Application of the 3-Loop FlexibleEFTHiggs Method to the MSSM and the NMSSM

TL;DR

This work extends the 3-loop FlexibleEFTHiggs hybrid approach to the MSSM and NMSSM to predict the light CP-even Higgs pole mass $M_h$ with high precision. It demonstrates robustness across highly non-degenerate SUSY spectra and provides a full NMSSM implementation with 3-loop matching and large-log resummation up to $\text{N}^3\text{LO}/\text{N}^3\text{LL}$ in the strong sector. Uncertainty estimation separates high-scale matching uncertainties and low-scale SM-side uncertainties, and the analysis identifies NMSSM-specific effects and regions where predictions remain reliable within about $0.7$ GeV. The work includes validation against MSSM results, comparison with other codes, and benchmark NMSSM scenarios illustrating practical applicability for Higgs mass predictions.

Abstract

We perform an extensive analysis of the light CP-even Higgs boson pole mass in the MSSM and its dependencies on various parameters based on the 3-loop FlexibleEFTHiggs hybrid calculation which is implemented and publicly avaiable since recently in FlexibleSUSY. Our focus lies on the study of the robustness of the approach in scenarios of highly non-degenerate SUSY mass spectra. Also, we present an improved Higgs mass calculation in the NMSSM based on the same approach, which is published in the new version 2.9.0 of FlexibleSUSY as well. The calculation provides a treatment in the full-model parametrization, leading to an advantageous resummation of QCD-enhanced terms in the stop-mixing parameter and includes important 2-loop contributions as well as 3-loop QCD contributions in the MSSM limit. We assess the reliability of this new calculation by applying it to several distinct NMSSM scenarios. In this context, special attention is devoted to the estimation of NMSSM-specific theory uncertainty.

Figures (10)

• Figure 1: Schematic illustration of the scales used in the refined Flex-ib-le-EFT-Higgs method.
• Figure 2: Left panel: Dependence of $M_h$ (and the stop pole masses $M_{\tilde{t}_{1,2}}$) on the gluino mass parameter $M_3$ with all other parameters fixed as in Eqs. \ref{['eq:parameter_condition_MSSM']}. The kink in the red uncertainty band at $M_3 \approx 8\,\text{TeV}$ originates from the high-scale uncertainty determined according to Eq. \ref{['eq:DMh_Qm']}. At that point, the maximal difference is no longer found at the upper end of the $Q_{\text{m}}$ variation interval but at the lower one. Right panel: Dependence of $M_h$ on the stop mixing parameter $x_t$ for different values of $M_3$. For $M_3 = 9\,\text{TeV}$, the uncertainty is drawn as light blue band.
• Figure 3: Left panel: $M_h$ as a function of the $\overline{\text{DR}}\xspace'\xspace$ CP-odd Higgs mass $m_A$. Right panel: $M_h$ as a function of the $\mu$-parameter. The parameters not varied are fixed to $M_S=3\,\text{TeV}$, $\tan\beta=20$ and $x_t=-2.2$. The green band (left panel) and the blue band (right panel) show the uncertainty $\Delta M_h$. The brown band corresponds to the measured Higgs mass with its experimental uncertainty.
• Figure 4: Left panel: $M_h$ as a function of $(m_{\tilde{q}})_{33}$ for different values of $x_t$. Right panel: $M_h$ as a function of $(m_{\tilde{u}})_{33}$ for different values of $x_t$. The parameters not varied are fixed to $M_S=3\,\text{TeV}$ and $\tan \beta = 12$. All curves are obtained from a 2-loop matching except the black dashed ones which represent the 3-loop results only for $x_t=2$. Also for $x_t=2$, the uncertainty bands (corresponding to the 2-loop calculation) are drawn in red (left panel) and teal (right panel), respectively. The brown band corresponds to the measured Higgs mass with its experimental uncertainty.
• Figure 5: Diagrams corresponding to contributions to $\hat{\lambda}(Q_{\text{m}})$ not covered by the calculation, where $\phi\in\{\Re(h_u^0), \Re(h_d^0)\}$. \ref{['fig:mis_lambda_2l']}: non-log-enhanced 2-loop contribution of $\mathcal{O}\!\left((\lambda+\kappa)^4\lambda^2\right)$. \ref{['fig:mis_lambda_3l_log']}: log-enhanced 3-loop contribution of $\mathcal{O}\!\left((\lambda+\kappa)^8\right)$. \ref{['fig:mis_lambda_3l']}: non-log-enhanced 3-loop contribution of $\mathcal{O}\!\left((\lambda+\kappa)^6\lambda^2\right)$. Recall that triple scalar couplings involving one $s$ contribute a vertex factor of $\mathcal{O}\!\left(\kappa + \lambda\right)$ and quartic vertices involving only $\phi$ a factor of $\mathcal{O}\!\left(\kappa^2/\lambda\right)$.