An Introduction To Quantum Field Theory - M. Peskin, D. Schroeder (Perseus, 1995) · Problem 7.3

Radiative Corrections: Some Formal Developments — M. Peskin, D. Schroeder (Perseus, 1995)

Physics — QFT · ward-identity-yukawa-qed

Problem

Consider a theory of elementary fermions that couple both to QED and to a Yukawa field

\phi

H_{\text{int}} = \int d^3x \frac{\lambda}{\sqrt{2}} \phi \,\bar{\psi}\psi + \int d^3x \, e \, A_\mu \, \bar{\psi}\gamma^\mu\psi.

(a) Verify that the contribution to

Z_1

from the vertex diagram with a virtual

\phi

equals the contribution to

Z_2

from the diagram with a virtual

\phi

. Use dimensional regularization. Is the Ward identity generally true in this theory? (b) Now consider the renormalization of the

\phi\bar{\psi}\psi

vertex. Show that the rescaling of this vertex at

q^2 = 0

is *not* canceled by the correction to

Z_2

. (It suffices to compute the ultraviolet-divergent parts of the diagrams.) In this theory, the vertex and field-strength rescaling give additional shifts of the observable coupling constant relative to its bare value.

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