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Resumming transverse observables for NNLO+PS matching in GENEVA

This paper presents a GENEVA framework for NNLO+PS matching of transverse observables by embedding N $^3$ LL $q_T$ resummation and introducing generalized $N$ -jettiness resummation up to NLL $'$ accuracy, demonstrated through a Higgs boson production generator in heavy-quark annihilation.

Original authors: Alessandro Gavardi, Rebecca von Kuk, Matthew A. Lim

Published 2026-10-09

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CC BY 4.0

Original authors: Alessandro Gavardi, Rebecca von Kuk, Matthew A. Lim

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 use massive machines like the Large Hadron Collider to smash protons together, hoping to catch fleeting glimpses of the fundamental building blocks of the universe. To make sense of the billions of collisions that occur every second, researchers rely on theoretical tools that predict what should happen with extreme precision. These tools must account for the fact that when particles collide, they do not just produce a single new particle; they often emit a spray of other particles, creating complex patterns of energy and motion. The most advanced predictions currently available combine two different ways of calculating these events: one method that is precise for the main collision but misses the fine details of the spray, and another that simulates the spray but is less precise for the main event. Merging these two approaches to get the best of both worlds is a major challenge, and the accuracy of the result depends heavily on how scientists choose to measure the "resolution" of the event—essentially, how they decide to group the resulting particles into distinct jets or bundles.

A team of researchers at the Deutsches Elektronen-Synchrotron (DESY) and other institutions has developed a new way to perform this merging for a specific and important process: the creation of a Higgs boson through the collision of heavy quarks, specifically bottom or charm quarks. In their work, published in the DESY report 25-068, they introduced a fresh method for measuring the transverse momentum of the particles involved. Instead of using the traditional way of measuring the mass-like spread of particles, which can be difficult to match with the computer simulations used to model particle showers, they used a measurement based on the sideways momentum of the particles. This change allowed them to create a more accurate computer generator that can simulate the production of a Higgs boson along with the surrounding spray of particles, achieving a level of precision known as next-to-next-to-leading order matched with a parton shower.

The core of this achievement lies in how the researchers handled the mathematics of resummation, a technique used to sum up an infinite number of small corrections that become important when particles move in specific directions. They combined a highly accurate calculation for the sideways momentum of the Higgs boson itself with a new, improved way of measuring the momentum of the single hardest jet of particles that might appear alongside it. By using a measurement that focuses on sideways momentum rather than mass, they ensured that their theoretical calculations aligned perfectly with the way modern computer programs simulate the emission of particles. This alignment is crucial because it prevents the simulation from introducing artificial errors or "artifacts" that could distort the final results. The researchers tested this new framework by simulating the production of a Higgs boson from the annihilation of bottom quarks and charm quarks, comparing their results against other established calculations to ensure everything was working correctly.

The results of their simulations showed that the new method works exceptionally well. When they compared their predictions for the total rate of Higgs boson production against independent calculations, the numbers matched almost perfectly, differing by less than one percent even under challenging conditions. They also verified that the distribution of the Higgs boson's sideways momentum was preserved accurately after the computer simulation added the complex details of the particle shower. This is a significant finding because it demonstrates that the new approach can maintain high precision while still allowing the simulation to generate realistic events that include the full spray of particles. The researchers found that the choice of how the simulation handles the recoil of particles—how momentum is balanced when a particle is emitted—matters greatly. By using a specific scheme that distributes this recoil within the color connections of the particles, they were able to keep the predictions for the Higgs boson's momentum stable and accurate, whereas other schemes introduced noticeable deviations.

Beyond the immediate success with the Higgs boson, this work opens the door for future studies of other processes where a color-neutral particle is produced alongside jets. The ability to use transverse momentum measures for all stages of the calculation suggests that this method could be extended to other important processes, such as the production of W or Z bosons, which are also color-neutral. The researchers noted that while their current work achieves a high level of accuracy, there is still room to improve the precision of the jet measurements further. They also highlighted that the new framework provides a solid foundation for estimating the theoretical uncertainties that always accompany such complex calculations. By offering a way to match high-precision theory with realistic event generation without the need for cumbersome workarounds, this study provides a powerful new tool for physicists who are trying to extract precise values for fundamental properties, such as the strength of the interaction between the Higgs boson and bottom quarks, from the data collected at the Large Hadron Collider.

Technical Summary: Resumming Transverse Observables for NNLO+PS Matching in GENEVA

Problem Statement
The construction of Next-to-Next-to-Leading Order plus Parton Shower (NNLO+PS) generators requires the selection of resolution variables to partition phase space into jet bins (0-jet, 1-jet, 2-jet, etc.). These variables must be resummed to high logarithmic accuracy to handle large logarithmic corrections near the kinematic boundaries. A significant challenge arises from the choice of these variables: while variables like $N$ -jettiness ( $T_N$ ) offer simple factorization structures in Soft-Collinear Effective Theory (SCET-I), their standard definition relies on invariant-mass-like measures. This complicates matching to modern parton shower algorithms, which are typically ordered in transverse momentum ( $p_T$ ). Conversely, using transverse momentum ( $q_T$ ) for 0-jet bin separation is well-established but requires a compatible variable for 1-jet bin separation that preserves the advantages of $p_T$ -ordered showers.

