Physical Foundations of Cosmology

4.3 Electroweak theory 163

W+ W−

νL

eL

⎯ νR

∼

e⎯L

Fig. 4.8.

PandCoperations applied separately and in the Standard Model these symmetries
are violated in a maximal possible way.
It is obvious that (4.40) is not invariant with respect to the replacementL↔
Rbecause parity operation converts left-handed neutrinos into nonexistent right-
handed neutrinos. Similarly, charge conjugation converts left-handed neutrinos into
nonexistent left-handed antineutrinos. However, the combined operationCP, which
interchanges left-handed particles with right-handed antiparticles, seems to be a
symmetry of the electroweak theory. In fact,CPis a symmetry of the Lagrangian
without quarks. As an example, let us find out what happens to the charged weak
interaction coupling term in (4.74),

g

√

2

(

(ν ̄LγμeL)Wμ++(e ̄LγμνL)Wμ−

)

, (4.87)

under CPtransformations. The first term here corresponds to the vertex in
Figure 4.8 and can be interpreted as describing left-handed electron and right-
handed antineutrino “annihilation” with the emission of aW−boson. Recall that
an arrow entering a vertex corresponds to a wave functionψwhile an outgoing line
corresponds to the conjugated functionψ. ̄ If the arrow coincides with the direction
of time, then the corresponding line describes the particle; otherwise it describes
the antiparticle. Hence theW+boson line entering the vertex in Figure 4.8 corre-
sponds to its antiparticle, that is, theW−boson. The wave functioneLdescribes the
left-handed electrone−L,and ̄νLcorresponds to the right-handed antineutrino ̃νR.
Under charge conjugationCall arrows on the diagram are reversed (Figure 4.9).
The right-handed antineutrino goes to the right-handed neutrino, ̃νR→νR,and the
left-handed electron converts to the left-handed positron,e−L →e+L↔e ̄R.Thus

g

√

2

(ν ̄LγμeL)Wμ+→

g

√

2

(e ̄RγμνR)Wμ−. (4.88)

After applying the P operation, this term coincides with the second term in
Lagrangian (4.87). Likewise, the second term in (4.87) converts to the first one.
Therefore expression (4.87) isCP-invariant. The reader can verify that the other
terms in (4.74) are alsoCP-invariant.