⚛️ phenomenology

Two Feynman integral families for vector boson fusion at NNLO QCD

This paper constructs canonical master integrals and $\mathrm{d}\log$ forms for planar and non-planar two-loop Feynman integral families relevant to vector boson fusion Higgs production at NNLO QCD, providing strategies to handle nested square roots and expressing the results in an algebraically independent function basis via numerical integration of differential equations.

Original authors: Linus Götzfried, Andreas von Manteuffel

Published 2026-09-30

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Linus Götzfried, Andreas von Manteuffel

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

High-energy physics seeks to understand the fundamental building blocks of the universe by smashing particles together at incredible speeds. When these collisions occur, they create a shower of new particles, some of which are familiar, like the Higgs boson, and others that are fleeting and elusive. To predict exactly what happens in these collisions, scientists rely on a mathematical framework called quantum chromodynamics, which describes how particles interact through the strong nuclear force. However, calculating these interactions is notoriously difficult. The equations are so complex that scientists often have to break them down into smaller, manageable pieces, solving for specific scenarios and then stitching the answers together. One of the most important ways the Higgs boson is created in particle accelerators is through a process called vector boson fusion, where two particles collide and exchange force-carrying particles to produce the Higgs. While scientists have a good grasp of the simplest ways this happens, the more complicated scenarios, involving extra layers of interaction, have remained out of reach. Understanding these complex details is crucial because future experiments will look for tiny deviations from standard predictions, which could hint at entirely new laws of physics.

A team of researchers at the University of Regensburg has taken a significant step forward by solving a long-standing mathematical puzzle required to describe these complex interactions. They focused on a specific type of calculation involving two loops of interaction, a level of detail known as next-to-next-to-leading order, which is necessary for the precision required by modern experiments. The challenge they faced was not just the sheer number of equations, but the nature of the shapes involved in the calculations. In the world of particle physics, these shapes are represented by diagrams that look like boxes or networks of lines. The researchers had to analyze two specific, highly complicated shapes: one that looks like a five-sided box with an extra line, and another that is a six-sided box twisted in a way that makes it impossible to draw flat on a piece of paper without lines crossing. These shapes represent the paths particles can take, and calculating the probability of these paths requires solving intricate integrals, which are essentially sums of infinite possibilities.

The difficulty in this work arose from the fact that the particles involved have mass, and the geometry of the collision creates a situation with seven independent variables that change the outcome. This complexity introduced a specific mathematical obstacle known as nested square roots. In simpler terms, the equations required taking the square root of a number that itself contained a square root. This structure is notoriously difficult to handle because it breaks many of the standard shortcuts scientists use to solve these problems. In previous attempts to solve similar problems, researchers had found that these nested roots made it impossible to write the equations in a clean, simplified form. However, the Regensburg team discovered that for these specific shapes, it was indeed possible to reorganize the math so that the equations became manageable. They constructed a new set of building blocks, or a basis, that allowed them to express the entire problem in a streamlined way, turning a chaotic mess of terms into a structured system that could be solved step-by-step.

To achieve this, the researchers had to develop new strategies for dealing with the nested square roots. They treated these difficult mathematical objects not as dead ends, but as features that could be understood by looking at how they behave when the variables change. They found that these roots followed specific patterns, similar to how a mirror reflects an image, and used these patterns to simplify the equations. By doing so, they were able to derive a set of rules, called differential equations, that describe how the values of these integrals change as the energy and angles of the collision change. These rules were written in a special format that made them much easier to solve numerically. The team then used powerful computers to integrate these equations, effectively tracing the path of the solution from a known starting point to the complex scenarios needed for the experiment.

The result of this work is a complete set of solutions for the two most difficult shapes involved in this type of particle collision. The researchers expressed their answers in terms of a common language of functions that are independent of each other, ensuring that the results are consistent no matter how the particles are arranged. They provided these solutions in a format that other scientists can use immediately to refine their predictions for the Large Hadron Collider. While this work solves the problem for two specific shapes, the team acknowledges that there are two other similar shapes that are even more complicated, likely involving even deeper layers of mathematical complexity. These remaining pieces are left for future work. Nevertheless, by cracking the code for these two shapes and providing a method to handle the nested square roots, the researchers have cleared a major hurdle. Their findings allow for a more precise understanding of how the Higgs boson is produced, bringing scientists closer to detecting the subtle signs of new physics hidden within the data.

