Measurements of the $\mathrm {p}\mathrm {p}\rightarrow \mathrm{Z}\mathrm{Z}$ production cross section and the $\mathrm{Z}\rightarrow 4\ell$ branching fraction, and constraints on anomalous triple gauge couplings at $\sqrt{s} = 13\,\text {TeV}$

Abstract

Four-lepton production in proton-proton collisions, $\mathrm {p}\mathrm {p}\rightarrow (\mathrm{Z}/ \gamma ^*)(\mathrm{Z}/\gamma ^*) \rightarrow 4\ell$, where $\ell = \mathrm {e}$ or $\mu$, is studied at a center-of-mass energy of 13$\,\text {TeV}$ with the CMS detector at the LHC. The data sample corresponds to an integrated luminosity of 35.9$\,\text {fb}^{-1}$. The ZZ production cross section, $\sigma (\mathrm {p}\mathrm {p}\rightarrow \mathrm{Z}\mathrm{Z}) = 17.2 \pm 0.5\,\text {(stat)} \pm 0.7\,\text {(syst)} \pm 0.4\,\text {(theo)} \pm 0.4\,\text {(lumi)} \text { pb}$, measured using events with two opposite-sign, same-flavor lepton pairs produced in the mass region $60< m_{\ell ^+\ell ^-} < 120\,\text {GeV}$, is consistent with standard model predictions. Differential cross sections are measured and are well described by the theoretical predictions. The Z boson branching fraction to four leptons is measured to be $\mathcal {B}(\mathrm{Z}\rightarrow 4\ell ) = 4.83 _{-0.22}^{+0.23} (stat)_{-0.29}^{+0.32} (syst) \pm 0.08 (theo) \pm 0.12 (lumi) \times 10^{-6}$ for events with a four-lepton invariant mass in the range $80< m_{4\ell } < 100\,\text {GeV}$ and a dilepton mass $m_{\ell \ell } > 4\,\text {GeV}$ for all opposite-sign, same-flavor lepton pairs. The results agree with standard model predictions. The invariant mass distribution of the four-lepton system is used to set limits on anomalous ZZZ and ZZ$\gamma$ couplings at 95% confidence level: $-0.0012<f_4^\mathrm{Z}<0.0010$, $-0.0010<f_5^\mathrm{Z}<0.0013$, $-0.0012<f_4^{\gamma }<0.0013$, $-0.0012<f_5^{\gamma }< 0.0013$.

1 Introduction

Measurements of diboson production at the CERN LHC allow precision tests of the standard model (SM). In the SM, $\mathrm{Z}\mathrm{Z}$ production proceeds mainly through quark-antiquark t- and u-channel scattering diagrams. In calculations at higher orders in quantum chromodynamics (QCD), gluon-gluon fusion also contributes via box diagrams with quark loops. There are no tree-level contributions to $\mathrm{Z}\mathrm{Z}$ production from triple gauge boson vertices in the SM. Anomalous triple gauge couplings (aTGC) could be induced by new physics models such as supersymmetry [1]. Nonzero aTGCs may be parametrized using an effective Lagrangian as in Ref. [2]. In this formalism, two $\mathrm{Z} \mathrm{Z} \mathrm{Z}$ and two $\mathrm{Z} \mathrm{Z} \gamma$ couplings are allowed by electromagnetic gauge invariance and Lorentz invariance for on-shell $\mathrm{Z}$ bosons. These are described by two CP-violating ($f_4^{\mathrm {V}}$) and two CP-conserving ($f_5^{\mathrm {V}}$) parameters, where ${\mathrm {V}} = \mathrm{Z}$ or $\gamma$.

Previous measurements of the ZZ production cross section by the CMS Collaboration were performed for pairs of on-shell $\mathrm{Z}$ bosons, produced in the dilepton mass range 60–120$\,\text {GeV}$  [3,4,5,6]. These measurements were made with data sets corresponding to integrated luminosities of 5.1$\,\text {fb}^{-1}$ at $\sqrt{s} = 7\,\text {TeV}$ and 19.6$\,\text {fb}^{-1}$ at $\sqrt{s} = 8\,\text {TeV}$ in the $\mathrm{Z}\mathrm{Z}\rightarrow 2\ell 2\ell ^{\prime \prime }$ and $\mathrm{Z}\mathrm{Z}\rightarrow 2\ell 2\nu$ decay channels, where $\ell = \mathrm {e}$ or $\mathrm {\mu }$ and $\ell ^{\prime \prime } = \mathrm {e}, \mathrm {\mu }$, or $\mathrm {\tau }$, and with an integrated luminosity of 2.6$\,\text {fb}^{-1}$ at $\sqrt{s} = 13\,\text {TeV}$ in the $\mathrm{Z}\mathrm{Z} \rightarrow 2\ell 2\ell '$ decay channel, where $\ell ' = \mathrm {e}$ or $\mathrm {\mu }$. All of them agree with SM predictions. The ATLAS Collaboration produced similar results at $\sqrt{s} = 7$, 8, and 13$\,\text {TeV}$  [7,8,9,10], which also agree with the SM. These measurements are important for testing predictions that were recently made available at next-to-next-to-leading order (NNLO) in QCD [11]. Comparing these predictions with data at a range of center-of-mass energies provides information about the electroweak gauge sector of the SM. Because the uncertainty of the CMS measurement at $\sqrt{s} = 13\,\text {TeV}$ [6] was dominated by the statistical uncertainty of the observed data, repeating and extending the measurement with a larger sample of proton-proton collision data at $\sqrt{s} = 13\,\text {TeV}$ improves the precision of the results.

The most stringent previous limits on $\mathrm{Z}\mathrm{Z}\mathrm{Z}$ and $\mathrm{Z}\mathrm{Z}\gamma$ aTGCs from CMS were set using the 7 and $8\,\text {TeV}$ data samples: $-0.0022<f_4^\mathrm{Z}<0.0026$, $-0.0023<f_5^\mathrm{Z}<0.0023$, $-0.0029<f_4^{\gamma }<0.0026$, and $-0.0026<f_5^{\gamma }<0.0027$ at 95% confidence level (CL) [4, 5]. Similar limits were obtained by the ATLAS Collaboration [12], who also recently produced limits using $13\,\text {TeV}$ data [10].

Extending the dilepton mass range to lower values allows measurements of $\left( \mathrm{Z}/\gamma ^{*}\right) \left( \mathrm{Z}/\gamma ^{*}\right)$ production, where $\mathrm{Z}$ indicates an on-shell $\mathrm{Z}$ boson or an off-shell $\mathrm{Z}^{*}$ boson. The resulting sample includes Higgs boson events in the $\mathrm{H} \rightarrow \mathrm{Z}\mathrm{Z}^{*} \rightarrow 2\ell 2\ell '$ channel, and rare decays of a single $\mathrm{Z}$ boson to four leptons. The $\mathrm{Z}\rightarrow \ell ^+\ell ^-\gamma ^{*} \rightarrow 2\ell 2\ell '$ decay was studied in detail at LEP [13] and was observed in pp collisions by CMS [6, 14] and ATLAS [15]. Although the branching fraction for this decay is orders of magnitude smaller than that for the $\mathrm{Z}\rightarrow \ell ^+\ell ^-$ decay, the precisely known mass of the $\mathrm{Z}$ boson makes the four-lepton mode useful for calibrating mass measurements of the nearby Higgs boson resonance.

