LETT:ER]~ AL NUOVO CIMENTO VOL. 25, N. 12 21 Luglio 1979

Relativistic Corrections to Weak Exchange Currents and the Muon-Proton Capture.

W . GRIMUS

Inst i tut ]iir Theoretische t)hysik der Universitdt Wien - Wien

H. ]3AIER

Inst i tut ]iir Radium]orschung und Kernphysik der Osterreich Akademie der Wissenscha/ten . Wien

(ricevuto il 15 Maggio 1979)

The effects of the axial-vector mesonic exchange currents are of crucial importance in some weak-interaction processes involving the l ightest complex nuclei (1.2). Most interest ing examples are the beta-decay of 3H and the muon-induced deuteron break-up (involving the weak muon capture in the deuteron). Quite recently IVANOV and TRUHLXK (2) derived all the relevant currents using the hard-pion approach and PCAC together with the vector-meson dominance (VMD) model. The conventional model of DAUTRY (1) referred to the soft-pion approach.

Covariant models were used by JAus (3) in a consistent t rea tment of both: bound- state wave functions and exchange currents in l ight nuclei. He used Blankenbeclcr and Sugar methods (4) together with a procedure developed by Mandclstam (5). Applying his formalism to the beta-decay of 5H he did not find any significant relativist ic corrections. SEROT (s) invest igated relat ivist ic effects to order (v/c) 2 in weak and electro- magnetic processes in nuclei neglecting exchange effects.

However he points out tha t relat ivist ic effects could become impor tant if second- class currents would contribute to some weak processes.

I t is well known that the older methods of FUKUDA, SAWADA and TAKETANI (7) and 0KVBO (s) provide a consistent and simple (but formally ra ther involved) descrip- tion of all electromagnetic mesonic exchange effects including the relat ivist ic cor- rections (9).

(1) F. DAUTRY, M. RHO a n d D. O. I:tISKA: Nucl. Phys., 264 A, 507 (1976). (~) E. IV)~NOV a n d E. TRUHLIK: ~Vuel. Phys., 316 A, 437, 451 (1979). (a) W. J A u s : Nucl. Phys., 2 7 1 A , 495 (1976); Helv. Phys. ,dcta, 49, 475 (1976). (4) R. BLANKENBECLER a n d R. SUGAR: Phys. Rev., 142, 1051 (1966). (s) S. 1KA~DELSTA~: Proc. Roy. Soc., 233 A, 248 (1955). (s) B. D. SEROT: Nucl. Phys., 308 A, 457 (1978). (~) N. F[YKUD.~, ]i~. SAWAD~ a n d $I. TAKETANI: Prog. Theor. Phys., 12, 156 (1954). (8) S. OKUBO: Prog. Theor. Phys., 12, 603 (1954). (~) 1VL G.~RI a n d I t . HYUGA: Zeits. Phys., 277 A, 291 (1976).

353

~ , ~ w. GRIMUS and ~r. BAIER

E v e n the exc i ta t ion of nucleonic resonance s ta tes (isobaric exci tat ions) can be hand led using the same methods . E x t e n d i n g the m e t h o d of F S T to weak processes we der ived some of the re la t iv is t ic correct ions to some of the exchange currents i nvo lved in the muon-pro ton capture process. We obta ined to order M -3 one-body currents , and the one-pion exchange cont r ibut ion to the recoi l graph, the wave funct ion re-ortho- normal iza t ion and the pa i r te rm. The formulae obta ined are col lected in the paper . To lowest order we reproduce the usual expressions.

The me thod as out l ined in the examples of the paper can be used to der ive:reIa t ivis t ic correct ions to all the o ther essent ial exchange graphs (heavy-boson exchange, p-T: exchange, isobaric exci ta t ion v ia one-boson exchange etc.). The re la t iv i s t ic cor rec ted currents give then exchange correct ions to the P r imakof f -Hami l ton ian for muon- p ro ton capture. The currents should consis tent ly be used wi th the wave funct ions for the nucleus der ived wi th in the same procedure. In a first test of the resul ts one would s tar t however us ing some real is t ic re la t iv i s t ic correc ted nuc lear -wave funct ions (such as t he re la t iv i s t ic wave funct ions obta ined w i t h t he me thod of HA~NE~, BAIleR 0~ in the case of the three-nucleon problem) and then calcula te the above-ment ioned weak-be ta decay or nmonie cap ture processes.

