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Measurement of the triple-differential cross section of Z+jet production in proton-proton collisions at $\sqrt{s}$ = 13 TeV

Using 138 fb $^{-1}$ of proton-proton collision data at $\sqrt{s}$ = 13 TeV collected by the CMS experiment, this paper presents a triple-differential measurement of the Z+jet production cross section that is unfolded for detector effects and compared with next-to-next-to-leading order QCD predictions to characterize the proton's partonic structure and determine the strong coupling.

Original authors: CMS Collaboration

Published 2026-10-01

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CC BY 4.0

Original authors: CMS Collaboration

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

The universe is built from a handful of fundamental particles, but the most abundant one, the proton, is a complex puzzle. It is not a solid marble but a seething cloud of smaller constituents called quarks and gluons, held together by the strong force. To understand how the universe works, physicists must map out exactly how these internal parts are arranged and how they move. This map is known as the parton distribution function. While scientists have long known that protons contain these parts, the precise details of their momentum and arrangement remain one of the great mysteries of modern physics. By smashing protons together at incredible speeds, researchers can probe this internal structure, watching how the debris flies apart to reveal the hidden architecture of matter.

In a recent study, the CMS collaboration at the Large Hadron Collider in Switzerland has taken a significant step forward in this mapping effort. They analyzed a vast collection of data from proton collisions recorded between 2016 and 2018, corresponding to an enormous amount of recorded activity. The researchers focused on a specific type of collision event where two protons smash together and produce a Z boson, a heavy particle that quickly decays into a pair of muons, accompanied by at least one jet of particles. This specific outcome is theoretically well-understood, making it an ideal laboratory for testing our knowledge of the proton's inner workings. By measuring the energy and direction of the muons and the jet with extreme precision, the team could reconstruct the conditions of the collision before the particles even formed.

The team measured the production rate of these events across three distinct dimensions simultaneously. They looked at the transverse momentum of the muon pair, which relates to the energy scale of the collision. They also measured the boost of the system, which indicates how unbalanced the initial momentum of the colliding parts was, and the separation in rapidity between the muon pair and the jet, which reveals the angle at which the particles scattered. This triple-differential approach allowed the scientists to isolate specific scattering angles and momentum fractions of the interacting parts inside the proton. To ensure their measurements reflected reality rather than the quirks of their detectors, they used sophisticated mathematical techniques to unfold the data, effectively removing the blurring effects of the instrument to reveal the true particle-level events.

When the researchers compared their unfolded results with the most advanced theoretical predictions available, they found a mix of agreement and tension. In the central regions of the collision, where the data is most precise, the measurements showed a discrepancy of up to twelve percent compared to the standard theoretical calculations, even after accounting for known corrections. This difference is larger than the margin of error in the measurement, suggesting that the current models of the proton's internal structure may be incomplete. In other regions of the data, where the number of events is smaller and the uncertainty is higher, the measurements were consistent with the predictions. The study also confirmed that the rate of these events drops significantly when the particles are scattered at large angles, a behavior that aligns with the known limitations of how quarks and gluons are distributed within the proton.

These findings provide a new and valuable constraint on the parton distribution functions, offering a fresh perspective on the proton's structure that complements previous studies. The results indicate that while our current understanding of the strong force is robust, there are still subtle details in how the proton's constituents interact that require further refinement. The data serves as a critical benchmark for future theories, helping to narrow down the possible configurations of the proton's internal landscape. As the field moves forward, these precise measurements will be essential for determining the strength of the strong coupling and for building a more complete picture of the fundamental building blocks of our universe.

Technical Summary: Measurement of the Triple-Differential Cross Section of Z+jet Production

Problem and Motivation
The production of a Z boson decaying into a muon pair in association with at least one jet ( $Z/\gamma^*(\to\mu\mu)+\text{jet} + X$ ) is a theoretically well-understood process that serves as a standard candle for testing perturbative quantum chromodynamics (pQCD). While previously measured in proton-antiproton collisions at the Tevatron and in proton-proton (pp) collisions at the LHC, this paper addresses the need for more granular constraints on the partonic structure of the proton. Specifically, the author aims to utilize a triple-differential measurement to probe the scattering angle in the partonic center-of-mass frame and the fractional momenta of the interacting partons. By subdividing the phase space into three specific kinematic variables, the analysis seeks to provide complementary information to dijet measurements, particularly regarding the decomposition of parton-parton subprocesses (e.g., quark-quark vs. gluon-quark scattering) which behave differently in Z+jet production compared to dijet production.

Methodology
The analysis utilizes proton-proton collision data collected by the CMS experiment at a center-of-mass energy of $\sqrt{s} = 13$ TeV during the 2016–2018 data-taking periods, corresponding to an integrated luminosity of 138 fb $^{-1}$ .

