CMS-PAS-SMP-24-003

CMS-PAS-SMP-24-003
Combined effective field theory interpretation of Higgs boson, electroweak vector boson, top quark, and multi-jet measurements
CMS Collaboration
26 September 2024

Abstract: Constraints on the Wilson coefficients (WCs) corresponding to dimension-six 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 multi-jet production. The measurements of the electroweak precision observables at LEP and SLC are also included and provide complementary constraints to those from CMS. The CMS measurements, using 36-138 fb $^{-1}$ of LHC proton-proton collision data at $\sqrt{s}=$ 13 TeV, 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 or, in the case of Higgs boson production, simplified template cross sections, which are subsequently parametrized in the WCs. Measurements targeting $\mathrm{t\bar{t}X}$ production model the SMEFT effects directly in the reconstructed observables. Individual constraints on 64 WCs, and constraints on 42 linear combinations of WCs, are obtained. In the case of the linear combinations, the 42 parameters are varied simultaneously.
Links: CDS record (PDF) ; CADI line (restricted) ;

Figures & Tables	Summary	Additional Tables	References	CMS Publications

Figures
png pdf	Figure 1: Examples of SM processes modified by the SMEFT operator $\mathcal{Q}_{\mathrm{W}}$ : $\mathrm{W}\gamma$ production (left), WW production (center), $\mathrm{H}\to\gamma\gamma$ decay (right). The WC $c_{\mathrm{W}}$ controls the strength of the interaction.
png pdf	Figure 1-a: Examples of SM processes modified by the SMEFT operator $\mathcal{Q}_{\mathrm{W}}$ : $\mathrm{W}\gamma$ production (left), WW production (center), $\mathrm{H}\to\gamma\gamma$ decay (right). The WC $c_{\mathrm{W}}$ controls the strength of the interaction.
png pdf	Figure 1-b: Examples of SM processes modified by the SMEFT operator $\mathcal{Q}_{\mathrm{W}}$ : $\mathrm{W}\gamma$ production (left), WW production (center), $\mathrm{H}\to\gamma\gamma$ decay (right). The WC $c_{\mathrm{W}}$ controls the strength of the interaction.
png pdf	Figure 1-c: Examples of SM processes modified by the SMEFT operator $\mathcal{Q}_{\mathrm{W}}$ : $\mathrm{W}\gamma$ production (left), WW production (center), $\mathrm{H}\to\gamma\gamma$ 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 Wilson coefficients that affect the $\mathrm{H}\to\gamma\gamma$ STXS bins. The top panel shows the measured values relative to the predictions in the SM.
png pdf	Figure 3: Relative effect of the linear SMEFT terms for the Wilson coefficients that affect the $\mathrm{W}\gamma$ , $\mathrm{Z}\to\nu\nu$ , and WW differential cross sections. The top panel shows the measured values relative to the predictions in the SM.
png pdf	Figure 4: Relative effect of the linear SMEFT terms for the Wilson coefficients that affect the $\mathrm{t} \overline{\mathrm{t}}$ differential cross sections. The top panel shows the measured values relative to the predictions in the SM.
png pdf	Figure 5: Relative effect of the linear SMEFT terms for the Wilson coefficients that affect the electroweak precision observables. The top panel shows the measured values relative to the predictions in the SM.
png pdf	Figure 6: Relative effect of the linear SMEFT terms for the Wilson coefficients that affect the inclusive jet differential cross sections in the rapidity bins $(0, 0.5)$ and $(0.5, 1)$ . The top panel shows the measured values relative to the predictions in the SM.
png pdf	Figure 7: Relative effect of the linear SMEFT terms for the Wilson coefficients that affect the inclusive jet differential cross sections in the rapidity bins $(1, 1.5)$ and $(1.5, 2)$ . The top panel shows the measured values relative to the predictions in the SM.
png pdf	Figure 8: Diagonal entries $H_{jj}^{p}$ 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 $H_{jj}^{p}$ correspond to larger 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.
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 lower panel shows the contribution of different input measurements to the total constraints. Note that 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 Wilson coefficient are obtained keeping the other coefficients fixed to 0. The lower panel shows the contribution of different input measurements to the total constraints. Note that 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: 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.
png pdf	Figure 14: Rotation matrix obtained by performing the PCA on the Hessian matrix of a reduced set of measurements, without the ${\mathrm{t}\overline{\mathrm{t}}} \mathrm{X}$ measurement included.
png pdf	Figure 15: Constraints on linear combinations of WCs, from both the hybrid likelihood fit and the $\mathcal{L}^{\text{simpl}}$ fit, excluding the ${\mathrm{t}\overline{\mathrm{t}}} \mathrm{X}$ analysis. The lower panel shows the contribution of different input measurements to the total constraints.
png pdf	Figure 16: Constraints on individual WCs, from both the hybrid likelihood fit and the $\mathcal{L}^{\text{simpl}}$ 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 lower panel shows the contribution of different input measurements to the total constraints.
png pdf	Figure A1: Observed likelihood scans for $c_{\mathrm{HQ}}^{3}$ , 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), while 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+quadratic curve with the $q^{68\%}$ line.
png pdf	Figure A1-a: Observed likelihood scans for $c_{\mathrm{HQ}}^{3}$ , 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), while 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+quadratic curve with the $q^{68\%}$ line.
png pdf	Figure A1-b: Observed likelihood scans for $c_{\mathrm{HQ}}^{3}$ , 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), while 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+quadratic curve with the $q^{68\%}$ line.
png pdf	Figure A1-c: Observed likelihood scans for $c_{\mathrm{HQ}}^{3}$ , 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), while 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+quadratic curve with the $q^{68\%}$ line.
png pdf	Figure A2: 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 intervals are generally compatible with each other, with only some small differences visible.

