Onshell neutral Higgs bosons in the NMSSM with complex parameters
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Abstract
The NexttoMinimal Supersymmetric Standard model (NMSSM) appears as an interesting candidate for the interpretation of the Higgs measurement at the LHC and as a rich framework embedding physics beyond the Standard Model. We consider the renormalization of the Higgs sector of this model in its \(\mathcal{CP}\)violating version, and propose a renormalization scheme for the calculation of onshell Higgs masses. Moreover, the connection between the physical states and the treelevel ones is no longer trivial at the radiative level: a proper description of the corresponding transition thus proves necessary in order to calculate Higgs production and decays at a consistent loop order. After discussing these formal aspects, we compare the results of our mass calculation to the output of existing tools. We also study the relevance of the onshell transition matrix in the example of the \(h_i\rightarrow \tau ^+\tau ^\) width. We find deviations between our full prescription and popular approximations that can exceed 10%.
1 Introduction
Since the discovery of a Higgslike particle with a mass around 125GeV by the ATLAS and CMS experiments [1, 2] at CERN, a lot of efforts have been made to reveal its nature as the particle responsible for electroweak symmetry breaking. While within the present experimental uncertainties the properties of the observed state are compatible with the predictions of the Standard Model (SM) [3] 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 softly broken supersymmetry (SUSY), which doubles the particle degrees of freedom by predicting two scalar partners for each SM fermion, as well as fermionic partners for all bosons—for reviews see [4, 5]. The NexttoMinimal Supersymmetric Standard Model (NMSSM) [6, 7] is a wellmotivated extension of the SM. In particular it provides a solution for the “\(\mu \) problem” [8] of the Minimal Supersymmetric Standard Model (MSSM), by naturally relating the \(\mu \) parameter to a dynamical scale of the Higgs potential [9, 10].
In contrast to the single Higgs doublet in the SM, the Higgs sector of the NMSSM contains two Higgs doublets (like the MSSM) and one Higgs singlet. After electroweak symmetry breaking the physical spectrum consists of five neutral Higgs bosons, \(h_i\) (\(i \in [1,5]\)), and the charged Higgs boson pair \(H^\pm \). Ever since the Higgs discovery, the possibility to interpret this signal in terms of an NMSSM (mostly) \(\mathcal{CP}\)even Higgs boson has been emphasized in Refs. [11, 12, 13, 14, 15, 16, 17, 18, 19, 20]. In particular, it has been argued that such a solution came with improved naturalness compared to the MSSM interpretation [21, 22, 23, 24, 25, 26, 27, 28, 29]. Moreover, several works have pointed out the possibility to accommodate deviations from a strict SM behavior in the diphoton rate, in Higgspair production or in associated production [30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47]. Admittedly, the viability of the extended NMSSM Higgs sector would be comforted by the detection of additional Higgs states. To this end, several search channels have been suggested, especially for states lighter than 125 GeV [48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65]. Another feature of the NMSSM phenomenology is the extended neutralino sector, due to the singlino.
In contrast to the situation in the MSSM, \(\mathcal{CP}\)violation can already occur at the tree level in the NMSSM Higgs sector [10, 17, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82]. While lowenergy observables place limits on such \(\mathcal{CP}\)violating scenarios [83], especially on MSSMlike phases [84], \(\mathcal{CP}\)violation beyond the SM appears as a wellmotivated requirement for a successful baryogenesis [85]. Correspondingly, several computer tools have been proposed in the past few years to promote the study of the \(\mathcal{CP}\)violating NMSSM: SPHENO [86, 87, 88, 89] and FlexibleSUSY [90, 91]—which employ SARAH [92, 93, 94, 95] in order to produce their modelfiles; FlexibleSUSY contains components from SoftSUSY [96, 97] and only the \(\mathcal{CP}\)conserving case is explicitly mentioned for both—as well as NMSSMCALC [98, 99] and NMSSMTools [100, 101, 102, 103].
