Effect of meson exchange corrections on allowed muon capture

Volume 106B, number 5

PHYSICS LETTERS

19 November 1981

EFFECT OF MESON EXCHANGE CORRECTIONS ON ALLOWED MUON CAPTURE R. PARTHASARATHY 1 International Centre for Theoretical Physics, Trieste, Italy

and V.N. SRIDHAR Matscience, Madras 600 020, India

Received 16 December 1980 Revised manuscript received 22 March 1981

Corrections to the impulse approximation calculation of the 7-u angular correlation coefficient and recoil nuclear polarization in allowed muon capture transitions, arising from meson exchange effects, are studied and the value for the induced pseudoscalar coupling consistent with both observables is found to be (13.3 + 3)gA, to a large extent free from nuclear wave function uncertainties. The study of nuclear muon capture is of considerable importance both for the understanding of nuclear structure and of the muon capture coupling constants, especially the induced pseudoscalar coupling. In particular the average recoil polarization of 12B(1+ ; gs)in muon capture by 12C (0+; gs), being insensitive to nuclear wave functions [1 ], provides reliable information about the induced pseudoscalar coupling constant gp [2]. Recently, we have studied [3] the gamma-neutrino angular correlation in muon capture by 28Si (0+; gs) and observed that the correlation coefficient/32 associated with the angular dependence (P"D(q'v), where P is the muon polarization, $ and ~ are unit vectors along the photon and neutrino directions, respectively, being to a large extent free from nuclear wave function uncertainties, also provides reliable information about gp when compared with the experimental data of the William and Mary group [4]. It is instructive to observe that these two observables, the average recoil polarization and/32, provide the following result: gp = (13.3 + 1.8)g A and (13.3_+3:5)g A, respectively, to a large extent free from nuclear wave function uncertainties, although the nuclei involved are different, namely 12C and I Permanent address: Matscience, Madras 600 020, India. 0 031-9163/81/0000-0000/$ 02.75 © 1981 North-Holland

28 Si. This common value of gp is slightly higher than the nficleon value given by PCAC and the reason for the small but definite enhancement is not clear [5]. However, the question arises as to how far can one rely upon the impulse approximation method, the basic ingredient in all these calculations. Recent studies [6] of the meson exchange currents provide a way of improving the impulse approximation (IA) specifically by the mesonic degrees of freedom in light nuclei and by the use of the soft pion theorem, which are totally absent in IA. A remarkable result is the electromagnetic process np ~ d7 in which the discrepancy between IA theory and experiment is removed only after including the meson exchange corrections, besides the role of the A(1232) isobar. Another significant result is the observation of Kubodera et al. [6] that if one looks into the time component of the axial vector current, for which soft pion arguments are as powerful as in the electromagnetic case, then meson exchange corrections are significant (O(1)) and can show up in angular correlation measurements in/3 decay or in the muon capture process 160.(0 +) 16N(0-) [6]. It is the purpose of this letter to examine how far these meson exchange effects (MEC) affect the 12B(l+; gs) recoil polarization and the angular correlation coef363

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ficient/32 in 28Si(0+; gs). It is rather obvious from the arguments of Kubodera et al. [6] that in allowed muon capture processes dominated by the space part of the IA axial current, MEC effects will be small. However, our calculations, although they explicitly confirm their argument, should be viewed as corrections to the impulse approximation rather than an indication for the MEC effects. We shall first discuss the necessary modification of the IA hamlltonian and then evaluate r2 and recoil nuclear polarization. For the sake of clarity we start with the FujiiPrimakoff hamiltonian but distinguish between the space and time components of the weak hadronic bare vector and axial vector currents. Such a distinction is not made for the strong interaction induced currents. Then using non-relativistic reduction the matrix element for the nuclear muon capture process in the impulse approximation is given by [7]

Q = (uvl~21uu) ,

(1)

where lu v) and lu u) are Dirac spinors for the muonneutrino and the muon, respectively, and a = ~(1 - - , "~)(c/~ 1 +a" c~2),

(2)

with

=gvf

fpl,

9tt2=gAf"7--gv

P7

-~.t(gpi, fi,.~i-gai, f i,..°

s gA gv f , x P'--M

- 1 --~

(3)

