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Space-time structure of weak neutral currents
Nuclear Physics Bl12 (1976) 387-399 © North-Holland Publishing Company
SPACE-TIME STRUCTURE OF WEAK NEUTRAL CURRENTS (11). Elastic scattering
C.F. ClIO and M. GOURDIN Laboratoire de Physique Th~orique et Hautes Energies, Paris *
Received 9 April 1976
The elastic scattering of neutrinos and antineutrinos on nucleons in which the initial and final nucleon polarizations are measured, is studied from the point of view of the space-time properties of the currents. Structure functions are defined and experimentally measurable quantities which can be usual to disentangle the Lorentz structure of the currents are given. Second class current effects are also examined.
1. Introduction In a previous paper [1 ] (which we shall refer to as paper 1 from now on), we have studied what information about the Lorentz structure of neutral currents one can obtain in inclusive neutrino and anti-neutrino reactions using a polarized target. It was found that as in the unpolarized case, the "confusion theorem" [21 holds; i.e. a mixture of S (scalar), P (pseudoscalar) and T (tensor) interactions can always simulate V (vector) and A (axial vector) interactions. In this paper, we study the neutrinonucleon and anti-nuetrino nucleon elastic scattering in which both the initial and final polarizations of the nucleons are measured. We realize that these types of experiments are very difficult to perform. On the other hand, these are among the cleanest and most straightforward experiments we can think of which can settle conclusively the question of the Lorentz structure of the weak neutral currents. If S, P and T are excluded, the relative amount of V and A can be quite easily determined [3]. The pN elastic scattering experiment measuring nucleon polarizations correlations has a further advantage in that it can be used to study second class current [4] effects. In the neutral current case, the study of second class currents can in turn shed some light on the question as to whether the initial and final neutrinos are identical [5]. Our treatment is valid for neutral current interactions as well as charged current interactions when the mass of the final lepton can be neglected. However, in the * Postal address: Universit~ Pierre et Marie Curie, Tour l 6-1 er &age, 4 Place J ussieu, 75230 Paris Cedex 05, France. 387
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CF. Cho, M. Gourdin / Weak neutral currents (II)
charged current case, it is probably easier to measure the final lepton polarization rather than measuring the nucleon polarizations [6] in order to study the Lorentz structure of the current. In sect. 2, we study the kinematics of uN elastic scattering when the target and recoil nucleon polarizations are measured. Structure functions are introduced and their general Lorentz structure and symmetry properties given. The experimentally measurable target and recoil nucleon asymmetry cross sections and the nucleon polarization correlations are expressed in terms of these structure functions. Our treatment here can be trivially generalized to the production of any hadrons with spin }. In this connection, we would like to mention that while the production of strange baryons by neutral currents is expected to be very small, there is still the possibility of producing baryons with new quantum numbers which decay weakly. In that case the polarization of the final hadron can be obtained from the angular distribution of the decay products. In sect. 3, we write down the most general form of the S, P, T, V and A currents, including second class currents, in uN elastic scattering. The structure functions can then be calculated explicitely in terms of the S, P, T, V and A form factors. This is done in the appendix. Using these results, we list in sect. 3 the experimentally measurable quantities which are most useful in probing the Lorentz structure and the second class current effects of the weak currents. Finally, in sect. 4, we discuss our results.
2. General discussion 2.1. K i n e m a t i c s
We study the reaction Q+N~'+N',
(1)
where ~ and 2' are leptons of momenta k and k' and N and N' are nucleons of mass M with momenta p and p'. £ is a neutrino or antineutrino and £' is either a neutral or charged lepton, q = k - k' is the m o m e n t u m transfer, s = - (k + p)2 is the square of the c.m. energy. We consider the general case where the weak current is a superposition of S, P, T, V and A Lorentz components. As in our study of inclusive neutrino reactions [1 ], we analyze the cross section by looking at the t-channel where the neutrino-antineutrino annihilation amplitudes proceed via spin-zero and spin-one exchanges [7]. It follows that the unpolarized cross section is a second-order polynomial in cos 0 t where 0 t is the t-channel c.m. scattering angle. In the local Fermi approximation where q2 is small compared to the square of the intermediate vector-meson mass, we get d°unp dq 2
_
G2q2M 2
27r(s - M2) 2
{ao(q2 ) + al(q2 ) cos 0 t + a2(q 2) cos20t ) ,
(2)
CF. Cho, M. Gourdin / Weak neutral currents (II)
389
where 2s - 2M 2 - q2 COS 0 t =
q2
2M[q2(1 + r~)] 1/2 '
r/
(3)
4M 2 ,
and e = cos20 C
for charged current case,
=1
for the neutral current case,
where 0 C is the Cabbibo angle. Also G is the conventional Fermi coupling constant. In eq. (2) as well as in the subsequent discussion, the lepton mass is neglected. To describe the polarization effects for the target and recoil nucleons, we introduce a space-like unit pseudovector N ( N ' ) orthogonal to p(p'), with N 2=N '2=1,
(4)
N.p=N'.p'=O.
The polarized cross section can then be written as do _ 1 d°unp {1 + N . A + N " A' + NuN'vCUU} dq2 2 dq2
(5)
The four-vector 2x(A') describes the various target (recoil) nucleon asymmetries, and the tensor Cuv describes the polarization correlations between the target and recoil nucleons. Because of constraints (4), there exists only three independent four-vectors orthogonal to p which we choose to be [1 ] 1
Q"
2M[r/(1 + n ) ] l/2 [qu
2npu] ,
1 ( k + k , ) u + p ' ( k + k ' ) [~qu+pu] Ku = 2M-~ 342(1 + r/) 1
,
(6)
T u = M- euv p apUQOK ° .
Similarly, we choose the three independent four-vectors orthogonal to p' to be K u, T u and Q'
=
1
t
2_M[rt(l + ~)11/2 [q, - 2~pu] .
(7)
We observe that N" Q and N ' . Q' carry no angular m o m e n t u m in the q direction and hence the orbital parts associated with them are again second-order polynomials in cos 0 t. But N . K, N ' - K, N . T and N ' . T all carry one unit of angular momentum in
390
CF. Cho, M. Gourdin / Weak neutral currents (II)
the q direction and the orbital parts associated with them can only be a first-order polynomial in cos 0 r The cross section with target polarization is therefore d°unp N" A =N" Q[bo(q2 ) + bl(q2 ) cos 0 t + b2(q 2) cos20t] dq 2
+N'K[Cl(q2)+c2(q2)cosOt]
+N'T[dl(q2)+d2(qZ)cosOt]
.
