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| CMS-SMP-24-013 ; CERN-EP-2026-184 | ||
| Combination of vector boson scattering measurements in leptonic final states in proton-proton collisions at $ \sqrt{s} = $ 13 TeV | ||
| CMS Collaboration | ||
| 2 August 2026 | ||
| Submitted to Physics Letters B | ||
| Abstract: A statistical combination of vector boson scattering measurements in proton-proton collisions at $ \sqrt{s} = $ 13 TeV is presented. The analysis is based on a dataset corresponding to an integrated luminosity of 138 fb$ ^{-1} $, collected with the CMS detector in 2016--2018. Events are selected to contain two vector bosons, with at least one decaying leptonically, and at least two additional jets with a high invariant mass and a large pseudorapidity separation. Signal rates are extracted for same-sign WW, opposite-sign WW, as well as for WZ and ZZ channels, and are compared with the standard model predictions. The combination provides a global view of vector boson scattering by simultaneously constraining all channels in a coherent framework, improving sensitivity beyond individual inputs, with all results in agreement with the standard model predictions. In addition, a charge-separated measurement is performed in which W bosons are distinguished according to their electric charge, allowing sensitivity to the production charge asymmetry. | ||
| Links: e-print arXiv:2608.01289 [hep-ex] (PDF) ; CDS record ; inSPIRE record ; HepData record ; CADI line (restricted) ; | ||
| Figures | |
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Figure 1:
The left side shows a representative schematic diagram for the EW-induced production of VBS involving massive vector bosons. The dashed circle encapsulates different contributions leading to the same final state, including triple and quartic gauge couplings, as well as Higgs boson exchange. On the right, the analyzed decay channels are grouped into same-sign and opposite-sign WW, and WZ, ZZ final states, including both leptonic and hadronic decays of the heavy vector bosons. In all cases, at least one vector boson is required to decay leptonically. Lepton lines in the $ \mathrm{W}^{\pm} $ decay vertices carry no arrows or charge labels, as a single diagram represents both the $ \mathrm{W^+} $ and $ \mathrm{W^-} $ decay modes. |
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Figure 2:
The signal fractions for each analysis contributing to the combination are presented, as well as the total expected signal yield for each channel. Rows correspond to the names of the analyses, while the $ x $-axis indicates the signal fraction. Different colors, as shown in the legend, represent the distinct signal parameters targeted for measurement. The left half of the figure displays the signal fractions for the six components, separated by the electric charge of the W bosons. The right half of the figure provides the equivalent information for the four components. All signal yields are obtained from MC simulations prior to the combined fit of the signal strengths and include all analysis regions. The uncertainties on these yields are not shown in the figure. |
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Figure 3:
The likelihood profiles from the combined fit for the various POIs listed in the legends vs. $ \mu = \sigma_{\text{obs}}/\sigma_{\text{SM}} $. The 68.3 ans 95.5% $ \text{CL}_\text{s} $ are drawn as dashed horizontal lines and are computed as $ -2\Delta \ln L = $ 1 and $ -2\Delta \ln L = $ 4 respectively. The left (right) figure shows the $ -2\Delta \ln L $ profiles for the 4-POIs (6-POIs) model. |
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Figure 3-a:
The likelihood profiles from the combined fit for the various POIs listed in the legends vs. $ \mu = \sigma_{\text{obs}}/\sigma_{\text{SM}} $. The 68.3 ans 95.5% $ \text{CL}_\text{s} $ are drawn as dashed horizontal lines and are computed as $ -2\Delta \ln L = $ 1 and $ -2\Delta \ln L = $ 4 respectively. The left (right) figure shows the $ -2\Delta \ln L $ profiles for the 4-POIs (6-POIs) model. |
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Figure 3-b:
The likelihood profiles from the combined fit for the various POIs listed in the legends vs. $ \mu = \sigma_{\text{obs}}/\sigma_{\text{SM}} $. The 68.3 ans 95.5% $ \text{CL}_\text{s} $ are drawn as dashed horizontal lines and are computed as $ -2\Delta \ln L = $ 1 and $ -2\Delta \ln L = $ 4 respectively. The left (right) figure shows the $ -2\Delta \ln L $ profiles for the 4-POIs (6-POIs) model. |
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Figure 4:
