An Introduction To Quantum Field Theory - M. Peskin, D. Schroeder (Perseus, 1995) · Problem 16.3
Quantization of Non-Abelian Gauge Theories — M. Peskin, D. Schroeder (Perseus, 1995)
Physics — QFT · non-abelian-gauge-counterterms
Problem
Counterterm relations. In Section 16.5, we computed the divergent parts of δ1, δ2, and δ3. It is a good exercise to compute the divergent parts of the remaining counterterms in Eq. (16.88) to one-loop order in the Feynman-'t Hooft gauge, and to explicitly verify that the counterterm relations (16.89) are consistent with the removal of ultraviolet divergences.
(a) The ghost counterterms are particularly easy to compute. Work out δ1c and δ2c, and show that the divergent part of their difference equals the divergent part of δ1−δ2. This gives a derivation of asymptotic freedom that is slightly easier than the one in Section 16.5.
(b) Compute the counterterm for the 3-gauge-boson vertex and verify the first equality in (16.89).
(c) Compute the counterterm for the 4-gauge-boson vertex and find, when the smoke clears, the second relation in (16.89).