Speaker
Description
Complete generalizations of Zamolodchikov's c-theorem to higher dimensions remains the holy grail in the study of renormalization group flows. While in the intervening years there has been success in proving weak and strong monotonicity theorems in a variety of dimensions and contexts, the strongest version, gradient flow, remains much more elusive. After a brief review of monotonicity theorems in general and gradient flow in particular, I will detail recent advances in the study of gradient flow in general scalar-fermion systems in d=4-ϵ. Perturbatively, gradient flow can be reduced to a set of constraint equations on the coefficients appearing in the beta function, and using recent results for the generic beta function I will show that these constraints are satisfied through all known loop orders only once one replaces beta by the so-called B function. Unlike in previously studied purely scalar systems, there exist fixed points at which the difference between beta and B is non-zero, and B must be introduced to correctly deal with fictitious limit cycles. The monotonic quantity appearing in this solution is connected to a weak monotonicity conjecture by Fei, Giombi, Klebanov and Tarnopolsky, suggesting that this conjecture can be strengthened.