Precise predictions for the Higgsboson masses in the NMSSM
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Abstract
The particle discovered in the Higgsboson searches at the LHC with a mass of about \(125 \, \mathrm{GeV}\) can be identified with one of the neutral Higgs bosons of the NexttoMinimal Supersymmetric Standard Model (NMSSM). We calculate predictions for the Higgsboson masses in the NMSSM using the Feynmandiagrammatic approach. The predictions are based on the full NMSSM oneloop corrections supplemented with the dominant and subdominant twoloop corrections within the Minimal Supersymmetric Standard Model (MSSM). These include contributions at \(\mathcal {O}{(\alpha _t \alpha _s, \alpha _b \alpha _s, \alpha _t^2,\alpha _t\alpha _b)}\), as well as a resummation of leading and subleading logarithms from the top/scalar top sector. Taking these corrections into account in the prediction for the mass of the Higgs boson in the NMSSM that is identified with the observed signal is crucial in order to reach a precision at a similar level as in the MSSM. The quality of the approximation made at the twoloop level is analysed on the basis of the full oneloop result, with a particular focus on the prediction for the Standard Modellike Higgs boson that is associated with the observed signal. The obtained results will be used as a basis for the extension of the code FeynHiggs to the NMSSM.
1 Introduction
The spectacular discovery of a boson with a mass around \(125 \, \mathrm{GeV}\) by the ATLAS and CMS experiments [1, 2] at CERN constitutes a milestone in the quest for understanding the physics of electroweak symmetry breaking. Any model describing electroweak physics needs to provide a state that can be identified with the observed signal. While within the present experimental uncertainties the properties of the observed state are compatible with the predictions of the Standard Model (SM) [3, 4], many other interpretations are possible as well, in particular as a Higgs boson of an extended Higgs sector.
One of the prime candidates for physics beyond the SM is supersymmetry (SUSY), which doubles the particle degrees of freedom by predicting two scalar partners for all SM fermions, as well as fermionic partners to all bosons. The most widely studied SUSY framework is the Minimal Supersymmetric Standard Model (MSSM) [5, 6], which keeps the number of new fields and couplings to a minimum. In contrast to the single Higgs doublet of the (minimal) SM, the Higgs sector of the MSSM contains two Higgs doublets, which in the \({\mathcal {CP}}\) conserving case leads to a physical spectrum consisting of two \({\mathcal {CP}}\)even, one \({\mathcal {CP}}\)odd and two charged Higgs bosons. The light \({\mathcal {CP}}\)even MSSM Higgs boson can be interpreted as the signal discovered at about \(125~\mathrm{GeV}\); see e.g. [7, 8].
Going beyond the MSSM, this model has a wellmotivated extension in the NexttoMinimal Supersymmetric Standard Model (NMSSM); see e.g. [9, 10] for reviews. The NMSSM provides in particular a solution for naturally associating an adequate scale to the \(\mu \) parameter appearing in the MSSM superpotential [11, 12]. In the NMSSM, the introduction of a new singlet superfield, which only couples to the Higgs and sfermion sectors, gives rise to an effective \(\mu \)term, generated in a similar way as the Yukawa mass terms of fermions through its vacuum expectation value. In the case where \({\mathcal {CP}}\) is conserved, which we assume throughout the paper, the states in the NMSSM Higgs sector can be classified as three \({\mathcal {CP}}\)even Higgs bosons, \(h_i\) (\(i = 1,2,3\)), two \({\mathcal {CP}}\)odd Higgs bosons, \(A_j\) (\(j = 1,2\)), and the charged Higgsboson pair \(H^\pm \). In addition, the SUSY partner of the singlet Higgs (called the singlino) extends the neutralino sector to a total of five neutralinos. In the NMSSM the lightest but also the second lightest \({\mathcal {CP}}\)even neutral Higgs boson can be interpreted as the signal observed at about \(125~\mathrm{GeV}\); see, e.g., [13, 14].
The measured mass value of the observed signal has already reached the level of a precision observable, with an experimental accuracy of better than \(300~\mathrm{MeV}\) [15], and by itself provides an important test for the predictions of models of electroweak symmetry breaking. In the MSSM the masses of the \({\mathcal {CP}}\)even Higgs bosons can be predicted at lowest order in terms of two SUSY parameters characterising the MSSM Higgs sector, e.g. \(\tan \beta \), the ratio of the vacuum expectation values of the two doublets, and the mass of the \({\mathcal {CP}}\)odd Higgs boson, \(M_A\), or the charged Higgs boson, \(M_{H^\pm }\). These relations, which in particular give rise to an upper bound on the mass of the light \({\mathcal {CP}}\)even Higgs boson given by the Zboson mass, receive large corrections from higherorder contributions. In the NMSSM the corresponding predictions are modified both at the tree level and the loop level. In order to fully exploit the precision of the experimental mass value for constraining the available parameter space of the considered models, the theoretical predictions should have an accuracy that ideally is at the same level of accuracy or even better than the one of the experimental value. The theoretical uncertainty, on the other hand, is composed of two sources, the parametric and the intrinsic uncertainty. The theoretical uncertainties induced by the parametric errors of the input parameters are dominated by the experimental error of the topquark mass (where the latter needs to include the systematic uncertainty from relating the measured mass parameter to a theoretically welldefined quantity; see e.g. [16, 17, 18]). However, the largest theoretical uncertainty at present arises from unknown higherorder corrections, as will be discussed below.
In the MSSM^{1} beyond the oneloop level, the dominant twoloop corrections of \({\mathcal {O}}(\alpha _t\alpha _s)\) [19, 20, 21, 22, 23, 24] and \({\mathcal {O}}(\alpha _t^2)\) [25, 26] as well as the corresponding corrections of \({\mathcal {O}}(\alpha _b\alpha _s)\) [27, 28] and \({\mathcal {O}}(\alpha _t\alpha _b)\) [27] are known since more than a decade. (Here we use \(\alpha _f = Y_f^2/(4\pi )\), with \(Y_f\) denoting the fermion Yukawa coupling.) These corrections, together with a resummation of leading and subleading logarithms from the top/scalar top sector [29] (see also [30, 31] for more details on this type of approach), a resummation of leading contributions from the bottom/scalar bottom sector [27, 28, 32, 33, 34, 35] (see also [36, 37]) and momentumdependent twoloop contributions [38, 39] (see also [40]) are included in the public code FeynHiggs [21, 29, 41, 42, 43, 44, 45]. A (nearly) full twoloop EP calculation, including even the leading threeloop corrections, has also been published [46, 47], which is, however, not publicly available as a computer code. Furthermore, another leading threeloop calculation of \({\mathcal {O}}(\alpha _t\alpha _s^2)\), depending on the various SUSY mass hierarchies, has been performed [48, 49], resulting in the code H3m (which adds the threeloop corrections to the FeynHiggs result up to the twoloop level). The theoretical uncertainty on the lightest \({\mathcal {CP}}\)even Higgsboson mass within the MSSM from unknown higherorder contributions is still at the level of about \(3~\mathrm{GeV}\) for scalar top masses at the \(\mathrm{TeV}\)scale, where the actual uncertainty depends on the considered parameter region [29, 43, 50, 51].
