Basis-independent methods for the two-Higgs-doublet model. III. The $CP$-conserving limit, custodial symmetry, and the oblique parameters $S$, $T$, $U$

In the standard model, custodial symmetry is violated by the hypercharge U(1) gauge interactions and the Yukawa couplings, while being preserved by the Higgs scalar potential. In the two-Higgs-doublet model (2HDM), the generic scalar potential introduces new sources of custodial symmetry breaking. We obtain a basis-independent expression for the constraints that impose custodial symmetry on the 2HDM scalar potential. These constraints impose $CP$-conservation on the scalar potential and vacuum, and in addition add one extra constraint on the scalar potential parameters. We clarify the mass degeneracies of the 2HDM that arise as a consequence of the custodial symmetry. We also provide a computation of the oblique parameters ($S$, $T$, and $U$) for the most general $CP$-violating 2HDM in the basis-independent formalism. We demonstrate that the 2HDM contributions to $T$ and $U$ vanish in the custodial symmetry limit, as expected. Using the experimental bounds on $S$ and $T$ from precision electroweak data, we examine the resulting constraints on the general 2HDM parameter space.

Figure 1

Scatter plots for $T$ as a function of $S$, with $m_1=117\pm 10\mathrm{GeV}$. The ellipses, representing the $1\sigma$ and $2\sigma$ contours allowed by precision electroweak data, are based on Ref. 19, with the parameter $U$ fixed to zero. Plot (a) shows the expanded view in the $S\mathrm{\text{−}}T$ plane, and plot (b) shows a close-up view of the allowed region.

Figure 2

The effect on $S$ and $T$ when $m_1$ is increased from 117 GeV to 500 GeV. Both $S$ and $T$ are zero at $m_1=m_{\varphi }=117\mathrm{GeV}$. When $m_1$ reaches 175 GeV, $S$ and $T$ exceed the boundary of the $2\sigma$ contour ellipse.

Figure 3

The effect on $S$ and $T$ when $m_{H^{\pm }}$ is varied by increasing $Z_3$, for $m_1=350\mathrm{GeV}$ and $m_{\varphi }=117\mathrm{GeV}$. In column (a), $Z_1=4$; in column (b), $Z_1=4\pi$. When $m_{H^{\pm }}$ is in the range 443–489 GeV [470–505 GeV], the points in column (a) [column (b)] fall within the $2\sigma$ contour ellipse.