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N $^{\mathbf{3}}$ LL + $\mathcal{O}(\alpha_s^2)$ predictions of lepton-jet azimuthal angular distribution in deep-inelastic scattering

This paper presents next-to-next-to-next-to-leading logarithmic (N $^{3}$ LL) resummation combined with $\mathcal{O}(\alpha_s^2)$ fixed-order corrections to predict lepton-jet azimuthal angular distributions in deep-inelastic scattering, offering a precision framework for probing the nucleon's three-dimensional structure and analyzing data from HERA and future electron-ion collider experiments.

Original authors: Shen Fang, Mei-Sen Gao, Hai Tao Li, Ding Yu Shao

Published 2026-08-28

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CC BY 4.0

Original authors: Shen Fang, Mei-Sen Gao, Hai Tao Li, Ding Yu Shao

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

To understand the inner life of matter, physicists look at how particles smash together at incredible speeds. When an electron collides with a proton, the electron acts like a high-powered probe, scattering off the tiny quarks and gluons inside the proton. By measuring how the electron and the resulting spray of particles, called a jet, move away from each other, scientists can map the three-dimensional structure of the proton. However, this map is not a simple photograph; it is a complex landscape where the particles are constantly jostling, emitting soft radiation, and interacting in ways that are difficult to predict. For decades, the tools used to calculate these interactions have been accurate enough for some tasks, but as new, more powerful particle colliders are built, the old tools are no longer precise enough to reveal the subtle details of how the proton is built.

A team of researchers has now taken a major step forward in this effort by calculating the behavior of these collisions with unprecedented precision. They focused on a specific measurement: the angle between the scattered electron and the leading jet of particles. In a perfect, simplified world, these two would fly off in exactly opposite directions. In reality, they are slightly misaligned due to the chaotic dance of particles inside the proton. The researchers developed a new mathematical framework to predict exactly how often this misalignment happens, combining two different methods of calculation to cover all possible scenarios. Their work provides a robust, high-precision guide for future experiments, ensuring that when new data arrives from upcoming facilities, scientists will have the right tools to interpret it correctly.

The core of this achievement lies in merging two distinct ways of looking at the same problem. One method is excellent at describing what happens when the electron and jet are nearly opposite each other, a situation where tiny, soft emissions of energy create large, confusing mathematical terms that must be summed up to infinite orders to get a correct answer. The other method is a standard, step-by-step calculation that works well when the particles are at wider angles but fails to capture the subtle, cumulative effects of those soft emissions. The team successfully combined these approaches, creating a single, seamless prediction that is accurate across the entire range of angles. They achieved a level of detail known as next-to-next-to-next-to-leading logarithmic accuracy, which is a significant leap forward in the hierarchy of precision calculations.

To make this calculation possible, the researchers had to define exactly what they meant by a "jet." In the messy environment of a collision, particles are constantly merging and splitting. The team used a specific set of rules, known as the anti-kT algorithm and the winner-take-all scheme, to group these particles into a single, well-defined jet. This choice was crucial because it simplified the complex mathematics, allowing them to ignore certain confusing effects that usually plague these calculations. By doing so, they could isolate the true signal of the proton's internal structure from the noise of the collision process. They also had to account for the fact that at very small angles, the mathematics can break down, so they introduced a smooth transition function to bridge the gap between their high-precision prediction for small angles and the standard calculation for larger angles.

The researchers tested their new predictions against data from past experiments at the HERA collider and projected what they would see at the future Electron-Ion Collider. They found that their high-precision model reduces the uncertainty in the predictions significantly compared to older methods. When they varied the parameters in their equations to test for stability, the results remained consistent, showing that their method is reliable. The study revealed that for the future Electron-Ion Collider, the corrections needed to match the high-precision theory with standard calculations are substantial, exceeding twenty percent in certain regions. This means that ignoring these corrections would lead to a distorted view of the proton's structure. For the older HERA data, the corrections were smaller but still necessary for a complete picture.

This work is not just a theoretical exercise; it is a practical toolkit for the next generation of discovery. The Electron-Ion Collider, currently under development, aims to create a three-dimensional image of the proton, revealing how quarks and gluons are arranged in space and momentum. Without the precise calculations provided by this study, the data from that machine would be difficult to interpret. The researchers have provided a clear, mathematically rigorous path forward, ensuring that when the first collisions occur at the new facility, the scientific community will be ready to extract the deepest secrets of nuclear matter. By refining the tools used to measure the angle between the electron and the jet, they have sharpened the lens through which we view the fundamental building blocks of the universe.