(8a)
And the cross section for recoil nucleon polarization is d°unp N'" A' = N " Q' [b~(q 2) + b](q 2) cos 0 t + b~(q 2) cos20t] dq 2
+ U ' . K [ C ' l ( q 2 ) + 'c2( q 2 ) cos Or] + N " T[d'l(q 2) + d~(q 2) cos Ot] .
(Sb)
Using similar arguments, we can write down the polarization correlation cross section as d°unp NuN'vCUV = (N'N')[eo(q2) + el(q 2) cos 0 t + ez(q 2) cos20t] dq 2
+ (IV. Q)(N'. K)[fl(q2 ) + f2(q 2) cos Ot] + ( N . K ) ( N ' . Q') [gl(q 2) +g2(q 2) cos Ot] + (N" Q)(N'. Q')[ho(q2 ) + hl(q2 ) cos 0 t + h2(q 2) cosZ0t] + (NoK)(N'.K)Jo(q 2) + (AT" Q)(N '° T)[ll(q2 ) +/z(q 2) cos at] + (N. T)(N'. Q') [pl (q 2) +p2(q 2) cos ot] + ( N , K ) ( N ' . T)ro(q2 ) + (N. T)(N' .K)oo(q2).
(9)
In the lab frame where the target is at rest, the vectors Q , K and Thave only space components: Q is a unit vector along the qlab direction in the scattering plane; K is orthogonal to the qlab direction in the scattering plane; T is normal to the scattering plane. Similarly, in the frame where the recoil nucleon is at rest, the vectors Q', K and T have only space components: Q' is a unit vector along the q direction in this frame; K is orthogonal to q in the scattering plane; T is normal to the scattering plane. Defining A = a 0 + a 1 cos 0 t + a 2 cos20t, we obtain the target asymmetries mea-
CF. Cho, M. Gourd& / Weak neutral currents (II}
391
sured in the lab frame AAQ =b O +b 1 c o s 0 t +b 2 c o s 2 0 t ,
A A K = r/-1/2 sin(Ot)(c I + c 2 cos Ot) , A A T = r/-1/2 sin(Ot)(dl + d 2 cos Ot),
(i0)
where AQ is the target asymmetry with N in the Q direction, etc. Note that 0 t is purely imaginary in the s-channel and we have defined sin 0 t = [cos20 t -- 1 ] 1/2
The recoil asymmetries can similarly be written in the recoil nucleon rest frame as *
cos0t •
cos20,, !
t
A A ~ = r/--l/2 sm(Ot)(e 1 + c2 cos Or), •
¢
t
AA~- = r/-1/2 sm(Ot)(d 1 + d 2 cos Ot).
(11)
We denote the nucleon polarization correlation as CNN,. So CQK is the polarization correlation in the target rest frame with N in the Q direction and N' in the K direction, etc. We then have
ACQQ, = (2r/+ 1)e 0 + h 0 + [(2r/+ 1)e 1 + h l ] cos0 t + [(2r/+ 1)e 2 + h 2 ] cos20 t ,
ACQK = r/-1/2 sin Ot(f 1 +f2 cos Ot), ACKQ , = r/-1/2 sin Ot(g 1 +g2 cos Ot) , ACKK = e0 +]0/r/+ el cos 0 t + (e 2 - / 0 / r / ) cos20t , ACTT = e 0 + e 1 cos 0 t + e 2 cos20t , ACQT = r/-1/2 sin Ot(l 1 + l 2 cos Ot) , ACTQ , = r/-1/2 sin Ot(P 1 +P2 cos Ot) ,
* In reactions where parity is conserved and if the polarizations of the beam particle are summed, then A~2,, A~( and A~e are simply related to AQ, AK and A T.
392
CF. Cho, M. Gourdin / Weak neutral currents ([[] A C K T = sin2O tro/rl ,
(12)
A C T K = sin2O tvo/rl .
Targets which are polarized perpendicular to the scattering plane or in the scattering plane are now available [8]. The polarization o f the recoil nucleon is normally measured b y allowing the nucleon to rescatter from a target, e.g.,carbon nucleus, and measure the azimuthal asymmetry in the rescattering [9]. We remark that here a nucleon polarized along its direction of motion cannot produce an azimuthal asymmetry when rescattered from an unpolarized target which serves as an analyzer. Hence the recoil nucleon polarization along the direction of motion may require special effort in the part of the experimentalists to measure [9]. Notice that the structure functions are functions o f only q2. From eq. (3), we see that cos 0 t is a function of s. One can therefore measure, at least in principle, all the
Table 1 ao al a2
SS, PP, TETE, TMT M STE, PTM TETE, TMT M
; ; ;
VV, AA VA W , AA
bo, b~) b 1, b'l b2, b~
SP, TET M STM, PTE TET M
; ; ;
VA VV, AA VA
Cl, c'l c2, c~
STM, PTE TET M
; ;
VV, AA VA
dl, d'~ d2, d~
STE, PTM TETE, TMTM
; ;
VA W , AA
eo el e2
SS, PP, TETE, TMT M STE, PTM TETE, TMTM
;
W , AA
;
VV, AA
fl f2
STE, PTM TETE, TMT M
; ;
VA VV, AA
gl g2
STE, PTM TETE, TMT M
; ;
VA VV, AA
ho h1 h2
SS, PP, TET E, TMT M STE, PTM TETE, TMT M
; ; ;
VV, AA VA W , AA
Jo
TETE, TMTM
;
W , AA
CF. Cho, M. Gourd& / Weak neutral currents (I1)
393
structure functions introduced here by varying the incident neutrino energy and studying the different polarization configurations. Also notice that the quantities A T, A'T, CQT, CTQ,, CKTand CTKm e a s u r e time reversal violation and that the structure functions di, li, Pi, ro and v0 are time-reversal violating structure functions. In this paper, we are not interested in the time-reversal violation effects. Therefore we shall ignore these quantities in our subsequent discussion.
