⚛️ phenomenology

A new approach to QCD final-state evolution in processes with massive partons

This paper introduces a new algorithm implemented in the Alaric code for simulating massive parton evolution in QCD, which utilizes azimuthal-dependent splitting functions and a symmetry-preserving kinematic mapping to achieve differentially accurate soft-gluon radiation, fully differential NLO matching, and NLL-accurate parton showers validated against experimental data.

Original authors: Benoit Assi, Stefan Höche

Published 2026-09-28

📖 3 min read🧠 Deep dive

CC BY 4.0

Original authors: Benoit Assi, Stefan Höche

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

In the high-energy world of particle physics, scientists smash protons together at speeds approaching that of light to recreate the conditions of the early universe. When these collisions occur, they produce a shower of new particles, many of which are heavy versions of the fundamental building blocks of matter, such as the bottom quark. To understand what happens in these collisions, researchers rely on computer simulations that act as a bridge between the mathematical theories of quantum mechanics and the actual data recorded by detectors. These simulations must track how particles fly apart and how they emit radiation, a process known as evolution. However, standard simulation tools often struggle when dealing with heavy particles because their mass changes the way they emit energy, creating a "dead cone" where radiation is suppressed. If the computer models do not account for this mass correctly, the predictions for complex events, like those involving multiple heavy quarks, can drift away from reality, making it difficult to spot new physics or measure known properties with high precision.

To address this challenge, Benoît Assi and Stefan Höche have developed a new algorithm designed to simulate the evolution of these massive partons with greater accuracy. Their work focuses on the final stage of a collision, where the particles are flying outward and emitting soft gluons, which are the carriers of the strong nuclear force. The researchers created a method that treats the emission of this radiation differently depending on the angle and the mass of the emitting particle. By carefully mapping the momentum of the particles after they split, they ensured that the simulation respects the symmetry of the physical process. A key innovation in their approach is how they handle the mathematical functions that describe the probability of a particle splitting. Instead of using complex, negative values that can cause numerical instability, they rearranged the equations so that the probabilities remain strictly positive. This ensures the computer simulation runs efficiently without needing to discard large numbers of calculated events, a common problem in this field.

The team implemented this new algorithm in a numerical code called Alaric, which is part of a larger framework used to generate particle collision events. They tested the system by simulating the production of hadrons from electron-positron collisions at an energy level of 91.2 GeV, a standard benchmark in particle physics. The results showed that their new method could accurately reproduce the rates at which jets of particles form, as well as the shapes of the events, matching data collected by previous experiments at the Large Electron-Positron collider. The simulations also successfully described how bottom quarks break apart into other particles, a process known as fragmentation, showing good agreement with measurements from several different experimental groups. While the current version of the algorithm does not yet include the effects of particle spin or the radiation that occurs before the collision, the authors have provided the necessary mathematical tools to match these simulations with higher-level theoretical calculations. This work lays the groundwork for more precise modeling of heavy quark processes, which will be essential for analyzing the massive amounts of data expected from the upcoming high-luminosity phase of the Large Hadron Collider.

Technical Summary: A New Approach to QCD Final-State Evolution in Processes with Massive Partons

Problem Statement
The production and evolution of massive partons, particularly heavy quarks like the bottom quark, are critical for precision physics at the Large Hadron Collider (LHC). Processes such as $t\bar{t}H$ and double Higgs boson production involve final states with multiple $b$ -jets, necessitating accurate modeling in Monte Carlo event generators. While fixed-order perturbative QCD and resummation techniques have been extensively studied for heavy quarks, fully differential simulations in the context of particle-level event generators remain challenging. Existing methods often struggle with the complexity of infrared counterterms required for Next-to-Leading Order (NLO) matching or fail to maintain strict positivity in splitting functions, which compromises the numerical efficiency of unweighted event generation. Furthermore, previous approaches to massive parton evolution often lacked a kinematic mapping that treats outgoing particles democratically while preserving Next-to-Leading Logarithmic (NLL) precision.

