On the symmetries between neutrino and antineutrino-nucleon elastic scattering

Pramana, Vol. 10, No. 4, April 1978, pp. 361-373, ~) printed in India.

On the symmetries between neutrino and antineutrino-nucleon elastic scattering

M T T E L I a n d R G T A K W A L E *

Department of Physics, Shivaji University, Kolhapur 416 004

*Department of Physics, University of Poona, Poona 411 007 MS received 2 September 1976; in final form 6 February 1978

Abstract. Various symmetry relations developed between neutrino-neutron and antineutrino-proton elastic scattering cross sections are surveyed and an identity between scattering amplitudes and a symmetry between cross sections of these processes established by considering CPT and G conjugation invariance of current matrix elements. A symmetry is obtained giving rise to a theorem on the nature of contribution of form factors to terms in the cross sections.

Keywords. Neutrino-nucleon scattering; polarization of particles; nucleon weak current form factors; scattering amplitude; scattering symmetries.

1. Introduction

Neutrinos experience only weak interactions a n d hence their processes are useful in studying the weak interaction properties such as hermiticity, C V C hypothesis, G- conjugation o f nucleon weak currents and thereby the properties o f first class and second class currents without the obscuring effects o f electromagnetic and strong interactions. The elastic scattering processes (hereinafter referred to as (I) and (11))

and

v - k n + l - b p (1)

~ + p ~ i + n ~ )

involve respectively np and pn currents which do not have the same structure.

Therefore, in principle, b o t h (I) a n d (II) should be studied independently of the other. C o m p u t a t i o n a l repetition o f this sort can be avoided if these processes have s y m m e t r y between their scattering amplitudes and hence between their cross sections.

During the last decade(I) and (II) have been related to each other by m a n y workers in a variety of ways each o f t h e m different, some incomplete or erroneous. It will be clear f r o m the review given in section 3 t h a t the earlier w o r k does not establish coherent symmetry between amplitudes a n d cross sections. The present p a p e r aims to establish such a s y m m e t r y between (I) a n d (II) based on C P T a n d G-transformation properties of particle currents by neglecting final state electromagnetic corrections.

361

362 M T Tell and R G Takwale

In section 2 we outline the mathematics used in establishing the symmetry. Section 4 is devoted to establish an identity between the amplitudes of (I) and (II) and a symmetry between their cross sections. By combining it with generalized Bell symmetry (Tell 1974) a symmetry relating a process to itself is obtained leading to a theorem on th~ contribution of form factors to the cross sections. In section 5, the symmetry developed in section 4 is compared with earlier symmetries and proper corrections are suggested in section 6.

2. Mathematical equipment

We take k 1, Pl, k2, P~ and s~, st, s 1, s~ respectively as four momenta and polarization four vectors (Okun 1965) of the particles in order of their occurrence in (I). The hamiltonian for (I) is taken as

H = (i/~/2) Jaja (1)

Ja = ~t, (P~, s2) Oa ~bn (Pl, Sl) (2)

Oa =•a (gv -? ga es) + iPa (fv )- fa en) -!- iqa (h v ! h a Ys) (2a)

P = Pl + P~, q -- P2 -- Pl : kl -- k2 (2b)

gv, A,./'v,a and hv, A are complex form factors and the functions o f q z. The leptonic current is given by

ja = ~, (kz, s,) la ~bp(k 1, s~) (3)

la = ~'a (1 + Ca)- (3a)

The nucleonic current (2) can be separated into first and second class parts of vector and axial vector currents (VI, n, AI, ii) as

v , : :

3. (gv y, + ie,fv) 4,.

Vn -= ~bp (iqa hv) ~b.