Within the NMSSM beyond the wellknown full oneloop results [52, 53, 54, 55] several codes exist that calculate the Higgs masses in the pure \(\overline{\mathrm {DR}}\) scheme with different contributions at the twoloop level. Amongst these codes SPheno [56, 57] incorporates the most complete results at the twoloop level, including SUSYQCD contributions from the fermion/sfermions of \({\mathcal {O}}(\alpha _t\alpha _s, \alpha _b\alpha _s)\), as well as pure fermion/sfermion contributions of \({\mathcal {O}}(\alpha _t^2, \alpha _b^2, \alpha _t\alpha _b, \alpha _\tau ^2, \alpha _\tau \alpha _b)\), and contributions from the Higgs/higgsino sector in the gaugeless limit of \({\mathcal {O}}(\alpha _\lambda ^2, \alpha _\kappa ^2, \alpha _\lambda \alpha _\kappa )\) [58] as well as mixed contributions from the latter two sectors of \({\mathcal {O}}(\alpha _\lambda \alpha _t, \alpha _\lambda \alpha _b)\). The included Higgs/higgsino contributions are genuine to the NMSSM, they are proportional to the NMSSM parameters \(\lambda ^2 = 4\pi \cdot \alpha _\lambda \) and \(\kappa ^2 = 4\pi \cdot \alpha _\kappa \). The tools FlexibleSUSY [59], NMSSMTools [60, 61] and SOFTSUSY [62, 63, 64] include NMSSM corrections of \({\mathcal {O}}(\alpha _t\alpha _s)\) and \({\mathcal {O}}(\alpha _b\alpha _s)\) supplemented by certain MSSM corrections. NMSSMCalc [54, 55, 65, 66] provides the option to perform the NMSSM Higgsmass calculation up to \({\mathcal {O}}(\alpha _t\alpha _s)\) with the \(\overline{\mathrm {DR}}\) renormalisation scheme applied to the top/stop sector, while in the electroweak sector at oneloop order onshell conditions are used. It has been noticed in a comparison of spectrum generators in the NMSSM that are currently publicly available that the numerical differences between the various codes can be very significant, often exceeding \(3~\mathrm{GeV}\) in the prediction of the SMlike Higgs even for the setup where all predictions were obtained within the \(\overline{\mathrm {DR}}\) renormalisation scheme [67]. While the sources of discrepancies between the different codes could be identified [67], a reliable estimate of the remaining theoretical uncertainties should of course also address issues related to the use of different renormalisation schemes. Beyond the pure \(\overline{\mathrm {DR}}\) scheme, so far only the code NMSSMCalc [54, 55, 65, 66] provides a prediction in a mixed OS/\(\overline{\mathrm {DR}}\) scheme, where genuine twoloop contributions in the NMSSM up to \({\mathcal {O}}(\alpha _t\alpha _s)\) have been incorporated. The resummation of logarithmic contributions beyond the twoloop level is not included so far in any of the public codes for Higgsmass predictions in the NMSSM. Accordingly, at present the theoretical uncertainties from unknown higherorder corrections in the NMSSM are expected to be still larger than for the MSSM.
Concerning the phenomenology of the NMSSM it is of particular interest whether this model can be distinguished from the MSSM by confronting Higgssector measurements with the corresponding predictions of the two models. In order to facilitate the identification of genuine NMSSM contributions in this context it is important to treat the predictions for the MSSM and the NMSSM within a coherent framework where in the MSSM limit of the NMSSM the stateoftheart prediction for the MSSM is recovered.
With this goal in mind, we seek to extend the public tool FeynHiggs to the case of the NMSSM. As a first step in this direction we present in this paper a full oneloop calculation of the Higgsboson masses in the NMSSM, where the renormalisation scheme and all parameters and conventions are chosen such that the wellknown MSSM result of FeynHiggs is obtained for the MSSM limit of the NMSSM. We supplement the full oneloop result in the NMSSM with all higherorder corrections of MSSMtype that are implemented in FeynHiggs, as described above. In our numerical evaluation we use our full oneloop result in the NMSSM to assess the quality of the approximation that we make at the twoloop level. We find that for a SMlike Higgs boson that is compatible with the detected signal at about \(125~\mathrm{GeV}\) this approximation works indeed very well. We analyse in this context which genuine NMSSM contributions are most relevant when going beyond the approximation based on MSSMtype higherorder corrections. We then apply our most accurate prediction including all higherorder contributions to four phenomenologically interesting scenarios. We compare our prediction both with the result in the MSSM limit and with the code NMSSMCalc [65]. We discuss in this context the impact of higherorder contributions beyond the ones of \({\mathcal {O}}(\alpha _t\alpha _s)\), which are not implemented in NMSSMCalc.
The paper is organised as follows. In Sect. 2 we describe our full oneloop calculation in the NMSSM, specify the renormalisation scheme that we have used and discuss the contributions that are expected to be numerically dominant at the oneloop level. The incorporation of higherorder contributions of MSSMtype is addressed in Sect. 3. Our numerical analysis for the prediction at the oneloop level, including a discussion of the quality of the approximation in terms of MSSMtype contributions, and for our most accurate prediction including higherorder corrections is presented in Sect. 4. The conclusions can be found in Sect. 5.
2 Oneloop result in the NMSSM
For the sectors that are identical for the calculation within the MSSM the conventions as implemented in FeynHiggs are used, as described in [44]. Therefore the present section is restricted to the quantities genuine to the NMSSM. For a more detailed discussion of the NMSSM; see e.g. [9].
2.1 The relevant NMSSM sectors
2.2 Renormalisation scheme
Parameters that do not enter the MSSM calculation are considered as genuine of the calculation in the NMSSM. Although the vacuum expectation value v is not a parameter genuine to the NMSSM, its appearance as an independent parameter is a specific feature of the NMSSM Higgsmass calculation; see below.
