- Email: [email protected]
On total muon capture rates and the average neutrino energy
1.D.I:
[
Nuclear Physics A201 (1973) 41 --48; (~) North-HollandPublishing Co., Amsterdam
I.E.4
[
Not to be reproduced by photoprint or microfilm without written permission from the publisher
ON T O T A L M U O N CAPTURE RATES AND T H E AVERAGE N E U T R I N O ENERGY J. B E R N A B E U
CERN, Geneva Received 28 September 1972 Abstract: A m e t h o d o f avoiding t h e u n c e r t a i n t y associated with the average n e u t r i n o energy v in the u s u a l closure a p p r o a c h to m u o n capture rates is discussed. Instead o f neglecting the kinematic dependence o n each particular c h a n n e l , the partial capture rate is a p p r o x i m a t e d by a first-order expansion a r o u n d v. A f t e r the s u m over the final states is p e r f o r m e d , the result is quite independent o f the specific value o f v. Application to 3He, 6Li a n d closed-shell nuclei is given, as an effective test o f the nuclear m o d e l s used.
1. Introduction
The experimental material accumulated on total muon capture rates in nuclei is rather abundant - see, for example, the results listed by Eckhause et al. 1). The original motivation for this study was to establish the basic assumptions of the theory of semi-leptonic weak interactions. This tool has been disregarded due to the uncertainty in the nuclear physics aspects, and the possibility of inferring some quantitative information on the weak form factors is restricted to experiments on hydrogen, deuterium and other selected partial transitions 2). The theory of total muon capture in nuclei has been developed especially by Primakoff 3) and by Luyten, Rood and Tolhoek 4). In the closure approximation, the rate can be written in terms of the expected values of two-body operators on the initial state, and one can think of using these processes as an effective tool for studying nuclear structure. However, one has to introduce in the closure approximation an average neutrino energy v and the result is very sensitive to the choice of v. The uncertainty on this parameter is so important that conclusions for the adequacy of the nuclear model used cannot be reached. Recently, Do Dang 5) has indicated the possibility of avoiding this difficulty by retaining in the dipole contribution of the matrix element a first-order expansion around the parameter v. This is applied after the factorization of the elastic form factor, and then the Yah dependence in each channel is just Yah, 4 corresponding to the dipole term in the limit Vab --, 0. In this paper it will be shown that the expansion of the effective Yah dependence - that is, for all possible multipoles and powers - is equivalent to a first-order expansion of the capture rate around v. Under this last form, the result is easily worked out to obtain independence of the specific value of v, within a large range of plausible values. 41
42
J. B E R N A B E U
In sect. 2 our method is clearly stated. This leads us to consider the study of two new terms besides the usual one appearing from the closure approximation. It is then pointed out how one of them can be calculated immediately from the "closure term", and the other one from a generalized Thomas-Reiche-Kuhn sum rule 6). In our case, this sum rule is expected to hold for nuclei in which spin-dependent forces can be ignored, and thus the application to closed-shell nuclei is given in sect. 3. In sect. 4, the cases of 3He and 6Li are studied, where L-S coupling is rather good. Finally, sect. 5 gives some discussion of the results obtained and our conclusions.
2. Theory In the usual notation the m u o n capture rate to a particular final state a --* b can be written in the form A(a --, b) = [G2v+2G2+(GA-Gp) 2] m 2 ( m ° 0 3 Ze'*ffAr(a ~ b),
(1)
2re 2
where the reduced capture rate A r is given in the impulse approximation by
J
+ B l ( b l ~ z}-eiV°b'xJaxjla)i2+Cl(b[ ~ z;e'V~b'X~a.~la)12}, J
(2)
J
and the sum and average over the final and initial polarizations are understood. The coefficients A, B and C can be written in terms of the effective coupling constants Gv, G A and Gp, and their values are 7) A = 0.19, B = 0.69, C = 0.12. The well-known results in the closure approximation is obtained by summing over the final states b, replacing Yah by an average value v. In terms of the correlation functions Ds, r, L introduced in ref. 7), the total capture rate is obtained as Ar(~ )
=
{A [-1 -- Z - 'Ds(v)] + B[1 - Z - 'D,r(V)] + C[-I - Z - 1Dt.(v)']},
(3)
where the dependence of A, B and C on v can be neglected. We turn our attention to eq. (2). Unlike the procedure of obtaining (3), the dependence on vau, i.e., on each particular channel, is not completely neglected. In terms of a parameter v we can write the identity yah - v[l +(V~b--V/V )] and the expansion of the nth power of V.b is V~b~
I+--[Vab--V
.