This paper reports a study of four-lepton production ($\mathrm {p}\mathrm {p} \rightarrow 2\ell 2\ell '$, where $2\ell$ and $2\ell '$ indicate opposite-sign pairs of electrons or muons) at $\sqrt{s} = 13\,\text {TeV}$ with a data set corresponding to an integrated luminosity of $35.9 \pm 0.9$ $\,\text {fb}^{-1}$ recorded in 2016. Cross sections are measured for nonresonant production of pairs of $\mathrm{Z}$ bosons, $\mathrm {p}\mathrm {p} \rightarrow \mathrm{Z}\mathrm{Z}$, where both $\mathrm{Z}$ bosons are produced on-shell, defined as the mass range 60–120$\,\text {GeV}$, and resonant $\mathrm {p}\mathrm {p} \rightarrow \mathrm{Z}\rightarrow 4\ell$ production. Detailed discussion of resonant Higgs boson production decaying to $\mathrm{Z}\mathrm{Z}^*$, is beyond the scope of this paper and may be found in Ref. [16].

2 The CMS detector

A detailed description of the CMS detector, together with a definition of the coordinate system used and the relevant kinematic variables, can be found in Ref. [17].

The central feature of the CMS apparatus is a superconducting solenoid of 6$\text { m}$ internal diameter, providing a magnetic field of 3.8$\text { T}$. Within the solenoid volume are a silicon pixel and strip tracker, a lead tungstate crystal electromagnetic calorimeter (ECAL), and a brass and scintillator hadron calorimeter, which provide coverage in pseudorapidity $| \eta | < 1.479$ in a cylindrical barrel and $1.479< | \eta | < 3.0$ in two endcap regions. Forward calorimeters extend the coverage provided by the barrel and endcap detectors to $|\eta | < 5.0$. Muons are measured in gas-ionization detectors embedded in the steel flux-return yoke outside the solenoid in the range $|\eta | < 2.4$, with detection planes made using three technologies: drift tubes, cathode strip chambers, and resistive plate chambers.

Electron momenta are estimated by combining energy measurements in the ECAL with momentum measurements in the tracker. The momentum resolution for electrons with transverse momentum $p_{\mathrm {T}} \approx 45\,\text {GeV}$ from $\mathrm{Z}\rightarrow \mathrm {e}^+\mathrm {e}^-$ decays ranges from 1.7% for nonshowering electrons in the barrel region to 4.5% for showering electrons in the endcaps [18]. Matching muons to tracks identified in the silicon tracker results in a $p_{\mathrm {T}}$ resolution for muons with $20<p_{\mathrm {T}} < 100\,\text {GeV}$ of 1.3–2.0% in the barrel and better than 6% in the endcaps. The $p_{\mathrm {T}}$ resolution in the barrel is better than 10% for muons with $p_{\mathrm {T}}$ up to 1$\,\text {TeV}$  [19].

3 Signal and background simulation

Signal events are generated with powheg  2.0 [20,21,22,23,24] at next-to-leading order (NLO) in QCD for quark-antiquark processes and leading order (LO) for quark-gluon processes. This includes $\mathrm{Z}\mathrm{Z}$, $\mathrm{Z}\gamma ^*$, $\mathrm{Z}$, and $\gamma ^*\gamma ^*$ production with a constraint of $m_{\ell \ell '} > 4\,\text {GeV}$ applied to all pairs of oppositely charged leptons at the generator level to avoid infrared divergences. The $\mathrm{g} \mathrm{g} \rightarrow \mathrm{Z}\mathrm{Z}$ process is simulated at LO with mcfm  v7.0 [25]. These samples are scaled to correspond to cross sections calculated at NNLO in QCD for $\mathrm{q}\overline{\mathrm{q}}\rightarrow \mathrm{Z}\mathrm{Z}$ [11] (a scaling K factor of 1.1) and at NLO in QCD for $\mathrm{g} \mathrm{g} \rightarrow \mathrm{Z}\mathrm{Z}$ [26] (K factor of 1.7). The $\mathrm{g} \mathrm{g} \rightarrow \mathrm{Z}\mathrm{Z}$ process is calculated to $\mathcal {O}\left( \alpha _s^3\right)$, where $\alpha _s$ is the strong coupling constant, while the other contributing processes are calculated to $\mathcal {O}\left( \alpha _s^2\right)$; this higher-order correction is included because the effect is known to be large [26]. Electroweak $\mathrm{Z}\mathrm{Z}$ production in association with two jets is generated with Phantom v1.2.8 [27].

A sample of Higgs boson events is produced in the gluon-gluon fusion process at NLO with powheg. The Higgs boson decay is modeled with jhugen 3.1.8 [28,29,30]. Its cross section is scaled to the NNLO prediction with a K factor of 1.7 [26].

Samples for background processes containing four prompt leptons in the final state, $\mathrm{t}\overline{\mathrm{t}} \mathrm{Z}$ and $\mathrm {W}\mathrm {W}\mathrm{Z}$ production, are produced with MadGraph 5_amc@nlo v2.3.3 [31]. The $\mathrm{q}\overline{\mathrm{q}}\rightarrow \mathrm {W}\mathrm{Z}$ process is generated with powheg.

Samples with aTGC contributions included are generated at LO with sherpa v2.1.1 [32]. Distributions from the sherpa samples are normalized such that the total yield of the SM sample is the same as that of the powheg sample.

The pythia  v8.175 [23, 33, 34] package is used for parton showering, hadronization, and the underlying event simulation, with parameters set by the CUETP8M1 tune [35], for all samples except the samples generated with sherpa, which performs these functions itself. The NNPDF 3.0 [36] set is used as the default set of parton distribution functions (PDFs). For all simulated event samples, the PDFs are calculated to the same order in QCD as the process in the sample.

The detector response is simulated using a detailed description of the CMS detector implemented with the Geant4 package [37]. The event reconstruction is performed with the same algorithms used for data. The simulated samples include additional interactions per bunch crossing, referred to as pileup. The simulated events are weighted so that the pileup distribution matches the data, with an average of about 27 interactions per bunch crossing.

4 Event reconstruction

All long-lived particles—electrons, muons, photons, and charged and neutral hadrons—in each collision event are identified and reconstructed with the CMS particle-flow (PF) algorithm [38] from a combination of the signals from all subdetectors. Reconstructed electrons [18] and muons [19] are considered candidates for inclusion in four-lepton final states if they have $p_{\mathrm {T}} ^\mathrm {e}> 7\,\text {GeV}$ and $|\eta ^\mathrm {e} | < 2.5$ or $p_{\mathrm {T}} ^\mathrm {\mu }> 5\,\text {GeV}$ and $|\eta ^\mathrm {\mu } | < 2.4$.

Lepton candidates are also required to originate from the event vertex, defined as the reconstructed proton-proton interaction vertex with the largest value of summed physics object $p_{\mathrm {T}} ^2$. The physics objects used in the event vertex definition are the objects returned by a jet finding algorithm [39, 40] applied to all charged tracks associated with the vertex, plus the corresponding associated missing transverse momentum [41]. The distance of closest approach between each lepton track and the event vertex is required to be less than 0.5$\text { cm}$ in the plane transverse to the beam axis, and less than 1$\text { cm}$ in the direction along the beam axis. Furthermore, the significance of the three-dimensional impact parameter relative to the event vertex, $\mathrm {SIP_{3D}}$, is required to satisfy $\mathrm {SIP_{3D}} \equiv | \mathrm {IP} / \sigma _\mathrm {IP} | < 10$ for each lepton, where $\mathrm {IP}$ is the distance of closest approach of each lepton track to the event vertex and $\sigma _\mathrm {IP}$ is its associated uncertainty.