An -- ~b~, (iPaf a ~ ) ~b.. (4)

The notations l/t, n, A~, n will be used in the text to denote the corresponding form factors as

Vl ~- gv, fv, VII =-- hv, ai ~ gA, ha, An -~-.fA" (5) The transition amplitude for (1) from eqs (I)--(3) then becomes

,4. = ( t p l H l v n ) = <p IJa I n ) ( l l j a Iv)

-= (i/~/2) ft, (p,, s~) Oa u. (Px' st) tT, (kz, st) la u e (k~, sp). (6)

Neutrino a n d antineutrino elastic scattering 363 Under PT-transformation, P T ( q a ) : q a . However, under T or P T , k 1 <--~ k2, Pt <' > P2 and hence qa changes sign under PT. Similarly sa is PT-odd. According to Muirhead (1968) and Okun (1965) the scattering amplitude As, (6) transforms under PTand charge conjugation C respectively as

where

P T ( A ~ ) = ( i / v ' 2 ) a . ( p ~ , - - s O O ' a u , ( p ~ , - - s 2 ) a v ( k j , - - s , ) l a ' ut(k~,--s,) (7)

and

(7a)

l'a = 7a (I--75) (7b)

c ( A . ) = (i/V2) ~. (p~. sO 0'~ v~ (p,, s.) ~ qq, s.) la' vz (k,, s,)

(8)

Combining the P T and C transformations (7) and (8) one gets the C P T transforma- tion of Av as

CPT(Av)=--(i/~2)vT,(p2,--s~)Oxv~(pl,--sl)vT(k2,--sl)lav~(kl,--su) (9) G-conjugation (Weinberg 1958) of the nucleonic part of (9) gives

i.e.

c [~p (p~,-s2) o~ ~.

(pl.-s01

GVI G -1 : V I , 6II,, G -I = - - lilt

GAI G -x : - AI, G A n G -l : A w

(11)

The cross section is given by lAy ]2 and it inw~lves terms like k i p 1, k l s 1, sls~, etc and the angle brackets ( p l p 2 k t s t ) =: iPl , p~. kta s t , ~,,ap. (plp2sxs2), etc. All the angle brackets are T-odd and the terms without them are T-even. The terms involving odd number of s, are PT-odd while those involving even number of them are PT-even. The P-parity nature of the terms follows from their behaviour under T and PT.

3. Earlier work on symmetries

3.1. L e e and Yang s y m m e t r y

Lee and Yang (1962) were the first to develop theorems for the functional form of the cross sections &to and dcr~ of (I) and (II) respectively by using the matrix elements given by eq. (6). They have given the expressions for cross sections in the laboratory system when the target nucleons are unpolarized and the final particles are longitu- dinally polarized (eq. 88 of Lee and Yang). These expressions can be summarised as follows:

364 M T Teli and R G Takwale

da~(st ' Sl)= a [AtXq_A2X-Xq_A3 Y ÷ A4 y-x t As]

d~(s 2, sl)-=~ [B~X+ B2X-a-I BaY-I B4Y -x I Bs]

Al=Al(Sl, Sl), B,=Bi(s~, Sl), i=1, 2, 3, 4, 5.

(12)

(13)

(13a) a=½(1 +vl) Kd(q 2) and Sx, s2, s I and s t are the helicities of the final state particles n, p, / and I in (I) and (II) while v I is the lepton velocity. The structure functions Ar and B~ are functions off, g and h and can be related with the Lee and Yang functions a + , b_ k, c, a ' + , b'+ and c' as

At~--a+, h~4~ : a _ , A3c v : b + , A4~2~ =b_,

A s = + ( - - ) c for Sl--L(R) (I4)

Al =A~ (s I, s I, gv, a, .['v.A, hv.a) and

a+ ~a+(s t, s l) gv.a, .fv.A, hv, A), etc. (14a)

Bt~zt-=a'-, B2c4~= a'+, Bz~I, b'+, B4~:=b'_

Bs=+(--)c'

Bt---Bl (s2, sl), a'+~--a':i:.(sa, sl), etc. (15a) Lee and Yang relate the functions a'+, b'± and c to the corresponding ones a±, b~

and c by the following (L Y) transformation operator obtained from hermiticity (eq.