(4)
v
This is analogous to the idea used in ref. 5), because v.b = m - ( E b - E ~) t, for the dependence of the unretarded dipole term, after the approximation of factorizing the t In m u o n capture we have the transition (A, Z ) ~ (A, Z - - l ) a n d t h e n the C o u l o m b energy difference Ec m u s t be included. This correction, m --+ m + E c , m a y be i m p o r t a n t for h e a v y nuclei. W e acknowledge T. Ericson for this remark.
TOTAL MUON CAPTURE RATES
43
elastic charge form factor; in this way, eq. (4) was used in the particular case of n = 4. We are going to apply (4) for each contribution of (2) when the exponential e ivab- x is developed in powers of V~b. It is obvious, and it can be proved straightforwardly, that this expansion is equivalent to approximating eq. (2) by the following form A r(a --* b) ~ Ar(a --* b) v.b=v+ dAr(a -+ b) vab=v(Vab__V). dYab
(5)
This equation for partial transitions will be our starting point for the following discussion. If in (5) we are doing the sum over the final states b we obtain
Ar =
l+(m-v)
dv
d-v( ~ (Eb-Ea)Ar(a~
b)l,.},
(6)
because the derivatives are calculated for Vab = V, and identifying ~ A r (a ~ b)lv = Ar(v) as given by eq. (3). The result of eq. (6) must be quite independent of the parameter v over a large zone of plausible values of the average neutrino energy. A specific value of v will not be needed. We can calculate Ar (v) once a nuclear model for the initial state is considered. The last term in eq. (6) is an energy-weighted sum rule, and we shall use the following theorem 6): in a many-body system described by the Hamiltonian H = T + V, an operator ~ which commutes with the potential energy V satisfies E (Eb--Ea)l(b] E f i l a ) l 2 = 1 (al ~', {Vif/+}YtViI,}la). b i 2M
(7)
By seeing the structure of our operators [eq. (2)] we expect to apply our considerations to nuclei in which spin-dependent forces can be ignored. This includes the closed-shell nuclei 4He, 160 and *°Ca, and the light nuclei with A < 9 [ref. 8)], especially 3He [ref. 9)] and 6Li [ref. lo)], where the coupling to L = 0 gives a very good description. Applying eq. (7) to the matrix elements (2), our final expression is
Ar = {l+(m-v) 3 "d} Ar(v)-2 " /vl rn (v)
(8)
3. Application to closed-shell nuclei
Here we are going to study the cases of *He, 160 and 4°Ca, using a double closedshell description with harmonic oscillator wave functions. The present situation is confused with regard to the adequacy of this simple model for describing the total muon capture rates. It is known 4) that, in the cases of t 60 and 4°Ca, the oscillator shell model, without the closure approximation, gives much too large capture rates. However, the use of v = 80 MeV [the value of the parameter as originally proposed by Primakoff3)] in eq. (3) gives reasonable agreement with the experimental results 7). We apply our method to this model in order to clarify the situation.