2.2. Lorentz structure of the structure functions The S, P, T, V, A amplitudes correspond to specific values of the jpC quantum numbers of the uv state in the t-channel. This correspondence is already given in paper I. As in paper I, the S, P, T, V, A contents of the structure functions are most easily studied by the partial-wave expansion of the t-channel helicity amplitudes. The essential point to keep in mind in the computation is that in the t-channel, S, P and T couple to the neutrinos only when they have the same helicity while Vand A couple to the neutrinos when they have opposite helicities. After a straightforward calculation, we obtain the results shown in table I. The parity properties of the structure functions can be read off directly from table 1. The result is parity conserving:
!
t
t
ao,a2, bl, bl, Cl, Cl,d2,d2, eo, e2,f2,g2, ho, h2, ]0 •
parity violating :
t
t
t
a l , b0, b0, b2, b 2, c 2, c 2, d l , d ~ , e l , f l , g l ,
hl •
3. Explicit Lorentz structure of currents and useful experimental quantities We assume that the incoming v(v-) is left (right) handed. This is essentially true for neutrinos coming from K and ~ decays. The most general matrix element for vN twobody scattering in the local approximation can then be written as
T = v'~c ~,qc')(1 + -rs)u~(~)~N,(p', X')(S +.P)uN (p, X) + X/~G -uv(k , )x/21 ou,,(1 + 75)u~(k)UN'(P, , ~.')TV"UN(P, ~.) + X/~-~GuQ,(k')Tv(1 + 75)uQ(k)lSN'(P', X')(V u +A•)UN(P, ~k), where [4]: S= F s ,
CF. Cho, M. Gourdin / Weak neutral currents (II)
394 P = 3'5Fp , 1
..
t
F
T u~, -N/r~fT °lay - gT(q u'Yv - qv'Y u) -- tIT(P ~Pv - P ~Pv) -- hT(')'U')" q')'v - 7v ')'° q"f u) '
V , = Tug v + i h v ( P + p ' ) u / 2 M + i k v q u / 2 M , A u = "y~3,5GA + ihAqu'~5/2M + ikA( p + p ' ) u 7 5 / 2 M .
(13)
In the vector current, an alternate set of form factors commonly used is GE--gv-(r/+l)h
V,
GM : g v ,
(14a)
Also for later convenience, we introduce the following set of form factors: t E = (q 2g T -- N / - ~ f T ) / ~ - 2 M , tM = - I T , TE = - [fT + 2X/-2M2(1 + r/)jT + 2X/-2MgT ] , (14b)
T M = 4X/-2MhT.
The terms k v , k A and h T are the so-called second class current form factors [4]. Assuming time-reversal invariance, all the form factors are real. The form factors defined in eq. (14) have special meaning in the t-channel: GA, G M, t E and T M are associated with the nucleon helicity flip amplitudes while kA, GI,;, t M and T E are associated with the nucleon helicity non-flip amplitudes. We can now calculate the structure functions in terms o f the form factors given in (12). Since the result is quite lengthy, we record it in the appendix, and we present here only some experimental quantities which are useful in disentangling the Lorentz structure o f the weak currents. As most of us believe that neutral current is a combination of V and A, our main concern is to see how we can exclude (S, P, T). Using the results given in the appendix, we get 2+a 2-a 0=4t 2+4~(l+r/)T 2,
(lSa)
(2r/+ 3)(e 0 + e2) + h 0 + h 2 + a 2 + a 0 = 4(1 + r~)F2 + 4r/T2; ,
(15b)
(2~, - 1)(e 0 + e2) + h 0 + h 2 + a 2 + a 0 = 4r~F2 + 4(1 + r~)t2M .
(15e)
(2r~+ 1 ) ( % - e 2 ) + h
0-h
Cb: Cho, M. Gourdin / Weak neutral currents (II}
395
These are all positive definite quantities. So if these quantities are zero (or nonzero), this will show conclusively that there are no (or there are) S, P and T contributions. The structure functions required in eq. (15) are el, h i and a i. The ai's are the easiest to measure. From eq. (2), we see that these are the structure functions for the unpolarized cross section. The ei's also should not be too difficult to measure. As seen from eq. (12), what is required is a target polarized perpendicular to the scattering plane and one measures the recoil nucleon polarization perpendicular to the scattering plane. This can be done, for example, by allowing the recoil nucleon to rescatter and one measures the resulting azimuthal asymmetry [9 ]. The structure functions h i are, however, more difficult to measure. One needs a target polarized in the scattering plane and one selects the polarization component in the recoil nucleon direction in the lab frame (i.e., the Q direction). Such targets are available. But one also must measure the polarization of the recoil nucleon in its direction o f motion (i.e. in the Q' direction *). As mentioned earlier, in rescattering type polarization measurements, a nucleon polarized along its direction of motion cannot produce an azimuthal asymmetry when rescattered from an unpolarized target. Therefore it may require special effort on the part of the experimentalists to measure h i ** Next we study second-class current contributions. Again using the results in the appendix, we find ~7-1/2(fl + g l ) = -4r/(1 + r?) 1/2 [kAG M - TMFp] , r / - l / 2 ( f 2 + g 2 ) = --4r/1/2( 1 + r/)[kAGA + TMtM] •
(16)
Recall that k A and T M are the second class current form factors. If these quantities are non-zero, we will have proved conclusively that the second class currents exist. We have checked that even if one does not assume time reversal invariance, all the above conclusions remain valid. Furthermore, a nil result for eq. (15a) also implies that the second class tensor current is zero. If eq. (16) also turns out to be zero experimentally, then we can conclude that at least the time-reversal conserving part of the second class axial current is zero. We notice, however, that the second class vector current k V does not appear in the results given in the appendix. This is due to the masslessness of the leptons. Therefore we cannot exclude second class vector currents with this type of experiment. Again we would like to remind the reader how the structure functions in eq. (16) are measured. The fi's are obtained using a target polarized in the scattering plane * The Q and Q' directions defined in the text are parallel to each other because the transformation from the target rest frame to the recoil nucleon rest frame is along p'. ** As remarked by Wolfenstein [9], one may use a magnetic field to rotate the proton spin.
396
CF. Cho, M. Gourdin / Weak neutral currents (II}
and one selects the polarization component in the recoil nucleon direction in the lab frame (i.e., the Q direction). One then measures the polarization of the recoil nucleon perpendicular to its direction of motion in the scattering plane. The gi's, o n the other hand, are obtained using a target polarized in the scattering plane and one selects the polarization component perpendicular to the initial nucleon direction in the recoil nucleon rest frame. One then measures the polarization of the recoil nucleon opposite to the initial nucleon direction in the recoil nucleon rest frame. Here we encounter the same experimental difficulties mentioned earlier in association with the measurement of hi's. Finally, we notice that if crossing is assumed to hold *, the relations between u and vinteractions are that structure functions with index 1 have the same magnitudes but opposite sign, while the rest of the structure functions have the same magnitudes and signs. If the neutral current interaction is purely (V, A), the violation of crossing would imply that the initial and final neutrinos are not identical [5 ].
