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0.06 0.08,
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(13.53)
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in excellent agreement with the standard model and sin Ow ::::: ~ (see Table 13.2).
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Fig. 13.5 Determination of the parameters Cv and CA of (13.46) by neutrino-electron data Figure is taken from Hung and Sakurai (1981). The absence of Pe data is because reactor beams only have iie .
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13.6 Electroweak Interference in e + e - Annihilation
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However, the best determinations of sin2 Ow come from the analyses of data for inclusive and exclusive neutrino-nucleon processes. A recent simultaneous analysis (Kim et al., 1981) of all available data gives sin2 Ow = 0.234 0.013,
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(13 .54)
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13.6 Electroweak Interference in e + e - Annihilation Shortly after Bludman in 1958 first speculated about the existence of a weak neutral current, Zel'dovich gave a very simple argument to estimate the size of the asymmetry arising from the interference of the electromagnetic amplitude 0lL EM - e 2 /k 2 with a small weak contribution. He predicted
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using G "" 1O-5/m~ and e 2 /4 T = 1/137. The high-energy e +e - colliding beam machines are an ideal testing ground for such interference effects. The e +e - annihilations can occur through electromagnetic (y) or weak neutral current (Z) interactions, see, for example, Fig. 13.6. With e beam energies of 15 GeV we have k 2 :::: s = (30 GeV)2, and so (13.55) predicts about a 10% effect, which is readily observable. To make a detailed prediction for the process e +e - -+ p, +P, -, we assume that the neutral current interaction is mediated by a Z boson with couplings given by (13.41). Using the Feynman rules (Section 6.17), the amplitudes 0lLy and 0lL z corresponding to the diagrams of Fig. 13.6 are
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(13.56)
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4C~: OW [fiy'( e~ - e~y5)p,1 ( g'o;2 ~~Mi) [eyo( e~ - e~y5)el,
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(13.57) where k is the four-momentum of the virtual y (or Z), S :::: k 2 With electron-muon universality, the superscripts on e v A are superfluous here, but we keep them so as to be able to translate the result~ directly to e+e--+ qq (where eq ell). We
Fig. 13.6 Electromagnetic and weak contributions to e + e- -+
JL+JL-.
Electroweak Interactions
(1ko)eYO
ignore the lepton masses, so the Dirac equation for the incident positron reads = 0 and the numerator of the propagator simplifies to gvo' Thus, (13.57) becomes
using (13.32) and (13.36) with p That is, we have chosen to write
1, and where (13.59)
CAy5 =
(c v - cA H(1 + y5) +(c v + cA H(1
y5).
The 1(1 y5) are projection operators, see (5.78), which enable 0lLz to be expressed explicitly in terms of right- and left-handed spinors. It is easier to 2 calculate 1000y + 0lLz 1 in this form. With definite electron and muon helicities, we can directly apply the results of the QED calculation of e - e +--+ p. - p. + given in Sections 6.5 and 6.6. For instance, do(eze% --+ P.LP.~) a2 ( )2 2 = - 1 + cos{) 11 + rcrcfl , drl 4s do(eze% --+p.ip.!) a2 ( )2 2 (13.60) = - 1 - cos () 11 + rc~cf I drl 4s [see (6.39) and (6.32)], where r is the ratio of the coefficients in front of the brackets in (13.58) and (13.56), that is,
s - Mi
Ii GMi
+ iMzrz
( s )
-;z ,
(13.61)
where we have included the finite resonance width r z, which is important for s - Mi [see X(E) of (2.56) multiplied by 1/(E + M)]. Expressions similar to (13.60) hold for the other two nonvanishing helicity configurations. To calculate the unpolarized e+e---+ p.+p.- cross section, we average over the four allowed L, R helicity combinations. We find
do drl
4s[Ao(1+cos 2 {))+AIcos{)],
(13.62)
where (assuming electron-muon universality
cr =
= c;)
d) 2
(13.63)