We develop tools for performing effective field theory (EFT) calculations in a manifestly gauge-covariant fashion. We clarify how functional methods account for one-loop diagrams resulting from the exchange of both heavy and light fields, as some confusion has recently arisen in the literature. To efficiently evaluate functional traces containing these "mixed" one-loop terms, we develop a new covariant derivative expansion (CDE) technique that is capable of evaluating a much wider class of traces than previous methods. The technique is detailed in an appendix, so that it can be read independently from the rest of this work. We review the well-known matching procedure to one-loop order with functional methods. What we add to this story is showing how to isolate one-loop terms coming from diagrams involving only heavy propagators from diagrams with mixed heavy and light propagators. This is done using a non-local effective action, which physically connects to the notion of "integrating out" heavy fields. Lastly, we show how to use a CDE to do running analyses in EFTs, \textit{i.e.} to obtain the anomalous dimension matrix. We demonstrate the methodologies by several explicit example calculations.
Three-loop $β$-functions and two-loop anomalous dimensions for MSSM regularized by higher covariant derivatives in an arbitrary supersymmetric subtraction scheme
Two theorems for the gradient expansion of relativistic hydrodynamics
这封信是专门提供证明的两个声明有关的梯度展开的相对论流体力学。第一种说法是\textit{the ordering of transverse derivatives is irrelevant in the gradient expansion of a non-conformal fluid}。第二种说法是\textit{the longitudinal projection of the Weyl covariant derivative can be eliminated in the gradient expansion of a conformal fluid}。这第二个陈述不适用于曲率张量。在共形的情况下，我们知道Weyl协变导数的排序在梯度展开中是无关的。