2.3 Reparametrisation of the electromagnetic coupling
2.4 Dominant contributions at oneloop order
Topologies and their order in terms of the couplings in the top/stop sector that contribute to the selfenergies of the \({\mathcal {CP}}\)even fields \(\phi _i\) at oneloop order in the gaugeless limit. The numbers 1 and 2 denote the doubletstates as external fields, while s denotes an external singlet. The internal lines depict either a top (solid) or a scalar top (dashed)
(i, j)  (12, 12)  (1, s)  (2, s)  (s, s) 

Order  \(\mathcal {O}{\left( Y_t^2\right) }\)  \(\mathcal {O}{\left( \lambda Y_t\right) }\)  \(\mathcal {O}{\left( \lambda Y_t\right) }\)  \(\mathcal {O}{\left( \lambda ^2\right) }\) 
Fields  Top/stop  Stop  Stop  Stop 
Topologies     
  

Since the corrections from the top/stop sector are usually the by far dominant ones, we start with a qualitative discussion of those contributions before we perform a numerical analysis in the following section. In the MSSM the leading corrections from the top/stop sector are commonly denoted as \({\mathcal {O}}(\alpha _t)\), indicating the occurrence of two Yukawa couplings \(Y_t\). In the limit where all other masses of the SM particles and the external momentum are neglected compared to the topquark mass, for dimensional reasons the correction to the squared Higgsboson mass furthermore receives a contribution proportional to \(m_t^2\). This gives rise to the wellknown coefficient \(G_F m_t^4\) of the leading oneloop contributions. In the NMSSM the formally leading contributions either are of \({\mathcal {O}}(Y_t^2)\) (involving two Yukawa couplings), of \({\mathcal {O}}(\lambda Y_t)\) (involving one Yukawa coupling), or of \({\mathcal {O}}(\lambda ^2)\) (involving no Yukawa coupling). The various contributions from the top/stop sector are summarised in Table 1. The contributions in the second column are the ones of MSSMtype, while the entries in the third through fifth column represent the genuine NMSSM corrections, involving only scalar tops.^{3}
3 Incorporation of higherorder contributions
Definition of the sample scenario, S. All dimensionful parameters are given in \(\mathrm{GeV}\). All \(\overline{\text {DR}}\)parameters are defined at \(m_t^{\overline{\text {MS}}}{\left( m_t\right) }\). All stopparameters are onshell parameters. As indicated by the superscript “(GUT)”, \(M_1\) is related to \(M_2\) by the usual GUT relation, \(M_1^{(\text {GUT})} = 5 s_\mathrm{w}^2/(3 c_\mathrm{w}^2) M_2\)
4 Numerical results
A particular goal of our numerical analysis is to test the kind of approximation in terms of MSSMtype contributions that we have used at the twoloop level. For this purpose a genuine NMSSM scenario will be studied, which gives rise to a SMlike Higgs with a predicted mass at the twoloop level of around \(125~\mathrm{GeV}\) and a singletlike Higgs field with a mass that can be above or below the one of the SMlike state. In order to investigate the influence of the extended Higgs and higgsino sector of the NMSSM compared to the MSSM the parameter \(\lambda \) will be varied. In the limit \(\lambda \rightarrow 0\) and constant \(\mu _{\text {eff}}\) all singlet fields decouple from the remaining field spectrum. Increasing the value of \(\lambda \) directly translates to increasing the influence of genuine NMSSMeffects. A detailed study of the oneloop result and the quality of approximations based on partial contributions will be presented here. In order to study the approximation of making the restriction of MSSMlike contributions beyond oneloop order at \(\mathcal {O}{(\alpha _t \alpha _s)}\), we will compare our result with the public tool NMSSMCalc [65], which incorporates the genuine NMSSMtype contributions of \(\mathcal {O}{\left( \alpha _t \alpha _s\right) }\) using a hybrid \(\overline{\mathrm {DR}}\)/onshell renormalisation scheme. While for the MSSM various other higherorder corrections are implemented in FeynHiggs, the corresponding contributions have not been taken into account in NMSSMCalc. We will compare in this context the numerical effect of the NMSSMtype contributions of \(\mathcal {O}{\left( \alpha _t \alpha _s\right) }\) as implemented in NMSSMCalc with the MSSMtype contributions of this order, and we will investigate the numerical impact of the MSSMtype corrections beyond \(\mathcal {O}{\left( \alpha _t \alpha _s\right) }\).
Definition of the analysed scenarios P1, P9 and A1. All dimensionful parameters are given in \(\mathrm{GeV}\). All \(\overline{\text {DR}}\)parameters are considered to be defined at \(m_t^{\overline{\text {MS}}}{\left( m_t\right) }\), and all stopparameters are considered to be onshell parameters. The remaining trilinear breaking parameters are chosen as \(A_f = 1500~\mathrm{GeV}\). The parameter \(\hat{m}_A\) is related to the charged Higgs \(M_{H^\pm }\) mass by Eq. (8)
Higgssector parameters  Sfermion and gaugino parameters  

\(\hat{m}_A\)  \(\tan {\beta }\)  \(\mu _{\text {eff}}\)  \(A_\kappa \)  \(\kappa \)  \(M_{\tilde{q}}\)  \(M_{\tilde{l}}\)  \(A_t\)  \(M_1\)  \(M_2\)  \(M_3\)  
P1  760  10  150  0  0.25  1750  300  \(4000\)  500  1000  3000 
P9  765  14  110  0  0.17  2050  400  \(4000\)  500  1000  3000 
\(M_{H^\pm }\)  \(\tan {\beta }\)  \(\mu _{\text {eff}}\)  \(A_\kappa \)  \(\kappa \)  \(M_{\tilde{q}}\)  \(M_{\tilde{l}}\)  \(A_t\)  \(M_1\)  \(M_2\)  \(M_3\)  

A1  1500  10  150  0  0.25  1750  300  \(4000\)  500  1000  3000 
4.1 Numerical scenarios and treatment of input parameters
In our study we will discuss four different scenarios. The first “sample scenario”, S, for our study is defined by the parameters given in Table 2. It has been chosen to exemplify typical features of NMSSM phenomenology and is well suited for studying the magnitude of the NMSSMcontributions and the behaviour of the employed approximation. The second and third scenario are the benchmark scenarios P1 and P9 defined in [79], where the parameter \(\lambda \) is varied. While the original motivation for these scenarios arising from the diphoton excess that was observed by ATLAS [80] and CMS [81] in the 2015 Run 2 data has not received support from the latest data, we use those scenarios here to serve as examples of possible NMSSM phenomenology in order to test to what extent the features visible for the “sample scenario” S also apply to completely different scenarios. The fourth scenario A1 is based on P1, but permits much larger values of \(\lambda \). The Higgssector parameters of P1, P9 and A1 are given in Table 3. Throughout our analysis the parameter \(\lambda \) is varied if not stated otherwise. We will show in our numerical discussion below that the qualitative features of the scenarios P1, P9 and A1 can be understood from the discussion of the “sample scenario”.
