Srednicki — Quantum Field Theory · Problem 15.2

Canonical Quantization of Scalar Fields — Mark Srednicki

Physics — QFT · dispersion-relations

Problem

Dispersion relations. Consider the exact

\Pi(k^2)

, with

\varepsilon = 0

. Assume that its only singularity is a branch point at

k^2 = -4m^2

, that it obeys eq. (15.17), and that

\Pi(k^2)

grows more slowly than

|k^2|^2

at large

|k^2|

. By recapitulating the analysis in the previous problem, show that

\Pi''(k^2) = \frac{2}{\pi} \int_{4m^2}^\infty ds\, \frac{\mathrm{Im}\,\Pi(-s-i\varepsilon)}{(k^2+s)^3} . \tag{15.19}

This is a twice subtracted dispersion relation. It gives

\Pi''(k^2)

throughout the complex

k^2

plane in terms of the values of the imaginary part of

\Pi(k^2)

along the branch cut.