CMS-SMP-24-003

CMS-SMP-24-003 ; CERN-EP-2025-035
Combined effective field theory interpretation of Higgs boson, electroweak vector boson, top quark, and multijet measurements
CMS Collaboration
3 April 2025
Submitted to Eur. Phys. J. C
Abstract: Constraints on Wilson coefficients (WCs) corresponding to dimension-6 operators of the standard model effective field theory (SMEFT) are determined from a simultaneous fit to seven sets of CMS measurements probing Higgs boson, electroweak vector boson, top quark, and multijet production. Measurements of electroweak precision observables at LEP and SLC are also included and provide complementary constraints to those from the CMS experiment. The CMS measurements, using LHC proton-proton collision data at $\sqrt{s}=$ 13 TeV, corresponding to integrated luminosities of 36.3 or 138 fb $^{-1}$ , are chosen to provide sensitivity to a broad set of operators, for which consistent SMEFT predictions can be derived. These are primarily measurements of differential cross sections which are parameterized as functions of the WCs. Measurements targeting $\mathrm{t}(\overline{\mathrm{t}})\mathrm{X}$ production directly incorporate the SMEFT effects through event weights that are applied to the simulated signal samples, which enables detector-level predictions. Individual constraints on 64 WCs, and constraints on 42 linear combinations of WCs, are obtained.
Links: CDS record ; CADI line (restricted) ;