44
J. BERNABEU
In these cases D s = D r = D L and the correlation functions have been calculated in ref. 7). We quote the results in the f o r m
A~(v)lz =
{1 - [ 1 +¼(½b 2 vZ)2]e-½b22v2},
Ar(v)13 = ( vm) 2{ l - [ l + ½ ( ½ b ~ v 2 ) 2
- T ~1( ½ b 32v ) +2 - 3~ ( 2 b1a v 12 ) 42 ] e - ½ b 3 2 " ~ } ,
(9)
where 1, 2 and 3 refer to 4He, 160 and 4°Ca, respectively, and bx = 1.39 fm, b z = 1.76 fm and b 3 = 1.99 fm, as determined from the experimental rms charge radii. /
/
0.20
035 /
....................... /
/
/.;../. ......... /
/
/
~
%
/
~//////////'2////// / / ~ / / e/ / / / / / / / / / / / / , /"//
/ / f s
//
'6o ~ o.lo
//
0.05
/ j /
/
I
I
I
I
0.6
0.7
0.8
09
1.0
x_-V Fig. 1. Reduced capture rates Ar for the closed-shell nuclei 4He, 160 and 4°Ca. Our predictions (solid lines) are compared to the v-dependence in the usual closure approach (broken lines) and to the experimental results. We obtain for A r [eq. (8)] the results displayed in fig. 1 (solid lines) where the experimental results Arl4 = 0.086+0.007, A~I2 = 0.111 +0.004,
Arl3 = 0.130_+0.001,
(10)
TOTAL MUON CAPTURE RATES
45
are also given for comparison. These ones are computed from the A listed by Eckhause 1) using the Zcff values of Ford and Wills 11). The broken lines show the v-dependence encountered in the usual closure approximation. Needless to say, fig. 1 clearly indicates that the description of 4He is sufficient to explain its capture rate, whereas this is not the case for x60 and 4°Ca; we obtain, in a closure approach to the problem, too large results in accordance with ref. 4) without the closure approximation. The agreement in ref. 7) is due to the particular value of v used. We do not claim that v = 80 MeV is not a good value, but that this value is not consistent with the oscillator model. Our method does not need the specific value of v to predict At; if it is preferred, the method determines, once a model for Ar(v) is chosen, the v-value in which this quantity has to be calculated. This value ofv corresponds to that one in which the two corrective terms in eq. (8) are cancelled. In this sense we can say that the oscillator shell model gives the results v t = 81 MeV,
Arll = 0.089,
v2
87 MeV,
Arl 2 = 0.165,
v3 = 88 MeV,
Ar[ 3 = 0.192,
=
(11)
to be compared with the experimental results (10). This result is easily understood because, as shown in ref. 12), the effects of SU(4) breaking are completely negligible in 4He whereas they are very important in a60 and 4°Ca. The effective excitation energy is different for the vector and axial terms, and this gives a reduction of the capture rate. Furthermore, we know 1 3 ) o f the existence of deformed 2p-2h admixtures in the ground state wave function of x60
4. The light nuclei 3He and 6Li The cases of 3He and 6Li are particularly interesting, because the behaviour of the correlation functions is very different from the one appearing for closed-shell nuclei. Furthermore, it is well known that the wave functions can be written in a realistic way using L-S coupling. Thus we expect the sum rule given in eq. (7) to work. We shall describe 3He as the coupling of three nucleons in s-wave to a spin S = -} with a symmetric spatial wave function of Gaussian kind. This has been proved to be very successful in many processes, and particularly for the partial transition 3He ~ 3H in muon capture 9). Under these conditions Ds(v ) = Dr(v ) = Dz(v ) with D s(O) = 1. The reduced capture rate in the closure approximation is then
A,(v) =
{1 --½e-'2/6~2},
(12)
with c~ = 0.356 fm -1 as determined from electron scattering x4). We see that the v-dependence is weaker than the other cases, and Ar (v) oc vz if v ~ 0.
46
J. B E R N A B E U
The nucleide 6Li is described xo) as a core (4He) with two nucleons in the Ip shell coupled to L = 0, S = 1. As it is a doubly odd nucleus, the relation D s (v) = D r (v) = DL (v) is no longer true t. This is violated by the contribution to the correlation functions coming from the expected value of the operators on the two external nucleons, in a way very similar to the well-known deuteron case x6). We have for aLi: D (v) = 212(s, s)+I2(p, p)+212(p, p)+EI2(s, p),
(13)
Dr(v) = OL(V) = 212(S, S) + ~tlZ(p, p) + ~}i22(p, p) + 212(s, p),
A~
/// /.
////////
I
0.6 3
s
I
Q6
017
I 0.8
I 0.9
1.0
x-=~. Fig. 2. Same as fig. 1, b u t for 6Li a n d aHe.
where I l (x, y) refer to the radial integrals
It(x, y) =
fo
dr r2Rx(r)j,(vr)R,(r).
