Two-loop QCD corrections to $H \rightarrow b + \bar{b} + g$ at higher powers in the dimensional regulator

This paper presents the computation of two-loop massless QCD corrections to the Higgs boson decay amplitude into a bottom quark pair and a gluon ($H \rightarrow b + \bar{b} + g$) expanded to higher orders in the dimensional regulator $\epsilon$, providing essential ingredients for future three-loop calculations of Higgs plus jet production at hadron colliders.

The Gist

Imagine the universe is a giant, high-stakes kitchen where particles are the ingredients. For decades, scientists have been trying to understand the recipe for the Higgs boson, a special particle that gives mass to everything else. They know the main ingredients, but they are trying to perfect the recipe by calculating the tiniest, most subtle interactions that happen when these particles collide.

This paper is like a team of master chefs (physicists) who have just finished a very specific, incredibly difficult step in refining that recipe.

The Main Dish: A Higgs Particle Breaking Apart

The scientists are looking at a specific event: a Higgs boson decaying (breaking apart) into three smaller pieces:

1. A bottom quark (a heavy type of particle).
2. An anti-bottom quark (its mirror twin).
3. A gluon (the "glue" that holds quarks together).

Think of this as a Higgs cookie shattering into two chocolate chips and a sprinkle of sugar.

The Problem: The "Blurry" Camera

In the world of quantum physics, calculating these interactions is like trying to take a photo of something moving incredibly fast. If you use a standard camera, the picture is blurry. To fix this, physicists use a mathematical trick called dimensional regularization.

Imagine you are trying to count the grains of sand on a beach, but the beach keeps changing size. To make the math work, the physicists pretend the beach exists in a slightly different number of dimensions (not just 3D, but $4 + \epsilon$ dimensions). The symbol $\epsilon$ (epsilon) represents this tiny, imaginary "extra" dimension.

Usually, physicists only care about the main result (the "zeroth power" of $\epsilon$). But to get the perfect recipe for future experiments, they need to know what happens in the "blurry" parts of the calculation too. They need to calculate the result not just for the main picture, but for the tiny, fuzzy edges of the photo, represented by higher powers of $\epsilon$ (like $\epsilon^1$, $\epsilon^2$, etc.).

What This Paper Did

The authors of this paper did the heavy lifting to calculate the two-loop corrections for this specific Higgs decay.

• The "Two-Loop" Analogy: Imagine you are trying to predict the path of a ball bouncing in a room.
  • Tree level (simple): You just throw the ball and watch it bounce once.
  • One-loop: You account for the ball hitting the wall and bouncing back.
  • Two-loop: You account for the ball hitting the wall, bouncing to the ceiling, hitting a fan, and then landing. It's a much more complex path with many more variables.
• The Achievement: Previous studies only calculated the "main path" (up to $\epsilon^0$). This paper calculated the path all the way through the "fuzzy edges" (up to $\epsilon^2$).

They used powerful computer programs (like QGRAF to draw the diagrams, Reduze and Kira to simplify the math, and FORM to crunch the numbers) to turn thousands of complex diagrams into a clean set of formulas.

Why It Matters (According to the Paper)

The paper states that these calculations are the "missing ingredients" needed for the next level of precision.

Think of it like building a skyscraper.

• The ground floor (current data) is solid.
• The second floor (Next-to-Next-to-Leading Order) is built.
• To build the third floor (Next-to-Next-to-Next-to-Leading Order, or N3LO), you need a specific type of steel beam that was missing.

This paper provides those steel beams. Specifically, they are needed to calculate the three-loop virtual corrections for when bottom quarks smash together to create a Higgs boson plus a jet (a spray of particles) at the Large Hadron Collider (LHC).

The Results

• The Math: They successfully extracted the "form factors" (the mathematical values describing the strength of the interaction) up to the second power of the dimensional regulator ($\epsilon^2$).
• The Speed: They found that calculating these higher powers takes significantly more computer time. Calculating the $\epsilon^2$ part took about 266 seconds per data point, while the simpler $\epsilon^0$ part only took 2 seconds. This is because the higher powers involve much more complex mathematical functions (called Goncharov polylogarithms).
• The Verification: They checked their work against known rules for how these particles should behave (infrared structure) and confirmed their results were correct.

Summary

In short, this paper doesn't discover a new particle or change how we use the Higgs boson today. Instead, it provides the ultra-precise mathematical blueprint required for physicists to perform the next generation of super-accurate calculations at the LHC. It ensures that when they look at the data from future experiments, their theoretical predictions are sharp enough to spot even the tiniest deviations from the Standard Model.

