Muon wave function and Coulomb propagator in radiative muon capture in nuclei

Nuclear Physics A345 (1980) 317-330 Not

to

@ North-Holland Publishing Co., Amsterdam

be reproduced by photoprint or microfilm without written permission from the publisher

MUON WAVE FUNCTION AND COULOMB PROPAGATOR IN RADIATIVE MUON CAPTURE IN NUCLEI P. CHRISTILLIN CERN,

Geneva, Switzerland

M. ROSA-CLOT

and

S. SERVADIO

INFN, Piss, Italy Received 14 May 1979 (Revised 28 January 1980)

Abstract: The electromagnetic effects connected with the muon radiating diagram in radiative muon capture in nuclei are discussed. The non-relativistic Schrodinger equation, reproducing with a very good accuracy the muonic ground-state properties, i.e., binding energies and effective charges Z,,, is shown to be adequate for the bound muon. The effects of the extended nuclear Coulomb field on the muon propagator are considered, and an explicit solution is worked out for the high-energy region of the photon spectrum.

1. Introduction This paper is concerned with the electromagnetic properties of bound muons in connection with radiative muon capture (RMC). Because of that, all the following statements will apply only to the 1s state, where the muon ends its cascade and is eventually weakly captured. Secondly, this work is instrumental to a proper treatment of RMC, and will hence not attempt a general solution of the electromagnetic problem but only give results in the domain of interest. As a matter of fact, both for experimental (neutron contamination) and theoretical reasons, see e.g. ref. ‘), one is limited to the high-energy part of the photon spectrum, typically for energies k s 60MeV. In that region, kinematical conditions enhance the contribution of the pseudoscalar coupling constant via radiation by the hadrons, which accounts for about one-half of the total. One is therefore in the position to test PCAC or eventually to learn something about coupling constant renormalization in nuclei 2*3). Of course, to that aim, the total amplitude must be correctly treated and particular care must therefore be devoted to the diagram responsible for the rest of the process, i.e., the radiation by the muon. Indeed an overall treatment of the muon electromagnetic features is lacking. In particular, as regards the muon propagator in the nuclear Coulomb field, only two attempts have been made to go beyond the free propagator. The first one extends the well-known results of Martin and Glauber 4, for the electron case to muons and has been adopted by Lobov ‘) for ‘*C and by Rood 317 Aumurt

1980

318

P. Christillin et al. / Radiative muon capture

et al. 6, for 4”Ca. It has to be regarded with suspicion because of the inherent approximation of a point-like nucleus. The second one, due to Rood et al. 7, is the proper solution of the equation obeyed by the muon Green function in an extended nucleus. It is, however, limited to 40Ca, and not easily extensible. All of the recent literature *), although limited to light nuclei where the above-mentioned approximation is rather justified, assumes a free muon propagator. In view of the general interest of the problem, we will therefore attempt a solution, for all nuclei, of the high-energy part of the photon spectrum. For orientation purposes in comparing radiative with ordinary electron capture, where the process takes place at the (almost point-like) nucleus, two effects, connected with the electron propagator, contribute to the spectrum with opposite sign. First, electrons can emit photons before undergoing the usual capture process at distances from the centre of the nucleus (where the wave function has its maximum value) up to the order of the range of the propagator. This therefore reduces the capture probability. On the other hand, the probability amplitude for an electron reaching the nucleus after photon emission is enhanced by the attractive electrostatic field. The net result has been found to be a decrease of the full radiative amplitude. In the muon case, because of the large ratio of muon to electron mass, the muon Bohr radius is some two hundred times smaller than the corresponding electron radius. Therefore, the muon orbit lying to a large extent inside medium heavy nuclei, it is clearly inadmissible to treat the nucleus as point-like by replacing the muon wave function by its value at the origin. As a matter of fact, the muon wave function is averaged over the nuclear volume and it is customary to express it in terms of an effective charge 2,~ of the protons participating in the process. Therefore, while the argument about the attraction due to the Coulomb interaction in the intermediate state remains valid, muons can emit photons outside such an average and then be captured inside, as in the electron case, and can also emit and then be captured at different places within this “effective” nuclear volume with a total effect of no obvious sign. This qualitative argument shows that the electromagnetic features cannot be straightforwardly taken over from the electron 4, to the muon case. For this reason the problem has been totally reformulated. Non-locality and Coulomb attraction have been considered separately and the net result has been proved not to drastically change the free case, because of partial cancellation between these two competing effects. Heavy nuclei are more affected. Of course, for the explicit calculation both an expression for the Coulomb propagator and the muon wave function are needed. As regards the latter the degree to which a non-relativistic wave function can be used in order to simplify calculations has been checked. All relevant quantities pertaining to the 1s wave muon, i.e., binding energy, 2,~ and average kinetic energy, have been explicitly computed using the solutions of the Schrodinger equation with a uniform sphere for the nuclear density. It has been