Lepton candidates are required to be isolated from other particles in the event. The relative isolation is defined as

$$\begin{aligned} R_\text {iso} = \bigg [ \sum _{\begin{array}{c} \text {charged} \\ \text {hadrons} \end{array}} \!\! p_{\mathrm {T}} \, + \, \max \big (0, \sum _{\begin{array}{c} \text {neutral} \\ \text {hadrons} \end{array}} \!\! p_{\mathrm {T}} \, + \, \sum _{\text {photons}} \!\! p_{\mathrm {T}} \, - \, p_{\mathrm {T}} ^\mathrm {PU} \big )\bigg ] \bigg / p_{\mathrm {T}} ^{\ell }, \end{aligned}$$

(1)

where the sums run over the charged and neutral hadrons and photons identified by the PF algorithm, in a cone defined by $\Delta R \equiv \sqrt{\smash [b]{\left( \Delta \eta \right) ^2 + \left( \Delta \phi \right) ^2}} < 0.3$ around the lepton trajectory. Here $\phi$ is the azimuthal angle in radians. To minimize the contribution of charged particles from pileup to the isolation calculation, charged hadrons are included only if they originate from the event vertex. The contribution of neutral particles from pileup is $p_{\mathrm {T}} ^\mathrm {PU}$. For electrons, $p_{\mathrm {T}} ^\mathrm {PU}$ is evaluated with the “jet area” method described in Ref. [42]; for muons, it is taken to be half the sum of the $p_{\mathrm {T}}$ of all charged particles in the cone originating from pileup vertices. The factor one-half accounts for the expected ratio of charged to neutral particle energy in hadronic interactions. A lepton is considered isolated if $R_\text {iso} < 0.35$.

The lepton reconstruction, identification, and isolation efficiencies are measured with a “tag-and-probe” technique [43] applied to a sample of $\mathrm{Z}\rightarrow \ell ^+\ell ^-$ data events. The measurements are performed in several bins of $p_{\mathrm {T}} ^{\ell }$ and $|\eta ^\ell |$. The electron reconstruction and selection efficiency in the ECAL barrel (endcaps) varies from about 85% (77%) at $p_{\mathrm {T}} ^{\mathrm {e}} \approx 10\,\text {GeV}$ to about 95% (89%) for $p_{\mathrm {T}} ^{\mathrm {e}} \ge 20\,\text {GeV}$, while in the barrel-endcap transition region this efficiency is about 85% averaged over all electrons with $p_{\mathrm {T}} ^{\mathrm {e}} > 7\,\text {GeV}$. The muons are reconstructed and identified with efficiencies above ${\sim }98\%$ within $|\eta ^{\mathrm {\mu }} | < 2.4$.

5 Event selection

The primary triggers for this analysis require the presence of a pair of loosely isolated leptons of the same or different flavors [44]. The highest $p_{\mathrm {T}}$ lepton must have $p_{\mathrm {T}} ^\ell > 17\,\text {GeV}$, and the subleading lepton must have $p_{\mathrm {T}} ^\mathrm {e}> 12\,\text {GeV}$ if it is an electron or $p_{\mathrm {T}} ^\mathrm {\mu }> 8\,\text {GeV}$ if it is a muon. The tracks of the triggering leptons are required to originate within 2 mm of each other in the plane transverse to the beam axis. Triggers requiring a triplet of lower-$p_{\mathrm {T}}$ leptons with no isolation criterion, or a single high-$p_{\mathrm {T}}$ electron or muon, are also used. An event is used if it passes any trigger regardless of the decay channel. The total trigger efficiency for events within the acceptance of this analysis is greater than 98%.

The four-lepton candidate selections are based on those used in Ref. [45]. A signal event must contain at least two $\mathrm{Z}/\gamma ^{*}$ candidates, each formed from an oppositely charged pair of isolated electron candidates or muon candidates. Among the four leptons, the highest $p_{\mathrm {T}}$ lepton must have $p_{\mathrm {T}} > 20\,\text {GeV}$, and the second-highest $p_{\mathrm {T}}$ lepton must have $p_{\mathrm {T}} ^\mathrm {e}> 12\,\text {GeV}$ if it is an electron or $p_{\mathrm {T}} ^\mathrm {\mu }> 10\,\text {GeV}$ if it is a muon. All leptons are required to be separated from each other by $\Delta R \left( \ell _1, \ell _2 \right) > 0.02$, and electrons are required to be separated from muons by $\Delta R \left( \mathrm {e}, \mu \right) > 0.05$.

Within each event, all permutations of leptons giving a valid pair of $\mathrm{Z}/\gamma ^{*}$ candidates are considered separately. Within each $4\ell$ candidate, the dilepton candidate with an invariant mass closest to 91.2$\,\text {GeV}$, taken as the nominal $\mathrm{Z}$ boson mass [46], is denoted $\mathrm{Z}_1$ and is required to have a mass greater than 40$\,\text {GeV}$. The other dilepton candidate is denoted $\mathrm{Z}_2$. Both $m_{\mathrm{Z}_1}$ and $m_{\mathrm{Z}_2}$ are required to be less than 120$\,\text {GeV}$. All pairs of oppositely charged leptons in the $4\ell$ candidate are required to have $m_{\ell \ell '} > 4\,\text {GeV}$ regardless of their flavor.

If multiple $4\ell$ candidates within an event pass all selections, the one with $m_{\mathrm{Z}_1}$ closest to the nominal $\mathrm{Z}$ boson mass is chosen. In the rare case of further ambiguity, which may arise in less than 0.5% of events when five or more passing lepton candidates are found, the $\mathrm{Z}_2$ candidate that maximizes the scalar $p_{\mathrm {T}}$ sum of the four leptons is chosen.

Additional requirements are applied to select events for measurements of specific processes. The $\mathrm {p}\mathrm {p} \rightarrow \mathrm{Z}\mathrm{Z}$ cross section is measured using events where both $m_{\mathrm{Z}_1}$ and $m_{\mathrm{Z}_2}$ are greater than 60$\,\text {GeV}$. The $\mathrm{Z}\rightarrow 4\ell$ branching fraction is measured using events with $80< m_{4\ell } < 100\,\text {GeV}$, a range chosen to retain most of the decays in the resonance while removing most other processes with four-lepton final states. Decays of the $\mathrm{Z}$ bosons to $\tau$ leptons with subsequent decays to electrons and muons are heavily suppressed by requirements on lepton $p_{\mathrm {T}}$, and the contribution of such events is less than 0.5% of the total $\mathrm{Z}\mathrm{Z}$ yield. If these events pass the selection requirements of the analysis, they are considered signal, while they are not considered at generator level in the cross section unfolding procedure. Thus, the correction for possible $\tau$ decays is included in the efficiency calculation.

