Two-loop top-Yukawa-coupling corrections to the charged Higgs-boson mass in the MSSM

The top-Yukawa-coupling enhanced two-loop corrections to the charged Higgs-boson mass in the real MSSM are presented. The contributing two-loop self-energies are calculated in the Feynman- diagrammatic approach in the gaugeless limit with vanishing external momentum and bottom mass, within a renormalization scheme comprising on-shell and $\overline{\text{DR}}$ conditions. Numerical results illustrate the effect of the $\mathcal{O}{(\alpha_t^2)}$ contributions and the importance of the two-loop corrections to the mass of the charged Higgs bosons.

omitting higher powers in the field components. Explicit expressions for the entries in the mass matrices are given in Ref. [28] for the general complex MSSM [the special case here is obtained for setting T A = 0 in those expressions]. Of particular interest for the correlation between the neutral CP -odd and the charged Higgs-boson masses are the entries for m 2 A and m 2 H ± , reading m 2 A = m 2 1 s 2 βn + m 2 2 c 2 βn + m 2 12 s 2βn − 1 4 (g 2 1 + g 2 2 )(v 2 1 − v 2 2 ) c 2βn , m 2 H ± = m 2 1 s 2 βc + m 2 2 c 2 βc + m 2 12 s 2βc − 1 4 (g 2 1 + g 2 2 )(v 2 1 − v 2 2 ) c 2βc + 1 2 g 2 2 (v 1 c βc + v 2 s βc ) 2 . (2.5) At lowest order, after applying the minimization conditions for the Higgs potential, the tadpole coefficients T h , T H vanish and the mass matrices become diagonal for β c = β n = β, yielding when α is chosen according to The Goldstone bosons G 0 and G ± remain massless.
In the following we focus on the modification of the relation (2.6) by higher-order contributions, which allows to derive the charged Higgs-boson mass in terms of the A-boson mass m A and the model parameters entering through quantum loops.

The charged Higgs-boson mass beyond lowest order
Beyond the lowest order, the entries of the mass matrix of the charged Higgs bosons are shifted by adding their corresponding renormalized self-energies. The higher-order corrected mass M H ± of the physical charged Higgs bosons, the pole mass, is obtained from the zero of the renormalized two-point vertex function, Therein,Σ H + H − p 2 denotes the renormalized self-energy for the charged Higgs bosons H ± , which we treat as a perturbative expansion, (2.10) At each loop order k, the renormalized self-energyΣ At the one-loop level the counterterm is given by and at the two-loop level by involving field-renormalization constants and genuine mass counterterms of one-and two-loop order; they are specified in Ref. [30], from where conventions and notations have been taken over and simplified to the case of the real MSSM. Whereas the one-loop self-energyΣ (1) H + H − (p 2 ) of the charged Higgs boson is completely known, at the two-loop level only results in the approximation for p 2 = 0 have become available, namely the O(α t α s ) corrections calculated earlier [27,39], and the two-loop Yukawa contributions O α 2 t which are presented in this paper. The evaluation of these terms is performed in the gaugeless limit and the bottom-quark mass set to zero (as done in Ref. [39]), thus yielding the top-Yukawa-coupling enhanced parts. Detailed analytical results of the two-loop self-energy and renormalization were published in Ref. [30]. The diagrammatic calculation of the self-energies and counterterms was performed with FeynArts [40], FormCalc [41], and TwoCalc [42]. The full list of Feynman diagrams of O α 2 t for the self-energy of the charged Higgs boson is illustrated in Fig. 1. Within our approximations for the two-loop part of the charged Higgs-boson self-energy, the two-loop counterterm (2.13) simplifes to The genuine mass counterterms δ (k) m 2 H ± are determined by Eq. (2.5) and setting β n = β c = β (see also Ref. [30]). In the gaugeless limit they are given by (for k = 1, 2) The other genuine mass counterterms are determined by the relation involving the tadpole counterterm δ (1) T H and the counterterm δ (1) t β for the renormalization of tan β.
In the real MSSM, the mass of the CP -odd Higgs boson m A is conventionally chosen as a free input parameter; it can thus be renormalized on-shell at each order. Accordingly, the corresponding renormalization conditions in our present approximation read in terms of the renormalized A-boson self-energy as follows,Σ (k) The unrenormalized self-energy Σ A corresponds to the Feynman diagrams depicted in Fig. 2. The counterterms in (2.18) at the one-loop and two-loop level read as follows, (2.19b) The one-loop non-diagonal mass counterterm δ (1) m 2 AG therein is given by (2.20) From the conditions (2.18) for k = 1, 2 the renormalization constants δ (k) m 2 A are determined and thus the mass counterterms δ (k) m 2 H + for the charged Higgs bosons in Eq. (2.16), required for the two-loop counterm (2.15) in the charged Higgs-boson self energy. All field-renormalization constants δ (k) Z {AA,AG,H ± H ± ,H ± G ± } are linear combinations of the basic field-renormalization constants δ (k) Z Hi for the two scalar doublets (2.2), as given in Ref. [30].
In addition to the mass counterterms δ (k) m 2 A , the independent renormalization constants required for renormalization of the charged Higgs-boson self-energy are: the field renormalization constants δ (1) Z Hi , the renormalization constant δ (1) t β for tan β, and the tadpole renormalization constants δ (1) T h , δ (1) T H (the two-loop field renormalization constants cancel in the renormalized self-energies in the p 2 = 0 approximation). Moreover, for the one-loop subrenormalization, we need the counterterms for the top quark and squark masses δ (1) m t , δ (1) mt 1 , δ (1) mt 2 and for the trilinear coupling δ (1) A t , as well as the counterterm for the bilinear coefficient of the superpotential, δ (1) µ. They are fixed in the same way as described in Ref. [30] and we do not repeat them here.
In this section we compute numerically the charged Higgs-boson mass M H ± in the real MSSM in terms of m A chosen as an input parameter. For this purpose, we combine in the renormalized charged Higgs-boson self-energy our new O α 2 t contribution described in the previous section with the already known complete one-loop term and the O(α t α s ) contribution, as the currently best approximation for (2.10). The resulting charged Higgs-boson mass M H ± is obtained via Eq. (2.9) with the help of FeynHiggs.
In the following numerical analysis we use the input parameters as listed in Tab. 1 (giving also those parameters not needed for the two-loop self-energies, but required for specifiying the input for the other terms in (3.1) and for FeynHiggs). The other parameters of the MSSM not contained in Table 1 are kept variable and are given in the figures. The quantities µ, t β and the Higgs fieldrenormalization constants are defined in the DR scheme at the scale m t (see also Ref. [30] for more details).