319

P. Christillin et al. / Radiative muon capfure

shown that these quantities vary at most by 1.5% with respect to those calculated by of the Dirac equation with more sophisticated forms of the nuclear density. Finally, the muon propagator in the finite-size nuclear Coulomb field has been treated and solved in a closed form, without recurring to the explicit wave functions summation. Along the lines of previous works, the equation satisfied by the Green function has been written down and shown to reduce in the energy domain considered to a scalar equation (which is, of course, not true in general), with the potential term as a correction to the free case. This has made it possible to try a kind of eikonal approximation for its solution. The plan of this work is as follows: (i) In sect. 2 the amplitude for the muon radiating diagram is explicitly written down and commented upon. (ii) Sect. 3 is devoted to the muon wave function and to muonic ground-state properties. (iii) In sect. 4 we give the solution of the Green function in the energy domain of interest. (iv) Finally, in sect. 5 the explicit calculations relevant to the muon spectrum will be performed for a sample of nuclei, and some comments will be made in sect. 6. Sens 9, and Ford and Wills lo) through numerical integration

2. Transition

amplitude

and rate for radiative

capture

Most previous calculations of RMC were based on an effective one-body hamiltonian derived from an appropriate set of Feynman diagrams. After the nonrelativistic reduction the amplitude is of the form KS=

$ O(i)S(x --xi), i=l

where O(i) are operators depending on the kinematical variables of the problem. Such a hamiltonian is strictly local if no momentum-dependent terms appear in O(i).

This is, for instance, not the case for the y-emission from the hadrons, as for example, in fig. la, where the propagator is usually expanded keeping linear terms in the nucleon momentum p. Neither is it for the y-emission from the muon as seen in fig. lb, where, however, terms in the muon momentum p could be safely dropped in deriving Hen for 4”Ca [ref. “)I.

Fig. 1. Different

graphs

contributing

to radiative muon capture: the muon, fig. lb.

radiation

by the hadrons,

fig. la and by

I? Christillin et al. / Radiative mum capture

320

Since we want a formulation also take

into account

intermediate is worth

valid for all nuclei, we resort to a formalism

the effect of the attractive

state, as well as the above-mentioned

remarking

that the range

of the muon

nuclear

Coulomb

non-locality.

which can field in the

In this connection,

propagator

being

it

of the order

of

= 2 fm for k 2 60 MeV, locality may appear justified in 40Ca where only a small variation in the overlap of the muon wave function on the nuclear volume ensues, whereas no a priori conclusion seems evident in 208Pb. The hadronic spinor structure of the matrix elements will be omitted here, being irrelevant to our discussion of fig. lb. Let us therefore M =

start by writing

d”x&(x)O&,,(x)

e -i”.X

down the corresponding d3x’k,y,(1

amplitude

+ YS)Glils-k(x, x’)e(*) * y e-‘k’“‘+g(x’),

(1) where the symbols have obvious meanings. GE,s_k(x, x’) represents the Green function of the muon in a Coulomb field, Els standing for the muon energy in the 1S orbit, i.e., muon mass minus binding energy E, and satisfies the equation [r.V-~4(E+eAo(x))+m,]G~(~,x’)=S(x-x’),

(2)

and the choice

(3)

will be made in the following, Along Green

standard

function

R standing

lines it is convenient

g&_k(x,

for the nuclear to re-express

radius.

eq. (1) via the second-order

x’) satisfying

[V’ + (E + eAo(x))2 - m 2 - (ey . VA~(X))Y~]~E(X, x’) = --6(x -x’) Then upon performing

an integration

by parts and using the Dirac equation

.

