Matching conditions and Higgs mass upper bounds revisited

Abstract

Matching conditions relate couplings to particle masses. We discuss the importance of one-loop matching conditions in Higgs and top-quark sector as well as the choice of the matching scale. We argue for matching scales $μ_{0,t} \simeq m_t$ and $μ_{0,H} \simeq max[ m_t, M_H ]$. Using these results, the two-loop Higgs mass upper bounds are reanalyzed. Previous results for $Λ\approx$ few TeV are found to be too stringent. For $Λ=10^{19}$ GeV we find $M_H < 180 \pm 4\pm 5$ GeV, the first error indicating the theoretical uncertainty, the second error reflecting the experimental uncertainty due to $m_t=175\pm6$ GeV.

Figures (6)

• Figure 1: (a) Values of $\mu_0$ and $M_H$ for which the one-loop Higgs matching correction $\delta_H(\mu_0)$, Eq. (\ref{['mchiggs']}), equals the values indicated next to the various contour lines. The top quark mass is taken to be 175 GeV. (b) Same plot, but the leading two-loop heavy-Higgs corrections NR have been added.
• Figure 2: Values of $\mu_0$ and $M_H$ for which top-quark matching correction $\delta_t(\mu_0)$, Eq. (\ref{['mctop']}), equals the values indicated next to the various contour lines. Results are shown using $m_t=165$ GeV (dotted), 175 GeV (solid), and 185 GeV (dashed).
• Figure 3: Choosing either one-loop or two-loop RG evolution and various cutoff conditions $\lambda_c(\Lambda)$, the maximally allowed value of $\lambda(M_Z)$ is given as a function of $g_t(M_Z)$. The cutoff condition $\lambda_c(\Lambda)$ is imposed at scales $\Lambda=10^3$ GeV (left plot) and $\Lambda=10^6,10^{10},10^{16}$ GeV (right plot).
• Figure 4: Choosing two-loop RG evolution and cutoff condition $\lambda_c(\Lambda)=\lambda_{\rm FP}/2$, the upper bound on $M_H$ is calculated. The running Higgs and Yukawa couplings, $\lambda(\mu)$ and $g_t(\mu)$, are fixed by the physical masses $M_H$ and $m_t$ using matching conditions with and without one-loop matching corrections. In addition, the Higgs matching scale is varied to be $\mu_{0,H}=M_H$ and $M_Z$. The top-quark mass is fixed at $m_t=175$ GeV, and $\mu_{0,t}=m_t$. The left plot shows the result for small values of $\Lambda$, the right plot extends up to values of $\Lambda=10^{19}$ GeV.
• Figure 5: The dependence of the upper $M_H$ bound on the top-quark mass. The $\overline{\rm MS}$ matching conditions with $\mu_{0,H}=M_H$ and $\mu_{0,t}=m_t$ are used in connection with two-loop RG evolution and cutoff condition $\lambda_c(\Lambda)=\lambda_{\rm FP}/2$. For low values of the embedding scale $\Lambda$, the $M_H$ upper bound is insensitive to the exact value of $m_t$. For large embedding scales there is a larger $m_t$ dependence. Without matching corrections (not shown), the top mass dependence is qualitatively the same.