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The Two Loop Crossed Ladder Vertex Diagram with Two Massive Exchanges

U. Aglietti, R. Bonciani, L. Grassi, E. Remiddi

TL;DR

The paper tackles the two-loop crossed ladder vertex diagram with two equal-mass exchanges by reducing it to three master integrals and solving a coupled differential system in the evolution variable $x=-s/m^2$ via power-series expansions around the singular points $0$, $8$, $-1$, and $\infty$. It builds full high-precision numerical representations of the MIs by matching local series at multiple centers, including improvements with Bernoulli variables, and confirms consistency with known large-momentum behavior and with the equal-mass sunrise in two dimensions. A key finding is that the topology requires elliptic integrals beyond harmonic polylogarithms, highlighting elliptic structures in a non-planar massive two-loop diagram. The results provide a fast, highly accurate computational framework for the master integrals $F_1$, $F_2$, and $F_3$ and establish important connections to established two-loop results, enabling precise electroweak form-factor calculations at two loops.

Abstract

We compute the (three) master integrals for the crossed ladder diagram with two exchanged quanta of equal mass. The differential equations obeyed by the master integrals are used to generate power series expansions centered around all the singular (plus some regular) points, which are then matched numerically with high accuracy. The expansions allow a fast and precise numerical calculation of the three master integrals (better than 15 digits with less than 30 terms in the whole real axis). A conspicuous relation with the equal-mass sunrise in two dimensions is found. Comparison with a previous large momentum expansion is made finding complete agreement.

The Two Loop Crossed Ladder Vertex Diagram with Two Massive Exchanges

TL;DR

The paper tackles the two-loop crossed ladder vertex diagram with two equal-mass exchanges by reducing it to three master integrals and solving a coupled differential system in the evolution variable via power-series expansions around the singular points , , , and . It builds full high-precision numerical representations of the MIs by matching local series at multiple centers, including improvements with Bernoulli variables, and confirms consistency with known large-momentum behavior and with the equal-mass sunrise in two dimensions. A key finding is that the topology requires elliptic integrals beyond harmonic polylogarithms, highlighting elliptic structures in a non-planar massive two-loop diagram. The results provide a fast, highly accurate computational framework for the master integrals , , and and establish important connections to established two-loop results, enabling precise electroweak form-factor calculations at two loops.

Abstract

We compute the (three) master integrals for the crossed ladder diagram with two exchanged quanta of equal mass. The differential equations obeyed by the master integrals are used to generate power series expansions centered around all the singular (plus some regular) points, which are then matched numerically with high accuracy. The expansions allow a fast and precise numerical calculation of the three master integrals (better than 15 digits with less than 30 terms in the whole real axis). A conspicuous relation with the equal-mass sunrise in two dimensions is found. Comparison with a previous large momentum expansion is made finding complete agreement.

Paper Structure

This paper contains 32 sections, 195 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Power Series Solution of the Differential Equations on the MIs
  3. Threshold Structure
  4. Reduction to Master Integrals
  5. The System of Differential Equations
  6. Master Integrals Close to Zero External Momentum
  7. Second-Order Differential Equation for $F_1(x)$
  8. Power-Series Solution of the Second-Order Differential Equation for $F_1(x)$
  9. Expansion Around $x \, = \, 0$ --- Small Momentum Expansion
  10. Homogeneous Equation
  11. Complete Equation
  12. Improved Expansion --- Bernoulli Variable
  13. Expansion Around $x \, = \, 8$
  14. Homogeneous Equation
  15. Complete Equation
  16. ...and 17 more sections

Figures (1)

  • Figure 1: Feynman diagram for the annihilation of a pair of massless fermions with the exchange of two massive quanta with equal mass $m$. The thin lines represent the massless fermions, while the thick lines represent the massive quanta. The outgoing dashed line represents the probe (for instance a $Z'$).