Srednicki — Quantum Field Theory · Problem 68.3

Canonical Quantization of Scalar Fields — Mark Srednicki

Physics — QFT · ward-identities-scalar-electrodynamics

Problem

Scalar electrodynamics. (Prerequisite: 65.) a) Consider the Fourier transform of

\langle 0|\mathrm{T}J^\mu(x)\varphi(y)\varphi^\dagger(z)|0\rangle

, where

J^\mu = -ieZ_2[\varphi^\dagger\partial^\mu\varphi - (\partial^\mu\varphi^\dagger)\varphi] - 2Z_1 e^2 A^\mu\varphi^\dagger\varphi \tag{68.15}

is the Noether current. You may assume that

Z_4 = Z_1^2/Z_2

(which is necessary for gauge invariance). Show that

(p'-p)_\mu V_3^\mu(p',p) = Z_2^{-1}Z_1 e\left[\tilde{\Delta}(p')^{-1} - \tilde{\Delta}(p)^{-1}\right], \tag{68.16}

where

V_3^\mu(p',p)

is the exact scalar-scalar-photon vertex function, and

\tilde{\Delta}(p)

is the exact scalar propagator. b) Use this result to show that

Z_1 = Z_2

in both the

\overline{\mathrm{MS}}

and OS renormalization schemes. c) Consider the Fourier transform of

\langle 0|\mathrm{T}J^\mu(x)A^\nu(w)\varphi(y)\varphi^\dagger(z)|0\rangle

. Show that

k_\mu V_4^{\mu\nu}(k,p',p) = Z_1^{-1}Z_4 e\left[V_3^\nu(p'-k,p) - V_3^\nu(p',p+k)\right], \tag{68.17}

where

V_4^{\mu\nu}(k,p',p)

is the exact scalar-scalar-photon-photon vertex function, with

k

the incoming momentum of the photon at the

\mu

vertex.