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Tseytlin has recently proposed that an action functional exists whosegradient generates to all orders in perturbation theory the RenormalizationGroup (RG) flow of the target space metric in the worldsheet sigma model. Thegradient is defined with respect to a metric on the space of coupling constantswhich is explicitly known only to leading order in perturbation theory, but atthat order is positive semi-definite, as follows from Perelman's work on theRicci flow. This gives rise to a monotonicity formula for the flow which isexpected to fail only if the beta function perturbation series fails toconverge, which can happen if curvatures or their derivatives grow large. Wetest the validity of the monotonicity formula at next-to-leading order inperturbation theory by explicitly computing the second-order terms in themetric on the space of coupling constants. At this order, this metric is foundnot to be positive semi-definite. In situations where this might spoilmonotonicity, derivatives of curvature become large enough for higher orderperturbative corrections to be significant.

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