Methodology
The authors implement a new combination of resolution variables within the Geneva framework to achieve NNLO+PS matching. The methodology involves two primary components:

$q_T$ Resummation for 0-Jet Bins: The authors utilize the transverse momentum ( $q_T$ ) of the colour-singlet system as the primary resolution variable. They embed $q_T$ resummation at Next-to-Next-to-Next-to-Leading Logarithmic (N3LL) accuracy, obtained via SCET-II and implemented in the SCETlib library. This replaces previous approaches that relied on the RadISH formalism.
Generalised One-Jettiness ( $T_1^{p_T}$ ) for 1-Jet Bins: For the separation of 1-jet and 2-jet bins, the authors introduce a generalised one-jettiness variable, denoted $T_1^{p_T}$ $T_{1}^{p_{T}}$ . Unlike standard $T_1$ $T_{1}$ , which uses invariant mass measures, $T_1^{p_T}$ $T_{1}^{p_{T}}$ employs transverse momentum-like measures for both the beam and jet regions.
- Observable Definition: The observable uses a conical measure for region assignment (equivalent to the anti- $k_T$ algorithm) and a boost-invariant generalisation of broadening for the jet region measurement.
- Resummation: The authors achieve NLL' (Next-to-Leading Logarithmic prime) resummation for $T_1^{p_T}$ . This is the first time this specific observable has been resummed beyond leading logarithmic order. The factorization structure is of the SCET-II type, involving hard, soft, beam, and jet functions.
- Splitting Mappings: To distribute the resummed spectrum over the phase space, the authors construct specific splitting mappings. A key innovation is a hybrid mapping that preserves the transverse momentum of the colour singlet (crucial for the $q_T$ resummation) and the rapidity of the hardest final-state parton (defining the jet axis). This ensures the resummation of $T_1^{p_T}$ does not spoil the accuracy of the $q_T$ spectrum.

Case Study and Implementation
The framework is tested on Higgs boson production via heavy-quark annihilation ( $b\bar{b} \to H$ and $c\bar{c} \to H$ ). The authors interface the SCETlib $q_T$ module directly with Geneva. They utilize the $p_T$ -ordered Pythia8 shower, employing the SpaceShower:dipoleRecoil=on option to minimize the impact of recoil on the colour-singlet momentum.

Key Results

Fixed-Order Validation: The inclusive cross-section predictions at NNLO agree with independent calculations from SusHi and MCFM within 1%, even for relatively large resolution cuts ( $q_T^{cut} = 10$ GeV). Excellent agreement is observed for the Higgs rapidity spectrum.
Resummation Validation: The $q_T$ spectrum generated by Geneva matches the standalone N3LL predictions from SCETlib exactly, confirming the correct implementation of the resummation.
$T_1^{p_T}$ Performance: The nonsingular remainder of the $T_1^{p_T}$ spectrum vanishes linearly as $T_1^{p_T} \to 0$ , confirming the correct logarithmic structure. The resummed distribution exhibits the expected Sudakov peak.
Showering Effects: When matched to the parton shower, the Higgs boson $q_T$ spectrum is numerically preserved, demonstrating that the $q_T$ -preserving mapping successfully decouples the shower recoil from the singlet momentum. For exclusive observables like the hardest jet rapidity and $p_T$ , showering induces expected modifications (e.g., depletion in the high- $p_T$ tail due to additional radiation), but the results remain consistent within scale uncertainties.

Significance and Claims
The paper claims to take a "very first step" toward achieving NNLO+PS matching for colour-singlet production in association with final-state jets. The primary significance lies in the successful combination of $q_T$ resummation (N3LL) and generalised $T_1^{p_T}$ resummation (NLL') within a single generator.

The authors emphasize that the use of transverse measures facilitates matching to $p_T$ -ordered showers, which is a prerequisite for future extensions to processes involving additional hard jets. While the current work focuses on the $b\bar{b}H$ and $c\bar{c}H$ channels, the methodology is presented as a generalizable framework for other colour-singlet processes. The paper modestly notes that while the formal accuracy of the showered events is limited by the LL accuracy of the shower, the matching procedure preserves the high-order accuracy of the underlying resummed distributions for the colour-singlet kinematics. Future work is identified as necessary to extend the $T_1^{p_T}$ resummation to NNLL' and to automate the switching between different resolution variables for uncertainty estimation.

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