- Email: [email protected]
Detailed W0 effects in e+e− → e+e−
Volume 55B, number 2
PttYSICS LETTERS
3 February 1975
D E T A I L E D W ° E F F E C T S IN e+e - ~ e + e R. BUDNY
Institute of Theoretical Physics, Department of Physics, Stanford University, Stanford, California 94305, USA Received 16 October 1974 The predicted cross section for the reaction e+e- ~ e÷e- is presented assuming one photon and W° exchange. Pure W° terms and the W° width are included. The reaction e+e - -* e+e - in storage rings is especially interesting for detecting and studying a W0, or weak neutral current, for several reasons. Since this has a relatively large cross section, many events will be available for analysis. Also the W0 coupling may be straightforward since only one type o f particle is involved. In this aspect the e+e - channel is complementary to the other purely leptonic channel, e+e - ~/~+/a-, where products of the e and/a couplings of the W0 can be measured. For instance, the possibility [1 ] that/a - e universality is violated by the W 0 can be checked by comparing the W ° effects in these two channels. Several papers [ 2 4 ] have already treated W0 effects in e+e- -* e+e - . The purpose of this paper is to present further details of these effects. The new details are the purely weak terms in the cross section and polarization effects. Pure weak terms become important when x/~becomes comparable to the mass of the W0. Thus they are needed to test whether resonances, such as the one newly discovered near V~ = 3.1 GeV, are W°'s. The cross section given below is for the polarization configuration where the spins of the beams are polarized colineady and transversely to the beam direction and where the final pair is in a helfcity state. The conditions where this choice is applicable are the following: a) Normally storage ring beams become transversely polarized and final helicities are not observed. When the cross section is summed on final helicities it predicts this case; b) It is advantageous and apparently feasible, but difficult, to rotate the direction of the beam polarization into the direction of the beam. The cross section also predicts this case if the semantics of initial and final states in the reaction is reversed; and c) The result also is applicable to the unlikely case that final helicities can be measured. It is easy and apparently not interesting to generalize the result to the case of arbitrary initial and final spin polarizations. The cross section results from the following amplitudes: e2-
'
-
t~
'
cl~ =--[-u(p_h_)~/~u(p_ ~_) o(p+~+)~/ o(p'+h+) - s +
I
t - M2
e2-
'
-
U(p'_h_)'y~ o(p +h+) o (P +~+)'t~u(P- ~- )
u ( p _ h _)~'u(gv + gA75 ) u (p_ ~_ ) v (p+ ~+)~' (gv + gA~'5 ) o (p+h+)
1 -~(p,_h_).yu(gv+gA~[5)-~(p,+h+)-~(p+~+).t~(g v +gA,/5)u(p_~_) " S -- M 2 The neutral weak form factors g v and gA are assumed to be real, but M is complex, to include the effects of the width of the W0. An expedient way o f calculating the square of the four amplitudes is to first square the sum of the two W0 amplitudes, and then to generate the one photon and photon-W 0 terms from this by setting the axial coupling at zero and the vector coupling at e. Research sponsored by the National Science Foundation grant NSF MPS 074-08916. 227
Volume 55B, number 2
PIIYSICS LETTERS
3 l:ebruary 1975
The kinematic variables are defined as follows: The scattering angle is 0 and the azimuthal angle (between the direction of transverse polarization and the scattering plane) is ~. The magnitude of the transverse, colinear spins of the initial e - and e + are ~_ and ~+, and their limiting values are ~_ = -~÷ = 0.924. Also, h and h+ are the helicites of the final e - and e +, s = ( p _ + p+)2 = 4E2, t = (p_ - p'_)2 = _ 2 E 2 ( 1 _ cos 0), R - s/e2(s - M2), and Q =- t/e2(t - M 2 ) , with e 2 = 47ra and M the mass of the W0. Terms of order me/E are neglected. The cross section in the CM system is 32s do =(l+h_h+)4B l+(1-h ~2
