model yields the same reduction of the /Ij decay amplitude as given by the calculations in finite nuclei of Refs. [lo] and [ 121. In particular we remark that the reduction of the decay amplitude in going from Se*’ to Te13’ is induced, in our framework, mostly by the q-dependence occurring in the QRPA calculation. Indeed, as shown in Fig. 2, the cancellation between the p-h and the p-p components of the amplitude is quite sensitive to q. In addition the A-dependence entailed by our approach leads to a more rapid variation of the amplitude with g,, in heavier nuclei. It is worth pointing out that our approach displays the mechanism for the reduction of the 2vpp decay amplitudes in a quite transparent way. Indeed, in our nuclear matter framework, the /I/? process involves only the neighborhood of the Fermi surface. Here the nuclear matter displays a dual spectrum: fermionic and bosonic at the same time. As often occurs in nature, these two aspects of the same physical reality tend to oppose each other: in this instance the contrast is particularly evident owing to the structure of the second order formula expressing the Bfi decay amplitude. This indeed allows the two modes of GT (spin-isospin) and spin pairing vibrations to compete in the intermediate state. In conclusion we remark that a reliable theory should be able to encompass both the single and the double fi decay processes. Accordingly we are at present exploring, in the context of asymmetric nuclear matter, where it is conveniently done, the predictions of our approach for the single /I decay amplitudes, letting g,, TABLE ( ~‘2”‘lPoh
III
T\f;),,(in lO*Oy)
Ge76
0.040.048
3241
Sea2
0.0384.047
1.0-1.6
T’*‘) l/2.erp(in 10zoy) >3” 1.0 + 0.4h 1.30 * 0.05’ 1.1’;;” lf2b 16.3 + 1.4‘
Note. The 88 decay amplitudes (lirst column) and half-lives (second column) in Ge76, Se8*, and Te”” evaluated in our nuclear matter QRPA are presented. The ranges of the theoretical results correspond to the values 0.42
94
ALBERICO
ET AL.
vary in the range previously found. Also we stress once more that the smallness of the amplitudes evaluated here is a direct consequence of large cancellations between larger contributions of opposite sign. In such a situation it is of relevance to consider other contributions to the process, which were previously expected to be negligible with respect to the simple two-nucleon mechanism. This is the case for the BP decays involving MEC and dd mechanisms, which can be conveniently dealt with in nuclear matter. Calculations on this subject are in progress and will appear in a forthcoming paper.
APPENDIX
In the normal nuclear matter framework we choose an initial state with vanishing energy. More specifically we write
2p- 2h, J” =O+
( i) = 12~ - 2h; 0, 0, Ti, Tz, >
x ( - 1 ) 1 - s*, - Sh2
(~ssp,;-sh,~
k,)
X (~~p2~-~h2)1~,2)(1T2,1~Z,~TjT,,)(-l)’-’h~-rh~ ’
<$p,t-
th,
1 1Z;,)(&,2$-
th2
1 lzz2>
XB(P,-k~)e(P2-k~)e(k~-h,)e(k~-hz)Li~,ri~~B,,d,,
1F),
(A.11
where latin letters refer to particles states, greek letters to particle-hole states, and 1F) is the Fermi sphere. In our convention t,, = tp, = - 4 (neutron states) and tkl = th2= 4 (proton states); thus the total isospin of the initial state is Ti = 2, Tz, = -2. With the same notations the intermediate isovector J” = 1 + state reads: (ph; 10r)=~(-l)1’2-m ph x (- l)WSh
($t,+-t,,)
ltzO)
(;sp$ - sh 1 lt~;,) O(p - kF) O(k, -h) +i,,
I F).
(A.21
One gets then, for the spin part S (the isospin is trivial and yields unity) of the second order matrix element of the Gamow-Teller operator,
8&=
C Ok, I ap I sk)Cb It, I fk) 4X k. k’
(A.3)
2V
X
A straightforward
DOUBLE BETA DECAY
(Fj ii~;Cik,‘+i,, ) F)(Fj
application
d~,8,,d~;8kz8~,ci~,ci,,rih, ) F).
95
(A.41
of Wick’s theorem leads then to the result s=2.
(A.5)
Note that all the states have been normalized to unity. Analogous calculations yield the same outcome for the superfluid system.
ACKNOWLEDGMENTS We wish to acknowledge interesting discussions on /?p decays with Professors T. W. Donnelly, A. Faessler, and A. Morales. This work was supported in part by Research Funds of the Minister0 della Pubblica Istruzione, Italy.
REFERENCES 1. For a general review on the &I decay, see, for instance: W. C. HAXTON AND G. J. STEPHENSON, JR., Prog. Part. Nucl. Phys. 12 (1984), 409; M. DOI, T. KOTANI, AND E. TAKASUGI, Prog. Theor. Phys. Suppl. 83 (1985), 1; J. D. VERGADOS, Phys. Rep. 133 (1986), 1. 2. E. FIORINI, Review talk given at the Eighth Workshop on Grand Unification, Syracuse, April 1987. 3. E. BELLO~I, 0. CREMONESI, E. FIORINI, C. LIGUORI, A. P~LLIA, P. P. SVERZELLATI, AND L. ZANOTTI, Nuovo Cimenro A 95 (1986), 1. 4. D. 0. CALDWELL, R. M. EISBERG, D. M. GRUMM, D. L. HALE, M. S. WITHERELL, F. S. GOIJLDING, D. A. LANDIS, N. W. MADDEN, D. F. MALONE, P. H. PEHL, AND A. R. SMITH, Phys. Rev. D 33 (1986), 2737. 5. F. T. AVIGNONE III, R. L. BRODZINSKI, J. C. EVANS, JR., W. K. HENSLEY, H. S. MILEY, AND J. H. REEVES, Phys. Rev. C34 (1986), 666. 6. 0. K. MANUEL, “Proc. Int. Symp. on Nucl. Beta Decays and Neutrino” (T. KOTANI, .H. EJIRI, AND E. TAKASUGI, Eds.), p. 71, World Scientific, Singapore, 1986. 7. T. KIRSTEN, E. HEUSER, D. KAETHER, J. OEHM, E. F’ERNICKA,AND H. RICHTER, “Proc. Int. Symp. on Nucl. Beta Decays and Neutrino” (T. KOTANI, H. EJIRI, AND E. TAKASUGI, Eds.), p. 81, World Scientific, Singapore, 1986. 8. S. R. ELLIOTT, A. A. HAHN, AND M. K. MOE, Phys. Rev. Len. 59 (1987), 2020. 9. P. VOGEL AND M. R. ZIRNBAUER,Phys. Rev. Letr. 57 (1986), 3148. 10. 0. CIVITARESE, A. FAESSLER, AND T. TOMODA, Phys. Left. 8194 (1987), 11. 11. A different mechanism for suppression of decay rates has been suggested by K. GROTZ AND H. V. KLAPWR, Nucl. Phys. A460 (1986), 395.
