Precise Predictions for Hadronic Higgs Decays

This paper presents precise NNLO QCD predictions, enhanced with NNLL resummation for specific observables, for hadronic Higgs decays into $b\bar{b}$ and $gg$ channels using a newly developed generalised antenna formalism to support future Higgs factory measurements.

The Gist

Here is an explanation of the paper "Precise Predictions for Hadronic Higgs Decays" using simple language and creative analogies.

The Big Picture: The "Higgs Factory"

Imagine the Large Hadron Collider (LHC) as a chaotic, noisy demolition derby. Scientists found the Higgs boson there, but it's hard to study its details because the environment is so messy with background noise.

Now, imagine a future machine called a "Higgs Factory" (like FCC-ee or CEPC). This is a pristine, silent laboratory. It smashes electrons and positrons together to create thousands of Higgs bosons in a very clean environment. Because the "noise" is gone, scientists can measure the Higgs with incredible precision—down to the "per mille" level (one part in a thousand).

The Problem: To make these measurements useful, we need a perfect theoretical map. If the experiment says "The Higgs decayed this way," but our theory predicts "It should have decayed that way," we might miss a new discovery. This paper provides that map.

The Two Main Ways the Higgs "Breaks"

When a Higgs boson decays (breaks apart), it usually turns into particles that fly out like shrapnel. The paper focuses on two main types of shrapnel:

1. The Heavy Quark Pair (The "Bottom" Mode): About 85% of the time, the Higgs turns into a bottom quark and an anti-bottom quark. Think of this as the Higgs splitting into two heavy, slow-moving bowling balls.
2. The Gluon Pair (The "Gluon" Mode): About 15% of the time, it turns into two gluons (particles that carry the strong force). Think of this as the Higgs splitting into two high-speed, invisible jets of energy.

The Challenge: The "Gluon Mode" is tricky. In the messy LHC environment, it gets lost in a sea of other gluons. But in the clean "Higgs Factory," we can see it. The goal of this paper is to predict exactly what these two modes look like so we can tell them apart.

The Tools: "Antennas" and "Jet Rates"

To predict these decays, the authors used a new mathematical tool called the "Generalised Antenna Formalism."

• The Analogy: Imagine trying to predict how a sound wave spreads when a speaker breaks. You need to know how the sound behaves in every direction. In particle physics, when particles fly apart, they emit "radiation" (gluons) that can mess up the calculation. The "Antenna" is a mathematical device that acts like a noise-canceling headset. It perfectly predicts and subtracts the "noise" (infrared singularities) so the scientists can hear the true signal.

They used this to calculate Jet Rates.

• The Analogy: Imagine throwing a handful of marbles (particles) onto a table. A "Jet Algorithm" is a rule that says, "If two marbles are close enough, glue them together."
  • If you glue them all together, you have a 1-Jet event.
  • If you leave them separate, you have a 5-Jet event.
  • The paper calculates exactly how often the Higgs decay results in 2, 3, 4, or 5 distinct "clumps" (jets) of particles.

The Results: What They Found

The team calculated these events up to a very high level of precision (NNLO). Here is what they discovered:

1. The "Back-to-Back" Problem: When particles fly apart perfectly opposite each other (back-to-back), simple math breaks down and gives impossible answers (like negative probabilities).

   • The Fix: They used Resummation. Imagine you are trying to count a crowd, but the crowd is so dense you can't see individuals. Instead of counting one by one, you estimate the density of the whole crowd. Resummation sums up all the tiny, messy interactions to give a smooth, realistic prediction even in the "dense" regions.

2. The Thrust Distribution: They looked at a shape called "Thrust," which measures how "cigar-shaped" or "pancake-shaped" the debris is.

   • The Finding: The "Bottom" mode and the "Gluon" mode look different! The Gluon mode tends to create a "fatter" spray of particles at certain angles, while the Bottom mode is more "slim."
   • Why it matters: By measuring the shape of the debris, scientists can tell if they are looking at the 85% mode or the 15% mode. This allows them to measure the Gluon mode precisely for the first time.

3. The Match: They combined their high-precision "fixed" math with the "resummation" math.

   • The Analogy: Think of it like stitching two different maps together. One map is great for the open highway (high energy, simple events), and the other is great for the winding city streets (low energy, complex events). They stitched them together perfectly so you have one seamless map that works everywhere.

The Conclusion

This paper is like providing a high-definition blueprint for future experiments.

• Before: Scientists had a blurry sketch of how the Higgs decays into gluons.
• Now: They have a crystal-clear, 3D model that predicts exactly how many jets will form and what shape the debris will take.

With this blueprint, when the "Higgs Factories" come online in the future, they won't just be collecting data; they will be able to spot tiny deviations that could reveal new physics beyond our current understanding of the universe. They can finally weigh the "Gluon Mode" with the same precision as the "Bottom Mode."

Technical Summary

Here is a detailed technical summary of the paper "Precise Predictions for Hadronic Higgs Decays" by Elliot Fox.

1. Problem Statement

The discovery of the Higgs boson at the LHC established its dominant decay channel ($H \to b\bar{b}$) but left the sub-dominant decay to gluons ($H \to gg$) unmeasured due to overwhelming irreducible QCD backgrounds in hadron-hadron collisions. Future electron-positron colliders (e.g., FCC-ee, CEPC) operating as "Higgs factories" offer a clean environment to measure these observables with per-mille precision.