Measurement of the VBS signal strengths ($ \mu $) and comparison with the SM predictions. The upper (lower) panel presents results considering four (six) independent signal strengths. In each panel, the left side shows the measured signal strengths, with thick (thin) black lines representing the 1 (2) standard deviation (SD) confidence intervals. The red and blue bands within the 1 standard deviation interval illustrate the systematic and statistical uncertainties, respectively. The right side displays the observed (black crosses) and expected (filled grey bars) statistical significance of the EW VBS signal relative to the SM background. The SSWW and $ \mathrm{W^+}\mathrm{W^+} $ significance bars are truncated, as their values exceed the axis range. |
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Figure 5:
Observed $ \log_{10}[S(\mu_i=1)/B] $ distributions for the VBS SM combination and four parameters model. From left to right, upper to lower, the filled white histograms represent the overall background contributions, whereas the blue histograms represent the signal yields for $ \mu_{\mathrm{SS}\mathrm{W}\mathrm{W}} $, $ \mu_{\mathrm{OS}\mathrm{W}\mathrm{W}} $, $ \mu_{\mathrm{W}\mathrm{Z}} $, and $ \mu_{\mathrm{Z}\mathrm{Z}} $. Observations are displayed as black markers with vertical bars representing the uncertainty. The post-fit 68.3% uncertainty interval on the overall background template is shown in pink. The distribution is binned as a function of the pre-fit value of $ \log_{10}[S(\mu_i=1)/B] $, obtained from the input template bins, and is filled with the post-fit yields of total signal and backgrounds. Each input bin is assigned to the signal process providing the largest (``max") contribution in that bin of the input templates, as indicated in the legend. |
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Figure 5-a:
Observed $ \log_{10}[S(\mu_i=1)/B] $ distributions for the VBS SM combination and four parameters model. From left to right, upper to lower, the filled white histograms represent the overall background contributions, whereas the blue histograms represent the signal yields for $ \mu_{\mathrm{SS}\mathrm{W}\mathrm{W}} $, $ \mu_{\mathrm{OS}\mathrm{W}\mathrm{W}} $, $ \mu_{\mathrm{W}\mathrm{Z}} $, and $ \mu_{\mathrm{Z}\mathrm{Z}} $. Observations are displayed as black markers with vertical bars representing the uncertainty. The post-fit 68.3% uncertainty interval on the overall background template is shown in pink. The distribution is binned as a function of the pre-fit value of $ \log_{10}[S(\mu_i=1)/B] $, obtained from the input template bins, and is filled with the post-fit yields of total signal and backgrounds. Each input bin is assigned to the signal process providing the largest (``max") contribution in that bin of the input templates, as indicated in the legend. |
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Figure 5-b:
Observed $ \log_{10}[S(\mu_i=1)/B] $ distributions for the VBS SM combination and four parameters model. From left to right, upper to lower, the filled white histograms represent the overall background contributions, whereas the blue histograms represent the signal yields for $ \mu_{\mathrm{SS}\mathrm{W}\mathrm{W}} $, $ \mu_{\mathrm{OS}\mathrm{W}\mathrm{W}} $, $ \mu_{\mathrm{W}\mathrm{Z}} $, and $ \mu_{\mathrm{Z}\mathrm{Z}} $. Observations are displayed as black markers with vertical bars representing the uncertainty. The post-fit 68.3% uncertainty interval on the overall background template is shown in pink. The distribution is binned as a function of the pre-fit value of $ \log_{10}[S(\mu_i=1)/B] $, obtained from the input template bins, and is filled with the post-fit yields of total signal and backgrounds. Each input bin is assigned to the signal process providing the largest (``max") contribution in that bin of the input templates, as indicated in the legend. |
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Figure 5-c:
Observed $ \log_{10}[S(\mu_i=1)/B] $ distributions for the VBS SM combination and four parameters model. From left to right, upper to lower, the filled white histograms represent the overall background contributions, whereas the blue histograms represent the signal yields for $ \mu_{\mathrm{SS}\mathrm{W}\mathrm{W}} $, $ \mu_{\mathrm{OS}\mathrm{W}\mathrm{W}} $, $ \mu_{\mathrm{W}\mathrm{Z}} $, and $ \mu_{\mathrm{Z}\mathrm{Z}} $. Observations are displayed as black markers with vertical bars representing the uncertainty. The post-fit 68.3% uncertainty interval on the overall background template is shown in pink. The distribution is binned as a function of the pre-fit value of $ \log_{10}[S(\mu_i=1)/B] $, obtained from the input template bins, and is filled with the post-fit yields of total signal and backgrounds. Each input bin is assigned to the signal process providing the largest (``max") contribution in that bin of the input templates, as indicated in the legend. |
