Modelling $b\bar b H$ production for the LHC at 13.6 TeV

This paper presents state-of-the-art Standard Model predictions for $b\bar bH$ production at the LHC's 13.6 TeV energy, providing updated cross sections and matched simulations in both five-flavour and four-flavour schemes while exploring their implications for Beyond-the-Standard-Model scenarios and Higgs coupling sensitivities.

The Gist

Imagine the Large Hadron Collider (LHC) as a giant, high-speed particle smasher. Its main job is to smash protons together to create new particles, most notably the Higgs boson, which is like the universe's "glue" that gives other particles mass.

For a long time, scientists have known how the Higgs is usually made: mostly by smashing two gluons (particles that hold protons together) into a loop of heavy top quarks. But there's another way the Higgs can be made, and this paper is all about understanding that specific, trickier method.

Here is a simple breakdown of what this paper does, using everyday analogies:

1. The Two Ways to Make a Higgs with Bottom Quarks

The paper focuses on a process called $b\bar{b}H$, where a Higgs boson is produced alongside a pair of bottom quarks (heavy cousins of the electron). There are two main ways nature does this, and the paper tries to figure out exactly how much of each happens:

• The "Tree-Level" Way ($y_b^2$): Imagine the Higgs is a ball being thrown. In this scenario, the Higgs is "radiated" off a bottom quark, kind of like a ball bouncing off a bat. This depends entirely on how strongly the Higgs talks to the bottom quark (the "bottom Yukawa coupling").
• The "Loop" Way ($y_t^2$): This is more like a magic trick. Two gluons smash together, create a temporary loop of heavy top quarks, and then spit out a Higgs and a pair of bottom quarks. Even though the bottom quarks are the ones we see at the end, the heavy top quark in the middle is doing the heavy lifting.

The Paper's Finding: In the Standard Model (our current best theory of physics), the "Loop" way (involving the top quark) is actually about twice as common as the "Tree" way (involving the bottom quark). This makes it very hard to measure the bottom quark's specific interaction because the top quark's contribution is hiding in the background.

2. The "Map" Problem: Two Different Schemes

To calculate these probabilities, physicists use two different "maps" or mathematical frameworks:

• The 5-Flavour Scheme (5FS): This treats bottom quarks as if they are massless and always present inside the proton (like a permanent resident). It's great for high-energy collisions but ignores the fact that bottom quarks have mass.
• The 4-Flavour Scheme (4FS): This treats bottom quarks as heavy particles that are created during the collision (like a guest arriving at a party). It accounts for their mass but misses some high-energy details.

The Old Problem: For years, these two maps gave different answers (discrepancies of 20–30%), leaving scientists confused about which one was right.
The New Solution: This paper presents brand-new, ultra-precise calculations (up to "NNLO" accuracy, which is like calculating a recipe with extreme precision) for both maps. They found that when you use this high level of precision, the two maps finally agree. The confusion is resolved.

3. The "Traffic Jam" of Particles (Parton Showers)

When particles smash, they don't just fly apart; they shower a cascade of other particles, like a traffic jam of debris. To simulate this, scientists use "Parton Showers."

• The paper compares two advanced computer programs, MiNNLOPS and Geneva, which act like different traffic simulators.
• They found that while the two programs use different logic to handle the traffic, they produce very similar results for the Higgs' speed and direction. This gives experimentalists (the people building the detectors) confidence that their simulations are reliable.

4. Looking for "New Physics" (BSM)

The paper also tested how these tools would work if the universe were slightly different (Beyond the Standard Model).

• Analogy: Imagine the bottom quark's "voice" (its interaction strength) gets much louder in a different universe.
• Result: The MiNNLOPS program was successfully adapted to simulate this scenario. It showed that if the bottom quark's interaction is enhanced, the Higgs production changes dramatically. This proves the tools are ready to help scientists hunt for new, exotic particles in the future.

5. The "Background Noise" Problem

The $b\bar{b}H$ process is a major "background noise" when scientists are trying to find Di-Higgs events (where two Higgs bosons are made at once).

• Analogy: If you are trying to hear a whisper (two Higgs bosons) in a noisy room, the $b\bar{b}H$ process is like someone constantly shouting in the background.
• The Paper's Contribution: By providing much more accurate calculations of this "shouting," the paper helps experimentalists subtract the noise more effectively, making it easier to hear the whisper of the double Higgs.