h+){(1-cos0)2B2+(l+cos0)2B3
+ ~1 ~2 sin20 [cos (2~b)B4 + sin (2O)B 5 ] } + (h_ - h+) {(1 + cos 0)2B 6 + ~1 ~2 sin20 [cos (2q~)B7 + sin (2q~)B8 ] }, where B 1 = IC112 ,
B3=½{IC3+C412+ If5 + C612},
B2= IC212 ,
B5=Im{C~(C3+C4-C5-C6)},
B6=½{1C3+C412-1C5+C612},
B 4 =Re(C~(C3+C4+C5+C6)}, B7=Re{C~(C3+C4-C5-C6)),
B 8 = Im {C:~(C3 - C5)},
Cl =t[ l + (g2 - g2 ) Q] , C4 =
l+(gv +gA)2R'
c2 -- 1 + C v - g )R,
C3 = t [1 + ( g v + g A ) 2 Q ] ' C 6 = 1 + ( g v - gA) 2R "
C5 = t [1+ ( g v - g A ) 2 Q ] '
The effect of summing helicities is to omit h _ and h÷ and to multiply the rest of the cross section by four. The square of the photon propagator was factored out of Q and R, so the Q and R independent parts of the B / s result from one photon exchange. The terms linear in Q and R resulted from one photon-W 0 interference, and the rest resulted from one W0 exchange*. When sit is set equal to zero, the cross section gives the prediction for W0 effects in e÷e - -~/a+p- [5]. A similar treatment o f W 0 effects in e e ~ e e will be given elsewhere [6]. Applications of the cross section in two areas are discussed in the following. The first area is the case of a light narrow width W0. For this application, the effects of the width are of interest since the peak and both sides are experimentally accessible. Then M is replaced by M o - i1-'/2. For a narrow resonance this means s -M 2 ReR
e2[(s-M2)2+(Mo F)2 ]
s2 IRI 2
e4[(s_M2o)2+(MoF)2 ]
For example, if the ~k(3 105) is a W0, then the Q terms are negligible and the R terms are large near v ~ = M 2. The cross section predicts angular distributions and correlations which can be compared with data to determine g v and gA" Radiative corrections need to be incorporated in detailed considerations such as the height and shape of the peak, but do not generally dominate the W0 effects. In the second application the mass of the W0 is assumed too large to be reached in existing storage rings. In this case higher order electromagnetic effects can mask out the W0 contributions to the cross section. One way around this is to look for terms in the cross section which are odd under symmetries conserved by electromagnetism. If the cross section were even under parity, charge conjugation, and time reversal, then the following relations would hold respectively: do(0, ¢, '~_, ~+, h _ , h+) = do(O, .Tr-~b, ~_, ~+, - h _ , - h + )
= do(O, - ~ , ,~+, ~_, h+, h_) -- do(O, -~,, ,~_, ~+, h_, h+). * The interference terms do not agree with ref. [2], specifically eq. (27) but do agree with those of ref. [3] (where the sign conver. tion ofg A is reversed). These results disagree with those in ref. [4]. 228
~/olume 55B, number 2
PHYSICS LETTERS
3 February 1975
The transformation properties of the coefficients o f B 1 - B 8 are even in the table. Thus the terms in the cross section with coefficients B 1 - B 4 are even under these symmetries. Higher order electromagnetic effects, which conserve parity and change conjugation, can contribute to, or simulate these terms (i.e., have a similar kinematic structure), but can not simulate the B 5 - B 7 terms. This means that measurement of any of the B 5 - B 7 terms would give relatively unambiguous information about the W0. The simplest polarization configurations to measure are those of unpolarized or transversely polarized beams with no final helicity detection. The one photon-W 0 prediction for this case is gotten by summing the cross section over h _ , h÷ = -+1. With unpolarized beams, the B1-B 3 terms can be measured; however, higher order electromagnetic contributions must be subtracted in order to determine g v , gA, and M. The one photon-two photon