QCD corrections for Pseudoscalar Higgs decay to 3 partons at higher orders in dimensional regulator

This paper presents a study of second-order QCD corrections for the decay of a pseudoscalar Higgs boson into three partons within an effective theory framework, providing essential results for predicting differential distributions of the Higgs boson in association with jets at hadron colliders.

The Gist

Here is an explanation of the paper, translated from complex physics jargon into everyday language using analogies.

The Big Picture: The "Ghost" Particle and the Three-Way Crash

Imagine the Large Hadron Collider (LHC) as the world's most powerful particle smasher. In 2012, scientists found the famous Higgs boson, a particle that gives mass to everything else. But in many theories (like Supersymmetry), there isn't just one Higgs; there's a whole family. One of these is a "sibling" called the Pseudoscalar Higgs (let's call it "A").

The big mystery is: Is "A" a "good twin" (CP-even) or a "bad twin" (CP-odd)? To figure this out, scientists need to watch how "A" behaves when it decays (breaks apart).

This paper is about calculating exactly what happens when the "A" particle crashes and breaks into three smaller pieces (called partons: either three gluons or a quark, an anti-quark, and a gluon).

The Problem: The "Blurry" Camera

In quantum physics, calculating these crashes is like trying to take a photo of a speeding bullet with a camera that has a blurry lens.

• The Blur: When you do the math, you get infinite numbers (divergences) because the particles are massless and moving at light speed.
• The Fix (Dimensional Regulator): To fix the blur, physicists use a mathematical trick called "Dimensional Regularization." Imagine the universe usually has 4 dimensions (3 space + 1 time). To fix the math, they temporarily pretend the universe has 4.00001 dimensions (or $4 + \epsilon$). This extra tiny bit of dimension acts like a "sharpening filter" that removes the infinities.

The goal of this paper is to calculate the "photo" of the crash not just once, but with extreme precision, looking at the picture through this "blurry lens" up to the second order (meaning they are looking at the second layer of detail in the math).

The Analogy: Baking a Cake with a Secret Recipe

Think of the Pseudoscalar Higgs ("A") as a special cake.

1. The Ingredients (Operators): The cake is made using two secret recipes (operators, $O_G$ and $O_J$). One recipe is standard, but the other involves a "chiral" ingredient (involving $\gamma_5$) that behaves weirdly in our 4-dimensional world, like a left-handed glove trying to fit on a right-handed hand.
2. The Bakers (Wilson Coefficients): The authors calculated exactly how much of each ingredient is needed. They found that the "chiral" ingredient needs a special adjustment (renormalization) to make the math work, otherwise, the cake falls apart.
3. The Crumbs (The Decay): When the cake is eaten (decays), it breaks into three crumbs ($ggg$ or $q\bar{q}g$). The authors calculated the exact shape and weight of these crumbs.

The Challenge: The "Math Monster"

The calculations involved in this paper are massive.

• The File Size: The authors mention that the raw math files were several gigabytes in size. Imagine trying to solve a Sudoku puzzle where the grid is the size of a city, and every square changes as you solve it.
• The Cleanup: They used powerful computer programs (like FORM and Mathematica) to simplify these giant equations. It's like taking a tangled ball of 10,000 yarn strings and finding a way to untangle them into neat, usable threads.
• The Result: They managed to simplify the math so that a modern computer could calculate the result in a few seconds, rather than days.

Why Does This Matter? (The "Jet" Connection)

You might ask, "Why do we care about a particle decaying into three pieces?"

The answer lies in crossing symmetry. In physics, if you know how a particle breaks apart (decay), you can mathematically flip the script to know how particles collide to create it.

• The Real Goal: The authors aren't just studying the decay; they are doing this to help predict what happens when we smash protons together at the LHC to create a Pseudoscalar Higgs plus a jet (a spray of particles).
• The Precision: Current predictions are like guessing the weather with a 50% chance of rain. These new calculations are like a satellite forecast telling you exactly when the rain will start and how hard it will pour. This is crucial for distinguishing the "bad twin" (Pseudoscalar) from the "good twin" (Standard Higgs).