595/187/l-7
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ET AL.
12. K. MUTO AND H. V. KLAPDOR, Phys. Left B201 (1988), 420. 13. A. FAESLER, talk given at the 10th Workshop on Particles and Nuclei, Heidelberg, West Germany, October 1987. 14. T. TOMODA AND A. FAESSLER,preprint, June 1987. 15. An evaluation of the (Ova/I) decay rates which is based on values of g,, determined from B+ decays is given by J. ENGEL, P. VOGEL AND M. R. ZIRNBALJER,preprint, June 1987; the theoretical predictions of this paper differ considerably from the conclusion of Ref. [14]. 16. P. VOGEL AND P. FISHER, Phys. Rev. C32 (1985), 1362; L. ZAMICK AND N. AUERBACH, Phys. Rev. C26 (1982), 2185; K. GROTZ, H. V. KLAPDOR, AND J. METZINGER, Phys. Left. B 132 (1983), 22. 17. A. FAZELY AND L. C. Lru, Phys. Rev. L&t. 57 (1986), 968. 18. L. P. GORKOV AND T. K. MELIK-BARKHUDAROV, Sm. Phys. JETP (Engl. Transl.) 13 (1961), 1018. 19. A. BOHR AND B. R. MOTTELSON, “Nuclear Structure,” Vol. II, Benjamin, New York, 1975. 20. A. L. FETTER AND J. D. WALEcKA,“Quantum Theory of Many-Particle Systems,” MC Craw-Hill, New York, 1971. 21. R. A. BROGLIA, A. MOLINARI, AND T. REGGE, Ann. Phys. (N.Y.) 109 (1977), 349.
OF PHYSICS
187, 79-96 (1988)
The 2v Double Beta Decay in Nuclear Matter W. M. ALBERICO,M. B. BARBARO,* A. BOTTINO, AND A. MOLINARI Dipartimento
di Fisica Teorica deN’Uniuersifri 10125 Torino, Italy; and Nazionale di Fisica Nucleare, Sezione Torino, Italy
di Torino,
Istituto
di Torino,
Received January 13, 1988
In the present paper we analyze the suppression mechanism of the (Zvfifl) amplitudes, in the framework of superfluid nuclear matter. We show that the interplay between spin-isospin modes and spin pairing vibrations in quenching the jj amplitude is particularly transparent in the context of the polarization propagator method. Our procedure is applied to finite nuclei of interest and our results are compared to other theoretical evaluations and to the experimental data. 0 1988 Academic Press. Inc.
I. INTRODUCTION
The hope of obtaining, from studies of double beta (BP) decays, through the neutrinoless channel (Ovp/?), reliable information about the neutrino properties (Majorana/Dirac nature, mass) and the strength of a possible V+ A term in the weak-interaction Lagrangian has been deceived to a large extent until very recently, because of serious discrepancies between experimental data of different groups and also on account of some disagreement between experimental results and theoretical lifetime evaluations, already at the level of the “standard” (2$/3) mode [ 11. In fact shell model calculations based on realistic nucleon-nucleon interactions appear to lead to an overestimate of the 2v/3b decay amplitude in medium-heavy nuclei. However, according to the most recent developments, some of the above-mentioned conflicts between experimental results seem now to be substantially resolved [2-S]. Furthermore, an interesting suggestion has been made [9, lo], which might lead to a significant suppression of the (2vfib) decay rate thus improving the agreement with the data. In this connection we recall that the nuclei of interest for the /?/? decay are, in general, suitably described in terms of quasi-particles; accordingly they are conveniently treated in the framework of the quasi-particle random phase approximation (QRPA). The latter, beyond the usual particle-hole (p-h) correla-
* Dottorato di Ricerca dell’Universit8 di Torino.
79 OOO3-4916/88 $7.50 5951187/l-6
Copyright 0 1988 by Academic Press, Inc. All rights of reproduction in any form reserved.
80
ALBERICO
ET AL.
