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Measurements of inclusive and differential cross sections for the Higgs boson production and decay to four-leptons in proton-proton collisions at $\sqrt{s}$ = 13 TeV

Using 138 fb $^{-1}$ of proton-proton collision data at 13 TeV collected by the CMS experiment, this paper presents measurements of inclusive and differential fiducial cross sections for Higgs boson production and decay to four leptons, all of which are consistent with Standard Model predictions.

Original authors: CMS Collaboration

Published 2026-07-28

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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

Imagine the universe as a giant, cosmic pinball machine. For decades, physicists have been trying to figure out the rules of the game: why do some particles have weight while others zip around at the speed of light? The answer lies in a mysterious, invisible field that fills all of space, like a thick, cosmic molasses. When particles move through this molasses, they get "stuck" a bit, which gives them mass. The key to proving this molasses exists was finding the particle that represents it: the Higgs boson. Think of the Higgs as the "mascot" of this field. When scientists smash particles together at incredible speeds, they sometimes create a Higgs, but it's incredibly shy and disappears almost instantly, breaking apart into other particles. To catch a glimpse of it, scientists have to look for specific "debris" patterns left behind, like finding a specific type of shell fragment to prove a cannonball was fired.

One of the most reliable ways to spot this Higgs mascot is when it decays into four charged particles called leptons (which are like tiny, electrically charged marbles: electrons or muons). This "four-lepton" path is special because it's very clean; the background noise is low, and the signal is sharp, making it a perfect laboratory to study the Higgs's personality. Scientists care about this because the Higgs isn't just a static object; it interacts with other particles in specific ways. By measuring exactly how often it appears and how it behaves when it breaks apart, physicists can check if our current rulebook for the universe (called the Standard Model) is perfect, or if there are hidden cracks that point to new, unknown physics.

This paper is a massive report card from the CMS experiment at CERN, a giant underground collider in Switzerland. The scientists took a huge pile of data—138 "inverse femtobarns" of proton-proton collisions, which is like collecting every single grain of sand from a massive beach to find a few specific, rare shells. They focused on the moment a Higgs boson is born and immediately turns into four leptons. Their main job was to measure the "fiducial cross section," which is a fancy way of saying: "How often does this specific event happen within the area our detector can actually see?" They found that this happens about 2.73 times for every trillion trillion collisions, with a tiny margin of error. This number matches the prediction from the Standard Model (2.86) almost perfectly, suggesting that, in this specific corner of the universe, the rules we know are holding up very well.

But the scientists didn't just count the events; they wanted to see the Higgs in motion. They measured how the Higgs behaves when it's moving fast (its transverse momentum) and how it's oriented when it's created. They also looked at the angles at which the four leptons fly out, which is like watching how a shattered vase falls to the floor to guess how it was thrown. They checked if the Higgs interacts with other particles (like bottom and charm quarks) in the way we expect, and they even looked for signs that the Higgs might be talking to itself (a "trilinear coupling"). The results? Everything they measured—from the speed of the Higgs to the angles of its decay products—lined up beautifully with the Standard Model predictions.

The paper also used these measurements to set strict limits on some wilder ideas. For instance, they tested if the Higgs's self-interaction strength (how much it pushes or pulls on itself) could be wildly different from what we think. Their data suggests it can't be too crazy; the value is likely between -5.4 and 14.9 times the standard value. Similarly, they checked if the Higgs's connection to bottom and charm quarks was weirdly strong or weak, finding that these connections are also behaving exactly as the Standard Model predicts. While they didn't discover a new particle or a new force in this specific analysis, they did something just as important: they tightened the net. By showing that the Higgs behaves exactly as predicted in so many different ways, they made it much harder for any "new physics" to hide in the shadows. It's a victory for the current theory, confirming that our map of the subatomic world is still accurate, even as we keep looking for the uncharted territories beyond it.

Technical Summary: Measurements of Inclusive and Differential Cross Sections for Higgs Boson Production and Decay to Four-Leptons in Proton-Proton Collisions at $\sqrt{s} = 13$ TeV

Problem and Motivation
The characterization of the Higgs boson ( $H$ ) remains a central objective in particle physics to validate the Standard Model (SM) and search for Beyond the Standard Model (BSM) physics. The decay channel $H \to ZZ \to 4\ell$ (where $\ell = e, \mu$ ) is a "golden channel" due to its fully reconstructible final state and high signal-to-background ratio. While previous measurements using Run 1 (7 and 8 TeV) and partial Run 2 (13 TeV) data established the boson's mass, spin, parity, and inclusive properties, this paper addresses the need for a more comprehensive characterization using the full Run 2 dataset. Specifically, it aims to measure inclusive and differential fiducial cross sections with reduced model dependence, probe kinematic observables sensitive to anomalous couplings, and utilize the transverse momentum ( $p_T$ ) distribution to constrain the Higgs trilinear self-coupling and couplings to bottom and charm quarks.

Methodology
The analysis utilizes proton-proton collision data recorded by the CMS experiment at the LHC during 2016, 2017, and 2018, corresponding to an integrated luminosity of $138 \text{ fb}^{-1}$ .