(14)
Using the harmonic oscillator potential well with parameter b = 1.98 fm to reproduce the experimental rms charge radius, (14) can be solved explicitly and we obtain:
Os(v) = e-~b2v'{3 + ~(½b2v2)2}'
(IS)
l( b2v2)2~ Dr(v) = DL(V) = e-½b2v2{~+4__ 9(½b2v2 ) ..i ~-,½ , ,"
Then the reduced capture rate in the closure approximation is
At(v) =
{1 -e-*b2v~[0.82+0.12(½b2v2)+0.05(½b2v2)2]},
(16)
* O n this point, c o m p a r e with the conclusions reached in ref. ~s). F o r an N = Z d o u b l y o d d nucleus, t h e partition (P, P', P") in the SU(4) s c h e m e is effectively (1,0, 0), but the a s s i g n m e n t o f " p h y s i c a l " q u a n t u m n u m b e r s is in this case P = ( K 3 + L a ) ...... P" = T = IT3! instead o f P ~_ P'.
TOTAL MUON CAPTURE RATES
47
and we observe that A r (v) oc v2 with v ~ 0. However, the reason for this dependence is completely different from the case of 3He; for the latter, the difference follows since N < Z while for 6Li it is due to the difference in the functions D s and D r. In fig. 2, our results for 3He and 6Li are given (solid lines) where the broken lines show the v-dependence given by (12) and (16). The experimental results Ar(aHe)
n g/'~! + 0.026
.......
0.040,
(17)
Zr(6Li) = 0.273+0.057, are obtained from the A quoted by Eckhause ~) using the values of Zeff given by K i m and Primakoff ~7). The agreement is satisfactory, showing that the description of the nuclear states is adequate. The point at which the two corrective terms in (8) are cancelled is localized in v(3He) = 93 MeV, Af(3He) = 0.49,
(18)
v(6Li) = 91 MeV,
Ar(6Li) = 0.31,
to be compared with the experimental results (17). 5. Conclusion
In this work we have discussed a method of avoiding the uncertainty associated with the average neutrino energy in the usual closure approach to muon capture rates. The crucial steps are summarized. Instead of neglecting the kinematic dependence on each particular channel, the capture rate is approximated by an expansion around a parameter v in the form given in eq. (5). The sum over the final states can be performed straightforwardly with the help of theorem (7). Our final expression (8) allows us to calculate the total reduced capture rate in terms of a nuclear model for the initial state which gives the usual function (3) in closure approximation. The results we obtain using eq. (8) are quite independent on the specific value ofv over a large range of plausible values. As we observe in figs. 1 and 2, it is interesting to notice that the point for which the two corrective terms are cancelled is in a zone of high stability. Thus the study of these processes rests as an effective test of the nuclear model used. Our comparison with the 3He and 6Li cases allows us to conclude that the method is successful, confirming that L - S coupling gives a good description of the capture rates. For the closed-shell nuclei, it is clear that 160 and 4°Ca present terms of SU(4) breaking and we rule out the simple harmonic oscillator shell model to explain the muon capture. For 4He, according with Cannata e t al. 12), the SU(4) breaking effect must be completely negligible. The author wishes to thank the C E R N Theoretical Study Division for hospitality, and the G I F T for financial support. He is indebted to J. S. Bell, T. E. O. Ericson and J. Hiifner for very valuable comments and discussions. A critical reading of the manuscript by T. E. O. Ericson is acknowledged.