The choice for the topquark mass in the loop contributions will be the pole mass \(m_t^{\text {OS}}\) for the comparison with NMSSMCalc and \(m^{\overline{\text {MS}}}_t(m_t)\) for the remaining studies. Using the \(\overline{\text {MS}}\) topquark mass allows us to include the resummation of leading and nexttoleading logarithms implemented in FeynHiggs. The renormalisation scale for the studies in this chapter will be fixed at the used value of the topquark mass.
4.1.1 Sample scenario S
The sample scenario S for our study is defined by the parameters given in Table 2. For values \(\lambda \gtrsim 0.32\) the mass of the lightest state becomes tachyonic at tree level for this scenario, and therefore the analyses will be performed only for values of \(\lambda \) up to 0.32.
The viability of the discussed scenario is tested with the full set of experimental data implemented in the tool HiggsBounds 4.1.3 [82, 83, 84, 85, 86]. In order to obtain the necessary input for HiggsBounds we made use of NMSSMTools 4.4.0 [9] and linked it with HiggsBounds. While our calculation assumes an onshell renormalised stop sector as in [44], the SLHA input file for NMSSMTools needs \(\overline{\mathrm {DR}}\)parameters for the stop sector. Thus a conversion from the onshell into the \(\overline{\mathrm {DR}}\) scheme is necessary for the parameters of the sample scenario given in Table 2. We only accounted for the dominant effect of these conversions that occurs for \(X_t = A_t  \mu _{\text {eff}}\cot {\beta }\) by applying the onshell to \(\overline{\mathrm {DR}}\) conversion outlined in [87]. We find that the scenario is in agreement with the experimental limits implemented in HiggsBounds 4.1.3.
4.1.2 Scenarios with \(A_\kappa = 0\) and very large \(A_t\)
The scenarios P1 and P9 are defined by the parameters given in Table 3. They are characterised in particular by the choice of \(A_\kappa = 0\) and very large (negative) \(A_t\). While in the original definition of [79] the values \(\lambda =0.1\) and \(\lambda =0.05\) were chosen for the scenarios P1 and P9, respectively, we vary the parameter \(\lambda \) here. We nevertheless refer to the scenarios as P1 and P9 also for other values of \(\lambda \) for simplicity.
In the scenario P1 for all values of \(\lambda \gtrsim 0.43\) the lightest Higgs state becomes tachyonic, for scenario P9 this is the case for \(\lambda \gtrsim 0.35\). The analyses will therefore be restricted to values of \(\lambda \lesssim 0.43\) for the scenario P1 and \(\lambda \lesssim 0.35\) for the scenario P9, respectively. The parameters entering at higher order are chosen as given in Table 3 in the same fashion as above.
4.1.3 Example of a scenario with large values of \(\lambda \)
The scenario A1 is based on P1, but with a substantially larger value of \(M_{H^\pm }\), which prevents tachyonic Higgs masses at the tree level even for large values of \(\lambda \). The parameters are given in the lower part of Table 3. Although we found that this scenario is in disagreement with experimental data from the Tevatron and the LHC Run 1 for \(\lambda \gtrsim 0.75\), it permits the analysis of the MSSMapproximation also for very large values of \(\lambda \).
4.2 Full results at twoloop order
4.2.1 Sample scenario S
The variation of the two masses with \(\lambda \) in the first row of Fig. 1 clearly shows a crossover type behaviour between the masses, which is correlated to their mixing character w.r.t. the singlet field and the doublet fields. For small values of \(\lambda \) the field \(h_1\) is doubletlike in this scenario and, based on the prediction incorporating all available higherorder corrections, can be identified with the signal that was detected at the LHC at about \(125\ \mathrm{GeV}\). The prediction for \(m_{h_1}\) varies only very little with \(\lambda \) in this region. The field \(h_2\), on the other hand, is predominantly singletlike in this parameter region, and its mass prediction falls steeply with increasing \(\lambda \). The crossover occurs at \(\lambda _\mathrm{c}^{(0)} \approx 0.26\) at tree level, at \(\lambda _\mathrm{c}^{(1)} \approx 0.22\) at oneloop order, and at \(\lambda _\mathrm{c}^{(2)} \approx 0.23\) at twoloop order. Above the crossover point the behaviour of the two masses and the admixture of the fields \(h_1\) and \(h_2\) in terms of singlet and doublet fields are reversed. The two fields are evenly mixed between singlet (i.e., genuine NMSSMtype) and doubletfield (i.e., MSSMtype) components for \(\lambda _\mathrm{c}^{(n)}\), with \(n = 0, 1, 2\). The heaviest \({\mathcal {CP}}\)even Higgs field, \(h_3\), is doubletlike in the depicted interval of \(\lambda \). As in the MSSM, the larger masses (of doubletlike fields) are affected by higherorder corrections to a lesser extent than the lighter states. Since at \(\lambda _\mathrm{c}^{(n)}\) the MSSMtype and genuine NMSSMtype contributions enter at equal footing, the SMlike state is most sensitive to genuine NMSSMtype contributions in the region of the crossover behaviour.
4.2.2 Scenario P1
The results for the scenario P1 are shown in the second row of Fig. 1. The lightest field is dominantly doubletlike, and the second lightest state is singletlike for the depicted values of \(\lambda \). The crossover region between the doublet and singletlike state is rather wide in this case and starts at \(\lambda \approx 0.2\). The crossover would occur for values of \(\lambda \) above 0.43, where the lightest field becomes tachyonic at the tree level (this parameter region is therefore not shown here). Thus, even for the largest value of \(\lambda \approx 0.43\) shown in the plot the lightest field is still dominantly doubletlike at all depicted orders. We therefore find that the qualitative behaviour in this scenario is very similar to the sample scenario, but the allowed range of \(\lambda \) is restricted to the region below the crossover point in this case. For small values of \(\lambda \) the lightest field can be identified with the signal that was detected at the LHC at about \(125\ \mathrm{GeV}\). The heaviest \({\mathcal {CP}}\)even Higgs field (not shown in the figure) remains doubletlike with a nearly constant mass of \(\approx \) \(760~\mathrm{GeV}\) for the depicted values of \(\lambda \).