(4) obeyed

by I&(X’), eq. (1) reads &f=

I

d3x&(x)Op&,(x)

e-i”‘r

I

d’x’E,y,,(l

+YS)gEls-k(x,

x [2p - ecA)+ ka . &(*)(A+ ys)J e.-ik.x’fiF(x’) , where p stands for the muon momentum

and A for the states of circular

x’) (5) polarization

*l. The pleasant feature, already noted by Rood et al., that the term y * VA” in the equation for gEls_k (x, x’) can be dropped with negligible error in the energy region of interest, renders the second-order Green function a scalar object.

321

P. Christillin et al. / Radiative muon capture

Thanks to the proof of the possibility of neglecting interference terms among large and small components (see the appendix for details) in the calculation of the trace, eq. (5) decouples into a spinor part and a coordinate part A?, upon which we focus. With the usual assumption that protons participate independently in F-capture, we have two kinds of contributions to the amplitude d3x’gE:,,_&

x’) e-~‘~““‘~~“‘llrr,(x’)la>,

(64

i=l

where [a) andjn) are the initial and any allowed final state respectively, the subscript / stands for leptons (Fig. lbj and h for hadrons (fig. la). These matrix elements are multiplied by appropriate coupling constant combinations; the factor eiUZhas been inserted in eq. (6a) for reasons which will become clear in the following. The contributions of the matrix elements to the rate, with the usual factorization hypothesis, read

(74

6%)

X

ld3$gEls- k(X, X’) e- ik”“-“‘$Jx’)~,(x)

+ h.c.!

(7c)

In the previous expressions ($&[*which represents the density of states for the muon should be understood as (g’i-f”). However, it will turn out from the results of the next section that a completely non-relativistic formulation is enough. p(x) stands for the nuclear proton density and the first factor in the r.h.s. of eqs. (7) is the usual one-body operator pertaining to radiative muon capture to be evaluated in some model. The above equations can therefore be regarded as our starting point in interpreting the effects due to the muon non-locality.

P. Chri~tillin et al. / Radiative nwon capture

322

3. Muonic ground-state

properties

This paragraph is devoted to the discussion of the properties of the bound muon 1s wave function. Effective charges and binding energies were calculated by Sens 9, and by Ford and Wills lo) by numerically solving the Dirac equation in the extended nuclear field. Because of the calculational complication of these solutions, we investigated the possibility of using non-relativistic wave functions and checked the accuracy of other partial calculations along this line. After completing the calculation we became aware’ that the fact that the muon is to a few per cent a non-relativistic object, which seems to be totally unrecognized in the recent literature, had already been pointed out in earlier papers [see e.g., ref. 12)in connection with muonic energy levels]. The agreement of our results with relativistic calculations 9*10)is extremely good, in accord with available explicit calculations of small components 13). As shown in the following, these ground-state properties turn out to be rather insensitive also to the details of the nuclear charge distribution. We now proceed to give the expressions of interest by generalizing to heavier nuclei the wave function solution obtained by Yano and Yano 14)for 40Ca*. The 1s radial wave equation for a uniformly charged nucleus of radius R with the potential given by eq. (3), is

_!_&&2y)

-eAo(x)@,(x)=E@,(x),

(8)

whose solutions, obeying the proper boundary conditions are

=$ e-s2X2’2M(a,1, p2,y2) ,

@in(X)

1 @in(R) cPout(x)= - -e

N @m(R)

N is the renormalization

bX/2u(1 -A, 2, bx) .

