⚛️ phenomenology

Holonomic techniques for massive 3-loop form factors: the gluonic case

This paper outlines the core strategy and challenges of computing massive three-loop gluonic form factors using holonomic techniques, delivering exact analytic representations in terms of MZVs and higher number space constants around $s=0$ and $s=\pm\infty$ to serve as a landmark benchmark for precision collider phenomenology and symbolic computation.

Original authors: J. Blümlein, A. De Freitas, P. Marquard, J. Obrovsky, C. Schneider

Published 2026-09-22

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: J. Blümlein, A. De Freitas, P. Marquard, J. Obrovsky, C. Schneider

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

Deep within the subatomic world, particles do not exist in isolation; they are constantly interacting, exchanging invisible messengers that bind the universe together. In the realm of the strong nuclear force, which holds the heart of atoms together, these interactions are described by a theory called Quantum Chromodynamics. To understand how heavy particles behave when struck by high-energy beams in particle colliders, physicists must calculate the precise "form factors" of these interactions. Think of a form factor as a detailed map of how a particle responds to a push or a pull; it tells scientists exactly how the internal structure of a heavy quark pair changes when a virtual boson, a carrier of force, passes through them. These maps are essential for interpreting the data from massive machines like the Large Hadron Collider, where the difference between a new discovery and a known background often lies in the tiniest decimal place. As experimental measurements become increasingly precise, the theoretical maps must be drawn with equal, if not greater, accuracy, pushing scientists to calculate these interactions with a level of detail that was previously unimaginable.

A team of researchers has now completed a monumental calculation of these form factors for a specific, highly complex scenario involving the exchange of gluons, the particles that carry the strong force. While previous work had mapped out similar interactions for quarks, the gluonic case presented a significantly more difficult challenge due to the sheer number of ways these particles can interact and the mathematical structures involved. The team successfully computed these contributions for three distinct types of currents—vector, axial-vector, scalar, and pseudoscalar—which represent different ways the particles can interact. Their work provides a complete, exact mathematical description of these interactions in two critical energy limits: when the energy is very low, near the point where the particles are created, and when the energy is extremely high. This achievement is not merely a numerical approximation but a full analytic representation, meaning the results are expressed as precise formulas rather than just a list of numbers, allowing physicists to predict outcomes with unprecedented reliability across the entire range of possible energies.

The path to this result required navigating a landscape of mathematical complexity that would overwhelm standard computing methods. The researchers began by breaking down the problem into thousands of smaller, manageable pieces known as Feynman diagrams, which represent the possible ways particles can scatter. Using advanced software, they reduced these thousands of diagrams into a set of 417 fundamental building blocks, known as master integrals. These blocks were then described by a system of differential equations, which act like a set of rules governing how the values change as the energy of the collision shifts. The team employed a technique called the large-moment method, which involves calculating the first several thousand terms of a series expansion for these values. For the most difficult parts of the calculation, they generated over 40,000 individual coefficients, some of which were so large that writing them out in full would require dozens of pages of text.

With these massive lists of numbers in hand, the researchers used powerful computer algorithms to detect hidden patterns. They searched for recurrence relations, which are rules that allow one number in the sequence to be predicted from the ones before it, and differential equations that describe the overall shape of the solution. This process was akin to finding a single, coherent melody hidden within a chaotic noise of millions of notes. For the most intricate part of the calculation, involving a specific color factor related to the strong force, the team had to guess a recurrence relation that spanned 92 steps, with coefficients so complex that a single constant term required over 8,000 digits to write down. Once these patterns were identified, the team solved the equations to express the results in terms of known mathematical constants and a few new, specialized numbers that arise from the geometry of the problem.

The final output is a complete analytic series that describes the behavior of these gluonic interactions from the lowest energies to the highest. Around the point where the particles are created, the results are expressed using multiple zeta values, a family of mathematical constants that appear frequently in quantum physics. At the high-energy limit, the description includes these same constants along with three additional, more exotic numbers that belong to a broader class of mathematical structures. The team also provided a way to smoothly connect these two extremes, ensuring that the description remains accurate and consistent across all possible energy levels. This work was achieved through massive parallel processing on supercomputers, utilizing thousands of processor cores to handle the immense computational load. The result stands as a rigorous benchmark for the field, demonstrating that even the most complex quantum interactions can be tamed with the right combination of advanced algorithms and symbolic computation, providing a solid foundation for future precision tests of the Standard Model.

Technical Summary: Holonomic Techniques for Massive 3-Loop Form Factors (The Gluonic Case)

Problem Statement
The paper addresses the computation of massive three-loop form factors for vector, axial-vector, scalar, and pseudoscalar currents within Quantum Chromodynamics (QCD). These quantities are essential for precision collider phenomenology, particularly for heavy-quark calculations involving virtual Higgs decays and heavy-flavor production. While the quarkonic case has been partially addressed in previous literature, the gluonic case—where gluons are exchanged exclusively between heavy quarks—presents significantly higher complexity. This complexity arises from the vast number of contributing Feynman diagrams and the intricate structures of the underlying Feynman integrals, which extend beyond standard harmonic polylogarithms. The primary challenge is to obtain complete analytic results for these contributions, specifically in the high-energy limit, which had not been fully achieved prior to this work.