6 Background estimate

The major background contributions arise from $\mathrm{Z}$ boson and $\mathrm {W}\mathrm{Z}$ diboson production in association with jets and from $\mathrm{t}\overline{\mathrm{t}}$ production. In all these cases, particles from jet fragmentation satisfy both lepton identification and isolation criteria, and are thus misidentified as signal leptons.

The probability for such objects to be selected is measured from a sample of $\mathrm{Z}+ \ell _\text {candidate}$ events, where $\mathrm{Z}$ denotes a pair of oppositely charged, same-flavor leptons that pass all analysis requirements and satisfy $| m_{\ell ^+\ell ^-} - m_{\mathrm{Z}} | < 10\,\text {GeV}$, where $m_\mathrm{Z}$ is the nominal $\mathrm{Z}$ boson mass. Each event in this sample must have exactly one additional object $\ell _\text {candidate}$ that passes relaxed identification requirements with no isolation requirements applied. The misidentification probability for each lepton flavor, measured in bins of lepton candidate $p_{\mathrm {T}}$ and $\eta$, is defined as the ratio of the number of candidates that pass the final isolation and identification requirements to the total number in the sample. The number of $\mathrm{Z}+ \ell _\text {candidate}$ events is corrected for the contamination from $\mathrm {W}\mathrm{Z}$ production and $\mathrm{Z}\mathrm{Z}$ production in which one lepton is not reconstructed. These events have a third genuine, isolated lepton that must be excluded from the misidentification probability calculation. The WZ contamination is suppressed by requiring the missing transverse momentum $p_{\mathrm {T}} ^\text {miss}$ to be below 25$\,\text {GeV}$. The $p_{\mathrm {T}} ^\text {miss}$ is defined as the magnitude of the missing transverse momentum vector ${\vec p}_{\mathrm {T}}^{\text {miss}}$, the projection onto the plane transverse to the beams of the negative vector sum of the momenta of all reconstructed PF candidates in the event, corrected for the jet energy scale. Additionally, the transverse mass calculated with ${\vec p}_{\mathrm {T}}^{\text {miss}}$ and the ${\vec {p}}_{\mathrm {T}}$ of $\ell _\text {candidate}$, $m_{\mathrm {T}} \equiv \sqrt{\smash [b]{(p_{\mathrm {T}} ^\ell + p_{\mathrm {T}} ^\text {miss})^2 - ({\vec {p}}_{\mathrm {T}} ^{\ell } + {\vec p}_{\mathrm {T}}^{\text {miss}})^2}}$, is required to be less than 30$\,\text {GeV}$. The residual contribution of $\mathrm {W}\mathrm{Z}$ and $\mathrm{Z}\mathrm{Z}$ events, which may be up to a few percent of the events with $\ell _\text {candidate}$ passing all selection criteria, is estimated from simulation and subtracted.

To account for all sources of background events, two control samples are used to estimate the number of background events in the signal regions. Both are defined to contain events with a dilepton candidate satisfying all requirements ($\mathrm{Z}_1$) and two additional lepton candidates $\ell ^{\prime +}\ell ^{\prime -}$. In one control sample, enriched in $\mathrm {W}\mathrm{Z}$ events, one $\ell ^{\prime }$ candidate is required to satisfy the full identification and isolation criteria and the other must fail the full criteria and instead satisfy only the relaxed ones; in the other, enriched in $\mathrm{Z}$+jets events, both $\ell ^{\prime }$ candidates must satisfy the relaxed criteria, but fail the full criteria. The additional leptons must have opposite charge and the same flavor ($\mathrm {e}^{\pm }\mathrm {e}^{\mp }, \mathrm {\mu }^{\pm }\mathrm {\mu }^{\mp }$). From this set of events, the expected number of background events in the signal region, denoted “$\mathrm{Z}+ \text {X}$” in the figures, is obtained by scaling the number of observed $\mathrm{Z}_1+\ell ^{\prime +}\ell ^{\prime -}$ events by the misidentification probability for each lepton failing the selection. It is found to be approximately 4% of the total expected yield. The procedure is described in more detail in Ref. [45].

In addition to these nonprompt backgrounds, $\mathrm{t}\overline{\mathrm{t}} \mathrm{Z}$ and $\mathrm {W}\mathrm {W}\mathrm{Z}$ processes contribute a smaller number of events with four prompt leptons, which is estimated from simulated samples to be around 1% of the expected $\mathrm{Z}\mathrm{Z} \rightarrow 4\ell$ yield. In the $\mathrm{Z}\rightarrow 4\ell$ selection, the contribution from these backgrounds is negligible. The total background contributions to the $\mathrm{Z}\rightarrow 4\ell$ and $\mathrm{Z}\mathrm{Z}\rightarrow 4\ell$ signal regions are summarized in Section 8.

7 Systematic uncertainties

The major sources of systematic uncertainty and their effect on the measured cross sections are summarized in Table 1. In both data and simulated event samples, trigger efficiencies are evaluated with a tag-and-probe technique. The ratio of data to simulation is applied to simulated events, and the size of the resulting change in expected yield is taken as the uncertainty in the determination of the trigger efficiency. This uncertainty is around 2% of the final estimated yield. For $\mathrm{Z}\rightarrow 4\mathrm {e}$ events, the uncertainty increases to 4%.

Table 1 The contributions of each source of systematic uncertainty in the cross section measurements. The integrated luminosity uncertainty, and the PDF and scale uncertainties, are considered separately. All other uncertainties are added in quadrature into a single systematic uncertainty. Uncertainties that vary by decay channel are listed as a range

The lepton identification, isolation, and track reconstruction efficiencies in simulation are corrected with scaling factors derived with a tag-and-probe method and applied as a function of lepton $p_{\mathrm {T}}$ and $\eta$. To estimate the uncertainties associated with the tag-and-probe technique, the total yield is recomputed with the scaling factors varied up and down by the tag-and-probe fit uncertainties. The uncertainties associated with lepton efficiency in the $\mathrm{Z}\mathrm{Z} \rightarrow 4\ell$ ($\mathrm{Z}\rightarrow 4\ell$) signal regions are found to be 6(10)% in the $4\mathrm {e}$, 3(6)% in the $2\mathrm {e}2\mu$, and 2(7)% in the $4\mu$ final states. These uncertainties are higher for $\mathrm{Z}\rightarrow 4\ell$ events because the leptons generally have lower $p_{\mathrm {T}}$, and the samples used in the tag-and-probe method have fewer events and more contamination from nonprompt leptons in this low-$p_{\mathrm {T}}$ region.

Uncertainties due to the effect of factorization ($\mu _\mathrm {F}$) and renormalization ($\mu _\mathrm {R}$) scale choices on the $\mathrm{Z}\mathrm{Z} \rightarrow 4\ell$ acceptance are evaluated with powheg and mcfm by varying the scales up and down by a factor of two with respect to the default values $\mu _\mathrm {F} = \mu _\mathrm {R} = m_{\mathrm{Z}\mathrm{Z}}$. All combinations are considered except those in which $\mu _\mathrm {F}$ and $\mu _\mathrm {R}$ differ by a factor of four. Parametric uncertainties (PDF$+ \alpha _s$) are evaluated according to the pdf4lhc prescription [47] in the acceptance calculation, and with NNPDF3.0 in the cross section calculations. An additional theoretical uncertainty arises from scaling the powheg $\mathrm{q}\overline{\mathrm{q}}\rightarrow \mathrm{Z}\mathrm{Z}$ simulated sample from its NLO cross section to the NNLO prediction, and the mcfm $\mathrm{g} \mathrm{g} \rightarrow \mathrm{Z}\mathrm{Z}$ samples from their LO cross sections to the NLO predictions. The change in the acceptance corresponding to this scaling procedure is found to be 1.1%. All these theoretical uncertainties are added in quadrature.