We wri te t he Hami l ton ian (see e.g. 01)) for t he weak capture of muons on pro tons in the fol lowing way :

G cos 0 [ iF~(q2) cr;~e qo + Fa(q2) (1) H t - - ~ ] r , ~ f l ' r - ~ . F ~ ( q a ) ~ + ~ - - ~ - q~. §

iG~(q ~) -~ Gl(q~)yaya ~- ~ - aaeYsqq + G~(q2) ]

, y s q ~ .

Ix

Here we use t he no ta t ion of Bjorken and Drel l 0~). H~ is an opera tor in nucleonic space. (v,, vv, v~) ~ are t he usual isospin-matr ices . 0: Cabibbo angle, G: Fermi- coupl ing constant . ~'L: weak leptonic 4-current , M respec t ive ly ix: mass of the nucleon respec t ive ly muon.

Fig. 1. - ~z-+p-+v~z+n.

fi=7o, a j = ? 0 ~ : Dirac-matr ices . Some fur ther convent ions fol low from the d iag ram fig. 1 :

q = p , _ p , q~ = _ ~ 2 ( i p / i x _ ~r + ~ o ) ) = _ o . s s i x ~ ,

M n respec t ive ly M , : mass of the neut ron respec t ive ly proton.

(10) E. J. ~[A3I~IEL and H. BXlER: Lett. Nuovo Cimento, 22, 587 (1978). (ll) H. PRIMAKOFF: in Muon-Physics, Vol. II, edited by V. W. HUGHES and C. S. Wu (New York, N.Y., 1975), p. 3. (~) J. D. BJORKEN and S. D. DRELL: Relativistische Quantenmechanik, BI-98/98a (~annheim, 1964).

RELATIVISTIC 00RRECTIONS TO Wl~AK EXCI-IA~GE CURRENTS ~TC. 355

The form factors E 3, G~(q ~-) appearing in (1) correspond to second-class currents. The weak-leptonic current ]L is given by the following expression:

1 1 (2) i ~ = ~-~ z+PCv(x), A = - ~ x+PaCv(,',),

a: Pauli-spin matrices operating in leptonic space (in the following spin matrices without indices are operating on the leptonic spin whereas indices indicate nucleonic spin),

P = 1 - - a . p , p = p ( v ~ ) / E ~ , E~: energy of the neutrino,

g~ respectively ~ symbolize 2-spinors of the v~ respectively ~-,

~(x) = r exp [- - i(l~ - - Ev) t - - i E v 1~" x] ,

r muon wave function. We will neglect the spatial variation of r and use instead of

(3) @(x) = r exp [ - - i(~t - - E~) t - - iE~ ft. x] .

Now the Foldy-Wouthuysen (FW) transformation (12,13) was applied to eliminate the negative energy components in the nucleonic part of the operator H~ to the order M -a.

To perform the transformation, we have to write H I as a sum of even (si) and odd (0x) operators:

(4)

/ / i = EI § 0 i ,

G cos 0 E~ - @(x) T - z + P [F1- - ~F~ ~(~. p § ~-. a) + flF3 § G1 ~" a - -

2

- - G 2 f l ( S E . p - - e Z . a)] C,

G cos 0 O~ - @(x) ~- Z + P [F1r162 a - - F2 fl(,~" P - - q~" a) §

2

__E V ~ t - E v F i ~ F i (q 2) Gi =_ Gi(q2) , ~ = 2 ~ v ' e - 2 ~ '

~ - -

For the pion-nucleon interaction only the pseudoscalar form was used. As a consequence we had to take

O x ~ = igflr5 4" "r

(is) It. tIYu(~h and M. GARI: Nucl. Phys., 274A, 333 (1976).

~ W. GRIMU8 and H. BAIER

The full Hamiltonian to which the FW transformation has to be applied is given by

(5) h = t im + E + 0

where O = a . p ~- O x ~- Ox= and E ---- E~. We also have taken into account the Barnhill-freedom (see e.g. (la,1~)) by decom-

posing 0 by a sum of two odd operators:

O = A - ] - B ,

The 2's arc arbitrary real numbers. In the following we use the abbreviations:

(6b) a = 21--~ ~, b = ~ - - ~ , c = 2 a - - 2 1 = - - a - - b .

As a result of the F W transformation we obtain the corrected Primakoff Hamiltonian Hp including terms of order M -a (corrections to the order M -~ have already been obtained (1~)) and terms containing the pion field (seagull terms).

0n ly components acting iu the positive-energy space are writ ten down and t = 0 used in the following results.