6. Listening to the "Whispers" of Light Quarks

Finally, the paper looked at even lighter quarks (like up, down, and charm).

• The Idea: Just as the bottom quark can make a Higgs, these lighter quarks can too, but their "voices" are incredibly faint.
• The Clue: The paper found that the speed (transverse momentum) of the Higgs boson acts like a fingerprint. Lighter quarks produce a Higgs that moves differently than heavier ones. By measuring the Higgs' speed very precisely, scientists might finally be able to "hear" these faint whispers and measure how the Higgs interacts with light quarks, which is currently a mystery.

Summary

In short, this paper is a masterclass in precision. It:

1. Fixed a long-standing disagreement between two different calculation methods.
2. Provided the most accurate "recipe" yet for how Higgs bosons are made with bottom quarks at the LHC's new energy level (13.6 TeV).
3. Created better tools to help scientists separate the "signal" (new discoveries) from the "noise" (standard background processes).
4. Showed how to use the speed of the Higgs to probe the interactions of lighter quarks.

It doesn't predict a new particle or a new technology; rather, it provides the high-definition map scientists need to navigate the LHC data and find what lies beyond our current understanding.

Technical Summary

Technical Summary: Modelling $b\bar{b}H$ Production for the LHC at 13.6 TeV

Problem Statement
The production of a Higgs boson in association with a bottom-quark pair ($b\bar{b}H$) is a critical process for the LHC, serving as a probe of the bottom Yukawa coupling ($y_b$) and a significant background for di-Higgs ($HH$) searches. Historically, theoretical predictions for this process have faced two primary challenges:

1. Flavour Scheme Discrepancy: Calculations can be performed in the Four-Flavour Scheme (4FS), where bottom quarks are massive and generated in the final state, or the Five-Flavour Scheme (5FS), where bottom quarks are treated as massless partons within the PDFs. While the 4FS captures mass effects essential for $b$-jet kinematics, it suffers from large logarithmic terms at higher orders. Conversely, the 5FS resums these logarithms but neglects power corrections in the bottom mass. Previous comparisons between NLO 4FS and NNLO 5FS predictions showed discrepancies of 20–30%, leading to significant theoretical uncertainty.
2. Missing Higher-Order Corrections: Accurate modeling of the $y_t^2$ (top-Yukawa induced) and $y_t y_b$ interference contributions, particularly in the context of $HH$ backgrounds and Beyond the Standard Model (BSM) scenarios, has been limited by the lack of Next-to-Next-to-Leading Order (NNLO) QCD corrections matched to parton showers (NNLO+PS) in the massive scheme.

Methodology
This report presents state-of-the-art predictions for $b\bar{b}H$ production at a center-of-mass energy of $\sqrt{s} = 13.6$ TeV, adhering to the LHC Higgs Working Group (LHCHWG) recommendations. The computational setup utilizes the PDF4LHC21 sets and consistent scale choices (typically of the order of the Higgs mass).

The methodology encompasses several advanced theoretical frameworks:

• Inclusive Cross Sections: Updated predictions are derived by interpolating existing NLO 4FS and NNLO 5FS matched results (using the NLO+NNLL$_{part}+y_b y_t$ approach) from 13 and 14 TeV to 13.6 TeV.
• Transverse Momentum Resummation: The Higgs transverse momentum ($p_T$) spectrum is calculated in the 5FS at N3LL' + approximate N3LO (aN3LO) accuracy using Soft-Collinear Effective Theory (SCET) to resum logarithms of $q_T/m_H$.
• Monte Carlo Simulations (NNLO+PS):
  • 5FS: A numerical comparison is performed between two independent NNLO+PS generators: MiNNLOPS and Geneva. Both methods match fixed-order NNLO calculations with parton showers but employ different philosophies (factorized Sudakov vs. additive matching with slicing).
  • 4FS: The report utilizes the MiNNLOPS framework extended to heavy-quark-associated colour-singlet production ($Q\bar{Q}F$). This allows for the first NNLO+PS predictions in the massive scheme. The two-loop virtual corrections are approximated via a small-mass expansion, retaining logarithmically enhanced and constant terms while dropping power-suppressed terms, utilizing full-colour two-loop amplitudes.
• BSM and Background Modeling: The MiNNLOPS generator is adapted for Minimal Supersymmetric Standard Model (MSSM) scenarios (specifically the $M_h^{125}$ scenario) to study heavy Higgs production. Furthermore, a consistent strategy is proposed to model the $b\bar{b}H$ background in $HH$ searches by combining the $y_t^2$ contribution (calculated at NLO+PS in 4FS) with inclusive Higgs production (from NNLOPS in 5FS, filtered to remove $b$-quarks) to avoid double counting.
• Light-Quark Contributions: The framework is extended to study Higgs production via light-quark fusion ($q\bar{q} \to H$), analyzing the sensitivity of the $p_T$ spectrum to light-quark Yukawa couplings ($\kappa_q$).