The largest uncertainty in the estimated background yield arises from differences in sample composition between the $\mathrm{Z}+ \ell _\text {candidate}$ control sample used to calculate the lepton misidentification probability and the $\mathrm{Z}+ \ell ^+\ell ^-$ control sample. A further uncertainty arises from the limited number of events in the $\mathrm{Z}+ \ell _\text {candidate}$ sample. A systematic uncertainty of 40% is applied to the lepton misidentification probability to cover both effects. The size of this uncertainty varies by channel, but is less than 1% of the total expected yield.

The uncertainty in the integrated luminosity of the data sample is 2.5% [48].

8 Cross section measurements

The distributions of the four-lepton mass and the masses of the $\mathrm{Z}_1$ and $\mathrm{Z}_2$ candidates are shown in Fig. 1. The expected distributions describe the data well within uncertainties. The SM predictions include nonresonant $\mathrm{Z}\mathrm{Z}$ predictions, production of the SM Higgs boson with mass 125$\,\text {GeV}$  [49], and resonant $\mathrm{Z}\rightarrow 4\ell$ production. The backgrounds estimated from data and simulation are also shown. The reconstructed invariant mass of the $\mathrm{Z}_1$ candidates, and a scatter plot showing the correlation between $m_{\mathrm{Z}_2}$ and $m_{\mathrm{Z}_1}$ in data events, are shown in Fig. 2. In the scatter plot, clusters of events corresponding to $\mathrm{Z}\mathrm{Z}\rightarrow 4\ell$, $\mathrm{Z}\gamma ^*\rightarrow 4\ell$, and $\mathrm{Z}\rightarrow 4\ell$ production can be seen.

Fig. 1

Distributions of (upper) the four-lepton invariant mass $m_{4\ell }$ and (lower) the dilepton invariant mass of all $\mathrm{Z}/\gamma ^*$ bosons in selected four-lepton events. Both selected dilepton candidates are included in each event. In the $m_{4\ell }$ distribution, bin contents are normalized to a bin width of 25$\,\text {GeV}$; horizontal bars on the data points show the range of the corresponding bin. Points represent the data, while filled histograms represent the SM prediction and background estimate. Vertical bars on the data points show their statistical uncertainty. Shaded grey regions around the predicted yield represent combined statistical, systematic, theoretical, and integrated luminosity uncertainties

Fig. 2

(Upper): the distribution of the reconstructed mass of $\mathrm{Z}_1$, the dilepton candidate closer to the nominal $\mathrm{Z}$ boson mass. Points represent the data, while filled histograms represent the SM prediction and background estimate. Vertical bars on the data points show their statistical uncertainty. Shaded grey regions around the predicted yield represent combined statistical, systematic, theoretical, and integrated luminosity uncertainties. (Lower): the reconstructed $m_{\mathrm{Z}_2}$ plotted against the reconstructed $m_{\mathrm{Z}_1}$ in data events, with distinctive markers for each final state. For readability, only every fourth event is plotted

The four-lepton invariant mass distribution below 100$\,\text {GeV}$ is shown in Fig. 3 (upper). Figure 3 (lower) shows $m_{\mathrm{Z}_2}$ plotted against $m_{\mathrm{Z}_1}$ for events with $m_{4\ell }$ between 80 and 100$\,\text {GeV}$, and the observed and expected event yields in this mass region are given in Table 2. The yield of events in the $4\mathrm {e}$ final state is significantly lower than in the $4\mathrm {\mu }$ final state because minimum $p_{\mathrm {T}}$ thresholds are higher for electrons than for muons, and inefficiencies in the detection of low-$p_{\mathrm {T}}$ leptons affect electrons more strongly than they affect muons.

Fig. 3

(Upper): the distribution of the reconstructed four-lepton mass $m_{4\ell }$ for events selected with $80< m_{4\ell } < 100\,\text {GeV}$. Points represent the data, while filled histograms represent the SM prediction and background estimate. Vertical bars on the data points show their statistical uncertainty. Shaded grey regions around the predicted yield represent combined statistical, systematic, theoretical, and integrated luminosity uncertainties. (Lower): the reconstructed $m_{\mathrm{Z}_2}$ plotted against the reconstructed $m_{\mathrm{Z}_1}$ for all data events selected with $m_{4\ell }$ between 80 and 100$\,\text {GeV}$, with distinctive markers for each final state

Table 2 The observed and expected yields of four-lepton events in the mass region $80< m_{4\ell } < 100\,\text {GeV}$ and estimated yields of background events, shown for each final state and summed in the total expected yield. The first uncertainty is statistical, the second one is systematic. The systematic uncertainties do not include the uncertainty in the integrated luminosity

The reconstructed four-lepton invariant mass is shown in Fig. 4 (upper) for events with two on-shell $\mathrm{Z}$ bosons. Figure 4 (lower) shows the invariant mass distribution for all $\mathrm{Z}$ boson candidates in these events. The corresponding observed and expected yields are given in Table 3.

Fig. 4

Distributions of (upper) the four-lepton invariant mass $m_{\mathrm{Z}\mathrm{Z}}$ and (lower) dilepton candidate mass for four-lepton events selected with both $\mathrm{Z}$ bosons on-shell. Points represent the data, while filled histograms represent the SM prediction and background estimate. Vertical bars on the data points show their statistical uncertainty. Shaded grey regions around the predicted yield represent combined statistical, systematic, theoretical, and integrated luminosity uncertainties. In the $m_{\mathrm{Z}\mathrm{Z}}$ distribution, bin contents are normalized to the bin widths, using a unit bin size of 50$\,\text {GeV}$; horizontal bars on the data points show the range of the corresponding bin

Table 3 The observed and expected yields of $\mathrm{Z}\mathrm{Z}$ events, and estimated yields of background events, shown for each final state and summed in the total expected yield. The first uncertainty is statistical, the second one is systematic. The systematic uncertainties do not include the uncertainty in the integrated luminosity

The observed yields are used to evaluate the $\mathrm {p}\mathrm {p} \rightarrow \mathrm{Z}\rightarrow 4\ell$ and $\mathrm {p}\mathrm {p} \rightarrow \mathrm{Z}\mathrm{Z} \rightarrow 4\ell$ production cross sections from a combined fit to the number of observed events in all the final states. The likelihood is a combination of individual channel likelihoods for the signal and background hypotheses with the statistical and systematic uncertainties in the form of scaling nuisance parameters. The fiducial cross section is measured by scaling the cross section in the simulation by the ratio of the measured and predicted event yields given by the fit.

The definitions for the fiducial phase spaces for the $\mathrm{Z}\rightarrow 4\ell$ and $\mathrm{Z}\mathrm{Z} \rightarrow 4\ell$ cross section measurements are given in Table 4. In the $\mathrm{Z}\mathrm{Z} \rightarrow 4\ell$ case, the $\mathrm{Z}$ bosons used in the fiducial definition are built by pairing final-state leptons using the same algorithm as is used to build $\mathrm{Z}$ boson candidates from reconstructed leptons. The generator-level leptons used for the fiducial cross section calculation are “dressed” by adding the momenta of generator-level photons within $\Delta R\left( \ell ,\gamma \right) < 0.1$ to their momenta.