The Takeaway

This paper is a mathematical blueprint.

1. The authors calculated the incredibly complex "blueprint" of how a hypothetical Pseudoscalar Higgs breaks into three pieces.
2. They fixed the "weird" math problems associated with 4-dimensional space.
3. They turned gigabytes of raw, messy equations into a clean, usable code that other scientists can plug into their simulations.

In short: They built a high-precision ruler. Now, when experimentalists at the LHC look for this mysterious particle, they have a much better ruler to measure it against, helping them decide if the "bad twin" Higgs actually exists.

Technical Summary

Here is a detailed technical summary of the paper "QCD corrections for Pseudoscalar Higgs decay to 3 partons at higher orders in dimensional regulator" by Pulak Banerjee et al.

1. Problem Statement and Motivation

The discovery of the Higgs boson at the LHC necessitates precise measurements of its CP properties and couplings. While the Standard Model (SM) Higgs is CP-even, many Beyond the Standard Model (BSM) scenarios, such as the Minimal Supersymmetric Standard Model (MSSM), predict a CP-odd pseudoscalar Higgs boson ($A$). Distinguishing between CP-even and CP-odd states is crucial for identifying the nature of the discovered Higgs.

• The Challenge: The dominant production mechanism for Higgs bosons at the LHC is gluon fusion, which receives large perturbative QCD corrections. To achieve high-precision theoretical predictions for $A$ production in association with a jet ($A + \text{jet}$) beyond Next-to-Next-to-Leading Order (NNLO), one requires specific higher-order virtual corrections.
• The Specific Gap: Calculating the NNLO and beyond for $A + \text{jet}$ requires the two-loop virtual amplitudes for the decay of the pseudoscalar into three partons ($A \to ggg$ and $A \to q\bar{q}g$) expanded up to $\mathcal{O}(\epsilon^2)$ in dimensional regularization ($d = 4 - 2\epsilon$). Prior to this work, these specific two-loop results for the pseudoscalar case were not fully available in the literature.

2. Theoretical Framework and Methodology

Effective Field Theory (EFT) Approach

The authors work within an EFT framework where the heavy top quark is integrated out (infinite mass limit). The interaction is described by an effective Lagrangian involving two operators:
$$\mathcal{L}_{eff}^A = \Phi_A(x) \left[ -\frac{1}{8} C_G O_G(x) - \frac{1}{2} C_J O_J(x) \right]$$

• Operators: $O_G$ involves the gluonic field strength, while $O_J$ is a chiral operator involving the axial current.
• Wilson Coefficients: $C_G$ and $C_J$ are expanded in powers of the strong coupling constant $\alpha_s$.
• $\gamma_5$ Handling: A critical technical challenge is the definition of $\gamma_5$ in $d \neq 4$ dimensions, as it does not anticommute with $\gamma_\mu$ in dimensional regularization. The authors adopt the 't Hooft-Veltman definition:
  $$\gamma_5 = \frac{i}{4!} \epsilon_{\nu_1 \nu_2 \nu_3 \nu_4} \gamma^{\nu_1} \gamma^{\nu_2} \gamma^{\nu_3} \gamma^{\nu_4}$$
  This requires careful treatment of the axial singlet current renormalization.

