⚛️ phenomenology

High-energy logarithms and precision at the LHC

This paper demonstrates that the breakdown of fixed-order perturbation theory in high-energy LHC processes, caused by large logarithms of $s/p_t^2$ , is resolved by resumming these logarithms, thereby providing accurate predictions for $Z$ +dijet production that match recent ATLAS measurements.

Original authors: Jeppe R. Andersen, Sebastian Jaskiewicz, Andreas Maier, Jennifer M. Smillie

Published 2026-09-30

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CC BY 4.0

Original authors: Jeppe R. Andersen, Sebastian Jaskiewicz, Andreas Maier, Jennifer M. Smillie

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

At the Large Hadron Collider, scientists smash protons together at energies never before reached, creating a shower of new particles that fly out in all directions. Among the most common outcomes are jets: tight sprays of particles that form when the fundamental building blocks of matter, called quarks and gluons, are knocked loose and then immediately recombine into new particles. To understand the laws of nature, physicists calculate how often these jets should appear and how they should behave using a method called perturbation theory. This approach works like a mathematical recipe: it starts with a simple, basic prediction and then adds layers of increasingly complex corrections to account for the messy reality of particle interactions. For decades, this method has been the gold standard for predicting what happens in these collisions. However, when the jets fly apart with a very large separation in energy or distance, the standard recipe begins to fail. The corrections become so large and unruly that the predictions turn negative or wildly unstable, suggesting that the mathematical tools being used are missing a crucial piece of the puzzle.

A team of researchers has now identified exactly why this breakdown happens and has provided a way to fix it. They focused on a specific process where a Z boson—a heavy particle that acts as a messenger of the weak nuclear force—is produced alongside two jets. In the standard calculations, as the distance between the two jets grows, a specific type of mathematical term, known as a high-energy logarithm, begins to dominate the result. These terms are not errors in the math itself, but rather a signal that the standard method of adding corrections one by one is insufficient. When the jets are far apart, these logarithmic terms grow so large that they overwhelm the basic prediction, causing the calculated probability of the event to drop below zero, which is physically impossible. The researchers found that the common practice of simply adjusting a parameter in the calculation to force the numbers to stay positive was not a true solution. While this adjustment might make the numbers look stable, it effectively hides the problem by changing the underlying physics in a way that removes the natural dependence on the strength of the interaction, leading to results that are mathematically consistent but physically misleading.

To solve this, the team turned to a technique called resummation. Instead of calculating corrections one by one and hoping they stay small, resummation treats the large logarithmic terms as a single, unified effect that must be summed up to infinity. Imagine trying to predict the path of a ball rolling down a hill where the wind keeps pushing it harder the further it goes; calculating the wind's effect step-by-step might eventually lead to a nonsensical prediction, but summing the wind's total influence from the start gives a clear, stable path. The researchers applied this method using a framework called High Energy Jets, which systematically accounts for these high-energy effects across all possible levels of complexity. They matched this all-order calculation with the standard, high-precision fixed-order calculations for the parts of the process where the high-energy effects are small. This hybrid approach allowed them to generate predictions that remain positive and stable, even when the jets are separated by vast distances.

When they compared these new predictions to real data collected by the ATLAS experiment at the Large Hadron Collider, the results were striking. The standard calculations, which rely on fixed-order methods, failed to describe the shape of the data for the distance between the jets, often predicting a sharp drop that the data did not show. In contrast, the new resummed predictions followed the slope of the experimental data perfectly, capturing the behavior of the particles across the entire range of energies studied. The researchers also found that the new method resolved other subtle issues, such as strange asymmetries in the uncertainty bands that had plagued previous calculations. By systematically including these high-energy logarithms, the team demonstrated that the breakdown in the standard theory was not a flaw in the laws of physics, but a limitation in the way the calculations were being performed. Their work provides a robust, accurate description of these high-energy collisions, ensuring that future measurements at the collider can be interpreted with the precision required to uncover new physics.

Technical Summary: High-energy logarithms and precision at the LHC

Problem Statement
The paper addresses a fundamental breakdown in fixed-order perturbation theory for processes involving two or more jets at the Large Hadron Collider (LHC), specifically when the dijet invariant mass ( $m_{j_1j_2}$ ) is large relative to the jet transverse momenta ( $p_t$ ). The authors identify that logarithmic terms of the form $\ln(m_{j_1j_2}^2/p_t^2)$ become dominant, causing several pathologies in standard Next-to-Leading Order (NLO) predictions:

Breakdown of the perturbative series: For distributions like $d\sigma/dm_{j_1j_2}$ , the NLO corrections become so large and negative that the cross-section prediction turns negative at high invariant masses (e.g., around 3 TeV for $Z+$ dijets), rendering the prediction unphysical.
Asymmetric scale variations: Even for observables not directly dependent on $m_{j_1j_2}$ (such as $p_{T,V}$ or azimuthal separations $\Delta\phi_{jj}$ ), the scale variation bands become highly asymmetric. This is driven by the large, uncancelled logarithmic corrections at high $m_{j_1j_2}$ which dominate the uncertainty estimation.
Inadequacy of ad-hoc scale choices: The customary practice of choosing a renormalisation scale $\mu_R = m_{j_1j_2}$ to stabilize the cross-section is shown to be a superficial fix. While it may yield positive results, it fundamentally alters the perturbative behavior. Specifically, in the high-energy limit, the product $\alpha_s(m_{j_1j_2}^2) \ln(m_{j_1j_2}^2/p_t^2)$ approaches a constant ( $1/b_0$ ), causing the dependence on the coupling $\alpha_s$ to vanish. This effectively removes the perturbative nature of the correction and masks the underlying logarithmic divergence rather than resolving it.