Event Selection: Candidates are selected requiring four well-identified, isolated leptons ( $4e$ , $4\mu$ , or $2e2\mu$ ) originating from the primary vertex. Events must contain two same-flavor, opposite-sign lepton pairs forming $Z$ boson candidates ( $Z_1$ and $Z_2$ ) with invariant masses within specific ranges ( $40 < m_{Z1} < 120$ GeV and $12 < m_{Z2} < 120$ GeV). The four-lepton invariant mass ( $m_{4\ell}$ ) is required to be between 105 and 160 GeV.
Fiducial Phase Space: To minimize theoretical model dependence, cross sections are measured within a fiducial phase space defined at the generator level. This region closely mimics the experimental acceptance, including requirements on lepton kinematics, isolation, and event topology.
Signal and Background Modeling: Signal samples for the five main production mechanisms (gluon fusion, vector boson fusion, associated production with vector bosons, and associated production with top quarks) are simulated at next-to-leading order (NLO) or higher. Irreducible backgrounds ($ZZ$ production) are estimated using simulation with NNLO corrections, while reducible backgrounds ( $Z$ +jets, $t\bar{t}$ +jets, etc.) are estimated using a data-driven "matrix method" based on lepton misidentification rates.
Statistical Extraction: Inclusive and differential cross sections are extracted via a maximum likelihood fit to the observed $m_{4\ell}$ distribution. The fit simultaneously accounts for signal and background shapes, treating the branching fractions of different final states as unconstrained parameters to enhance model independence. Detector effects are unfolded to the fiducial phase space using response matrices derived from simulation.
Observables: The analysis measures cross sections differentially in bins of:
- Production kinematics: Higgs transverse momentum ( $p_T^H$ ), rapidity ( $|y_H|$ ), jet multiplicity ( $N_{jets}$ ), and jet kinematics ( $p_T^{j1}, p_T^{j2}, m_{jj}, \Delta\eta_{jj}, \Delta\phi_{jj}$ ).
- Decay kinematics: Invariant masses of $Z$ candidates ( $m_{Z1}, m_{Z2}$ ) and five angular variables ( $\theta^*, \theta_1, \theta_2, \Phi, \Phi_1$ ).
- Matrix element discriminants: Six kinematic discriminants sensitive to anomalous HVV couplings ( $a_3, a_2, \Lambda_1, \Lambda_{Z\gamma}$ ) and interference terms.
- Double-differential measurements: Combinations of the above observables to maximize phase space coverage.

Key Contributions

Precision Measurements: This work presents the first measurements of inclusive and differential fiducial cross sections for $H \to ZZ \to 4\ell$ using the full Run 2 dataset ( $138 \text{ fb}^{-1}$ ).
Systematic Uncertainty Reduction: A new method for estimating systematic uncertainties in electron efficiency measurements is introduced, utilizing a root-mean-square approach of alternative tag-and-probe fit variations. This reduces the leading systematic uncertainty (electron reconstruction and selection efficiency) by approximately 40% compared to previous analyses.
Comprehensive Kinematic Coverage: The paper provides a complete set of differential measurements, including six double-differential cross sections, covering the entire fiducial phase space and separating same-flavor and different-flavor final states to account for interference effects.
New Constraints: The analysis derives constraints on the Higgs trilinear self-coupling ( $\kappa_\lambda$ ) and the coupling modifiers for bottom ( $\kappa_b$ ) and charm ( $\kappa_c$ ) quarks directly from the $p_T^H$ distribution, offering a complementary approach to previous constraints derived from simplified template cross sections (STXS) or combined decay channels.

Results

Inclusive Cross Section: The measured inclusive fiducial cross section is $\sigma_{\text{fid}} = 2.73 \pm 0.26 \text{ fb}$ (statistical and systematic uncertainties combined), which is in agreement with the SM expectation of $2.86 \pm 0.15 \text{ fb}$ . The total uncertainty is reduced by 15% compared to previous measurements, primarily due to the improved electron efficiency uncertainty.
Differential Measurements: All differential cross sections measured as functions of $p_T^H$ , $|y_H|$ , jet observables, and decay angles are consistent with SM predictions within uncertainties. The $p$ -values for the compatibility of data with SM predictions range from 0.05 to 0.99.
Anomalous Couplings: The measurements of matrix element discriminants sensitive to anomalous HVV couplings show no significant deviation from the SM hypothesis. The data favor the SM predictions over various BSM scenarios.
Coupling Constraints:
- Trilinear Self-Coupling ( $\kappa_\lambda$ ): The observed 95% confidence level (CL) interval is $-5.4 < \kappa_\lambda < 14.9$ , with an expected interval of $-7.6 < \kappa_\lambda < 17.7$ .
- Bottom and Charm Couplings ( $\kappa_b, \kappa_c$ ): Assuming the branching fraction depends on these couplings, the observed 95% CL limits are $-1.1 < \kappa_b < 1.1$ and $-5.3 < \kappa_c < 5.2$ . These constraints supersede previous results reported for the $H \to ZZ \to 4\ell$ channel alone.

Significance
The paper claims that these results provide a comprehensive and model-independent characterization of the Higgs boson production and decay in the $4\ell$ channel. By utilizing the full Run 2 dataset and improved calibration techniques, the analysis achieves a significant reduction in systematic uncertainties, particularly regarding electron identification. The differential measurements offer a detailed test of the SM predictions across various kinematic regimes, while the derived constraints on the trilinear self-coupling and quark couplings represent an important step toward understanding the Higgs potential and its interactions with fermions. All results are consistent with the Standard Model, reinforcing the current theoretical framework while setting stringent limits on potential BSM physics effects in this specific decay channel.

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