An Introduction To Quantum Field Theory - M. Peskin, D. Schroeder (Perseus, 1995) · Problem 19.3

Perturbation Theory Anomalies — M. Peskin, D. Schroeder (Perseus, 1995)

Physics — QFT · perturbation-theory-anomalies

Problem

19.3 Computation of anomaly coefficients. (a) Consider a product

r_1 \times r_2

SU(n)

representations, which is decomposed into irreducible representations as in (15.98). Using the explicit form of the generators given in (15.99), show that the anomaly coefficients satisfy

d_1 A(r_2) + d_2 A(r_1) = \sum_i A(r_i).

(b) As we saw in Problem 15.5, the two-index symmetric and antisymmetric tensors form irreducible representations of

SU(n)

, which we will call

s

and

a

, respectively. In

SU(3)

, the representation

a

is three-dimensional. Show that it is equivalent to the

\bar{3}

. Compute the anomaly coefficients for

a

and

s

, making use of the identity in part (a). (c) Since

SU(n)

has a unique three-index symmetric tensor

d^{abc}

which is already nonvanishing in an

SU(3)

subgroup, we can compute the anomaly coefficient in

SU(n)

by restricting our attention to three generators in this subgroup. By decomposing

SU(n)

representations into

SU(3)

representations, compute the anomaly coefficients for

a

and

s

SU(n)

and derive Eq. (19.141). Find the anomaly coefficient of the

j

-index totally antisymmetric tensor representation of

SU(n)

. Why does the result always vanish when

2j = n

Related theory

Anomalies In Quantum Field Theory

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