Perturbative analysis of the gradient flow in non-abelian gauge theories

Abstract

The gradient flow in non-abelian gauge theories on $ {\mathbb{R}^4} $ is defined by a local diffusion equation that evolves the gauge field as a function of the flow time in a gauge-covariant manner. Similarly to the case of the Langevin equation, the correlation functions of the time-dependent field can be expanded in perturbation theory, the Feynman rules being those of a renormalizable field theory on $ {\mathbb{R}^4} \times \left[ {0,\infty } \right) $ . For any matter multiplet and to all loop orders, we show that the correlation functions are finite, i.e. do not require additional renormalization, once the theory in four dimensions is renormalized in the usual way. The flow thus maps the gauge field to a one-parameter family of smooth renormalized fields.