A. Zee, Quantum Field Theory in a Nutshell · Problem IV.7.5

Field Theory Redux

Physics — QFT

Problem

There are literally NN ways of deriving the anomaly. Here is another. Evaluate Λλμν(k1,k2)=(−1)i3∫d4p(2π)4tr γλγ51p̸−q̸−mγν1p̸−̸k1−mγμ1p̸−m+{μ,k1↔ν,k2}\Lambda^{\lambda\mu\nu}(k_1, k_2) = (-1)i^3 \int \frac{d^4p}{(2\pi)^4} \text{tr } \gamma^\lambda\gamma^5 \frac{1}{\not{p} - \not{q} - m}\gamma^\nu \frac{1}{\not{p} - \not{k}_1 - m}\gamma^\mu \frac{1}{\not{p} - m} + \{\mu, k_1 \leftrightarrow \nu, k_2\} in the massive fermion case not by brute force but by first using Lorentz invariance to write Λλμν(k1,k2)=ϵλμνσk1σA1+...+ϵμνστk1σk2τk2λA8\Lambda^{\lambda\mu\nu}(k_1, k_2) = \epsilon^{\lambda\mu\nu\sigma} k_{1\sigma}A_1 + ... + \epsilon^{\mu\nu\sigma\tau} k_{1\sigma} k_{2\tau} k_2^\lambda A_8 where Ai≡Ai(k12,k22,q2)A_i \equiv A_i(k_1^2, k_2^2, q^2) are eight functions of the three Lorentz scalars in the problem. You are supposed to fill in the dots. By counting powers as in chapters III.3 and III.7 show that two of these functions are given by superficially logarithmically divergent integrals while the other six are given by perfectly convergent integrals. Next, impose Bose statistics and vector current conservation 0=k2νΛλμν0 = k_{2\nu}\Lambda^{\lambda\mu\nu} to show that we can avoid calculating the superficially logarithmically divergent integrals. Compute the convergent integrals and then evaluate qλΛλμν(k1,k2)q_\lambda\Lambda^{\lambda\mu\nu}(k_1, k_2).

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