Tables
png pdf	Table 1: SMEFT operators studied in this analysis, following the definitions of Ref. [8], where $(q,u,d)$ denote quarks of the first two generations, $(Q,t,b)$ quarks of the third generation, and $(l,e,\nu)$ leptons of all three generations. $H$ is the Higgs doublet, $D$ represents a covariant derivative, $X = G, W, B$ denotes a vector boson field strength tensor. $\Psi$ represents fermion fields, with $L$ and $R$ indicating left- and right-handed fermions. An indication of the sensitivity of the input analyses to the different operators is given in Fig. 8.
png pdf	Table 2: Summary of input analysis characteristics
png pdf	Table 3: SM parameters used in the event generation to derive the SMEFT parameterizations [66].
png pdf	Table B1: Expected and observed 95% CL limits on linear combinations of Wilson coefficients from the Hybrid fit with ${\mathrm{t}\overline{\mathrm{t}}} \mathrm{X}$ .
png pdf	Table B2: Expected and observed individual 95% CL limits on Wilson coefficients from the Hybrid fit with ${\mathrm{t}\overline{\mathrm{t}}} \mathrm{X}$ .

Summary

A combined effective field theory interpretation of CMS data has been presented. This interpretation is based on a simultaneous fit of seven sets of CMS measurements that probe Higgs boson, electroweak vector boson, top quark, and multi-jet production, and also incorporates electroweak precision observable measurements 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 effective field theory basis considered in this note, the combined interpretation constrains 64 Wilson coefficients (WCs) individually. Simultaneous constraints were set on 42 linear combinations of WCs. The 95% confidence intervals range from around

$\pm$ 0.002 to

$\pm$ 10 for the constraints on the linear combinations of WCs, while for the individual WCs the constraints range from

$\pm$ 0.003 to

$\pm$ 20. 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, for example for

$c_{\mathrm{W}}$ . In the fit that constrains the linear combination of WCs, The p-value for the compatibility with the SM is found to be 1.7%.