4. Summary and conclusions In order to settle unambiguously the question of the existence or non-existence of S, P and T in neutral currents, we studied the uN two-body scattering with both target and recoil nucleon polarizations measured. Structure functions are introduced to describe nucleon polarization correlations. We have studied the Lorentz structure of these structure functions and we have listed the experimentally measurable quantities which can be used to conclusively determine whether (S, P, T) exist. If (S, P, T) are excluded, the relative amount of V and A will be quite easily determined. One only needs to look at the p (or y) distribution in the unpolarized cross section [3]. On the other hand, if (S, P, T) are found to exist, one can again study the formulae given in the appendix to see how the polarization experiment may be used to determine whether (V, A) exist in neutral currents as well as the relative amount of S, P and T components. We have also shown how the nucleon polarization correlation experiment can help to determine whether second class current exist. In the case of neutral currents, this problem is related to the question as to whether the initial and final neutrinos are identical [5 ]. Our results also apply to charged current processes. However, in this case, it is probably easier to measure the polarization of the final charged lepton in order to study the Lorentz structure of the weak charged current [6]. Nucleon polarization correlation experiments in vN elastic scattering can undoubtedly clarify some of the fundamental problems in neutral currents. At the same
* Crossing is valid if the time reversal transformation between the amplitudes for u + u ~ N + N and tlae amplitudes for N + N -~ v + ~-is unique.
CF. Cho, M. Gourdin / Weak neutral currents (IiJ
time, we realize that this type of experiment leave this challenge to the experimentalists.
391
is very difficult to carry out. We shall
Appendix Using the matrix element given in eq. (13), we can calculate the structure functions introduced in eq. (9) in terms of the S, P, T, V and A form factors. Our result is
al
=2[q(1
n2
=
tq)]1/2[2GMGA+FSTE+FptMl
>
G; t qG& t (1 + T-/)G; + VT; + (1 + v)t; + t; + 7)(1 + v)T$
+ 77(1 + s)k2A 3 b, = 2 [n( 1 + $1 ‘I2 [G,G,
+ FsFp + TMtE + ~,GE ] 9
bb = 2[7)(1 •t q)] 1/2 [-G,G,
b, = 2nGi b; = -2nG; b2=2[7(1
+ F,F,
+ 2(1 + n)G2, + 2(1 + a)FsfM + 2r&T~
+Q)]‘/2[GMGA+TEtM
-
+ T,t,
TMtE-G,k,l + TMtE - G&l
1 = 2n112 [GMG, + FptF], + 2~l/~(l
1)-m, 1 = 2~~11~[-GMGE
+ F,t,]
>
7
+ 2(l + @‘Sth4 + bF,T,
- 2(1 + n)G;
b; = 2[17(1 + n)] ‘/2[-GMGA +/2,
- Tf,,f, + k,G,l
+ n)[FSTM
>
> >
+kAGAl 2
+ 2~$‘~(1 +n)t-FsT~
,+/2,2 =2(1 t#/2[GEGA
+tMtE]
,-1/2c> =2(1 tn)1/2[GEGA
- tMfE] +2n(l
(217t l)eo t ho = -QG~ - (1 + n)Gi
+2q(1
+k,G~l a
+q)1’2[T~T&G&Al 9 +q)“2[TET~
-GM~AI y
- Gi + tg + (1 + @$R + M’s
398
CF. Cho, M, Gourd& / Weak neutral currents (II) (2r/+ 1)e 1 + h 1 = 2 [r/(1 + r/)] 1/2 [ _ 2 G M G A + F s T E + FptM ] ,
(2r/+ 1)e 2 + h 2 = - ~ O 2 - ( l
+rl)G2A + G 2 - t
2 + t i T 2 + ( I + r/)t2M
+ r/(l + r/)k2A -- r/(l + r/)T2M , "0-1/2/1 = 2(1 + r/) 112 [GEG A - F s t E - rlkAO M + r/TMFp] , r u 1 / Z f 2 = 2r/1/2 [GEG M - t E T E - (1 + r~)kAG A -- (1 + r/)TMtM] , r u 1 / 2 g 1 = 2(1 + r~)1/2 [ - G E G A + F s t E - rlkAG M + ~ T M F p ] , r/-1/2g2 = 2r/1/2 [ - G E G M + t E T E - (1 + r/)kAG A - (1 + r/)TMtM] , e0 - jo/rl = riG 2 - (1 + r/)G 2 - G 2 + t 2 + (1 + r/)F 2 - ~F 2 + r/(1 + r/)k 2
- r/(1 + ~7)T2 , e 1 = 2[~(1 + B)] 1/2 [ F s T E _ FptM ] ,
e 0 +/'0/r~ = - r / G 2M + ( I + r/)G~ + G ~ - t 2 E + r/T2 - (I + r~)t2M - 77(I + r/)k2A + r/(l + r/)T 2 , e 0=-r/G2M + ( I +r/)G 2 - G +
t 2 + ( i +r/)F 2 - r / F 2 + r / ( l +r/)k 2
+
e2 =riG 2 -(I -
2-
+~7)G 2 + G 2 + t 2 + r / T 2 - ( I
+rl)t 2 - r/(l + r/)k2A
+
We have checked that the result here agrees with the t-channel partial-wave expansion results.
References [1] C.F. Cho and M. Gourdin, Nucl. Phys. Bl12 (1976) 365. [2] S.P. Rosen, Proc. 4th Int. Conf. on neutrino physics and astrophysics, Downingtown, 1974, ed. C. Baltay (ALP, New York, 1974) p. 5; B. Kayser et al., Phys. Letters 52B (1974) 385. [31 R.L. Kingsley, F. Wilczek and A. Zee, Phys. Rev. D10 (1974) 2216. [4] S. Wemberg, Phys. Rev. 112 (1958) 1375.
C.F. Cho, M. Gourd& / Weak neutral currents (II) [5] L. Wolfenstein, Nucl. Phys. B91 (1975) 95; B.R. Kim and R. Rodenberg, Phys. Rev. D10 (1974) 2234. [61 T.P. Cheng and Wu-KiTung, Phys. Rev. D3 (1971) 733. [71 M. Gourdin, Rapporteur's talk, EPS Conf., Palermo, June 1975. [8] M. Borghini, Proc. 6th Rencontre de Moriond, Meribel-les-AUues, 1971, p. 145. [9] L. Wolfenstein, Phys. Rev. 96 (1954) 1654.