@a> Pb)

factor which has been calculated numerically,

p4 = m,Ze”fR3,

62=8m,(E[,

A = 2m,Ze’/b,

and h4, U are Kummer functions 16).For fixed R and Z i.e., for a given nucleus, the

t We thank $ Analogous

Professor T. E. 0. Ericson for bringing wave functions had also been obtained

it to our attention. for ‘He, 6Li and “C in ref. Is).

323

P. Christillin et al. / Radiative muon capture

energy eigenvalue is obtained by matching inside and outside derivatives at x = R, obeying -p=R=+$l=R=a

M(a + I,$, p2R2) M(a, t, P2R2)

=-$R-b~(i-~)

(10)

which we have solved numerically. This then uniquely determines cP~= @@(x,R, Z) which we need to calculate the effective charge defined as Z‘& =7ra:(p)=7rai

I

d3xdx)~@,Jx,

R Z>\=.

(11)

Here a0 stands for the muon Bohr radius l/m(,(~, where (Y= e2 = &, yielding for a point-like nucleus [4(x) - 4(O), j d3xp(x) = Z] the expected Z,, = Z. The evaluation of Z,e according to eq. (11) has been carried out with a step function density according to which

z:, =A

ha)

FjR dxx21a1,(x, R o

R, Z)\=

.

(12)

Since all the properties we are interested in depend on the muon at the nucleus, and finite-size effects perturb the point-like solution in such a way that the 1s energy level, and hence the 2p-1S energy difference, is primarily a measure of (r2), we have chosen the radius R of the homogeneous sphere, so as to reproduce, through R = 2d(r2>, the mean square radius of experimental electron scattering data “). However, in order to compare our analysis with the Ford-Wills results and to test the sensitivity of our predictions at the same time, another radius R has also been used in such a way as to reproduce the same (r2) obtained with the charge distributions adopted in their work “). Results are reported in table 1 for the binding energy E, the effective charge Zefi, the kinetic energy K and the “effective” kinetic energy, defined as

for a significant sample of nuclei. When two columns appear, the first one simply reproduces the Ford-Wills results, whereas the second carries ours for the two sets of R’s. As a further check the model-independent analysis based on generalized moments “) has been used. From the same transition energy as above which determines a corresponding generalized moment, the equivalent radius Rk is extracted. In its determination, the vacuum polarization, nuclear polarization, and electron screening, are taken into account. In the existing cases it is seen to be on the whole very similar to R corresponding to Ford-Wills analysis but with such tiny differences that cause practically no ensuing changes in the numerical results.

324

P. ~hristillin

et al. / Radiative TABLE

Ground-state IRI

“nucleus,

Binding

energy

properties

muon capture

1 of muonic

E

atoms

Z e”

(K)

(&a)

3.389 3.163

0.100

0.10 0.10

5.72

5.61 5.61

0.097 0.098

0.0000 0.0000

RO

3.681 3.491

0.177

0.18 0.18

7.49

7.41 7.42

0.173 0.173

0.0001 0.0001

‘%i

4.261 4.000

0.532

0.53 0.54

12.22

12.21 12.25

0.502 0.503

0.0021 0.0017

“Ca

4.812 4.450

1.047

1.05 1.06

16.15

16.12 16.24

0.935 0.936

0.0117 0.0098

*sNi

5.131 4.970

1.952

1.95 1.96

20.66

20.56 20.66

1.621 1.626

0.0472 0.443

42Mo

5.662 5.660

3.936

3.90 3.91

26.37

26.17 26.17

2.848 2.848

0.2038 0.2037

‘“Sn

5.982 5.960

5.198

5.15 5.16

28.64

28.41 28.44

3.443 3.445

0.3529 0.3517

@Cd

6.625 6.580

7.456

7.37 7.40

31.34

30.98 31.06

4.247 4.259

0.7008 0.6991

“W

7.011 6.970

9.102

8.98 9.01

32.76

32.33 32.41

4.754 4.759

0.9860 0.9882

s2Pb

7.023 7.100

10.590

10.47 10.42

34.18

33.87 33.70

5.185 5.147

1.2289 1.2197

7.466 7.490

12.175

12.02 11.99

34.94

34.48 34.42

5.489 5.483

1.5351 1.5288

92

U

In the different entries the first Column reproduces Ford-Wills results, whereas the second shows our results for different values of the nuclear radius. For details, see text. R is in fm and energies in MeV.