Figures & Tables	Summary	References	CMS Publications

Figures
png pdf	Figure 1: Example Feynman diagrams of modifications of SM processes by the SMEFT operator $\mathcal{Q}_{\mathrm{W}}$ : $\mathrm{W}\gamma$ production (left), WW production (centre), $\mathrm{H}\to\gamma\gamma$ decay (right). The WC $c_{\mathrm{W}}$ controls the strength of the interaction.
png pdf	Figure 1-a: Example Feynman diagrams of modifications of SM processes by the SMEFT operator $\mathcal{Q}_{\mathrm{W}}$ : $\mathrm{W}\gamma$ production (left), WW production (centre), $\mathrm{H}\to\gamma\gamma$ decay (right). The WC $c_{\mathrm{W}}$ controls the strength of the interaction.
png pdf	Figure 1-b: Example Feynman diagrams of modifications of SM processes by the SMEFT operator $\mathcal{Q}_{\mathrm{W}}$ : $\mathrm{W}\gamma$ production (left), WW production (centre), $\mathrm{H}\to\gamma\gamma$ decay (right). The WC $c_{\mathrm{W}}$ controls the strength of the interaction.
png pdf	Figure 1-c: Example Feynman diagrams of modifications of SM processes by the SMEFT operator $\mathcal{Q}_{\mathrm{W}}$ : $\mathrm{W}\gamma$ production (left), WW production (centre), $\mathrm{H}\to\gamma\gamma$ decay (right). The WC $c_{\mathrm{W}}$ controls the strength of the interaction.
png pdf	Figure 2: Relative effect of the linear SMEFT terms for the WCs that affect the Higgs STXS cross sections and the $\mathrm{H}\to\gamma\gamma$ branching fraction. The parameters $c_j/\Lambda^2$ are set to different multiples of 1 TeV $^{-2}$ to ensure the effect of all WCs can be visualized on the same $y$ axis scale. The upper panel shows the measured values and their uncertainties relative to the predictions in the SM. As these are measurements of the cross sections times branching fraction, no measurement is displayed in the rightmost bin (labelled `` $\mathrm{H}\to\gamma\gamma$ '').
png pdf	Figure 3: Relative effect of the linear SMEFT terms for the WCs that affect the $\mathrm{W}\gamma$ , $\mathrm{Z}\to\nu\nu$ , and WW differential cross sections. The parameters $c_j/\Lambda^2$ are set to different multiples of 1 TeV $^{-2}$ to ensure the effect of all WCs can be visualized on the same $y$ axis scale. The upper panel shows the measured values and their uncertainties relative to the predictions in the SM.
png pdf	Figure 4: Relative effect of the linear SMEFT terms for the WCs that affect the $\mathrm{t} \overline{\mathrm{t}}$ differential cross sections. The parameters $c_j/\Lambda^2$ are set to different multiples of 1 TeV $^{-2}$ to ensure the effect of all WCs can be visualized on the same $y$ axis scale. The upper panel shows the measured values and their uncertainties relative to the predictions in the SM.
png pdf	Figure 5: Relative effect of the linear SMEFT terms for the WCs that affect the inclusive jet differential cross sections in the rapidity bins $(0, 0.5)$ and $(0.5, 1)$ . The parameters $c_j/\Lambda^2$ are set to different multiples of 1 TeV $^{-2}$ to ensure the effect of all WCs can be visualized on the same $y$ axis scale. The upper panel shows the measured values and their uncertainties relative to the predictions in the SM.
png pdf	Figure 6: Relative effect of the linear SMEFT terms for the WCs that affect the inclusive jet differential cross sections in the rapidity bins $(1, 1.5)$ and $(1.5, 2)$ . The parameters $c_j/\Lambda^2$ are set to different multiples of 1 TeV $^{-2}$ to ensure the effect of all WCs can be visualized on the same $y$ axis scale. The upper panel shows the measured values and their uncertainties relative to the predictions in the SM.
png pdf	Figure 7: Relative effect of the linear SMEFT terms for the WCs that affect the EWPO [36,37]. The parameters $c_j/\Lambda^2$ are set to different multiples of 1 TeV $^{-2}$ to ensure the effect of all WCs can be visualized on the same $y$ axis scale. The upper panel shows the measured values and their uncertainties relative to the predictions in the SM.
png pdf	Figure 8: Diagonal entries $H_{jj}^{\alpha}$ of the Hessian matrix evaluated for each input channel. These indicate which of the input channels are expected to be the most sensitive to any given operator. Larger values of $H_{jj}^{\alpha}$ correspond to higher sensitivity.
png pdf	Figure 9: Rotation matrix obtained by performing the PCA on the Hessian matrix of the full set of measurements, including the $\mathrm{t}(\overline{\mathrm{t}})\mathrm{X}$ analysis. Only matrix coefficients with absolute value ${\geq}$ 0.05 are displayed.
png pdf	Figure 10: Constraints on linear combinations of WCs, for the hybrid fit including the $\mathrm{t}(\overline{\mathrm{t}})\mathrm{X}$ analysis. The shaded areas correspond to the expected 95% confidence intervals, the thick and thin bars to the observed 68% and 95% confidence intervals, respectively. The lower panel shows the contribution of different input measurements to the total constraints. The constraints are scaled by powers of 10 to ensure the constraints on all 42 eigenvectors can be visualized on the same $y$ axis scale.
png pdf	Figure 11: Constraints on individual WCs, for the hybrid fit including the $\mathrm{t}(\overline{\mathrm{t}})\mathrm{X}$ analysis. The constraints for each WC are obtained keeping the other coefficients fixed to 0. The shaded areas correspond to the expected 95% confidence intervals, the thick and thin bars to the observed 68% and 95% confidence intervals, respectively. The lower panel shows the contribution of different input measurements to the total constraints. The constraints are scaled by powers of 10 to ensure the constraints on all 64 WCs can be visualized on the same $y$ axis scale.
png pdf	Figure 12: The 95% CL lower limits on the scales $\Lambda_j$ for the indicated values of the WCs $c_j$ .
png pdf	Figure 13: Constraints on individual WCs, showing both the constraints considering only linear terms in the SMEFT parameterization and those considering both linear and quadratic terms. The constraints for each WC are obtained keeping the other coefficients fixed to 0. The shaded areas correspond to the expected 95% confidence intervals, the thick and thin bars to the observed 68% and 95% confidence intervals, respectively. The constraints are scaled by powers of 10 to ensure the constraints on all 64 WCs can be visualized on the same $y$ axis scale.
png pdf	Figure A1: Rotation matrix obtained by performing the PCA on the Hessian matrix of a reduced set of input measurements, $\mathrm{H}\to\gamma\gamma$ , $\mathrm{W}\gamma$ , $\mathrm{Z}\to\nu\nu$ , and WW. Only matrix coefficients with absolute value $\geq$ 0.05 are displayed.
png pdf	Figure A2: Constraints on linear combinations of WCs, using a reduced set of input measurements, $\mathrm{H}\to\gamma\gamma$ , $\mathrm{W}\gamma$ , $\mathrm{Z}\to\nu\nu$ , and WW. The shaded areas correspond to the expected 95% confidence intervals, the thick and thin bars to the observed 68% and 95% confidence intervals, respectively. The lower panel shows the contribution of different input measurements to the total constraints. The constraints are scaled by powers of 10 to ensure the constraints on all eigenvectors can be visualized on the same $y$ axis scale.
png pdf	Figure A3: Constraints on individual WCs, using a reduced set of input measurements, $\mathrm{H}\to\gamma\gamma$ , $\mathrm{W}\gamma$ , $\mathrm{Z}\to\nu\nu$ , and WW. The constraints for each WC are obtained keeping the other coefficients fixed to 0. The shaded areas correspond to the expected 95% confidence intervals, the thick and thin bars to the observed 68% and 95% confidence intervals, respectively. The lower panel shows the contribution of different input measurements to the total constraints. The constraints are scaled by powers of 10 to ensure the constraints on all WCs can be visualized on the same $y$ axis scale.
png pdf	Figure B1: Rotation matrix obtained by performing the PCA on the Hessian matrix of a reduced set of measurements, excluding the $\mathrm{t}(\overline{\mathrm{t}})\mathrm{X}$ measurement. Only matrix coefficients with absolute value $\geq$ 0.05 are displayed.
png pdf	Figure B2: Constraints on linear combinations of WCs, from the simplified likelihood fit, excluding the $\mathrm{t}(\overline{\mathrm{t}})\mathrm{X}$ analysis. The shaded areas correspond to the expected 95% confidence intervals, the thick and thin bars to the observed 68% and 95% confidence intervals, respectively. The lower panel shows the contribution of different input measurements to the total constraints. The constraints are scaled by powers of 10 to ensure the constraints on all eigenvectors can be visualized on the same $y$ axis scale.
png pdf	Figure B3: Constraints on individual WCs, from the simplified likelihood fit, excluding the $\mathrm{t}(\overline{\mathrm{t}})\mathrm{X}$ analysis. The constraints for each WC are obtained keeping the other coefficients fixed to 0. The shaded areas correspond to the expected 95% confidence intervals, the thick and thin bars to the observed 68% and 95% confidence intervals, respectively. The lower panel shows the contribution of different input measurements to the total constraints. The constraints are scaled by powers of 10 to ensure the constraints on all WCs can be visualized on the same $y$ axis scale.
png pdf	Figure C1: Observed likelihood scans for $c^{(3)}_{\mathrm{HQ}}$ , where linear terms dominate (upper left); $c_{\mathrm{l}\mathrm{u}}$ , where quadratic terms dominate (upper right); and $c_{\mathrm{W}}$ , where quadratic and linear terms both contribute (lower). The results with quadratic terms included in the parameterization (solid black line) and with linear terms only (dashed blue line) are shown. The solid grey lines indicate the sets of points $q^{68\%}$ (lower line) and $q^{95\%}$ (upper line), whereas the dashed grey lines denote the test statistic values used to determine the 68% and 95% confidence intervals in the asymptotic approximation. The confidence intervals shown on the figures are the 68% confidence intervals extracted from the intersection of the linear-plus-quadratic curve with the $q^{68\%}$ line.
png pdf	Figure C1-a: Observed likelihood scans for $c^{(3)}_{\mathrm{HQ}}$ , where linear terms dominate (upper left); $c_{\mathrm{l}\mathrm{u}}$ , where quadratic terms dominate (upper right); and $c_{\mathrm{W}}$ , where quadratic and linear terms both contribute (lower). The results with quadratic terms included in the parameterization (solid black line) and with linear terms only (dashed blue line) are shown. The solid grey lines indicate the sets of points $q^{68\%}$ (lower line) and $q^{95\%}$ (upper line), whereas the dashed grey lines denote the test statistic values used to determine the 68% and 95% confidence intervals in the asymptotic approximation. The confidence intervals shown on the figures are the 68% confidence intervals extracted from the intersection of the linear-plus-quadratic curve with the $q^{68\%}$ line.
png pdf	Figure C1-b: Observed likelihood scans for $c^{(3)}_{\mathrm{HQ}}$ , where linear terms dominate (upper left); $c_{\mathrm{l}\mathrm{u}}$ , where quadratic terms dominate (upper right); and $c_{\mathrm{W}}$ , where quadratic and linear terms both contribute (lower). The results with quadratic terms included in the parameterization (solid black line) and with linear terms only (dashed blue line) are shown. The solid grey lines indicate the sets of points $q^{68\%}$ (lower line) and $q^{95\%}$ (upper line), whereas the dashed grey lines denote the test statistic values used to determine the 68% and 95% confidence intervals in the asymptotic approximation. The confidence intervals shown on the figures are the 68% confidence intervals extracted from the intersection of the linear-plus-quadratic curve with the $q^{68\%}$ line.
png pdf	Figure C1-c: Observed likelihood scans for $c^{(3)}_{\mathrm{HQ}}$ , where linear terms dominate (upper left); $c_{\mathrm{l}\mathrm{u}}$ , where quadratic terms dominate (upper right); and $c_{\mathrm{W}}$ , where quadratic and linear terms both contribute (lower). The results with quadratic terms included in the parameterization (solid black line) and with linear terms only (dashed blue line) are shown. The solid grey lines indicate the sets of points $q^{68\%}$ (lower line) and $q^{95\%}$ (upper line), whereas the dashed grey lines denote the test statistic values used to determine the 68% and 95% confidence intervals in the asymptotic approximation. The confidence intervals shown on the figures are the 68% confidence intervals extracted from the intersection of the linear-plus-quadratic curve with the $q^{68\%}$ line.
png pdf	Figure C2: Summary of best fit values and confidence intervals extracted with the asymptotic approximation (grey lines) and with the pseudo-experiment-based method described in this appendix (black lines). The constraints are scaled by powers of 10 to ensure the constraints on all WCs can be visualized on the same $y$ axis scale. The intervals are generally compatible with each other, with only some small differences visible.