Table 4 Fiducial definitions for the reported cross sections. The common requirements are applied for both measurements

The measured cross sections are

$$\begin{aligned} \begin{aligned} \sigma _{\text {fid}}&(\mathrm {p}\mathrm {p} \rightarrow \mathrm{Z}\rightarrow 4\ell ) \\&= 31.2 _{-1.4}^{+1.5} \,\text {(stat)} _{-1.9}^{+2.1} \,\text {(syst)} \pm 0.8 \,\text {(lumi)} \text { fb},\\ \sigma _{\text {fid}}&(\mathrm {p}\mathrm {p} \rightarrow \mathrm{Z}\mathrm{Z} \rightarrow 4\ell ) \\&= 40.9 \pm 1.3 \,\text {(stat)} \pm 1.4 \,\text {(syst)} \pm 1.0 \,\text {(lumi)} \text { fb}. \end{aligned} \end{aligned}$$

(2)

The $\mathrm {p}\mathrm {p} \rightarrow \mathrm{Z}\rightarrow 4\ell$ fiducial cross section can be compared to $27.9^{+1.0}_{-1.5} \pm 0.6\text { fb}$ calculated at NLO in QCD with powheg using the same settings as used for the simulated sample described in Section 3, with dynamic scales $\mu _\mathrm {F} = \mu _\mathrm {R} = m_{4\ell }$. The uncertainties correspond to scale and PDF variations, respectively. The $\mathrm{Z}\mathrm{Z}$ fiducial cross section can be compared to $34.4^{+0.7}_{-0.6} \pm 0.5\text { fb}$ calculated with powheg and mcfm using the same settings as the simulated samples, or to $36.0_{-0.8}^{+0.9}$ computed with matrix at NNLO. The powheg and matrix calculations used dynamic scales $\mu _\mathrm {F} = \mu _\mathrm {R} = m_{4\ell }$, while the contribution from mcfm was computed with dynamic scales $\mu _\mathrm {F} = \mu _\mathrm {R} = 0.5 m_{4\ell }$.

The $\mathrm {p}\mathrm {p} \rightarrow \mathrm{Z}\rightarrow 4\ell$ fiducial cross section is scaled to $\sigma (\mathrm {p}\mathrm {p} \rightarrow \mathrm{Z}) \mathcal {B} (\mathrm{Z}\rightarrow 4\ell )$ using the acceptance correction factor $\mathcal {A} = 0.125 \pm 0.002$, estimated with powheg. This factor corrects the fiducial $\mathrm{Z}\rightarrow 4\ell$ cross section to the phase space with only the 80–100$\,\text {GeV}$ mass window and $m_{\ell ^+\ell ^-} > 4\,\text {GeV}$ requirements, and also includes a correction, $0.96 \pm 0.01$, for the contribution of nonresonant four-lepton production to the signal region. The uncertainty takes into account the interference between doubly- and singly-resonant diagrams. The measured cross section is

$$\begin{aligned} \sigma (\mathrm {p}\mathrm {p} \rightarrow \mathrm{Z}) \mathcal {B}(\mathrm{Z}\rightarrow 4\ell ) = 249 \pm 11 (stat) _{-15}^{+16} (syst) \pm 4 (theo) \pm 6 (lumi) fb \end{aligned}$$

(3)

The branching fraction for the $\mathrm{Z}\rightarrow 4\ell$ decay, $\mathcal {B}(\mathrm{Z}\rightarrow 4\ell )$, is measured by comparing the cross section given by Eq. (3) with the $\mathrm{Z}\rightarrow \ell ^+\ell ^-$ cross section, and is computed as

$$\begin{aligned} \mathcal {B}(\mathrm{Z}\rightarrow 4\ell ) = \frac{\sigma (\mathrm {p}\mathrm {p} \rightarrow \mathrm{Z}\rightarrow 4\ell )}{\mathcal {C}^{\text {60--120}}_{\text {80--100}} \, \sigma (\mathrm {p}\mathrm {p} \rightarrow \mathrm{Z}\rightarrow \ell ^+\ell ^-) / \mathcal {B}(\mathrm{Z}\rightarrow \ell ^+\ell ^-)}, \end{aligned}$$

(4)

where $\sigma (\mathrm {p}\mathrm {p} \rightarrow \mathrm{Z}\rightarrow \ell ^+\ell ^-) = 1870 _{-40}^{+50}\text { pb}$ is the $\mathrm{Z}\rightarrow \ell ^+\ell ^-$ cross section times branching fraction calculated at NNLO with fewz v2.0 [50] in the mass range 60–120$\,\text {GeV}$. Its uncertainty includes PDF uncertainties and uncertainties in $\alpha _s$, the charm and bottom quark masses, and the effect of neglected higher-order corrections to the calculation. The factor $\mathcal {C}^{\text {60--120}}_{\text {80--100}} = 0.926 \pm 0.001$ corrects for the difference in $\mathrm{Z}$ boson mass windows and is estimated using powheg. Its uncertainty includes scale and PDF variations. The nominal $\mathrm{Z}$ to dilepton branching fraction $\mathcal {B}(\mathrm{Z}\rightarrow \ell ^+\ell ^-)$ is 0.03366 [46]. The measured value is

$$\begin{aligned} \mathcal {B}(\mathrm{Z}\rightarrow 4\ell ) = 4.83 _{-0.22}^{+0.23} (stat) _{-0.29}^{+0.32} (syst) \pm 0.08 (theo) \pm 0.12 (lumi) \times 10^{-6} \end{aligned}$$

(5)

where the theoretical uncertainty includes the uncertainties in $\sigma (\mathrm {p}\mathrm {p} \rightarrow \mathrm{Z}) \mathcal {B} (\mathrm{Z}\rightarrow \ell ^+\ell ^-)$, $\mathcal {C}^{\text {60--120}}_{\text {80--100}}$, and $\mathcal {A}$. This can be compared with $4.6 \times 10^{-6}$, computed with MadGraph 5_amc@nlo, and is consistent with the CMS and ATLAS measurements at $\sqrt{s} = 7, 8,$ and 13$\,\text {TeV}$  [6, 14, 15].

The total $\mathrm{Z}\mathrm{Z}$ production cross section for both dileptons produced in the mass range 60–120$\,\text {GeV}$ and $m_{\ell ^+\ell ^{\prime -}} > 4\,\text {GeV}$ is found to be

$$\begin{aligned} \sigma (\mathrm {p}\mathrm {p} \rightarrow \mathrm{Z}\mathrm{Z}) = 17.5 _{-0.5}^{+0.6} \,\text {(stat)} \pm 0.6 \,\text {(syst)} \pm 0.4 \,\text {(theo)} \pm 0.4 \,\text {(lumi)} \text { pb}. \end{aligned}$$

(6)

The measured total cross section can be compared to the theoretical value of $14.5^{+0.5}_{-0.4} \pm 0.2\text { pb}$ calculated with a combination of powheg and mcfm with the same settings as described for $\sigma _{\text {fid}} (\mathrm {p}\mathrm {p} \rightarrow \mathrm{Z}\mathrm{Z} \rightarrow 4\ell )$. It can also be compared to $16.2^{+0.6}_{-0.4}$ $\text { pb}$, calculated at NNLO in QCD via matrix v1.0.0_beta4 [11, 51], or $15.0^{+0.7}_{-0.6} \pm 0.2$ $\text { pb}$, calculated with mcfm at NLO in QCD with additional contributions from LO $\mathrm{g} \mathrm{g} \rightarrow \mathrm{Z}\mathrm{Z}$ diagrams. Both values are calculated with the NNPDF3.0 PDF sets, at NNLO and NLO, respectively, and fixed scales set to $\mu _\mathrm {F} = \mu _\mathrm {R} = m_\mathrm{Z}$.