interference contributions have been calculated for unpolarized beams [7]. They depend on the detection abilities of the experiment and in SPEAR II they are expected to be on the order of 10%. In storage rings with higher energies it may be easier tO separate the higher order electromagnetic effects from W0 effects since the former dominate small angle scattering and grow like ins, whereas the latter tend to dominate large angle scattering and grow faster with s (initially like s to order s/M2). Examples of the magnitude of the photon-W 0 interference were given in ref. [3] assuming x/s = 28 GeV (a typical energy for the next round of storage rings being planned) and using the Weinberg model (for g v , gA, and )14). The pure W 0 effects are very small i f M is as large as the Weinberg model predicts. For this case the change due to the W0 in the differential cross section, i.e. 6(s, O, c~, ~1~2, h _ , h+) defined by d , (onephoton) (higherorder em) o _
uo
~--~= (1 +6)~-~
do
+d-~
'
is at most about 2% for favored values o f g v , gA, and M ( ~ 75 GeV) regardless of whether the beam is transversely polarized or unpolarized. One way to study the W0 would be to measure the angular dependence of 6 and then to fit this with g v , gA, and M. Another would be to measure the dependence of 8 on ~_ ~+ which grows with the age of the beam from 0 to - ( 0 . 9 2 4 ) 2 (except at depolarizing resonances), and then to fit Aft. This method may involve less systematic errors since it does not require comparison of event rates at different angles. A more difficult polarization configuration to measure is the longitudinal case which can be obtained using magnets to rotate the transverse polarization into the direction of the beam. The one photon-W 0 prediction for this case is gotten by multiplying the right side of the cross section by four and setting ~_~+ = 0 and h _ = P _ and h+ = - P ÷ . Since the initial transverse configuration has equal but opposite polarizations (if both beams have the same age), any magnetic field acting equally on both beams will result in antiparaUel spins, i.e., h_ ---h+ where the terms with B 6 - B $ vanish. Furthermore, in the approximation h _ = h+ = +-1, all the s channel exchange amplitudes vanish. The weak corrections to the cross section, do/d~2 = (o~2/s)B 1 , are proportional to g 2 _ g2A and have a large coefficient for moderate and large scattering angles. In practice one expects Ih_ ~ h+l < 0.924, but 5 could still be large. As an example, in the Weinberg model with 30 ° Weinberg angle ( g v = 0, gA = 0.17, M = 87 GeV), V~= 10 GeV, h _ =h+ = +0.924, and 0 = 140 °, the prediction is 8 L = - 0 . 0 0 7 . For comparison, the largest unpolarized and transversely polarized changes occur at 6 = 110 ° where 5 u = - 0 . 0 0 3 and fiT = --0.001. Thus a longitudinally polarized beam would be advantageous in this case. It is very desirable to measure at least one of the B s - B 8 terms since they indicate violations of parity, charge conjugation or time reversal invariance**. A plausible way to measure B 6 is to depolarize one of the transversely polarized beams before rotating the spins into the longitudinal direction [8]. This has the effect of setting one helicity at zero and the other at P(-<< 0.924). With such a beam one could measure a corresponding 5, however, ** Observation of the term with B 7 and Bs requires measurement of three spins which is nearly impossible since final electron helicities are very difficult to measure (as would also be the helicities of high energy ~'s from the reaction/~+/~- ~ t~+t~-). In the case of e+e- ~ u+~- there is a remote possibility of measuring B 7 and B s (with s]r = Q = 0). 229
Volume 55B, number 2
PItYSICS LFTTERS
3 February 1975
Table 1 Discrete symmetries of the terms in the cross section.
