⚛️ high-energy theory

Graded transcendental functions: an application to four-point amplitudes with one off-shell leg

This paper introduces a systematic method for constructing minimal bases of graded transcendental functions to simplify scattering amplitudes by minimizing spurious features, which is then applied to evaluate three-loop $Hggg$ and $Hgq\bar{q}$ amplitudes and compute a three-point form factor in planar $\mathcal{N}=4$ sYM.

Original authors: Thomas Gehrmann, Johannes Henn, Petr Jakubčík, Jungwon Lim, Cesare Carlo Mella, Nikolaos Syrrakos, Lorenzo Tancredi, William J. Torres Bobadilla

Published 2026-10-19

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CC BY 4.0

Original authors: Thomas Gehrmann, Johannes Henn, Petr Jakubčík, Jungwon Lim, Cesare Carlo Mella, Nikolaos Syrrakos, Lorenzo Tancredi, William J. Torres Bobadilla

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 subatomic world, particles do not simply bounce off one another like billiard balls; they interact through a complex web of forces that can be described by mathematical structures known as scattering amplitudes. These amplitudes are the fundamental recipes that physicists use to predict the outcome of high-energy collisions, such as those occurring inside the Large Hadron Collider. To make these predictions with the precision required by modern experiments, scientists must calculate these interactions with extreme accuracy, often accounting for effects that happen at multiple levels of complexity simultaneously. This involves solving intricate integrals, which are mathematical tools used to sum up all the possible ways particles can exchange energy and momentum. As the calculations reach higher levels of complexity, the mathematical expressions describing them grow so large and unwieldy that they become nearly impossible to manage, filled with redundant terms that eventually cancel each other out but obscure the underlying physics.

A team of researchers has developed a new method to tame this complexity, specifically for calculations involving a Higgs boson interacting with three other particles. By organizing the mathematical functions used in these calculations into a carefully graded hierarchy, they were able to strip away unnecessary clutter and identify the essential building blocks of the solution. This approach allowed them to compute the three-loop corrections for the production of a Higgs boson alongside a jet of particles, a process crucial for testing the Standard Model of particle physics. Their work not only provides the first direct analytic calculation of a specific quantum field theory quantity in a highly symmetric version of the theory but also reveals that the mathematical functions required to describe these real-world particle collisions are far simpler than previously thought, dropping out entire classes of complicated terms that were expected to appear.

The core of this achievement lies in how the researchers handled the "alphabet" of mathematical functions. In these calculations, the solutions are built from a set of basic letters that can be combined in various sequences to form complex words, much like how letters form words in a language. For decades, scientists assumed that as calculations became more precise, these alphabets would grow to include increasingly strange and complicated letters, including those involving square roots of kinematic variables. However, the team discovered that for the specific process of a Higgs boson decaying into three particles, the vast majority of these complicated letters never actually make it into the final physical result. Instead, they found that the physical amplitude resides in a much smaller, more restricted space of functions.

To reach this conclusion, the researchers first mapped out every possible integral topology, or geometric arrangement of particle paths, that could contribute to the process. They then derived a set of canonical differential equations for these integrals, which describe how the values change as the energy and momentum of the particles vary. This step revealed the full alphabet of letters available at this level of complexity, including new types of letters that had never been seen before in three-loop calculations. These new letters included quadratic forms and square roots that complicated the mathematical landscape. The team then constructed a minimal basis of functions, a curated list of the most efficient mathematical tools needed to describe the system, and rotated this basis to separate the physically relevant parts from the spurious ones.

By applying this graded basis, they were able to show that many of the new, complicated letters appear only in intermediate steps of the calculation and are forced to vanish in the final physical answer. Specifically, they found that functions containing certain square-root letters and hyperbolic forms do not survive in the finite part of the amplitude, which is the part that remains after removing infinities. This means that the final description of the Higgs boson interaction is much cleaner than the raw mathematical machinery suggests. The researchers verified this by checking the results against numerical simulations and by comparing them to a related quantity in a highly symmetric theoretical model known as N=4 super Yang-Mills theory. In this symmetric model, they performed the first direct analytic calculation of a three-point form factor, confirming that it shares the same simple alphabet as the more complex Higgs interaction.

The implications of this work extend beyond just simplifying the math. The researchers identified that the new letters appearing in the calculation only show up in diagrams with eight or more internal lines, and even then, they are constrained by strict rules about how they can be ordered. They found that the new quadratic letters only ever appear as the fourth integration kernel in the sequence, a restriction that eliminates a huge number of potential terms. This discovery allows physicists to bypass the most computationally expensive parts of the calculation, known as integration-by-parts reductions, by focusing only on the minimal set of functions that actually matter. The team also demonstrated how to convert the remaining complicated integrals into a form that is explicitly real and single-valued in the physical region, making them suitable for direct numerical evaluation.

In the end, the study provides a complete set of master integrals for the leading-color contribution to the three-loop Higgs-plus-jet amplitude. It confirms that the analytic structure of this complex process is governed by a surprisingly simple set of rules, where the most exotic mathematical features are systematically filtered out. This work not only delivers the necessary ingredients for the next generation of precision predictions at the Large Hadron Collider but also offers a new blueprint for how to approach even more complex calculations in the future. By organizing the mathematical landscape into a graded hierarchy, the researchers have shown that nature's most intricate interactions can often be described with a elegance that is hidden beneath layers of computational noise.