Technical Summary: Two Feynman Integral Families for Vector Boson Fusion at NNLO QCD

Problem Statement
Vector boson fusion (VBF) is a dominant Higgs-boson production mode at the Large Hadron Collider (LHC). While standard kinematic cuts allow for a clean separation from other processes (like Higgs-Strahlung) dominated by factorisable contributions, future measurements with relaxed cuts require precise calculations of non-factorisable quantum chromodynamics (QCD) corrections. These corrections first appear at next-to-next-to-leading order (NNLO) and involve complex five-point kinematics with full color exchange between incoming partons. A major obstacle to exact NNLO predictions is the evaluation of the complicated two-loop Feynman integrals arising in these non-factorisable topologies. Specifically, this work addresses the planar penta-box and non-planar hexa-box families, which depend on seven independent kinematic variables (six Mandelstam invariants and one internal mass) and involve nested square roots, complicating the derivation of canonical differential equations.

Methodology
The authors employ the method of canonical differential equations to compute the $\epsilon$ -expansion of the master integrals. The workflow involves several key technical steps:

Integral Reduction and Basis Construction: Using integration-by-parts (IBP) reduction via Laporta's algorithm (implemented in Reduze 2 and Finred), the authors reduce the families to a set of master integrals (111 for the penta-box, 121 for the hexa-box). They construct a canonical basis where the differential equations take the $\epsilon$ -factorised form $d\vec{I} = \epsilon \sum_k \alpha_k A_k \vec{I}$ .
Handling Nested Square Roots: A significant portion of the methodology focuses on the penta-box subtopology (depicted in Figure 5) which requires nested square roots ( $R_{1a}, R_{1b}$ $R_{1 a}, R_{1 b}$ ) to achieve a canonical form. The authors develop strategies to:
- Derive master integrals involving these roots by adapting known results from similar topologies (e.g., $t\bar{t}H$ production).
- Determine the symbol alphabet (the set of $d\log$ arguments) for these cases. They utilize Galois theory to analyze the transformation properties of the integrals under sign changes of the roots, leading to a "duplet" structure for the letters involving nested roots.
- Reconstruct the symbolic form of the letters by integrating partial derivatives numerically and using integer relation algorithms (PSLQ) to identify minimal polynomials, thereby confirming the existence of $d\log$ forms.
Permutation Closure: The calculation is performed for all 24 permutations of the massless external legs. The authors construct a permutation-closed alphabet and a common basis of algebraically independent transcendental functions to express the $\epsilon$ -expanded integrals, eliminating redundancies between different permutations.
Numerical Evaluation: The differential equations are solved numerically using the Odeint library. To handle the complex kinematics and avoid singularities, the authors employ an automated analytic continuation algorithm (based on solving univariate polynomial equations for branch cut crossings) to transport solutions from a complex base point to the physical scattering region.

Key Results

Canonical Basis Construction: The authors successfully construct a canonical basis for both the planar penta-box and non-planar hexa-box families. This is the first time a canonical basis has been constructed for non-planar Feynman integrals depending on seven scales.
d-log Forms: Despite the presence of nested square roots, the authors demonstrate that the differential equations for these topologies can be expressed entirely in $d\log$ form. They provide explicit strategies for deriving the corresponding letters, which involve a specific algebraic structure related to the Galois group of the nested roots.
Function Basis: The $\epsilon$ -expanded integrals are expressed in terms of a common basis of algebraically independent functions (transcendental functions of weights 1 through 4).
Numerical Benchmarks: The paper provides benchmark values for the almost-canonical top-sector master integrals at a specific phase-space point, achieving relative error estimates of less than $10^{-16}$ using quadruple precision.
Ancillary Files: The results, including the integral basis, differential equations, and numerical values for all permutations, are provided in computer-readable formats in the ancillary files.

Significance and Claims
The paper claims to provide the necessary theoretical infrastructure for calculating the non-factorisable NNLO QCD corrections to Higgs-boson production via vector boson fusion. By solving the problem of nested square roots in a seven-variable kinematic setup, the authors enable the evaluation of these previously intractable integrals.

The authors explicitly state that their results allow for compact representations of amplitudes and the analytical cancellation of ultraviolet and infrared poles in finite remainders. They note that the strategies developed for handling nested square roots are expected to be generally applicable to similar problems involving massive propagators, which are becoming increasingly common in higher-order calculations.

The work is modest regarding its scope: it addresses only two of the four integral families required for the complete NNLO VBF calculation (the penta-box and hexa-box shown in the first row of Figure 1). The authors acknowledge that the remaining two double-pentagon families (second row of Figure 1) are expected to be more difficult, potentially involving further nested square roots and elliptic structures, and are left for future work.

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