399
SPACE-TIME STRUCTURE OF WEAK NEUTRAL CURRENTS (11). Elastic scattering
C.F. ClIO and M. GOURDIN Laboratoire de Physique Th~orique et Hautes Energies, Paris *
Received 9 April 1976
The elastic scattering of neutrinos and antineutrinos on nucleons in which the initial and final nucleon polarizations are measured, is studied from the point of view of the space-time properties of the currents. Structure functions are defined and experimentally measurable quantities which can be usual to disentangle the Lorentz structure of the currents are given. Second class current effects are also examined.
1. Introduction In a previous paper [1 ] (which we shall refer to as paper 1 from now on), we have studied what information about the Lorentz structure of neutral currents one can obtain in inclusive neutrino and anti-neutrino reactions using a polarized target. It was found that as in the unpolarized case, the "confusion theorem" [21 holds; i.e. a mixture of S (scalar), P (pseudoscalar) and T (tensor) interactions can always simulate V (vector) and A (axial vector) interactions. In this paper, we study the neutrinonucleon and anti-nuetrino nucleon elastic scattering in which both the initial and final polarizations of the nucleons are measured. We realize that these types of experiments are very difficult to perform. On the other hand, these are among the cleanest and most straightforward experiments we can think of which can settle conclusively the question of the Lorentz structure of the weak neutral currents. If S, P and T are excluded, the relative amount of V and A can be quite easily determined [3]. The pN elastic scattering experiment measuring nucleon polarizations correlations has a further advantage in that it can be used to study second class current [4] effects. In the neutral current case, the study of second class currents can in turn shed some light on the question as to whether the initial and final neutrinos are identical [5]. Our treatment is valid for neutral current interactions as well as charged current interactions when the mass of the final lepton can be neglected. However, in the * Postal address: Universit~ Pierre et Marie Curie, Tour l 6-1 er &age, 4 Place J ussieu, 75230 Paris Cedex 05, France. 387
388
CF. Cho, M. Gourdin / Weak neutral currents (II)
charged current case, it is probably easier to measure the final lepton polarization rather than measuring the nucleon polarizations [6] in order to study the Lorentz structure of the current. In sect. 2, we study the kinematics of uN elastic scattering when the target and recoil nucleon polarizations are measured. Structure functions are introduced and their general Lorentz structure and symmetry properties given. The experimentally measurable target and recoil nucleon asymmetry cross sections and the nucleon polarization correlations are expressed in terms of these structure functions. Our treatment here can be trivially generalized to the production of any hadrons with spin }. In this connection, we would like to mention that while the production of strange baryons by neutral currents is expected to be very small, there is still the possibility of producing baryons with new quantum numbers which decay weakly. In that case the polarization of the final hadron can be obtained from the angular distribution of the decay products. In sect. 3, we write down the most general form of the S, P, T, V and A currents, including second class currents, in uN elastic scattering. The structure functions can then be calculated explicitely in terms of the S, P, T, V and A form factors. This is done in the appendix. Using these results, we list in sect. 3 the experimentally measurable quantities which are most useful in probing the Lorentz structure and the second class current effects of the weak currents. Finally, in sect. 4, we discuss our results.
2. General discussion 2.1. K i n e m a t i c s
We study the reaction Q+N~'+N',
(1)
where ~ and 2' are leptons of momenta k and k' and N and N' are nucleons of mass M with momenta p and p'. £ is a neutrino or antineutrino and £' is either a neutral or charged lepton, q = k - k' is the m o m e n t u m transfer, s = - (k + p)2 is the square of the c.m. energy. We consider the general case where the weak current is a superposition of S, P, T, V and A Lorentz components. As in our study of inclusive neutrino reactions [1 ], we analyze the cross section by looking at the t-channel where the neutrino-antineutrino annihilation amplitudes proceed via spin-zero and spin-one exchanges [7]. It follows that the unpolarized cross section is a second-order polynomial in cos 0 t where 0 t is the t-channel c.m. scattering angle. In the local Fermi approximation where q2 is small compared to the square of the intermediate vector-meson mass, we get d°unp dq 2
_
G2q2M 2
27r(s - M2) 2
{ao(q2 ) + al(q2 ) cos 0 t + a2(q 2) cos20t ) ,
(2)
CF. Cho, M. Gourdin / Weak neutral currents (II)
389
where 2s - 2M 2 - q2 COS 0 t =
q2
2M[q2(1 + r~)] 1/2 '
r/
(3)
4M 2 ,
and e = cos20 C
for charged current case,
=1
for the neutral current case,
where 0 C is the Cabbibo angle. Also G is the conventional Fermi coupling constant. In eq. (2) as well as in the subsequent discussion, the lepton mass is neglected. To describe the polarization effects for the target and recoil nucleons, we introduce a space-like unit pseudovector N ( N ' ) orthogonal to p(p'), with N 2=N '2=1,
(4)
N.p=N'.p'=O.
The polarized cross section can then be written as do _ 1 d°unp {1 + N . A + N " A' + NuN'vCUU} dq2 2 dq2
(5)
The four-vector 2x(A') describes the various target (recoil) nucleon asymmetries, and the tensor Cuv describes the polarization correlations between the target and recoil nucleons. Because of constraints (4), there exists only three independent four-vectors orthogonal to p which we choose to be [1 ] 1
Q"
2M[r/(1 + n ) ] l/2 [qu
2npu] ,
1 ( k + k , ) u + p ' ( k + k ' ) [~qu+pu] Ku = 2M-~ 342(1 + r/) 1
,
(6)
T u = M- euv p apUQOK ° .
Similarly, we choose the three independent four-vectors orthogonal to p' to be K u, T u and Q'
=
1
t
2_M[rt(l + ~)11/2 [q, - 2~pu] .
(7)
We observe that N" Q and N ' . Q' carry no angular m o m e n t u m in the q direction and hence the orbital parts associated with them are again second-order polynomials in cos 0 t. But N . K, N ' - K, N . T and N ' . T all carry one unit of angular momentum in
390
CF. Cho, M. Gourdin / Weak neutral currents (II)
the q direction and the orbital parts associated with them can only be a first-order polynomial in cos 0 r The cross section with target polarization is therefore d°unp N" A =N" Q[bo(q2 ) + bl(q2 ) cos 0 t + b2(q 2) cos20t] dq 2
+N'K[Cl(q2)+c2(q2)cosOt]
+N'T[dl(q2)+d2(qZ)cosOt]
.