4.2.3 Scenario P9
The results for the scenario P9 are shown in the third row of Fig. 1. Similarly to scenario P1 the variation of the two masses with \(\lambda \) follows the behaviour of the sample scenario. The interval in which the crossover behaviour occurs is larger than in the sample scenario, but smaller than in scenario P1. The crossover occurs at \(\lambda _\mathrm{c}^{(0)} > 0.34\) at tree level, at \(\lambda _\mathrm{c}^{(1)} \approx 0.25\) at oneloop order, and at \(\lambda _\mathrm{c}^{(2)} \approx 0.26\) at twoloop order. It thus lies within the displayed \(\lambda \) range if loop corrections are taken into account. While as before the character of the lightest field \(h_1\) changes from dominantly doubletlike to dominantly singletlike when \(\lambda \) is increased through the crossover region (and vice versa for \(h_2\)), \(h_1\) retains a doublet admixture of more than 40% even for \(\lambda \) values above the crossover region in this scenario. Because of the sizeable admixture in this region, \(m_{h_1}\) and \(m_{h_2}\) each receive significant selfenergy contributions from both the singlet and the doublet fields. The heaviest \({\mathcal {CP}}\)even Higgs field (not shown in the figure) remains doubletlike with a nearly constant mass of \(\approx \) \(750~\mathrm{GeV}\) for the depicted values of \(\lambda \).
4.2.4 Scenario A1
The results for the scenario A1 are shown in Fig. 2. For values \(\lambda \lesssim 0.75\) the variation of the masses with \(\lambda \) follows the behaviour of the sample scenario. In this region the lightest field is doubletlike and the second lightest field is singletlike. The \(\lambda \) values for which a crossover behaviour occurs are at the treelevel \(\lambda _\mathrm{c}^{(0)} \approx 0.75\), at oneloop order \(\lambda _\mathrm{c}^{(1)} \approx 0.70\) and at twoloop order \(\lambda _\mathrm{c}^{(2)} \approx 0.62\). For larger values of \(\lambda \) the lightest field \(h_1\) obtains a singlet admixture of roughly 70%, and the nextto lightest field \(h_2\) obtains a doublet admixture of the same size. A doubletlike Higgs field with a mass close to \(125~\mathrm{GeV}\) can be realised only for values of \(\lambda \) smaller than \(\lambda _\mathrm{c}\) in this scenario. The heaviest \({\mathcal {CP}}\)even Higgs field remains doubletlike with a mass increasing from nearly 1500 to \(1580~\mathrm{GeV}\) for the displayed values of \(\lambda \). In the following we will omit the discussion of the heaviest \({\mathcal {CP}}\)even Higgs field, since it receives only very small twoloop contributions.
4.3 Numerically leading contributions at the oneloop level
For the prediction in the MSSM the top/stopsector contributions are numerically leading. In the studied scenarios, given in Tables 2 and 3, the genuine NMSSMcorrections are suppressed w.r.t. the corresponding MSSMlike stopcorrections since \(\lambda \lesssim \lambda _\mathrm{max} < Y_t\), where \(\lambda _\mathrm{max} = 0.32, 0.43, 0.35\) in the three scenarios, see the discussion in Sect. 2.4. Thus, the genuine NMSSM corrections from this sector are expected to be subleading.
In order to study the impact of the genuine NMSSM contributions we compare the approximation based on the leading MSSMtype oneloop corrections in the gaugeless limit of \(\mathcal {O}{\left( Y_t^2\right) }\), labelled “\(t/\tilde{t}\text {MSSM}\)” in Fig. 3, with the one where the genuine NMSSM corrections of \(\mathcal {O}{\left( \lambda Y_t, \lambda ^2\right) }\) are incorporated.
4.3.1 Sample scenario
The sharp increase of the corrections of \(\mathcal {O}{(\lambda Y_t, \lambda ^2)}\) for the highest values of \(\lambda \) that is visible for the light singletlike field in the upper left plot of Fig. 3 indicates that the approximation for the stop sector of making the restriction of the MSSMtype contributions becomes questionable for the singletlike state in this region. However, as shown in the upper right plot of Fig. 3, in this parameter region the stop sector as a whole ceases to provide a reliable approximation of the full oneloop contributions. In the right plot the difference between the full result and the approximation based on the leading MSSMtype contributions from the top/stop sector, \(\Delta {m_{h_i}} =  m_{h_i}^{(\text {1L})}  m_{h_i}^{t/\tilde{t}\text {MSSM}} \), is shown together with \(\Delta {m_{h_i}} =  m_{h_i}^{(\text {1L})}  m_{h_i}^{t/\tilde{t}\text {MSSM} + \mathrm{HG}}\), where in the latter case the leading MSSMtype contributions from the top/stop sector are supplemented by the contribution from the Higgs–higgsino and gauge/gaugino sectors. While for the singletlike state the deviation between the leading contributions from the top/stop sector and the full oneloop result becomes huge for the largest values of \(\lambda \), reaching the level of \(20~\mathrm{GeV}\), the deviations stay small, far below the level of \(1~\mathrm{GeV}\), if the leading contributions from the top/stop sector are supplemented by the contributions from the Higgs/higgsino and gauge–gaugino sectors. This result for the singletlike state can be understood from the fact that the gauge couplings of the singletlike state are heavily suppressed and that therefore the leading contributions for large \(\lambda \) arise from the Higgs and higgsino sector. Thus, improving on the approximation of MSSMtype contributions in the stop sector requires the incorporation of the contributions from the Higgs and higgsino sector, while the genuine NMSSM contributions in the stop sector are of minor significance in this context.