It is apparent how the agreement of our ground-state property calculations with relativistic predictions is excellent, i.e., always better than 1.5%, and to a very great extent insensitive to charge density details. Having accurately reproduced all the relevant 1s properties we can confidently pursue our calculations in the non-relativistic limit.

4. The muon propagator For both theoretical and experimental reasons we are interested in the spectrum only for k a 60 MeV. Hence we will search for a solution of gEls_ k(x, x’) only in that domain. This simplifies things quite a lot: first of all because the term y * VAo, with the assumed form of a non-point-like A 0, can be dropped with negligible error ‘>, thus making

our Green

function

a scalar object.

Straightforward

estimates

show that

325

P. Christillin et al. / Radiative muon capture

one can also neglect, although its inclusion would not cause any problem, the Ai term, so that our equation reads in the energy range considered [V” - (2m, -~-k)(k+~)+2e(rn,-~-k)A~(~)]g~,~_~(x,~’)=-S(~-~‘).

(13)

From now.on the symbol m2 = (2m, -F - k)(k + E) will be used. To recover the usual free formulation it is straightforward to show that this is obtained by neglecting the intermediate Coulomb field and the momentum distribution of the muon wave function. As a matter of fact, under the first condition, one has the familiar g(x

,

e-m’x-x”

=TL

x,)

(14)

47r IX--x’I ’

and assuming, for instance, 41L(x’)- e-ip’X’, as regards the second, the relevant term in eq. (7a) becomes d3Xtg(X,

xfj

e-ip.r'

e -i&,(x’-x)

1

1

=

m’+(k-p)‘-2mFk-2p-

k’

(15)

The standard expression for the rate in terms of an effective hamiltonian and averaged wave function is immediately derived. Let us now come to the approximate closed solution of eq. (13). A crucial remark is that m2 can be seen to be rather constant for k a 60 MeV and straightforward order of magnitude estimates show that the “potential” term 2e(m, -F - k)A&) is a correction of = 20% at most. This means that the parameter l/m I=L 2 fm practically determines the range of the propagator. Roughly the point x’ at which the photon is emitted (see fig. 2) cannot lie outside the nucleus of radius R (where the muon has to come to be actually captured) by more than l/m. We can therefore try a solution modifying the unperturbed gE,S_t (x,x’) by an appropriate “phase” factor [in the sense of a small variation with respect to e -m’r-x”//~ - ~‘1, ref. ‘“)I which gives IX--x’l

I0

eAo(y)ds

I

(16)

with symbols as explained in fig. 2. It is straightforward to prove that eq. (16) satisfies the defining eq. (13) up to (the neglected) e2Ai of Ol(Za/m,R)*/ and ey * VA0 of

m s

X-

Y

f

x’

--

R

Fig. 2 Quantities pertaining to the solution of the muon Green functions.

P. Chtistillin et al. / Radiative muon capture

326

0~Zcu/(m,R)2~, therefore completely in line with our approximations. Physically this amounts to a kind of an eikonal solution or in other words, that the particle is little deflected by the interaction from its straight-line trajectory. The integral in the above expression yields e

Ix’--XI Ao(y)ds=~~(lx-x’l)-e(x’-R)~~(~~)+e(x’-R)~~(so),

(17)

I0

where s0 is the positive root of R =JXr2+S~-22x’s”cos

8, (I8a)

2

F2(sj=

-ZCY In

s +x’ 2-2x’S cos e+s-x’cos

8

(18b)

x’(1 -cos e)

Eqs. (16), (17) and (18) represent, therefore, the muon propagator in the nuclear electrostatic field for energies greater than 60 MeV. It is worth stressing that since the propagator is a scalar object, the photon polarization will not be altered in this energy domain. Therefore emitted photons will remain mainly rightly polarized because of the term a - E(~)(A + -y~j in eq. (5). The extent to which the other term CL’&(‘) can be neglected will be discussed in the appendix.

