Associated $ZH$ production in gluon fusion process at NLO+NLL

This paper presents precise NLO+NLL QCD calculations for associated $ZH$ production via gluon fusion at the LHC, demonstrating that threshold resummation enhances the cross-section by approximately 20% and significantly reduces scale uncertainties compared to fixed-order NLO results, while also combining these findings with Drell-Yan type calculations to achieve the most accurate predictions for hadron collisions.

The Gist

Imagine the Large Hadron Collider (LHC) as a massive, high-speed particle racetrack where protons zoom around and smash into each other. When they collide, they sometimes create a "Higgs boson" (a particle that gives other particles mass) alongside a "Z boson" (a carrier of the weak force). This specific event is called associated ZH production.

For a long time, physicists have been able to predict how often this happens using a set of rules called Quantum Chromodynamics (QCD). However, there are two main ways these particles can be created:

1. The "Quark" Way: Two quarks (inside the protons) smash together. This is the main, well-understood path.
2. The "Gluon" Way: Two gluons (the glue holding quarks together) smash together. This path is trickier because it involves a complex loop of heavy particles (top quarks) that acts like a hidden bridge.

The Problem: The "Fuzzy" Prediction

Think of the "Gluon Way" as trying to predict the weather in a stormy region. The standard predictions (called NLO, or Next-to-Leading Order) are okay, but they have a big "fog" around them. This fog represents uncertainty.

In this paper, the authors say that for the Gluon Way, the uncertainty in their predictions was about 20%. That's like a weather forecaster saying, "It will rain between 20% and 40% of the time." That's not very helpful if you're trying to build a house!

The Solution: Adding "Resummation"

The authors decided to clean up this fog. They used a mathematical technique called Resummation (specifically NLO+NLL).

The Analogy:
Imagine you are listening to a radio station that is full of static (noise).

• The Old Method (NLO): You turn up the volume to hear the music, but the static gets louder too. You can hear the song, but you aren't sure if that crackle is part of the music or just interference.
• The New Method (NLO+NLL): You put on a pair of noise-canceling headphones. You still hear the music, but the static is significantly reduced. You can now hear the details much more clearly.

In physics terms, the "static" is the threshold logarithms—mathematical terms that get huge and messy when the particles are moving at specific speeds. The authors calculated these messy terms and added them to their prediction, effectively "canceling out" the noise.

What They Found

The paper presents two major discoveries:

1. The "Exact" Top Quark Mass Matters:
   Previous studies often approximated the heavy top quark as being infinitely heavy to make the math easier. The authors did the hard work of calculating the exact mass of the top quark.

   • The Result: Near a specific energy level (where the energy equals twice the mass of a top quark), the old "approximate" math was wrong. It missed a peak in the data. The new, exact math shows a sharp spike in production that the old math smoothed over.

2. The Numbers Got Better (and Bigger):

   • More Production: When they added the "noise-canceling" math, the total predicted number of ZH events went up by about 20% at the LHC's current energy. It turns out the Gluon Way happens more often than the old, fuzzy math suggested.
   • Less Fog (Uncertainty): While the total number went up, the "fog" (uncertainty) got smaller.
     • At high energies (3000 GeV), the uncertainty dropped from 20% down to 12%.
     • This means physicists can now trust their predictions much more when looking for new physics or measuring the properties of the Higgs boson.

The "Z-Radiated" Surprise

The authors also looked at a specific type of diagram where the Z boson is "radiated" from a loop of particles. They found that these specific diagrams act like a turbo-boost. At very high energies, these diagrams make the production rate jump significantly higher than expected, creating a massive "K-factor" (a ratio showing how much the prediction changes).

The Final Picture

The authors combined their new, precise "Gluon Way" calculations with the existing, highly precise "Quark Way" calculations.

• The Result: They now have the most precise map ever created for ZH production in proton collisions.
• Why it matters: By reducing the uncertainty from 20% to 12%, they have cleared the fog. This allows experimentalists at the LHC to look for tiny deviations in the data that might signal new, undiscovered physics, rather than just seeing the "fog" of calculation errors.

In short: The authors took a messy, uncertain prediction about how particles collide, added a sophisticated mathematical filter to clean up the noise, and found that the collision happens more often and is much more predictable than we thought, especially when accounting for the exact weight of the heavy top quark.

Technical Summary

Technical Summary: Associated ZH Production in Gluon Fusion at NLO+NLL

Problem Statement
The precise measurement of the Higgs boson's Yukawa coupling to bottom quarks ($y_b$) is critical for understanding fermion mass origins and constraining new physics. The associated production of a Higgs boson with a Z boson ($ZH$), where the Higgs decays to $b\bar{b}$ and the Z to leptons, provides a clean signal due to the suppression of QCD backgrounds. While the quark-antiquark ($q\bar{q}$) initiated Drell-Yan-like process is well-understood up to N$^3$LO+N$^3$LL, the gluon-fusion ($gg \to ZH$) channel remains a significant source of theoretical uncertainty.

