Renormalization Group Running of the Minimal Leptophilic Dark Matter Model toward a UV Completion

TL;DR

This work examines the renormalization group evolution of a minimal leptophilic dark matter model in which a SM-singlet fermion $N$ couples to right-handed charged leptons via a charged scalar $S^{\pm}$. By computing one- and two-loop RGEs for the Yukawa couplings and scalar quartics, the authors show that the sizable Yukawas needed to reproduce the observed relic density push the theory toward a Landau pole and can destabilize the scalar potential, limiting perturbativity and vacuum stability up to high energies. Stabilizing the potential by increasing quartic couplings can itself trigger nonperturbative behavior, yielding a constrained maximal cutoff scale $\Lambda_{\mathrm{cut}}^{\mathrm{max}}$ and strong sensitivity to initial conditions $\lambda_S(m_S)$ and $\lambda_{SH}(m_S)$. Consequently, Planck-scale consistency generally requires relatively light spectra, with $m_N$ and $m_S$ below about $350$ GeV (roughly $20~\mathrm{GeV} \lesssim m_N \lesssim 350~\mathrm{GeV}$, $150~\mathrm{GeV} \lesssim m_S \lesssim 350~\mathrm{GeV}$), and motivates UV completions below the Planck scale; these findings provide theoretical guidance for building more fundamental models and imply testable collider implications for future lepton colliders.

Abstract

We study the renormalization group running of the coupling constants in a minimal leptophilic dark matter model in which a Standard Model singlet fermion, acting as the dark matter (DM) candidate, couples exclusively to right-handed charged leptons via a new charged scalar mediator. Reproduction of the observed thermal relic abundance of the DM candidate requires sizable Yukawa couplings, and such sizable Yukawa couplings can significantly affect the renormalization group evolution of the model parameters. We examine the conditions that the model remains perturbative and the vacuum stability is maintained up to high-energy scales. We find that the parameter space is severely constrained to ensure the perturbativity and the vacuum stability up to the Planck scale. In particular, the masses of the dark matter and the charged scalar mediator should be smaller than about 350 GeV and can be tested by future collider experiments. The lower bound of the dark matter mass that is larger than a few GeV is also obtained by perturbativity.

Figures (6)

• Figure 1: The value of $\Lambda_{\mathrm{cut}}^{\mathrm{max}}$ as a function of the input Yukawa coupling $y_3$ at $\mu=300$ GeV. The solid (red) curve shows the numerical solution of the one-loop RGE for $y_3$ in the case that $N$ only couples to the tau lepton, while the dashed (blue) curve shows the approximate solution given in Eq. \ref{['eq:yNRGE:approx']}. The two curves are almost identical.
• Figure 2: The RG running of the scalar quartic couplings $\lambda_H$, $\lambda_S$, and $\lambda_{SH}$, and the mass squared parameters $\mu_H^2$ and $\mu_S^2$ in the case that $m_N=200$ GeV, $m_S=250$ GeV, $\lambda_S(m_S)=0.5$. The panels (a), (b), (c), and (d) correspond to the initial values of $\lambda_{SH}(m_S)=0.0, 0.3, 0.45, 0.8$, respectively. In each panel, the dashed curves show the running with the one-loop beta functions, while the solid curves show the running with the two-loop beta functions.
• Figure 3: The RG running of the scalar quartic couplings $\lambda_H$, $\lambda_S$, and $\lambda_{SH}$, and the mass squared parameters $\mu_H^2$ and $\mu_S^2$ in the case that $m_N=200$ GeV, $m_S=250$ GeV, $\lambda_S(m_S)=0$. The panels (a) and (b) correspond to the initial values of $\lambda_{SH}(m_S)=0.5$ and $0.8$, respectively. In each panel, the dashed curves show the running with the one-loop beta functions, while the solid curves show the running with the two-loop beta functions.
• Figure 4: The RG running of the scalar quartic couplings $\lambda_H$, $\lambda_S$, and $\lambda_{SH}$, and the mass squared parameters $\mu_H^2$ and $\mu_S^2$ in the case that $m_N=300$ GeV, $m_S=450$ GeV, $\lambda_{SH}(m_S)=0.5$. The panels (a) and (b) correspond to the initial values of $\lambda_{S}(m_S)=1.60$ and $1.65$, respectively. In each panel, the dashed curves show the running with the one-loop beta functions, while the solid curves show the running with the two-loop beta functions.
• Figure 5: Contour plot of the energy scale where the perturbative effective theory description breaks down. on the plane of $\lambda_{SH}(m_S)$ and $\lambda_S(m_S)$ for fixed values of $m_N$ and $m_S$. In each figure, the red curves show the contour of the scale where one of the quartic couplings $\lambda_H$, $\lambda_S$, and $\lambda_{SH}$ become nonperturbative (i.e., ${}^{\exists}\lambda_i > 4\pi$). The blue and green curves show the scale where $\lambda_H<0$ and $\lambda_S<0$, respectively. The purple curve shows the contour that $\mu_S^2<0$ at $10^{19}~\mathrm{GeV}$ and $\mu_S^2<0$ occurs at the lower scale in the bottom-left side range of the curve. The cyan shaded region shows the range where the effective theory description is valid up to $10^{12}~\mathrm{GeV}$, and the yellow shaded region shows the range where the effective theory description is valid up to $10^{19}~\mathrm{GeV}$.