Tables
png pdf	Table 1: The SMEFT operators studied in this analysis, following the definitions of Ref. [9], where $(q,u,d)$ denote quark fields of the first two generations, $(Q,t,b)$ quark fields of the third generation, and $(l,e,\nu)$ lepton fields of all three generations. The Higgs doublet field is indicated by $H$ ; $D$ represents a covariant derivative; $\Box$ is the d'Alembert operator; $X = G, W, B$ denotes a vector boson field strength tensor; $p,r$ are flavour indices. Fermion fields are represented by $\psi$ , with $L$ and $R$ indicating left- and right-handed fermion fields.
png pdf	Table 2: Summary of input analysis characteristics. The observables are defined in the following sections and the experimental likelihood is defined in Section 6.
png pdf	Table 3: The SM parameters used in the event generation to derive the SMEFT parameterizations [80].
png pdf	Table D1: Expected and observed 95% CL limits on linear combinations of WCs from the hybrid fit with the full set of input measurements, in units of TeV $^{-2}$ .
png pdf	Table D2: Expected and observed individual 95% CL limits on WCs from the hybrid fit with the full set of input measurements, in units of TeV $^{-2}$ . This table shows the 32 WCs with the strongest expected constraints, when considering the fit with linear terms only.
png pdf	Table D3: Expected and observed individual 95% CL limits on WCs from the hybrid fit with the full set of input measurements, in units of TeV $^{-2}$ . This table shows the 32 WCs with the weakest expected constraints, when considering the fit with linear terms only.

