401
22.5. Fermion masses
in the W rest frame, measured with respect to a direction parallel (antiparallel)
to the ¯
p(p) beam. The expected form (1 + cos θ
∗ )
2 is followed very closely.
e
In summary, we may say that the early discovery experiments provided
remarkably convincing confirmation of the principal expectations of the GSW
theory, as outlined in section 22.3.
We now consider some further aspects of the theory.
22.5 Fermion masses
22.5.1 One generation
The fact that the SU(2) L gauge group acts only on the L components of the
fermion fields immediately appears to create a fundamental problem as far
as the masses of these particles are concerned; we mentioned this briefly at
the end of section 19.6. Let us recall first that the standard way to introduce
the interactions of gauge fields with matter fields (e.g. fermions) is via the
covariant derivative replacement
∂
μ
→ D
μ
≡ ∂
μ + igτ · W
μ /2
(22.119)
for SU(2) fields W
μ acting on t = 1/2 doublets. Now it is a simple exercise
(compare problem 18.3) to check that the ordinary ‘kinetic’ part of a free
Dirac fermion does not mix the L and R components of the field:
¯
¯
¯
ψ ˆ / ∂ψ ˆ = ψ ˆ
R / ∂ψ R + ψ ˆ
L / ∂ψ L .
(22.120)
Thus we can in principle contemplate ‘gauging’ the L and the R components
differently. Of course, in the case of QCD (cf (18.39)) the replacement ∂ / →/ D
was made equally in each term on the right-hand side of (22.120). But this was
because QCD conserves parity, and must therefore treat L and R components
the same. Weak interactions are parity violating, and the SU(2) L covariant
derivative acts only in the second term of (22.120). On the other hand, a
Dirac mass term has the form
¯ ˆ ˆ
¯ ˆ ˆ
−m(ψ L ψ R + ψ R ψ L )
(22.121)
(see equation (18.41) for example), and it precisely couples the L and R components. It is easy to see that if only ψ ˆ L is subject to a transformation, then
(22.121) is not invariant. Thus mass terms for Dirac fermions will explicitly
break SU(2) L . The same is also true for Majorana fermions which might
describe the neutrinos.
This kind of explicit breaking of the gauge symmetry cannot be tolerated,
in the sense that it will lead, once again, to violations of unitarity, and then of
22.5. Fermion masses
in the W rest frame, measured with respect to a direction parallel (antiparallel)
to the ¯
p(p) beam. The expected form (1 + cos θ
∗ )
2 is followed very closely.
e
In summary, we may say that the early discovery experiments provided
remarkably convincing confirmation of the principal expectations of the GSW
theory, as outlined in section 22.3.
We now consider some further aspects of the theory.
22.5 Fermion masses
22.5.1 One generation
The fact that the SU(2) L gauge group acts only on the L components of the
fermion fields immediately appears to create a fundamental problem as far
as the masses of these particles are concerned; we mentioned this briefly at
the end of section 19.6. Let us recall first that the standard way to introduce
the interactions of gauge fields with matter fields (e.g. fermions) is via the
covariant derivative replacement
∂
μ
→ D
μ
≡ ∂
μ + igτ · W
μ /2
(22.119)
for SU(2) fields W
μ acting on t = 1/2 doublets. Now it is a simple exercise
(compare problem 18.3) to check that the ordinary ‘kinetic’ part of a free
Dirac fermion does not mix the L and R components of the field:
¯
¯
¯
ψ ˆ / ∂ψ ˆ = ψ ˆ
R / ∂ψ R + ψ ˆ
L / ∂ψ L .
(22.120)
Thus we can in principle contemplate ‘gauging’ the L and the R components
differently. Of course, in the case of QCD (cf (18.39)) the replacement ∂ / →/ D
was made equally in each term on the right-hand side of (22.120). But this was
because QCD conserves parity, and must therefore treat L and R components
the same. Weak interactions are parity violating, and the SU(2) L covariant
derivative acts only in the second term of (22.120). On the other hand, a
Dirac mass term has the form
¯ ˆ ˆ
¯ ˆ ˆ
−m(ψ L ψ R + ψ R ψ L )
(22.121)
(see equation (18.41) for example), and it precisely couples the L and R components. It is easy to see that if only ψ ˆ L is subject to a transformation, then
(22.121) is not invariant. Thus mass terms for Dirac fermions will explicitly
break SU(2) L . The same is also true for Majorana fermions which might
describe the neutrinos.
This kind of explicit breaking of the gauge symmetry cannot be tolerated,
in the sense that it will lead, once again, to violations of unitarity, and then of