tions, introduces particle-particle (p-p) correlations as well; it so happens that a destructive interference between these two forces arises in the evaluation of the j?fi decay rates [ 111. Although the inclusion of the p-p correlations provides a remarkable suppression mechanism for the (2v/?/I) decay rate, yet one should realize that: (i) The agreement between theory and experiment occurs through a drastic cancellation of large numbers and appears, as a consequence, to critically depend upon the (p-p) interaction strength, g,,, whose range, as determined from 6’ decay, appears to encompass the values at which the (2v/?p) decay rates vanish [12-13 J. Thus, for the (2v/?B) decay lifetimes, the theoretical calculations yield lower bounds, small enough to be compatible with the experimental data, rather than definite predictions. Accordingly, an alternative view could be to exploit the sensitivity of the (2vB/?) amplitudes to the p-p strength in order to fix g,, from the Dp decay experimental data and then to employ this value as an input for the calculation of the (Ov@) nuclear amplitudes [ 14-151. (ii) Since the phenomenon under investigation entails strong cancellations, other effects [such as meson exchange currents (MEC) and the delta-delta mechanism (Ad)], which were commonly believed to represent small contributions to the (2vjIj3) rates, could play a non-negligible role and should then be taken into proper account. These effects might turn out to be particularly relevant in connection with the above-mentioned determination of g,,. Our aim, in the present paper, is to shed further light on the suppression mechanism of the (2$/I) decay rates and to lay the groundwork for the inclusion of the contributions mentioned in (ii). For this purpose nuclear matter offers a convenient framework. In spite of its simple geometry this system, as we shall illustrate in the following, allows us to substantially reproduce the general trend of the results of accurate finite nuclei calculations [S-lo, 121, in both the RPA framework and in the QRPA framework. Therefore we believe nuclear matter to be a testing ground reliable enough for the inclusion of further contributions to the fi/? decay, namely the above-mentioned effects associated, e.g., with the MEC. The latter indeed are more easily dealt with in this system than in finite nuclei. Also the A,, isobar, which specifically affects the spin-isospin channel considered here, is suitably inserted in the RPA framework of nuclear matter. We consider the response of the latter to the Gamow-Teller (GT) operator (the only one we shall take into account in the present work, since it dominates over the Fermi operator) in the scheme of the polarization propagator method. When one introduces the pairing interaction and nuclear matter becomes superfluid, the polarization propagator unfolds two components: one associated with the particle-hole (p-h) and the other with the particle-particle (pp) elementary excitations. The competition between these two terms, especially when the p-h and p-p interactions are switched on, is displayed by our formalism in a very trans-
2~ DOUBLEBETA
81
DECAY
parent way and accounts for the reduction of the fib decay amplitude through a cancellation occurring between the spin-isospin (Gamow-Teller) and what may be defined as a spin pairing vibration mode. Furthermore this approach appears particularly flexibile in order to follow the evolution of the process under investigation both as a function of the momentum associated with the first order transition leading to the intermediate nucleus and of the temperature. This may bear some relevance for double charge exchange reactions, particularly those sensitive to the spin-isospin degrees of freedom, as well as for heavy ion physics, where the temperature plays a decisive role. At zero temperature and zero momentum the total diamagnetism of the noninteracting nuclear matter entails the vanishing of the fib decay amplitude. The interaction shifts this diamagnetism at a finite q = 4 (when T= 0) or at a finite T= T (when q = 0). This can be viewed as a generalization of the Meissner effect to include the isospin degrees of freedom as well as the spin degrees of freedom. For temperatures different from T (at zero momentum) the amplitude for ,6p decay is no longer vanishing, but is still quite depressed, with respect to an independent particle model prediction, by pairing effects and by long range correlations, suitably described in QRPA. The same situation occurs for non-zero momentum (at T= 0), but now, as already mentioned, it is the interplay between the p-h and p-p channels (in other words, between the spinisospin mode and the spin pairing vibration), which yields large cancellations in the j3p amplitude. We illustrate these ideas in the present paper, which is organized as follows: in Section II we recall the general expression for the (2vbp) decay half-life and find its relationship to the spin-isospin polarization propagator for normal nuclear matter. In Section III we introduce spin pairing correlations and consider the polarization propagator for the superfluid system; the flp decay amplitudes are then obtained in both the free case and in the QRPA framework. Finally in Section IV we present the results of our calculations and discuss them in connection with previous work.
II. GENERAL FORMALISM Let us start from the general expression of the (2v@?) decay half-life for the o++o+ transitions; following the notations of Doi et al. [l] it reads (11.1) where
Came CO,+IICn~n+~rrI/N,(l+))(N,(l+)IlC,t,+a,II Eo - (lP)(Mi + M,) m, being the electron mass, E, the intermediate
0:)
*, tII2j
state energy, Mi (M,) the mass of
82
ALBERICO
ET AL.
the initial (final) nucleus, t + the isospin raising operator and the sum C, referring to the nucleons in the nucleus. Finally Go= is a phase-space factor defined and tabulated in Doi et al. [I]. For a free Fermi gas, Eq. (11.2) can be factorized into a spin part and a dynamical one as follows:
Before evaluating both factors we remark that symmetric nuclear matter may be thought of (in the phase space) as two Fermi spheres, one for neutrons and one for protons, with the same radius k,. Accordingly, we conceive our initial state as made up by two J” = I+ neutron-particle, proton-hole excitations coupled to J” = O+, lying very close to the Fermi surface. Once the first weak transition has taken place, the intermediate state, in a Fermi gas scheme, is an isovector J” = 1 + neutron-particle, proton-hole state (lp - lh). Then (see Appendix) one gets, for the spin factor,
Moreover, neglecting the energy difference between the initial and final systems, the following expression for the dynamical factor d(q) holds, W,lhl)~(lpl -k’)f(q2)s d(q) = c p,h tl=p2/2hf- ?w/2M 52 -2
=--
Re
n”(q,
0
=
_ q>P
b
O)f(q2),
0
where M is the nucleon mass, Q is the volume enclosing the system, andf(q2) axial-vector nucleon form factor,
is the
