$SO(N)$ models and Higgs extensions

Abstract

We discuss previous studies on $SO(N)$ linear sigma models (L$\sigma$M) and some limits of phenomenological interest. These models suffer a spontaneous symmetry breaking (SSB) down to $SO(N-1)$, with the appearance of an associated vacuum expectation value (vev) $f$, a heavy scalar degree of freedom (dof) with mass $M$ and $N-1$ massless Nambu-Goldstone bosons (NGB). These models are of a high interest for beyond Standard Model extensions where the Higgs boson is identified with a pseudo Nambu-Goldstone boson (pNGB) that appears in the $SO(N)/SO(N-1)$ SSB. It gains a non-zero mass $m$ due to soft explicit $SO(N)$ symmetry breaking (ExSB) terms in the Lagrangian. In particular, we will focus on the soft breaking pattern $SO(N)\stackrel{\rm ExSB}{\longrightarrow}SO(4)\times SO(P)\stackrel{\rm SSB}{\longrightarrow} SO(3)\times SO(P-1)$, with $4+P=N$, e.g., via new beyond Standard Model (BSM) gauge boson loops. The $SO(4)/SO(3)$ are the electroweak (EW) chiral/custodial groups and the associated SSB is exactly the Standard Model (SM) one, giving mass to the $W^\pm$ and $Z$ gauge bosons while avoiding large corrections to the oblique $T$ parameter. The comparison of this type of models with the current phenomenological situation, close to the SM ($m=0.125$~TeV, EW vev $v=0.246$~TeV, $M> \mathcal{O}($TeV$)$, $g_{hWW}\approx g_{hWW}^{SM}$) sets important constraints on the L$\sigma$M parameters: there is a very small mixing between the heavy and light L$\sigma$M massive scalars and the pNGB $h$ is essentially SM-like, the low-energy effective field theory (EFT) couplings are very close to the SM ones, and a large hierarchy $\xi=\frac{v^2}{f^2}\ll 1$ is needed in these L$\sigma$M near the $SO(N)$ limit (and $\xi$ much smaller than a certain ratio $\frac{\lambda_2}{\lambda_1}$ of quartic L$\sigma$M couplings in the general case). Likewise, we note the existence of strongly coupled scenarios with a hierarchy $m^2 \sim v^2 \ll f^2 \ll M^2$.