Asymptotic expansion of Feynman integrals near threshold

Abstract

We present general prescriptions for the asymptotic expansion of massive multi-loop Feynman integrals near threshold. As in the case of previously known prescriptions for various limits of momenta and masses, the terms of the threshold expansion are associated with subgraphs of a given graph and are explicitly written through Taylor expansions of the corresponding integrands in certain sets of parameters. They are manifestly homogeneous in the threshold expansion parameter, so that the calculation of the given Feynman integral near the threshold reduces to the calculation of integrals of a much simpler type. The general method is illustrated by two-loop two-point and three-point diagrams. We discuss the use of the threshold expansion for problems of physical interest, such as the next-to-next-to-leading order heavy quark production cross sections close to threshold and matching calculations and power counting in non-relativistic effective theories.

Figures (4)

• Figure 1: One-loop vertex integral. Solid (wavy) lines denote massive (massless) propagators.
• Figure 2: Diagrammatic representation of the threshold expansion for $I_2$. Solid lines on the left-hand side denote massive propagators, wavy lines denote massless propagators. On the right-hand side a solid line with '$\pm'$ denotes (hard) 'on-shell' propagators of form $1/(k^2\pm q.k)$ for line momentum $k\pm q/2$, a wavy line stands for a hard massless line. The dashed line denotes a potential massive line and the dotted line a potential massless line. The zigzagged line is an ultrasoft massless line.
• Figure 3: The box graph. Solid lines massive propagators, wavy lines massless.
• Figure 4: Examples of two-loop three-point graphs. Solid (wavy) lines denote massive (massless) lines.