Abstract
In the context of electroweak precision observables the W boson mass, being highly sensitive to loop corrections of new physics, plays a crucial role. The accuracy of the measurement of the W boson mass has been significantly improved over the last years (particularly by the Tevatron results) and further improvement of the experimental accuracy is expected from future LHC and ILC measurements. In order to fully exploit the precise experimental determination, an accurate theoretical prediction for the W boson mass in models beyond the SM is of central importance. In this chapter we present the currently most precise prediction of the W boson mass in the MSSM with complex parameters and in the NMSSM, including the full one-loop result and the relevant available higher order corrections of SM and SUSY type. The evaluation of the W boson mass is performed in a very flexible framework, which facilitates the extension to other models beyond the SM. The size of the contribution of the various SUSY sectors in both models is studied in detail. Performing a detailed parameter scan in the MSSM, we investigate the impact of limits from direct SUSY searches as well as from the Higgs discovery on the W boson mass prediction in the MSSM. Assuming hypothetical future scenarios, we discuss the impact of the W boson mass prediction on the MSSM parameter space. A significant part of this chapter concerns the discussion of genuine NMSSM contributions to the W boson mass.
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- 1.
Note that in our convention A is used to denote both the photon and the \(\mathcal{CP}\)-odd Higgs in the MSSM. In the context of Higgs decays we denote the photon \(\gamma \).
- 2.
- 3.
In the procedure that was applied in Ref. [50], \(\Delta r_{\text {ferm}}^{(\alpha ^2)} + \Delta r_{\text {bos}}^{(\alpha ^2)}\) could only be evaluated at \(M_W^\mathrm{SM}\). As mentioned already before, we will include the SM higher order corrections also in the (N)MSSM \(M_W\) prediction. Using the fit formula for \(\Delta r_{\text {ferm}}^{(\alpha ^2)} + \Delta r_{\text {bos}}^{(\alpha ^2)}\) allows us to evaluate these contributions (in each iteration step) at the particular (N)MSSM value for \(M_W\).
- 4.
Since the complete one-loop results for \(\Delta r\) in the SM and in the NMSSM are used in Eq. (5.25), this splitting has an impact only from the two-loop level onwards.
- 5.
One could also include the term \(\Delta \alpha \Delta r_{\text {rem}}\), which is however numerically small and not included.
- 6.
We plan to provide a Fortran code also for the NMSSM.
- 7.
The code FeynHiggs includes (amongst others) predictions for the anomalous magnetic moment of the muon and electric dipole moments of electron, neutron and mercury. Additionally it provides the information whether a parameter point corresponds to a colour-breaking minimum. NMSSMTools contains a list of theoretical and experimental constraints, e.g. constraints from collider experiments (such as LEP mass limits on SUSY particles), B-physics and astrophysics. More details on the constraints included in NMSSMTools can be found in Refs. [94, 99].
- 8.
The Mathematica code is linked to a Fortran driver program, calling the other programs (FeynHiggs for the MSSM, NMSSMTools for the NMSSM and HiggsBounds for both models). In the NMSSM case the calculation of the SUSY particle masses and the tree-level Higgs masses is also included in the Fortran driver.
- 9.
Since this limit is not applied everywhere we will comment on whether a LHC mass limit on squarks and gluinos is considered, when discussing a specific analysis.
- 10.
R-parity violation is not discussed in this thesis.
- 11.
The lower limit of 4 MeV corresponds to the SM uncertainty, which one gets in the decoupling limit of the MSSM. For the upper limit of 9 MeV very light SUSY particles were considered, which are not in agreement with the current limits anymore. Taking the experimental bounds into account the (maximal) uncertainty from missing higher orders should be considerably reduced.
- 12.
For the parameters given in Eq. (5) in Ref. [37] and \(M_{H^{\text {SM}}}= 100 \,\mathrm {GeV}\).
- 13.
- 14.
It should be noted that a similar kind of feature would occur even if one restricted the predicted value for \(M_h\) in the MSSM to the same region as the range adopted for \(M_{H^{\text {SM}}}\). This is caused by the fact that the additional theoretical uncertainties from unknown higher-order corrections affecting the prediction for \(M_h\) in the MSSM, which are not present in the SM where \(M_{H^{\text {SM}}}\) is a free input parameter, essentially lead to a broadening of the allowed range of \(M_h\) in the MSSM as compared to \(M_{H^{\text {SM}}}\).