This measurement agrees with the previously published cross section measured by CMS at 13$\,\text {TeV}$  [6] based on a 2.6$\,\text {fb}^{-1}$ data sample collected in 2015:

$$\begin{aligned} \sigma (\mathrm {p}\mathrm {p} \rightarrow \mathrm{Z}\mathrm{Z}) = 14.6 ^{+1.9}_{-1.8} \,\text {(stat)} _{-0.5}^{+0.3} \,\text {(syst)} \pm 0.2 \,\text {(theo)} \pm 0.4 \,\text {(lumi)} \text { pb}. \end{aligned}$$

(7)

The two measurements can be combined to yield the “2015 + 2016 cross section”

$$\begin{aligned} \sigma (\mathrm {p}\mathrm {p} \rightarrow \mathrm{Z}\mathrm{Z}) = 17.2 \pm 0.5 \,\text {(stat)} \pm 0.7 \,\text {(syst)} \pm 0.4 \,\text {(theo)} \pm 0.4 \,\text {(lumi)} \text { pb}. \end{aligned}$$

(8)

The combination was performed once considering the experimental uncertainties to be fully correlated between the 2015 and 2016 data sets, and once considering them to be fully uncorrelated. The results were averaged, and the difference was added linearly to the systematic uncertainty in the combined cross section.

The total $\mathrm{Z}\mathrm{Z}$ cross section is shown in Fig. 5 as a function of the proton-proton center-of-mass energy. Results from CMS [3, 4] and ATLAS [7, 8, 10] are compared to predictions from matrix and mcfm with the NNPDF3.0 PDF sets and fixed scales $\mu _\mathrm {F} = \mu _\mathrm {R} = m_\mathrm{Z}$. The matrix prediction uses PDFs calculated at NNLO, while the mcfm prediction uses NLO PDFs. The uncertainties are statistical (inner bars) and statistical and systematic added in quadrature (outer bars). The band around the matrix predictions reflects scale uncertainties, while the band around the mcfm predictions reflects both scale and PDF uncertainties.

Fig. 5

The total ZZ cross section as a function of the proton-proton center-of-mass energy. Results from the CMS and ATLAS experiments are compared to predictions from matrix at NNLO in QCD, and mcfm at NLO in QCD. The mcfm prediction also includes gluon-gluon initiated production at LO in QCD. Both predictions use NNPDF3.0 PDF sets and fixed scales $\mu _\mathrm {F} = \mu _\mathrm {R} = m_\mathrm{Z}$. Details of the calculations and uncertainties are given in the text. The ATLAS measurements were performed with a $\mathrm{Z}$ boson mass window of 66–116$\,\text {GeV}$, and are corrected for the resulting 1.6% difference. Measurements at the same center-of-mass energy are shifted slightly along the horizontal axis for clarity

The measurement of the differential cross sections provides detailed information about $\mathrm{Z}\mathrm{Z}$ kinematics. The observed yields are unfolded using the iterative technique described in Ref. [52]. Unfolding is performed with the RooUnfold package [53] and regularized by stopping after four iterations. Statistical uncertainties in the data distributions are propagated through the unfolding process to give the statistical uncertainties on the normalized differential cross sections.

The three decay channels, $4\mathrm {e}$, $4\mathrm {\mu }$, and $2\mathrm {e}2\mathrm {\mu }$, are combined after unfolding because no differences are expected in their kinematic distributions. The generator-level leptons used for the unfolding are dressed as in the fiducial cross section calculation.

The differential distributions normalized to the fiducial cross sections are presented in Figs. 6, 7, 8 for the combination of the 4$\mathrm {e}$, 4$\mathrm {\mu }$, and 2$\mathrm {e}$2$\mathrm {\mu }$ decay channels. The fiducial cross section definition includes $p_{\mathrm {T}} ^{\ell }$ and $|\eta ^{\ell } |$ selections on each lepton, and the 60–120$\,\text {GeV}$ mass requirement, as described in Table 4 and Sect. 4. Figure 6 shows the normalized differential cross sections as functions of the mass and $p_{\mathrm {T}}$ of the $\mathrm{Z}\mathrm{Z}$ system, Fig. 7 shows them as functions of the $p_{\mathrm {T}}$ of all $\mathrm{Z}$ bosons and the $p_{\mathrm {T}}$ of the leading lepton in each event, and Fig. 8 shows the angular correlations between the two $\mathrm{Z}$ bosons. The data are corrected for background contributions and compared with the theoretical predictions from powheg and mcfm, MadGraph 5_amc@nlo and mcfm, and matrix. The bottom part of each plot shows the ratio of the measured to the predicted values. The bin sizes are chosen according to the resolution of the relevant variables, while also keeping the statistical uncertainties at a similar level in all bins. The data are well reproduced by the simulation except in the low $p_{\mathrm {T}}$ regions, where data tend to have a steeper slope than the prediction.

Fig. 6

Differential cross sections normalized to the fiducial cross section for the combined 4$\mathrm {e}$, 4$\mathrm {\mu }$, and 2$\mathrm {e}$2$\mathrm {\mu }$ decay channels as a function of mass (left) and $p_{\mathrm {T}}$ (right) of the $\mathrm{Z}\mathrm{Z}$ system. Points represent the unfolded data; the solid, dashed, and dotted histograms represent the powheg+mcfm, MadGraph 5_amc@nlo+mcfm, and matrix predictions for $\mathrm{Z} \mathrm{Z}$ signal, respectively, and the bands around the predictions reflect their combined statistical, scale, and PDF uncertainties pythia  v8 was used for parton showering, hadronization, and underlying event simulation in the powheg, MadGraph 5_amc@nlo, and mcfm samples. The lower part of each plot represents the ratio of the measured cross section to the theoretical distributions. The shaded grey areas around the points represent the sum in quadrature of the statistical and systematic uncertainties, while the crosses represent the statistical uncertainties only

Fig. 7

Normalized $\mathrm{Z}\mathrm{Z}$ differential cross sections as a function of the $p_{\mathrm {T}}$ of (upper) all $\mathrm{Z}$ bosons and (lower) the leading lepton in $\mathrm{Z}\mathrm{Z}$ events. Other details are as described in the caption of Fig. 6

Fig. 8

Normalized $\mathrm{Z}\mathrm{Z}$ differential cross sections as a function of (upper) the azimuthal separation of the two $\mathrm{Z}$ bosons and (lower) $\Delta R$ between the $\mathrm{Z}$-bosons. Other details are as described in the caption of Fig. 6

Figure 9 shows the normalized differential four-lepton cross section as a function of $m_{4\ell }$, subject only to the common requirements of Table 4. This includes contributions from the $\mathrm{Z}$ and Higgs boson resonances and continuum $\mathrm{Z}$ $\mathrm{Z}$ production.