Methodology
The authors propose a systematic solution based on the resummation of high-energy logarithms to all orders in the strong coupling constant, utilizing the High Energy Jets (HEJ) framework.

Theoretical Basis: The approach relies on the factorisation of scattering amplitudes in the Multi-Regge Kinematics (MRK) limit, where $s \gg |t|$ . The high-energy logarithms arise from the colour-octet exchange channel between scattered partons. These logarithms originate from both virtual corrections (Lipatov Ansatz) and real emissions (Lipatov vertex).
Resummation Scheme: The paper implements an all-order resummation of these logarithmic terms. The calculation includes:
- Leading Logarithmic (LL) accuracy for standard configurations.
- Next-to-Leading Logarithmic (NLL) configurations, specifically including central quark-antiquark pairs and unordered gluon emissions, denoted as LL+.
- Matching to fixed-order NLO results. For distributions where the high-energy logarithm is not dominant (e.g., $p_{T,V}$ ), the HEJ prediction is multiplicatively matched to NLO bin-by-bin. For the $m_{j_1j_2}$ distribution, a matching factor is derived from the ratio of total cross-sections between NLO and HEJ-expanded-to-NLO.
Process Studied: The methodology is applied to the production of a $Z$ boson in association with dijets ( $pp \to (Z \to e^+e^-) + jj$ ) at $\sqrt{s} = 13$ TeV, comparing against recent ATLAS measurements [1].

Key Contributions and Results

Diagnosis of Fixed-Order Failure: The paper explicitly demonstrates that the negative cross-sections and asymmetric scale variations observed in NLO predictions for $Z+$ dijets are direct consequences of the unresummed high-energy logarithms. It shows that these issues persist even without central jet vetoes, though the magnitude of the logarithmic coefficient is enhanced by such vetoes.
Critique of Scale Setting: The authors provide a detailed argument against the choice $\mu_R = m_{j_1j_2}$ . They show that while this choice suppresses the coupling $\alpha_s$ , it promotes the UV logarithm to a leading status and accidentally cancels the high-energy logarithm's coefficient, leading to a result that is independent of $\alpha_s$ in the high-energy limit. This is described as "unperturbative" behavior.
Validation of Resummation:
- $m_{j_1j_2}$ Distribution: The HEJ predictions, which include the all-order resummation, yield a positive, physically meaningful cross-section across the entire kinematic range. The resummed result corrects the slope of the distribution, which fixed-order NLO fails to describe.
- Comparison to Data: When matched with Leading Order (LO) electroweak contributions (calculated separately), the HEJ+EW prediction provides an accurate description of the ATLAS data for $d\sigma/dm_{j_1j_2}$ in the region where the QCD component dominates ( $m_{j_1j_2} < 1.52$ TeV).
- Other Observables: For distributions like $d\sigma/dp_{T,V}$ and $d\sigma/d\Delta\phi_{jj}$ , the HEJ predictions (matched to NLO) are very similar to pure NLO predictions in the central region. However, the paper notes that the HEJ framework inherits the asymmetric scale variation of the NLO matching. This asymmetry is identified as a "masquerading" effect of the high-energy logarithms affecting the NLO input, suggesting that future NNLO corrections would be required to fully resolve the scale variation in these specific observables.

Significance and Claims
The paper claims that high-precision theoretical predictions for standard LHC observables involving multi-jet final states require not just higher fixed-order corrections (e.g., NNLO) but also a systematic treatment of logarithmically enhanced contributions arising from high-energy limits.

Universality: The source of the perturbative breakdown is identified as universal for all processes with two or more jets involving a colour-octet exchange (including $W$ , $Z$ , $\gamma$ , and Higgs production).
Resolution: The systematic resummation of high-energy logarithms is presented as the necessary solution to restore perturbative stability and physical positivity to cross-section predictions at large invariant masses.
Practical Impact: The authors demonstrate that the HEJ framework, when matched to NLO, successfully describes current ATLAS data for $Z+$ dijets using a standard renormalisation scale ( $\mu_R = \hat{H}_T/2$ ) throughout, avoiding the need for the ad-hoc, physically questionable choice of $\mu_R = m_{j_1j_2}$ .

The work concludes that while fixed-order expansions are insufficient in the high-energy regime, the combination of all-order resummation and fixed-order matching provides a robust framework for precision phenomenology at the LHC.

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