with m, = 0.85 GeV. Formula (11.5) provides an interesting connection between the /3/i decay amplitude and the real part of the (static) free particle-hole polarization propagator at zero temperature, no. This is in line with the well-known relationship between the various susceptibilities of atomic nuclei and Re P '(4, 0) in the appropriate spin-isospin channels. Accordingly, in our model, the /$!l decay amplitude can be viewed as the paramagnetic susceptibility of nuclear matter (both in its normal and superconducting states). We notice that there is no explicit momentum dependence in the GT operator of formula (11.2); actually the momentum associated with the p-h excitation should be related to the first weak transition leading to the (virtual) intermediate isovector 1+
2V DOUBLE
BETA
DECAY
83
state; we shall see later that the interesting values of q range between 0.04 and 0.05 fm-i. However, the momentum dependence (at zero frequency) of the free polarization propagator 17’ is rather smooth for values of q up to 1 fm-‘. The expression (11.5) for the fl/? amplitude refers to the crude description of a non-interacting Fermi gas. In order to treat nuclear matter more realistically, we should dress with correlations the real part of the static particle-hole polarization propagator n(q, 0). To do this we remind the reader that RPA correlations, induced by the strong spin-isospin force, are essential in accounting, e.g., for the measured GT strength in nuclei. The corresponding RPA polarization propagator obeys the standard Dyson’s equation nRPA(q) = no(q) + no(q) J$,(q) ~RPAtqh
(11.6)
which, in nuclear matter and neglecting the exchange matrix elements of the p-h force, is algebraic and yields IIRPA( q) =
Accordingly,
no(q) 1 - V,,(q) no(q)’
(11.7)
one gets for the j?p amplitude, dRPA(q)
= -
T
Re URPA(q, O)f(q2)
0
(11.8) got% 0) f (q2) 1 - f$dq) flO(% 0)
[notice that for zero frequency and finite momentum no(q) is a real quantity]. For the p-h interaction in the spin-isospin channel, we include here only a short range repulsive component which, according to the Landau-Migdal parametrization, reads (11.9) where the factor 4 corresponds to the ph matrix elements of the (a, . (r2)(r1 . r2) operators, rz(q2) is the nNN form factor and the Landau-Migdal parameter g’ is given in pionic units. Indeed, in the cz channel, the rr- and p-meson exchange forces (V, and V,) should be included as well, but their direct matrix elements are negligible at small q. We also remind the reader that the parameter g’ approximately accounts (in the limit q -+ 0) for the exchange ph matrix elements of V, and I’,. Notice that, owing to the repulsive nature of the ph force, the denominator in (11.8) is larger than unity, thus reducing the free amplitude: the nuclear forces oppose the flipping of the nucleonic spins.
84
ALBERICOETAL.
Thus far we have only considered intermediate nucleon-particle nucleon-hole states, but the GT operator, responsible for b* decay, can also induce the transition from the initial state to a A,,-particle, nucleon-hole state. Obviously such a state would be far off-shell (by about 300 MeV) and for this reason it has been frequently neglected. Moreover, although the single B decay matrix elements leading to 1AN-‘) states are different from zero, the B/I matrix elements with a direct A,,hole contribution vanish [ 161. However, the RPA correlations in the spin-isospin channel are significantly altered when the A,,-hole polarization propagator, Z7A, is taken into account as well. This is easily done in nuclear matter, by replacing Z7’ in the denominator of (11.8) with the sum of the free NN-’ and AN-’ propagators: aiRPA(
-
t
Re
0
n,cs, 0) fW). i 1- ~,lmC~N(cl~ 0) + fl‘A% 011
(11.10)
Notice that ZZd does not appear in the numerator since it would imply a direct AN-’ contribution to the /@I amplitude. In concluding this section we remind the reader that the A,, can offer additional mechanisms for the O+ + O’pg decay, providing one takes into account two-body processes with intermediate A,, states [AA mechanism] [17]. This item, however, goes beyond the present RPA treatment. III.
PAIRING CORRELATIONS
It has been pointed out by several authors [9-121 that pairing correlations tend to reduce the BP decay amplitude, since it is more difficult for the single intermediate decay to occur, breaking a pair of correlated nucleons. This situation, where particles with opposite spins, namely (Nr, N 1 ), are paired by the attractive interaction Vpair, is described by replacing the (unstable) Fermi sea with the BCS ground state through the well-known canonical transformation to quasi-particle operators, which diagonalizes the pairing interaction. The latter modifies the singleparticle spectrum and occupation probability according to the substitutions WEk=&TZ
(111.1) (111.2)
where ,u~ is the chemical potential of nuclear matter and A the energy gap. Now, although real nuclei are at zero temperature in the ground state (recall, however, that considerably high temperatures may be reached in relativistic heavy ion reactions) it is convenient here to treat the superfluid nuclear matter within the temperature Green’s function formalism.
2~ DOUBLE
BETA
85
DECAY
We have then solved the following gap equation 1 = g&J(O) j,“‘“” J&tanh(‘y),
(111.3)
where fi = l/k,T, T being the absolute temperature, ho,, = 0.49 pF [18], N(0) z A4k,/2n2h2 is the density of states at the Fermi surface, and g, = 120.4 MeV fm3, the strength of the pairing force Vpair. This value of g, corresponds to the parameter G = 28.84/A MeV frequently used in finite nuclei c191. The above equation allows the determination of the gap as a function of T; it stays almost constant for low temperatures and then drops to zero at the critical temperature T,, where the transition from the superfluid to the normal phase occurs. For nuclear matter (k, = 1.36 fm-‘) Tc turns out to be close to 0.8 MeV, if a nucleon effective mass larger than the free one (M,, = 1.15 M) is used. Indeed, at very low q values both the particles and the holes are close to the Fermi surface, where any realistic energy dependent mean field yields an increased nucleon mass, due to the coupling of the single particle motion with the nuclear surface vibrations. Let us then introduce the free quasi-particle-quasi-hole temperature polarization propagator [20] no,&% a) = @$q,
(111.4)
0) + ~~,“(q, 01,
where
X
Ly(q,w)=~
2m5 X
1
1
(III.Sa)
ho+E+-Ep+iq-fiw-E++E-+iq 4
--o+d2
(U+K
Cl-f(E+)-f(Ep)l
1 r%zo--E+-Ep+iv]-ko+E++E_+iq
1 >’
(IILSb)
In the above formulas the symbol + implies a wavenumber (p f q/2),
and f(E) is the modified Fermi function
S(E) = -1 :P
(111.7)
86
ALBERICO
For a better understanding
ET AL.
of (111.4) it is worth noticing that for temperatures
T> T,, where the gap vanishes, one has
24:=a B(k - kF)
(III.8a)
v; * 8(k, -k);
(III.8b)
this helps in recognizing particle-hole or particle-particle contributions. In particular, terms proportional to U: ~5, v: ~2, or UC v: stem from the convolution of two single particle temperature Green’s functions-
4a(x, x’) = - (0 I T C$&)
(111.9)
&AT(x’)llO>
whereas terms with the product (u, U_ u, v- ) stem from particle number nonconserving Green’s functions, 8 or St, which are specific for the superfluid phase and are defined as follows:
Ejdx> x’ ) = - (0 I T C$&) Thus the typical combination quasi-particle (or quasi-hole)
(111.10)
&qW)l lo>.
