Statistical approach to Higgs boson couplings in the standard model effective field theory

We perform a parameter fit in the standard model effective field theory (SMEFT) with an emphasis on using regularized linear regression to tackle the issue of the large number of parameters in the SMEFT. In regularized linear regression, a positive definite function of the parameters of interest is added to the usual cost function. A cross-validation is performed to try to determine the optimal value of the regularization parameter to use, but it selects the standard model (SM) as the best model to explain the measurements. Nevertheless as proof of principle of this technique we apply it to fitting Higgs boson signal strengths in SMEFT, including the latest Run-2 results. Results are presented in terms of the eigensystem of the covariance matrix of the least squares estimators as it has a degree model-independent to it. We find several results in this initial work: the SMEFT predicts the total width of the Higgs boson to be consistent with the SM prediction; the ATLAS and CMS experiments at the LHC are currently sensitive to non-resonant double Higgs boson production. Constraints are derived on the viable parameter space for electroweak baryogenesis in the SMEFT, reinforcing the notion that a first order phase transition requires fairly low-scale beyond the SM physics. Finally, we study which future experimental measurements would give the most improvement on the global constraints on the Higgs sector of the SMEFT.

Figure 1

(Left:) Cross-validation test to determine the optimal value of $\kappa$. The average ${\chi }^2$ per number of measurements is shown as a function of $\kappa$. The blue circles and orange squares correspond to the training (${\chi }_t^2/n_t$) and validation (${\chi }_v^2/n_v$) data sets, respectively. The cross-validation selects the SM ($\kappa \to \infty$) as the best model to explain the measurements. (Right:) The same as on the left, but with an artificial BSM signal injected. Here all $tth$ signal strength central values have been fixed to 3.0 with their uncertainties left unchanged. In this case, the cross-validation selects $\kappa \approx 1$ as the best model as it has the lowest (average) value of ${\chi }_v^2/n_v$.

Figure 2

One sigma limits ${\sigma }_k$ on the $k=1\dots 12$ eigenvectors $W_k$ of the covariance matrix for the least squares estimators. The blue circles and orange squares are the results of 12-parameter, regularized fits with $\kappa =1$ and ${10}^{-2}$, respectively. The green diamonds correspond to an unregularized, 10-parameter least squares fit.

Figure 3

(Left:) Preferred parameter space in the $c_g-c_{\gamma }$ plane based on the criteria $\mathrm{\Delta }{\chi }^2=2.30$ (darker shading) and 6.18 (lighter shading). The central values of $c_{\gamma }$ and $c_g$ depend on the assumption of what additional parameters may be nonzero and how large they can be, while the variances of and correlation between $c_{\gamma }$ and $c_g$ are the same in all cases. See the text for details about the five scenarios. (Right:) In contrast, when the same fits are presented in terms of the eigenvectors $W_{11}$ and $W_{12}$, perfect agreement between the five cases is found. This indicates the eigensystem has a degree of model independence to it.

Figure 4

Constraints on the parameter space relevant for EW baryogenesis in the SMEFT. A first-order phase transition occurs in the wedge bounded by the red lines [84], and viable parameter space still exists. The purple ellipses are the result of a regularized fit with $\kappa =10$, which is not consistent with a first-order phase transition at the one sigma level. A regularization parameter of 10 approximately corresponds to an effective scale of 800 GeV, reinforcing the notion that successful EW baryogenesis requires fairly low-scale BSM physics [80, 85].

Figure 5

The same as Fig. 2, but with the results of the pseudoinverse fit included as the red triangles. The eigenvalues of the blind directions are zero when using the pseudoinverse with regularization, and none of the other eigenvalues have regulator dependence.

Figure 6

Results of regularized fits with $\kappa ={10}^{-2}$ using only Run-1 results (orange squares), only Run-2 results (green diamonds), and both runs (blue circles). Many of the bounds are now driven by the Run-2 results, but the contributions from the Run-1 measurements are still important.

Figure 7

The same as Fig. 3 but in the $c_g-c_t$ plane (left), and the $W_{11}-W_5$ plane (right). The correlation between the eigenvectors in the different scenarios is not perfect. Nevertheless the agreement between the different scenarios is better in the eigenvector basis.

Figure 8

Bounds on double Higgs production, $\mu (pp\to hh)=\sigma (pp\to hh)/{\sigma }_{\mathrm{SM}}(pp\to hh)$, when all 12 parameters are allowed to be nonzero as a function of the regularization parameter, $\kappa$. The blue curve gives the best fit values, and the darker and lighter shaded regions are allowed at 1 and $2\sigma$, respectively. The dashed line is the experimental upper limit from CMS [75].