In this work, we specialize in the \(Z_3\)conserving version of the NMSSM, characterized by a scaleinvariant superpotential. The main effort of our project consists in analyzing radiative corrections in the Higgs sector of the \(\mathcal{CP}\)violating NMSSM. To serve this purpose, we elaborated a FeynArts [104, 105] model file and a set of Mathematica routines for the evaluation of the Higgs masses and wavefunction normalization matrix at full oneloop order and beyond. These should serve as a basis for a future inclusion of the \(\mathcal{CP}\)violating NMSSM in the FeynHiggs [106, 107, 108, 109, 110, 111, 112] package—originally designed for precise calculations of the masses, decays, and other properties of the Higgs bosons in the \(\mathcal{CP}\)conserving or violating MSSM. A first step in this direction is represented by Ref. [113], centering on the \(\mathcal{CP}\)conserving NMSSM. In the current paper, we expand this project further. We follow the general methodology of FeynHiggs, relying on a Feynmandiagrammatic calculation of radiative corrections, which employs FeynArts [104, 105], FormCalc [114] and LoopTools [114]. Our chosen renormalization scheme differs somewhat from earlier proposals [88, 113, 115]. In particular, in our renormalization scheme, the electromagnetic coupling e—which is related to the finestructure constant \(\alpha = e^2/(4\pi )\)—is defined in terms of the Fermi constant \(G_F\) measured in muon decays.
In Sect. 2, we shall introduce relevant notations and describe the renormalization procedure underpinning our model file for the \(\mathcal{CP}\)violating NMSSM. In this section we also describe our implementation of higherorder corrections in the Higgs sector. A numerical evaluation of our results follows in Sect. 3, where we will validate our calculation by a comparison with public codes. We will also insist on the relevance of the fieldrenormalization matrix for a consistent evaluation of the Higgs decays at the oneloop level, before a short conclusion in Sect. 4.
2 Higgs masses and mixing in the \(\mathcal{CP}\)violating NMSSM
After a few general remarks concerning our notations and conventions, we present the renormalization conditions that we employ in our calculation. There, we focus on effects beyond the MSSM in the Higgs and higgsino sectors, since we otherwise align with the conventions of FeynHiggs, described in [109]. Then we discuss how to formally extract the loopcorrected Higgs masses and the wavefunction normalization factors.
2.1 Conventions and relations at the tree level
where the masses of the W and Z bosons are denoted by \(M_W\) and \(M_Z\), respectively, and the phases combine to \(\zeta _1 = \xi  2\,\xi _s + \phi _\lambda  \phi _\kappa \), \(\zeta _2 = \xi + \xi _s + \phi _{A_\lambda } + \phi _\lambda \) and \(\zeta _3 = 3\,\xi _s + \phi _{A_\kappa } + \phi _\kappa \). The expressions of Eq. (2.10) make plain that the tadpole coefficients can substitute the five parameters \(m_1^2\), \(m_2^2\), \(m_S^2\), \(\phi _{A_\lambda }\) and \(\phi _{A_\kappa }\), so that the latter will not be regarded as free parameters in the following. Finally, the tadpole coefficients in the (treelevel) mass basis, \(\mathbf {T}_{h} = (T_{h_1},T_{h_2},T_{h_3},T_{h_4},T_{h_5},0)\), where the zero denotes the vanishing tadpole coefficient of the Goldstone mode, are obtained by \(\mathbf {T}_h = \mathbf {U}_{n} \mathbf {T}\). The minimization of \(V_{\text {H}}\) at the chosen Higgs VEVs is guaranteed through the condition that all tadpole coefficients \(\mathbf {T}\) vanish at the tree level.
2.2 Renormalization of the Higgs potential
In the past, radiative corrections to the Higgs masses of the \(\mathcal{CP}\)conserving NMSSM have been considered in the effective potential approach, see e.g. Refs. [116, 117, 118, 119, 120, 121, 122, 123, 124]. This topic has also been analyzed from the perspective of a diagrammatic expansion, including radiative corrections from part or the full set of the particle content of the NMSSM: see Refs. [113, 125, 126]. Both procedures have also been employed for the \(\mathcal{CP}\)violating case: contributions to the effective potential have been discussed in Refs. [17, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 102], while contributions using the diagrammatic approach have been presented in Refs. [88, 115].