92 of them):

L Y- :[gv.A---+g*v,a, .fv----~f*v, hv--'-~--h* v, .fA----~--f*a, hA----~+h*A]

(16) Thus the Lee and Yang (L Y) relations between a'+ and a :, etc. are

a'±(s S, sl)----L Y (a4-)=a+ (st, s 1, g*v,a,.f*v, --h'v, --f*a,

h'a)

By using eqs (14)-(17) the L Y relations between A~ and Bt then beeome

BI~ :A'2~1~ , B j : A ' j , j = 3 , 4, 5 for Sl=L, S l = - - S l : R (18)

and

where

Ba~4~---A'4~3~, B~=A'j, j = l , 2, 5 for Sl=R Sl=--Sl=L (19)

A';=L YA~-Aj (s 1, s 1,

g*g,A' f*g' --h*g'

Neutrino and antineutrino elastic scattering 365 We note from (18) and (19) that the L Y relations do not relate B 1 directly to A1, B 2 to Az when s / = L and also B s to A 3 and Ba to d 4 when S l = R . The lack of such a direct symmetry between the respective coefficients of X, X -1, Y and y-1 in eqs (12) and (13) does not, therefore, allow one to obtain d% by a mere application of L Y (16) to dcru (12) unless X and X -1 or Y and y-1 are also interchanged as per the case of lepton polarization in it. When this is done, one has the following L Y symmetry between &r o and d~o

do~(s2, Sl)=dcru(sl, --Sl, g'v,,4, f ' v , --h'v, - - f ' A , h'A, X-X, X, Y, y-x)

for Sl=L, s t = --Sl=R, and (21)

dar,(sz, Sl)=dao(sl, --Sl, g*V,A,f*v, --h'v, - - f ' a , h*.4, X, X -1, y - l , y)

for sl=R, s t = - - s t = L . (21a)

3.2. Adler symmetry

Starting with the matrix elements Ao(6), Adler (1963) evaluated covariant expression for dory for the production of polarized nucleons and leptons. For obtaining d ~ , he changed the sign of ~'5 in both the leptonic and nucleonic parts of (6). The result of this mechanism, as Adler obtains, is that d~rv and da~ differ only in the signs of particle polarization four vectors and in the signs of T-odd (angle bracket) terms.

Separating d ~ , ~ into T-even (do +) and T-odd (do-) parts as

d~ro, i =do+p,o ~ d~r-o, ~.

(22)

Adler's symmetry can be written as

da_ -t= = -4- do 4- ( _ s t , __Sl, gv, a, v,a, hv, a)

(23)

where the upper sign is for do + and the lower one is for do- and where s 1 and s I are respectively the polarization four vectors o f p and l in (1).

Because of the assumed equality of the form factors in the matrix elements of (I) and (II') by Adler, eq. (23) gives equal polarization independent cross sections in disagreement with the cross sections of Marshak et al (1969).* It then appears that the Adler symmetry (23) is not complete.

3.3. Sarkar's symmetry

Sarkar (1966) following the method of Adler considered the process (I) to include target polarization s 2. He, however, neglected the lepton mass and so the form factors hv, A are absent in his expressions. Sarkar's symmetry is, therefore the following d<,~ --- -~ d<~ ~ (--s,, --s~, gv, A' fv, A)" (24) By splitting the cross section into functions of particle momenta and form factors

*Here replacements of hA by --fA and of fa by hA is to be made in order to agree with our notations, Also compare the results with those given by Smith (1972), his eqs (3.18) and (3.22).

