⚛️ phenomenology

Soft gluon resummation for gluon fusion $ZH$ production

This paper presents a comprehensive phenomenological analysis of soft and next-to-soft gluon resummation effects on the gluon fusion $ZH$ production process at the 13.6 TeV LHC, providing next-to-leading logarithmic predictions for the total cross-section and invariant mass distribution to aid future experimental comparisons.

Original authors: Goutam Das, Chinmoy Dey, M. C. Kumar, Kajal Samanta

Published 2026-01-28

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Goutam Das, Chinmoy Dey, M. C. Kumar, Kajal Samanta

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the Large Hadron Collider (LHC) as a massive, high-speed particle racetrack. Scientists smash protons together to create new particles, hoping to catch a glimpse of the Higgs boson, the particle that gives other particles their mass. One specific race they are watching is the "ZH event," where a Higgs boson is created right next to a Z boson (another heavy particle).

For a long time, physicists thought they understood this race perfectly. They knew that most of the time, the Higgs and Z bosons are born when two quarks (tiny building blocks inside the proton) collide and annihilate each other. This is like two cars crashing head-on to create a new object.

However, this paper reveals that there is a "ghostly" side of the race that was previously being ignored or only roughly estimated.

The "Ghostly" Gluon Fusion

Inside the proton, there are also particles called gluons (the glue holding quarks together). Sometimes, two gluons collide to create the Higgs and Z boson. This is called "gluon fusion."

Think of the quark collision as a loud, obvious explosion. The gluon collision is like a quiet, invisible whisper. For a long time, scientists thought this whisper was too faint to matter. But this paper shows that at the high energies of the LHC, this whisper is actually quite loud—it contributes about 20% of the total events in certain energy ranges. Ignoring it would be like trying to count the total number of people at a concert but forgetting to count the people standing in the back row.

The Problem: The "Soft" Noise

The authors focus on a specific problem with calculating these gluon collisions. When these particles interact, they often emit "soft gluons." Imagine you are trying to hear a specific conversation in a noisy room. The "soft gluons" are like the background hum, the rustling of clothes, and the distant chatter.

In physics calculations, this background noise creates huge mathematical errors, especially when the particles are moving slowly (near the "threshold"). It's like trying to measure the exact weight of a feather while a wind tunnel is blowing on it. The standard calculations (called "fixed-order") get messy and unreliable because they try to count every single gust of wind individually, leading to huge uncertainties.

The Solution: Resummation (Tuning the Radio)

To fix this, the authors used a technique called resummation.

Imagine you are listening to a radio station with static.

Standard Calculation: You try to write down every single crackle and pop of the static. It's impossible to get a clear signal, and your notes are full of errors.
Resummation: Instead of counting every pop, you tune the radio to filter out the static and focus on the music. You group all the "soft" background noise together and treat it as a single, predictable effect.

The paper uses advanced mathematical tools (like "cusp anomalous dimensions," which are essentially universal rules for how these particles behave) to "tune the radio." They calculated the effect of this background noise not just once, but to a very high level of precision (called "Next-to-Leading Logarithmic" accuracy).

What They Found

The Whisper is Loud: When they applied this "tuning" to the gluon fusion process, they found the total number of Higgs-Z events increased significantly. The "soft" noise actually adds a massive amount of weight to the prediction.
Better Precision: By including this noise properly, the uncertainty in their predictions dropped. Before, they were guessing the result with a margin of error of about 20%. After "tuning the radio," the uncertainty dropped to around 15% or less in many cases.
The Shape of the Race: They didn't just count the total number of events; they looked at how the energy was distributed. They found that the "soft" noise changes the shape of the distribution, especially at high energies. It's like realizing that the crowd isn't just standing still; they are swaying in a specific pattern that changes the overall vibe of the concert.

The Big Picture

The authors combined their new, more accurate calculation for the "gluon whisper" with the existing, very accurate calculations for the "quark crash."

The result is a complete, high-definition map of the ZH production process at the LHC. They provide a "total score" (the total cross-section) and a detailed breakdown of how the energy is distributed.

Why does this matter?
The paper claims that by providing these precise numbers, experimentalists at the LHC (like the ATLAS and CMS teams) can now compare their real-world data against a much sharper theoretical target. If the real data doesn't match this new, precise prediction, it might be a sign of "New Physics"—something beyond our current understanding of the universe. But if it does match, it confirms our current theories are solid.

In short, this paper took a messy, noisy calculation involving invisible particles, cleaned up the static, and gave physicists a much clearer picture of how the Higgs boson is born alongside the Z boson.

Technical Summary: Soft Gluon Resummation for Gluon Fusion $ZH$ Production

Problem Statement
The associated production of a Higgs boson ( $H$ ) and a $Z$ boson ($ZH$) at the Large Hadron Collider (LHC) is a critical channel for probing Higgs couplings to weak gauge bosons and constraining Beyond the Standard Model (BSM) scenarios. While the dominant production mechanism is the Drell-Yan (DY) type quark-antiquark annihilation ( $q\bar{q} \to Z^* \to ZH$ ), which has been computed up to next-to-next-to-next-to-leading order (N $^3$ LO), the gluon fusion channel ( $gg \to ZH$ ) becomes phenomenologically significant starting at next-to-next-to-leading order (N $^2$ LO).