2.3 Renormalization conditions at the oneloop order
For the parameters \(M_{H^\pm }\), \(M_W\), \(M_Z\) and \(\tan {\beta }\), which enter the oneloop calculation of the Higgs masses in the MSSM as well, we follow the renormalization prescription outlined in Ref. [109]: the onshell renormalization scheme is employed for the gauge boson masses, \(M_Z\) and \(M_W\), and the charged Higgs mass \(M_{H^\pm }\), while the parameter \(\tan {\beta }\) is renormalized \(\overline{\mathrm {DR}}\).
The remaining independent parameters and the fieldrenormalization constants are renormalized \(\overline{\mathrm {DR}}\). We present a detailed description of the \(\overline{\mathrm {DR}}\) renormalization conditions that we apply. The actual cancellation of UVdivergences, that we recover at the diagrammatic level, represents a nontrivial check for the validity of the FeynArts model file employed for our calculation.
 For \(\delta {\left \lambda \right }\) and \(\delta {\phi _{\lambda }}\) we impose the \(\overline{\mathrm {DR}}\) renormalization condition of Eq. (2.20) on the vertices \({\Gamma _{\tilde{S}\tilde{H}^\phi _1^+}^{(0)} = \lambda }\) and \({\Gamma _{\tilde{S}\tilde{H}^+\phi _1^}^{(0)} = \lambda ^*}\), which yields$$\begin{aligned} \frac{\delta \left \lambda \right }{\left \lambda \right }&= \frac{1}{2}\left\{ \left. \frac{\Gamma _{\tilde{S}\tilde{H}^\phi _1^+}^{(1)} }{ \Gamma _{\tilde{S}\tilde{H}^\phi _1^+}^{(0)}}\right _{\text {div}} + \frac{1}{2}\left( \delta Z_{\tilde{S}} + \delta Z_{\tilde{H}^\pm }^{\text {R}} + \delta Z_{\mathcal {H}_1} \right) \right\} \nonumber \\&\quad + \text {c.c.}\,, \end{aligned}$$(2.21a)$$\begin{aligned} \delta \phi _\lambda&= \frac{1}{2{i}}\left\{ \left. \frac{\Gamma _{\tilde{S}\tilde{H}^\phi _1^+}^{(1)} }{ \Gamma _{\tilde{S}\tilde{H}^\phi _1^+}^{(0)}}\right _{\text {div}} + \frac{1}{2}\left( \delta Z_{\tilde{S}} + \delta Z_{\tilde{H}^\pm }^{\text {R}} + \delta Z_{\mathcal {H}_1} \right) \right\} \nonumber \\&\quad + \text {c.c.} \end{aligned}$$(2.21b)
 For \(\delta \xi \) we impose the renormalization condition of Eq. (2.20) on the vertices \({\Gamma _{\tilde{S}\tilde{H}^+\phi _2^}^{(0)} = \lambda \,\mathrm {e}^{{i}\,\xi }}\) and \({\Gamma _{\tilde{S}\tilde{H}^\phi _2^+}^{(0)} = \lambda ^*\,\mathrm {e}^{{i}\,\xi }}\). The counterterm reads(2.22)
 We fix the renormalization constant \(\delta \xi _s\) for the phase \(\xi _s\) by applying Eq. (2.20) on the vertices \(\Gamma _{\tilde{H}^, \tilde{H}^+\phi _s}^{(0)} = \lambda \,\mathrm {e}^{{i}\,\xi _s}/\sqrt{2}\) and \({\Gamma _{\tilde{H}^+, \tilde{H}^\phi _s}^{(0)} = \lambda ^*\,\mathrm {e}^{{i}\, \xi _s}/\sqrt{2}}\), which yields$$\begin{aligned} \delta \xi _s&= \bigg \{\frac{1}{2{i}}\bigg [ \frac{\Gamma _{\tilde{H}^, \tilde{H}^+\phi _s}^{(1)} }{ \Gamma _{\tilde{H}^, \tilde{H}^+\phi _s}^{(0)}}\bigg _{\text {div}} + \frac{1}{2}\bigg ( \delta Z_{\tilde{H}^\pm }^{\text {L}}\nonumber \\&\quad + \delta Z_{\tilde{H}^\pm }^{\text {R}} + \delta Z_{\phi _s} \bigg )\bigg ] + \text {c.c.} \bigg \}  \delta \phi _\lambda \,. \end{aligned}$$(2.23)
