and the fields are described by an expansion similar to (2.45), except that the charged field is not self-conjugate. We have

where $^'(1) are the plane wave solutions for the W boson. The weak current is a sum of hadron and lepton parts:

'This effective coupling constant is gw cos 0yv in the notation of Sec. 15.4.

The lepton part has the famous vector — axial vector (V — A) structure,

where there are believed now to be only three terms in the sum (9.131), one for each generation of leptons (see the table in Appendix D). In the examples discussed in this section, we will be concerned only with the first two generations, consisting of the electron and the electron neutrino, denoted i/e, and the muon and muon neutrino, denoted (The muon is often denoted by p, but we will use the subscript "muon" in order to avoid confusion with the vector index /i.) Note that the coupling (9.131) only involves left-handed neutrinos (and right-handed antineutrinos) and therefore violates parity. There is no role for right-handed neutrinos (or left-handed antineutrinos) in the electroweak interactions, and whether or not they exist, along with the question of whether or not the mass of the neutrino is exactly zero, is currently not known.

The most fundamental definition of the hadronic current is in terms of quark fields. However, for phenomenological applications we can express these currents directly in terms of the composite hadrons which are observed in the laboratory. Among these will be contributions from the nucleón, the pion, and other hadrons, but in this section we will discuss the pion contribution only, which can be written

where 0 is the positively charged pion field and fn is the famous pion decay constant. Since the pion and W fields have dimensions of mass, the constant fn must also have dimensions of mass, and as defined here its value turns out to be u = 93.0 MeV . (9.133)

In Chapter 13 we will show how the current (9.132) can be obtained from a particular model, and more generally such a current can be justified directly from the quark structure of the pion, but now we will simply assume that a current of the form (9.132) exists, and use it to study pion decay.

Using the interaction Lagrangian (9.126) with the currents (9.131) and (9.132), the second order matrix element for tt~ decay is

where is the initial state with four-momentum q and {pi| is the final state of an outgoing electron with momentum p and an electron antineutrino with momentum I. The decay is illustrated in Fig. 9.8. We are forced to consider the decay into leptons because energy-momentum conservation would not permit the decay of a it" into a real W boson unless its mass were identical to the mass

where Mw is the mass of the boson and = (En,kn), with En = \jMw + Hence, going to continuum normalization and using (4.77) to eliminate the 6-functions give immediately

To obtain this result, we used (4.77), with n = 0 for the spatial components and with n = 2 for the (0,0) part. This is the general form for the propagator of a massive vector meson.

Now we must reduce the matrix element

Since ipUc destroys electron neutrinos and creates electron antineutrinos, this matrix element describes the decay

For the pion at rest, this matrix element becomes (letting L -+ 2n)

Inserting (9.141) and (9.138) into (9.134), doing the integration over x\, x2, and k, and extracting the .M-matrix from the result give

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