4.3.2 Scenario P1
The difference between the mass predictions in the two approximations in the top/stop sector is plotted as a function of \(\lambda \) for \(m_{h_1}\) and \(m_{h_2}\) in the left plot of the second row in Fig. 3. As in Fig. 1, the qualitative behaviour is similar to the one in the sample scenario, while the allowed range in \(\lambda \) in P1 is restricted to the region below the crossover point. The impact of the genuine NMSSM corrections is even smaller in this case than in the sample scenario, amounting to less than \(100~\mathrm{MeV}\) for both lighter \({\mathcal {CP}}\)even Higgs fields. In the right plot of the second row the difference between the full result and the approximation based on the leading MSSMtype contributions from the top/stop sector and the leading MSSMtype contributions from the top/stop sector supplemented by the contribution from the Higgshiggsino and gauge/gaugino sectors are shown. By supplementing the partial oneloop results with the Higgs/higgsino/gaugeboson/gaugino contributions the mass prediction for the doubletlike state is improved by \(\approx \) \(1.5\ \mathrm{GeV}\). As explained above for the sample scenario, the difference between the approximate mass prediction and the full oneloop result for the doubletlike state is mainly due to MSSMtype subleading contributions in the top/stop sector. The different variation with \(\lambda \) in the right plot as compared to the sample scenario is related to the much wider crossover region in this case (starting at about \(\lambda = 0.2\)). The large deviation encountered in the sample scenario for the singletlike state above the crossover region is obviously not present in scenario P1, as the latter one is confined to \(\lambda \) values below the crossover region.
4.3.3 Scenario P9
Main calculational differences between NMSSMCalc and our result (labelled NMSSMFeynHiggs) in the setup used for the comparison in Sect. 4.4. The difference at oneloop order is caused only by the different renormalisation of the electric charge, described in Sect. 2.3. At twoloop order the codes in this setup only differ by the genuine NMSSM contributions of \(\mathcal {O}{\left( Y_t\lambda \alpha _s, \lambda ^2 \alpha _s\right) }\). The twoloop MSSM corrections beyond \(\mathcal {O}{\left( \alpha _t \alpha _s\right) }\) and the resummation of logarithms are switched off in NMSSMFeynHiggs for the comparison in Sect. 4.4
NMSSMCalc  NMSSMFeynHiggs  

One loop  \(\alpha _{\text {em}}{\left( M_Z \right) }\) renormalised  \(\leftrightarrow \)  \(\alpha _{\text {em}}{\left( M_Z \right) }\) reparametrised 
Two loop  NMSSM \(\mathcal {O}{\left( \alpha _t \alpha _s\right) }\)  \(\leftrightarrow \)  MSSM \(\mathcal {O}{\left( \alpha _t \alpha _s\right) }\) 
4.3.4 Scenario A1
For the scenario A1 the corresponding analysis is shown in Fig. 4. As in Figs. 1 and 2, the qualitative behaviour for values close to and below the crossover region is similar to the sample scenario. Up to values \(\lambda \approx 0.7\) the genuine NMSSMtype corrections from the top/stop sector are of similar size as for the sample scenario, amounting up to \(\approx \) \(100~\mathrm{MeV}\) for the doubletlike field \(h_1\) field with a mass close to \(125~\mathrm{GeV}\). For the singletlike field the NMSSMtype contributions from the top/stop sector amount up to \(\approx \) \(500~\mathrm{MeV}\) in this region. The NMSSMtype contributions from the top/stop sector increase sharply for the singletlike field \(h_1\) at values above \(\lambda _\mathrm{c}\), amounting up to \(4~\mathrm{GeV}\) for the largest values for \(\lambda \), and become tiny for the doubletlike field \(h_2\), staying well below \(20~\mathrm{MeV}\).
As before we observe also for this scenario with very large values of \(\lambda \) that other contributions beyond the leading MSSMtype contributions from the top/stop sector are numerically much more important than the leading genuine NMSSMtype contributions from the top/stop sector. As can be seen in the right plot of Fig. 4, the difference between the leading MSSMtype contributions from the top/stop sector and the full oneloop result amounts to about \(8~\mathrm{GeV}\) for the doubletlike field \(h_1\) in the region where \(\lambda \lesssim 0.5\). Supplementing the leading MSSMtype contributions from the top/stop sector with the Higgs/higgsino/gaugeboson/gaugino contributions improves the prediction by about 1–2 GeV. As before, the remaining difference in this parameter region is mainly caused by subleading contributions from the top/stop sector. For the singletlike state \(h_2\) the discrepancy between the full oneloop result and the leading MSSMtype contributions from the top/stop sector becomes very significant for increasing \(\lambda \), reaching about \(12~\mathrm{GeV}\) for \(\lambda \approx 0.45\). This large effect is caused by the contributions of the Higgs/higgsino/gaugeboson/gaugino sectors. Incorporating those contributions reduces the discrepancy below the level of \(100~\mathrm{MeV}\). For \(\lambda \gtrsim 0.5\) the discrepancy between the full oneloop result and the leading MSSMtype contributions from the top/stop sector becomes huge for \(h_1\). The same is true for \(h_2\) for very large values of \(\lambda \) above 1. This huge effect is again caused by the contributions of the Higgs/higgsino/gaugeboson/gaugino sectors. Incorporating those contributions reduces the discrepancy to the level of 3–5 GeV. Accordingly, even for this extreme scenario the top/stop sector is well described by just the MSSMtype contributions in those regions of the parameter space where the top/stop sector itself provides an adequate approximation of the full oneloop result. For the highest values of \(\lambda \) in this scenario the contributions beyond the top/stop sector are huge, demonstrating the necessity to use in this case a complete result incorporating also the contributions from the Higgs/higgsino and gaugeboson/gaugino sectors.
4.3.5 Conclusion
As a result of the comparison performed in this section the MSSMtype top/stopsector contributions of \({\mathcal {O}}(Y_t^2)\) have been verified as the leading oneloop contributions to MSSMlike fields. The genuine NMSSM top/stopsector contributions of \({\mathcal {O}}(\lambda Y_t, \lambda ^2)\) have the largest impact on singletlike fields for large values of \(\lambda \), where, however, an approximation based only on contributions from the fermion/sfermion sector is in any case insufficient. Our analysis at the oneloop level therefore shows that approximating the result for the top/stop sector by the leading MSSMtype contributions turns out to work well in the parameter regions where the top/stop sector itself yields a reasonable approximation of the full result. These findings provide a strong motivation for applying the same kind of approximation also at the twoloop level. For the description of singletlike fields in the region of large values of \(\lambda \) we have demonstrated the importance of incorporating also the contributions from the Higgs/higgsino and gaugeboson/gaugino sectors.
