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where M x is the mass of the boson and where we have neglected the masses of the fermions.
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Hint Use (6.93) to show that after summing over the fermion and averaging over the boson spins,
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where k, k' are the four-momenta of the fermions. Work in the boson rest frame. Use (4.37). EXERCISE 13.3 Assuming the standard model coupling, show that
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see Exercise 13.2. Given that sin2 Ow numerical value of the partial width.
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EXERCISE 13.4 Calculate the partial widths of the three decay modes Z -+ e +e -, uu, dd. Hence, predict the total width of the Z in the standard model, assuming sin2 Ow = ~ and M z = 90 GeV. Do not forget color. . EXERCISE 13.5 Repeat Exercise 13.3 for the W+ -+ e+Pe decay mode; take M w = 80 GeV. EXERCISE 13.6 Calculate the partial widths of the two decay modes W+ -+ du, su; use (12.102) and (12.103). Predict the total width of the W+ in the standard model. 13.5 Neutrino-Electron Scattering The pp'e- and pp'e- elastic scattering processes can only proceed via a neutral current interaction, see Fig. 13.3. The current-current form of the invariant amplitude for the process pp'e - -+ pp'e - is analogous to (12.84) for Pq -+ Pq scattering:
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where GN = pG ::::: G [see (12.88)]. In fact, assuming electron-muon universality, we shall find that the four elastic scattering processes, pp'e -, pp'e -, Pee -, and Pee -, can all be explained in terms of the two parameters c v == c~ and cA == CA. EXERCISE 13.7 If pe -+ pe scattering proceeds by Z exchange, show that (13.46) is obtained from the Feynman rules using (13.41) as the vertex factor. In particular, use the expression for the boson propagator (see Section 6.17) to verify that (13.46) is valid provided the four-momentum transfer q is such that Iq 2 1 Mi. Given that (13.46) is of identical form to that for Pq -+ Pq scattering, we may use the results of Section 12.10 to obtain the pp'e- -+ pp'e- cross section. We
Fig. 13.3 The neutral current
pp'e- -+ pp'e-
13.5 Neutrino-Electron Scattering
therefore have [see (12.90) and (12.91)]
da(PILe) dy
G~s [
(C v + C ) +(C v - cA) (1- y) . A
Carrying out the y integration from 0 to 1 gives
a ( PILe
For iiILe- elastic scattering,
_ a ( PILe
G~S(2 PILe ) =);- C v
2) + C v CA + CA
CA -+ -CA
in (13.47), and so
_) G~s ( 2 2) PILe =);- Cv - CVCA + C . A
EXERCISE 13.8 Equation (13.47) is valid if m 2 Is 1. If the electron mass m is not ignored, show that the extra contribution to (13.47) is
The process Pee--+ Pee- offers the intriguing possibility of studying charged current and neutral current interference, see Fig. 13.4. The amplitude for diagram (a) is ~NC of (13.46) with P = Pe . For diagram (b); we have
~cc =
~ [e yIL{l
- y5)Pe ][ iieYIL{1 - y5)e],
where the minus sign relative to (13.46) arises from interchange of the outgoing leptons [see (6.9)]. We may use the Fierz reordering theorem, see, for example, Bailin (1982), to rewrite (13.50) as
(iiey IL{l- y5)Pe )( eYIL{1- y5)e).
Fig. 13.4 The neutral and charged current
p~- -+ p~-
Electroweak Interactions
EXERCISE 13.9 Show that (13.51) follows from (13.50). The invariance under reordering of the spinors is an important property of the V-A interaction. The effect of reordering in the scalar product of bilinear covariants is, in general, much more involved. The answer is the Fierz theorem. To obtain the amplitude ~(Pee- -+ Pee-), we add the amplitudes (~NC and for the two diagrams of Fig. 13.4. If we take p = 1, then G N = G [see (12.88)], and we find ~ = ~ NC + ~ cc is given by (13.46) with
~ CC)
-+ C v
+ 1,
Thus, the Pee - and Pee - elastic scattering cross sections are in tum given by (13.48) and (13.49) with these replacements. It is customary to present the results of a given neutrino-electron cross section measurement as an ellipse of possible values of C v and cA in the c v ' cA plane. Recent results are shown in Fig. 13.5. The three "experimental" ellipses mutually intersect to give two possible solutions. The cA dominant solution is
cA = -0.52 0.06,