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Untangling the IBP Equations

Junhan W. Liu, Alexander Mitov

TL;DR

This work develops a systematic diagonalization framework for Integration-by-Parts (IBP) identities, introducing diagonal, matrix-diagonal, and triangular reformulations to expose recurrence structure and sector organization. The diagonal form yields higher-order recurrences whose order matches the number of master integrals, while the matrix-diagonal form maps integrals directly to masters; the triangular form provides strictly lower-triangular systems with smaller coefficients, enabling efficient, back-substitution-free solving. The authors present an algorithm to construct these reformulations, determine the integral sets, and compute the required coefficients using shifted IBP equations and finite-field linear algebra. Benchmark studies on two-loop topologies show the triangular approach competes with, and in some implementations surpasses, syzygy-based methods, especially when using fast coefficient evaluation with Fermat; the methods hold promise for analytic solutions and for handling multivariate recurrences, potentially benefiting Mellin-space formulations and dimensional-shift strategies. Overall, the work offers a versatile and scalable toolkit to solve IBP identities more efficiently and to illuminate the analytic structure of loop integrals.

Abstract

In this work, we present an algorithm for the diagonalization of the Integration-by-Parts (IBP) equations. Diagonalized IBP equations are indispensable for reducing loop integrals with high numerator powers to master integrals and for solving IBP identities in closed analytic form. A prime example is provided by multivariate Mellin representations of loop amplitudes and cross sections. The extension of these methods to other multivariate recurrence relations is also discussed. As a by-product of our diagonalization procedure, we show how the IBP equations can be cast into an efficient, fully triangular form that is well suited for computer implementation. Several complicated topologies have been computed.

Untangling the IBP Equations

TL;DR

This work develops a systematic diagonalization framework for Integration-by-Parts (IBP) identities, introducing diagonal, matrix-diagonal, and triangular reformulations to expose recurrence structure and sector organization. The diagonal form yields higher-order recurrences whose order matches the number of master integrals, while the matrix-diagonal form maps integrals directly to masters; the triangular form provides strictly lower-triangular systems with smaller coefficients, enabling efficient, back-substitution-free solving. The authors present an algorithm to construct these reformulations, determine the integral sets, and compute the required coefficients using shifted IBP equations and finite-field linear algebra. Benchmark studies on two-loop topologies show the triangular approach competes with, and in some implementations surpasses, syzygy-based methods, especially when using fast coefficient evaluation with Fermat; the methods hold promise for analytic solutions and for handling multivariate recurrences, potentially benefiting Mellin-space formulations and dimensional-shift strategies. Overall, the work offers a versatile and scalable toolkit to solve IBP identities more efficiently and to illuminate the analytic structure of loop integrals.

Abstract

In this work, we present an algorithm for the diagonalization of the Integration-by-Parts (IBP) equations. Diagonalized IBP equations are indispensable for reducing loop integrals with high numerator powers to master integrals and for solving IBP identities in closed analytic form. A prime example is provided by multivariate Mellin representations of loop amplitudes and cross sections. The extension of these methods to other multivariate recurrence relations is also discussed. As a by-product of our diagonalization procedure, we show how the IBP equations can be cast into an efficient, fully triangular form that is well suited for computer implementation. Several complicated topologies have been computed.

Paper Structure

This paper contains 16 sections, 58 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Diagonalizing the IBP identities
  3. Equations in diagonal form
  4. Sector structure and reduction of the order of diagonal recurrences
  5. Equations in matrix-diagonal form
  6. Motivation for equations in triangular form
  7. Equations in triangular form
  8. The algorithm
  9. Determining the set of integrals
  10. Determining the coefficients
  11. Implementation details
  12. Benchmark results for the triangular approach
  13. Conclusions
  14. Contiguous Relations
  15. Fibonacci recurence in matrix form
  16. ...and 1 more sections

Figures (5)

  • Figure 1: The one-loop box topology.
  • Figure 2: An illustration of the "evolution" of the diagonal (left) and matrix-diagonal (right) equations.
  • Figure 3: Flowchart of the rank-based decision process.
  • Figure 4: Two-loop topologies: off-shell di-boson (left), three jet planar $C_1$ (center) and three jet non-planar $B_1$ (right).
  • Figure 5: The sector structure for $_2F_1$ (left) and the one-loop-box topology (right); the shaded regions are the ones with two boundary conditions and the remaining are the one with one boundary condition (applies to integer-valued indices).