Threshold resummation for gluon fusion $ZH$ production at the LHC

This paper presents precise calculations for the $ZH$ production cross-section and invariant mass distribution at the LHC by improving both quark- and gluon-initiated subprocesses through leading and sub-leading soft gluon threshold resummation within the QCD framework.

The Gist

Imagine the Large Hadron Collider (LHC) as the world's most powerful particle smasher. Inside its massive ring, protons collide at nearly the speed of light, creating a chaotic explosion of energy that occasionally births new, rare particles. One of the most important "babies" we are looking for is a pair of particles: a Z boson and a Higgs boson. Finding them together helps us understand the fundamental rules of the universe and look for clues about physics beyond what we currently know.

However, predicting exactly how often these pairs appear is incredibly difficult. It's like trying to predict the exact number of raindrops that will hit a specific leaf during a storm, when the wind is blowing, the clouds are shifting, and the rain is falling in complex patterns.

The Problem: The "Noise" of Soft Gluons

In the world of particle physics, when particles collide, they don't just bounce off each other; they often emit tiny, invisible messengers called gluons. Most of the time, these gluons are energetic and easy to track. But near the "edge" of the collision energy (what physicists call the "threshold"), a huge number of very low-energy, "lazy" gluons are emitted. These are called soft gluons.

Think of these soft gluons like a crowd of people whispering in a library. Individually, a whisper is quiet. But if thousands of people whisper at the same time, it creates a deafening roar that drowns out the main conversation. In our physics calculations, these "whispers" (soft gluons) create huge mathematical errors called logarithms. If we ignore them, our predictions for how many Z-Higgs pairs will be created are wildly inaccurate, especially when the particles are moving fast (high energy).

The Solution: "Resummation" (The Noise Cancelling Headphones)

The authors of this paper, a team of theoretical physicists, have developed a sophisticated method to fix this. They call it Threshold Resummation.

Imagine you are trying to listen to a specific song on the radio, but there is static interference.

• Standard Calculation (Fixed Order): This is like trying to listen to the song while ignoring the static. You get a rough idea of the melody, but the sound is fuzzy and the volume keeps jumping up and down unpredictably.
• Resummation: This is like putting on noise-cancelling headphones. Instead of ignoring the static, the headphones actively analyze the noise pattern and generate an "anti-noise" signal to cancel it out perfectly.

The paper focuses on two types of "noise":

1. Soft-Virtual (SV) Resummation: This handles the main "whispers" (the loudest part of the static).
2. Next-to-Soft (NSV) Resummation: This is the new, advanced feature. It catches the very faint whispers that the first method missed. It's like upgrading from standard noise-cancelling headphones to a high-end, AI-powered version that filters out even the faintest background hum.

What They Did

The team focused on a specific way the Z-Higgs pair is made: Gluon Fusion.

• Usually, Z-Higgs pairs are made when two quarks (the building blocks of protons) smash together. This is the "main road."
• However, sometimes two gluons (the force carriers inside the proton) smash together to make the pair. This is a "side road."
• For a long time, physicists thought the "side road" was too small to matter. But the authors show that because there are so many gluons inside a proton, this side road is actually a major highway, especially at high energies.

They used their "noise-cancelling" math to calculate exactly how many Z-Higgs pairs should be produced via this gluon route, including the tricky soft-gluon effects.

The Results: A Clearer Picture

When they applied their new math, they found:

• Bigger Numbers: The "noise" was actually hiding a lot of extra production. When they turned on the "noise cancellation," the predicted number of Z-Higgs pairs increased significantly (by up to 40% in some high-energy scenarios).
• Less Guesswork: Before this, the predictions had a huge "fuzziness" (uncertainty) because of the uncalculated soft gluons. By resumming these effects, they reduced the uncertainty from about 20% down to 15% or less. It's like going from a blurry photo to a high-definition image.
• Better Tools for Experimenters: The ATLAS and CMS experiments at the LHC are currently measuring these collisions. By giving them these precise, "noise-free" predictions, the experimentalists can now compare their real-world data against a much sharper target. If the real data still doesn't match the prediction, it might mean we've discovered something truly new (New Physics).

In a Nutshell

This paper is about cleaning up the mathematical "static" that was making our predictions for particle collisions fuzzy. By using advanced techniques to account for the collective "whispers" of soft gluons, the authors have provided a much clearer, more accurate map of where and how often the Z-Higgs pair is born. This helps the world's best physicists know exactly what to look for in their giant particle smashers, bringing us one step closer to understanding the deepest secrets of the universe.

Technical Summary

Here is a detailed technical summary of the paper "Threshold resummation for gluon fusion $ZH$ production at the LHC" by Goutam Das et al.

1. Problem Statement

The production of a $Z$ boson in association with a Higgs boson ($ZH$) is a critical process for testing the Standard Model (SM) and probing Beyond-the-Standard-Model (BSM) physics, particularly regarding the top-quark Yukawa coupling and $CP$ structure.

• Dominant vs. Sub-dominant Channels: At the LHC, $ZH$ production is primarily driven by the Drell-Yan (DY) type quark-antiquark annihilation ($q\bar{q} \to Z^* \to ZH$), which has been calculated up to Next-to-Next-to-Next-to-Leading Order (N3LO).
• The Gluon Fusion Challenge: The gluon fusion channel ($gg \to ZH$), while suppressed by two powers of the strong coupling constant ($\alpha_s^2$) relative to the DY channel, becomes phenomenologically significant at Next-to-Next-to-Leading Order (N2LO) due to the high gluon luminosity at the LHC.
• Theoretical Limitation: Fixed-order perturbative QCD calculations for the gluon fusion subprocess suffer from large threshold logarithms arising from soft gluon emissions near the kinematic threshold ($z \to 1$). These logarithms spoil the convergence of the perturbative series, making fixed-order predictions (even at NLO) unreliable, especially in the high invariant mass region.
• Gap: While soft-virtual (SV) resummation is well-established, the incorporation of next-to-soft (NSV) logarithms for the gluon fusion $ZH$ process had not been fully explored to the same extent as for Higgs or DY processes.