MSSM input SM input
The shifts in the charged Higgs-boson mass resulting from the O α 2 t contributions are in general small. In Fig. 3 the dependence of M H ± on the Higgs-sector input parameter m A and on the thirdgeneration soft-breaking squark mass parameter mt ≡ mq 3 = mt L = mt R is depicted, showing a decreasing size of the two-loop mass shift (red) for increasing values of both variables. The upper section of the figure shows the charged Higgs-boson mass as obtained at the one-loop level (dashed), and with the inclusion of the O(α t α s ) contributions (green) and also the O α 2 t terms (blue). The lower section of Fig. 3 shows the mass shift originating solely from the O α 2 t two-loop part. Thereby, the O α 2 t corrections appear as negative, thus diminishing the two-loop contribution of O(α t α s ). In total, the two-loop terms still yield a positive shift upon the one-loop result for M H ± . Fig. 4 contains the charged Higgs-boson mass M H ± , together with the two-loop shift of O α 2 t , for a typical low-m H scenario (left) [43] and for a scenario with heavier H ± (right), versus the Higgsino mass µ. For large values of µ, the charged Higgs-boson mass M H ± decreases, but the mass shift ∆M H ± resulting from the O α 2 t contributions becomes more sizeable, reaching 1 GeV and more for the low M H ± case. In the scenario shown in the right panel of Fig. 4

Conclusions
We have calculated the two-loop O α 2 t contributions to the mass M H ± of the charged Higgs boson when derived from the A-boson mass m A as an on-shell input parameter within the real, CP -conserving, MSSM and combined them with the complete one-loop and the two-loop O α 2 t contributions. We have presented numerical studies for scenarios of current phenomenological interest and discussed the effects of the various two-loop terms.
The O α 2 t two-loop corrections appear with opposite sign and smaller size with respect to the O(α t α s ) contributions; in combination, the two-loop terms yield a positive shift to the mass of the charged Higgs boson as calculated at one-loop order. This shift in M H ± can be at the level of several GeV for small m A and tanβ, and thus of a size that may be relevant for the LHC (and a future electron-positron collider), especially for the interesting region of a low A-boson mass.
The set of two-loop corrections considered here are expected to be particularly relevant in parameter ranges of the real MSSM where the top-Yukawa terms provide a good approximation to the complete one-loop result, i.e. in particular for relatively low values of tanβ and m A . Our results for the charged Higgs-boson mass will become part of the Fortran code FeynHiggs.