Threshold expansion of Feynman diagrams within a configuration space technique

TL;DR

The paper develops a configuration-space method to obtain near-threshold expansions for n-line water melon Feynman diagrams, yielding explicit spectral-density expansions and a generalized resummation for a very small mass. By decomposing the problem into analytic and non-analytic parts and exploiting the large-x behavior of propagators, it provides closed-form or hypergeometric representations for the threshold expansions that agree with known exact results. The approach is universal for arbitrary n, masses, and space-time dimensions, and includes practical examples: equal-mass topologies, strongly asymmetric mass arrangements, and convolution-based resummations. This work enhances both the accuracy and efficiency of threshold analyses in perturbative quantum field theory with wide-ranging applications.

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.

Figures (7)

• Figure 1: Representation of $\Pi^+(p)$ as vacuum bubble with added line. The cross denotes an arbitrary number of derivatives to the specified line.
• Figure 2: Various results for the spectral density for $n=3$ equal masses in $D=4$ space-time dimensions in dependence on the threshold parameter $E/M$. Shown are the exact solution obtained by using Eq. (\ref{['exactrho4']}) (solid curve) and threshold expansions for different orders taken from Eq. (\ref{['pidas430']}) (dashed to dotted curves).
• Figure 3: Various solutions for the spectral density for two masses $m$ and $m_0\ll m$ and $D=4$ space-time dimensions. Shown are the exact solution which is obtained by using Eq. (\ref{['exactrho4']}) (solid curve), the pure threshold expansions using Eq. (\ref{['pidas42t']}) (dotted curves), and the solutions for the resummation of the smallest mass contributions like in Eq. (\ref{['pipas42t']}) (dashed curves), both expansions from the first up to the fourth order in the asymptotic expansion. For the pure threshold expansion the order is indicated explicitly.
• Figure 4: The same as Fig. \ref{['fig3']} where the spectral density is normalized to the leading order expression of the pure threshold expansion.
• Figure 5: The spectral density for the sunset diagram in $D=4$ space-time dimensions with a tiny mass $m_0$, normalized to the general power behaviour. Shown are the exact result obtained by using Eq. (\ref{['exactrho4']}) (solid curve), the threshold expansion according to Eq. (\ref{['pidas43t']}) (dotted curves), and the result for the resummation of the smallest mass contributions according to Eq. (\ref{['pipas43t']}) (dashed curves).