5. Corrections

to the free case limit: results

We proceed now to the calculation of Gep=j

d3xp(x)(j

GhT = 1 d’x&)Ij

d3x’gE,S_k(x, dj e’*‘“-“‘~~(x)~~(x’)+h.c.l

To that aim the usual propagator

e

d3x’gE,s_k(X, x’) eilr”lr-X’)@F(xr))2,

decomposition

.

x’>x

$“go(2n+l)

I

Jr

-K"+l,2hj

2mx

(20)

16)will be used

-m,r-x’i

Jq==(

(1%

J- -1,+r,~(mx9~,(cos 2mx’

(21) eXXS)

x)x’

327

P. ~hrist~lli~ et al. / Radialice muon capture

and the property

notations

cos &,+= f - i?, cos 8’ = d * 2’.

Exploiting

the

well-known

= I0

0’) sin 8’ de’ = 2(i)“j”(kx’) .= e ikx’cosO’Pn(~~~

(22)

with the analogous definitions, k,(mx) = Grf tfansfo~s

&K+t,2(mx), d-

i,(mx) =

d- &r.+3,2(mxl,

into

Gee = 1 d3xp(x)( $, (2~ + 1) ~I;l’[i”t~~)QIL,(x)+~j~(x)k.(~x)l~}, (23) GhP= 1 d’xp(xl[ nE, (2n + l)j.(kx)$

~~(x)[i,(rirx)4.(x)+~i”(x)k,(Ax)l),

(24) where @k,(x) = Irn dx’xf2&(tizx’&, (kx’)Qj,(x’) , x

(25) @i,,(X) =

xdx’x’2i~(~x’)~~(~x’~~~(x~).

i0

For the actual evaluation the same density form as in sect. 3 will be used. The results obtained by numerically evaluating the above.expressions for k = 60, 70,80, and 90 MeV for some typical nuclei are given in table 2. Reported results are the ratio of Gpe and G,,P to [1/(2m,J~)~]]&]&, in order to have immediately the correction to the free case. For illustration purposes also the effect of the non-locality of the muon wave function alone on Get has been reported in the line A@. This has been obtained by normalizing our eikonal solution at every point x to [l/(2& (x)k + k2)2]j&jzv to single out this effect. The influence of the Coulomb field attraction is given, as a matter of fact, by (2~~k)2/(2~(x)k +k2)2 and this quantity (see above) can be easily checked not to have a linear behaviour in k, as a function of E and R. This explains the slight structure which can be traced in our final results for Ger and 6. t Eq. (21) has been used also for the actual propagator eq. (16). As a matter of fact, because of the already commented upon fact that ~x’--x~~ I/ m, it can be explicitly checked that the exact form can be approximated to the per cent by

+

eA&))(x

-x*1) =exp(-&lx-x’])

4alx -x’j This is of great help in simplifying the final numerical calculations.

4&

--x’I

P. Christillin et al. / Radiative

328

TABLE Results

for the muon

normalized

propagator

in the extended

muon capture

2

nuclear

field for the quantities

Gee; GhP (see text),

(-1 to the free case [1/(2m,k)‘]~4,~~,, f or various nuclei; in the column A@, the effect of sole non-locality

