Quantum Field Theory and the Standard Model by Schwartz M.D. z-lib.org · Problem 7.4

Feynman rules — Schwartz M.D. z-lib.org

Physics — QFT · mass-as-interaction-perturbation

Problem

We made a distinction between kinetic terms, which are bilinear in fields, and interactions, which have three or more fields. Time evolution with the kinetic terms is solved exactly as part of the free Hamiltonian. Suppose, instead, we only put the derivative terms in the free Hamiltonian and treated the mass as an interaction. So,

H_0 = \frac{1}{2}\phi\Box\phi, \quad H_{\mathrm{int}} = \frac{1}{2}m^2\phi^2. \tag{7.125}

(a) Draw the (somewhat degenerate looking) Feynman graphs that contribute to the 2-point function

\langle 0|T\{\phi(x)\phi(y)\}|0\rangle

using only this interaction, up to order

m^6

. (b) Evaluate the graphs. (c) Sum the series to all orders in

m^2

and show you reproduce the propagator that would have come from taking

H_0 = \frac{1}{2}\phi\Box\phi + \frac{1}{2}m^2\phi^2

. (d) Repeat the exercise classically: Solve for the massless propagator using an external current, perturb with the mass, sum the series, and show that you get the same answer as if you included the mass to begin with.

Related theory

Mass To Mass Stoichiometry

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