Srednicki — Quantum Field Theory · Problem 14.3

Canonical Quantization of Scalar Fields — Mark Srednicki

Physics — QFT · dimensional-regularization-tensor-integrals

Problem

a) Show that

\int d^d q\, q^\mu f(q^2) = 0 , \tag{14.52}

\int d^d q\, q^\mu q^\nu f(q^2) = C_2\, g^{\mu\nu} \int d^d q\, q^2 f(q^2) , \tag{14.53}

and evaluate the constant

C_2

in terms of

d

. Hint: use Lorentz symmetry to argue for the general structure, and evaluate

C_2

by contracting with

g_{\mu\nu}

. b) Similarly evaluate

\int d^d q\, q^\mu q^\nu q^\rho q^\sigma f(q^2)