for Gpp is calculated

60

0.99

1.00

0.95

0.96

0.98

0.90

0.91

0.95

0.83

0.88

0.93

0.82

70

0.99

1.00

0.96

0.96

0.98

0.91

0.92

0.95

0.86

0.89

0.94

0.85

80

0.99

1.00

0.97

0.96

0.98

0.93

0.92

0.96

0.89

0.90

0.94

0.88

90

0.99

1.00

0.98

0.96

0.98

0.94

0.92

0.96

0.91

0.90

0.94

0.90

By inspection some trends are obvious. First, GhCis essentially given by the square root.of Gpp as might naively be expected. Second, both quantities are nearly constant in the energy range considered. Heavy nuclei are.much more affected than light ones where practically no corrections show up, but even in the former case, e.g., for lead, the correction amounts to at most, 10%. Our numerical calculations have been checked to reproduce alternatively in the case of constant 4, and ri? = m, the free propagator result to O.l%, and with the accuracy already discussed in sect. 3 the normalizing quantity Zen. As a final remark, let us stress that the way results are quoted depends, of course, on the chosen normalization, i.e., strictly free cases*. In conclusion, the combined effect of Coulomb attraction and muon non-locality has been shown to be much less relevant than expected from a naive extrapolation to high-2 of the point-like case.

6. Conclusions An adequately approximate solution has been worked out for the ground-state muon wave function and the muon propagator in the extended nuclear field for K Z=60 MeV, explicitly calculated for all nuclei through simple analytical expressions. These quantities, relevant for the study of RMC, have respectively proved to be essentially non-relativistic and not to differ markedly from the free particle case. We thank Professor T. E. 0. Ericson for a critical reading of the manuscript. $ In this connection let us also remark and in contrast with ref. ‘).

that we use throughout

Els = m,

-

E

in accordance

with ref. “)

P. Christillin

muon capture

et al. / Radiatioe

329

Appendix Let us consider

the spin structure

of eq. (5),

M = o&& = O,U”Y,U + Ydk a . E’*‘(A+ yj) + 211.’ and its contribution

to the differential

E(*)]&,(X)

,

rate. As far as the lepton part is concerned

have to calculate Tr LL:. Keeping only the first term in the square bracket the properties of the neutrino wave function we can easily get Tr LL;

= Tr &-y,(a =Tr

The explicit

(A.1)

* E(+))&(x)[&~,(cT

$,(~)(a.

* ~(.‘))a,b,(x)]~

(A.3

E(+))$~(x) .

~(-))-y,,lfy~(u~

form of the 1s muon wave function

we

and using

is

where

$‘2

=

C m-=+1/2

(1,

i,

4li-

m,

m,

9

YL~~~-~x~,

being usual Pauli spinors, i.e., the well-known combination of S- and P-waves in large and small components. It is therefore clear that they are connected in eq. (A.2) but that this interference actually disappears in the rate because of the angular integration over the muon angle. [Such an argument applies also for the case in which the lepton part in eq. (A.l) enters Tr LL; with the ordinary lepton current.] It might survive only in the presence of a momentum operator in the matrix element. Such is the case if one keeps also the second term in the square bracket of eq. (A.l). ,y”’

In this connection

some comments

are in order.

First, in the comparison

with the

photon energy l/k, the momentum pertaining to the transition is actually the “effective” momentum (average over the nuclear volume), which in the case of an extended nucleus is considerably smaller than its mean value over all space (sect. 3). Moreover, with respect to the dominant u * E term, a further factor f/g intervenes. Since our non-relativistic expectation values of ground state properties differ from the relativistic calculations by some per cent at most, even ascribing all this effect to the wave function and nothing to the unsophisticated form of the charge density, we have a definite indication that this f/g ratio is at most, of the order of 10%. We feel therefore reassured to a few per cent in keeping only the first term in eq. (A.l) and in disregarding interferences. This is valid also for the spinor part relative to muon spin photon angular correlation where extra momentum-dependent terms are absent. In principle, it is to be observed that the presence of the p * E term spoils, however, the proof of Fearing’s theorem 21) that asymmetry and photon polarization differ

330

P. Christillin et al. / Radiative mw_m capture

from 1 by terms of 0(1/M*). In practice both this effect and the influence of the Coulomb propagator do not alter sensitively the above mentioned free nucleon predictions. References 1) 2) 3) 4) 5) 6) 7) 8)

9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21)

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