Fig. 9

The normalized differential four-lepton cross section as a function of the four-lepton mass, subject only to the common requirements of Table 4. SM $\mathrm{g} \mathrm{g} \rightarrow \mathrm{H} \rightarrow \mathrm{Z}\mathrm{Z}^*$ production is included, simulated with powheg. Other details are as described in the caption of Fig. 6

9 Limits on anomalous triple gauge couplings

The presence of aTGCs would increase the yield of events at high four-lepton masses. Figure 10 presents the distribution of the four-lepton reconstructed mass of events with both $\mathrm{Z}$ bosons in the mass range 60–120$\,\text {GeV}$ for the combined $4\mathrm {e}$, 4$\mathrm {\mu }$, and $2\mathrm {e}2\mathrm {\mu }$ channels. This distribution is used to set the limits on possible contributions from aTGCs. Two simulated samples with nonzero aTGCs are shown as examples, along with the SM distribution simulated by both sherpa and powheg.

Fig. 10

Distribution of the four-lepton reconstructed mass for the combined $4\mathrm {e}$, 4$\mathrm {\mu }$, and $2\mathrm {e}2\mathrm {\mu }$ channels. Points represent the data, the filled histograms represent the SM expected yield including signal and irreducible background predictions from simulation and the data-driven background estimate. Unfilled histograms represent examples of aTGC signal predictions (dashed), and the sherpa SM prediction (solid), included to illustrate the expected shape differences between the sherpa and powheg predictions. Vertical bars on the data points show their statistical uncertainty. The sherpa distributions are normalized such that the SM sample has the same total yield as the powheg sample predicts. Bin contents are normalized to the bin widths, using a unit bin size of 50$\,\text {GeV}$; horizontal bars on the data points show the range of the corresponding bin. The last bin includes the “overflow” contribution from events at masses above 1.2$\,\text {TeV}$

The invariant mass distributions are interpolated from the sherpa simulations for different values of the anomalous couplings in the range between 0 and 0.015. For each distribution, only one or two couplings are varied while all others are set to zero. The measured signal is obtained from a comparison of the data to a grid of aTGC models in the $(f_{4}^\mathrm{Z}, f_{4}^\gamma )$ and $(f_{5}^\mathrm{Z}, f_{5}^\gamma )$ parameter planes. Expected signal values are interpolated between the 2D grid points using a second-degree polynomial, since the cross section for the signal depends quadratically on the coupling parameters. A binned profile likelihood method, Wald Gaussian approximation, and Wilk’s theorem are used to derive one-dimensional limits at a 95% confidence level (CL) on each of the four aTGC parameters, and two-dimensional limits at a 95% CL on the pairs ($f_4^\mathrm{Z}$, $f_4^\gamma$) and ($f_5^\mathrm{Z}$, $f_5^\gamma$) [46, 54, 55]. When the limits are calculated for each parameter or pair, all other parameters are set to their SM values. The systematic uncertainties described in Section 7 are treated as nuisance parameters with log-normal distributions. No form factor is used when deriving the limits so that the results do not depend on any assumed energy scale characterizing new physics. The constraints on anomalous couplings are displayed in Fig. 11. The curves indicate 68 and 95% confidence levels, and the solid dot shows the coordinates where the likelihood reaches its maximum. Coupling values outside the contours are excluded at the corresponding confidence levels. The limits are dominated by statistical uncertainties.

Fig. 11

Two-dimensional observed 95% CL limits (solid contour) and expected 68 and 95% CL limits (dashed contour) on the $\mathrm{Z}\mathrm{Z}\mathrm{Z}$ and $\mathrm{Z}\mathrm{Z}\gamma$ aTGCs. The upper(lower) plot shows the exclusion contour in the $f_{4(5)}^\mathrm{Z}, f_{4(5)}^\gamma$ parameter planes. The values of couplings outside of contours are excluded at the corresponding confidence level. The solid dot is the point at which the likelihood is at its maximum. The solid lines at the center show the observed one-dimensional 95% CL limits for $f_{4,5}^\gamma$ (horizontal) and $f_{4,5}^\mathrm{Z}$ (vertical). No form factor is used

The observed one-dimensional 95% CL limits for the $f_4^{\mathrm{Z},\gamma }$ and $f_5^{\mathrm{Z},\gamma }$ anomalous coupling parameters are:

$$\begin{aligned} \begin{aligned} -0.0012<f_4^\mathrm{Z}< 0.0010 ,\quad -0.0010<f_5^\mathrm{Z}< 0.0013 , \\ \quad -0.0012<f_4^{\gamma }< 0.0013 ,\quad -0.0012<f_5^{\gamma }< 0.0013 . \end{aligned} \end{aligned}$$

(9)

These are the most stringent limits to date on anomalous $\mathrm{Z}\mathrm{Z}\mathrm{Z}$ and $\mathrm{Z}\mathrm{Z}\gamma$ trilinear gauge boson couplings, improving on the previous strictest results from CMS [5] by factors of two or more and constraining the coupling parameters more than the corresponding ATLAS results [10].

One way to impose unitarity on the aTGC models is to restrict the range of four-lepton invariant mass used in the limit calculation. The limits will then depend on the “cutoff” value used. The computation of the one-dimensional limits is repeated for different maximum allowed values of $m_{4\ell }$, and the results are presented in Fig. 12 as a function of this cutoff.

Fig. 12

Expected and observed one-dimensional limits on the four aTGC parameters, as a function of an upper cutoff on the invariant mass of the four-lepton system. No form factor is used

10 Summary

A series of measurements of four-lepton final states in proton-proton collisions at $\sqrt{s} = 13\,\text {TeV}$ have been performed with the CMS detector at the CERN LHC. The measured $\mathrm {p}\mathrm {p} \rightarrow \mathrm{Z}\mathrm{Z}$ cross section is $\sigma (\mathrm {p}\mathrm {p}\rightarrow \mathrm{Z}\mathrm{Z}) = 17.2 \pm 0.5 \,\text {(stat)} \pm 0.7 \,\text {(syst)} \pm 0.4 \,\text {(theo)} \pm 0.4 \,\text {(lumi)} \text { pb}$ for $\mathrm{Z}$ boson masses in the range $60< m_{\mathrm{Z}} < 120\,\text {GeV}$. The measured branching fraction for $\mathrm{Z}$ boson decays to four leptons is $\mathcal {B}(\mathrm{Z}\rightarrow 4\ell ) = 4.83 _{-0.22}^{+0.23} (stat)_{-0.29}^{+0.32} (syst) \pm 0.08 (theo) \pm 0.12 (lumi) \times 10^{-6}$ for four-lepton mass in the range $80< m_{4\ell } < 100\,\text {GeV}$ and dilepton mass $m_{\ell \ell } > 4\,\text {GeV}$ for all oppositely charged same-flavor lepton pairs. Normalized differential cross sections were also measured. All results agree well with the SM predictions. Improved limits on anomalous $\mathrm{Z} \mathrm{Z} \mathrm{Z}$ and $\mathrm{Z} \mathrm{Z} \gamma$ triple gauge couplings were established, the most stringent to date.