RF+ entering into (111.4) corresponds to a two Green’s function, which obviously vanishes for
T> Tc.
Let us now consider the behaviour of (111.4) at zero frequency (as in the previous section) both at 1q ) = 0 as a function of the temperature and at T= 0 as a function of momentum. For zero momentum the quantities indicated with suffixes + or coincide (e.g., U, = U- = u,), hence E$j’(q, 0) identically vanishes at any temperature. The function n!$(q, 0), instead, can be recast in the following form: l7~c”(q+O,o=O;
(111.11)
T)=,&%$
At zero temperature the expression (III.1 1) vanishes, thus entailing the total diamagnetism of the superfluid matter, and then it slowly increases with the temperature; above T,, it merges into the normal temperature-dependent particle-hole polarization propagator in the q -0 limit. Notice that the factor (u; + vi)*, not explicitly written in (III.1 1) since it is unity, still contains a p-h and a p-p contribution (the latter being associated with the double product). For the purposes of the present paper, the limit T + 0, q finite, is more interesting. In this case, as already pointed out, J7!$‘(q, o = 0; T= 0) = 0, whereas I7”,;(q,
which is conveniently
w=O;
T=O)=
-
-u-v+)’ sdp(u+v(2n)3
E,+E-
split into a p-h and a p-p contribution,
Zi’&Jh(q, w = 0; T= 0) = -
’
(111.12)
according to
dp cu:v2 +u?v$) s (2,rr)3 E++Ep
(111.13a)
87
2V DOUBLE BETA DECAY
and
Izos’cpp(q,o=o; T=0)=2@&,v-;~~+.
(111.13b)
+
Figure 1 illustrates the behaviours of both Z70S,Cph and @,pp as well as their sum as functions of q [actually they are multiplied by - (Q/2)f(q2) in order to obtain the corresponding /?b amplitudes]. The amplitude for a normal fluid [Eq. (IIS)] is also reported, for comparison. One sees that the p-h and p-p contributions exactly cancel at q = 0, but then the p-h term prevails and approaches the free normal amplitude, although it is always smaller than the latter as a result of the pairing correlations. At this stage we should explore the most interesting effect of RPA correlations in the superfluid system. As suggested by several authors [9-121 this may be achieved in the quasi-particle random phase approximation. In principle, the corresponding equation for the polarization propagator (111.12) would mix the p-h and p-p channels; however, it has been shown [21] that if the gap d is much smaller than the Fermi energy and the density of the energy levels close to the Fermi surface is
-+%)/A 0.000
I“’ r I” n I I8 ’ I’1 _....._________....................................................................,.....~
0.004
_---
__--
__--
_----
--
_---
-_--
_----
--
_---
0.002
0.000
-0.002
_--0
_--QII tt 1 1 II II I 1 I I 1 I 1 1 I I -. -. 0.05
0.1
0.15
0.2
rm-’ FIG. 1. Behaviour of ZZ& (real part of) versus 4 (continuous line), together with its J75j$‘” and l7$‘r components (dashed lines). The curves correspond to A = 1.48 MeV, the value obtained by solving Eq. (111.3) with the strength of the force, g,, and the cutoff, tiw,, given in the text. The density of states at the Fermi surface is evaluated with an enhanced nucleon effective mass M,s= 1.15 h4. Note how f7& results from opposite contributions. Re ITo for a normal Fermi gas is also displayed (dotted line).
88
ALBERICO
ET AL.
constant (which is the case in our model), the QRPA equations for the p-h and the p-p polarization propagators decouple as follows:
In the above
ng;A’ph(q)
= n$;7Ph(q) + h?&“(q)
v,,(q) n,R,‘A’ph(q)
(111.14)
z7p.yq)
= @iy(q)
V,,(q)
(111.15)
eqUatiOnS
Vph
+ @pyq)
is given by (11.9) and VP, is written in the form
V,,(q) = -22 where matrix g,, in The QRPA
ITip-yq).
P?I
(111.16)
gppW2),
g,, is the strength of the particle-particle interaction and the factor - 2 is the element of the (a, .c2)(t, . r2) operators. We will comment on the value of the next section. solution of Eqs. (111.14) and (111.15) is straightforward and yields, for the j?fi amplitude, J&p(q)
= dig-yq)
(111.17)
+ dg-yq),
where dg-(q
)=-
0
5
Re
Ggh(q, 0) f (q2) i 1 - v,,(q) @,ph(q, O)
(III. 18a)
y
Re
@p(q, 0) 1 V,,(q) n$yq, i
(IIL18b)
and &;:A*
pp(q) = 0
0) “Oq2).
The behaviour of (111.17) as a function of q obviously depends on the precise values of g’ and g,, ; as a general remark we observe that, since both g’ and g,, are positive quantities, one has Vph > 0 and Vpp < 0 [see Eqs. (11.9) and (111.16)]. In turn, Zjzh ~0 whereas Z7‘&‘P > 0 so that the RPA correlations produce a quenching of the corresponding amplitudes in both channels. However, for values of g,, N g’, the Vpp is weaker than VP,, by a factor of 2; hence the RPA pp amplitude is less reduced than the corresponding ph amplitude and the total cancellation of the two components, which occurs at q = 0 in the free case, is displaced toward finite values of the momentum (which are the ones of interest for finite nuclei), as we shall see in the next section. For the sake of illustration, the QRPA amplitudes (111.18) are displayed, together with their sum, in Fig. 2, for g,, =g’ (the physical values for these parameters will be discussed later ).