Although the gluon fusion subprocess is suppressed by two powers of the strong coupling constant ( $\alpha_s^2$ ) relative to the DY channel, it is compensated by the large gluon luminosity at the LHC. Previous calculations indicate that the gluon fusion contribution yields a correction of approximately 7% relative to the NLO DY contribution at leading order (LO), with scale uncertainties around 25%. Furthermore, fixed-order calculations for this loop-induced process suffer from large threshold logarithms arising from soft gluon emissions, which degrade the reliability of predictions, particularly in the high invariant mass region. While soft gluon resummation has been established for the DY channel and the Higgs production process, a comprehensive analysis of soft and next-to-soft (NSV) gluon resummation specifically for the gluon fusion $ZH$ process, including the invariant mass distribution, was lacking.

Methodology
The authors employ a theoretical framework based on QCD factorization to analyze the $ZH$ production cross section. The hadronic cross section is decomposed into parton distribution functions (PDFs) and partonic coefficient functions, which are separated into soft-virtual (SV) and regular (REG) components. The SV component captures the leading singular terms in the threshold limit ( $z \to 1$ , where $z = Q^2/\hat{s}$ ), while the REG component includes subleading contributions.

Key methodological steps include:

Resummation Formalism: The authors utilize a formalism in Mellin $N$ -space to resum large logarithms. They incorporate both SV and NSV resummation up to next-to-leading logarithmic (NLL) accuracy. The exponents $G_{N}^{SV}$ and $G_{N}^{NSV}$ are constructed using universal cusp anomalous dimensions and splitting kernels.
Universality and Approximation: To handle the complexity of the full top-quark mass dependence in the NLO corrections, the authors adopt a "Born-improved" approach. They use the exact LO result (which includes full top-quark mass dependence via triangle and box diagrams) and rescale it by the NLO K-factor calculated in the infinite top-mass limit (Effective Field Theory, EFT). This approach is justified by the observation that the exact LO captures the bulk shape of the invariant mass distribution, and the NLO correction's impact on the shape is minimal compared to the overall normalization.
Matching: The resummed results are matched with fixed-order calculations (NLO for the gluon channel, N $^3$ LO for the DY channel) using the minimal prescription to avoid double-counting logarithmic terms.
Phenomenological Setup: Numerical analyses are performed for proton-proton collisions at $\sqrt{S} = 13.6$ TeV, utilizing the PDF4LHC21_40 set. The renormalization and factorization scales are set to the invariant mass of the $ZH$ system ( $Q$ ).

Key Contributions

First NLL Resummation for Gluon Fusion $ZH$: The paper provides the first calculation of soft and next-to-soft gluon resummation for the gluon fusion $ZH$ subprocess at the NLL level, covering both the total cross section and the differential invariant mass distribution.
NSV Effects: The authors extend the analysis to include next-to-soft (NSV) threshold effects, estimating their impact on the cross section and distributions at the same level of accuracy.
Comprehensive Predictions: The study combines the resummed gluon fusion results with the state-of-the-art N $^3$ LO+N $^3$ LL results for the DY channel and other sub-percent contributions (bottom quark annihilation, top-quark loop radiation) to provide complete predictions for $pp \to ZH$ at $\mathcal{O}(\alpha_s^3)$ .
Validation: The authors validate their in-house resummation code by reproducing known results for the total resummed cross section of $ZH$ production and inclusive Higgs production, ensuring consistency with existing literature.

Results

Perturbative Convergence: The inclusion of SV resummation significantly improves perturbative convergence. The NLO correction to the gluon fusion channel is found to be approximately 100% of the LO prediction. The addition of NLL resummation further enhances the cross section, reaching roughly 2.8 times the LO value in the high invariant mass region ( $Q \approx 3000$ GeV).
NSV Impact: The NSV resummation contributes an additional 12–15% correction over the NLO+NLL result across most of the invariant mass range.
Uncertainty Reduction:
- Scale Uncertainty: In the high invariant mass region, SV resummation reduces the seven-point scale uncertainty at NLO by a few percent (from ~20% to ~15%). However, NSV resummation does not show a similar reduction in the low invariant mass region.
- PDF Uncertainty: PDF uncertainties increase with $Q$ , reaching ~~10% in the high- $Q$ region due to poor constraints at large- $x$ . Resummation offers only marginal improvements (~~0.8%) in PDF uncertainties compared to fixed-order results.
Total Cross Section: For the total $ZH$ production cross section at 13.6 TeV, the inclusion of resummation effects increases the prediction by approximately 2.7% compared to the fixed-order result. The total scale uncertainty for the combined process at N $^3$ LO+N $^3$ LL is found to be around 2.98%.

Significance and Claims
The paper claims that its primary significance lies in providing comprehensive, high-precision theoretical predictions for the $ZH$ production process that are essential for comparing with upcoming experimental data from the LHC and future hadron colliders. By incorporating soft and next-to-soft gluon resummation, the authors demonstrate that the gluon fusion channel, while sub-dominant, makes a non-negligible contribution (around 20% of the DY channel near the top-pair threshold) and requires higher-order corrections for reliable phenomenological analysis.

The authors modestly note that while the theoretical uncertainties in the high invariant mass region are well-controlled, further precision requires higher-order corrections beyond NLO for the gluon fusion channel, as the current scale uncertainties remain the dominant source of error. The work serves as a necessary step toward reducing theoretical errors to match the precision of experimental measurements for Higgs coupling studies.