366 M T Teli and R G Takwale

which are independent of particle polarization, .4, dependent on s~ and s~ separately, B(sl) and C(s2) and simultaneously D(s 1, s~), Sarkar wrote

a ~ = a + B(sl) + C(s~) + D(sl, s~)

(25)

A ~ A ( k 1, k 2, Pl, Pa, gv, a, fv, a), etc. (26)

In terms of the following transformations

ga --> --ga, Pl < >P~, MI<---->Mg, st -+ --sz

(27)

Sarkar related C(s2) with B(sl) in the same process (I) or (II) as

C(s~)--B(--s~, k 1, k~, p~, Pl, 312, 341, gv, --ga, fV, A)" (28) Eq. (27) will be regarded as Sarkar's internal symmetry operation.

3.4. Pals symmetry

The matrix elements (6) used by Pais (1971, 1972) for (I) and (II) differ in the sign of Y5 only in the leptonie part. As a result, the expressions of dcrv and dcr~ differ from each other in the signs of the polarization of the considered final particles and in the sign of the T-odd terms and in addition, all the axial vector form factors have opposite signs. Thus the Pais symmetry is

dor,+:+dccufl=(P, n, q, - - s 1, --s 2, g v , f v , hv, --ga, --fA, --h~) (29) P-:Pt + P2, n = k 1 + k 2, q = k t - - k ~ : P 2 - P t . (29a) The Pals symmetry gives correct signs of ga and h a but not of.fa when compared with the polarization independent expressions of do~ given by Marshak et al (1969) where h v changes sign w h i l e f a (i.e. h a of Marshak e t a l ) does not and hence it is also in error.

3.5. Wolfenstein's theorems

Wolfenstein (1972) derived two theorems for relating dc,o with d~ v and one theorem for relating a process to itself. The theorems are in terms of ten bilinear terms aa obtained from Vt,~i and A~,u which are classified in the following four groups.

AIAI: AIAI, VIii, AnAII, ~iiVii, AIAH: AIAH, VIVII,

VIAI" I/'IAI, VIIAII, VIAIi: VtAm AIVIt. (30)

By applying hermitieity, lepton crossing and simultaneous exchange of initial and final momenta of the particles in (I), Wolfenstein obtains theorem 1

Neutrino and antineutrino elastic scattering 367 Theorem 1: In going from process (I) to the (II), observables which are P-even and exchange even (odd) and P-odd and exchange odd (even) satisfy

a v = ~ aa ~ for AIAI and AIAII

: =F aa ~ for VIAt and VIAn. (31)

By applying C-conjugation to (I) and G-conjugation to the nucleon current he obtains theorem 2

Theorem 2: In going from process

(1)

(II)

aV = i _Gt a . ~ for AIAI a n d VIAII

= T a~ ~ for VIA 1 and AIAn. (32)

Adding theorems 1 and 2 he gets theorem 3 rel~,ting a process to itself.

Theorem 3: For either reaction (I) or (II), observables which are T-even (odd) and exchange even (odd) must satisfy

a~ = 0 for AIAII and VIAII (33)

whereas for observables which are T-even (odd) and exchange odd (even)

a a - - 0 for AIAI and VIAl

(34)

Wolfenstein's theorems, though general, contain the following weak points:

(i) No symmetry is implied between scattering amplitudes of (I) and (II).

(ii) Lack ot maximum symmetry with minimum factors thus entailing lengthy and difficult statements.

(iii) Their execution is tedious.

3.6. Bell symmetry

A symmetry following from hermiticity and CPT invariance of spin averaged lepton tensors in (I) has been obtained between the unpolarized cross sections d% and dcr~

in terms of Mandelstam variables s, t, u (Bell 1963, Smith 1972). This is stated as

d ~ (s, t, u) = dcrv (u, t, s)

(35)

and shows that while going from (I) to (II) s and u exchange and hence (s-u) changes sign.

