Top-associated Higgs-boson production using perturbative fragmentation functions at next-to-leading-order

This paper demonstrates that the perturbative fragmentation function approximation for top-associated Higgs-boson production yields reliable results at LHC energies using a hybrid top-quark mass prescription, while the zero-mass prescription remains viable only for specific channels at LHC energies or for the full process at a 100 TeV collider, alongside a discussion of challenges in extending the formalism to NNLO.

The Gist

Imagine you are trying to predict how a specific, heavy object (a Higgs boson) behaves when it's created alongside two other very heavy objects (top quarks) in a massive particle smash-up, like the ones happening at the Large Hadron Collider (LHC).

This paper is essentially a "quality control" report for a specific mathematical shortcut physicists use to make these predictions easier.

The Problem: The "Heavy Suit" vs. The "Speeding Bullet"

In the world of particle physics, the top quark is incredibly heavy (like a bowling ball), while the Higgs boson is a bit lighter but still substantial. When these particles are created, they usually move slowly. However, sometimes, due to the chaos of the collision, the Higgs boson gets kicked out with enormous speed (high energy).

When the Higgs moves this fast, it often flies out almost in a straight line, right next to one of the top quarks. It's like a bowling ball (the top) throwing a smaller ball (the Higgs) so hard that they are practically glued together in their direction of travel.

Calculating exactly how this happens with all the heavy weights included is like trying to solve a 10,000-piece puzzle where every piece is made of lead. It's accurate, but it takes a supercomputer an eternity to finish.

The Shortcut: The "Fragmentation" Trick

To save time, physicists use a "shortcut" called perturbative fragmentation functions.

Think of this like a two-step process:

1. The Zero-Weight Assumption: First, they pretend the top quark has no weight at all (it's a feather). This makes the math incredibly fast and easy.
2. The "Re-Weighting": Then, they apply a correction factor (the fragmentation function) to account for the fact that the top quark is actually heavy.

The paper asks: "Does this shortcut work well enough to trust the results?"

The Two Methods Tested

The authors tested two different ways of applying this shortcut:

1. The "Zero-Mass" Approach (ZMTQ)
This is the pure shortcut. You pretend the top quark is weightless, do the math, and apply the correction.

• The Result: At the current LHC energy (13 TeV), this method is a bit shaky. It works okay for some types of collisions (quark-antiquark), but it fails to predict the "tail" of the distribution (the very high-speed Higgs events) accurately in other types (gluon-gluon collisions). It's like using a map designed for a bicycle to drive a truck; it works on the main road, but you'll get lost on the rough terrain.
• The Future: However, the authors found that if we build a much bigger collider in the future (100 TeV), this shortcut becomes very reliable. The higher the energy, the better the shortcut works.

2. The "Hybrid" Approach
This is a smarter version of the shortcut. It takes the "Zero-Mass" calculation but mixes in the exact heavy-weight math for the parts where the Higgs isn't moving as fast. It's like using a bicycle map for the flat parts of the road but switching to a truck manual for the steep hills.

• The Result: This method works beautifully, even at current LHC energies. It captures the heavy weight of the top quark correctly while still keeping the calculation fast. The authors found that the errors in this method are tiny (around 1-2%), making it a reliable tool for now.

The "Double-Counting" Trap

The paper also discusses a tricky logical problem. If you require a specific outcome (like "we must see a top quark pair"), you have to be careful not to count the same event twice.
Imagine you are counting cars in a parking lot. If you count "red cars" and then count "cars with a specific license plate," you might accidentally count a "red car with that license plate" twice. The authors had to write a very careful set of rules to ensure they didn't double-count these rare, high-speed events. They found that while this gets complicated at very high levels of precision (beyond what they tested), for the current level of accuracy, the rules are manageable.

The Bottom Line

• The Shortcut Works: The mathematical framework used to simplify these heavy-particle calculations is valid.
• Pick the Right Tool: If you are working at current LHC energies, you must use the "Hybrid" method (mixing exact and approximate math) to get reliable results. The pure "Zero-Mass" shortcut is too inaccurate for current energies but will be perfect for future, higher-energy colliders.
• Why it Matters: This gives physicists confidence that they can use these faster calculations to study the Higgs boson and look for new physics without waiting for supercomputers to run the full, heavy-weight calculations for every single scenario.

In short: The paper proves that with the right "mix-and-match" strategy, we can predict how heavy particles behave at high speeds without needing a supercomputer to solve the impossible puzzle every time.

Technical Summary

Technical Summary: Top-associated Higgs-boson production using perturbative fragmentation functions at next-to-leading-order

Problem Statement
The production of a Higgs boson in association with a top-antitop pair ($t\bar{t}H$) is a critical process for measuring the top Yukawa coupling and searching for Beyond the Standard Model (BSM) physics. While fixed-order calculations exist up to Next-to-Next-to-Leading Order (NNLO) in various approximations (e.g., soft-Higgs approximation), the high-transverse-momentum ($p_{T,H}$) tail of the Higgs spectrum presents a specific challenge. In this kinematic regime, the Higgs boson can become quasi-collinear to one of the top quarks, generating large logarithms of the form $\ln(m_t/p_{T,H})$. These logarithms can spoil the perturbative convergence of the fixed-order expansion.

