**10.3 Field-strength renormalization in ϕ4 theory.** The two-loop contribution to the propagator in ϕ4 theory involves the three diagrams shown in (10.31). Compute the first of these diagrams in the limit of zero mass for the scalar field, using dimensional regularization. Show that, near d=4, this diagram takes the form:
(diagram)=−ip2⋅12(4π)4λ2[−ϵ1+logp2+⋯],
with ϵ=4−d. The coefficient in this equation involves a Feynman parameter integral that can be evaluated by setting d=4. Verify that the second diagram in (10.31) vanishes near d=4. Thus the first diagram should contain a pole only at ϵ=0, which can be canceled by a field-strength renormalization counterterm.