The Bell symmetry (35) has been extended (Tell 1974) for (1) and (II) involving nucleons of unequal masses and polarization of all the particles by using the fact that

368 M T Tell and R G Takwale

under CPT, the polarization of four vector changes sign and hence the exchange of s and u should be accompanied with the exchange of nucleon masses M 1 and Mz and their polarizations with sign change. The Bell symmetry generalized in this way then is the following

d~o (kl, ka, p~, Ps, Mx, Ms, s t, st, ss)

= d % (k 1, ks,--ps,--pl, Ms, M 1 , - - s l , - - s s , - - s l ) (36) or da n (s, t, u, M 1, Ms, s t , sl, s2)

= d % (u, t, s, Ms, M 1 , - - s 1, ss,--sl). (37) The six types of symmetries between d~v and doo surveyed above are completely different from each other with the exceptions of the equivalence between the Adler and Sarkar symmetries (23) and (24) and a partial similarity between the Adler and Pais symmetries (23) and (29), the latter differing from the former two only in the sign of axial vector form factors. However, all the three are in error. The Lee and Yang symmetries (21) and (21a) are obtained under special circumstances viz. by considering (I) and (II) to produce only longitudinally polarized leptons and nucleons in laboratory system of the target. The generalized Bell symmetry (36) or (37) is quite simpler than Wolfenstein's theorems which require knowledge of the transfor- mation properties of the observables under P, P T and exchange of initial and final momenta.

4. Further symmetries

We now wish to establish an identity between the amplitudes Av and A~ of (I) and (IT) respectively and a symmetry between their cross sections based on the C P T and G-conjugation properties of the particle current matrix elements by neglecting final state electromagnetic corrections.

4.1. An amplitude identity

Statement: If A, is

given

then

Ap = (i/v'2) ft, (Ps, sz) Oa u,, (Pl, sl) ftl (k~, sl) ta u u (k 1, su)

= A~ (kl, k~, PI, P~, s~, s 1, s 1, s~, Vi,u, AI,II)

(38)

(39)

A~ = (i/v'2) a. (p2,--s2) O'aup (p~,--s 1) -~1 (k2'sl) I~ % (kl,--s p) (40) : A~, (kl, kz, Pl, P2,--sv,--Sl,--sl, --s~, VI,-- VII,--AI,An) (41) with O'a as given by eq. (7a).

N e u t r i n o a n d a n t i n e u t r i n o elastic s c a t t e r i n g 369 On squaring the amplitude A~(41) we get the following symmetry between the cross sections

do ~ 4- ==dop 5= (ks, kz, Ps, Pa, 3/11, M s , - - s v , - - s l, - - s l , --s2, VI, - - V n , - - A I , A u (42) --: do v ~ (s, t, u, M 1, M 2 , - s ~ , - - s l , - - s l , - - s 2, V I , - - V n , - - / l l , AII). (42a) It is clear from (41) and (42) that the symmetry between the amplitudes is carried over to the cross sections. This coherence between the amplitude identity and the cross sectional symmetry is not present in the symmetries discussed in section 3.

The proof of (40) and hence of (41) directly follows from the combined operation of C P T (9) and the G-conjugation (10) on Av(6).

4.2. l n t e r n a l s y m m e t r y

Let us denote by S 1 the generalized Bell symmetry operation which gives (36) when operated on dov. The operator S 1 is then

$1 : [Pl ----> --P~, P2 ----> - P l , M1 ~-"--> M~, sp • > --sv,

S l - ' - - + - - S l , S t ~ " - > - - S 2 , S~ ---'--> - - S l ] . (43)

Similarly the operator for the symmetry (42) is

S z : s 1 ~--> --s~, Vn ----> - - VII, A t ' > - - A I (44) with sj : s . , s I, s 1, s2.

Now if we apply S t to d% we get d~r~ and when the latter is further subjected to Sz we get back dop. This is also true for obtaining d ~ from itself by the combined operation o f S~ and $2. The combined operation which relates a cross section to itself is thus

S : S 1 S ~ = [Pl ----> --P2, P2 ~ - > - - P l , M I <---> M2, ss <-- • > s~,

Vnt > -- VII, -//l - > --AI ]. (45)

We then have the following internal symmetry.

