Standard Model and Beyond · Problem 4.12

The Strong Interactions and QCD

Physics — Particle Physics · anomaly-coefficient-su-m

Problem

(a) Calculate the anomaly coefficient for a left-chiral fundamental representation of

SU(m)

in terms of the

d_{ijk}

defined in (3.26). (b) Consider a field

\psi_{ab}

that transforms as the (reducible) direct product of two

SU(m)

fundamentals. It follows from (3.118) that the corresponding representation matrices are

\left(L^i_D\right)_{ab;cd} = \left(L^i_{ac}\delta_{bd} + \delta_{ac}L^i_{bd}\right)

, which can conveniently be written in direct product notation as

L^i_D = L^i \otimes I + I \otimes L^i

, where

I

is the

m \times m

identity. Calculate the anomaly coefficient for a left-chiral field transforming as

\psi_{ab}

. Hint:

\mathrm{Tr}\, A \otimes B = \mathrm{Tr}\, A\, \mathrm{Tr}\, B

, and

(A \otimes B)(C \otimes D) = AC \otimes BC