The decoupling limit in the Georgi-Machacek model

We study the most general scalar potential of the Georgi-Machacek model, which adds isospin-triplet scalars to the Standard Model (SM) in a way that preserves custodial SU(2) symmetry. We show that this model possesses a decoupling limit, in which the predominantly-triplet states become heavy and degenerate while the couplings of the remaining light neutral scalar approach those of the SM Higgs boson. We find that the SM-like Higgs boson couplings to fermion pairs and gauge boson pairs can deviate from their SM values by corrections as large as $\mathcal{O}(v^2/M_{\text{new}}^2)$, where $v$ is the SM Higgs vacuum expectation value and $M_{\text{new}}$ is the mass scale of the predominantly-triplet states. In particular, the SM-like Higgs boson couplings to $W$ and $Z$ boson pairs can decouple much more slowly than in two Higgs doublet models, in which they deviate from their SM values like $\mathcal{O}(v^4/M_{\text{new}}^4)$. Furthermore, near the decoupling limit the SM-like Higgs boson couplings to $W$ and $Z$ pairs are always larger than their SM values, which cannot occur in two Higgs doublet models. As such, a precision measurement of Higgs couplings to $W$ and $Z$ pairs may provide an effective method of distinguishing the Georgi-Machacek model from two Higgs doublet models. Using numerical scans, we show that the coupling deviations can reach 10% for $M_{\text{new}}$ as large as 800 GeV.

Figure 1

Constraints on the $({\lambda }_3,{\lambda }_4)$ and $({\lambda }_5,{\lambda }_2)$ planes from perturbative unitarity. The constraints from $x_1^{\pm }$ and $x_2^{\pm }$ are the maximum allowed ranges obtained by setting the couplings not shown on the figure axes to zero.

Figure 2

The boundary of the region in the $(\zeta ,\omega )$ plane that is populated by taking all possible combinations of field values. The region enclosed by the curve is populated.

Figure 3

Constraints on the $({\lambda }_3,{\lambda }_4)$ plane from perturbative unitarity, as in Fig. 1, together with the bounded-from-below (BFB) constraints ${\lambda }_4>-\frac{1}{3}{\lambda }_3$ and ${\lambda }_4>-{\lambda }_3$.

Figure 4

Six projections of the accessible region in the space of $\zeta ,\omega ,\sigma ,\rho$. The accessible region is that enclosed within the outermost lines (including dashed lines). The solid line is the path traced out by the parametrization of Eq. (55), with values of $\theta \in [0,2\pi )$ as marked. Our desired vacuum, with $⟨{\chi }^0⟩=⟨{\xi }^0⟩=v_{\chi }$, corresponds to $\theta =a$ for positive $v_{\chi }$ and $\theta =\pi +a$ for negative $v_{\chi }$.

Figure 5

The values of $\zeta$, $-\omega$, $-\sigma$, and $-\rho$ as a function of $\theta$, in the parametrization of Eq. (55). The vertical dotted lines correspond to the desired minima at $\theta =a$ and $\theta =\pi +a$.

Figure 6

Top: mass spectrum of the model as a function of $\sqrt{{\mu }_3^2}$ for cases A (left) and B (right). Bottom: mass splittings $m_i-\sqrt{{\mu }_3^2}$ for the heavy scalars as a function of ${\mu }_3$ for cases A (left) and B (right). In the bottom plots the colored (light) curves show the exact tree-level masses while the black curves are the associated expansion formulas from Eq. (58). For $m_3$ and $m_5$, the expansion formula curves are almost identical to the exact curves.

Figure 7

Top: dependence of the triplet vev $v_{\chi }$ (shown normalized to $v=246\mathrm{GeV}$) as a function of ${\mu }_3$ in cases A (left) and B (right). Bottom: mixing angle $\sin \alpha$ that controls the composition of the light Higgs boson $h={\varphi }^{0,r}\cos \alpha -H_1^{0\prime }\sin \alpha$, shown as a function of ${\mu }_3$ in cases A (left) and B (right). In all plots the solid red (light) line is the exact curve, while the dashed black (dark) line is the corresponding expansion formula from Eqs. (59) and (60).

Figure 8

The light Higgs coupling modification factors ${\kappa }_V$ (upper dashed curves) and ${\kappa }_f$ (lower dot-dashed curves) as a function of ${\mu }_3$, for cases A (left) and B (right). The colored (light) curves are the exact results while the black (dark) curves show the corresponding expansion formulas as in Eq. (61).

Figure 9

Top: light Higgs coupling modification factor ${\kappa }_{\gamma }$ as a function of ${\mu }_3$, for cases A (left) and B (right). Bottom: light Higgs coupling modification factor $\mathrm{\Delta }{\kappa }_{\gamma }$, comprising only the contributions from the non-SM charged scalars in the loop, as a function of ${\mu }_3$. In all plots the solid red (light) line shows the exact one-loop result, while the dashed black (dark) line is the expansion formula as discussed in the text.

Figure 10

Top: light Higgs coupling modification factor ${\kappa }_{Z\gamma }$ as a function of ${\mu }_3$, for cases A (left) and B (right). Bottom: light Higgs coupling modification factor $\mathrm{\Delta }{\kappa }_{Z\gamma }$, comprising only the contributions from the non-SM charged scalars in the loop, as a function of ${\mu }_3$. In all plots the solid red (light) line shows the exact one-loop result, while the dashed black (dark) line is the expansion formula as discussed in the text.

Figure 11

The light Higgs coupling modification factors ${\kappa }_V$ (upper) and ${\kappa }_f$ (lower) as a function of the mass of the lightest of the new scalars, $M_{\text{new}}=\min (m_H,m_3,m_5)$. The right panels show a closeup of the region of small coupling deviations.

Figure 12

As in Fig. 11 but for ${\kappa }_{\gamma }$ (upper) and ${\kappa }_{Z\gamma }$ (lower).