Event Selection: Events are selected requiring at least two oppositely charged muons with $p_T > 29$ GeV and $|\eta| < 2.4$ , forming a Z boson candidate with an invariant mass within $\pm 20$ GeV of the nominal Z mass ($91.1876$ GeV). The Z boson candidate must have $p_T^Z > 25$ GeV. Additionally, at least one jet with $p_T > 20$ GeV and $|y| < 2.4$ is required. Overlap removal is applied between muons and jets ( $\Delta R < 0.4$ ).
Observables: The cross section is measured differentially in three variables:
1. The transverse momentum of the Z boson ( $p_T^Z$ ).
2. The boost of the Z boson and leading-jet system in the laboratory frame ( $y_b = \frac{1}{2}|y_Z + y_{\text{jet}}|$ ).
3. Half of the absolute rapidity separation between the Z boson and the leading jet ( $y^* = \frac{1}{2}|y_Z - y_{\text{jet}}|$ ).
  These variables are chosen because they correlate directly with the energy scale, initial-state parton momentum imbalance, and scattering angle, respectively.
Unfolding: To correct for detector effects (resolution, acceptance, and efficiency), a simultaneous three-dimensional unfolding is performed using the TUNFOLD package. A response matrix is constructed from simulation (MADGRAPH5_aMC@NLO interfaced with PYTHIA 8) mapping particle-level truth to reconstructed observables. The unfolding is validated using alternative simulation models (POWHEG MINNLOPS + PYTHIA 8) to estimate systematic uncertainties related to the unfolding model.
Backgrounds and Systematics: Backgrounds from top quark production, dibosons, and single top quarks are simulated and subtracted. Systematic uncertainties include jet energy scale and resolution (JES/JER), integrated luminosity, pileup modeling, muon reconstruction/trigger efficiencies, and theoretical uncertainties in the background normalization.

Key Contributions

First Triple-Differential Measurement: This work presents the first measurement of the Z+jet cross section unfolded simultaneously in three dimensions ( $p_T^Z$ , $y_b$ , $y^*$ ).
Phase Space Coverage: The measurement covers a wide range of kinematics, with $p_T^Z$ extending up to 1000 GeV (truncated at 250 GeV in specific high- $y^*$ bins due to statistics) and $y^*$ and $y_b$ ranging from 0 to 2.5.
Theoretical Comparison: The unfolded results are compared against:
- Monte Carlo event generators at Next-to-Leading Order (NLO) in pQCD (MADGRAPH5_aMC@NLO and POWHEG MINNLOPS).
- Fixed-order calculations at Next-to-Next-to-Leading Order (NNLO) in pQCD (using NNLOJET), corrected for Next-to-Leading Order (NLO) electroweak (EW) effects and nonperturbative (NP) corrections (hadronization and multiparton interactions).
Subprocess Decomposition Analysis: The paper analyzes the relative contributions of different parton-parton subprocesses (e.g., $gg$, $gq$, $qq$) at NNLO, highlighting how the $y^*$ variable isolates specific kinematic regimes where certain subprocesses are suppressed or enhanced.

Results

Agreement with NLO MC: The unfolded data shows good agreement with NLO MC predictions (MADGRAPH5_aMC@NLO and POWHEG MINNLOPS) across the phase space, with POWHEG MINNLOPS providing a slightly better description in the high- $y^*$ region.
Discrepancy with NNLO Predictions: When compared to NNLO pQCD predictions corrected for EW and NP effects, discrepancies are observed. Notably, in the central rapidity region (where experimental precision is highest, $\sim 1.5\%$ ), the theory underestimates the measured cross section by up to 12%. This discrepancy is not fully covered by the combined experimental and theoretical uncertainties.
Kinematic Dependence: In the highly boosted ( $y_b > 1$ ) and forward-backward (large $y^*$ ) regions, where experimental uncertainties increase (up to 8–13%), the observed differences between data and theory become less significant. The data confirms the predicted decrease in cross section for larger $y^*$ , consistent with parton distribution functions (PDFs) that disfavor the kinematic regime for leading-order parton-parton processes.
Uncertainties: The total experimental uncertainty ranges from 1.5% in the central region to 8% in the boosted regime. At low $p_T^Z$ , the uncertainty is dominated by the jet energy scale; at high $p_T^Z$ , it is dominated by statistical limitations.

Significance
The paper claims that this measurement provides valuable information for characterizing the partonic structure of the proton and determining the strong coupling constant. The triple-differential nature of the measurement offers a unique probe of the scattering angle and parton momentum fractions, providing complementary constraints to existing dijet data. The observed discrepancies between the precise central-region data and NNLO predictions suggest that current theoretical models or PDF sets may require further refinement. The author concludes that these data have the potential to further constrain parton distribution functions and the strong coupling in combined fits with LHC dijet data, particularly given the distinct subprocess decomposition of Z+jet production compared to dijet production.

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