⚛️ phenomenology

Pseudoscalar Higgs boson decay to three parton amplitudes at NNLO to higher orders in the dimensional regulator

This paper presents the first calculation of second-order corrections for pseudo-scalar Higgs boson decay into three partons ( $A \to ggg$ and $A \to q\bar{q}g$ ) expanded to higher orders in the dimensional regulator within an effective theory framework, providing finite amplitude pieces essential for predicting NNLO differential distributions and future three-loop cross sections at hadron colliders.

Original authors: Pulak Banerjee, Chinmoy Dey, M. C. Kumar, V. Ravindran

Published 2026-01-30

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Pulak Banerjee, Chinmoy Dey, M. C. Kumar, V. Ravindran

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

Imagine the universe as a giant, high-stakes billiard game. The "balls" are tiny particles, and the "table" is the space they move through. Scientists at the Large Hadron Collider (LHC) smash these particles together to see what happens, hoping to understand the rules of the game.

One of the most important "balls" they found is the Higgs boson, which gives other particles their mass. But there's a twist: the Standard Model (the rulebook of physics) predicts a specific type of Higgs, but there might be a "cousin" particle called a pseudo-scalar Higgs (let's call it "A"). This cousin is a bit different; it has a different "spin" or personality (called CP-odd) compared to the original.

Here is what this paper does, explained simply:

1. The Goal: Predicting the "Explosion"

When a heavy particle like the Higgs or "A" decays, it doesn't just vanish; it bursts into smaller pieces called partons (which are like tiny shrapnel, specifically gluons and quarks).

The scientists wanted to calculate exactly what happens when this "A" particle decays into three pieces (like three billiard balls flying off).
They wanted to do this calculation with extreme precision, going far beyond the standard "rough draft" math. They aimed for the NNLO level (Next-to-Next-to-Leading Order). Think of this as moving from a sketch to a photorealistic 3D render.

2. The Problem: The Math Gets Messy

To get this level of precision, the math gets incredibly complicated.

The Dimensional Regulator (The "Magic Dimension"): In quantum physics, calculations often blow up and give infinite answers. To fix this, physicists pretend the universe has a slightly different number of dimensions (like 4.0001 instead of 4). This is called the "dimensional regulator" (denoted by the symbol $\epsilon$ ).
The Challenge: Usually, scientists only calculate the main part of the answer. But to get the next level of precision (NNNLO, or the "Next-to-Next-to-Next" level), they need the "leftover" parts of the math that usually get thrown away. They needed to calculate the answer not just for the main part, but also for the parts that depend on these extra dimensions ( $\epsilon^1, \epsilon^2$ , etc.).
The Analogy: Imagine you are baking a cake. Usually, you just care about the cake itself. But to make a perfect cake later, you need to know exactly how much flour, sugar, and heat were lost during the mixing process. This paper calculates those "lost ingredients" with extreme detail.

3. The Solution: A New Recipe

The team (Banerjee, Dey, Kumar, and Ravindran) did the following:

Effective Theory: Since the top quark (a heavy particle) is too heavy to track directly in every step, they used a "shortcut" method called Effective Field Theory. It's like describing a heavy truck by its weight and speed rather than tracking every bolt on the engine.
The Two-Loop Calculation: They performed a "two-loop" calculation. In physics diagrams, a "loop" is a path a particle takes that circles back on itself. Doing this twice (two loops) is like solving a maze where you have to trace two different paths simultaneously.
Handling the "Gamma-5" Monster: A specific mathematical tool used to describe the "A" particle's spin (called $\gamma_5$ ) behaves strangely in these extra dimensions. The team had to apply a special "fix" (renormalization) to make the math consistent, ensuring the laws of physics didn't break.

4. The Result: A Digital Blueprint

After doing the heavy lifting with complex algebra and supercomputers:

They produced the first-ever calculation of these specific decay amplitudes expanded to higher orders in the dimensional regulator.
They didn't just leave the math on paper. They turned these massive, complex formulas into a computer code (written in FORTRAN-95).
The "Optimization": The raw math was so huge it would take a computer hours to run a single calculation. The team used special software to "compress" and "optimize" the code, making it fast enough to be used in real-time simulations (Monte Carlo generators) that physicists use to predict what the LHC will see.

5. Why It Matters (According to the Paper)

The paper states that these results are a crucial missing piece for predicting how often a pseudo-scalar Higgs is produced alongside a "jet" (a spray of particles) at the LHC.

Currently, we have predictions up to a certain level of accuracy (NNLO).
To get even more precise predictions (N3LO), physicists need the "leftover" math pieces that this paper provides.
By providing these pieces, the authors are handing the community the tools needed to build a more accurate map of the universe, helping to confirm if this "pseudo-scalar" particle actually exists and how it behaves.

In summary: This paper is a massive mathematical and computational achievement. It calculates the "fine print" of a particle decay that was previously unknown, fixes the weird math rules that break in higher dimensions, and packages the result into a fast, usable tool so other scientists can use it to hunt for new physics.