(4)

f (.,.p,)o

where li,o i and Pi are the nucleon unit, Pauli spin and momentum operators and the superscripts 0 and s stand for the contributions due to time and space part of the bare currents, respectively. To this IA expression we add the MEC contributions fi la Kubodera et al. [6]. We neglect the MEC contributions to the vector current but only consider the time component of the ME axial current coming from the sea-gull graph 364

19 November 1981

[6] neglecting a-particle contributions. Although the Gamow-Teller transitions considered here are dominantly described by the space part of the IA axial current [the gA f ~ts term in eq. (4)], the MEC corrections to this space part are believed to l~e very small. The non-relativistic structure of the sea-gull term is F ~ 1e~a.rr7 and goes like O(1) for the time part and O(P/M) for the space part [6]. It is argued that there is no reason to believe that pion exchange can adequately describe the space part [8]. In refs. [8] and [9], the space part is related to the n-nuclear scattering amplitude with a description in terms of isobar particle-nucleon hole states, in nuclei. The result of these analyses is the redefinition o f g A as gA/(1 + a), where ~ is the polarizability of the nuclear medium. The presence of the factor (1 + a) in gA is called the Lorentz-Lorenz effect and was first discussed by Ericson et al. [10] as a short-range effect. Although gA is quenched by 25% on nuclear matter, microscopic calculations [9] reveal that the quenching in light nuclei is very small, ~1%. For quantitative purposes, we have studied the effect of the quenching o f g n on/32 and recoil polarization. Our results show that with a 10% decrease o f g A in the space part, the recoil polarization decreases by less than 1% while/32 increases by less than 2%. (Typically for gA = --1.2 gv, Pav(12B(l+)) = 0.5784,/32 = 0.1906, for gA = --1.1 gv, they are 0.5768 and 0.1944, and for gA = --1.0 g v , they are 0.5749 and 0.1987, respectively.) Therefore these effects on Pav and/32 are very small; so we neglect such corrections. However, in view of the unclear description of the space part of the MEC axial current correction, we refrain from emphasizing the second class induced tensor coupling constant. The explicit form of the time component of the MEC correction to the IA axial vector current is given in ref. [6]. Following ref. [8] let us denote the nuclear matrix elements of the time component of the IA axial current (one-body operator) and ME axial current (two-body operator) by and respectively. (.40) then corresponds to the terms in eq. (4)with gA and the integrals with the superscript 0. Let us denote the ratio of GtOEc)/(AOA ) by F, a measure of the effect of MEC corrections to IA. The evaluation of F consists in evaluating (AOEc) with specific nuclear wave functions. The advantage of writing ~ 2 as in eq. (4) will be in seeing that MEC corrections affect the nucleon momentum dependent

(AOA) (AOEc),

Volume 106B, number 5

PHYSICS LETTERS

term v f ( ~ i ' P i ) 0 directly. This is the reason why the 160(0+)-+ 16N(0-) partial capture rate, which is very sensitive to nucleon momentum dependent terms, is in turn sensitive to MEC corrections. This was first pointed out by Rood [11 ] in an entirely different context (not MEC corrections) of the importance of nucleon momentum dependent terms. We thus see from eq. (4) that any observable sensitive to the momentum dependent terms will be a candidate for detecting the effect of MEC corrections. Having added the MEC corrections, we can dispense with the superscripts and absorb F in the coupling constants. Then eq. (4) is simply

el,

%= M

1.4

*'~ m ÷o .-

P

,.× fp,

1.1

1.(]

.E

without MEC cerrection(IA)

0.9

with 5 0 % MEC correction o.e

m

E x perlment [4]

0.7

o

(5)

t

gA

gv

19 November 1981

f <,,,'pi)