48
J. BERNABEU References
1) 2) 3) 4) 5) 6) 7) 8) 9) 10)
11) 12) 13) 14) 15) 16) 17)
M. Eckhause, R. T. Siegel, R. E. Welsh and T. A. Filippas, Nucl. Phys. 11 (1966) 575 E. Zavattini, CERN report, 1972; Muon Physics, to be published H. Primakoff, Rev. Mod. Phys. 31 (1959) 802 R. J. Luyten, H. P. C. Rood and H. A. Tolhoek, Nucl. Phys. 41 (1963) 236 G. Do Dang. Phys. Lett. 38B (1972) 397 T. E. O. Ericson and M. P. Locher, Nucl. Phys. A148 (1970) 1 J. S. Bell and Ch. LIewellyn Smith, Nucl. Phys. B28 (1971) 317 S. Cohen and D. Kurath, Nucl. Phys. 73 (1965) 1; A101 (1967) 1; A141 (1970) 145 R. Pascual and P. Pascual, Nuovo Cim. 44B (1966) 434 A. Lodder and C. C. Jomber, Phys. Lett. 15 (1965) 245; F. Roig and P. Pascual, G I F T preprint 4/72 (1972); N. C. Mukhopadhyay, Phys. Lett. 40B (1972) 157 K. W. Ford and J. G. Wills, Nucl. Phys. 35 (1962) 295 F. Cannata, R. Leonardi and M. Rosa Clot, Phys. Lett. 321} (1970) 6 K. H. Purser, W. P. Alford, D. Cline, H. W. Fulbright, H. E. Gove and M. S. Krick, Nucl. Phys. A132 (1969) 75 L. L Schiff, Phys. Rev. 133 (1964) B802 P. Hrasko, Phys. Lett. 28B (1969) 470 J. Bernabeu and P. Pascual, Nuovo Cim. 10A (1972) 61 C. W. Kim and H. Primakoff, Phys. P,ev. 140 (1965) B566
[
Nuclear Physics A201 (1973) 41 --48; (~) North-HollandPublishing Co., Amsterdam
I.E.4
[
Not to be reproduced by photoprint or microfilm without written permission from the publisher
ON T O T A L M U O N CAPTURE RATES AND T H E AVERAGE N E U T R I N O ENERGY J. B E R N A B E U
CERN, Geneva Received 28 September 1972 Abstract: A m e t h o d o f avoiding t h e u n c e r t a i n t y associated with the average n e u t r i n o energy v in the u s u a l closure a p p r o a c h to m u o n capture rates is discussed. Instead o f neglecting the kinematic dependence o n each particular c h a n n e l , the partial capture rate is a p p r o x i m a t e d by a first-order expansion a r o u n d v. A f t e r the s u m over the final states is p e r f o r m e d , the result is quite independent o f the specific value o f v. Application to 3He, 6Li a n d closed-shell nuclei is given, as an effective test o f the nuclear m o d e l s used.
1. Introduction
The experimental material accumulated on total muon capture rates in nuclei is rather abundant - see, for example, the results listed by Eckhause et al. 1). The original motivation for this study was to establish the basic assumptions of the theory of semi-leptonic weak interactions. This tool has been disregarded due to the uncertainty in the nuclear physics aspects, and the possibility of inferring some quantitative information on the weak form factors is restricted to experiments on hydrogen, deuterium and other selected partial transitions 2). The theory of total muon capture in nuclei has been developed especially by Primakoff 3) and by Luyten, Rood and Tolhoek 4). In the closure approximation, the rate can be written in terms of the expected values of two-body operators on the initial state, and one can think of using these processes as an effective tool for studying nuclear structure. However, one has to introduce in the closure approximation an average neutrino energy v and the result is very sensitive to the choice of v. The uncertainty on this parameter is so important that conclusions for the adequacy of the nuclear model used cannot be reached. Recently, Do Dang 5) has indicated the possibility of avoiding this difficulty by retaining in the dipole contribution of the matrix element a first-order expansion around the parameter v. This is applied after the factorization of the elastic form factor, and then the Yah dependence in each channel is just Yah, 4 corresponding to the dipole term in the limit Vab --, 0. In this paper it will be shown that the expansion of the effective Yah dependence - that is, for all possible multipoles and powers - is equivalent to a first-order expansion of the capture rate around v. Under this last form, the result is easily worked out to obtain independence of the specific value of v, within a large range of plausible values. 41
42
J. B E R N A B E U
In sect. 2 our method is clearly stated. This leads us to consider the study of two new terms besides the usual one appearing from the closure approximation. It is then pointed out how one of them can be calculated immediately from the "closure term", and the other one from a generalized Thomas-Reiche-Kuhn sum rule 6). In our case, this sum rule is expected to hold for nuclei in which spin-dependent forces can be ignored, and thus the application to closed-shell nuclei is given in sect. 3. In sect. 4, the cases of 3He and 6Li are studied, where L-S coupling is rather good. Finally, sect. 5 gives some discussion of the results obtained and our conclusions.