First we state the Primakof~ Hamiltonian

(7)

(8a)

(8b)

H p = ho% h i § h~% h 3,

G cos 0 h o -- qS(x)v-Z+ p [Gv -}- Gxa. ao~ -~ G e ~ - / ~ ] $,

2

av= ~+d--~ ~176 F~-- ~--~o~ F~+ 1-~d, F~+

(14) M. V. B~.R~HILL I I I : Nucl. Phys., 131 A, 106 (1969). ('~) J . L. FRI~.R: Nucl. Phys., 87, 407 (1966).

RELATIVISTIC CORRECTIONS TO WEAK EXCHANGE CURRENTS ETC. 357

(9a) Mh I -- G cos 0

2 ~(x)v-z+P[gPfi 'P+g~247247215 +

(9b)

[ gP

ga

gA =

gx =

g p p =

gAP =

gAO =

1 1 (1 1 )

1 1 ( 1 1 ) ( 1 - ($2)F~-- ~ ( ( ~ 2 O~)F2__ ~ ~G 1 ~_ a

( 1 1 ) 1

( ~ ) ( 1 ) ( ) ~ ( ) ( ~ 1 ) ~eFl+ e--~O "~ ~ - - OG,-- ~ - - ~ g~+a ~-~ ~ ,

(10) M2h2 OcosO [(1 1 )

1 -~ ~ (~F2 + G1-- eG2)a'pa,v'p + ~Flli'pa'p +

(~ ~ 1 , ) + - - ~ - - ~ F3 + ~ 0 F 1 - - G~) a.v- a + ~ ~(F~ + G~-- G~) ax" P p2 +

a ] + ~ ( - oFl(~.(~,~o• + ,~x.p~.p~.p-,~.,w,(~.p)~) + ~a,,,or.p~.p) ~,

(11) 1 Gcos0

Mah3 2 2 ~(x)v-;I+P[Fla'P "~ Gla~

In the 2~'r:-weak interaction (Seagull) terms, the pion field may be el iminated by using the FST-method (7) leading to the pair current. To get this current to the order M -a one needs the F W reduced AeT:-interaction to the order M -2 (13):

g (12) Hi(i) = -- (~j.V.,~)(4(xj). ~ ) - (1 + c ) ~ (o~.p~, $(x~).~},

p j = - - V j (momentum of the nucleon). V..~ acts only on ~(x), {, } denotes the anti- commutator , j labels the nucleons.

3 5 8 W. GRIMUS a n d ~ . B A I ~ R

Using the general expression of the pair current, as given by GaRI and HYVGA (9) we obtain the pair contribution to the weak capture Hamiltonian:

(13a) H , ~ = i G c o s O \ 2 M ] 4:~ ~ ( x D z+P - - r

(13b)

1) ) ( , 1 )

+ --(1 + b)~qF~--ef~+ ( 1 - - c ) ~ + OG~ ,,~.,~+

( 1 1 ~ )

1 - - c

( , 1 ,) § ( l § 2 4 7 2 4 7 1 ~ . f i - -

1

V~ = - - 2 F a ~ § (1 - -c ) Gli a X a~.,

m pion mass, a~ is the transposed of aj, (~ x xk)- = } ((xr • l:~,)l--i('~j )< '~k)2), Xjk = X j - - Xk, rjk = ]X~kl, n~k = xjk/r~k,

(14)

exp [-- x] % ( z ) = - - ,

~Jl(x) = (1 + x) %(x),

(33) ~ ( x ) = 1 + - + ~/o(x) ,

x

J - ~ z ( x ) = - - 6~,t~ ~4(x)/x~" + x , , x ~ ~ 4 ( x ) / x ~ .

Now we add the weak recoil and wave function re-orthonormalization currents. These contributions cancel each other to order M -~ giving a first nonvanishing expression

RELiTIVISTIC CORRECTIONS TO WEAK EXCHANGE CI/RRENTS ETC. 359

in the order M -3. Our result is given in the following formulae:

(15a)

(15b)

(1 +c)m~ g ~2 ~/l(mrj# )

�9 [%. ~(=~-. ,~a~.p~-- ~- n~=~'pj), ho~

G cos 0 h0~ - - ,2 0 ~(XJ)T7 Z + p [ F 1 ~- F3~- GI(~~ {~]] ~ ,

[, ]: denotes the commutator . The final Hamil tonian is then the sum of all the terms obtained and all other exchange

corrections (which we will derive in a forthcoming paper applying the same methods)

(16) H = E ~p(]) ~- /~'l)a, ir -~ HtocoII+WFR "~ . . . .