where G~, = (gp - F g A - g v - gM) v/2M, and g~ = F g A. We shall now examine the effect of F on/32. The quantity F has been calculated for A = 12 in ref. [6] and forA = 16 in ref. [12]. It is nearly a uniform positive value ~0.5 indicating that it is qualitatively independent of nuclear transition. To have a quantitative view, we vary F and study the effects. We have shown [3] that among the three g a m m a neutrino angular correlation coefficients in polarized muon capture by 28Si(0+; gs), the coefficient/32 associated with the "),-v angular dependence (P'~)('~'i,) is nuclear model insensitive and fortunately this has been measured rather more accurately than the other two. The detailed expression for/32 can be found in ref. [3] and we do not repeat it here. We have used the particle-hole wave functions of Donnelly and Walker [13] for 28Si. In fig. 1 we give the variation of/32 and gp without (IA) and with 50% meson exchange correction, corresponding to F = 0.5 (although we have studied F = 0.2 to 1, we present here only the result for F = 0.5 following refs. [6] and [11]). From this figure it is clear that the MEC corrections are indeed small as expected in an allowed muon capture transition. However, the MEC corrections decrease the numerical value of/32 up to gp ~ 10g A and then enhance it uniformly. The effect of increasing F from 0.5 to 1 is found to be negligible in/32 . By comparing with experiment [4] we find two sets o f g p / g A values: set I: (-6.65 + 4.3) in IA and (-9.1 +- 3.1) with 50% MEC;

a

'~ I

0.6

0.5

0.4

o.zl -10

I -5

I

I

I

I

I

I

I

0

5

10

15

20

25

30

g4g, Fig. i. set II: (12.5 + 5) in IA and (12.9 -+ 3.9) with 50% MEC. Of the two sets, set I contradicts the PCAC value. We shall resort to recoil nuclear polarization and then decide which set is to be used. In calculating the 12B(l÷ ; gs) recoil polarization, a correction due to the gamma decay of 12B(1-;2.62 MeV) is necessary [14]. Let us call the corrected recoil polarization the effective average recoil polarization. While the transition 12C(0÷; gs) -+ 12B(l+ ; gs) is an allowed process, the transition 12 C (0 + ; gs) ~ 12B( 1-; 2.62 MeV) is a forbidden one and the induced pseudoscalar interaction does not contribute to this process. It is also independent of the time component of the axial current. This would mean that the MEC corrections are negligible here. However, owing to CVC and observing the importance of MEC corrections in the electromagnetic process np -~ dT, we have examined the effect of MEC corrections coming from the space part of the MEC and found it to be very negligible. It is then straightforward to compute the effective 12 B(1 +; gs) recoil polarization for various values of F. In fig. 2, we show the variation of this with gp without 365

Volume 106B, number 5

PHYSICS LETTERS

0.7 without MEC correction

~

with 5 0 % MEC correction

~ i

(IA)

/'/'/'J'77"TExperiment[2]

0.6-

~

19 November 1981

mate m a y be interpreted as due to the presence of a second class axial current in the impulse approximation, o f magnitude g T ~ (5.5 + 3 ) g A a s g p a n d g T occur in the linear combination gp + gT in muon capture. The authors are grateful to Professor A. Ramakrishnan for constant encouragement. One o f them (R.P.) would like to thank Professor Abdus Salam, the International Atomic Energy Agency and UNESCO for hospitality at the International Centre for Theoretical Physics, Trieste.

0.5--

/~ O.4

References 0.3

-10

-5

0

gp/g,

5

10

15

20

Fig. 2.

and with 50% MEC corrections. Here the MEC corrections decrease the recoil polarization. By comparing with experiment, we find gp = (13.62 + 2 . 1 ) g A , a value nearly independent of MEC corrections (see fig. 2). In this calculation we have employed the wave functions for 12B o f Gillet and Vinh Mau [15], Elliot and Flowers [15] and Donnelly and Walker [13] and they all give nearly the same value. The results shown in fig. 2 correspond to that of the Gillet and Vinh Mau [15] p a r t i c l e - h o l e wave functions. The result o f the recoil nuclear polarization allows us to choose between the two sets o f values for gp obtained in the analysis o f the angular correlation coefficient/~2 and the result is set II. F r o m the results o f the analysis o f the angular correlation coefficient/32 i n / a - + 28Si(0+ ; gs) ~ 28Al*(1+ ; 2202 keV) + vu; 28A1" (1 +; 2202 keV) ~ 28A1(0 + ; 973 keV) + 7 and the recoil polarization of 12B(l+ ; gs), after correcting for the gamma decay o f 12B(1 - ; 2.62 MeV), we conclude the following: the numerical value o f the induced pseudoscalar coupling is gp = (13.3 + 3 ) g A , a value to a large extent free from nuclear wave function uncertainties and obtained after correcting the impulse approximation with 50% meson exchange corrections through the time part of the bare axial vector current. The MEC corrections to the space part o f the IA axial current do n o t change our conclusions. The slightly higher value of ga from the naive PCAC esti366

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