2. Theory In the usual notation the m u o n capture rate to a particular final state a --* b can be written in the form A(a --, b) = [G2v+2G2+(GA-Gp) 2] m 2 ( m ° 0 3 Ze'*ffAr(a ~ b),
(1)
2re 2
where the reduced capture rate A r is given in the impulse approximation by
J
+ B l ( b l ~ z}-eiV°b'xJaxjla)i2+Cl(b[ ~ z;e'V~b'X~a.~la)12}, J
(2)
J
and the sum and average over the final and initial polarizations are understood. The coefficients A, B and C can be written in terms of the effective coupling constants Gv, G A and Gp, and their values are 7) A = 0.19, B = 0.69, C = 0.12. The well-known results in the closure approximation is obtained by summing over the final states b, replacing Yah by an average value v. In terms of the correlation functions Ds, r, L introduced in ref. 7), the total capture rate is obtained as Ar(~ )
=
{A [-1 -- Z - 'Ds(v)] + B[1 - Z - 'D,r(V)] + C[-I - Z - 1Dt.(v)']},
(3)
where the dependence of A, B and C on v can be neglected. We turn our attention to eq. (2). Unlike the procedure of obtaining (3), the dependence on vau, i.e., on each particular channel, is not completely neglected. In terms of a parameter v we can write the identity yah - v[l +(V~b--V/V )] and the expansion of the nth power of V.b is V~b~
I+--[Vab--V
.
(4)
v
This is analogous to the idea used in ref. 5), because v.b = m - ( E b - E ~) t, for the dependence of the unretarded dipole term, after the approximation of factorizing the t In m u o n capture we have the transition (A, Z ) ~ (A, Z - - l ) a n d t h e n the C o u l o m b energy difference Ec m u s t be included. This correction, m --+ m + E c , m a y be i m p o r t a n t for h e a v y nuclei. W e acknowledge T. Ericson for this remark.
TOTAL MUON CAPTURE RATES
43
elastic charge form factor; in this way, eq. (4) was used in the particular case of n = 4. We are going to apply (4) for each contribution of (2) when the exponential e ivab- x is developed in powers of V~b. It is obvious, and it can be proved straightforwardly, that this expansion is equivalent to approximating eq. (2) by the following form A r(a --* b) ~ Ar(a --* b) v.b=v+ dAr(a -+ b) vab=v(Vab__V). dYab
(5)
This equation for partial transitions will be our starting point for the following discussion. If in (5) we are doing the sum over the final states b we obtain
Ar =
l+(m-v)
dv
d-v( ~ (Eb-Ea)Ar(a~
b)l,.},
(6)
because the derivatives are calculated for Vab = V, and identifying ~ A r (a ~ b)lv = Ar(v) as given by eq. (3). The result of eq. (6) must be quite independent of the parameter v over a large zone of plausible values of the average neutrino energy. A specific value of v will not be needed. We can calculate Ar (v) once a nuclear model for the initial state is considered. The last term in eq. (6) is an energy-weighted sum rule, and we shall use the following theorem 6): in a many-body system described by the Hamiltonian H = T + V, an operator ~ which commutes with the potential energy V satisfies E (Eb--Ea)l(b] E f i l a ) l 2 = 1 (al ~', {Vif/+}YtViI,}la). b i 2M
(7)
By seeing the structure of our operators [eq. (2)] we expect to apply our considerations to nuclei in which spin-dependent forces can be ignored. This includes the closed-shell nuclei 4He, 160 and *°Ca, and the light nuclei with A < 9 [ref. 8)], especially 3He [ref. 9)] and 6Li [ref. lo)], where the coupling to L = 0 gives a very good description. Applying eq. (7) to the matrix elements (2), our final expression is
Ar = {l+(m-v) 3 "d} Ar(v)-2 " /vl rn (v)
(8)
3. Application to closed-shell nuclei
Here we are going to study the cases of *He, 160 and 4°Ca, using a double closedshell description with harmonic oscillator wave functions. The present situation is confused with regard to the adequacy of this simple model for describing the total muon capture rates. It is known 4) that, in the cases of t 60 and 4°Ca, the oscillator shell model, without the closure approximation, gives much too large capture rates. However, the use of v = 80 MeV [the value of the parameter as originally proposed by Primakoff3)] in eq. (3) gives reasonable agreement with the experimental results 7). We apply our method to this model in order to clarify the situation.