At Leading Order (LO), the $gg \to ZH$ process is loop-induced and non-Drell-Yan in nature. Previous studies often relied on the Born-improved Effective Field Theory (EFT) approach, which assumes infinite top-quark mass ($m_t \to \infty$) and vanishing bottom-quark mass. This approximation fails to capture exact top-mass effects, particularly in the invariant mass region near $Q \approx 2m_t$, and exhibits large unphysical scale uncertainties (approximately 25% at LO). Furthermore, while Next-to-Leading Order (NLO) corrections with exact mass dependence have been studied, the impact of threshold resummation (NLL) on these exact mass-dependent results had not been fully explored. The lack of precise predictions for the $gg \to ZH$ channel limits the ability to extract the $Zb\bar{b}$ coupling and distinguish new physics scenarios.

Methodology
The authors present a calculation of the $pp \to ZH$ process at Next-to-Leading Order (NLO) matched with Next-to-Leading Logarithmic (NLL) threshold resummation accuracy. The methodology involves several key components:

1. Exact Mass Dependence: Unlike previous EFT-based studies, this work retains the exact top-quark mass dependence in the virtual amplitudes. The two-loop virtual corrections are taken from the ggxy package, while real-emission amplitudes are computed using a combination of Recola2 (for loop-induced processes) and MadGraph5_aMC@NLO (for tree-level processes), with specific filters applied to isolate the $gg \to ZH$ channel and remove Drell-Yan-like contributions.
2. Subtraction and Validation: The NLO cross-sections are computed using an in-house implementation of the Catani-Seymour dipole subtraction method. The authors validate their code by comparing total cross-sections and invariant mass distributions against the public ggxy package, finding agreement at the per-mille level.
3. Threshold Resummation: Large soft-virtual (SV) logarithms arising near the production threshold ($z \to 1$) are resummed to NLL accuracy in Mellin space. The resummed exponent $G_N$ includes universal coefficients dependent on the initiating partons (gluons) and process-dependent coefficients ($g_0$).
4. Matching: The resummed results are matched to the full fixed-order NLO results using the minimal prescription to avoid double-counting. The final prediction combines the resummed gluon-fusion contribution with the Drell-Yan type contribution (calculated at N$^3$LO+N$^3$LL) to provide a complete picture of $pp \to ZH$.
5. Scale Variation: Uncertainties are estimated using the standard 7-point scale variation method, varying renormalization ($\mu_R$) and factorization ($\mu_F$) scales within $[Q/2, 2Q]$.

Key Contributions and Results

• First NLO+NLL with Exact Mass: This work provides the first NLO+NLL predictions for the $gg \to ZH$ channel that include exact top-quark mass dependence in the virtual amplitudes.
• Cross-Section Enhancement: At the LHC energy of 13.6 TeV, the inclusion of NLO+NLL resummation increases the inclusive cross-section by approximately 20% compared to the fixed-order NLO result. Specifically, the total cross-section increases by ~21% over the fixed-order result.
• Differential Distributions and Scale Uncertainty:
  • The invariant mass distribution ($d\sigma/dQ$) shows significant differences between the exact NLO and Born-improved EFT approaches, particularly around $Q \approx 2m_t$, where the Born-improved approach fails to capture the peak structure.
  • For differential distributions at high invariant masses ($Q = 3000$ GeV), the unphysical scale uncertainty for the fixed-order NLO result is approximately 20%. The resummed (NLO+NLL) result reduces this uncertainty to roughly 12%.
  • The reduction in uncertainty is primarily driven by an improved dependence on the renormalization scale ($\mu_R$) after resummation, although the factorization scale ($\mu_F$) uncertainty shows a slight enhancement in some regions.
• Z-Radiated Diagrams: The study analyzes "Z-radiated" diagrams (where the Z boson is emitted from an external fermion line). These diagrams contribute significantly to the K-factor in the high-$Q$ region, reaching values as high as 7.74 at NLO, which are further enhanced to 8.42 after resummation.
• Combined Precision: By combining the N$^3$LO+N$^3$LL Drell-Yan results with the new NLO+NLL gluon-fusion results, the authors present the most precise QCD prediction for the $ZH$ invariant mass distribution currently available.

Significance
The paper claims that these results represent a necessary step toward reducing theoretical uncertainties in Higgs physics, specifically for the $gg \to ZH$ channel which contributes about 6% to the total cross-section at 13 TeV. The authors emphasize that the large scale uncertainties in LO and approximate NLO calculations often fail to capture the size of higher-order corrections, making the NLO+NLL calculation with exact mass dependence essential for precision phenomenology.

The work highlights that while the Born-improved EFT approach is useful, it misses critical top-mass effects near the $2m_t$ threshold. By providing exact results, the authors enable more accurate constraints on the $Zb\bar{b}$ coupling and offer a robust theoretical baseline for experimental analyses at the LHC, where systematic uncertainties in $ZH$ measurements are currently larger than in $WH$ measurements due to the gluon-fusion contribution. The study concludes that combining fixed-order and resummed results across different channels yields the state-of-the-art precision for $ZH$ production in hadron collisions.