3 The Standard Model of Electroweak Interactions
57
down by powers of o(m 2
W /m 2
t ). In fact, the longitudinal W is dominant in the final
state because h t >> g 2 . Similarly the equivalence theorem can be applied to find
the dominant terms at large
√
s for the crosssection e + e − → W
+
L W
−
L , or the leading
contribution in the limit m H >> m V to the width for the decay (H → V V ).
The formalism of the R ξ gauges is also very useful in proving that spontaneously
broken gauge theories are renormalizable. In fact, the non singular behaviour of
propagators at large momenta is very suggestive of the result. Nevertheless to
prove it is by far not a simple matter. The fundamental theorem that in general a
gauge theory with spontaneous symmetry breaking and the Higgs mechanism is
renormalizable was proven by ’t Hooft and Veltman [27, 28].
For a chiral theory like the SM an additional complication arises from the
existence of chiral anomalies. But this problem is avoided in the SM because the
quantum numbers of the quarks and leptons in each generation imply a remarkable
(and, from the point of view of the SM, mysterious) cancellation of the anomaly,
as originally observed in Ref. [29]. In quantum field theory one encounters an
anomaly when a symmetry of the classical lagrangian is broken by the process of
quantization, regularization and renormalization of the theory. Of direct relevance
for the EW theory is the Adler-Bell-Jackiw (ABJ) chiral anomaly [30]. The classical
lagrangian of a theory with massless fermions is invariant under a U(1) chiral
transformations ψ = e iγ 5 θ ψ. The associated axial Noether current is conserved
at the classical level. But, at the quantum level, chiral symmetry is broken due to the
ABJ anomaly and the current is not conserved. The chiral breaking is produced by a
clash between chiral symmetry, gauge invariance and the regularization procedure.
The anomaly is generated by triangular fermion loops with one axial and two
vector vertices (Fig. 3.9). For example, for the Z the axial coupling is proportional
to the third component of weak isospin t 3 , while the vector coupling is proportional
to a linear combination of t 3 and the electric charge Q. Thus in order for the chiral
anomaly to vanish all traces of the form tr{t 3 QQ}, tr{t 3 t 3 Q}, tr{t 3 t 3 t 3 } (and also
tr{t + t − t 3 } when charged currents are also included) must vanish, where the trace
is extended over all fermions in the theory that can circulate in the loop. Now all
these traces happen to vanish for each fermion family separately. For example take
tr{t 3 QQ}. In one family there are, with t 3 = +1/2, three colours of up quarks with
Fig. 3.9 Triangle diagram
that generates the ABJ
anomaly
A
V
V
57
down by powers of o(m 2
W /m 2
t ). In fact, the longitudinal W is dominant in the final
state because h t >> g 2 . Similarly the equivalence theorem can be applied to find
the dominant terms at large
√
s for the crosssection e + e − → W
+
L W
−
L , or the leading
contribution in the limit m H >> m V to the width for the decay (H → V V ).
The formalism of the R ξ gauges is also very useful in proving that spontaneously
broken gauge theories are renormalizable. In fact, the non singular behaviour of
propagators at large momenta is very suggestive of the result. Nevertheless to
prove it is by far not a simple matter. The fundamental theorem that in general a
gauge theory with spontaneous symmetry breaking and the Higgs mechanism is
renormalizable was proven by ’t Hooft and Veltman [27, 28].
For a chiral theory like the SM an additional complication arises from the
existence of chiral anomalies. But this problem is avoided in the SM because the
quantum numbers of the quarks and leptons in each generation imply a remarkable
(and, from the point of view of the SM, mysterious) cancellation of the anomaly,
as originally observed in Ref. [29]. In quantum field theory one encounters an
anomaly when a symmetry of the classical lagrangian is broken by the process of
quantization, regularization and renormalization of the theory. Of direct relevance
for the EW theory is the Adler-Bell-Jackiw (ABJ) chiral anomaly [30]. The classical
lagrangian of a theory with massless fermions is invariant under a U(1) chiral
transformations ψ = e iγ 5 θ ψ. The associated axial Noether current is conserved
at the classical level. But, at the quantum level, chiral symmetry is broken due to the
ABJ anomaly and the current is not conserved. The chiral breaking is produced by a
clash between chiral symmetry, gauge invariance and the regularization procedure.
The anomaly is generated by triangular fermion loops with one axial and two
vector vertices (Fig. 3.9). For example, for the Z the axial coupling is proportional
to the third component of weak isospin t 3 , while the vector coupling is proportional
to a linear combination of t 3 and the electric charge Q. Thus in order for the chiral
anomaly to vanish all traces of the form tr{t 3 QQ}, tr{t 3 t 3 Q}, tr{t 3 t 3 t 3 } (and also
tr{t + t − t 3 } when charged currents are also included) must vanish, where the trace
is extended over all fermions in the theory that can circulate in the loop. Now all
these traces happen to vanish for each fermion family separately. For example take
tr{t 3 QQ}. In one family there are, with t 3 = +1/2, three colours of up quarks with
Fig. 3.9 Triangle diagram
that generates the ABJ
anomaly
A
V
V