Quartic coupling unification in the maximally symmetric 2HDM

We consider the maximally symmetric two-Higgs doublet model (MS-2HDM) in which the so-called Standard Model (SM) alignment can be naturally realized as a consequence of an accidental SO(5) symmetry in the Higgs sector. This symmetry is broken (i) explicitly by renormalization-group (RG) effects and (ii) softly by the bilinear scalar mass term $m_{12}^2$. We find that in the MS-2HDM all quartic couplings can unify at large RG scales ${\mu }_X\sim {10}^{11}–10^{20}\mathrm{GeV}$. In particular, we show that quartic coupling unification can take place in two different conformally invariant points, where all quartic couplings vanish. We perform a vacuum stability analysis of the model in order to ensure that the electroweak vacuum is sufficiently long-lived. The MS-2HDM is a minimal and very predictive extension of the SM governed by only three additional parameters: the unification scale ${\mu }_X$, the charged Higgs mass $M_{h^{\pm }}$ (or $m_{12}^2$), and $\tan \beta$, which allow one to determine the entire Higgs sector of the model. In terms of these input parameters, we present illustrative predictions of misalignment for the SM-like Higgs-boson couplings to the $W^{\pm }$ and $Z$ bosons and, for the first time to our knowledge, to the top and bottom quarks.

Figure 1

The RG evolution of the quartic couplings ${\lambda }_{1,2,3,4}$ and the gauge and Yukawa couplings from the threshold scale $M_{h^{\pm }}=500\mathrm{GeV}$ up to their first quartic coupling unification scale ${\mu }_X^{(1)}={10}^{11}\mathrm{GeV}$ for $\tan \beta =50$.

Figure 2

The same as in Fig. 1, where the RG evolution extended up to the second quartic coupling unification point ${\mu }_X^{(2)}={10}^{18}\mathrm{GeV}$ is shown.

Figure 3

The scalar mass spectrum of the MS-2HDM with $m_{12}^2\approx 70^2{\mathrm{GeV}}^2$ and $\tan \beta =50$. The dashed and the solid curves show the one-loop and the two-loop RGEs of the scalar masses, respectively.

Figure 4

Sets of quartic coupling unification points in the $(\tan \beta ,{\log }_{10}\mu )$ plane, for charged Higgs-boson masses $M_{h^{\pm }}=500\mathrm{GeV}$, 1 TeV, 10 TeV, and 100 TeV. The dashed and solid curves show the sets of low-scale and high-scale quartic coupling unification points, ${\mu }_X^{(1)}$ and ${\mu }_X^{(2)}$, respectively.

Figure 5

Numerical estimates of the misalignment parameter $|1-g_{HVV}^2|$ pertinent to the $HVV$ coupling (with $V=W^{\pm },Z$) as functions of the RG scale $\mu$, for a low-scale quartic coupling unification scenario, assuming $M_{h^{\pm }}=500\mathrm{GeV}$ and $\tan \beta =2$, 5, 20, 35, and 50.

Figure 6

The same as in Fig. 5, but for a high-scale quartic coupling unification scenario.

Figure 7

The misalignment parameters $|1-g_{Htt}^2|$ (left panel) and $|1-g_{Hbb}^2|$ (right panel) versus the RG scale $\mu$ for a low-scale quartic coupling unification scenario, assuming $M_{h^{\pm }}=500\mathrm{GeV}$ and $\tan \beta =2$, 5, 20, 35, and 50.

Figure 8

The same as in Fig. 7, but for a high-scale quartic coupling unification scenario.

Figure 9

The profile of the effective potential along the $H$- and $h$-field directions, with $\tan \beta =50$ and $M_h^{\pm }=500\mathrm{GeV}$. The potential along $H(h)$ is getting negative at scales around $\sim {10}^{11}\mathrm{GeV}$ and reaches a new minimum at roughly $\sim {10}^{18}\mathrm{GeV}$ ($\sim {10}^{15}\mathrm{GeV}$).