Inverse Mass Expansions from Worldline Path

Higher order coefficients of the inverse mass expansion of one–loop effective actions are obtained from a one–dimensional path integral representation. For the evaluation of the path integral with Wick contractions a suitable Green function has to be chosen. We consider the case of a massive scalar loop in the background of both a scalar potential and a (non–abelian) gauge field. For the pure scalar case the method yields the coefficients of the expansion in a minimal set of basis terms whereas complicated ordering problems arise in gauge theory. An appropriate reduction scheme is discussed. In recent years, the inverse mass expansion of one–loop effective actions has been applied in fields as different as the calculation of quark determinants, bubble nuc-leation during the electroweak phase transition or baryon number violation by in-stanton or sphaleron processes. A variety of approaches and results can be found in the literature [1]. The actual computation of higher order coefficients, however, suffers from practical limitations and was worked out only to order O(T 6) for the ungauged theory [2] and O(T 5) for the gauged case [3]. In our approach, instead of using conventional heat–kernel techniques we start from a representation of the one–loop effective action as a one–dimensional path integral. For the evaluation of the path integral we make use of recent progress in calculating one–loop amplitudes inspired by string theory [4,5,6], which enables us to compute the inverse mass expansion to high orders. Formally, the one-loop effective action for a scalar particle with mass m in the background of a gauge field A µ (abelian or non–abelian) and a matrix–valued scalar potential V is obtained as the determinant of a fluctuation operator M , which can be written in the Schwinger proper time formalism as