⚛️ phenomenology

Transverse-momentum resummation at mixed QCD $\otimes$ QED NNLL accuracy for Z boson production at hadron colliders

This paper presents a calculation of the transverse momentum distribution for neutral charged bosons at hadron colliders, performing resummation of simultaneous QCD and QED initial-state radiation effects up to mixed NNLL accuracy and demonstrating that these mixed contributions induce percent-level corrections to pure QCD predictions.

Original authors: Andrea Autieri, Stefano Camarda, Leandro Cieri, Giancarlo Ferrera, German Sborlini

Published 2026-01-26

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CC BY 4.0

Original authors: Andrea Autieri, Stefano Camarda, Leandro Cieri, Giancarlo Ferrera, German Sborlini

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

Imagine you are trying to listen to a specific conversation in a very noisy, crowded room. This is what particle physicists do when they study the Z boson, a tiny particle produced in collisions at giant machines like the Large Hadron Collider (LHC). To understand the Z boson, they need to know exactly how much "sideways push" (transverse momentum) it has when it's created.

However, the room is incredibly noisy. There are two main sources of noise:

The "Strong" Noise (QCD): This is like a massive, chaotic crowd shouting and pushing. It comes from the strong nuclear force, which is the most powerful force in the subatomic world.
The "Electromagnetic" Noise (QED): This is like a smaller, but still annoying, group of people whispering and jostling. It comes from the electromagnetic force (electricity and magnetism).

The Problem: Too Much Noise at Low Energy

When the Z boson is created with very little sideways push, the "noise" from the crowd becomes overwhelming. The standard mathematical tools physicists use (called "fixed-order calculations") start to break down because the noise terms get so huge they cancel out the signal. It's like trying to hear a whisper when a jet engine is roaring right next to you.

To fix this, physicists use a technique called resummation. Think of this as a sophisticated noise-canceling headphone algorithm. Instead of trying to calculate every single shout and whisper individually, the algorithm groups them together and predicts the overall "hum" of the noise, allowing the signal to come through clearly.

The Breakthrough: Mixing the Two Noises

For a long time, physicists treated these two types of noise separately. They would calculate the "Strong Noise" very precisely and then add a small correction for the "Electromagnetic Noise."

This paper, "Transverse-momentum resummation at mixed QCD⊗QED NNLL accuracy," does something new. It builds a hybrid noise-canceling system that listens to both the Strong and Electromagnetic noises simultaneously and calculates how they interact with each other.

The authors have upgraded their "headphones" to a new level of precision called NNLL (Next-to-Next-to-Leading Logarithmic).

Previous models were like listening to the crowd and the whisperers separately.
This new model understands that the crowd's shouting might change how the whisperers behave, and vice versa.

What They Found

The researchers used a computer program called DYTurbo to run these new calculations for two different "rooms":

The LHC (13 TeV): A massive, high-energy collider in Europe.
The Tevatron (1.96 TeV): An older collider in the US.

Here is what they discovered, using simple terms:

The Effect is Small but Real: When they added this new "mixed noise" calculation to their predictions, the results changed by about 1%. In the world of high-energy physics, where measurements are incredibly precise, a 1% shift is significant. It's the difference between guessing the weight of a car and actually weighing it on a scale.
The Shape Changes: The new calculation makes the "sideways push" distribution slightly "harder." Imagine a bell curve (the shape of the data). The new math suggests the Z boson is slightly more likely to have a bit more sideways energy than previously thought, especially at the edges of the curve.
Stability: The new method is more stable. When they tweaked the settings of their calculation (like turning the volume up or down slightly to check for errors), the results didn't swing wildly. This gives them more confidence that their prediction is correct.
The "Quiet" Room Effect: At the Tevatron (the older, smaller collider), the "Strong Noise" (QCD) is naturally quieter because there are fewer gluons (the particles causing the strong noise) involved. Because the background noise is lower, the "Electromagnetic Noise" (QED) and the mixed effects stand out more clearly there than at the LHC.

The Bottom Line

The authors have built a more precise mathematical tool to predict how Z bosons behave when they are produced in particle collisions. By finally accounting for how the strong force and electromagnetic force "talk" to each other in the background noise, they have reduced the uncertainty in their predictions.

This isn't about building a new machine or curing a disease; it's about calibration. Just as a musician needs to tune their instrument perfectly before a concert, physicists need these ultra-precise predictions to ensure that when they measure the mass of the W boson or the strength of the strong force, they aren't being misled by the "noise" of the calculation itself. They have simply turned the volume down on the uncertainty.

Technical Summary: Transverse-momentum resummation at mixed QCD⊗QED NNLL accuracy for Z boson production at hadron colliders

Problem Statement
High-precision measurements at the Large Hadron Collider (LHC) and the Tevatron require theoretical predictions for Drell-Yan (DY) processes that match experimental accuracy. While fixed-order perturbative QCD calculations are reliable at high transverse momentum ( $q_T \sim M$ ), they break down in the region $q_T \ll M$ due to large logarithmic terms of the form $\alpha_S^n \log^m(M^2/q_T^2)$ . These terms must be resummed to all orders.

Although resummation formalisms exist for pure QCD (up to N $^4$ LL) and mixed QCD $\otimes$ QED at lower orders (NLL+NLO), a consistent treatment of mixed QCD $\otimes$ QED logarithms at next-to-next-to-leading logarithmic (NNLL) accuracy has been lacking. Given that the strong and electromagnetic couplings are related by $\alpha_S^2 \sim \alpha$ , neglecting mixed higher-order terms introduces uncertainties that may become relevant for precision physics, such as the extraction of the $W$ boson mass and the strong coupling constant.

