where the second expression neglects terms of order ml/M^ = 10"6.
The Feynman diagram for this process has already been drawn in Fig. 9.8. This evaluation illustrates some new Feynman rules:
9.10 ELECTROWEAK DECAYS 277
for each W —> e + ve weak vector, where ¡j, is the polarization index carried by the W. • a factor of
for each interaction in which a 7r~ of momentum q turns into a virtual W~ with polarization index )i. Rule 2: a factor of i (g"" - k»kv/M2)
M2 - k2 - it for each internal line describing the propagation of a spin one boson with mass M, momentum k, and polarization indices /i and v. Rule 3: multiply from the right by v(p, A) for each outgoing antifermion with momentum p and spin (or helicity) A.
Using these rules (and Rule 0) it is easy to reconstruct the result (9.142). The quantity gl^/M^ is related to the Fermi coupling constant G,
The Fermi constant G was introduced when the weak interactions were first described as a current-current interaction of the form a
Since the boson mass Mw is very large, the lowest order results obtained from ,9.126) and (9.145) are equivalent as long as the identification (9.144) is made. However, if one tries to calculate higher order corrections, one finds that (9.145) is unrenormalizable (see Chapter 16) while (9.126) can be renormalized if the theory is converted into a gauge theory. These issues will be discussed further in Chapter 15.
For now, we take the result (9.144) and compute the decay rate of the pion. Summing over the final spin states and integrating over all momenta, as in Eq. (9.19), give d3pd3e (27r)4<54(p + t — q)
To evaluate the spin sum over \M\2, we first simplify M by using the Dirac equation and the fact that the neutrino mass is zero. Hence
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