 The absolute value and phase of \(\kappa \) are renormalized by \(\delta {\left \kappa \right }\) and \(\delta {\phi _{\kappa }}\). We fix both renormalization constants by applying Eq. (2.20) on the vertex \({\Gamma _{\tilde{S}\tilde{S}\phi _s}^{(0)} = \sqrt{2}\,\kappa \,\mathrm {e}^{{i}\,\xi _s}}\), which yields$$\begin{aligned} \frac{\delta \left \kappa \right }{\left \kappa \right }&= \frac{1}{2}\left\{ \left. \frac{\Gamma _{\tilde{S}\tilde{S}\phi _s}^{(1)} }{ \Gamma _{\tilde{S}\tilde{S}\phi _s}^{(0)}}\right _{\text {div}} + \frac{1}{2}\left( 2\delta Z_{\tilde{S}} + \delta Z_{\phi _s} \right) \right\} + \text {c.c.}\,, \end{aligned}$$(2.24a)$$\begin{aligned} \delta \phi _\kappa&= \left\{ \frac{1}{2{i}}\left[ \left. \frac{\Gamma _{\tilde{S}\tilde{S}\phi _s}^{(1)} }{ \Gamma _{\tilde{S}\tilde{S}\phi _s}^{(0)}}\right _{\text {div}} + \frac{1}{2}\left( 2\delta Z_{\tilde{S}} + \delta Z_{\phi _s} \right) \right] + \text {c.c.} \right\}  \delta \xi _s. \end{aligned}$$(2.24b)
 The parameter \(\mu _{\text {eff}}\) could be renormalized in the onshell scheme for one of the charginos or neutralinos [130, 131, 132]. However, such schemes cannot be stabilized over the whole parameter space (due to masscrossings). We thus prefer to apply the \(\overline{\mathrm {DR}}\) condition$$\begin{aligned} \delta \mu _{\text {eff}}&= \mu _{\text {eff}}\left( \frac{\delta \lambda }{\lambda }+\frac{1}{2}\delta Z_S\right) \,. \end{aligned}$$(2.25)
 In order to fix \(\delta {\left A_{\kappa }\right }\), we impose the renormalization condition of Eq. (2.20) on the vertex \(\Gamma _{\phi _s\phi _s\phi _s}^{(0)} = \sqrt{2}\,\kappa \left( \frac{6\,\kappa \,\mu _{\text {eff}}}{\lambda } + A_\kappa \cos {\zeta _3}\right) \), [\(\zeta _3\) was defined after Eq. (2.10)]. It reads$$\begin{aligned}&\delta A_\kappa = A_\kappa \left( \frac{\delta {\kappa }}{\kappa } + \frac{\delta {\cos {\zeta _3}}}{\cos {\zeta _3}}\right)  \frac{6\,\kappa \,\mu _{\text {eff}}}{\lambda \,\cos {\zeta _3}} \left( \frac{\delta \mu _{\text {eff}}}{\mu _{\text {eff}}}  \frac{\delta \lambda }{\lambda } + \frac{2\,\delta \kappa }{\kappa }\right)  \frac{\delta \Gamma _{\phi _s\phi _s\phi _s}}{\sqrt{2}\,\kappa \cos {\zeta _3}},\end{aligned}$$(2.26a)where we used the oneloop relation \(\delta {\cos {\zeta _3}} = \sin {\zeta _3}\delta {\zeta _3}\), and \(\delta v\) is not an independent counterterm, but a quantity depending on the counterterms to the electroweak parameters$$\begin{aligned}&\delta \zeta _3 = \left( \frac{\delta \kappa }{\kappa } + \frac{2\,\delta \lambda }{\lambda }  \frac{\delta \mu _{\text {eff}}}{\mu _{\text {eff}}} +\frac{\delta \sin {\left( 2\beta \right) }}{\sin {\left( 2\beta \right) }} {+} \frac{\delta \sin {\zeta _1}}{\sin {\zeta _1}} {+} \left. \frac{2\,\delta v}{v}\right _{\text {div}}\right) \cos {\zeta _3}\sin {\zeta _3} {+} \left[ \frac{\delta \Gamma _{\phi _s\phi _s\phi _s}}{\sqrt{2}\,\kappa } {+} \frac{6\,\kappa \,\mu _{\text {eff}}}{\lambda }\left( \frac{\delta \mu _{\text {eff}}}{\mu _{\text {eff}}}\right. \right. \nonumber \\&\left. \left.  \frac{\delta \lambda }{\lambda } + \frac{2\,\delta \kappa }{\kappa }\right) \right] \frac{\sin {\zeta _3}}{A_\kappa } + \left( \left. \delta T_{\chi _s}\right _{\text {div}}  \frac{\lambda \,v\,\cos {\beta }}{\mu _{\text {eff}}}\left. \delta T_{\chi _1}\right _{\text {div}}\right) \frac{\lambda ^2\,\cos {\zeta _3}}{ \sqrt{2}\,\kappa \,A_\kappa \,\mu _{\text {eff}}^2}, \end{aligned}$$(2.26b)$$\begin{aligned} \delta v= & {} v \left( \frac{\delta M_W}{M_W} + \frac{\delta s_{\text {w}}}{s_{\text {w}}}  \delta Z_e\right) \,,\nonumber \\ \delta s_{\text {w}}= & {} \frac{c_{\text {w}}^2}{s_{\text {w}}}\left( \frac{\delta M_Z}{M_Z}  \frac{\delta M_W}{M_W}\right) . \end{aligned}$$(2.27)

all the renormalized Higgs selfenergies are UVfinite, for arbitrary values of the momentum,

all the vertexdiagram amplitudes of a Higgs state decaying to SMparticles or a pair of charginos/neutralinos are UVfinite,

the UVdivergences of the counterterms to gauge couplings, superpotential parameters or soft terms are consistent with the corresponding oneloop beta functions (see e.g. Refs. [6, 133]),

in the \(\mathcal{CP}\)conserving limit, our parameters and couplings are identical to the findings of a previously developed model file [113],

in the MSSM limit, we have found agreement of the values of all our couplings with their counterparts in the complex MSSM, obtained with the model file of [134].

we checked that \(\phi _{\lambda }+\xi +\xi _s\) and \(\phi _{\kappa }+3\,\xi _s\) were the only relevant combinations of the phases \(\phi _{\lambda }\), \(\phi _{\kappa }\), \(\xi \) and \(\xi _s\) at the level of amplitudes,

finally, we checked explicitly, that the counterterms \(\delta {\phi _\lambda }\), \(\delta {\phi _\kappa }\), \(\delta \xi \) and \(\delta \xi _s\) vanish when all NMSSM contributions are included, as pointed out in Ref. [115]. This can also be placed in the perspective of the \(\beta \)functions [133, 135, 136, 137, 138, 139]: the phases from the superpotential parameters have no scaledependence (at least up to twoloop order); since \(\xi \) and \(\xi _s\) are spurious degrees of freedom, we could expect their counterterms to present the same vanishing behaviors as \(\delta {\phi _\lambda }\) and \(\delta {\phi _\kappa }\).
2.4 Quark Yukawa couplings
The Yukawa couplings of the top and bottom quarks, \(Y_t\) and \(Y_b\), have a sizable impact on radiative corrections to the Higgs masses. We present our prescriptions in this subsection.
The top Yukawa coupling \(Y_t=\sqrt{2\sqrt{2}\,G_F}\,m_t/\sin \beta \) is defined by the onshell top mass \(m_t\).
For the bottom quark, we employ the running \(\overline{\mathrm {DR}}\) bottommass of the SM (containing oneloop QCD corrections), \(\overline{m}_b\), at the scale \(m_t\) [140]. Additionally, we subtract the possibly large \(\tan \beta \)enhanced oneloop contributions to \(\overline{m}_b\)—induced by gaugino–squark and higgsino–squark loops—from the numerical definition of \(Y_b\) at the tree level: \(Y_b=\sqrt{2\sqrt{2}\,G_F}\,\overline{m}_b/[\cos \beta \,1+\Delta _b ]\), where \(\Delta _b\) is discussed in e.g. Refs. [98, 140, 141, 142, 143, 144, 145, 146].