89
2~ DOUBLEBETADECAY
1~‘~~1~1”1~“” 0.002 -____.....________.__.......................,....,,,,...,,,,,,..,.,.,.,....,,~,,,....~,,.,. _----____-----___----0.001 7 g
0.000
-0.001 __--0.002 --
_---
_---
_---
__--
-
tl~~~l,,,~l,“‘l”‘~0 0.05
9 0.1
0.15
_ 0.2
fm-’ FIG. 2. The same as in Fig. 1, but with QRPA correlations present. The p-h and p-p interaction strengths are g’ = gpp = 0.7.
IV. RESULTS AND CONCLUSIONS In this section we present the results of our calculations, both for the normal and for the superfluid system. Although the present approach refers to infinite nuclear matter, we shall nevertheless apply it to a few nuclei, which are of particular interest for /?/I decays from the experimental point of view, namely Ge76, Se’*, Te13’, and Xe’36. Accordingly, our method is expected to yield the general trend of the amplitude versus the mass number A, but it is certainly not expected to account for specific features related to a given nucleus. In fact, in our model nuclear structure enters into the expression for the amplitude through the following three characteristic quantities: the Fermi momentum k,, the momentum q at which the amplitude is evaluated, and the mass number A, which is used to replace the nuclear volume Q through the customary relationship 52 = A/pF, OF being the average nuclear density. The latter is taken as the mean value of the Fermi distribution
PO(A) PF(r’ A)= 1 +exp((r-R(A))/a) with A po(A)= (47cR3/3)[1 + (uTc/R(A))‘]
(IV.2)
90
ALBERICO
ET AL.
In the above a=0.54fm is the surface thickness and R(A) the nuclear radius R, given by the usual expression [19] R=
(IV.3)
For the Fermi momentum (which in nuclear matter is 1.36 fm-‘), we utilize the average value, depending upon A, which can be deduced from PF itself: F,(A)
= [T
(IV.4)
p,~]“~.
We have then extracted the characteristic values of q for the different nuclei considered here from the corresponding energy differences of their typical single weak transitions; we have found q values ranging between 0.04 and 0.06 fm-‘. Let us first consider the normal (i.e., non-superfluid) system: we remind the reader that for q, say up to 1 fm-‘, the static polarization propagator (both in the free and in the RPA cases) has a very mild momentum dependence. Therefore the differences between the various nuclei essentially reflect, in this instance, the different sizes and kF values. In Table I we report the results of our calculation for 1M&)/p,/ for a noninteracting system (first row) and for the RPA correlated matter, both without (second row) and with (third row) the inclusion of the A,,-hole polarization propagator. For the Landau-Migdal parameter g’ we employ here the value 0.7, which in the RPA framework accounts fairly well for the giant GT resonance measured by (p, n) experiments. One clearly sees that the RPA correlations significantly reduce the free amplitudes, typically by a factor of 3; the inclusion of the d,, entails an additional 10% reduction. However the values of Table I do not match the available experimental data on Se** and Te13’, which require (see Table III, for reference) much smaller B/? decay amplitudes. Let us then consider the superfluid matter. By looking at Fig. 1, one realizes that TABLE Nucleus
I
Ge76
Se*?
Te”O
Xe’36
4 (fm-‘1
0.048
0.051
0.067
0.052
Free RPA RPA
0.568 0.203 0.179
0.609 0.217 0.191
0.927 0.325 0.285
0.966 0.339 0.297
(A331
Nore. Values of 1Mg;;Zp,,I. The RPA calculation is reported both without (second row) and with (third row) the inclusion of d-hole intermediate states; the q values utilized for the different nuclei are also reported here.
2V DOUBLE
BETA
DECAY
91
in the range of the q values pertinent to the /?/I decays considered in the present paper (see Table I) the pairing correlations sizably reduce the p/I amplitude, already in the absence of RPA correlations. However, as was pointed out in Section III, an even more pronounced quenching can be obtained within the QRPA framework. In order to evaluate (111.17) one would need, as an input, the value of the p-p interaction strength, g,, . In principle, the latter should stem from a realistic NN force and indeed a microscopic G-matrix calculation of g,,, to be utilized in a shell model framework, has been carried out (see Haxton and Stephenson in [ 1 ] ). Here, however, we are utilizing effective residual interactions, which need to be fixed on the basis of some experimental constraints. Indeed the value of the p-h strength g’ has been determined from the giant GT resonance in a fairly unambiguous way; similarly the p-p strength can be extracted from the available data on fl’ decays in several nuclei [9, 121, but the resulting uncertainties on g,, are too large to allow precise predictions on the 2vb/? amplitudes. In view of this fact, we adopt here the heuristic attitude of considering g,, as a free parameter. Our 2vpfl amplitudes, evaluated according to (111.17), for the four nuclei under consideration, are displayed in Fig. 3 as a function of g,, (we remind the reader that an additional parameter characterizes, here, the different nuclei, namely the energy gap, A; instead of solving Eq. (111.3) we use, for finite systems, the value 12/fi MeV [19]). Also shown in the figure are the ranges of the experimental values, corresponding to the recent measurements of /?/I decay in Ses2
FIG. 3. The 2$/3 amplitudes versus the strength of the particle-particle interaction g,, are shown for Ge76 (dashed line), Ses2 (dotted line), TelW (dotdashed line), and Xe’36 (continuous line). They have been obtained by adapting the nuclear matter amplitude to suit the various nuclei according to the simple prescription given in the text. Also displayed (by vertical bars) are the experimental results for Seg2 and Tei3’ reported in Table III.