Parity Change Time
B1
B2
B3
B4
B5
B6
B7
B8
+ + +
+ + +
+ + +
+ + +
-
÷
_ +
+ + -
the terms B I - B 4 contribute as well as B 6 . If a comparison is made with this done first to one then the other charge, the term with B 6 changes sign and those with B1-B 4 remain constant. Thus a succinct indication o f weak effects would be a non-zero result for the asymmetry defined by do(h_ = t ' _ , h÷ = 0) - d a ( h _ = 0, h÷ -- e ÷ ) A---
do(h_ = P _ ,
h+ = O) + d o ( h _ = O, h+ = P+) "
The prediction for this is (P_ + P+) (1 + cos 0)2B6 2[B 1 + (1 - cos 0)2B2 + (1 + cos 0)2B3 ] + (P_ - P+) (1 + cos 0)2B6 " The limit of A for small s[M 2, where (s/t)Q -, R and R --, - s / e 2 M 2 , is simply A --, 2 (P_ + P+) (1 + cos 0) 3 (1 - cos O) g v g A s (3 + cos20) 2 e2M 2 If the beams can be polarized in the difficult longitudinal parallel configuration h _ ~ - h + , then another asymmetry, defined by
A'=_
do(h_ = e _ , h+ = - e ÷ ) - do(h_ -- - e _ , h÷ = e÷ )
do(h_ =P_,h+ =-e+)
+ do(h_ = - P _ , h + = e + ) '
would be an interesting indication o f weak effects. The prediction for this is (P_ + P+) (1 + cos 0)2B 6 4(1 - P _ P + ) B 1 + (1 + P _ P + ) [ ( 1 - cos 0)2B2 + (1 + cos 0)2B3 ]' The small s/M 2 limit o f this was given in ref. [3] along with examples o f its magnitude at x ~-= 28 GeV. A ' can be of the order o f five times greater than 5 u and 6 T for favored values o f the Weinberg angle. In the limit o f small s/M 2 , A '/,4 depends only on cos 0 and varies betwen 2 and 6 when P _ = P÷ = 0.924, so longitudinal polarization in the difficult parallel configuration has advantages over longitudinal polarization in the easy antiparaUel configuration. In a sense, measurements o f A and A ' would complement measurements o f 6 L since A and A ' vanish when gvgA = 0 and 16 L I is largest when gA or g v = 0. Unfortunately, A and A ' may be very susceptable to systematic errors since they require comparisons o f cross sections in cases where drastic operations have been performed on the beam (presumably in separate runs). In conclusion, the reaction e+e - --, e+e - should provide decisive information about the W0. Measurements at various angles, energies, and polarization configurations can specify g v , g A ' and M.
References
[1] S. Berman, D. Cline, 1974 PEP Summer Study Proc., December 1974. [2] D. Dicus, Phys. Rev. D$ (1973) 890. 230
Volume 55B, number 2 [3] [4] [5] [6] [7] [8]
PHYSICS LETTERS
3 I'ebruary 1975
R. Budny and A. McDonald, Phys. Rev, Comments, November 1974. C.H. Llewellyn-Smith and D.V. Nanopoulos, Nucl. Phys. B78 (1974) 205. R. Budny, Phys. Lett. 45B (1973) 340. R. Budny, to be published. F.A. Berends, K.J.F. Gaemers, and R. Gastmans, Nucl. Phys. B68 (1974) 541. W. Toner, Rutherford Lab report RL-74-102.
231
PttYSICS LETTERS
3 February 1975
D E T A I L E D W ° E F F E C T S IN e+e - ~ e + e R. BUDNY
Institute of Theoretical Physics, Department of Physics, Stanford University, Stanford, California 94305, USA Received 16 October 1974 The predicted cross section for the reaction e+e- ~ e÷e- is presented assuming one photon and W° exchange. Pure W° terms and the W° width are included. The reaction e+e - -* e+e - in storage rings is especially interesting for detecting and studying a W0, or weak neutral current, for several reasons. Since this has a relatively large cross section, many events will be available for analysis. Also the W0 coupling may be straightforward since only one type o f particle is involved. In this aspect the e+e - channel is complementary to the other purely leptonic channel, e+e - ~/~+/a-, where products of the e and/a couplings of the W0 can be measured. For instance, the possibility [1 ] that/a - e universality is violated by the W 0 can be checked by comparing the W ° effects in these two channels. Several papers [ 2 4 ] have already treated W0 effects in e+e- -* e+e - . The purpose of this paper is to present further details of these effects. The new details are the purely weak terms in the cross section and polarization effects. Pure weak terms become important when x/~becomes comparable to the mass of the W0. Thus they are needed to test whether resonances, such as the one newly discovered near V~ = 3.1 GeV, are W°'s. The cross section given below is for the polarization configuration where the spins of the beams are polarized colineady and transversely to the beam direction and where the final pair is in a helfcity state. The conditions where this choice is applicable are the following: a) Normally storage ring beams become transversely polarized and final helicities are not observed. When the cross section is summed on final helicities it predicts this case; b) It is advantageous and apparently feasible, but difficult, to rotate the direction of the beam polarization into the direction of the beam. The cross section also predicts this case if the semantics of initial and final states in the reaction is reversed; and c) The result also is applicable to the unlikely case that final helicities can be measured. It is easy and apparently not interesting to generalize the result to the case of arbitrary initial and final spin polarizations. The cross section results from the following amplitudes: e2-