2.5 Higgs masses at higher orders
We note that the twoloop \(O(\alpha _b\alpha _s)\) contributions to the Higgs selfenergies are not included in our calculation. Still, as we employ the running bottom mass in the definition of \(Y_b\) entering \(\left. \hat{\varvec{\Sigma }}^{(\text {1L})}_{hh}{(k^2)} \right ^{\text {NMSSM}}\), we expect that the missing twoloop piece is numerically subleading [151, 152, 153].
2.6 Wavefunction normalization factors: the matrix \(\mathbf {Z}^{\mathrm{mix}}\)
The determination of \(\mathbf {Z}^\mathrm{{mix}}\) in terms of the eigenstates of \(\hat{\varvec{\Delta }}^{1}_{hh}{(k^2)}\) is numerically easier to handle than its determination via Eq. (2.33). Applying the two defining conditions Eqs. (2.36) and (2.37) to the expression of Eq. (2.33), one can verify that both definitions of \(\mathbf {Z}^\mathrm{{mix}}\) are identical.

The first approach consists in freezing the momentum to \(k^2=0\) in the selfenergy of Eq. (2.28). This assumption is known as the effective potential approximation. In this approach the inverse propagator matrix \(\mathbf {\Delta }_{hh}^{1}{(k^2=0)}\), as given by Eq. (2.28), is diagonalized by a simple orthogonal matrix \(\mathbf {U}^0\), which approximates \(\mathbf {Z}^\mathrm{{mix}}\).

Another choice consists in replacing the momentum dependence of the selfenergy in Eq. (2.28) by \([\hat{\varvec{\Sigma }}_{hh}{(k^2)}]_{ij}\rightarrow [\hat{\varvec{\Sigma }}_{hh}{((m_{h_i}^2+m_{h_j}^2)/2)}]_{ij}\) (given in the basis of the treelevel mass states). This procedure aims at more closely mimicking the actual values of the selfenergies involved in the mass calculation. In this approach the inverse propagator is also diagonalized by an orthogonal matrix \(\mathbf {U}^m\).
3 Numerical analysis
In this section we present the results of our Higgs mass calculation and compare them with the output of public tools for several \(\mathcal{CP}\)violating scenarios. We also investigate the relevance of the matrix \(\mathbf {Z}^\mathrm{{mix}}\) for transition amplitudes in the example of the oneloop corrected decays of one Higgs field into a tau/antitau pair, \(h_i\rightarrow \tau ^+\tau ^\).
The choice for the topquark mass is \(m_t = 173.2\) GeV. Throughout this section all \(\overline{\mathrm {DR}}\) parameters are defined at \(m_t\), and all stopparameters are onshell parameters.
From the point of view of the Higgs phenomenology we test the scenarios presented in this section with the full set of experimental constraints and signals implemented in the public tools HiggsBounds4.3.1 [159, 160, 161, 162, 163, 164] and HiggsSignals1.3.1 [164, 165].