Methodology
The authors extend the transverse-momentum resummation formalism (originally developed for QCD in impact parameter space) to include simultaneous QCD and QED initial-state radiation (ISR). The calculation is performed in Mellin and impact parameter ( $b$ ) space, where the cross-section is factorized into a hard-virtual coefficient and a Sudakov-like form factor.

Key methodological steps include:

Formalism Extension: The resummed form factor $W$ is expanded in a double perturbative series in $\alpha_S$ and $\alpha$ . The exponent $G'_N$ and the hard-virtual coefficient $H'_N$ are constructed to include pure QED, pure QCD, and mixed QCD $\otimes$ QED terms.
New Ingredients Calculation: To achieve mixed NNLL accuracy, the authors computed three previously unknown functions: $g'^{(2,1)}_N$ $g_{N}^{' (2, 1)}$ , $g'^{(1,2)}_N$ $g_{N}^{' (1, 2)}$ , and $g'^{(3)}_N$ $g_{N}^{' (3)}$ .
- The mixed terms $g'^{(2,1)}$ and $g'^{(1,2)}$ were derived from the integral representation of the universal form factors, utilizing mixed cusp and collinear anomalous dimensions ( $A^{(2,1)}$ , $A^{(1,2)}$ , $\bar{B}^{(1,1)}$ , etc.).
- The pure QED NNLL term $g'^{(3)}$ was obtained via Abelianization of the corresponding QCD function.
- The evolution of the couplings $\alpha_S$ and $\alpha$ was treated simultaneously using coupled Renormalization Group Equations (RGEs) including mixed $\beta$ -function coefficients up to the required order.
Implementation: The calculated coefficients were implemented in the numerical code DYTURBO.
Phenomenological Setup: Predictions were generated for $Z/\gamma^*$ production at $\sqrt{s} = 13$ TeV (LHC) and $\sqrt{s} = 1.96$ TeV (Tevatron). The study utilized the NNPDF40.QED parton distribution functions and varied renormalization, factorization, and resummation scales to estimate theoretical uncertainties.

Key Contributions

First NNLL Mixed Resummation: This work presents the first calculation of mixed QCD $\otimes$ QED resummation at NNLL accuracy for the neutral current DY process.
Derivation of Anomalous Dimensions: The paper provides explicit expressions for the mixed cusp anomalous dimensions ( $A^{(2,1)}$ , $A^{(1,2)}$ ) and the mixed collinear coefficients required for the resummation, derived from first principles and verified against known QCD results via Abelianization.
Consistent Coupling Evolution: The formalism consistently incorporates the running of both QCD and QED couplings, including their mutual mixing effects at NNLO in the perturbative expansion.

Results
The authors analyzed the impact of these mixed corrections on the $q_T$ , invariant mass ( $M_{inv}$ ), and rapidity ( $y$ ) distributions:

Transverse Momentum ( $q_T$ ):
- At the LHC, mixed NLL and NNLL corrections are of the order of a few percent. Compared to pure QCD NNLL predictions, mixed corrections make the spectrum harder for $q_T > 5$ GeV.
- The mixed NLL and NNLL results are very similar, differing only at the permille level, suggesting good stability.
- At the Tevatron, where gluon-induced processes are suppressed, the relative impact of QED and mixed corrections is enhanced (reaching $\sim 1-2\%$ ).
- Scale variation bands indicate that uncertainties are primarily driven by pure QCD contributions; the inclusion of mixed terms does not degrade the convergence of the perturbative series.
Invariant Mass and Rapidity:
- Around the $Z$ -boson peak, LL QED corrections reduce the cross-section by $\sim 0.2\%$ , while mixed NLL and NNLL corrections increase it by $\sim 0.1\%$ and $\sim 0.2\%$ , respectively.
- Rapidity distributions show similar qualitative behavior with corrections of order $0.1\%$ .
Fixed vs. Running Coupling: An appendix investigates the effect of fixing the QED coupling. While the qualitative behavior remains similar, the running coupling treatment is necessary for a consistent theoretical framework.

Significance and Claims
The paper claims that providing accurate theoretical predictions for the DY $q_T$ spectrum is essential for the current high-precision physics program at colliders. The authors state that they have "gone beyond the current precision frontier" by calculating mixed QCD $\otimes$ QED contributions at NNLL accuracy for the first time.

The significance of this work lies in:

Stabilization: The inclusion of higher-order mixed terms helps stabilize the theoretical predictions, particularly in the intermediate $q_T$ region.
Precision: The percent-level effects observed in the $q_T$ spectrum are relevant for precision measurements of Standard Model parameters (e.g., $W$ mass, strong coupling) and for reducing theoretical uncertainties in detector calibration and luminosity monitoring.
Completeness: By providing the full set of ingredients for NNLL mixed resummation, the work enables more reliable phenomenological descriptions of neutral boson production, ensuring that theoretical errors do not limit the interpretation of high-luminosity LHC data.

The authors remain modest, noting that while the effects are at the percent level, they are crucial for matching the "impressive level of precision" achieved by modern experiments. They do not claim to solve all theoretical uncertainties but rather to provide a necessary step toward a fully consistent mixed QCD $\otimes$ QED description.

The Problem: Too Much Noise at Low Energy

The Breakthrough: Mixing the Two Noises

What They Found

The Bottom Line

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