However, to capitalize on this experimental precision, theoretical predictions must match the experimental accuracy. Current challenges include:

• Differential Precision: Existing calculations often lack Next-to-Next-to-Leading Order (NNLO) accuracy for differential observables (like jet rates and event shapes) distinguishing between the Yukawa-induced $H \to q\bar{q}$ and the loop-induced $H \to gg$ channels.
• Perturbative Breakdown: In the limit of small event-shape variables (e.g., back-to-back jets), fixed-order perturbation theory breaks down due to large logarithmic terms ($\alpha_s^n L^m$), leading to unphysical predictions (e.g., negative cross-sections).
• Interference: While $H \to q\bar{q}$ and $H \to gg$ are treated as independent channels in the massless limit, precise quantification of their differences requires rigorous control over higher-order corrections.

2. Methodology

The authors employ a state-of-the-art theoretical framework combining fixed-order calculations with all-order resummation:

• Effective Field Theory (EFT): The $H \to gg$ channel is treated in an EFT where heavy quarks (top) are integrated out, treating them as infinitely massive. Light quarks are treated as massless in the calculation (except for Yukawa couplings), ensuring no interference between the $q\bar{q}$ and $gg$ channels to all orders in perturbation theory.
• Matrix Elements:
  • Tree-level amplitudes for up to 5 partons.
  • One-loop amplitudes for up to 4 partons.
  • Two-loop amplitudes for 3 partons (recomputed for $H \to q\bar{q}$ and adapted from existing literature for $H \to gg$).
• Infrared Subtraction: The Generalised Antenna Subtraction Method is used to handle infrared singularities. This approach constructs subtraction terms using idealized antenna functions derived from specific infrared limits, significantly improving efficiency over traditional methods.
• Numerical Integration: The finite remainders are integrated using the NNLOjet Monte Carlo event generator, adapted to handle generalised antenna functions.
• Resummation and Matching:
  • Large logarithmic corrections in the limit of small event shapes ($\tau \to 0$) are resummed to Next-to-Next-to-Leading Logarithmic (NNLL) accuracy using the ARES formalism.
  • The resummed results are matched to the fixed-order NNLO predictions using the log-R matching scheme. This ensures a smooth transition between the resummation-dominated region (small $\tau$) and the fixed-order dominated region (large $\tau$).

3. Key Contributions

• First NNLO Differential Predictions: The paper presents the first NNLO predictions for jet rates and event shapes (specifically Thrust) for both $H \to b\bar{b}$ and $H \to gg$ channels.
• Generalised Antenna Formalism: Demonstrates the successful application of the newly developed generalised antenna formalism to Higgs decays, streamlining the construction of subtraction terms.
• NNLO+NNLL Matching: Provides fully matched predictions (NNLO fixed-order + NNLL resummation) for the Thrust distribution, valid across the entire kinematic range from the back-to-back limit to the multi-jet region.
• Channel Differentiation: Quantifies the distinct kinematic behaviors of the $b\bar{b}$ and $gg$ decay modes, which is crucial for disentangling them in future collider data.

4. Results

• Jet Rates ($R_n$):
  • Calculated $n$-jet rates ($n=2,3,4,5$) differential in the Durham resolution parameter $y_{cut}$.
  • Convergence: For large $y_{cut}$, both channels show good perturbative convergence (scale variation bands overlap).
  • Divergence: For small $y_{cut}$, fixed-order predictions break down. The $H \to gg$ channel exhibits earlier onset of large logarithms and unphysical negative predictions for $y_{cut} < 0.001$.
  • Sensitivity: At large $y_{cut}$, the 3-jet rate is sensitive to the $gg$ channel (~25% contribution at $y_{cut}=0.1$), but this contribution vanishes as $y_{cut} \to 0.001$.
• Thrust Distribution ($\tau$):
  • Sudakov Shoulder: A distinct feature at $\tau = 1/3$ where the LO distribution is zero, and NNLO corrections are enhanced.
  • Channel Composition: In the region $\tau \in [0.1, 0.3]$, the $b\bar{b}$ channel contributes ~70% and $gg$ ~30%. For $\tau > 0.3$, the $gg$ contribution increases significantly (>33%).
  • Resummation Necessity: Fixed-order predictions for small $\tau$ become negative (unphysical) for the $gg$ channel. Resummation restores physicality and predictive power.
• Matched Predictions (NNLO+NNLL):
  • The matched distributions show restored perturbative convergence (overlapping uncertainty bands) across the full $\tau$ range.
  • The peak of the distribution is higher and shifted to larger $\tau$ values for the $gg$ channel compared to the $b\bar{b}$ channel.
  • For $\tau < e^{-4.24} \approx 0.0145$, the $gg$ contribution effectively vanishes, leaving only $b\bar{b}$ and $c\bar{c}$ contributions.

5. Significance

This work provides the essential theoretical infrastructure required for the "Higgs factory" era of particle physics. By delivering precise, differential NNLO+NNLL predictions:

1. Experimental Readiness: It enables future experiments (FCC-ee, CEPC) to extract the $H \to gg$ branching ratio and test the Standard Model's heavy quark loop structure with unprecedented precision.
2. Background Suppression: The ability to distinguish kinematic features between $b\bar{b}$ and $gg$ decays via event shapes allows for better background rejection and signal isolation.
3. Methodological Advancement: The successful application of the generalised antenna formalism to Higgs decays sets a precedent for future high-precision QCD calculations in other processes.

In summary, the paper bridges the gap between the high-precision experimental capabilities of future lepton colliders and the theoretical requirements needed to interpret Higgs boson properties, specifically focusing on the elusive gluonic decay mode.