3.1 Comparison with FeynHiggs in the MSSM limit
We consider a region in the parameter space of the NMSSM with the following characteristics: \(\lambda =\kappa =10^{5}\), \(\tan \beta =10\), \(m_{H^{\pm }}=500\) GeV, \(\mu _{\text {eff}}=250\) GeV, \(A_{\kappa }=100\) GeV; the sfermion soft masses are set to the universal value of 1.5 TeV and the sfermion trilinear couplings to a value of 0.5 TeV, with the exception of the third generation parameters \(A_t=A_b=2.5\) TeV; the gaugino masses are chosen as follows: \(2M_1=M_2=M_3/5=0.5\) TeV. We then vary the phase \(\phi _{A_t}\). A variation of \(\phi _{A_t}\) (or of any MSSMlike phase) in such a naive direction is of limited phenomenological interest, since in this case limits from Electric Dipole Moments (EDMs) are violated almost as soon as \(\mathcal{CP}\), see e.g. Ref. [84]. In the following we dismiss this issue, however, and allow \(\phi _{A_t}\) to vary over its full range. Indeed, we are only interested in comparing our results with those of FeynHiggs. Due to the largely SMlike properties of the state with a mass close to 125 GeV, our scenario appears to retain characteristics that are compatible with the experimental data implemented in HiggsBounds and HiggsSignals, over the whole range of \(\phi _{A_t}\). The results for the Higgs masses are displayed in Fig. 1. We observe a near perfect agreement between our results (solid curves) and those of FeynHiggs (squares) with differences of order MeV. This agreement is expected, since we closely follow the procedure for the renormalization and processing of the MSSMlike input of FeynHiggs. Moreover, due to the small values for \(\lambda \) and \(\kappa \), deviations induced by genuine NMSSM effects remain negligible. The results for the elements of the matrix \(\mathbf {Z}^\mathrm{{mix}}\) are displayed in Fig. 2. Again, we find a very good agreement between our results and FeynHiggs with differences below 1 Open image in new window for the modules.
3.2 Comparison in the \(\mathcal{CP}\)conserving limit
We now turn away from the MSSM limit. Our mass calculation can be confronted to the routines presented in [113] in the \(\mathcal{CP}\)conserving case. Both approaches employ an identical renormalization scheme in this limit, with the exception of the electroweak VEV, which receives a \(\overline{\mathrm {DR}}\) renormalization in [113] while we parametrize v in terms of \(M_W\), \(M_Z\) and e (see Eq. (2.27)). However, in [113] the input for v is obtained via a reparametrization from our scheme (the scheme using \(\alpha (M_Z)\) as input is also considered), as explained in section 2.3 of that reference. Therefore, both mass predictions are directly comparable and the mismatch between them should be understood as an effect of twoloop electroweak order, due to the approximations in the reparametrization used by [113].
We then focus on the comparison with the masses predicted by [113]. The plot on the lefthand side of Fig. 3 illustrates a general agreement between our calculation (solid curves) and the results of [113] (squares). On the righthand side of Fig. 3, we display the mass differences between the two procedures, which are due to differences of twoloop order induced by the reparametrization used by [113]. We observe vanishing effects in the MSSM limit, while the mass differences eventually reach \(\mathcal {O}{\left( 40\,\text {MeV}\right) }\) for \(\lambda \simeq 0.16\). This can be understood in the following fashion: the leading effect originates in the Higgs mass matrix at the tree level, where an explicit dependence on v appears only^{4} through terms of the form \(\lambda \,v\) and \(\kappa \,v\) (quadratically for the doublet and singlet mass entries, and linearly for the doublet–singlet mixing). These terms are processed differently in both approaches: in [113] v is regarded as an independent \(\overline{\mathrm {DR}}\) parameter, while in our calculation v is a dependent quantity that is expressed in terms of the independent parameters \(M_W\), \(M_Z\) and e. While the reparametrization of [113] should restore the agreement between the two procedures, neglected effects of twoloop electroweak order in this reparametrization result in a small mismatch. Since the terms that convey this mismatch come with prefactors \(\lambda \) or \(\kappa \), the difference vanishes in the MSSM limit (\(\lambda ,\kappa \rightarrow 0\)). Moreover, in the regime under consideration, where \(\tan \beta \gg 1\), it is possible to understand why the mass of the \(\mathcal{CP}\)odd singlet (blue curve) is largely insensitive to the mismatch: terms \(\propto (\lambda \,v)^2\) in the \(\mathcal{CP}\)odd singlet mass entry are suppressed as \(1/\tan \beta \). Additionally, leading oneloop radiative corrections of \(\mathcal {O}{\left( \alpha _t\right) }\) induce further dependence on the processing of v. However, these corrections are suppressed for the points of Fig. 3, as the stops are relatively light.
On the whole, the numerical mismatch with the procedure of [113] is very minor, which places our current code in the direct continuity of this earlier work.