(8a)
And the cross section for recoil nucleon polarization is d°unp N'" A' = N " Q' [b~(q 2) + b](q 2) cos 0 t + b~(q 2) cos20t] dq 2
+ U ' . K [ C ' l ( q 2 ) + 'c2( q 2 ) cos Or] + N " T[d'l(q 2) + d~(q 2) cos Ot] .
(Sb)
Using similar arguments, we can write down the polarization correlation cross section as d°unp NuN'vCUV = (N'N')[eo(q2) + el(q 2) cos 0 t + ez(q 2) cos20t] dq 2
+ (IV. Q)(N'. K)[fl(q2 ) + f2(q 2) cos Ot] + ( N . K ) ( N ' . Q') [gl(q 2) +g2(q 2) cos Ot] + (N" Q)(N'. Q')[ho(q2 ) + hl(q2 ) cos 0 t + h2(q 2) cosZ0t] + (NoK)(N'.K)Jo(q 2) + (AT" Q)(N '° T)[ll(q2 ) +/z(q 2) cos at] + (N. T)(N'. Q') [pl (q 2) +p2(q 2) cos ot] + ( N , K ) ( N ' . T)ro(q2 ) + (N. T)(N' .K)oo(q2).
(9)
In the lab frame where the target is at rest, the vectors Q , K and Thave only space components: Q is a unit vector along the qlab direction in the scattering plane; K is orthogonal to the qlab direction in the scattering plane; T is normal to the scattering plane. Similarly, in the frame where the recoil nucleon is at rest, the vectors Q', K and T have only space components: Q' is a unit vector along the q direction in this frame; K is orthogonal to q in the scattering plane; T is normal to the scattering plane. Defining A = a 0 + a 1 cos 0 t + a 2 cos20t, we obtain the target asymmetries mea-
CF. Cho, M. Gourd& / Weak neutral currents (II}
391
sured in the lab frame AAQ =b O +b 1 c o s 0 t +b 2 c o s 2 0 t ,
A A K = r/-1/2 sin(Ot)(c I + c 2 cos Ot) , A A T = r/-1/2 sin(Ot)(dl + d 2 cos Ot),
(i0)
where AQ is the target asymmetry with N in the Q direction, etc. Note that 0 t is purely imaginary in the s-channel and we have defined sin 0 t = [cos20 t -- 1 ] 1/2
The recoil asymmetries can similarly be written in the recoil nucleon rest frame as *
cos0t •
cos20,, !
t
A A ~ = r/--l/2 sm(Ot)(e 1 + c2 cos Or), •
¢
t
AA~- = r/-1/2 sm(Ot)(d 1 + d 2 cos Ot).
(11)
We denote the nucleon polarization correlation as CNN,. So CQK is the polarization correlation in the target rest frame with N in the Q direction and N' in the K direction, etc. We then have
ACQQ, = (2r/+ 1)e 0 + h 0 + [(2r/+ 1)e 1 + h l ] cos0 t + [(2r/+ 1)e 2 + h 2 ] cos20 t ,
ACQK = r/-1/2 sin Ot(f 1 +f2 cos Ot), ACKQ , = r/-1/2 sin Ot(g 1 +g2 cos Ot) , ACKK = e0 +]0/r/+ el cos 0 t + (e 2 - / 0 / r / ) cos20t , ACTT = e 0 + e 1 cos 0 t + e 2 cos20t , ACQT = r/-1/2 sin Ot(l 1 + l 2 cos Ot) , ACTQ , = r/-1/2 sin Ot(P 1 +P2 cos Ot) ,
* In reactions where parity is conserved and if the polarizations of the beam particle are summed, then A~2,, A~( and A~e are simply related to AQ, AK and A T.
392
CF. Cho, M. Gourdin / Weak neutral currents ([[] A C K T = sin2O tro/rl ,
(12)
A C T K = sin2O tvo/rl .
Targets which are polarized perpendicular to the scattering plane or in the scattering plane are now available [8]. The polarization o f the recoil nucleon is normally measured b y allowing the nucleon to rescatter from a target, e.g.,carbon nucleus, and measure the azimuthal asymmetry in the rescattering [9]. We remark that here a nucleon polarized along its direction of motion cannot produce an azimuthal asymmetry when rescattered from an unpolarized target which serves as an analyzer. Hence the recoil nucleon polarization along the direction of motion may require special effort in the part of the experimentalists to measure [9]. Notice that the structure functions are functions o f only q2. From eq. (3), we see that cos 0 t is a function of s. One can therefore measure, at least in principle, all the
Table 1 ao al a2
SS, PP, TETE, TMT M STE, PTM TETE, TMT M
; ; ;
VV, AA VA W , AA
bo, b~) b 1, b'l b2, b~
SP, TET M STM, PTE TET M
; ; ;
VA VV, AA VA
Cl, c'l c2, c~
STM, PTE TET M
; ;
VV, AA VA
dl, d'~ d2, d~
STE, PTM TETE, TMTM
; ;
VA W , AA
eo el e2
SS, PP, TETE, TMT M STE, PTM TETE, TMTM
;
W , AA
;
VV, AA
fl f2
STE, PTM TETE, TMT M
; ;
VA VV, AA
gl g2
STE, PTM TETE, TMT M
; ;
VA VV, AA
ho h1 h2
SS, PP, TET E, TMT M STE, PTM TETE, TMT M
; ; ;
VV, AA VA W , AA
Jo
TETE, TMTM
;
W , AA
CF. Cho, M. Gourd& / Weak neutral currents (I1)
393
structure functions introduced here by varying the incident neutrino energy and studying the different polarization configurations. Also notice that the quantities A T, A'T, CQT, CTQ,, CKTand CTKm e a s u r e time reversal violation and that the structure functions di, li, Pi, ro and v0 are time-reversal violating structure functions. In this paper, we are not interested in the time-reversal violation effects. Therefore we shall ignore these quantities in our subsequent discussion.