92
ALBERICO ET AL.
and Te13’ [6-81; f or both nuclei there exists a range of g,, values (0.42 6 g,, < 0.51) which render our theory compatible with the experiments. It is remarkable that the general trend of our p/3 amplitude as a function of g, agrees with the calculations of Refs. [9, 10, 123; the different slope has to be essentially ascribed to the differences in the Fermi surface of a nucleus with respect to the Fermi surface of nuclear matter. Moreover the interplay between p-p and p-h components of the amplitude, which is most apparent in the framework of the polarization propagator, seems to act somewhat differently within the QRPA equations for the nuclear wavefunctions. In order to compare our results with previous ones [9, 10, 121 without reference to this cancellation mechanism induced by the p-p interaction, we report in Table II the results of Refs. [9, 10, 121 at g,,,=O and our evaluation of the /3/I RPA amplitude in the superfluid system omitting the influence of the p-p channel. To achieve this goal, we either set g,, = 0 or wash out completely the p-p contribution (namely we set 17$2” = 0). Indeed it is not clear whether the g,, = 0 calculations of Refs. [9, 10, 121 include the p-p contribution at the free level. Indeed this second instance seems to provide results which are in closer agreement with the ones of Refs. [9, 10, 121. It is thus justified to utilize our superfluid matter approach for an evaluation of the /I/l decay amplitudes and half-lives in Ge76, Ses2, and Te13’. Both spin-isospin and pairing correlations are included, by utilizing the full RPA p-h and p-p polarization propagators [ Eqs. (III. 17) and (III. 18)]. One should recall that we express both g’ and g,, in pionic units, at variance TABLE Nucleus H-S C-F-T (g,,=O) M-K (g,,=O) Normal RPA (g’=O.7) Superfluid RPA (g,,=O) Superfluid RPA Ep = 0
II
Ge76
Se*?
Te”O
Xe’36
0.139
0.095
0.114
0.330
0.270
0.205
0.345
0.292
0.173
0.203
0.217
0.325
0.339
0.098
0.098
0.082
0.123
0.156
0.169
0.268
0.272
Note. Values of 1M,$)/&(, The first three rows are extracted from Haxton and Stephenson (H-S) [l], Civitarese, Faessler, and Tomoda (C-F-T) [lo], and Muto and Klapdor (M-K) [12], respectively. In the fourth row our RPA value for normal matter is shown, whereas in the last two rows the RPA evaluations in the superfluid matter are given, once with g,,=O only and once without the whole p-p contribution.
2V
93
DOUBLE BETA DECAY
with previous works [9, 10, 123; in the latter the commonly utilized value of g’ is close to unity. Thus, to facilitate the comparison, we point out that, at fixed g’ = 0.7, we use values of the ratio g,,/g’ in the range 0.6 < g,,/g’ < 0.73, which are rather close to the ones recommended by other authors [ 121. The results of our calculation are presented in Table III, where the pfl decay amplitudes (first column) and half-lives (second column) in Ge76, SeE2, and Te13’ are compared with the experiments. The theoretical values correspond to the abovediscussed range of gpp. One sees that the present model yields the same reduction of the /Ij decay amplitude as given by the calculations in finite nuclei of Refs. [lo] and [ 121. In particular we remark that the reduction of the decay amplitude in going from Se*’ to Te13’ is induced, in our framework, mostly by the q-dependence occurring in the QRPA calculation. Indeed, as shown in Fig. 2, the cancellation between the p-h and the p-p components of the amplitude is quite sensitive to q. In addition the A-dependence entailed by our approach leads to a more rapid variation of the amplitude with g,, in heavier nuclei. It is worth pointing out that our approach displays the mechanism for the reduction of the 2vpp decay amplitudes in a quite transparent way. Indeed, in our nuclear matter framework, the /I/? process involves only the neighborhood of the Fermi surface. Here the nuclear matter displays a dual spectrum: fermionic and bosonic at the same time. As often occurs in nature, these two aspects of the same physical reality tend to oppose each other: in this instance the contrast is particularly evident owing to the structure of the second order formula expressing the Bfi decay amplitude. This indeed allows the two modes of GT (spin-isospin) and spin pairing vibrations to compete in the intermediate state. In conclusion we remark that a reliable theory should be able to encompass both the single and the double fi decay processes. Accordingly we are at present exploring, in the context of asymmetric nuclear matter, where it is conveniently done, the predictions of our approach for the single /I decay amplitudes, letting g,, TABLE ( ~‘2”‘lPoh
III
T\f;),,(in lO*Oy)
Ge76
0.040.048
3241
Sea2
0.0384.047
1.0-1.6
T’*‘) l/2.erp(in 10zoy) >3” 1.0 + 0.4h 1.30 * 0.05’ 1.1’;;” lf2b 16.3 + 1.4‘
Note. The 88 decay amplitudes (lirst column) and half-lives (second column) in Ge76, Se8*, and Te”” evaluated in our nuclear matter QRPA are presented. The ranges of the theoretical results correspond to the values 0.42
94
ALBERICO
ET AL.
vary in the range previously found. Also we stress once more that the smallness of the amplitudes evaluated here is a direct consequence of large cancellations between larger contributions of opposite sign. In such a situation it is of relevance to consider other contributions to the process, which were previously expected to be negligible with respect to the simple two-nucleon mechanism. This is the case for the BP decays involving MEC and dd mechanisms, which can be conveniently dealt with in nuclear matter. Calculations on this subject are in progress and will appear in a forthcoming paper.
