The Higgs mass coincidence problem: why is the higgs mass $m_H^2=m_Z m_t$?

Abstract

On the light of the recent LHC boson discovery, we present a phenomenological evaluation of the ratio $\rho_t=m_Z m_t/m_H^2$, from the LHC combined $m_H$ value, we get ($ (1\sigma)$) $$\rho_t^{(exp)}= 0.9956\pm 0.0081.$$ This value is close to one with a precision of the order $\sim 1\%$. Similarly we evaluate the ratio $\rho_{Wt}=(m_W + m_t)/(2 m_H)$. From the up-to-date mass values we get $\rho_{Wt}^{(exp)}= 1.0066\pm 0.0035\; (1\sigma).$ The Higgs mass is numerically close (at the $1\%$ level) to the $m_H\sim (m_W+m_t)/2$. From these relations we can write any two mass ratios as a function of, exclusively, the Weinberg angle (with a precision of the order of $1\%$ or better): \begin{eqnarray} \frac{m_i}{m_j}&\simeq & f_{ij}(\theta_W),\quad i,j=W,Z,H,t. \end{eqnarray} For example:$m_H/m_Z \simeq 1+\sqrt{2} s_{\theta_W/2}^2$, $m_H/m_t c_{\theta_W} \simeq 1-\sqrt{2}s_{ \theta_W/2}^2$. In the limit $\cos\theta_W\to 1$ all the masses would become equal $m_Z=m_W=m_t=m_H$. We review the theoretical situation of this ratio in the SM and beyond. In the SM these relations are rather stable under RGE pointing out to some underlying UV symmetry. In the SM such a ratio hints for a non-casual relation of the type $\lambda \simeq \kappa \left (g^2+{g'}{}^2\right )$ with $\kappa\simeq 1+o(g/g_t)$. Moreover the existence of relations $m_i/m_j \simeq f_{ij}(\theta_W)$ could be interpreted as a hint for a role of the $SU(2)_c$ custodial symmetry, together with other unknown mechanism. % Without a symmetry at hand to explain then in the SM, it arises a Higgs mass coincidence problem, why the ratios $\rho_t,\rho_{Wt}$ are so close to one, can we find a mechanism that naturally gives $m_H^2=m_Z m_t$, $2m_H= m_W+m_t$?.