Metric for gradient renormalization group flow of the worldsheet sigma model beyond first order
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 semidefinite, 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 nexttoleading order in perturbation theory by explicitly computing the secondorder terms in the metric on the space of coupling constants. At this order, this metric is found not to be positive semidefinite. In situations where this might spoil monotonicity, derivatives of curvature become large enough for higherorder perturbative corrections to be significant.
 Publication:

Physical Review D
 Pub Date:
 August 2007
 DOI:
 10.1103/PhysRevD.76.045001
 arXiv:
 arXiv:0705.0827
 Bibcode:
 2007PhRvD..76d5001O
 Keywords:

 11.10.Hi;
 Renormalization group evolution of parameters;
 High Energy Physics  Theory;
 General Relativity and Quantum Cosmology;
 Mathematics  Differential Geometry
 EPrint:
 15 pages