Technical Summary: Pseudo-scalar Higgs Decay to Three Parton Amplitudes at NNLO to Higher Orders in Dimensional Regulator

Problem Statement
The precise prediction of pseudo-scalar Higgs boson ( $A$ ) production in association with a jet at hadron colliders requires calculations beyond the Next-to-Next-to-Leading Order (NNLO) in Quantum Chromodynamics (QCD). Specifically, the computation of the Next-to-Next-to-Next-to-Leading Order (N3LO) cross-section necessitates the underlying lower-order virtual corrections expanded to higher powers of the dimensional regulator $\epsilon$ (where $d = 4 - 2\epsilon$ ). While the two-loop results for the Standard Model Higgs + 1 jet process expanded to $\mathcal{O}(\epsilon^2)$ have been recently computed, the corresponding contributions for the pseudo-scalar Higgs boson were lacking. This gap hinders the full N3LO computation for pseudo-scalar production, which is crucial for reducing scale uncertainties and probing the CP properties of the Higgs sector in Beyond the Standard Model (BSM) scenarios, such as the Minimal Supersymmetric Standard Model (MSSM).

Methodology
The authors perform a two-loop computation for the decay of a pseudo-scalar Higgs boson into three partons ( $A \to ggg$ and $A \to q\bar{q}g$ ) within an Effective Field Theory (EFT) framework, where the heavy top quark is integrated out. The calculation proceeds as follows:

Theoretical Framework: The interaction is described by an effective Lagrangian involving two operators, $O_G$ and $O_J$ , coupled via Wilson coefficients $C_G$ and $C_J$ . The calculation utilizes the 't Hooft-Veltman definition of $\gamma_5$ to handle chiral quantities in $d \neq 4$ dimensions.
Diagram Generation and Algebraic Manipulation: Feynman diagrams are generated using QGRAF. The resulting amplitudes are processed using in-house FORM codes to perform Dirac and $SU(N)$ color algebra.
Integral Reduction: The massive number of scalar Feynman integrals are reduced to a minimal set of Master Integrals (MIs) using the public package KIRA, which implements the Laporta algorithm via Integration-by-Parts (IBP) and Lorentz invariance identities. The authors utilize MIs analytically computed in recent work [68], expressed in terms of Generalised Polylogarithms (GPLs), extending previous results that were limited to Harmonic Polylogarithms (HPLs) at lower orders in $\epsilon$ .
Renormalization:
- UV Renormalization: The coupling constant and composite operators are renormalized. Special attention is paid to the renormalization of the axial singlet current ( $O_J$ ), requiring a finite renormalization constant $Z_5^s$ to restore Ward identities and handle the mixing between operators $O_G$ and $O_J$ .
- IR Regularization: The infrared (IR) singularities are verified against the universal subtraction operators predicted by Catani [69] and generalized by Becher and Neubert [95].
Numerical Implementation: The finite parts of the amplitudes are simplified using Mathematica and optimized using FORM (employing Horner schemes and Monte-Carlo tree search) to generate efficient FORTRAN-95 routines. These routines are designed to interface with any Monte Carlo phase space generator.

Key Contributions

First Computation of Higher-Order $\epsilon$ Terms: The paper presents, for the first time, the $\mathcal{O}(\epsilon^1)$ and $\mathcal{O}(\epsilon^2)$ terms of the matrix elements at NNLO, as well as the $\mathcal{O}(\epsilon^3)$ and $\mathcal{O}(\epsilon^4)$ terms at NLO, for the processes $A \to ggg$ and $A \to q\bar{q}g$ .
Analytical Identities: The authors provide a comprehensive set of analytical identities mapping Harmonic Polylogarithms (HPLs) to Generalised Polylogarithms (GPLs), facilitating the comparison with previous results and the evaluation of the new amplitudes.
Validation: The results are validated by checking the IR pole structure against Catani's predictions and by analytically mapping the finite parts to earlier computations [76], finding perfect agreement.
Public Code: The finite pieces of the amplitudes are implemented in optimized FORTRAN-95 routines and made available as ancillary files, ready for use in phenomenological studies.

Results

Numerical Performance: The authors report the CPU time required to evaluate the optimized matrix elements per phase space point. At NNLO, the evaluation time for $\mathcal{O}(\epsilon^2)$ terms is approximately 18–25 seconds per point, significantly longer than the $\mathcal{O}(\epsilon^4)$ terms at NLO (which take $\sim$ 1 second). The optimization process for the $\mathcal{O}(\epsilon^2)$ NNLO results required approximately 16 hours due to the increased complexity and the appearance of higher-weight GPLs.
Magnitude of Contributions: The real part of the $\mathcal{O}(\epsilon^2)$ contribution for the gluonic channel ( $S_{ggg}$ ) is observed to be approximately 100 times larger than that of the quark-gluon channel ( $S_{q\bar{q}g}$ ).
Cross-Channel Relations: The results are derived with crossing symmetry relations in mind, making them directly applicable to the calculation of the pseudo-scalar Higgs boson produced in association with a jet ( $pp \to A + \text{jet}$ ).

Significance
The paper claims that these results are a necessary ingredient for the full three-loop (N3LO) cross-section calculation for pseudo-scalar Higgs production. By providing the amplitudes expanded to higher orders in the dimensional regulator, the work enables the cancellation of IR singularities at the N3LO level. The authors emphasize that while the SM Higgs boson has been established as CP-even, the search for CP-odd scalar particles remains crucial. The availability of these higher-order corrections is essential for reducing theoretical uncertainties in the differential distributions (such as transverse momentum) of pseudo-scalar Higgs bosons at the LHC, thereby improving the precision of tests for BSM physics. The work does not propose new experimental setups but provides the theoretical infrastructure required to interpret future high-precision collider data.

1. The Goal: Predicting the "Explosion"

2. The Problem: The Math Gets Messy

3. The Solution: A New Recipe

4. The Result: A Digital Blueprint

5. Why It Matters (According to the Paper)

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