Methodology: The Large-Moment Holonomic Approach
The authors employ a strategy based on the "large-moment method," enhanced by advanced analytic continuation and computer algebra. The calculation proceeds through five distinct steps:

Master Integral Generation: Using Integration-by-Parts (IBP) reduction techniques (specifically the software package Crusher), the form factors are reduced to a linear combination of 417 master integrals (MIs). These MIs satisfy a coupled system of first-order linear differential equations. The authors compute power series expansions (in terms of the dimensionless variable $s = q^2/m^2$ or $x$ ) for these MIs up to high orders in the dimensional regularization parameter $\varepsilon$ .
Form Factor Series Construction: The truncated power series of the master integrals are substituted back into the linear combination to generate the series coefficients for the form factors. These coefficients are linear combinations of specific constants (including Riemann zeta values $\zeta_n$ , polylogarithms, and color factors).
Guessing Recurrences and Differential Equations: For the finite lists of rational coefficients generated, the authors use automated guessing algorithms (via the ore_algebra package in Sage) to conjecture linear recurrence relations and linear differential equations satisfied by the series.
Solving Factorizable Systems: For components where the derived equations factorize into first-order operators, the recurrences are solved in closed form using iterative sums (harmonic sums), and the series are converted into iterative integrals.
Analytic Continuation: For non-factorizable contributions, the authors employ Frobenius' method and extensions to construct high-order local series expansions around regular and singular points (including $s \to \pm \infty$ , $s=4$ , and $s=16$ ). These overlapping expansions are matched to achieve analytic continuation across the full kinematic range $s \in (-\infty, \infty)$ .

Key Technical Challenges and Scale
The paper highlights the extreme computational scale required for the gluonic case, which necessitated new technological advancements:

Coefficient Generation: To guess the necessary recurrences, the authors required tens of thousands of rational coefficients. For the most complex constant contribution (the unit constant), 26,000 coefficients were needed. For specific color factors like $C_F^3 \zeta_5$ , over 25,000 values were required.
Recurrence Size: The guessed linear recurrence for the axial-vector form factor projection $g_1$ with color factor $C_F^3$ involves 92 terms (up to $F(n+91)$ ) with polynomial coefficients of degree up to 4,078. The printed form of this recurrence would occupy approximately 650,000 A4 pages.
Differential Equations: Guessing the linear differential equations required even more coefficients than the recurrences (e.g., ~37,700 values for a 102nd-order equation).
Data Magnitude: The rational numbers involved are massive; the 26,000th coefficient alone requires 18 printed pages (over 160,000 decimal digits for the numerator and 150,000 for the denominator).
Computational Resources: These calculations were performed using massive parallel processing on the MACH-2 supercomputer (1,728 cores, 20 TB shared memory) at JKU Linz and DESY servers.

Results
The authors achieved the following results:

Exact Analytic Representation: They provided an exact analytic representation of the form factors around $s=0$ in terms of Multiple Zeta Values (MZVs).
High-Energy Expansion: For the first time, a complete analytic series expansion around $s = \pm \infty$ $s = \pm \infty$ was derived. This expansion depends on MZVs and three specific constants from higher number spaces:
- $\tilde{\kappa}_1$ and $\tilde{\kappa}_2$ : Cyclotomic constants.
- $\tilde{\kappa}_3$ : A constant belonging to the class of quadratic form induced polylogarithms.
Analytic Continuation: The work enables reliable analytic continuations across the entire physical region $s \in (-\infty, \infty)$ , including critical points like the threshold ( $s=4$ ) and pseudo-threshold ( $s=16$ ).
High-Precision Data: Expansion coefficients at $s = \pm \infty$ were determined with an accuracy up to 4,000 digits, allowing for the identification of the constants via the PSLQ algorithm.

Significance and Claims
The paper positions this work as a landmark benchmark for modern symbolic computation in particle physics. The authors claim that the gluonic case is significantly more complex than the quarkonic case, requiring sophisticated computer algebra techniques and automated guessing algorithms that push the boundaries of current capabilities. The successful derivation of these results demonstrates the power of combining:

Cutting-edge differential equation methods.
Large-scale PSLQ integer-relation searches.
State-of-the-art symbolic summation methods.

The results serve as a vital input for precision QCD phenomenology and provide a fully analytic framework for massive three-loop form factors in the gluonic sector, a feat previously unattained. The work validates the large-moment method as a robust tool for solving record-sized recurrence and differential equation systems in quantum field theory.

Technical Summary: Holonomic Techniques for Massive 3-Loop Form Factors (The Gluonic Case)

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