Key Contributions and Results

1. Resolution of the 4FS/5FS Discrepancy:
   The most significant result is the agreement between the 4FS and 5FS predictions at NNLO accuracy. While NLO 4FS and NNLO 5FS previously differed by 20–30%, the inclusion of NNLO corrections in the 4FS (via MiNNLOPS) and the negative NNLO corrections in the 5FS bring the two schemes into agreement within 10% or better for the inclusive cross section. This resolves the long-standing tension regarding the appropriate flavour scheme for $b\bar{b}H$.

2. NNLO+PS Generator Comparison:
   The numerical comparison between MiNNLOPS and Geneva in the 5FS shows excellent agreement for inclusive observables and the Higgs rapidity distribution. For the transverse momentum spectrum, differences are observed in the transition region ($p_T \sim 30\text{--}90$ GeV) depending on the scale choice in Geneva (cumulant vs. spectrum). When the spectrum scale choice is applied in Geneva, the two generators agree within their respective scale uncertainties across the entire $p_T$ range. Both generators show good agreement with the analytic N3LL' resummed predictions.

3. Massive Scheme Predictions:
   The first NNLO+PS predictions in the 4FS are presented. These results demonstrate that NNLO corrections increase the inclusive cross section by approximately 30% relative to NLO in the 4FS, significantly reducing scale uncertainties. The 4FS predictions are shown to be superior for observables involving identified $b$-jets, particularly for exclusive bins with two or more $b$-jets, where the 5FS accuracy effectively degrades to LO.

4. $HH$ Background Modeling:
   In the context of $HH \to b\bar{b}\gamma\gamma$ searches, the report quantifies the $y_t^2$ and $y_b^2$ contributions. The $y_t^2$ component is found to be larger than the $y_b^2$ component in inclusive regions but becomes comparable in kinematic regions sensitive to the trilinear Higgs coupling. The study highlights that previous modeling of the $y_t^2$ background using inclusive gluon-fusion generators (NNLOPS) with $g \to b\bar{b}$ splitting leads to double counting and large uncertainties (up to 100%). The proposed 4FS combination strategy (NLO+PS $b\bar{b}H$ + filtered NNLOPS inclusive $H$) offers a more coherent description.

5. Light-Quark Yukawa Sensitivity:
   The analysis of $q\bar{q} \to H$ production reveals that the shape of the Higgs $p_T$ spectrum is highly sensitive to the initial-state quark flavour. Light-quark channels (e.g., $c\bar{c}$) exhibit softer spectra compared to the dominant gluon-fusion channel, offering a potential handle to constrain light-quark Yukawa couplings, particularly the charm coupling, which remains weakly constrained.

Significance
The paper claims that the inclusion of NNLO QCD corrections in the massive 4FS, matched to parton showers, finally resolves the discrepancy between the 4FS and 5FS schemes for $b\bar{b}H$ production. This achievement substantially reduces the theoretical uncertainty in modeling the $b\bar{b}H$ signal and background. By providing matched NNLO+PS predictions in both schemes and a consistent framework for handling the $y_t^2$ background in $HH$ searches, the work offers the necessary tools for precision measurements at the LHC Run 3 and the High-Luminosity LHC. Additionally, the extension to light-quark fusion and BSM scenarios demonstrates the versatility of the MiNNLOPS framework for future phenomenological studies.