Generalised cusp anomalous dimension in β−deformed super Yang Mills theory

Abstract

In this work we study the cusp anomalous dimension of the marginally deformed N = 4 sYM theory. We find the expression of the cusp anomalous dimension both at the weak and strong coupling limits. On the gravity side we partially map the system of equations to that of undeformed N = 4 sYM, by defining in the internal space an effective angle ∆θ ef f which depends on the deformation parameter. The cusp anomalous dimension then can be read from that of the N = 4 sYM theory, by using this effective angle instead of the usual angle of the undeformed 5-sphere. In the weak coupling regime we find no β dependence up to two loops. Based on these results we conjecture that for any value of λ the cusp anomalous dimension is given by the N = 4 sYM result, using the coupling dependent effective angle. As a consequence of this, we derive a BPS-like condition between the angle of the AdS space and the ∆θ ef f of the deformed sphere which when satisfied, nullifies the cusp anomalous dimension.

On marginal deformations and non-integrability

“…It is worth remarking that due to the fact that the complex deformed background has an overall conformal factor in the AdS part of the metric and in the complex deformed 5-sphere, as well as a B-field having components along several directions, its system of string equations has major differences compared to that of the real β deformed [23] and to that of the undeformed theory. A recent realization of this appeared in the computation of the generalized cusp anomalous dimension [26] where in the gravity side for real β parameter the system of string equations can be partially mapped to the undeformed N = 4 sYM, while for the complex β this is not possible. Another computation pointing out such differences, is the Lax pair construction for the real β deformed background, which is crucially based on a correspondence between the deformed and undeformed string equations [20,27], while this mapping can not be made in the complex β deformed background.…”