S d o ~= : do 5= (46)

o r

d ~ ± -~(kl, k2, Ps, P',, M1, M2, sv, s t, ss, s.z, 1/i. 11, AI, n)

= do ± _~',~ -i- (ks, kz, --P2, - - P l , M~, M j , su, s t, s2, sl, Vl, - - I'll, - - A I , Al~). (47) The internal symmetry operator S (45) relates the term dependent on s~ [i.e. B(s~) in eq. (25)] to that dependent on s,, [i.e. C(s2)] and hence it gives the advantage of eva- luating say, C(s2) from B ( s l ) and vice versa.

370 Thus

M T Teli and R G Takwale

C(s2, kl, k2, Pl, Ps, M1, M2, s , , st, VI, II, AI,

II)

- B(S2, hi, ks, --P2, --Pl, Ms, M1, Sv, st, VI, -- VII, --AI, An). (48) This internal symmetry (48) is equivalent to that used by Sarkar, eq. (28) since the functions B and C and also A and D in (26) involve the terms like k i p 1 • k , p 1, Pl pg. • k 1 sl, etc. which transform to the same respective terms under the operator S (45) as well as under the transformation (27). This shows that the transformation (45) is exactly equivalent to that given by eq. (27). From the internal symmetry we obtain the following theorem.

Theorem." Irrespective of whether T-invariance holds or not, for both the processes (I) and (II) observables which are even under the operation S (45) must satisfy

b~ : 0 for AI Au, VI AI (49)

while for those odd under S

d~ = 0 for A I A[, VI An (49a)

Proof: Let us write day, ~ (as Wolfenstein 1972) (suffixes v, ~, dropped)

a<'+

(Z

Cl

da = Ai AI, Pi All, ba : AI AII, Pl A| (51)

f ? = f ? (kt, k~,pl, ps, M 1, Ms, ss). (51a)

Under Sa, da ---> d,. and b~ ---> --b~ while '4-

', D= ~ = :,= = D f (k,. ~,.-ps.-p,. M,. M..-,..-s..-s,,-,,)

(52)

Hence under S, ¢q. (47) becomes

i.e.

Now from (52) we have

(54)

Neutrino and antineutrino elastic scattering Hence S has eigenvalues A = 4- 1 so that

f : is either even (;t = -t-1) or odd (A = --1) under S (47), and hence

(55) then give

[ +

The theorem then follows from (56).

371

(55)

Eqs (53) and

(56)

5. Comparison with earlier symmetries

We now show that the LY symmetries (eqs (18) and (19)) and the S~ symmetry (42) or (44) give identical results (eqs (59) and (59a) below). S~ gives the following symmetry between the Lee and Yang functions Al and Bl when applied to (12) to obtain (13).

S,(s2, s t) : S~A, : A, (--s 1, --s t, gv, f v, --hv, --g A , f a, ha). (57) Consider the case of (--sj) = L and s t -- R and take B~ = B 1. Then the combina- tion ofeqs (14), (15), (18) and (19) gives eq. (58) below while eqs (14), (15) and (57) give eq. (58a) below.

B 1 : a'_ : L Y(A~) = L Y(a_) : a_(s 1, --sl, g*v. A, - - h ' v , - - f ' a , h'A) (58) B 1 = S~A1 : Sza+ : a+(--s 1, --st, gv, --gA, fv, A, --by, A)" (58a) Eqs (58) and (58a) then give

a_(sl, L, g ' v , f ' v , - - h ' v , --f*A, g*a, h'A)

-~- a+(--s t, L, gv, f v , --hv, --gA, fa, --ha) (59)

Similar relations for b+ and c are

bj (s t, L, g ' v , f ' v , - - h ' v , g'A, - - f ' A , h'a)

--bj (--s 1, L, gv, f v, --hv, --g A, f a, --hA) (59a)

bj = b+, b_ and c.

Equations (59)-(59a) can be easily verified by using the Lee and Yang expressions (their eqs (90) and (91)) of a + , b± and c.