- 15.
In all plots in Fig. 5.12 one can see a small gap between the MSSM points for \(m_{\tilde{t}_1}> 1900 \,\mathrm {GeV}\) and the SM line. This is an artefact of the chosen scan ranges: in this region the mass-splitting between \(\tilde{t}_1\) and \(\tilde{t}_2\) is small, and \(m_h\) does not reach values up to \({\sim } 126\,\mathrm {GeV}\). The \(M_W\) value approached in the decoupling limit therefore corresponds to the SM prediction for a lower Higgs mass value.
- 16.
If the Higgs sector contains an additional singlet, as in the NMSSM, it is possible to have a SM-like second-lightest Higgs, while the charged Higgs boson can be much heavier in this case, see e.g. Ref. [119].
- 17.
See also Ref. [127] for a recent analysis investigating constraints on the scalar top sector.
- 18.
The exact parameters we use are \(m_t=173.2\), \(\tan \beta =15\), \(\mu = 350 \,\mathrm {GeV}\), \(M_{\tilde{L}/\tilde{E}_{1,2}}=500 \,\mathrm {GeV}\), \(M_{\tilde{L}/\tilde{E}_{3}}=1000 \,\mathrm {GeV}\), \(M_{\tilde{Q}/\tilde{U}/\tilde{D}_{1,2}}=1500 \,\mathrm {GeV}\), \(M_{\text {SUSY}}=M_{\tilde{Q}_{3}}=M_{\tilde{U}_{3}}=M_{\tilde{D}_{3}}=500/300 \,\mathrm {GeV}\) (see description in text), \(A_t=|A_t|\,\exp (i\phi _{A_t})\) with \(|A_t|=2\,M_{\text {SUSY}}+\mu /\tan \beta \), \(A_b=A_t\), \(A_{\tau }=0\), \(M_2=350 \,\mathrm {GeV}\), \(M_3=1500 \,\mathrm {GeV}\) and \(M_A=800 \,\mathrm {GeV}\). All parameters apart from \(A_{t/b}\) are chosen real.
- 19.
The \(X_t\) parameter that we plot here is the on-shell parameter. As described in Sect. 5.4.2 the on-shell value is transformed into a \(\overline{\mathrm {DR}}\) value, which is used as input for NMSSMTools to calculate the Higgs masses. All numerical values given for \(X_t\) in this section refer to the on-shell parameters.
- 20.
From here on we will leave out the subscript ‘eff’ for the \(\mu \) parameter in the NMSSM.
- 21.
The splitting between the sbottoms is determined by \(X_b = A_b - \mu \tan \beta \). For the chosen \(\tan \beta \) and \(\mu \) values it is smallest for \(X_t = 2100 \,\mathrm {GeV}\) (\(\implies X_b = -1890 \,\mathrm {GeV}\)) and is largest for \(X_t = -2100 \,\mathrm {GeV}\) (\(\implies X_b = -6090 \,\mathrm {GeV}\)).
- 22.
The terms in the second line decrease first in the range \(|X_t| = 0 - 300 \,\mathrm {GeV}\), however the sum of the terms in the second and third line is getting larger for all \(|X_t|\) values.
- 23.
For one specific \(\tan \beta \) value around 4, the contribution from the additional tree-level terms seems to cancel the one from doublet-singlet mixing, for all values of \(\lambda \). Analytic confirmation of this cancellation is in progress.
- 24.
The difference in the predictions for the lightest \(\mathcal{CP}\)-even Higgs masses in the MSSM and the NMSSM, which we subtract this way, includes both the difference between the different mass relations in the MSSM and the NMSSM, as well as the “technical” difference between the FeynHiggs and the NMSSMTools evaluation.
- 25.
Neglecting experimental bounds one can have light \(\mathcal{CP}\)-Higgs bosons with a small singlet component, which would give large contributions to \(M_W\). However this possibility will not be discussed here.
- 26.
In this scenario the lightest neutralino has a small singlino component.
- 27.
This framework was developed first for the analysis presented in the next chapter, which has been published in Ref. [87]. In this thesis we decided to describe this framework in the context of the NMSSM \(M_W\) analysis, which we present first.
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Zeune, L. (2016). The W Boson Mass in the SM, the MSSM and the NMSSM. In: Constraining Supersymmetric Models . Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-22228-8_5
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