Technical Summary

Technical Summary: Two-loop QCD corrections to H → b + b̄ + g at higher powers in the dimensional regulator

Problem Statement
The paper addresses the need for higher-order perturbative Quantum Chromodynamics (QCD) calculations to support precision phenomenology at the Large Hadron Collider (LHC). Specifically, it targets the computation of two-loop massless QCD corrections to the amplitude of the Higgs boson decay into a bottom quark pair and a gluon ($H \to b\bar{b}g$). While previous studies [62, 63] provided these amplitudes only up to the leading order in the dimensional regulator $\epsilon$ (i.e., $O(\epsilon^0)$), achieving Next-to-Next-to-Next-to-Leading Order (N3LO) accuracy for Higgs production via bottom-quark annihilation ($b\bar{b} \to H$) requires knowledge of two-loop amplitudes expanded to higher powers of $\epsilon$ (specifically up to $O(\epsilon^2)$). These higher-order terms are essential ingredients for the three-loop virtual corrections required for N3LO predictions.

Methodology
The authors employ a projection method to extract form factors from the scattering amplitude. The process is defined as $H(q) \to b(p_1) + \bar{b}(p_2) + g(p_3)$, with massless bottom quarks retained in the loops but coupled to the Higgs via the Yukawa coupling $\lambda$.

1. Tensor Decomposition: The amplitude is decomposed into two independent Lorentz-invariant form factors, $A_1$ and $A_2$, based on Lorentz symmetry and gauge invariance.
2. Diagram Generation and Reduction: Feynman diagrams are generated using QGRAF. The authors utilize in-house FORM routines for Lorentz contractions and color algebra. The resulting scalar integrals are mapped to two families (planar and non-planar) using Reduze and reduced to Master Integrals (MIs) using Kira via Integration-by-Parts (IBP) and Lorentz invariance identities.
3. Master Integrals: The MIs are expressed in terms of dimensionless variables ($x, y, z$) and expanded in powers of $\epsilon$ using Goncharov Polylogarithms (GPLs). The authors utilize known results for these integrals from literature [65, 75, 76].
4. Renormalization and Subtraction: The unrenormalized form factors undergo Ultraviolet (UV) renormalization in the $\overline{\text{MS}}$ scheme for both the strong coupling constant and the Yukawa coupling. Infrared (IR) divergences are then subtracted using Catani's subtraction operators to isolate the finite parts of the form factors.
5. Expansion: The calculation is performed to obtain the one-loop results up to $O(\epsilon^4)$ and the two-loop results up to $O(\epsilon^2)$.

Key Contributions and Results

• Higher-Order $\epsilon$ Terms: The primary contribution is the first computation of the two-loop amplitudes for $H \to b\bar{b}g$ expanded up to $O(\epsilon^2)$, and one-loop results up to $O(\epsilon^4)$.
• Analytic Continuation: The results for $H \to b\bar{b}g$ can be analytically continued to obtain amplitudes for the crossed processes relevant to Higgs production at hadron colliders: $b\bar{b} \to Hg$, $bg \to Hb$, and $\bar{b}g \to H\bar{b}$.
• Numerical Behavior: The authors provide numerical evaluations of the form factors $A_1$ and $A_2$ across specific phase-space regions. They observe that the computational time required for evaluating the form factors increases significantly (by an order of magnitude) for higher powers of $\epsilon$ (e.g., from 2.0s for $\epsilon^0$ to 266.0s for $\epsilon^2$ for a single phase-space point). This is attributed to the presence of higher-weight GPLs in the coefficients of higher $\epsilon$ powers.
• Validation: The results are verified to reproduce the expected universal IR structure predicted by Catani's operators, showing complete agreement with independent calculations available in the literature up to $O(\epsilon^0)$.

Significance
The paper positions these amplitudes as "necessary ingredients" for computing the three-loop virtual corrections to bottom-quark annihilation to Higgs plus jet production ($b\bar{b} \to H + \text{jet}$) at hadron collisions. The authors note that while the dominant contribution to Higgs+jet production comes from gluon fusion, the bottom annihilation channel contributes approximately 3% and is crucial for precision studies, particularly given the high-luminosity data expected from the LHC. By providing the two-loop amplitudes to higher powers of $\epsilon$, this work enables the extension of current Higgs+jet results to N3LO accuracy, which is required to match the anticipated experimental precision of $\sim 1\%$. The authors modestly claim that these results provide "essential ingredients" for future high-precision (three-loop) calculations in bottom-quark-initiated channels.