44
J. BERNABEU
In these cases D s = D r = D L and the correlation functions have been calculated in ref. 7). We quote the results in the f o r m
A~(v)lz =
{1 - [ 1 +¼(½b 2 vZ)2]e-½b22v2},
Ar(v)13 = ( vm) 2{ l - [ l + ½ ( ½ b ~ v 2 ) 2
- T ~1( ½ b 32v ) +2 - 3~ ( 2 b1a v 12 ) 42 ] e - ½ b 3 2 " ~ } ,
(9)
where 1, 2 and 3 refer to 4He, 160 and 4°Ca, respectively, and bx = 1.39 fm, b z = 1.76 fm and b 3 = 1.99 fm, as determined from the experimental rms charge radii. /
/
0.20
035 /
....................... /
/
/.;../. ......... /
/
/
~
%
/
~//////////'2////// / / ~ / / e/ / / / / / / / / / / / / , /"//
/ / f s
//
'6o ~ o.lo
//
0.05
/ j /
/
I
I
I
I
0.6
0.7
0.8
09
1.0
x_-V Fig. 1. Reduced capture rates Ar for the closed-shell nuclei 4He, 160 and 4°Ca. Our predictions (solid lines) are compared to the v-dependence in the usual closure approach (broken lines) and to the experimental results. We obtain for A r [eq. (8)] the results displayed in fig. 1 (solid lines) where the experimental results Arl4 = 0.086+0.007, A~I2 = 0.111 +0.004,
Arl3 = 0.130_+0.001,
(10)
TOTAL MUON CAPTURE RATES
45
are also given for comparison. These ones are computed from the A listed by Eckhause 1) using the Zcff values of Ford and Wills 11). The broken lines show the v-dependence encountered in the usual closure approximation. Needless to say, fig. 1 clearly indicates that the description of 4He is sufficient to explain its capture rate, whereas this is not the case for x60 and 4°Ca; we obtain, in a closure approach to the problem, too large results in accordance with ref. 4) without the closure approximation. The agreement in ref. 7) is due to the particular value of v used. We do not claim that v = 80 MeV is not a good value, but that this value is not consistent with the oscillator model. Our method does not need the specific value of v to predict At; if it is preferred, the method determines, once a model for Ar(v) is chosen, the v-value in which this quantity has to be calculated. This value ofv corresponds to that one in which the two corrective terms in eq. (8) are cancelled. In this sense we can say that the oscillator shell model gives the results v t = 81 MeV,
Arll = 0.089,
v2
87 MeV,
Arl 2 = 0.165,
v3 = 88 MeV,
Ar[ 3 = 0.192,
=
(11)
to be compared with the experimental results (10). This result is easily understood because, as shown in ref. 12), the effects of SU(4) breaking are completely negligible in 4He whereas they are very important in a60 and 4°Ca. The effective excitation energy is different for the vector and axial terms, and this gives a reduction of the capture rate. Furthermore, we know 1 3 ) o f the existence of deformed 2p-2h admixtures in the ground state wave function of x60
4. The light nuclei 3He and 6Li The cases of 3He and 6Li are particularly interesting, because the behaviour of the correlation functions is very different from the one appearing for closed-shell nuclei. Furthermore, it is well known that the wave functions can be written in a realistic way using L-S coupling. Thus we expect the sum rule given in eq. (7) to work. We shall describe 3He as the coupling of three nucleons in s-wave to a spin S = -} with a symmetric spatial wave function of Gaussian kind. This has been proved to be very successful in many processes, and particularly for the partial transition 3He ~ 3H in muon capture 9). Under these conditions Ds(v ) = Dr(v ) = Dz(v ) with D s(O) = 1. The reduced capture rate in the closure approximation is then
A,(v) =
{1 --½e-'2/6~2},
(12)
with c~ = 0.356 fm -1 as determined from electron scattering x4). We see that the v-dependence is weaker than the other cases, and Ar (v) oc vz if v ~ 0.
