An Introduction To Quantum Field Theory - M. Peskin, D. Schroeder (Perseus, 1995) · Problem 19.4
Perturbation Theory Anomalies — M. Peskin, D. Schroeder (Perseus, 1995)
Physics — QFT · perturbation-theory-anomalies
Problem
19.4 Large fermion mass limits. In the text, we derived the Adler-Bell-Jackiw and trace anomalies by the use of dimensional regularization. As an alternative, one could imagine deriving these results using Pauli-Villars regularization. In that technique, one regularizes the value of a fermion loop integral by subtracting the value of the same loop diagram computed with fermions Ψ of large mass M. The parameter M plays the role of the cutoff and should be taken to infinity at the end of the calculation. The anomalies arise because some pieces of the diagrams computed for very heavy fermions do not disappear in the limit M→∞. These nontrivial M→∞ limits are interesting in their own right and can have physical applications (for example, in part (c) of the Final Project for Part III).
(a) Show that the Adler-Bell-Jackiw anomaly equation is equivalent to the following large-mass limit of a fermion matrix element between the vacuum and a two-photon state:
M→∞lim{⟨p,k∣2iMΨˉγ5Ψ∣0⟩}=−2π2e2ϵανβλpαϵν∗(p)kβϵλ∗(k).(b) Show that the trace anomaly, at one-loop order, is equivalent to the following large-mass limit:
M→∞lim{⟨p,k∣MΨˉΨ∣0⟩}=+6π2e2[p⋅kϵ∗(p)⋅ϵ∗(k)−p⋅ϵ∗(k)k⋅ϵ∗(p)].(c) Show that the matrix element in part (a) is ultraviolet-finite before the M→∞ limit is taken. Evaluate the matrix element explicitly at one-loop order and verify the limit claimed in part (a).
(d) The matrix element in part (b) has a potential ultraviolet divergence. However, show that the coefficient of (p⋅ϵ∗(k)k⋅ϵ∗(p)) is ultraviolet-finite, and that the rest of the expression is determined by gauge invariance. Compute the full matrix element using dimensional regularization as a gauge-invariant regulator and verify the result claimed in part (b).