'
-
t~
'
cl~ =--[-u(p_h_)~/~u(p_ ~_) o(p+~+)~/ o(p'+h+) - s +
I
t - M2
e2-
'
-
U(p'_h_)'y~ o(p +h+) o (P +~+)'t~u(P- ~- )
u ( p _ h _)~'u(gv + gA75 ) u (p_ ~_ ) v (p+ ~+)~' (gv + gA~'5 ) o (p+h+)
1 -~(p,_h_).yu(gv+gA~[5)-~(p,+h+)-~(p+~+).t~(g v +gA,/5)u(p_~_) " S -- M 2 The neutral weak form factors g v and gA are assumed to be real, but M is complex, to include the effects of the width of the W0. An expedient way o f calculating the square of the four amplitudes is to first square the sum of the two W0 amplitudes, and then to generate the one photon and photon-W 0 terms from this by setting the axial coupling at zero and the vector coupling at e. Research sponsored by the National Science Foundation grant NSF MPS 074-08916. 227
Volume 55B, number 2
PIIYSICS LETTERS
3 l:ebruary 1975
The kinematic variables are defined as follows: The scattering angle is 0 and the azimuthal angle (between the direction of transverse polarization and the scattering plane) is ~. The magnitude of the transverse, colinear spins of the initial e - and e + are ~_ and ~+, and their limiting values are ~_ = -~÷ = 0.924. Also, h and h+ are the helicites of the final e - and e +, s = ( p _ + p+)2 = 4E2, t = (p_ - p'_)2 = _ 2 E 2 ( 1 _ cos 0), R - s/e2(s - M2), and Q =- t/e2(t - M 2 ) , with e 2 = 47ra and M the mass of the W0. Terms of order me/E are neglected. The cross section in the CM system is 32s do =(l+h_h+)4B l+(1-h ~2
h+){(1-cos0)2B2+(l+cos0)2B3
+ ~1 ~2 sin20 [cos (2~b)B4 + sin (2O)B 5 ] } + (h_ - h+) {(1 + cos 0)2B 6 + ~1 ~2 sin20 [cos (2q~)B7 + sin (2q~)B8 ] }, where B 1 = IC112 ,
B3=½{IC3+C412+ If5 + C612},
B2= IC212 ,
B5=Im{C~(C3+C4-C5-C6)},
B6=½{1C3+C412-1C5+C612},
B 4 =Re(C~(C3+C4+C5+C6)}, B7=Re{C~(C3+C4-C5-C6)),
B 8 = Im {C:~(C3 - C5)},
Cl =t[ l + (g2 - g2 ) Q] , C4 =
l+(gv +gA)2R'
c2 -- 1 + C v - g )R,
C3 = t [1 + ( g v + g A ) 2 Q ] ' C 6 = 1 + ( g v - gA) 2R "
C5 = t [1+ ( g v - g A ) 2 Q ] '
The effect of summing helicities is to omit h _ and h÷ and to multiply the rest of the cross section by four. The square of the photon propagator was factored out of Q and R, so the Q and R independent parts of the B / s result from one photon exchange. The terms linear in Q and R resulted from one photon-W 0 interference, and the rest resulted from one W0 exchange*. When sit is set equal to zero, the cross section gives the prediction for W0 effects in e÷e - -~/a+p- [5]. A similar treatment o f W 0 effects in e e ~ e e will be given elsewhere [6]. Applications of the cross section in two areas are discussed in the following. The first area is the case of a light narrow width W0. For this application, the effects of the width are of interest since the peak and both sides are experimentally accessible. Then M is replaced by M o - i1-'/2. For a narrow resonance this means s -M 2 ReR
e2[(s-M2)2+(Mo F)2 ]
s2 IRI 2
e4[(s_M2o)2+(MoF)2 ]
For example, if the ~k(3 105) is a W0, then the Q terms are negligible and the R terms are large near v ~ = M 2. The cross section predicts angular distributions and correlations which can be compared with data to determine g v and gA" Radiative corrections need to be incorporated in detailed considerations such as the height and shape of the peak, but do not generally dominate the W0 effects. In the second application the mass of the W0 is assumed too large to be reached in existing storage rings. In this case higher order electromagnetic effects can mask out the W0 contributions to the cross section. One way around this is to look for terms in the cross section which are odd under symmetries conserved by electromagnetism. If the cross section were even under parity, charge conjugation, and time reversal, then the following relations would hold respectively: do(0, ¢, '~_, ~+, h _ , h+) = do(O, .Tr-~b, ~_, ~+, - h _ , - h + )
= do(O, - ~ , ,~+, ~_, h+, h_) -- do(O, -~,, ,~_, ~+, h_, h+). * The interference terms do not agree with ref. [2], specifically eq. (27) but do agree with those of ref. [3] (where the sign conver. tion ofg A is reversed). These results disagree with those in ref. [4]. 228
~/olume 55B, number 2
PHYSICS LETTERS
3 February 1975
