A Metric for Gradient RG Flow of the Worldsheet Sigma Model Beyond First Order

TL;DR

This paper investigates the validity of a proposed gradient flow metric for the worldsheet sigma model's RG flow beyond first order, finding that at second order the metric's positivity is not guaranteed, affecting monotonicity.

Contribution

It explicitly computes second-order corrections to the metric on coupling space, testing the proposed monotonicity formula beyond leading order.

Findings

Second-order metric is not positive semi-definite.

Monotonicity may fail when curvature derivatives grow large.

Higher order corrections become significant in certain regimes.

Abstract

Tseytlin has recently proposed that an action functional exists whose gradient generates to all orders in perturbation theory the Renormalization Group (RG) flow of the target space metric in the worldsheet sigma model. The gradient is defined with respect to a metric on the space of coupling constants which is explicitly known only to leading order in perturbation theory, but at that order is positive semi-definite, as follows from Perelman's work on the Ricci flow. This gives rise to a monotonicity formula for the flow which is expected to fail only if the beta function perturbation series fails to converge, which can happen if curvatures or their derivatives grow large. We test the validity of the monotonicity formula at next-to-leading order in perturbation theory by explicitly computing the second-order terms in the metric on the space of coupling constants. At this order, this metric is found not to be positive semi-definite. In situations where this might spoil monotonicity, derivatives of curvature become large enough for higher order perturbative corrections to be significant.