Regular Seminar Ian Jack (Liverpool)
The a-theorem expressing the monotonicity of renormalisation group flows in four (and other even) dimensions is now well-accepted. There is a function (the a-function) generating the RG beta-functions as a gradient flow via a positive-definite metric. However the standard definition of the a-function in terms of the trace anomaly of the energy-momentum tensor does not work in odd dimensions. In this talk we focus on the gradient-flow property of the a-function and show that a function with similar properties can be constructed order-by-order in three dimensions. We start by reviewing the a-function in even dimensions from a gradient-flow standpoint. Then we discuss our explicit calculations in three dimensions. Finally we present some progress towards relating our results to the F-function which has been shown to have the expected monotonicity properties at fixed points.