372 M T T e l i a n d R G T a k w a l e

Wolfenstein's theorem 2 also follows from $2 and hence from the amplitude identity (41).

Let r --- (r i, re) where

r i = (1, sls~, s i p J. s2pk, ( p l p ~ s j s , ) . . . . )

and

r 2 = (s i Pa, s2Pj, ( P l P , P k S j ) . . . . )

be the rows respectively of the PT-even and PT-odd elements. Let d~ and ba (51) form a column. Taking A as the matrix of elements which are functions of Mandel- stam variables s, t, u and of particle masses only, we write d(r, as

ao.

r=)r A,I A, ] [ao] (60)

L.,I3 [ A4 j b,,"

Then applying the symmetry operator S,, to (60) we obtain, since A does not change,

and this is Wolfenstein's theorem 2.

6. Conclusions

From the foregoing discussion it is concluded that the symmetry S s developed in section 4 is quite general and simpler in its form as compared with Wolfenstein's theorems and it gives results (eqs 48, 59a and 61) consistent with those obtained by earlier workers thereby strengthening the validity of the amplitude identity proposed in eqs (40) and (4l). The symmetry S 2 when combined with the generalized Bell symmetry (36) gives a theorem on the contribution of form factors to the cross sec- tions. The symmetry S 2 is complete improvement over the symmetries of Adler, Sarkar and Pais. The generalized Bell symmetry S 1 (36) exchanges nucleon polariza- tion while S~ does not. As a result (i) S 2 gives cross section d ~ from dc~p for (II) with polarized nucleons in the same states as those in (I) while the generalized Bell sym- metry gives d~v for polarized nucleons in the states opposite to those in (I). For example, if (I) involves unpolarized neutron targets and polarized final state protons then S 2 gives d~r~ for (II) involving unpolarized proton targets and polarized final state neutrons while S t (36) gives &r~ for (II) involving polarized proton targets and unpolarized final state neutrons. S t and S~, therefore, do not give identical results.

(ii) If both the initial and final state nucleons are simultaneously polarized then St and S~ give identical results.

In view of the accuracy of the symmetry S~, it follows that in Adler's (and also in Sarkar's) equations of d~o the signs of the form factors h v, g.4 and h a should be

Neutrino and antineutrino elastic scattering 373 changed along with the sign change o f all the T-odd (angle bracket) terms. Similarly, the equations o f d ~ o f Pals (1971) require sign change in all the T-odd t e r m s a n d o f h v a n d f a , the sign change o f f a required to correct the minus sign used by Pais.

As a result the terms w 5 and w 6 o f Pais (his eq. 2.24) should change sign.

Acknowledgements

The authors wish to t h a n k Prof. M R Bhiday a n d D r R N Patil for interest in o u r work. One o f us ( M T T ) offers sincere thanks to the Shivaji University, K o l h a p u r f o r financial assistance under U G C ' s travel grant scheme a n d to the Physics D e p a r t - ment, P o o n a University for generous hospitality.

References

Adler S L 1963 Nuovo Cimento 30 I020

Bell C S 1963 CERN NDA Neutrino Seminars (CERN 63-67)

Marshak R E, Riazuddin and Ryan C P 1969 Theory of Weak Interactions in Particle Physics (New York: Wiley Interscience) pp. 315-16

Muirhead H 1968 The Physics of Elementary Particles (Oxford: Pergamon Press) ch. 5.

Okun L B 1965 Weak Interactions of Elementary Particles (Oxford: Pergamon Press) p. 59 and 75.

Pals A 1971 Ann. Phys. 63 36

Pals A 1972 Ann. Phys. 69 604 (erratum) Sarkar S 1966 Acta. Phys. 21 211 Smith C H L 1972 Phys. Reg. C3 261-379

Tell M T 1974 Ph.D. thesis University of Poona Ch. 4 Weinberg S 1958 Phys. Rev. 112 1375

Wolfenstein L 1972 Ann. Phys. 71 569

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