2.2. Lorentz structure of the structure functions The S, P, T, V, A amplitudes correspond to specific values of the jpC quantum numbers of the uv state in the t-channel. This correspondence is already given in paper I. As in paper I, the S, P, T, V, A contents of the structure functions are most easily studied by the partial-wave expansion of the t-channel helicity amplitudes. The essential point to keep in mind in the computation is that in the t-channel, S, P and T couple to the neutrinos only when they have the same helicity while Vand A couple to the neutrinos when they have opposite helicities. After a straightforward calculation, we obtain the results shown in table I. The parity properties of the structure functions can be read off directly from table 1. The result is parity conserving:
!
t
t
ao,a2, bl, bl, Cl, Cl,d2,d2, eo, e2,f2,g2, ho, h2, ]0 •
parity violating :
t
t
t
a l , b0, b0, b2, b 2, c 2, c 2, d l , d ~ , e l , f l , g l ,
hl •
3. Explicit Lorentz structure of currents and useful experimental quantities We assume that the incoming v(v-) is left (right) handed. This is essentially true for neutrinos coming from K and ~ decays. The most general matrix element for vN twobody scattering in the local approximation can then be written as
T = v'~c ~,qc')(1 + -rs)u~(~)~N,(p', X')(S +.P)uN (p, X) + X/~G -uv(k , )x/21 ou,,(1 + 75)u~(k)UN'(P, , ~.')TV"UN(P, ~.) + X/~-~GuQ,(k')Tv(1 + 75)uQ(k)lSN'(P', X')(V u +A•)UN(P, ~k), where [4]: S= F s ,
CF. Cho, M. Gourdin / Weak neutral currents (II)
394 P = 3'5Fp , 1
..
t
F
T u~, -N/r~fT °lay - gT(q u'Yv - qv'Y u) -- tIT(P ~Pv - P ~Pv) -- hT(')'U')" q')'v - 7v ')'° q"f u) '
V , = Tug v + i h v ( P + p ' ) u / 2 M + i k v q u / 2 M , A u = "y~3,5GA + ihAqu'~5/2M + ikA( p + p ' ) u 7 5 / 2 M .
(13)
In the vector current, an alternate set of form factors commonly used is GE--gv-(r/+l)h
V,
GM : g v ,
(14a)
Also for later convenience, we introduce the following set of form factors: t E = (q 2g T -- N / - ~ f T ) / ~ - 2 M , tM = - I T , TE = - [fT + 2X/-2M2(1 + r/)jT + 2X/-2MgT ] , (14b)
T M = 4X/-2MhT.
The terms k v , k A and h T are the so-called second class current form factors [4]. Assuming time-reversal invariance, all the form factors are real. The form factors defined in eq. (14) have special meaning in the t-channel: GA, G M, t E and T M are associated with the nucleon helicity flip amplitudes while kA, GI,;, t M and T E are associated with the nucleon helicity non-flip amplitudes. We can now calculate the structure functions in terms o f the form factors given in (12). Since the result is quite lengthy, we record it in the appendix, and we present here only some experimental quantities which are useful in disentangling the Lorentz structure o f the weak currents. As most of us believe that neutral current is a combination of V and A, our main concern is to see how we can exclude (S, P, T). Using the results given in the appendix, we get 2+a 2-a 0=4t 2+4~(l+r/)T 2,
(lSa)
(2r/+ 3)(e 0 + e2) + h 0 + h 2 + a 2 + a 0 = 4(1 + r~)F2 + 4r/T2; ,
(15b)
(2~, - 1)(e 0 + e2) + h 0 + h 2 + a 2 + a 0 = 4r~F2 + 4(1 + r~)t2M .
(15e)
(2r~+ 1 ) ( % - e 2 ) + h
0-h
Cb: Cho, M. Gourdin / Weak neutral currents (II}
395
These are all positive definite quantities. So if these quantities are zero (or nonzero), this will show conclusively that there are no (or there are) S, P and T contributions. The structure functions required in eq. (15) are el, h i and a i. The ai's are the easiest to measure. From eq. (2), we see that these are the structure functions for the unpolarized cross section. The ei's also should not be too difficult to measure. As seen from eq. (12), what is required is a target polarized perpendicular to the scattering plane and one measures the recoil nucleon polarization perpendicular to the scattering plane. This can be done, for example, by allowing the recoil nucleon to rescatter and one measures the resulting azimuthal asymmetry [9 ]. The structure functions h i are, however, more difficult to measure. One needs a target polarized in the scattering plane and one selects the polarization component in the recoil nucleon direction in the lab frame (i.e., the Q direction). Such targets are available. But one also must measure the polarization of the recoil nucleon in its direction o f motion (i.e. in the Q' direction *). As mentioned earlier, in rescattering type polarization measurements, a nucleon polarized along its direction of motion cannot produce an azimuthal asymmetry when rescattered from an unpolarized target. Therefore it may require special effort on the part of the experimentalists to measure h i ** Next we study second-class current contributions. Again using the results in the appendix, we find ~7-1/2(fl + g l ) = -4r/(1 + r?) 1/2 [kAG M - TMFp] , r / - l / 2 ( f 2 + g 2 ) = --4r/1/2( 1 + r/)[kAGA + TMtM] •
(16)
Recall that k A and T M are the second class current form factors. If these quantities are non-zero, we will have proved conclusively that the second class currents exist. We have checked that even if one does not assume time reversal invariance, all the above conclusions remain valid. Furthermore, a nil result for eq. (15a) also implies that the second class tensor current is zero. If eq. (16) also turns out to be zero experimentally, then we can conclude that at least the time-reversal conserving part of the second class axial current is zero. We notice, however, that the second class vector current k V does not appear in the results given in the appendix. This is due to the masslessness of the leptons. Therefore we cannot exclude second class vector currents with this type of experiment. Again we would like to remind the reader how the structure functions in eq. (16) are measured. The fi's are obtained using a target polarized in the scattering plane * The Q and Q' directions defined in the text are parallel to each other because the transformation from the target rest frame to the recoil nucleon rest frame is along p'. ** As remarked by Wolfenstein [9], one may use a magnetic field to rotate the proton spin.
396
CF. Cho, M. Gourdin / Weak neutral currents (II}
and one selects the polarization component in the recoil nucleon direction in the lab frame (i.e., the Q direction). One then measures the polarization of the recoil nucleon perpendicular to its direction of motion in the scattering plane. The gi's, o n the other hand, are obtained using a target polarized in the scattering plane and one selects the polarization component perpendicular to the initial nucleon direction in the recoil nucleon rest frame. One then measures the polarization of the recoil nucleon opposite to the initial nucleon direction in the recoil nucleon rest frame. Here we encounter the same experimental difficulties mentioned earlier in association with the measurement of hi's. Finally, we notice that if crossing is assumed to hold *, the relations between u and vinteractions are that structure functions with index 1 have the same magnitudes but opposite sign, while the rest of the structure functions have the same magnitudes and signs. If the neutral current interaction is purely (V, A), the violation of crossing would imply that the initial and final neutrinos are not identical [5 ].
