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

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

2007-05-06

arxiv.org/abs/0705.0827v3

Phys.Rev.D76:045001,2007

Physics

High Energy Physics

High Energy Physics - Theory

15 pages; Erroneous sentence in footnote 14 removed; this version therefore supersedes the published version (our thanks to De

Scientific paper

10.1103/PhysRevD.76.045001

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.

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