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IBIS: Inverse BInomial sum Solver

The paper introduces IBIS, an efficient FORM program that solves specific classes of inverse binomial sums arising in higher-loop Feynman parameter integrals by deriving new recursion relations to express results in terms of analytic S-sums, significantly outperforming existing general-purpose tools.

Original authors: Paul A. J. W. van Hoegaerden, Coenraad B. Marinissen, Wouter J. Waalewijn

Published 2026-08-20

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CC BY 4.0

Original authors: Paul A. J. W. van Hoegaerden, Coenraad B. Marinissen, Wouter J. Waalewijn

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the world of particle physics, scientists act as cosmic detectives, trying to understand the fundamental building blocks of the universe by smashing particles together at incredible speeds. To make sense of the debris from these collisions, they rely on a set of rules called quantum field theory, which predicts how particles interact. However, these predictions are rarely simple. To get the level of precision required by modern experiments, physicists must calculate effects that involve particles appearing and disappearing in complex loops. These calculations are notoriously difficult, often producing mathematical expressions so vast and tangled that even the most powerful computers struggle to untangle them. The goal is to turn these messy, infinite possibilities into clean, precise numbers that can be compared with real-world data from machines like the Large Hadron Collider.

A specific type of mathematical knot often appears in these high-level calculations. When physicists try to simplify the equations describing these particle loops, they frequently encounter sums that involve a peculiar mix of factorials and alternating signs. These are known as inverse binomial sums. While general mathematical tools exist to handle many types of sums, these specific knots have proven stubborn. They are like a specific, intricate pattern in a vast tapestry that standard scissors cannot cut without unraveling the whole design. For years, researchers had to rely on slow, general-purpose methods to solve them, or they had to simplify their physical models to avoid these difficult terms entirely. This limitation meant that some of the most precise theoretical predictions were either too slow to compute or simply out of reach.

A team of researchers has now developed a specialized tool designed specifically to cut through these knots. They created a computer program called IBIS, which stands for Inverse BInomial sum Solver. Instead of trying to solve every possible mathematical puzzle with one giant, slow machine, IBIS is built like a master key for this specific type of lock. The researchers discovered a set of clever rules, or recursions, that allow the computer to break down these complicated sums into smaller, simpler pieces. By doing this step-by-step, the program can rewrite the difficult sums into a standard format that is much easier to handle. The result is a method that is dramatically faster than previous approaches. In tests, the program solved complex sums that would take other software hours or days to process in less than a second.

The power of this new tool lies in its efficiency and its ability to keep the answers in a useful form. When the program solves a sum, it does not just give a single number; it provides an answer that remains flexible, keeping the variable that represents the number of steps in the calculation. This allows physicists to use the result in further calculations without losing precision. The researchers tested their method on sums that involve up to six layers of complexity, a level of difficulty that is common in the most advanced theoretical work. They found that their program could handle these cases effortlessly, whereas general-purpose tools often stalled or took an unreasonably long time. This speed is crucial because it opens the door to calculations that were previously too slow to be practical, potentially allowing physicists to refine their predictions for particle collisions with unprecedented accuracy.

The development of IBIS represents a significant step forward in the toolkit available to theoretical physicists. It does not change the laws of physics, but it changes how quickly and accurately scientists can apply those laws to real-world problems. By automating the solution of these specific mathematical hurdles, the researchers have removed a bottleneck that was slowing down progress in the field. This means that in the near future, theorists will be able to compute higher-order corrections to particle interactions more reliably, providing sharper predictions for experiments. As the field moves toward even more precise measurements, tools like IBIS will be essential for ensuring that the theoretical side of the equation keeps pace with the experimental data, helping to uncover the deeper secrets of the universe.

Technical Summary of IBIS: Inverse BInomial sum Solver

Problem Statement
In high-order perturbative calculations within Quantum Chromodynamics (QCD), particularly for multi-loop Feynman integrals, the evaluation of complex expressions often relies on Mellin-Barnes (MB) representations. These representations convert Feynman parameter integrals into contour integrals involving ratios of Gamma functions. Upon closing the contours, these integrals yield infinite sums over residues. A specific and challenging class of sums arising in this context involves "inverse binomial sums," characterized by terms of the form $1/\binom{n}{j}$ multiplied by harmonic sums or other nested structures. While general-purpose symbolic summation tools exist (such as Sigma and EvaluateMultiSums), they are often computationally inefficient for the specific high-weight inverse binomial sums encountered in modern collider physics calculations. The authors identify a need for a specialized, high-speed solver capable of expressing these sums in terms of analytic S-sums, maintaining dependence on the symbolic upper limit of the summation.

