An Introduction To Quantum Field Theory - M. Peskin, D. Schroeder (Perseus, 1995) · Problem 7.2

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

Physics — QFT · ward-identity-regularization

Problem

Alternative regulators in QED. In Section 7.5, we saw that the Ward identity can be violated by an improperly chosen regulator. Let us check the validity of the identity Z1=Z2Z_1 = Z_2, to order α\alpha, for several choices of the regulator. We have already verified that the relation holds for Pauli-Villars regularization. (a) Recompute δZ1\delta Z_1 and δZ2\delta Z_2, defining the integrals (6.49) and (6.50) by simply placing an upper limit Λ\Lambda on the integration over ℓE\ell_E. Show that, with this definition, δZ1≠δZ2\delta Z_1 \neq \delta Z_2. (b) Recompute δZ1\delta Z_1 and δZ2\delta Z_2, defining the integrals (6.49) and (6.50) by dimensional regularization. You may take the Dirac matrices to be 4×44 \times 4 as usual, but note that, in dd dimensions, gμνγμγν=d.g_{\mu\nu}\gamma^\mu\gamma^\nu = d. Show that, with this definition, δZ1=δZ2\delta Z_1 = \delta Z_2.

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