The transformation properties of the coefficients o f B 1 - B 8 are even in the table. Thus the terms in the cross section with coefficients B 1 - B 4 are even under these symmetries. Higher order electromagnetic effects, which conserve parity and change conjugation, can contribute to, or simulate these terms (i.e., have a similar kinematic structure), but can not simulate the B 5 - B 7 terms. This means that measurement of any of the B 5 - B 7 terms would give relatively unambiguous information about the W0. The simplest polarization configurations to measure are those of unpolarized or transversely polarized beams with no final helicity detection. The one photon-W 0 prediction for this case is gotten by summing the cross section over h _ , h÷ = -+1. With unpolarized beams, the B1-B 3 terms can be measured; however, higher order electromagnetic contributions must be subtracted in order to determine g v , gA, and M. The one photon-two photon interference contributions have been calculated for unpolarized beams [7]. They depend on the detection abilities of the experiment and in SPEAR II they are expected to be on the order of 10%. In storage rings with higher energies it may be easier tO separate the higher order electromagnetic effects from W0 effects since the former dominate small angle scattering and grow like ins, whereas the latter tend to dominate large angle scattering and grow faster with s (initially like s to order s/M2). Examples of the magnitude of the photon-W 0 interference were given in ref. [3] assuming x/s = 28 GeV (a typical energy for the next round of storage rings being planned) and using the Weinberg model (for g v , gA, and )14). The pure W 0 effects are very small i f M is as large as the Weinberg model predicts. For this case the change due to the W0 in the differential cross section, i.e. 6(s, O, c~, ~1~2, h _ , h+) defined by d , (onephoton) (higherorder em) o _
uo
~--~= (1 +6)~-~
do
+d-~
'
is at most about 2% for favored values o f g v , gA, and M ( ~ 75 GeV) regardless of whether the beam is transversely polarized or unpolarized. One way to study the W0 would be to measure the angular dependence of 6 and then to fit this with g v , gA, and M. Another would be to measure the dependence of 8 on ~_ ~+ which grows with the age of the beam from 0 to - ( 0 . 9 2 4 ) 2 (except at depolarizing resonances), and then to fit Aft. This method may involve less systematic errors since it does not require comparison of event rates at different angles. A more difficult polarization configuration to measure is the longitudinal case which can be obtained using magnets to rotate the transverse polarization into the direction of the beam. The one photon-W 0 prediction for this case is gotten by multiplying the right side of the cross section by four and setting ~_~+ = 0 and h _ = P _ and h+ = - P ÷ . Since the initial transverse configuration has equal but opposite polarizations (if both beams have the same age), any magnetic field acting equally on both beams will result in antiparaUel spins, i.e., h_ ---h+ where the terms with B 6 - B $ vanish. Furthermore, in the approximation h _ = h+ = +-1, all the s channel exchange amplitudes vanish. The weak corrections to the cross section, do/d~2 = (o~2/s)B 1 , are proportional to g 2 _ g2A and have a large coefficient for moderate and large scattering angles. In practice one expects Ih_ ~ h+l < 0.924, but 5 could still be large. As an example, in the Weinberg model with 30 ° Weinberg angle ( g v = 0, gA = 0.17, M = 87 GeV), V~= 10 GeV, h _ =h+ = +0.924, and 0 = 140 °, the prediction is 8 L = - 0 . 0 0 7 . For comparison, the largest unpolarized and transversely polarized changes occur at 6 = 110 ° where 5 u = - 0 . 0 0 3 and fiT = --0.001. Thus a longitudinally polarized beam would be advantageous in this case. It is very desirable to measure at least one of the B s - B 8 terms since they indicate violations of parity, charge conjugation or time reversal invariance**. A plausible way to measure B 6 is to depolarize one of the transversely polarized beams before rotating the spins into the longitudinal direction [8]. This has the effect of setting one helicity at zero and the other at P(-<< 0.924). With such a beam one could measure a corresponding 5, however, ** Observation of the term with B 7 and Bs requires measurement of three spins which is nearly impossible since final electron helicities are very difficult to measure (as would also be the helicities of high energy ~'s from the reaction/~+/~- ~ t~+t~-). In the case of e+e- ~ u+~- there is a remote possibility of measuring B 7 and B s (with s]r = Q = 0). 229
Volume 55B, number 2
PItYSICS LFTTERS
3 February 1975
Table 1 Discrete symmetries of the terms in the cross section.
