Relation between worldline Green functions for scalar two-loop diagrams

We discuss a relation between two-loop bosonic worldline Green functions which are obtained by Schmidt and Schubert in two different parametrizations of a twoloop worldline. These Green functions are transformed into each other by some transformation rules based on reparametrizations of the proper time and worldline modular parameters. 1 Fellow of the Danish Research Academy, [email protected] One of interesting and important theoretical aspects of quantum field theories, in particular of gauge theories, is the Bern-Kosower formalism [1] which provides a reformulation of one-loop Feynman amplitudes. Their idea is very natural that amplitudes in ordinary field theory may be reproduced by the infinite string tension limit of superstring amplitudes. It results in a new set of rules instead of Feynman rules and enables several calculations: five point gluon [2], four point graviton amplitudes [3] and so on [4]. The advantage of this formalism is to get rid of handling a vast number of Feynman diagrams and of taking care of gauge cancelation of diagrams in gauge invariant theories. It might clarify underlying structure, which we have not been recognized yet in gauge theories; for example, various useful informations have been developed in worldline approaches [5]-[7]. Extension of this formalism to multi-loop diagrams has also been investigated recently [11],[12],[13] generalizing Strassler’s approach [8] to the Bern-Kosower rules. Strassler derived the worldline Green functions [9] of spinor, scalar and gauge fields for one-loop diagrams rewriting one-loop effective actions as path integrals of (supersymmetric) worldline actions. In one-loop case, the modular parametrization of a loop is unique and we can define the worldline Green functions in a unique way. However, this situation changes in multi-loop cases because we have a variety of choices of the parametrization due to node points, i.e., internal vertices. Schmidt and Schubert have actually obtained two expressions for the two-loop worldline Green function in the φ theory [11]. They explicitly checked that amplitudes derived from these respective Green functions coincide with at least three or four point functions from the Feynman rule calculations. It is apparent that this kind of equivalence check between worldline Green functions in different forms becomes difficult in more general situations such as higher loop diagrams and a large number of external legs. In addition, we can not recognize which parametrization is natural or convenient one in a complicated case, and hence it is useful to know how to convert the Green functions to a differently parametrized form. It is therefore important to find a clear connection between multi-loop Green functions defined on a differently parametrized worldline. In this report, we discuss a relationship between two-loop bosonic worldline Green 1 functions proposed by Schmidt and Schubert in the scalar φ theory. First let us survey the vacuum amplitude at one-loop level in two different ways which are instructive to get an insight into two-loop case. One is given by the path integral for a worldline action with cyclic boundary condition and the other may be given by sewing two propagators of path integral expression [8],[9]; I (1) 0 = ∫ ∞