The Higgs and top mass coincidence problem

IFT, Dept. of Physics, U. Murcia, Murcia, SPAIN

^a e-mail: etl@um.es

Published online: 29 May 2015

Abstract

On the light of the recent LHC boson discovery, we present a phenomenological evaluation of the ratio ρ_t = m_Zm_t/m²_H, from the LHC combined m_H value, we get ((1σ)) $${\rho }_t{}^{(exp)}=0.9956\pm \mathrm{0.0081.}$$

This value is close to one with a precision of the order ∼ 1%. Similarly we evaluate the ratio ρ_Wt = (m_W + m_t)/(2m_H). From the up-to-date mass values we get ρ^(exp)_wt = 1.0066 ± 0.0035 (1σ). The Higgs mass is numerically close (at the 1% level) to the m_H ∼ (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): $$\frac{m_i}{m_j}\simeq f_{ij}({\theta }_W),i,j=W,Z,H,t.$$(1)

For example: m_H/m_Z ≃ 1 + √2s²_θW/2, m_H/m_tc_θW ≃ 1 − √2s²_θW/2. In the limit cos θ_W → 1 all the masses would become equal m_Z = m_W = m_t = m_H. It is tempting to think that such a value, it is not a mere coincidence but, on naturalness grounds, a signal of some more deeper symmetry. In a model independent way, ρ_t can be viewed as the ratio of the highest massive representatives of the spin (0, 1/2, 1) SM and, to a very good precision the LHC evidence tell us that m_s=1m_s=1/2/m²_s=0 ≃ 1. Somehow the “lowest” scalar particle mass is the geometric mean of the highest spin 1, 1/2 masses.

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 λ ≃ κ(g² + g^′2) with κ ≃ 1 + o(g/g_t). Moreover the existence of relations m_i/m_j ≃ f_ij(θ_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 ρ_t, ρ_Wt are so close to one, can we find a mechanism that naturally gives m²_H = m_Zm_t, 2m_H = m_W + m_t ?.

PACS:14.80.Bn,14.80.Cp.