4.4 Comparison with NMSSMCalc
Mass of the lightest \({\mathcal {CP}}\)even Higgs fields obtained in the MSSM limit with NMSSMCalc and NMSSMFeynHiggs with the reparametrisation to \(\alpha {(M_Z)}\). Both codes yield the identical results in this limit
Sample scenario  Scenario P1  Scenario P9  Scenario A1  

\(\frac{m_{h_1}^\mathrm{}}{\mathrm{GeV}}\) MSSM limit  
Two loop  116.902  109.579  115.155  109.685 
One loop  140.742  115.154  152.526  151.293 
In a first step the one and twoloop results of NMSSMCalc and NMSSMFeynHiggs have been compared in the MSSM limit, where \(\lambda \) and \(\kappa \) vanish simultaneously. Both the effects of the different renormalisation schemes and the reparametrisation have to vanish in this limit and thus the results have to be identical. The one and twoloop results for the mass of the lightest \({\mathcal {CP}}\)even field obtained in this limit with both codes, given in Table 5, are in agreement with each other with a precision of better than \(1~\mathrm{MeV}\) for each scenario (the same holds in this limit also for the predictions for the other neutral Higgs bosons).
4.4.1 Sample scenario
4.4.2 Scenarios P1, P9 and A1
The comparisons between NMSSMFeynHiggs and NMSSMCalc for the scenarios P1 and P9 are shown in Fig. 6 in the first and second row, while the scenario A1 is shown in the lower row. For better illustration we plot here the size of the twoloop contributions, \(\Delta {m_{h_i}} = m_{h_i}^\mathrm{1L}  m_{h_i}^\mathrm{2L}\), as obtained with the two codes as a function of \(\lambda \). As for the sample scenario, the main effect in the comparison arises from a slight relative shift in \(\lambda \) between the predictions of the two codes. At the oneloop level this shift amounts typically to \(\Delta {\lambda _\mathrm{c}} \approx 10^{4}\) (the corresponding plots are not shown here since the curves for the predictions of the two codes would be essentially indistinguishable). For the twoloop contributions displayed in Fig. 6 one can see that the genuine NMSSMtype twoloop corrections that are implemented in NMSSMCalc give rise to a slightly different dependence on \(\lambda \), which becomes visible for large values of \(\lambda \).
As discussed above, in the P1 scenario the displayed range of \(\lambda \) corresponds to the region below the crossover point. For the sample scenario we found in this region a slight increase in the absolute difference between the results; see Fig. 5. The difference between the twoloop contributions shown in Fig. 6 is seen to follow a similar pattern. For \(h_1\) (upper left plot) the difference between the two contributions exceeds the level of \(0.5~\mathrm{GeV}\) for the highest values of \(\lambda \) that are possible in this scenario because of the steep slope of the curves (which are slightly shifted in \(\lambda \) with respect to each other) in this region. The dominantly doubletlike state \(h_1\) has a significant singlet admixture in this region, which increases up to more than 30% for the highest \(\lambda \) values. It should be noted that such a large singlet admixture severely worsens the compatibility of the state \(h_1\) with the observed Higgs signal at about \(125~\mathrm{GeV}\) (independently of its mass, which is incompatible with the signal in this part of the plot; see Fig. 1). The differences are smaller for the (dominantly singletlike) state \(h_2\) (upper right plot) and reach a significant level only for \(\lambda \) values that are close to the boundary of the allowed range.
For the scenario P9, where above the crossover region a relatively large admixture of more than 40% between the doubletlike and the singletlike state occurs, the differences stay relatively small over the whole displayed range of \(\lambda \) both for \(h_1\) (middle left plot) and \(h_2\) (middle right plot). The largest deviations occur for the dominantly singletlike state \(h_1\) (with a sizeable doublet admixture) for the highest values of \(\lambda \) above the crossover region, where the twoloop contributions differ from each other by up to \(0.8~\mathrm{GeV}\).
For the scenario A1, where above the crossover region a relatively large admixture of more than 30% between the doubletlike and the singletlike state occurs, the differences nevertheless stay small over the whole displayed range of \(\lambda \) both for \(h_1\) (lower left plot) and \(h_2\) (lower right plot). Even for the largest values of \(\lambda \) the difference between the two contributions remains below \(0.26~\mathrm{GeV}\). Our analysis shows that even for this extreme scenario with very high values of \(\lambda \) the genuine NMSSMtype twoloop corrections that are only implemented in NMSSMCalc are of minor numerical significance. From our analysis at the oneloop level, on the other hand, it is expected that the twoloop contributions beyond the fermion/sfermion sector are very important in this parameter region, so that the theoretical uncertainties of both codes are expected to be rather large in this region.
4.4.3 Conclusion
4.5 Impact of additional corrections beyond \(\mathcal {O}{\left( \alpha _t \alpha _s\right) }\)
While the genuine NMSSM twoloop corrections of \(\mathcal {O}(Y_t \lambda \alpha _s, \lambda ^2\alpha _s )\) induce small effects, as discussed in the previous section, the MSSM twoloop corrections beyond \(\mathcal {O}{(\alpha _t \alpha _s)}\) and the resummation of large logarithms can result in a shift for the mass of the light doubletlike field of several \(\mathrm{GeV}\). In order to quantify the impact of the additional MSSMcontributions of \(\mathcal {O}{(\alpha _t^2, \alpha _b \alpha _s, \alpha _t \alpha _b)}\) and the resummation of logarithms, which are incorporated in NMSSMFeynHiggs, the results with and without these corrections are plotted as functions of \(\lambda \) in Figs. 7 and 8 for the discussed scenarios. Here the oneloop \(\overline{\mathrm {MS}}\)value of the topquark, \(m_t^{\overline{\mathrm {MS}}}{(m_t)}\), is used in the loop contributions. A sizeable shift of about 3–8 \(\mathrm{GeV}\) can be observed for the mass of the doubletlike field. As expected, the impact of the MSSMtype twoloop contributions on the mass prediction for the singletlike field remains small. In comparison with the contributions discussed in the previous section we find that the effect of the additional corrections beyond \(\mathcal {O}{(\alpha _t\alpha _s)}\) can exceed the numerical impact of the genuine NMSSMcorrections of \(\mathcal {O}{(Y_t \lambda \alpha _s, \lambda ^2 \alpha _s)}\) by more than one order of magnitude.
5 Conclusions
We have presented predictions for the Higgsboson masses in the NMSSM obtained within the Feynmandiagrammatic approach. They are based on the full NMSSM oneloop corrections supplemented with the dominant and subdominant twoloop corrections of MSSMtype, including contributions at \(\mathcal {O}{\left( \alpha _t \alpha _s, \alpha _b \alpha _s, \alpha _t^2,\alpha _t\alpha _b\right) }\), as well as a resummation of leading and subleading logarithms from the top/scalar top sector. In order to enable a direct comparison with the corresponding results in the MSSM, the renormalisation scheme and all parameters and conventions have been chosen such that the wellknown MSSM result of the code FeynHiggs is recovered in the MSSM limit of the NMSSM.