APPENDIX
In the normal nuclear matter framework we choose an initial state with vanishing energy. More specifically we write
2p- 2h, J” =O+
( i) = 12~ - 2h; 0, 0, Ti, Tz, >
x ( - 1 ) 1 - s*, - Sh2
(~ssp,;-sh,~
k,)
X (~~p2~-~h2)1~,2)(1T2,1~Z,~TjT,,)(-l)’-’h~-rh~ ’
<$p,t-
th,
1 1Z;,)(&,2$-
th2
1 lzz2>
XB(P,-k~)e(P2-k~)e(k~-h,)e(k~-hz)Li~,ri~~B,,d,,
1F),
(A.11
where latin letters refer to particles states, greek letters to particle-hole states, and 1F) is the Fermi sphere. In our convention t,, = tp, = - 4 (neutron states) and tkl = th2= 4 (proton states); thus the total isospin of the initial state is Ti = 2, Tz, = -2. With the same notations the intermediate isovector J” = 1 + state reads: (ph; 10r)=~(-l)1’2-m ph x (- l)WSh
($t,+-t,,)
ltzO)
(;sp$ - sh 1 lt~;,) O(p - kF) O(k, -h) +i,,
I F).
(A.21
One gets then, for the spin part S (the isospin is trivial and yields unity) of the second order matrix element of the Gamow-Teller operator,
8&=
C Ok, I ap I sk)Cb It, I fk) 4X k. k’
(A.3)
2V
X
A straightforward
DOUBLE BETA DECAY
(Fj ii~;Cik,‘+i,, ) F)(Fj
application
d~,8,,d~;8kz8~,ci~,ci,,rih, ) F).
95
(A.41
of Wick’s theorem leads then to the result s=2.
(A.5)
Note that all the states have been normalized to unity. Analogous calculations yield the same outcome for the superfluid system.
ACKNOWLEDGMENTS We wish to acknowledge interesting discussions on /?p decays with Professors T. W. Donnelly, A. Faessler, and A. Morales. This work was supported in part by Research Funds of the Minister0 della Pubblica Istruzione, Italy.
REFERENCES 1. For a general review on the &I decay, see, for instance: W. C. HAXTON AND G. J. STEPHENSON, JR., Prog. Part. Nucl. Phys. 12 (1984), 409; M. DOI, T. KOTANI, AND E. TAKASUGI, Prog. Theor. Phys. Suppl. 83 (1985), 1; J. D. VERGADOS, Phys. Rep. 133 (1986), 1. 2. E. FIORINI, Review talk given at the Eighth Workshop on Grand Unification, Syracuse, April 1987. 3. E. BELLO~I, 0. CREMONESI, E. FIORINI, C. LIGUORI, A. P~LLIA, P. P. SVERZELLATI, AND L. ZANOTTI, Nuovo Cimenro A 95 (1986), 1. 4. D. 0. CALDWELL, R. M. EISBERG, D. M. GRUMM, D. L. HALE, M. S. WITHERELL, F. S. GOIJLDING, D. A. LANDIS, N. W. MADDEN, D. F. MALONE, P. H. PEHL, AND A. R. SMITH, Phys. Rev. D 33 (1986), 2737. 5. F. T. AVIGNONE III, R. L. BRODZINSKI, J. C. EVANS, JR., W. K. HENSLEY, H. S. MILEY, AND J. H. REEVES, Phys. Rev. C34 (1986), 666. 6. 0. K. MANUEL, “Proc. Int. Symp. on Nucl. Beta Decays and Neutrino” (T. KOTANI, .H. EJIRI, AND E. TAKASUGI, Eds.), p. 71, World Scientific, Singapore, 1986. 7. T. KIRSTEN, E. HEUSER, D. KAETHER, J. OEHM, E. F’ERNICKA,AND H. RICHTER, “Proc. Int. Symp. on Nucl. Beta Decays and Neutrino” (T. KOTANI, H. EJIRI, AND E. TAKASUGI, Eds.), p. 81, World Scientific, Singapore, 1986. 8. S. R. ELLIOTT, A. A. HAHN, AND M. K. MOE, Phys. Rev. Len. 59 (1987), 2020. 9. P. VOGEL AND M. R. ZIRNBAUER,Phys. Rev. Letr. 57 (1986), 3148. 10. 0. CIVITARESE, A. FAESSLER, AND T. TOMODA, Phys. Left. 8194 (1987), 11. 11. A different mechanism for suppression of decay rates has been suggested by K. GROTZ AND H. V. KLAPWR, Nucl. Phys. A460 (1986), 395.
595/187/l-7
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ALBERICO
ET AL.
12. K. MUTO AND H. V. KLAPDOR, Phys. Left B201 (1988), 420. 13. A. FAESLER, talk given at the 10th Workshop on Particles and Nuclei, Heidelberg, West Germany, October 1987. 14. T. TOMODA AND A. FAESSLER,preprint, June 1987. 15. An evaluation of the (Ova/I) decay rates which is based on values of g,, determined from B+ decays is given by J. ENGEL, P. VOGEL AND M. R. ZIRNBALJER,preprint, June 1987; the theoretical predictions of this paper differ considerably from the conclusion of Ref. [14]. 16. P. VOGEL AND P. FISHER, Phys. Rev. C32 (1985), 1362; L. ZAMICK AND N. AUERBACH, Phys. Rev. C26 (1982), 2185; K. GROTZ, H. V. KLAPDOR, AND J. METZINGER, Phys. Left. B 132 (1983), 22. 17. A. FAZELY AND L. C. Lru, Phys. Rev. L&t. 57 (1986), 968. 18. L. P. GORKOV AND T. K. MELIK-BARKHUDAROV, Sm. Phys. JETP (Engl. Transl.) 13 (1961), 1018. 19. A. BOHR AND B. R. MOTTELSON, “Nuclear Structure,” Vol. II, Benjamin, New York, 1975. 20. A. L. FETTER AND J. D. WALEcKA,“Quantum Theory of Many-Particle Systems,” MC Craw-Hill, New York, 1971. 21. R. A. BROGLIA, A. MOLINARI, AND T. REGGE, Ann. Phys. (N.Y.) 109 (1977), 349.