One-loop amplitudes for $t\bar{t}j$ and $t\bar{t}\gamma$ productions at the LHC through $\mathcal{O}(\epsilon^2)$

This paper presents analytic expressions for one-loop QCD helicity amplitudes for $t\bar{t}j$ and $t\bar{t}\gamma$ production at the LHC through $\mathcal{O}(\epsilon^2)$, expressed in terms of pentagon functions with rational coefficients in momentum-twistor variables, to facilitate NNLO QCD computations.

The Gist

Imagine the Large Hadron Collider (LHC) as the world's most powerful particle smasher. When scientists crash protons together, they are trying to recreate the conditions of the early universe. One of the most interesting things they look for is the "top quark," a particle so heavy it's like the heavyweight champion of the subatomic world.

This paper is about creating a highly detailed, mathematical "instruction manual" for predicting what happens when these top quarks are produced alongside other particles, specifically a jet (a spray of smaller particles) or a photon (a particle of light).

Here is a breakdown of what the authors did, using simple analogies:

1. The Goal: Building a Better Blueprint

Scientists want to predict exactly how often these collisions happen and what the particles look like when they fly apart. To do this, they use a set of rules called Quantum Chromodynamics (QCD).

• The Problem: The current rules are good, but the LHC is becoming so precise that the old rules aren't detailed enough. To match the LHC's precision, scientists need to calculate these collisions with extreme accuracy (called "NNLO").
• The Missing Piece: To get this high accuracy, you need to know the "one-loop" calculations (a specific level of complexity in the math) up to a very high degree of precision. Think of it like baking a cake: if you want a perfect cake, you can't just measure the flour roughly; you need to measure it to the milligram. This paper provides those milligram-level measurements for the top quark collisions.

2. The Method: The "Pentagon" Toolbox

The math involved in these collisions is incredibly messy. If you tried to write out the full equations on a piece of paper, it would be longer than the entire Encyclopedia Britannica.

• The Analogy: Imagine trying to describe a complex 3D sculpture. You could describe every single atom, or you could describe it using a set of standard building blocks.
• The Solution: The authors invented a set of standard building blocks called "Pentagon Functions." Instead of writing out the massive, messy equations every time, they expressed the results as a combination of these standard blocks.
  • It's like saying, "The shape of this cloud is 3 parts 'fluffy,' 2 parts 'wispy,' and 1 part 'dark.'"
  • By using these blocks, the math becomes much shorter, cleaner, and easier to handle.

3. The Process: Solving the Puzzle

The authors had to figure out exactly how these "Pentagon Functions" behave under different conditions.

• The Map: They drew a map of all the possible ways the particles can move (called "kinematics").
• The Engine: They used a method called "differential equations" to figure out how the functions change as the particles move.
• The Computer Power: The equations were too hard for a human to solve by hand. The authors used powerful computers and a technique called "finite-field arithmetic."
  • Analogy: Imagine trying to solve a giant Sudoku puzzle. Instead of writing down every number, the computer checks if the numbers work in a specific, simplified "world" (a finite field) to figure out the pattern. Once the pattern is found, they translate it back into the real, complex math.

4. The Result: A New Reference Guide

The paper presents the final results in two main forms:

1. Analytic Expressions: The "recipe" written in the language of the Pentagon Functions. This allows other scientists to plug in different numbers and get answers without redoing the hard math.
2. Numerical Benchmarks: They tested their recipe at a specific point in the "universe" of particle collisions to show that it works. They compared their numbers with other existing tools to prove they are correct.

Why This Matters (According to the Paper)

The authors state that this work is a necessary step to build the "two-loop" calculations (the next, even more complex level of math) required for the next generation of LHC experiments.

• The Metaphor: If the LHC is a high-speed camera taking pictures of the universe, this paper provides the sharper lens needed to see the details clearly. Without this "lens," the pictures would be blurry, and scientists might miss subtle signs of new physics.

In summary: The authors have created a streamlined, highly accurate mathematical toolkit for predicting how top quarks behave when produced with jets or light. They did this by breaking down complex equations into manageable "Pentagon" building blocks and using advanced computer techniques to solve the resulting puzzles. This toolkit is essential for future, ultra-precise experiments at the LHC.

Technical Summary

Technical Summary: One-loop amplitudes for $t\bar{t}j$ and $t\bar{t}\gamma$ productions at the LHC through $O(\epsilon^2)$

Problem Statement
Theoretical predictions for top-quark pair production in association with a jet ($t\bar{t}j$) or a photon ($t\bar{t}\gamma$) at the Large Hadron Collider (LHC) are currently available up to Next-to-Leading Order (NLO) accuracy. However, to match the unprecedented precision of experimental measurements in current and upcoming LHC runs, predictions must be provided at Next-to-Next-to-Leading Order (NNLO) in Quantum Chromodynamics (QCD). Achieving NNLO accuracy requires the computation of two-loop scattering amplitudes for $2 \to 3$ processes. A critical prerequisite for constructing these two-loop hard functions is the availability of one-loop amplitudes expanded up to $O(\epsilon^2)$ in the dimensional regularisation parameter $\epsilon$. While automated packages exist for calculating the finite part ($O(\epsilon^0)$) of one-loop amplitudes, they do not provide the higher-order $\epsilon$ terms necessary for the subtraction of ultraviolet (UV) and infrared (IR) singularities and the extraction of the finite remainder in NNLO computations. This paper addresses the gap by providing analytic expressions for the full-colour one-loop helicity amplitudes for $pp \to t\bar{t}j$ and $pp \to t\bar{t}\gamma$ processes through $O(\epsilon^2)$.