In our phenomenological analysis we have first investigated a scenario where depending on the value of \(\lambda \) either the lightest or the nexttolightest neutral Higgs state can be identified with a SMlike Higgs boson at about \(125~\mathrm{GeV}\). Furthermore we have investigated two scenarios (originally proposed in a different context; see the discussion in Sect. 4.1) where larger values of \(\lambda \) than in the sample scenario can be realised, and sizeable admixtures between singlet and doubletlike states can occur also outside of the “crossover” region. The lightest neutral Higgsstate can be identified with a SMlike Higgs boson at about \(125~\mathrm{GeV}\) in both scenarios for low and moderate values of \(\lambda \). As expected, the state that can be identified with the observed Higgs signal at about \(125~\mathrm{GeV}\) is doubletlike in all cases, i.e. it receives only relatively small contributions from the singlet state of the NMSSM. In order to investigate the impact of the various contributions for even higher values of \(\lambda \), we have furthermore analysed another variation of these scenarios in which values of \(\lambda \) up to \(\lesssim 1.5\) can be realised. The inclusion of the higherorder contributions which are known for the MSSM is crucial for all scenarios in order to obtain an accurate prediction for the mass spectrum.
We have investigated different approximations at the oneloop level in comparison with our full oneloop result for the NMSSM. We have found that the approximation of the result for the top/stop sector in terms of the leading MSSMtype contributions works well in the parameter regions where the top/stop sector itself yields a reasonable approximation of the full result. It therefore appears to be well motivated to make use of this approximation at the twoloop level. The genuine NMSSM top/stopsector contributions of \({\mathcal {O}}(Y_t\lambda , \lambda ^2)\) can be significant for singletlike fields if \(\lambda \) is large. For such large values of \(\lambda \), however, the improvement achieved by including those genuine NMSSM contributions from the top/stop sector is by far overshadowed by the fact that contributions from the Higgs and higgsino sector become more and more important for a singletlike Higgs field.
We have compared our predictions with the public code NMSSMCalc for onshell parameters in the top/stop sector. For the purpose of this comparison we have done an appropriate reparametrisation of the electromagnetic coupling constant, and we have switched off the twoloop corrections beyond the ones of \(\mathcal {O}{(\alpha _t \alpha _s)}\) as well as the resummation of leading and subleading logarithms in our code. After those adaptations the predictions of the two codes only differ in the charge renormalisation at the oneloop level and in the genuine NMSSM top/stopsector contributions of \({\mathcal {O}}(Y_t\lambda \alpha _s, \lambda ^2\alpha _s)\) at the twoloop level. Since these differences arise only from contributions beyond the MSSM, agreement between the predictions of the two codes is expected in the MSSM limit of the NMSSM. We have indeed found that the results obtained with the two codes perfectly agree with each other in this case. For the case of the NMSSM we have compared the predictions of the two codes as a function of \(\lambda \). We have found that the differences stay small over the whole range of \(\lambda \), with a maximum absolute difference in the mass of the singlet or the doubletlike state below \(1~\mathrm{GeV}\) in the considered scenarios. The difference is mainly caused by the different treatment of the charge renormalisation at the oneloop level, while the effect of the genuine NMSSM top/stopsector contributions of \({\mathcal {O}}(Y_t\lambda \alpha _s, \lambda ^2\alpha _s)\) is found to be generally smaller except for the highest values of \(\lambda \) that can be realised in the scenarios. The impact of the genuine NMSSM top/stopsector contributions of \({\mathcal {O}}(Y_t\lambda \alpha _s, \lambda ^2\alpha _s)\) turned out to be small even in parameter regions where the dominantly doubletlike state has a singlet admixture of more than 30%. A more detailed comparison between the two codes will be presented in a forthcoming publication.
As a final step of our numerical analysis we have investigated the impact of the MSSMcorrections beyond \(\mathcal {O}{(\alpha _t \alpha _s)}\) and the resummation of large logarithms that are incorporated in our code but not in NMSSMCalc. While those corrections are small for the mass of a dominantly singletlike state, they amount to an effect of 3–8 GeV for the mass of the doubletlike state in the considered scenarios. This is typically more than an order of magnitude larger than the corresponding effect of the genuine NMSSMcorrections of \(\mathcal {O}{(Y_t \lambda \alpha _s, \lambda ^2 \alpha _s)}\).
The results presented in this paper will be used as a basis for the extension of the code FeynHiggs to the NMSSM. Our analysis has revealed that for singletlike states in the parameter region of very high values of \(\lambda \) twoloop corrections beyond the fermion/sfermion sector are expected to be sizeable. In order to reduce the theoretical uncertainties in this parameter region the incorporation of twoloop contributions from the Higgs/higgsino and gaugeboson/gaugino sectors will be desirable. Partial results of this kind have only been obtained in a pure \(\overline{\mathrm {DR}}\) scheme up to now. We leave a more detailed discussion of this issue to future work.
Footnotes
 1.
As mentioned above, we focus in this paper on the case of real parameters, i.e. the \({\mathcal {CP}}\)conserving case.
 2.
 3.
We discuss here only the Higgsboson selfenergies. However, the same line of argument can be made for the tadpole contributions.
 4.
 5.
Details on the calculation of the renormalised selfenergy contributions will be presented in a future publication.
 6.
 7.
More recent updates of FeynHiggs contain additional contributions that, however, do not significantly modify the results of our present investigation.
 8.
We thank Kathrin Walz for providing a modified version of NMSSMCalc for this feature.
 9.
Notes
Acknowledgements
We want to thank Ramona Gröber, Margarete Mühlleitner, Heidi Rzehak, Oscar Stål and Kathrin Waltz for interesting discussions, useful input and interfaces with their codes. We thank Melina Gomez Bock, Rachid Benbrik and Pietro Slavich for helpful communication. The work of S.H. is supported in part by CICYT (grant FPA 201340715P) and by the Spanish MICINN’s ConsoliderIngenio 2010 Program under grant MultiDark CSD200900064. The authors acknowledge support by the DFG through the SFB 676 “Particles, Strings and the Early Universe”. This research was supported in part by the European Commission through the “HiggsTools” Initial Training Network PITNGA2012316704.
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