Threshold expansion of Feynman diagrams within a configuration space technique

Abstract

The near threshold expansion of generalized sunset-type (water melon) diagrams with arbitrary masses is constructed by using a configuration space technique. We present analytical expressions for the expansion of the spectral density near threshold and compare it with the exact expression obtained earlier using the method of the Hankel transform. We formulate a generalized threshold expansion with partial resummation of the small mass corrections for the strongly asymmetric case where one particle in the intermediate state is much lighter than the others.

On the evaluation of a certain class of Feynman diagrams in x-space: Sunrise-type topologies at any loop order

“…[37,38,39]) one has To generalize the expansion and extend it to the region of E ∼ M one has to treat the smallest mass exactly. In this case one can use a method which we call the resummation of small mass effects [40]. This method will be explained in the following.…”

“…The configuration space technique is best suited for the topology of the diagram. It considerably reduces the complexity of the calculation and allows for new qualitative and quantitative results for this important particular class of Feynman integrals [25,37,38,39,40,41]. Configuration space techniques can be used to verify known results obtained within other techniques both analytically and numerically [42,43,44,45] and to investigate some general features of Feynman diagram calculation [46,30,39,47,48,49,50,51].…”

Numerical evaluation of the general massive 2-loop sunrise self-mass master integrals from differential equations

“…(7). For the necessary initial conditions we use the values of the MI in the special points, where the differential equations simplify, allowing the analytic calculation of the MI; but as also the coefficients of the derivatives vanish there, to start the numerical evaluation from that points the values of the first derivatives must be provided as well.…”

The threshold expansion of the 2-loop sunrise self-mass master amplitudes

“…With the method established in [5] and [6], further, the expansion around threshold was obtained in [7], even if a complete analytical result was given there only for the case of equal masses. With the configuration space technique the expansion around threshold was investigated also in [8]. The expansion around one of the pseudothresholds (the others are straightforwardly given by a cyclic permutation of the masses) was obtained in analytical form in [9].…”