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University of California at San Diego – Department of Physics – Prof. John McGreevy Quantum Field Theory C (215C) Spring 2013 Assignment 3 Posted April 23, 2013 Due 11am, Thursday, May 2, 2013 Relevant reading: Zee, Chapter IV.3. Zee, Chapter III.3 also overlaps with recent lecture material. Problem Set 3 1. Propagator corrections in a solvable field theory. Consider a theory of a scalar field in D dimensions with action S = S0 + S1 where Z S0 = dD x and 1 ∂µ φ∂ µ φ − m20 φ2 2 Z 1 dD x δm2 φ2 . 2 We have artificially decomposed the mass term into two parts. We will do perturbation theory in small δm2 , treating S1 as an ‘interaction’ term. We wish to show that the organization of perturbation theory that we’ve seen lecture will correctly reassemble the mass term. S1 = − (a) Write down all the Feynman rules for this perturbation theory. (b) Determine the 1PI two-point function in this model. (c) Show that the (geometric) summation of the propagator corrections correctly produces the propagator that you would have used had we not split up m20 + δm2 . 2. Coleman-Weinberg potential. (a) [Zee problem IV.3.4] What set of Feynman diagrams are summed by the ColemanWeinberg calculation? [Hint: expand the logarithm as a series in V 00 /k 2 and associate a Feynman diagram with each term.] 1 (b) [Zee problem IV.3.3] Consider a massless fermion field coupled to a scalar field φ by a coupling gφψ̄ψ in D = 1 + 1. Show that the one loop effective potential that results from integrating out the fluctuations of the fermion has the form 2 φ 1 2 (gφ) log VF = 2π M2 after adding an appropriate counterterm. 2