Parity Change Time
B1
B2
B3
B4
B5
B6
B7
B8
+ + +
+ + +
+ + +
+ + +
-
÷
_ +
+ + -
the terms B I - B 4 contribute as well as B 6 . If a comparison is made with this done first to one then the other charge, the term with B 6 changes sign and those with B1-B 4 remain constant. Thus a succinct indication o f weak effects would be a non-zero result for the asymmetry defined by do(h_ = t ' _ , h÷ = 0) - d a ( h _ = 0, h÷ -- e ÷ ) A---
do(h_ = P _ ,
h+ = O) + d o ( h _ = O, h+ = P+) "
The prediction for this is (P_ + P+) (1 + cos 0)2B6 2[B 1 + (1 - cos 0)2B2 + (1 + cos 0)2B3 ] + (P_ - P+) (1 + cos 0)2B6 " The limit of A for small s[M 2, where (s/t)Q -, R and R --, - s / e 2 M 2 , is simply A --, 2 (P_ + P+) (1 + cos 0) 3 (1 - cos O) g v g A s (3 + cos20) 2 e2M 2 If the beams can be polarized in the difficult longitudinal parallel configuration h _ ~ - h + , then another asymmetry, defined by
A'=_
do(h_ = e _ , h+ = - e ÷ ) - do(h_ -- - e _ , h÷ = e÷ )
do(h_ =P_,h+ =-e+)
+ do(h_ = - P _ , h + = e + ) '
would be an interesting indication o f weak effects. The prediction for this is (P_ + P+) (1 + cos 0)2B 6 4(1 - P _ P + ) B 1 + (1 + P _ P + ) [ ( 1 - cos 0)2B2 + (1 + cos 0)2B3 ]' The small s/M 2 limit o f this was given in ref. [3] along with examples o f its magnitude at x ~-= 28 GeV. A ' can be of the order o f five times greater than 5 u and 6 T for favored values o f the Weinberg angle. In the limit o f small s/M 2 , A '/,4 depends only on cos 0 and varies betwen 2 and 6 when P _ = P÷ = 0.924, so longitudinal polarization in the difficult parallel configuration has advantages over longitudinal polarization in the easy antiparaUel configuration. In a sense, measurements o f A and A ' would complement measurements o f 6 L since A and A ' vanish when gvgA = 0 and 16 L I is largest when gA or g v = 0. Unfortunately, A and A ' may be very susceptable to systematic errors since they require comparisons o f cross sections in cases where drastic operations have been performed on the beam (presumably in separate runs). In conclusion, the reaction e+e - --, e+e - should provide decisive information about the W0. Measurements at various angles, energies, and polarization configurations can specify g v , g A ' and M.
References
[1] S. Berman, D. Cline, 1974 PEP Summer Study Proc., December 1974. [2] D. Dicus, Phys. Rev. D$ (1973) 890. 230
Volume 55B, number 2 [3] [4] [5] [6] [7] [8]
PHYSICS LETTERS
3 I'ebruary 1975
R. Budny and A. McDonald, Phys. Rev, Comments, November 1974. C.H. Llewellyn-Smith and D.V. Nanopoulos, Nucl. Phys. B78 (1974) 205. R. Budny, Phys. Lett. 45B (1973) 340. R. Budny, to be published. F.A. Berends, K.J.F. Gaemers, and R. Gastmans, Nucl. Phys. B68 (1974) 541. W. Toner, Rutherford Lab report RL-74-102.
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