15 The renormalization Group

Modern Quantum Field Theory 6015.4 Exercises1. Verify that the one-loop propagator in a massless ϕ 3 theory (251) is consistent with the the Callan Symanzikequation.2. Consider the case of a massless ϕ 3 theory, and suppose that there is a non-trivial fixed point α ∗ where β(α ∗ ) = 0.Show that the Callan Symanzik equation for the propagator is solved by a power of the momentum which isdictated by the field anomalous dimension γ ϕ .3. In the ϕ 3 theory in six dimensions, define the renormalization factor in the on-shell scheme by ϕ 0 (x) = 1ZOS 2ϕ ϕ OS (x) whereϕ 0 is the bare field and ϕ OS the field renormalized on-shell. The one-loop result forthis renormalization factor has been computed in lecture 10, eq. (175). Similarly, define the analogous relationϕ 0 (x) = ZϕMS12ϕ MS (x) intheMS.(a) Compute the residue R MS of the propagator in the MS scheme, as defined in (246) in terms of therenormalization factors ZϕOS and Zϕ MS .(b) Use this to determine R MS at one-loop and compare with (255).(c) Obtain eq. (240) and explain how the Feynman rules change in the MS scheme compared to the on-shellscheme and why.