Methodology
The authors present a new algorithm for massive parton evolution, implemented in the numerical code Alaric, which is integrated into the Sherpa event generation framework. The methodology is built upon several key theoretical and technical components:

Soft-Collinear Matching via Partial Fractioning:
The core of the algorithm involves deriving soft-gluon radiation splitting functions from the massive eikonal factor. Unlike previous methods that may result in negative splitting functions, the authors employ a partial fractioning technique on the complete soft eikonal. This decomposes the angular radiator function into strictly positive terms, ensuring the probability density interpretation is maintained and maximizing the efficiency of the Monte Carlo algorithm. The soft components are matched to the quasi-collinear limit using spin-averaged DGLAP splitting functions.
Kinematic Mappings:
Two distinct kinematic mappings are introduced to handle different regimes of radiation:
- Radiation Kinematics (Soft Evolution): A mapping is constructed where the momenta of the "mother" particle and a spectator are mapped to the final state momenta. This mapping reflects the symmetry of the final state in soft-gluon radiation and utilizes a reference spectator to define the azimuthal angle, ensuring Lorentz invariance of the emission phase space.
- Splitting Kinematics (Collinear Evolution): For purely collinear splittings (specifically gluons into quarks), a democratic kinematic mapping is proposed. This treats outgoing particles symmetrically and is designed to preserve NLL precision. The mapping is constructed to be both collinear-safe and NLL-safe, meaning it preserves the topological features of previously simulated radiation.
Phase Space Factorization and Counterterms:
The authors derive the differential phase-space factorization for $n+1$ particle states in $D = 4-2\epsilon$ dimensions. They compute the integrated infrared counterterms required for NLO matching (MC@NLO style). This involves calculating angular integrals of the partial fractioned soft eikonal using recent results for massive angular integrals in dimensional regularization. The authors separate the splitting functions into soft-enhanced parts (handled by the eikonal) and purely collinear remainders, allowing for a consistent treatment of sub-leading logarithms.
NLL Precision Analysis:
The paper analyzes the impact of the new kinematic mappings on sub-leading logarithmic corrections. By demonstrating that the mappings do not introduce spurious Lorentz transformations that alter existing momenta beyond the required order, the authors argue that the algorithm maintains NLL accuracy. The analysis confirms that the "dead-cone" effect is naturally incorporated via the modified integration boundaries and splitting functions.

Key Contributions

Algorithm Development: A new parton-shower algorithm for massive partons that utilizes partial fractioning of the soft eikonal to ensure strictly positive splitting functions.
NLO Matching: The computation of integrated infrared counterterms specifically for the Alaric parton-shower model, enabling NLO matching for processes involving massive quarks.
Democratic Kinematics: The introduction of a kinematic mapping for collinear gluon-to-quark splittings that treats outgoing particles democratically, preserving NLL precision.
Analytic Structure: A detailed discussion of the analytic structure of the parton shower in the NLL limit for massive partons, including the derivation of the relevant phase-space factors and anomalous dimensions.

Results
The authors present a first comparison of the Alaric algorithm with experimental data from LEP ( $e^+e^- \to \text{hadrons}$ at $\sqrt{s} = 91.2$ GeV). The comparisons include:

Differential Jet Rates: Predictions for 2-jet, 3-jet, 4-jet, and 5-jet rates using the Durham algorithm, showing good agreement with JADE and OPAL data.
Event Shapes: Comparisons for Thrust, Total Jet Broadening, C-Parameter, and Aplanarity against ALEPH data.
Fragmentation Functions: Predictions for the $b$ -quark fragmentation function compared to data from ALEPH, DELPHI, OPAL, and SLD.

The results indicate that the Alaric algorithm, when combined with standard hadronization models (Pythia 6.4), provides predictions of quality similar to existing state-of-the-art models (such as Dire) and agrees well with experimental data. The authors note that hadronization effects dominate predictions at very low jet resolution scales ( $y \sim 10^{-4}$ ).

Significance and Claims
The paper claims to provide a robust framework for simulating massive parton evolution that balances theoretical precision with numerical efficiency. By ensuring strictly positive splitting functions, the algorithm avoids the inefficiencies associated with negative weights in Monte Carlo simulations. The computation of the necessary counterterms is a crucial step toward enabling NLO matching for heavy-quark processes, a task that has been complicated by the ambiguous definitions of infrared counterterms in previous schemes.

The authors modestly frame their work as a foundational step. They explicitly state that the current implementation does not account for spin correlations or dominant sub-leading color effects, which are necessary for the algorithm to be on par with simulations used in current experiments. Furthermore, the extension to initial-state radiation and higher-order (NNLO) matching is identified as future work. The paper positions the derived all-orders expressions and angular integrals as components that can be utilized for future NNLO matching efforts, contributing to the precision goals of the high-luminosity LHC.

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