Problems
253
coupled to gauge fields cannot be tolerated. As we shall see in chapter 20,
and is already evident from (18.48), axial currents are indeed present in weak
interactions and they are coupled to the W
± , Z
0 gauge fields. Hence, if this
theory is to be satisfactory at the quantum level, all anomalies must somehow
cancel away. That this is possible rests essentially on the observation that the
anomaly (18.133) is independent of the mass of the circulating fermion. Thus
cancellations are in principle possible between quark and lepton ‘triangles’
in the weak interaction case. Bouchiat et al. (1972) were the first to point
out that, for each generation of quarks and leptons, the anomalies will cancel
between quarks and leptons if the fractionally charged quarks come in three
colours. The condition that anomalies cancel in the gauged currents of the
Standard Model is the remarkably simple one (Ryder 1996, p384):
N c (Q u + Q d ) + Q e = 0
(18.135)
where N c is the number of colours and Q u , Q d and Q e are the charges (in units
of e) of the ‘u’, ‘d’, and ‘e’ type fields in each generation. Clearly (18.135)
is true for each generation of the Standard Model; the condition indicates
a remarkable connection, at some deep level, between the facts that quarks
come in three colours and have charges which are 1/3 fractions. The Standard
Model provides no explanation for this connection. Anomaly cancellation is a
powerful constraint on possible theories (’t Hooft 1980, Weinberg 1996 section
22.4).
Problems
18.1 Verify (18.24)–(18.26).
18.2 Verify (18.28)–(18.30).
18.3 Show that L q of (18.12) can be written as (18.39).
18.4 Show that the rate for π
+
→ μ
+ ν μ , calculated from the lowest-order
matrix element (18.52), is given by (18.53).
18.5 Verify the transformation equations (18.76) and (18.77).
(a) Consider a Lagrangian L ˆ (φ ˆ r , ∂ μ φ ˆ r ) where the φ ˆ r could be either bosonic
or fermionic fields. Let the fields transform by an infinitesimal local
(x-dependent) transformation
φ ˆ r → φ ˆ r − i∈ α (x)T
α φ ˆ s (sum on s).
rs
Show that the change in L ˆ may be written as
δ ˆ ˆ j
αμ (x)∂ μ ∈ α (x) + ∈ α (x)∂ μ ˆ j
αμ (x)
L =
18.6
253
coupled to gauge fields cannot be tolerated. As we shall see in chapter 20,
and is already evident from (18.48), axial currents are indeed present in weak
interactions and they are coupled to the W
± , Z
0 gauge fields. Hence, if this
theory is to be satisfactory at the quantum level, all anomalies must somehow
cancel away. That this is possible rests essentially on the observation that the
anomaly (18.133) is independent of the mass of the circulating fermion. Thus
cancellations are in principle possible between quark and lepton ‘triangles’
in the weak interaction case. Bouchiat et al. (1972) were the first to point
out that, for each generation of quarks and leptons, the anomalies will cancel
between quarks and leptons if the fractionally charged quarks come in three
colours. The condition that anomalies cancel in the gauged currents of the
Standard Model is the remarkably simple one (Ryder 1996, p384):
N c (Q u + Q d ) + Q e = 0
(18.135)
where N c is the number of colours and Q u , Q d and Q e are the charges (in units
of e) of the ‘u’, ‘d’, and ‘e’ type fields in each generation. Clearly (18.135)
is true for each generation of the Standard Model; the condition indicates
a remarkable connection, at some deep level, between the facts that quarks
come in three colours and have charges which are 1/3 fractions. The Standard
Model provides no explanation for this connection. Anomaly cancellation is a
powerful constraint on possible theories (’t Hooft 1980, Weinberg 1996 section
22.4).
Problems
18.1 Verify (18.24)–(18.26).
18.2 Verify (18.28)–(18.30).
18.3 Show that L q of (18.12) can be written as (18.39).
18.4 Show that the rate for π
+
→ μ
+ ν μ , calculated from the lowest-order
matrix element (18.52), is given by (18.53).
18.5 Verify the transformation equations (18.76) and (18.77).
(a) Consider a Lagrangian L ˆ (φ ˆ r , ∂ μ φ ˆ r ) where the φ ˆ r could be either bosonic
or fermionic fields. Let the fields transform by an infinitesimal local
(x-dependent) transformation
φ ˆ r → φ ˆ r − i∈ α (x)T
α φ ˆ s (sum on s).
rs
Show that the change in L ˆ may be written as
δ ˆ ˆ j
αμ (x)∂ μ ∈ α (x) + ∈ α (x)∂ μ ˆ j
αμ (x)
L =
18.6