Technical Summary: Graded Transcendental Functions: An Application to Four-Point Amplitudes with One Off-Shell Leg

Problem Statement
The computation of multi-loop scattering amplitudes in quantum field theory relies heavily on the evaluation of Feynman integrals, typically reduced to a set of master integrals (MIs) via integration-by-parts (IBP) identities. These MIs are often solved using differential equations (DEs) expanded in the dimensional regulator $\epsilon$ . A persistent challenge is the "swell" of expression size at intermediate stages, caused by the inclusion of redundant transcendental functions, spurious branch cuts, and terms that eventually cancel out in the physical amplitude. Furthermore, the disconnect between the definition of transcendental functions (often derived from general integral topologies) and the specific constraints of a physical amplitude (such as the absence of certain letters or specific weight structures) leads to inefficient representations. This is particularly acute for three-loop four-point amplitudes with one off-shell leg, such as the Higgs-plus-jet ($H+jet$) production in the heavy-top limit and the form factor of the operator $\text{Tr}(\phi^2)$ in planar $\mathcal{N}=4$ super-Yang-Mills (sYM) theory.

Methodology
The authors develop a systematic framework for constructing a minimal basis of graded transcendental functions tailored to a specific scattering amplitude. The methodology proceeds in three main stages:

Construction of a Minimal Subspace: Starting from formal solutions for all master integral topologies expressed as Chen iterated integrals, the authors organize the appearing functions by their analytic properties (e.g., symbol alphabet, letter adjacency). By analyzing the linear dependencies among the coefficients of these iterated integrals across the master integrals, they identify a minimal subspace $M_w$ that spans the solutions. This is achieved by row-reducing a matrix $S$ formed by the coordinates of the formal solutions.
Grading and Rotation: The minimal basis is further rotated to separate functions into a physically relevant subspace $A_{\parallel, w}$ and an orthogonal complement $A_{\perp, w}$ . Functions in $A_{\perp, w}$ possess "spurious" features (e.g., specific alphabet letters or adjacency violations) that must vanish in the physical amplitude. By ensuring these features appear only in the orthogonal complement, the method avoids complex cancellations and enforces physical constraints (such as the absence of new letters in pole structures) directly at the basis level.
Boundary Condition Fixing without Full Functional Representation: To fix the boundary constants for the canonical DEs without requiring a full analytic solution in the multi-dimensional kinematic plane, the authors employ a method of kinematic slicing. They identify one-dimensional paths (slices) along the singular surfaces (thresholds and pseudo-thresholds) where the alphabet simplifies to linear forms, allowing solutions in terms of multiple polylogarithms (MPLs). By matching these partial solutions at the intersections of the slices and transporting the information to a common base point, they fix all boundary constants analytically.

Key Contributions and Results

Canonical Bases for Three-Loop Integrals: The authors derive canonical differential equations for all integral families relevant to the leading-colour three-loop $H+jet$ amplitudes (including non-planar topologies) and the $\text{Tr}(\phi^2)$ form factor in planar $\mathcal{N}=4$ sYM. This includes the identification of three new categories of alphabet letters appearing at three loops: parabolic and hyperbolic quadratic letters, and algebraic square-root letters.
Direct Analytic Computation of $\mathcal{N}=4$ Form Factor: The paper presents the first direct analytic computation of the three-loop planar part of the $\text{Tr}(\phi^2)$ form factor in $\mathcal{N}=4$ sYM. The result confirms the maximal transcendentality principle, showing that the finite remainder contains only the six linear letters known from lower loops, despite the master integrals containing new letters. The result is expressed in terms of a single "trident function" (a specific linear combination of the minimal basis) and agrees with previous bootstrapped results and numerical evaluations.
Analytic Structure of Leading-Colour $H+jet$ Amplitudes: Applying the graded basis to the QCD $H+jet$ amplitudes, the authors determine the analytic structure of the finite remainder without performing a full, expensive IBP reduction of the entire amplitude.
- They identify that hyperbolic and square-root letters drop out of the finite remainder entirely.
- They find that parabolic quadratic letters appear only in the $H \to q\bar{q}g$ channel and in lower-weight functions in the $H \to ggg$ channel.
- The new letters appear only in integrals with 8 or more internal propagators. By performing numerical IBP reductions that set integrals with fewer propagators to zero, they isolate the contributions of the new letters.
- The resulting amplitude is expressed with remarkably simple rational coefficients, and the authors verify that the adjacency conditions derived from the $G_2$ cluster algebra are satisfied.
Real MPL Representation: The authors demonstrate how to convert the iterated integrals involving quadratic letters into explicitly real multiple polylogarithms in the Euclidean decay region by utilizing a rational change of variables that linearizes the indices.

Significance and Claims
The paper claims that the proposed method of graded transcendental functions significantly simplifies the computation of high-loop amplitudes by:

Eliminating Redundancy: It constructs a basis where spurious features are absent by construction, avoiding the need for post-hoc cancellations.
Enabling Partial Reductions: It allows for the determination of the analytic structure of an amplitude (specifically the part involving new letters) by restricting the IBP reduction to only the relevant topologies, bypassing the need for a full reduction of the entire amplitude.
Constraining Finite Parts: It leverages the known pole structure and lower-loop properties to constrain the finite part of the amplitude, even before the full integrand is known.
Verification of Conjectures: The work provides direct analytic confirmation of the maximal transcendentality principle for the $\text{Tr}(\phi^2)$ form factor and validates new adjacency conditions for the alphabet of three-loop integrals.

The authors emphasize that while the method was applied to $H+jet$ and $\mathcal{N}=4$ form factors, the framework is general and applicable to any scattering amplitude where a formal solution for the master integrals is available. The results pave the way for more efficient numerical evaluations and future bootstrapping efforts by providing a compact, physically motivated function space.

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