2. Methodology

The authors employ a QCD factorization framework to improve theoretical predictions for the $ZH$ invariant mass distribution and total cross-section.

• Factorization and Decomposition:
  The hadronic cross-section is decomposed into Parton Distribution Functions (PDFs) and a partonic coefficient function. The coefficient function is split into:

  1. Singular Part ($\Delta_{SV}$): Contains distributions like $\delta(1-z)$ and plus-distributions $[\ln^k(1-z)/(1-z)]_+$, dominant near the threshold.
  2. Regular Part ($\Delta_{REG}$): Finite terms, further decomposed into NSV terms and regular non-singular terms.

• Resummation Formalism:

  • Mellin Space: The resummation is performed in Mellin $N$-space, where the threshold limit $z \to 1$ maps to $N \to \infty$. In this space, plus-distributions become logarithms of $N$.
  • Exponentiation: The resummed cross-section is expressed as an exponential of two functions: $G_{SV}$ (resumming Soft-Virtual logarithms) and $G_{NSV}$ (resumming Next-to-Soft-Virtual logarithms).
  • Accuracy Levels: The authors calculate up to Next-to-Leading Logarithmic (NLL) accuracy for both SV and NSV terms. The NSV formalism utilizes a factorization framework involving process-dependent hard functions, mass factorization kernels, and soft functions.

• Matching:
  To avoid double-counting, the resummed results are matched with fixed-order calculations using the minimal prescription.

  • Fixed Order: The authors use a Born-improved NLO approach. They calculate the NLO $K$-factor in the Effective Field Theory (EFT) limit (where $m_t \to \infty$) and multiply it by the exact top-mass dependent Leading Order (LO) result. This captures the correct kinematic shape of the invariant mass distribution while maintaining computational feasibility.
  • Combination: The final result combines the resummed gluon fusion contribution with the DY-type contribution (using N3LO+N3LL results from literature) to provide a complete $pp \to ZH$ prediction at $\mathcal{O}(\alpha_s^3)$.

• Parameters:

  • Center-of-mass energy: $\sqrt{S} = 13.6$ TeV.
  • PDFs: PDF4LHC21_40.
  • Masses: $M_H = 125.2$ GeV, $m_t = 172.57$ GeV (pole mass).

3. Key Contributions

1. First NSV Resummation for Gluon Fusion $ZH$: The paper extends the resummation formalism to include Next-to-Soft (NSV) logarithms specifically for the gluon-initiated $ZH$ subprocess, a significant step beyond standard SV resummation.
2. Born-Improved NLO Implementation: The authors propose and utilize a Born-improved NLO scheme that balances the complexity of full top-mass dependence at higher orders with the need for accurate kinematic shapes in the invariant mass distribution.
3. Comprehensive Uncertainty Analysis: A detailed breakdown of theoretical uncertainties (scale and PDF) is provided for both SV and NSV resummed results, comparing them against fixed-order predictions.
4. Complete $\mathcal{O}(\alpha_s^3)$ Prediction: The work provides a unified prediction for $pp \to ZH$ by combining the improved gluon fusion channel (with resummation) and the high-precision DY channel.

4. Key Results

• Perturbative Convergence:

  • The fixed-order NLO corrections are large, contributing up to 100% of the LO cross-section in the gluon fusion channel.
  • SV Resummation: At NLL accuracy, SV resummation adds an additional 19.4% correction over NLO in the low-$Q$ region.
  • NSV Resummation: Including NSV effects provides a further 35.3% correction over NLO in the low-$Q$ region.
  • In the high invariant mass region ($Q \approx 3000$ GeV), the NLO+NLL corrections enhance the LO result by a factor of ~2.8.

• Scale Uncertainty Reduction:

  • Fixed-order NLO scale uncertainty is approximately 20%.
  • SV resummation (NLO+NLL) reduces this to ~15%.
  • NSV resummation provides a marginal further improvement, reducing uncertainties by a few percent (specifically 5.0% for SV and 1.4% for NSV at high $Q$).

• PDF Uncertainties:

  • Both SV and NSV resummations show marginal improvements in PDF uncertainties (approx. 0.8%) compared to fixed-order NLO.
  • PDF uncertainty increases with invariant mass ($Q$), while theoretical (scale) uncertainty is largest near the top-quark threshold and decreases at higher masses.

• EFT vs. Exact Theory:

  • While the total cross-section difference between EFT and Exact (full top-mass) theories is ~30%, the difference in the invariant mass distribution is significant only near and above the top-quark threshold. Below the threshold, the theories are comparable.

5. Significance

• Precision Physics: This work provides the most precise theoretical predictions for the gluon fusion contribution to $ZH$ production to date. This is essential for interpreting upcoming high-luminosity LHC data from ATLAS and CMS.
• BSM Sensitivity: By reducing theoretical uncertainties (particularly scale uncertainties) through NSV resummation, the paper enhances the sensitivity to deviations from the SM, allowing for tighter constraints on the top-quark Yukawa coupling and potential BSM scenarios.
• Methodological Advancement: The successful application of NSV resummation to a process involving massive quark loops (top quark) and a vector boson validates the universality of the NSV formalism and paves the way for similar calculations in other complex processes.
• Experimental Utility: The provided invariant mass distributions and uncertainty bands serve as direct inputs for experimental analyses aiming to measure $ZH$ production rates and kinematic properties with high precision.