Methodology
The paper introduces IBIS ("Inverse BInomial sum Solver"), a program implemented in the Form computer algebra system. The core methodology relies on deriving and implementing a set of recursive relations to reduce the complexity of inverse binomial sums. The reduction process transforms sums containing inverse binomial factors and harmonic sums into a basis of S-sums (a generalization of harmonic sums with analytic dependence on the upper limit).

The recursive reduction is achieved through three primary technical steps:

Telescoping: The authors set up telescoping series with respect to the upper limit $n$ of the sum. By expressing the difference $G(k, i) - G(k, i-1)$ (where $G$ represents the inverse binomial sum), they derive relations that express the original sum in terms of sums with lower complexity.
Synchronization: A significant technical hurdle is that terms within the summand often have arguments that are not aligned (e.g., the binomial coefficient depends on $i-1$ while the harmonic sum depends on $i-j$ ). The authors develop synchronization techniques to rewrite these terms so that all components share a consistent argument structure, allowing them to be identified as standard inverse binomial sums or S-sums. This involves handling shifted denominators and aligning indices through algebraic manipulation.
Reduction of Complexity: The recursion is designed to systematically reduce the complexity of the sums. Complexity is defined by the number of harmonic sums (indices) and the initial exponent of the denominator. The algorithm reduces a sum with $k$ harmonic sums to sums with $k-1$ or $k-2$ harmonic sums, or reduces the initial exponent, ensuring the recursion terminates.

The implementation handles a specific basis set of inverse binomial sums, including those with fixed or alternating signs and containing zero, one, or two harmonic sums. The program utilizes existing Form packages like Summer and XSummer for the manipulation of S-sums and Z-sums, particularly for reducing products of S-sums and converting between different sum types.

Key Contributions

Development of IBIS: The authors provide a Form package specifically optimized for solving inverse binomial sums up to weight 6.
New Recursion Relations: The paper derives several new recursion relations based on telescoping series and synchronization techniques. These relations allow for the analytic expression of inverse binomial sums in terms of S-sums.
Handling of Shifted Denominators: The authors present a method to iteratively reduce shifted denominators (e.g., $(j+c)^k$ ) to the standard form $(j)^k$ , enabling the solver to handle a broader range of inputs arising from MB representations.
Validation: The correctness of the recursions is validated through three methods: checking the conservation of extended weight and extended product weight across recursion steps, and comparing numerical evaluations of the derived S-sums against direct numerical summation of the original inverse binomial sums for a representative set of cases (including 124 distinct cases for sums with two harmonic sums).

Results

Performance: Benchmarking demonstrates that IBIS is orders of magnitude faster than general-purpose tools like Sigma and EvaluateMultiSums for the targeted class of sums. Sums up to weight 6 can be computed in less than a second. The speed advantage is attributed to the pre-derived, hardcoded recursion relations in Form, whereas general tools must derive recursions on the fly.
Scalability: While execution time scales exponentially with the extended weight due to the nature of the fraction splitting in the recursions, the specialized nature of IBIS keeps absolute runtimes low for the weights relevant to current high-order calculations.
Application Examples: The paper demonstrates the utility of IBIS by solving specific nested sums derived from a three-loop two-point Feynman integral (a ladder diagram). These examples include double and triple sums involving inverse binomial factors and harmonic sums, which are successfully reduced to S-sums and subsequently evaluated to known constants (zeta values and polylogarithms).

Significance and Scope
The authors position IBIS as a specialized tool that fills a gap in the analytical toolkit for high-order perturbative calculations. By providing a fast method to resolve inverse binomial sums, the tool aims to broaden the scope of feasible multi-loop calculations in collider physics.

The paper is modest regarding its scope, explicitly noting that IBIS cannot yet handle sums that combine binomial coefficients with inverse binomial coefficients (as seen in one of the examples in the introduction, eq. 1.5), which still require general-purpose solvers like Sigma. However, the authors suggest that the approach taken here could be adapted to solve similar classes of binomial sums, potentially yielding similar performance gains. The work is presented as a starting point for studying these sums and a contribution to the efficiency of higher-loop calculations, rather than a complete solution for all possible summation problems in quantum field theory.

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