Two-loop corrections to the $\rho$ parameter in Two-Higgs-Doublet Models

Models with two scalar doublets are among the simplest extensions of the Standard Model which fulfill the relation $\rho = 1$ at lowest order for the $\rho$ parameter as favored by experimental data for electroweak observables allowing only small deviations from unity. Such small deviations $\Delta\rho$ originate exclusively from quantum effects with special sensitivity to mass splittings between different isospin components of fermions and scalars. In this paper the dominant two-loop electroweak corrections to $\Delta\rho$ are calculated in the $CP$-conserving THDM, resulting from the top-Yukawa coupling and the self-couplings of the Higgs bosons in the gauge-less limit. The on-shell renormalization scheme is applied. With the assumption that one of the $CP$-even neutral scalars represents the scalar boson observed by the LHC experiments, with standard properties, the two-loop non-standard contributions in $\Delta\rho$ can be separated from the standard ones. These contributions are of particular interest since they increase with mass splittings between non-standard Higgs bosons and can be additionally enhanced by $\tan\beta$ and $\lambda_5$, an additional free coefficient of the Higgs potential, and can thus modify the one-loop result substantially. Numerical results are given for the dependence on the various non-standard parameters, and the influence on the calculation of electroweak precision observables is discussed.