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Figure 5-d:
Observed $ \log_{10}[S(\mu_i=1)/B] $ distributions for the VBS SM combination and four parameters model. From left to right, upper to lower, the filled white histograms represent the overall background contributions, whereas the blue histograms represent the signal yields for $ \mu_{\mathrm{SS}\mathrm{W}\mathrm{W}} $, $ \mu_{\mathrm{OS}\mathrm{W}\mathrm{W}} $, $ \mu_{\mathrm{W}\mathrm{Z}} $, and $ \mu_{\mathrm{Z}\mathrm{Z}} $. Observations are displayed as black markers with vertical bars representing the uncertainty. The post-fit 68.3% uncertainty interval on the overall background template is shown in pink. The distribution is binned as a function of the pre-fit value of $ \log_{10}[S(\mu_i=1)/B] $, obtained from the input template bins, and is filled with the post-fit yields of total signal and backgrounds. Each input bin is assigned to the signal process providing the largest (``max") contribution in that bin of the input templates, as indicated in the legend. |
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Figure 6:
Summary plots of the two-dimensional fits for all pairs of POIs $ \mu = \sigma_{\text{obs}}/\sigma_{\text{SM}} $. The 68.3% $ \text{CL}_\text{s} $, drawn as solid lines, are computed from the likelihood profiles as $ -2\Delta \ln L < $ 2.3. The upper-left panel shows the 68.3% CL intervals for all pairs of parameters of interest in the 4-POIs model, with the legend listing first the parameter on the x axis, followed by that on the y axis. The results for the 6-POIs model are split across three panels: the upper-right panel shows all pairs involving WV combinations (WW and WZ), the lower-left panel shows all combinations of WW and WZ parameters, and the lower-right panel shows all pairs involving ZZ. In all cases, all parameters of interest other than the two under consideration are profiled in the maximum likelihood fit, together with the nuisance parameters. |
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Figure 6-a:
Summary plots of the two-dimensional fits for all pairs of POIs $ \mu = \sigma_{\text{obs}}/\sigma_{\text{SM}} $. The 68.3% $ \text{CL}_\text{s} $, drawn as solid lines, are computed from the likelihood profiles as $ -2\Delta \ln L < $ 2.3. The upper-left panel shows the 68.3% CL intervals for all pairs of parameters of interest in the 4-POIs model, with the legend listing first the parameter on the x axis, followed by that on the y axis. The results for the 6-POIs model are split across three panels: the upper-right panel shows all pairs involving WV combinations (WW and WZ), the lower-left panel shows all combinations of WW and WZ parameters, and the lower-right panel shows all pairs involving ZZ. In all cases, all parameters of interest other than the two under consideration are profiled in the maximum likelihood fit, together with the nuisance parameters. |
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Figure 6-b:
Summary plots of the two-dimensional fits for all pairs of POIs $ \mu = \sigma_{\text{obs}}/\sigma_{\text{SM}} $. The 68.3% $ \text{CL}_\text{s} $, drawn as solid lines, are computed from the likelihood profiles as $ -2\Delta \ln L < $ 2.3. The upper-left panel shows the 68.3% CL intervals for all pairs of parameters of interest in the 4-POIs model, with the legend listing first the parameter on the x axis, followed by that on the y axis. The results for the 6-POIs model are split across three panels: the upper-right panel shows all pairs involving WV combinations (WW and WZ), the lower-left panel shows all combinations of WW and WZ parameters, and the lower-right panel shows all pairs involving ZZ. In all cases, all parameters of interest other than the two under consideration are profiled in the maximum likelihood fit, together with the nuisance parameters. |
png pdf |
Figure 6-c:
Summary plots of the two-dimensional fits for all pairs of POIs $ \mu = \sigma_{\text{obs}}/\sigma_{\text{SM}} $. The 68.3% $ \text{CL}_\text{s} $, drawn as solid lines, are computed from the likelihood profiles as $ -2\Delta \ln L < $ 2.3. The upper-left panel shows the 68.3% CL intervals for all pairs of parameters of interest in the 4-POIs model, with the legend listing first the parameter on the x axis, followed by that on the y axis. The results for the 6-POIs model are split across three panels: the upper-right panel shows all pairs involving WV combinations (WW and WZ), the lower-left panel shows all combinations of WW and WZ parameters, and the lower-right panel shows all pairs involving ZZ. In all cases, all parameters of interest other than the two under consideration are profiled in the maximum likelihood fit, together with the nuisance parameters. |