46
J. B E R N A B E U
The nucleide 6Li is described xo) as a core (4He) with two nucleons in the Ip shell coupled to L = 0, S = 1. As it is a doubly odd nucleus, the relation D s (v) = D r (v) = DL (v) is no longer true t. This is violated by the contribution to the correlation functions coming from the expected value of the operators on the two external nucleons, in a way very similar to the well-known deuteron case x6). We have for aLi: D (v) = 212(s, s)+I2(p, p)+212(p, p)+EI2(s, p),
(13)
Dr(v) = OL(V) = 212(S, S) + ~tlZ(p, p) + ~}i22(p, p) + 212(s, p),
A~
/// /.
////////
I
0.6 3
s
I
Q6
017
I 0.8
I 0.9
1.0
x-=~. Fig. 2. Same as fig. 1, b u t for 6Li a n d aHe.
where I l (x, y) refer to the radial integrals
It(x, y) =
fo
dr r2Rx(r)j,(vr)R,(r).
(14)
Using the harmonic oscillator potential well with parameter b = 1.98 fm to reproduce the experimental rms charge radius, (14) can be solved explicitly and we obtain:
Os(v) = e-~b2v'{3 + ~(½b2v2)2}'
(IS)
l( b2v2)2~ Dr(v) = DL(V) = e-½b2v2{~+4__ 9(½b2v2 ) ..i ~-,½ , ,"
Then the reduced capture rate in the closure approximation is
At(v) =
{1 -e-*b2v~[0.82+0.12(½b2v2)+0.05(½b2v2)2]},
(16)
* O n this point, c o m p a r e with the conclusions reached in ref. ~s). F o r an N = Z d o u b l y o d d nucleus, t h e partition (P, P', P") in the SU(4) s c h e m e is effectively (1,0, 0), but the a s s i g n m e n t o f " p h y s i c a l " q u a n t u m n u m b e r s is in this case P = ( K 3 + L a ) ...... P" = T = IT3! instead o f P ~_ P'.
TOTAL MUON CAPTURE RATES
47
and we observe that A r (v) oc v2 with v ~ 0. However, the reason for this dependence is completely different from the case of 3He; for the latter, the difference follows since N < Z while for 6Li it is due to the difference in the functions D s and D r. In fig. 2, our results for 3He and 6Li are given (solid lines) where the broken lines show the v-dependence given by (12) and (16). The experimental results Ar(aHe)
n g/'~! + 0.026
.......
0.040,
(17)
Zr(6Li) = 0.273+0.057, are obtained from the A quoted by Eckhause ~) using the values of Zeff given by K i m and Primakoff ~7). The agreement is satisfactory, showing that the description of the nuclear states is adequate. The point at which the two corrective terms in (8) are cancelled is localized in v(3He) = 93 MeV, Af(3He) = 0.49,
(18)
v(6Li) = 91 MeV,
Ar(6Li) = 0.31,
to be compared with the experimental results (17). 5. Conclusion
In this work we have discussed a method of avoiding the uncertainty associated with the average neutrino energy in the usual closure approach to muon capture rates. The crucial steps are summarized. Instead of neglecting the kinematic dependence on each particular channel, the capture rate is approximated by an expansion around a parameter v in the form given in eq. (5). The sum over the final states can be performed straightforwardly with the help of theorem (7). Our final expression (8) allows us to calculate the total reduced capture rate in terms of a nuclear model for the initial state which gives the usual function (3) in closure approximation. The results we obtain using eq. (8) are quite independent on the specific value ofv over a large range of plausible values. As we observe in figs. 1 and 2, it is interesting to notice that the point for which the two corrective terms are cancelled is in a zone of high stability. Thus the study of these processes rests as an effective test of the nuclear model used. Our comparison with the 3He and 6Li cases allows us to conclude that the method is successful, confirming that L - S coupling gives a good description of the capture rates. For the closed-shell nuclei, it is clear that 160 and 4°Ca present terms of SU(4) breaking and we rule out the simple harmonic oscillator shell model to explain the muon capture. For 4He, according with Cannata e t al. 12), the SU(4) breaking effect must be completely negligible. The author wishes to thank the C E R N Theoretical Study Division for hospitality, and the G I F T for financial support. He is indebted to J. S. Bell, T. E. O. Ericson and J. Hiifner for very valuable comments and discussions. A critical reading of the manuscript by T. E. O. Ericson is acknowledged.
48
J. BERNABEU References
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