Technical Summary: N3LL + O(α²s) Predictions of Lepton-Jet Azimuthal Angular Distribution in Deep-Inelastic Scattering

Problem Statement
The paper addresses the need for high-precision theoretical predictions for the azimuthal angular distribution (decorrelation) between the scattered lepton and the leading jet in deep-inelastic scattering (DIS). This observable serves as a critical probe of the three-dimensional structure of the nucleon, specifically providing access to transverse-momentum dependent (TMD) parton distribution functions (PDFs). While previous studies have achieved next-to-next-to-leading logarithmic (NNLL) accuracy or fixed-order calculations, there is a requirement for predictions that combine next-to-next-to-next-to-leading logarithmic (N3LL) resummation with fixed-order corrections at $\mathcal{O}(\alpha_s^2)$ to reduce theoretical uncertainties and ensure reliability across different kinematic regimes, particularly for future Electron-Ion Collider (EIC) and HERA data analyses.

Methodology
The authors employ the Soft-Collinear Effective Theory (SCET) framework to derive the N3LL resummation formula. Key methodological choices include:

Jet Definition: Jets are defined in the laboratory frame using the anti- $k_T$ clustering algorithm with a radius parameter $R=1$ and the winner-take-all (WTA) recombination scheme. The WTA scheme is crucial as it renders the jet axis insensitive to soft emissions, thereby removing non-global logarithms (NGLs) and simplifying the factorization formula.
Factorization: The azimuthal decorrelation is described by a TMD factorization formula involving a hard function ( $H$ ), TMD beam functions ( $B_{q/p}$ ), a jet function ( $J_q$ ), and a soft function ( $S$ ). The formula is evolved from natural scales to a common scale to resum large logarithms of the form $\ln(\delta\phi)$ .
Fixed-Order Matching: To cover the region where the azimuthal angle $\delta\phi$ is not small, the resummation results are matched with fixed-order calculations at $\mathcal{O}(\alpha_s^2)$ . The singular part of the fixed-order cross-section is derived analytically from the resummation formula, while the non-singular part is calculated using the NLOJET++ event generator.
Jet Function Constant: A specific challenge addressed is the determination of the two-loop constant term in the jet function. While previously extracted numerically from the Event2 generator, the authors perform an independent cross-check by using NLOJET++ to generate fixed-order predictions for back-to-back dijet production in $e^+e^-$ collisions, comparing these with the singular distribution from the resummation expansion to extract the constant.
Matching Scheme: An additive matching scheme is utilized, incorporating a transition function $t(\delta\phi)$ to smoothly interpolate between the resummed region (small $\delta\phi$ ) and the fixed-order region (large $\delta\phi$ ).

Key Contributions

N3LL + $\mathcal{O}(\alpha_s^2)$ Prediction: The paper presents the first N3LL resummation combined with $\mathcal{O}(\alpha_s^2)$ fixed-order corrections for the lepton-jet azimuthal decorrelation in DIS within the lab frame.
Validation of Jet Function Constants: The authors provide an independent numerical determination and cross-check of the two-loop constant term in the jet function, a necessary ingredient for N3LL accuracy that was previously only available via Event2.
Comprehensive Uncertainty Analysis: The study includes a detailed assessment of theoretical uncertainties arising from renormalization scale variations ( $\mu_h$ and $\mu_b$ ) and the choice of the transition point $r$ in the matching procedure.
Phenomenological Predictions: Numerical predictions are provided for both the EIC (with specific kinematic cuts: $E_e=18$ GeV, $E_p=275$ GeV) and HERA ( $E_e=27.5$ GeV, $E_p=920$ GeV), comparing N3LL results with NNLL and pure NLO predictions.

Results

Convergence and Uncertainty: The comparison between NNLL and N3LL results shows that the perturbative uncertainties (represented by scale variation bands) are significantly reduced at N3LL accuracy. The bands for N3LL and NNLL overlap almost entirely, indicating perturbative convergence.
Matching Corrections: In the additive matching scheme, significant matching corrections are observed. For the EIC, these corrections exceed 20% for $\delta\phi \geq 0.3$ , while for HERA, they are milder (around 10% for $\delta\phi < 0.4$ ).
Non-Singular Contributions: The non-singular contributions (the difference between full NLO and singular NLO) show a substantial cancellation between LO and NLO terms, though this leads to larger relative scale uncertainties for the NLO non-singular distribution.
Transition Dependence: The results are shown to be stable with respect to variations in the transition parameter $r$ , with default values chosen as $r=0.16$ for EIC and $r=0.18$ for HERA.
Divergence Handling: The NLO prediction exhibits divergent behavior as $\delta\phi \to 0$ , whereas the N3LL + $\mathcal{O}(\alpha_s^2)$ prediction remains finite and smooth, correctly resumming the large logarithms in the back-to-back limit.

Significance
The paper claims that these results provide a robust framework for precision studies of Quantum Chromodynamics (QCD) and the internal structure of nucleons through jet observables in DIS. By reducing theoretical uncertainties through high-order resummation and fixed-order matching, the predictions are particularly significant for:

The analysis of existing HERA data.
The preparation for and interpretation of forthcoming Electron-Ion Collider (EIC) experiments.
The extraction of TMD PDFs and the study of the nucleon's 3D imaging.

The authors emphasize that their approach, which successfully interpolates between resummation and fixed-order regimes using the WTA scheme, establishes a foundation for future work involving power corrections and higher-order resummation techniques.

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