4. Summary and conclusions In order to settle unambiguously the question of the existence or non-existence of S, P and T in neutral currents, we studied the uN two-body scattering with both target and recoil nucleon polarizations measured. Structure functions are introduced to describe nucleon polarization correlations. We have studied the Lorentz structure of these structure functions and we have listed the experimentally measurable quantities which can be used to conclusively determine whether (S, P, T) exist. If (S, P, T) are excluded, the relative amount of V and A will be quite easily determined. One only needs to look at the p (or y) distribution in the unpolarized cross section [3]. On the other hand, if (S, P, T) are found to exist, one can again study the formulae given in the appendix to see how the polarization experiment may be used to determine whether (V, A) exist in neutral currents as well as the relative amount of S, P and T components. We have also shown how the nucleon polarization correlation experiment can help to determine whether second class current exist. In the case of neutral currents, this problem is related to the question as to whether the initial and final neutrinos are identical [5 ]. Our results also apply to charged current processes. However, in this case, it is probably easier to measure the polarization of the final charged lepton in order to study the Lorentz structure of the weak charged current [6]. Nucleon polarization correlation experiments in vN elastic scattering can undoubtedly clarify some of the fundamental problems in neutral currents. At the same
* Crossing is valid if the time reversal transformation between the amplitudes for u + u ~ N + N and tlae amplitudes for N + N -~ v + ~-is unique.
CF. Cho, M. Gourdin / Weak neutral currents (IiJ
time, we realize that this type of experiment leave this challenge to the experimentalists.
391
is very difficult to carry out. We shall
Appendix Using the matrix element given in eq. (13), we can calculate the structure functions introduced in eq. (9) in terms of the S, P, T, V and A form factors. Our result is
al
=2[q(1
n2
=
tq)]1/2[2GMGA+FSTE+FptMl
>
G; t qG& t (1 + T-/)G; + VT; + (1 + v)t; + t; + 7)(1 + v)T$
+ 77(1 + s)k2A 3 b, = 2 [n( 1 + $1 ‘I2 [G,G,
+ FsFp + TMtE + ~,GE ] 9
bb = 2[7)(1 •t q)] 1/2 [-G,G,
b, = 2nGi b; = -2nG; b2=2[7(1
+ F,F,
+ 2(1 + n)G2, + 2(1 + a)FsfM + 2r&T~
+Q)]‘/2[GMGA+TEtM
-
+ T,t,
TMtE-G,k,l + TMtE - G&l
1 = 2n112 [GMG, + FptF], + 2~l/~(l
1)-m, 1 = 2~~11~[-GMGE
+ F,t,]
>
7
+ 2(l + @‘Sth4 + bF,T,
- 2(1 + n)G;
b; = 2[17(1 + n)] ‘/2[-GMGA +/2,
- Tf,,f, + k,G,l
+ n)[FSTM
>
> >
+kAGAl 2
+ 2~$‘~(1 +n)t-FsT~
,+/2,2 =2(1 t#/2[GEGA
+tMtE]
,-1/2c> =2(1 tn)1/2[GEGA
- tMfE] +2n(l
(217t l)eo t ho = -QG~ - (1 + n)Gi
+2q(1
+k,G~l a
+q)1’2[T~T&G&Al 9 +q)“2[TET~
-GM~AI y
- Gi + tg + (1 + @$R + M’s
398
CF. Cho, M, Gourd& / Weak neutral currents (II) (2r/+ 1)e 1 + h 1 = 2 [r/(1 + r/)] 1/2 [ _ 2 G M G A + F s T E + FptM ] ,
(2r/+ 1)e 2 + h 2 = - ~ O 2 - ( l
+rl)G2A + G 2 - t
2 + t i T 2 + ( I + r/)t2M
+ r/(l + r/)k2A -- r/(l + r/)T2M , "0-1/2/1 = 2(1 + r/) 112 [GEG A - F s t E - rlkAO M + r/TMFp] , r u 1 / Z f 2 = 2r/1/2 [GEG M - t E T E - (1 + r~)kAG A -- (1 + r/)TMtM] , r u 1 / 2 g 1 = 2(1 + r~)1/2 [ - G E G A + F s t E - rlkAG M + ~ T M F p ] , r/-1/2g2 = 2r/1/2 [ - G E G M + t E T E - (1 + r/)kAG A - (1 + r/)TMtM] , e0 - jo/rl = riG 2 - (1 + r/)G 2 - G 2 + t 2 + (1 + r/)F 2 - ~F 2 + r/(1 + r/)k 2
- r/(1 + ~7)T2 , e 1 = 2[~(1 + B)] 1/2 [ F s T E _ FptM ] ,
e 0 +/'0/r~ = - r / G 2M + ( I + r/)G~ + G ~ - t 2 E + r/T2 - (I + r~)t2M - 77(I + r/)k2A + r/(l + r/)T 2 , e 0=-r/G2M + ( I +r/)G 2 - G +
t 2 + ( i +r/)F 2 - r / F 2 + r / ( l +r/)k 2
+
e2 =riG 2 -(I -
2-
+~7)G 2 + G 2 + t 2 + r / T 2 - ( I
+rl)t 2 - r/(l + r/)k2A
+
We have checked that the result here agrees with the t-channel partial-wave expansion results.
References [1] C.F. Cho and M. Gourdin, Nucl. Phys. Bl12 (1976) 365. [2] S.P. Rosen, Proc. 4th Int. Conf. on neutrino physics and astrophysics, Downingtown, 1974, ed. C. Baltay (ALP, New York, 1974) p. 5; B. Kayser et al., Phys. Letters 52B (1974) 385. [31 R.L. Kingsley, F. Wilczek and A. Zee, Phys. Rev. D10 (1974) 2216. [4] S. Wemberg, Phys. Rev. 112 (1958) 1375.
C.F. Cho, M. Gourd& / Weak neutral currents (II) [5] L. Wolfenstein, Nucl. Phys. B91 (1975) 95; B.R. Kim and R. Rodenberg, Phys. Rev. D10 (1974) 2234. [61 T.P. Cheng and Wu-KiTung, Phys. Rev. D3 (1971) 733. [71 M. Gourdin, Rapporteur's talk, EPS Conf., Palermo, June 1975. [8] M. Borghini, Proc. 6th Rencontre de Moriond, Meribel-les-AUues, 1971, p. 145. [9] L. Wolfenstein, Phys. Rev. 96 (1954) 1654.
399