Calculation Strategy

1. Processes Studied:
   • $A \to ggg$ (Three gluons)
   • $A \to q\bar{q}g$ (Quark-antiquark-gluon)
2. Loop Order: The authors computed one-loop and two-loop amplitudes.
3. Dimensional Expansion: The results were expanded in the dimensional regulator $\epsilon$ up to:
   • $\mathcal{O}(\epsilon^4)$ for one-loop amplitudes.
   • $\mathcal{O}(\epsilon^2)$ for two-loop amplitudes.
   • Note: The $\mathcal{O}(\epsilon^2)$ terms in the two-loop amplitude are essential for the NNLO prediction of the $A+\text{jet}$ cross-section via crossing symmetry.
4. Computational Tools:
   • Symbolic Manipulation: Used in-house FORM codes for Dirac algebra and $SU(N)$ color manipulations.
   • Integral Reduction: Utilized MultivariateApart to simplify Integration by Parts (IBP) identities.
   • Master Integrals: The results were expressed in terms of rational coefficients and Master Integrals (analytically computed up to $\mathcal{O}(\epsilon^2)$ using Generalized Polylogarithms) taken from previous literature [13].
   • Numerical Implementation: Simplified analytical results were converted into FORTRAN-95 routines for integration with Monte Carlo phase-space generators.

Renormalization and Regularization

• UV Renormalization: Addressed the renormalization of the strong coupling constant and the composite operators $O_G$ and $O_J$. Special attention was paid to the renormalization of the axial singlet current ($Z_5^s$) to satisfy Ward identities, followed by the $\overline{MS}$ operator renormalization ($Z_{MS}^s$).
• IR Regularization: Infrared (IR) singularities arising from massless particles were regulated using dimensional regularization. The authors verified that the IR poles factorize correctly according to universal QCD predictions (Catani's formula), ensuring the cancellation of divergences when combined with real emission contributions.

3. Key Results

• Analytical Expressions: The paper presents the first analytical computation of the two-loop matrix elements for $A \to ggg$ and $A \to q\bar{q}g$ expanded to $\mathcal{O}(\epsilon^2)$.
• Validation:
  • The finite parts of the matrix elements were compared with existing results in Ref. [26] (which covered the $A \to ggg$ case at lower orders or different contexts).
  • Table 1 in the paper demonstrates excellent agreement between the authors' $\mathcal{O}(\epsilon^0)$ results and Ref. [26] for various phase-space points ($y, z$).
  • The authors also provided higher-order terms ($\mathcal{O}(\epsilon)$ and $\mathcal{O}(\epsilon^2)$) which are new and necessary for higher-order phenomenology.
• Numerical Performance:
  • The optimized FORTRAN routines are highly efficient.
  • Execution Time: For the $ggg$ channel at $\mathcal{O}(\epsilon^0)$, evaluation takes $\sim 5 \mu s$ per phase-space point. At $\mathcal{O}(\epsilon^2)$, it takes $\sim 19 s$.
  • Magnitude: The $\mathcal{O}(\epsilon^2)$ contribution for the $ggg$ channel is found to be approximately 100 times larger than that for the $q\bar{q}g$ channel, highlighting the dominance of the gluonic mode in these corrections.

4. Significance and Impact

• Enabling NNLO+ Predictions: The primary significance of this work is providing the necessary "ingredients" (the two-loop virtual amplitudes expanded to $\mathcal{O}(\epsilon^2)$) to calculate the differential distributions for Pseudoscalar Higgs production in association with a jet at NNLO in QCD.
• BSM Phenomenology: These results allow for more precise theoretical predictions for the LHC, which are critical for distinguishing the CP nature of the Higgs boson. Without these higher-order corrections, theoretical uncertainties would be too large to definitively identify a CP-odd state.
• Methodological Advancement: The successful handling of the 't Hooft-Veltman $\gamma_5$ scheme in two-loop calculations for chiral operators demonstrates a robust methodology for future high-order QCD calculations involving pseudoscalars.
• Open Source: The authors have made their numerical codes available (via Ref. [15]), facilitating immediate use by the phenomenological community for Monte Carlo simulations.

In summary, this paper bridges a critical gap in high-order QCD calculations for BSM Higgs physics, providing the essential virtual corrections required to push precision predictions for pseudoscalar Higgs production to the NNLO level.