FIRE 7: Automatic Reduction with Modular Approach

TL;DR

FIRE7 addresses the computational bottlenecks of integration-by-parts reduction by introducing a modular arithmetic workflow that performs reductions at finite-field primes and reconstructs analytic coefficients. The approach is supported by single- and multi-probe binaries, MPI parallelization, and an automated reconstruction pipeline, complemented by tools to manipulate IBP tables and combine linear combinations. Key contributions include a robust presolve step that streamlines IBP identities, flexible orderings to optimize the Laporta solve, and the ability to reduce combinations directly, with significant speedups (up to ~50x) demonstrated in modular benchmarks and analytic benchmarks for multi-loop diagrams. The practical impact is a more scalable, automation-friendly IBP reduction framework suitable for large-scale perturbative QFT computations, with broad utility from signer workflows to high-performance computing environments.

Abstract

FIRE7 is a major update to the FIRE program for integration-by-parts (IBP) reduction of Feynman integrals. A large part of improvements is related to the automatic reduction and reconstruction with the modular arithmetic approach, while the performance of the classical rational polynomial approach is also significantly increased. An improved presolve algorithm performs Gaussian elimination to simplify IBP identities before substituting numerical indices as in the Laporta algorithm. Various new command line tools are included to facilitate tasks such as applying an IBP reduction table to reduce a loop integrand as a linear combination of individual integrals.

Figures (4)

• Figure 1: The nonplanar double box diagram with massive legs. The legs $p_1$, $p_2$ and $p_3$ share the same mass, while $p_4$ has a different mass. The list of seven propagator denominators and two ISPs are $(k_1-k_2-p_4)^2$, $(k_1+p_1+p_2)^2$, $(k_1-p_4)^2$, $(k_1-k_2)^2$, $k_2^2$, $(k_2+p_1)^2$, $(k_2+p_1+p_2)^2$, $(k_1+p_1)^2$, and $(k_2-p_4)^2$.
• Figure 2: The nonplanar double pentagon diagram. The list of eight propagator denominators and three ISPs are $l_1^2$, $(l_1-p_5)^2$, $(l_1-p_5-p_1)^2$, $(l_2-p_4-p_2)^2$, $(l_2-p_4)^2$, $(l_2)^2$, $(l_1+l_2)^2$, $(l_1+l_2+p_3)^2$, $(l_2+p_5)^2$, $(l_1+p_4)^2$, and $(l_1+p_4+p_2)^2$.
• Figure 3: The soft-expanded 4-loop quadruple box diagram. The massless vertical lines in the middle are considered as soft. The expansion linearizes the massive propagators shown as horizontal lines. The massive momenta running through the top horizontal line and bottom horizontal line are approximately $m_1 u_1$ and $m_2 u_2$, respectively, where $u_1$ and $u_2$ are normalized velocities. The thirteen propagators are nine ISPs are $-2 k_1 \cdot u_2$, $-2(k_1+k_2) \cdot u_2$, $-2(k_1+k_2+k_3) \cdot u_2$, $-2(k_1+k_2+k_3+k_4) \cdot u_2$, $2 k_1 \cdot u_1$, $2(k_1+k_2) \cdot u_1$, $2(k_1+k_2+k_3) \cdot u_1$, $2(k_1+k_2+k_3+k_4) \cdot u_1$, $k_1^2$, $k_2^2$, $k_3^2$, $k_4^2$, $(q-k_1-k_2-k_3-k_4)^2$, $k_1 \cdot q$, $k_2 \cdot q$, $k_3 \cdot q$, $k_1 \cdot k_2$, $k_1 \cdot k_3$, $k_1 \cdot k_4$, $k_2 \cdot k_3$, $k_2 \cdot k_4$, and $k_3 \cdot k_4$.
• Figure 4: The three-loop forward-scattering diagram with massive internal lines, with $q_1^2=q_2^2=0$, $(q_1+q_2)^2 = s$. The list of 10 propagators are $l_3^2 - m^2$, $(l_2+l_3)^2 - m^2$, $l_1^2 - m^2$, $(l_3-q_2)^2 - m^2$, $(l_2+l_3+q_1)^2 - m^2$, $(l_1+q_2)^2-m^2$, $(l_1+l_2+q_1+q_2)^2-m^2$, $(l_1+l_2+q_2)^2 - m^2$, $l_2^2$, and $(l_1-l_3+q_2)^2$. The two ISPs are $(l_1+l_2+l_3)^2$ and $(l_1-q_1)^2$.