Srednicki — Quantum Field Theory · Problem 15.1

Canonical Quantization of Scalar Fields — Mark Srednicki

Physics — QFT · lehmann-kallen-spectral-representation

Problem

In this problem we will verify the result of problem 13.1 to

O(\alpha)

. a) Let

\Pi_{\rm loop}(k^2)

be given by the first line of eq. (14.32), with

\varepsilon > 0

. Show that, up to

O(\alpha^2)

corrections,

A = \Pi'_{\rm loop}(-m^2) . \tag{15.14}

Then use Cauchy's integral formula to write this as

A = \oint \frac{dw}{2\pi i} \frac{\Pi_{\rm loop}(w)}{(w + m^2)^2} , \tag{15.15}

where the contour of integration is a small counterclockwise circle around

-m^2

in the complex

w

plane. b) By examining eq. (14.32), show that the only singularity of

\Pi_{\rm loop}(k^2)

is a branch point at

k^2 = -4m^2

. Take the cut to run along the negative real axis. c) Distort the contour in eq. (15.15) to a circle at infinity with a detour around the branch cut. Examine eq. (14.32) to show that, for

\varepsilon > 0

, the circle at infinity does not contribute. The contour around the branch cut then yields

A = \int_{-\infty}^{-4m^2} \frac{dw}{2\pi i} \frac{1}{(w+m^2)^2} \left[\Pi_{\rm loop}(w+i\epsilon) - \Pi_{\rm loop}(w-i\epsilon)\right] , \tag{15.16}

where

\epsilon

is infinitesimal (and is not to be confused with

\varepsilon = 6-d

). d) Examine eq. (14.32) to show that the real part of

\Pi_{\rm loop}(w)

is continuous across the branch cut, and that the imaginary part changes sign, so that

\Pi_{\rm loop}(w+i\epsilon) - \Pi_{\rm loop}(w-i\epsilon) = -2i\,\mathrm{Im}\,\Pi_{\rm loop}(w-i\epsilon) . \tag{15.17}

e) Let

w = -s

in eq. (15.16) and use eq. (15.17) to get

A = -\frac{1}{\pi} \int_{4m^2}^\infty ds\, \frac{\mathrm{Im}\,\Pi_{\rm loop}(-s-i\epsilon)}{(s-m^2)^2} . \tag{15.18}

Use this to verify the result of problem 13.1 to

O(\alpha)

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