Srednicki — Quantum Field Theory · Problem 29.2

Canonical Quantization of Scalar Fields — Mark Srednicki

Physics — QFT · effective-field-theory-phi3-cutoff-beta-function

Problem

Consider

\varphi^3

theory in six euclidean spacetime dimensions, with a cutoff

\Lambda_0

and lagrangian

\mathcal{L} = \tfrac{1}{2}Z(\Lambda_0)\partial_\mu\varphi\partial_\mu\varphi + \tfrac{1}{6}Z^{3/2}(\Lambda_0)g(\Lambda_0)\varphi^3 \,. \tag{29.40}

We assume that we have fine-tuned to keep

m^2(\Lambda) \ll \Lambda^2

, and so we neglect the mass term. a) Show that

\begin{aligned} Z(\Lambda) &= Z(\Lambda_0)\left(1 - \tfrac{1}{2}g^2(\Lambda_0)\frac{d}{dk^2}\left[\int_\Lambda^{\Lambda_0}\frac{d^6\ell}{(2\pi)^6}\frac{1}{(k+\ell)^2\ell^2}\right]_{k^2=0} + \ldots\right), \\ g(\Lambda) &= \frac{Z^{3/2}(\Lambda_0)}{Z^{3/2}(\Lambda)}g(\Lambda_0)\left(1 + g^2(\Lambda_0)\int_\Lambda^{\Lambda_0}\frac{d^6\ell}{(2\pi)^6}\frac{1}{(\ell^2)^3} + \ldots\right). \end{aligned}

Hint: note that the tree-level propagator is

\tilde{\Delta}(k) = [Z(\Lambda_0)k^2]^{-1}

. b) Use your results to compute the beta function

\beta(g(\Lambda)) \equiv \frac{d}{d\ln\Lambda}g(\Lambda) \,, \tag{29.41}

and compare with the result in section 27.