A. Zee, Quantum Field Theory in a Nutshell · Problem III.7.1

Field Theory Redux

Physics — QFT

Problem

Calculate

\Pi_{\mu\nu}(q)

using dimensional regularization. The procedure is to start with (6), evaluate the trace in

N_{\mu\nu}

, shift the integration momentum from

p

l

, and so forth, proceeding exactly as in the text, until you have to integrate over the loop momentum

l

. At that point you "pretend" that you are living in

d

-dimensional spacetime, so that the term like

l_\mu l_\nu

N_{\mu\nu}

, for example, is to be effectively replaced by

(1/d)g_{\mu\nu}l^2

. The integration is to be performed using (III.1.15) and various generalizations thereof. Show that the form (3) automatically emerges when you continue to

d = 4

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