Summary

A standard model effective field theory (SMEFT) interpretation of data collected by the CMS experiment has been presented. This combined interpretation is based on a simultaneous fit of seven sets of CMS measurements that probe Higgs boson, electroweak vector boson, top quark, and multijet production, and also incorporates measurements of electroweak precision observables from LEP and SLC. These input measurements were chosen to obtain sensitivity to a broad set of SMEFT operators. Out of 129 operators in the SMEFT basis considered in this paper, the combined interpretation constrains 64 Wilson coefficients (WCs) individually. The constraints are provided for both linear-only and linear-plus-quadratic parameterizations. Simultaneous constraints are set on 42 linear combinations of WCs. In the fit that constrains the linear combinations of WCs, the

$p$ -value for the compatibility with the standard model is 1.7%. When excluding the inclusive jet measurement from the combination, the

$p$ -value is 26%. The 95% confidence intervals range from around

$\pm$ 0.002 to

$\pm$ 10 TeV

$^{-2}$ for the constraints on the linear combinations of WCs, whereas for the individual WCs the constraints range from

$\pm$ 0.003 to

$\pm$ 20 TeV

$^{-2}$ . These constraints are also translated into lower limits on the probed energy scale of new physics

$\Lambda$ , for given values of the WCs. This combined interpretation yields improved constraints with respect to single-analysis results from CMS.

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