png pdf |
Figure 6-d:
Summary plots of the two-dimensional fits for all pairs of POIs $ \mu = \sigma_{\text{obs}}/\sigma_{\text{SM}} $. The 68.3% $ \text{CL}_\text{s} $, drawn as solid lines, are computed from the likelihood profiles as $ -2\Delta \ln L < $ 2.3. The upper-left panel shows the 68.3% CL intervals for all pairs of parameters of interest in the 4-POIs model, with the legend listing first the parameter on the x axis, followed by that on the y axis. The results for the 6-POIs model are split across three panels: the upper-right panel shows all pairs involving WV combinations (WW and WZ), the lower-left panel shows all combinations of WW and WZ parameters, and the lower-right panel shows all pairs involving ZZ. In all cases, all parameters of interest other than the two under consideration are profiled in the maximum likelihood fit, together with the nuisance parameters. |
| Tables | |
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Table 1:
Summary of the analyses considered in the combination. The symbol $ \ell $ denotes either an electron or muon, $ {\tau}_{\mathrm{h}} $ a hadronically decaying $ \tau $ lepton, and $ {\mathrm{j}} $ is the shorthand notation for a jet. |
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Table 2:
Summary of the phase space regions entering the combined fit, with the corresponding observables and number of bins. Two-dimensional observables are indicated by a colon separating their two components. Unless the data-taking year is explicitly specified in the subregion column, separate templates are constructed for each year of data taking. DY denotes Drell-Yan processes. The b-tag (b-veto) category requires at least one (zero) b-tagged jets [81] in the event. For WV and ZV, the resolved (merged) region targets events with four small-radius jets (one large-radius jet accompanied by two small-radius jets). DNN classifiers are labeled by the corresponding region name only when distinct models are used across analysis regions. |
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Table 3:
The signal strength $ \mu = \sigma_{\text{obs}}/\sigma_{\text{SM}} $, where $ \sigma_{\text{obs}} $ and $ \sigma_{\text{SM}} $ are the observed and SM predicted cross sections respectively, is reported for each VBS production mode (columns). Statistical significances $ \sigma $ (in standard deviations) are also reported. The upper block reports the results for the model with four signal strengths while the lower block reports the same results but splitting the signal strengths for the W boson electric charge. The expected results are reported within parentheses. For the expected signal strengths, only the ${\pm}$1 standard deviation uncertainty is reported as the central value is assumed to be 1. |
| Summary |
| The most comprehensive statistical combination of vector boson scattering (VBS) processes to date, integrating results from seven distinct analyses using CMS data, has been presented. Unlike previous single-channel measurements, this global combination accounts for the correlations between all VBS processes involving massive vector bosons (W, Z), providing a unique and rigorous stress test of the standard model (SM) electroweak (EW) sector. The targeted VBS production modes are all charge combinations of WW, WZ, and ZZ, with WW further split into same-sign (SSWW) and opposite-sign (OSWW) channels. Each process is studied in final states where both vector bosons decay leptonically into electrons, muons, or tau leptons, as well as in final states where one vector boson decays hadronically. The EW signal strengths are measured via a simultaneous fit of all channels in two separate configurations: one merging W boson electric charges with four free parameters ($ \mu_{\mathrm{SS}\mathrm{W}\mathrm{W}} $, $ \mu_{\mathrm{OS}\mathrm{W}\mathrm{W}} $, $ \mu_{\mathrm{W}\mathrm{Z}} $, $ \mu_{\mathrm{Z}\mathrm{Z}} $) and another separating W boson charges with six parameters ($ \mu_{\mathrm{W^+}\mathrm{W^+}} $, $ \mu_{\mathrm{W^-}\mathrm{W^-}} $, $ \mu_{\mathrm{W^+}\mathrm{W^-}} $, $ \mu_{\mathrm{W^+}\mathrm{Z}} $, $ \mu_{\mathrm{W^-}\mathrm{Z}} $, $ \mu_{\mathrm{Z}\mathrm{Z}} $). Simultaneous one- and two-dimensional fits were performed for all VBS EW production components. Overall, all results are found to be consistent with the SM predictions. This combination provides a simultaneous constraint of all VBS processes involving massive vector bosons, including charge-separated channels, enabling a coherent test of the EW gauge structure in VBS that was not accessible in previous single-channel measurements. It improves the precision of VBS measurements and provides a benchmark for future VBS studies at the LHC. |
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