Methodology
The authors employ a multi-stage computational framework combining finite-field arithmetic, tensor decomposition, and the method of pentagon functions:

1. Helicity Amplitude Construction: The calculation is performed in the 't Hooft-Veltman (tHV) scheme. The authors decompose partial amplitudes into independent 4-dimensional tensor structures. Massive spinors for the top quarks are defined using a reference direction to express helicity amplitudes in terms of massless momenta and spinor products. The helicity sub-amplitudes are extracted by evaluating the contracted partial amplitudes at specific choices of reference vectors and inverting the resulting linear system.
2. Finite-Field Reconstruction: To manage the algebraic complexity of intermediate expressions, the authors utilize finite-field methods. Feynman diagrams are generated using QGRAF, followed by colour decomposition. The resulting contracted partial amplitudes are reduced to a minimal set of Master Integrals (MIs) using Integration-By-Parts (IBP) identities (generated via NEATIBP and solved via FINITEFLOW). The rational coefficients multiplying the MIs are reconstructed analytically from their numerical evaluations over finite fields.
3. Pentagon Function Basis: The MIs are expressed in terms of a basis of algebraically independent components known as "pentagon functions." These functions correspond to the $\epsilon$-expanded components of the MIs. The authors derive canonical Differential Equations (DEs) for these functions in $d\log$ form. The alphabet of the DEs consists of 283 independent letters, including 29 algebraic square roots.
4. Numerical Evaluation: The DEs are solved numerically using the generalised power series expansion method. The authors utilize two software packages: DIFFEXP (Mathematica) and the newly released LINE (C/C++). Boundary conditions are determined by evaluating the MIs at specific phase-space points using the AMFLOW package with 70-digit precision. The solution is propagated to arbitrary physical phase-space points.
5. Renormalisation and Finite Remainder: The bare amplitudes are mass-renormalised and fully UV-renormalised by adding appropriate counterterms. The finite remainder is extracted by subtracting universal IR singularities, utilizing the MS scheme.

Key Contributions

• Analytic Expressions: The paper provides the first analytic expressions for the full-colour one-loop helicity amplitudes for $t\bar{t}j$ and $t\bar{t}\gamma$ production expanded through $O(\epsilon^2)$.
• Pentagon Function Basis: The authors construct a specific basis of pentagon functions for these processes, expressing the amplitudes as linear combinations of these functions with rational coefficients written in terms of momentum-twistor variables.
• Differential Equations: Canonical DEs for the pentagon functions are derived and solved numerically. The authors provide the DEs, boundary values, and permutation rules for the functions in ancillary files.
• Software Implementation: The work demonstrates the application of the LINE package for solving DEs involving square roots, showing it to be approximately twice as fast as DIFFEXP for this specific system of equations.
• Benchmark Results: Numerical benchmark results for colour- and helicity-summed squared matrix elements are provided for all relevant partonic channels ($gg \to t\bar{t}g$, $q\bar{q} \to t\bar{t}g$, $gg \to t\bar{t}\gamma$, $q\bar{q} \to t\bar{t}\gamma$) and validated against OPENLOOPS up to $O(\epsilon^0)$.

Results
The authors successfully computed the helicity amplitudes for processes involving top-quark pairs and massless particles (gluons, photons, light quarks). The analytic expressions for the rational coefficients of the pentagon functions were reconstructed, with the numerator degrees significantly reduced through univariate partial fraction decomposition. The numerical evaluation of the amplitudes at a specific physical phase-space point confirmed the validity of the results against existing NLO tools. The paper notes that while the rational coefficients for $t\bar{t}j$ and $t\bar{t}\gamma$ share some structural similarities, a full extraction of relations between the two processes at the level of primitive amplitudes was not performed.

Significance and Claims
The authors state that their results are essential for constructing the two-loop hard functions required for NNLO QCD computations of $t\bar{t}j$ and $t\bar{t}\gamma$ production. By providing amplitudes through $O(\epsilon^2)$, they enable the analytic subtraction of UV and IR divergences and the extraction of finite remainders, which are otherwise inaccessible with tools limited to $O(\epsilon^0)$.

The paper claims that expressing the amplitudes in terms of a pentagon-function basis offers a compact framework that simplifies the final expressions and improves numerical efficiency. Furthermore, the authors position this work as a stepping stone for future two-loop computations. They note that while leading-colour two-loop amplitudes for $t\bar{t}j$ involve elliptic structures, the pentagon-function basis constructed here (which separates polylogarithmic and non-polylogarithmic contributions) may provide a useful strategy for handling the more complex functional spaces expected in two-loop calculations, including those for $t\bar{t}\gamma$. The authors modestly conclude that while the basis facilitates future work, the full realization of two-loop computations remains a subject for future research.