The monodromy rings of a class of self-energy graphs

Abstract

The monodromy rings of self-energy graphs, with two vertices and an arbitrary number of connecting lines, are determined.

The monodromy rings of one loop Feynman integrals

“…Secondly there is the question as to whether the success of our methods for the single loop integrals depends on their special functional form (they are sums of Spence functions). In our opinion the present calculations, and those of [1], show that the key to the determination of the monodromy of a Feynman integral is the determination of the fundamental group. The fundamental groups of single loop graphs are particularly simple.…”

“…This paper is the second of a series whose general aims were outlined in the introduction to the first paper [1]. In this paper we make a systematic study of the Feynman integral for a general one loop graph in an arbitrary space-time dimension; we classify the possible paths of analytic continuation, label the determinations of the function over a fixed base point, and obtain explicit formulae for the action of analytic continuation on the vector space spanned by these determinations.…”

“…The present account is self contained, but for the reader who is familiar with [2], we proceed to give a brief comparison. In the present paper we introduce the auxiliary complex parameters also used in [1] and [3]. This makes a technical difference in the construction of the representation since, for general values of the parameters, the Cutkosky-Steinman relations do not hold nevertheless, we are able to determine the representation by exploiting more fully the consequences of the Picard-Lefschetz theorem.…”

“…In this paper (as in [1]) we consider all possible analytic continuations i.e. we consider all the parameters which enter into the integrals, including the internal and external masses, as complex variables.…”

New String Amplitudes From Old Fermat (Hyper)surfaces

“…Subsequently, much more sophisticated cohomological analysis of the whole problem was developed by Pham [25] whose results were brought to perfection by Milnor [26] and Brieskorn [27]. In physics literature the results of Pham were carefully analyzed in two fundamental papers by Ponzano, Regge, Speer and Westwater [28]. Unfortunately, subsequent development of particle physics went into different direction(s).…”