While perturbative fragmentation functions (PFFs) provide a well-established framework to resum such logarithms for heavy-quark production, their applicability to $t\bar{t}H$ production had not been rigorously demonstrated in a realistic setup beyond Leading Order (LO). A specific complication arises because the factorisation theorem for single-inclusive Higgs production assumes a massless top quark, whereas the physical process requires a massive top pair in the final state. The authors investigate under which conditions the PFF formalism can reliably describe $t\bar{t}H$ production at Next-to-Leading Order (NLO) in QCD.

Methodology
The authors employ the perturbative fragmentation formalism, which factorises the cross section into a hard scattering process (calculated with massless partons) and fragmentation functions describing the collinear emission of the Higgs from a top quark. The study evaluates two distinct prescriptions for handling the top-quark mass ($m_t$):

1. Zero-Mass Top-Quark (ZMTQ) Prescription: This approach sets $m_t = 0$ everywhere in the calculation, relying entirely on the factorisation theorem to capture mass effects via fragmentation functions and matching coefficients. The authors compute the "direct" contribution (where the Higgs is produced in the hard process) by approximating the massless limit using a small, non-zero top mass ($m_t^{small} = 10$ GeV) to minimize power corrections while maintaining numerical stability.
2. Hybrid Prescription: This method restores the exact $m_t$ dependence for the case where the Higgs mass $m_H = 0$ in the direct contribution. It effectively adds the leading power corrections in $m_t$ by taking the difference between the exact massive calculation (at $m_H=0$) and its factorised approximation. This isolates the impact of $m_t$ power corrections from those of $m_H$.

To handle the technical challenges of calculating massless $t\bar{t}H$ production, where the 3-particle final state introduces NLO-like singularities at LO and NNLO-like singularities at NLO, the authors utilize the STRIPPER subtraction scheme. They extended this framework to accommodate massless scalar particles (the Higgs). For the LO calculation, they developed a semi-analytic approach for the $q\bar{q}$ channel and a customised numerical subtraction scheme for the $gg$ channel to avoid "missed-binning" effects common in scalar subtraction schemes. For the NLO calculation, they relied on the factorisation theorem to approximate the fully massless direct contribution, as extending STRIPPER to NNLO with scalar sectors was not pursued.

Key Contributions

• NLO Verification of PFFs: This work provides the first demonstration of the perturbative fragmentation formalism for $t\bar{t}H$ production at NLO.
• Implementation of Scalar Subtraction: The authors implemented a scalar subtraction scheme within STRIPPER to handle massless Higgs bosons in the presence of massless top quarks, addressing the specific singularity structures of the $t\bar{t}H$ process.
• Comparison of Prescriptions: The paper systematically compares the ZMTQ and Hybrid prescriptions against fully massive calculations at NLO, quantifying the size of neglected power corrections.
• Kinematic Limits Analysis: The study analyzes the validity of the approximation across different center-of-mass energies (13 TeV and 100 TeV) and different partonic channels ($q\bar{q}$ and $gg$).

Results

• ZMTQ Prescription:
  • At LHC energies (13 TeV), the ZMTQ approximation is reliable only in the $q\bar{q}$ channel (agreement within 1-2% in the bulk spectrum). In the $gg$ channel, the approximation fails significantly (discrepancies up to 30%) due to large power corrections.
  • At a future 100 TeV collider, the ZMTQ prescription becomes viable for the full $pp \to t\bar{t}H$ process, with power corrections becoming small enough to yield reliable results in both channels.
  • At NLO, the numerical uncertainties in the ZMTQ approach are dominated by large cancellations between the direct massless calculation (using $m_t = 10$ GeV) and fragmentation contributions, leading to Monte Carlo errors of 10-20%.
• Hybrid Prescription:
  • The hybrid prescription, which includes leading $m_t$ power corrections, yields reliable results at both 13 TeV and 100 TeV.
  • In the hybrid approach, the approximation reproduces the fully massive calculation down to the percent level (1-2%) across a wide range of the $p_{T,H}$ spectrum in both the $q\bar{q}$ and $gg$ channels at NLO.
  • The results indicate that the dominant power corrections in the general factorisation theorem arise from top-quark mass effects, while Higgs mass power corrections are less pronounced.

Significance
The paper concludes that while the standard zero-mass factorisation theorem is insufficient for precise $t\bar{t}H$ predictions at LHC energies (particularly in the $gg$ channel), the hybrid prescription offers a robust and reliable method for describing the high-$p_T$ tail of the Higgs spectrum at NLO. This validates the use of perturbative fragmentation functions for resumming large logarithms in this process, provided that the leading top-mass power corrections are accounted for. The work establishes a foundation for future resummed predictions of $t\bar{t}H$ production, which are essential for precision measurements of the top Yukawa coupling at high transverse momenta. The authors also note that extending this formalism to NNLO introduces significant complications regarding soft-collinear emissions and colour coherence, which remain open challenges.