Additional Tables
png pdf	Additional Table 1: Linear parameterizations for the STXS $\mathrm{q}\mathrm{q}\mathrm{H}$ bins.
png pdf	Additional Table 2: Linear parameterizations for the STXS $\mathrm{g}\mathrm{g}\mathrm{H}$ bins.
png pdf	Additional Table 3: Linear parameterizations for the STXS VH leptonic bins (1 lepton).
png pdf	Additional Table 4: Linear parameterizations for the STXS VH leptonic bins (2 leptons).
png pdf	Additional Table 5: Linear parameterizations for the STXS $\mathrm{t}\overline{\mathrm{t}}\mathrm{H}$ bins
png pdf	Additional Table 6: Linear parameterizations for the STXS $\mathrm{t}\mathrm{H}$ bins
png pdf	Additional Table 7: Linear parameterizations for the H decay modes.
png pdf	Additional Table 8: Linear parameterizations for the $\mathrm{Z}\to\nu\nu$ analysis.
png pdf	Additional Table 9: Linear parameterizations for the WW analysis.
png pdf	Additional Table 10: Linear parameterizations for the $\mathrm{W}\gamma$ analysis.
png pdf	Additional Table 11: Linear parameterizations for the inclusive jet measurement, for the bins with $\|y\| <$ 0.5 and $p_{\mathrm{T}} <$ 967.
png pdf	Additional Table 12: Linear parameterizations for the inclusive jet measurement, for the bins with $\|y\| <$ 0.5 and $p_{\mathrm{T}} >$ 967.
png pdf	Additional Table 13: Linear parameterizations for the inclusive jet measurement, for the bins with $< 0.5 < \|y\| <$ 1.0 and $p_{\mathrm{T}} <$ 967 GeV.
png pdf	Additional Table 14: Linear parameterizations for the inclusive jet measurement, for the bins with $< 0.5 < \|y\| <$ 1.0 and $p_{\mathrm{T}} >$ 967 GeV.
png pdf	Additional Table 15: Linear parameterizations for the inclusive jet measurement, for the bins with $< 1.0 < \|y\| <$ 1.5 and $p_{\mathrm{T}} <$ 967 GeV.
png pdf	Additional Table 16: Linear parameterizations for the inclusive jet measurement, for bins with $< 1.0 < \|y\| <$ 1.5 and $p_{\mathrm{T}} >$ 967 GeV.
png pdf	Additional Table 17: Linear parameterizations for the inclusive jet measurement, for bins with $< 1.5 < \|y\| <$ 2.0 and $p_{\mathrm{T}} <$ 967 GeV.
png pdf	Additional Table 18: Linear parameterizations for the inclusive jet measurement, for bins with $< 1.5\|y\| <$ 2.0 and $p_{\mathrm{T}} >$ 967 GeV.
png pdf	Additional Table 19: Linear parameterizations for the $\mathrm{t}\overline{\mathrm{t}}$ measurement, for bins with $M_{\mathrm{t}\overline{\mathrm{t}}} <$ 1000 GeV.
png pdf	Additional Table 20: Linear parameterizations for the $\mathrm{t}\overline{\mathrm{t}}$ measurement, for bins with $M_{\mathrm{t}\overline{\mathrm{t}}} >$ 1000 GeV.
png pdf	Additional Table 21: Linear parameterizations for the electroweak precision observables.
png pdf	Additional Table 22: Definitions of eigenvectors in the measurement including the $\mathrm{t}\overline{\mathrm{t}}\mathrm{X}$ analysis and their eigenvalues. Definitions of EV1-17.
png pdf	Additional Table 23: Definitions of eigenvectors in the measurement including the $\mathrm{t}\overline{\mathrm{t}}\mathrm{X}$ analysis and their eigenvalues. Definitions of EV18-31.
png pdf	Additional Table 24: Definitions of eigenvectors in the measurement including the $\mathrm{t}\overline{\mathrm{t}}\mathrm{X}$ analysis and their eigenvalues. Definitions of EV32-42.
png pdf	Additional Table 25: Definitions of eigenvectors in the measurement including the $\mathrm{t}\overline{\mathrm{t}}\mathrm{X}$ analysis and their eigenvalues. Definitions of EV43-56.
png pdf	Additional Table 26: Definitions of eigenvectors in the measurement including the $\mathrm{t}\overline{\mathrm{t}}\mathrm{X}$ analysis and their eigenvalues. Definitions of EV57-64.
png pdf	Additional Table 27: Definitions of eigenvectors in the measurement without the $\mathrm{t}\overline{\mathrm{t}}\mathrm{X}$ analysis included and their eigenvalues. The eigenvectors are labelled $\mathrm{EV}n'$ to distinguish them from the eigenvectors in the combination that does include the $\mathrm{t}\overline{\mathrm{t}}\mathrm{X}$ inputs. Definitions of EV1'-EV16'.
png pdf	Additional Table 28: Definitions of eigenvectors in the measurement without the $\mathrm{t}\overline{\mathrm{t}}\mathrm{X}$ analysis included and their eigenvalues. The eigenvectors are labelled $\mathrm{EV}n'$ to distinguish them from the eigenvectors in the combination that does include the $\mathrm{t}\overline{\mathrm{t}}\mathrm{X}$ inputs. Definitions of EV17'-EV31'.
png pdf	Additional Table 29: Definitions of eigenvectors in the measurement without the $\mathrm{t}\overline{\mathrm{t}}\mathrm{X}$ analysis included and their eigenvalues. The eigenvectors are labelled $\mathrm{EV}n'$ to distinguish them from the eigenvectors in the combination that does include the $\mathrm{t}\overline{\mathrm{t}}\mathrm{X}$ inputs. Definitions of EV32'-EV42'.
png pdf	Additional Table 30: Definitions of eigenvectors in the measurement without the $\mathrm{t}\overline{\mathrm{t}}\mathrm{X}$ analysis included and their eigenvalues. The eigenvectors are labelled $\mathrm{EV}n'$ to distinguish them from the eigenvectors in the combination that does include the $\mathrm{t}\overline{\mathrm{t}}\mathrm{X}$ inputs. Definitions of EV43'-EV55'.

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