3.3 Comparison with NMSSMCALC
NMSSMCALC is particularly suitable for a comparison with our calculation, since its mixed \(\overline{\mathrm {DR}}\)/onshell renormalization scheme is relatively close to the one that we use.^{5} Yet, we note several differences between the prescriptions implemented by NMSSMCALC and the procedure that we have outlined in Sect. 2 (defining our “default” calculation). First, NMSSMCALC applies a renormalization scheme for the electric charge employing \(\alpha (M_Z)\) as input—whereas we decided to define \(\alpha \) via its relation to \(G_F\). Then the input parameters in the stop sector are defined in the \(\overline{\mathrm {DR}}\) scheme in NMSSMCALC—while we employ onshell definitions. Additionally, we resum large \(\tan \beta \) effects from our definition of the bottom Yukawa, contrarily to the Higgsmass calculation of NMSSMCALC. Finally, NMSSMCALC includes only \(\mathcal {O}{\left( \alpha _t\alpha _s\right) }\) corrections at the twoloop order—where we consider \(\mathcal {O}{\left( \alpha _t^2\right) }\) effects as well. However, the twoloop \(\mathcal {O}{\left( \alpha _t\alpha _s\right) }\) contributions of NMSSMCALC are exhaustive in the NMSSM (including corrections for the selfenergies with at least one external singlet field)—whereas ours are obtained in the MSSM approximation.
These observations mean that our mass calculation is not directly (at least, not quantitatively) comparable to the predictions of NMSSMCALC, since, of the items listed above, the first few produce a deviation relative to the scheme, while the later ones generate a mismatch of higher orders. Consequently, several adjustments need to be performed in order to make a comparison meaningful and control the sources of deviations. Thus, NMSSMCALC has been adjusted in view of accepting onshell input in the stop sector.^{6} Moreover, we also establish a “modified” version of our routines that attempts to mimic the choices of NMSSMCALC—i.e. employing \(\alpha (M_Z)\), discarding large\(\tan \beta \) effects for \(Y_b\) and subtracting \(\mathcal {O}{\left( \alpha _t^2\right) }\) corrections—although we cannot currently include \(\mathcal {O}{\left( \alpha _t\alpha _s\right) }\) corrections beyond the MSSM, so that this effect should control the difference of our modified version with NMSSMCALC. Beyond this comparison with NMSSMCALC, we will also try to quantify the magnitude of the other higherorder effects that distinguish our “default” result from NMSSMCALC.
First, we consider the regime of the NMSSM with low \(\tan \beta \) and large \(\lambda \). This region in parameter space is wellknown for maximizing the specific NMSSM treelevel contributions to the mass of the SMlike Higgs state as well as for stimulating singlet–doublet mixing effects and other genuine aspects of the NMSSM phenomenology. We employ the following parameters: \(\lambda =0.7\), \(\kappa =0.1\), \(\tan \beta =2\), \(M_{H^{\pm }}=1170\) GeV, \(\mu _{\text {eff}}=500\) GeV, \(A_{\kappa }=70\) GeV; the soft masses are taken as in Fig. 1 with the exception of the squarks of the third generation, for which the soft masses and trilinear couplings are set to 500 and 100 GeV, respectively. In the regime under consideration genuine NMSSM effects are indeed sufficient to produce a SMlike state in the observed mass range without requiring large top/stop corrections. We vary the phase \(\phi _{\kappa }\) (we restrict ourselves to a range where the treelevel squared Higgs masses remain positive). We note that, contrarily to MSSMlike phases, the phases from the singlet sector are allowed a wide range of variation without conflicting with the measured EDM [83, 169].
The results for the mass prediction are presented in Fig. 4. At vanishing \(\phi _{\kappa }\) the mass of the SMlike state (in blue) is somewhat low, \(m_{h_2} \sim 120\) GeV, so that this point in parameter space has a very marginal agreement with the observed characteristics of the Higgs state. For nonvanishing \(\phi _{\kappa }\), however, a \(\mathcal{CP}\)violating mixing with the lighter pseudoscalar singlet (in red) develops: this effect increases the mass of the light mostly \(\mathcal{CP}\)even state \(h_2\) but affects its otherwise SMlike properties only in a subleading way. Consequently, we recover an excellent agreement with the LHC results—as tested by HiggsSignals and HiggsBounds—for e.g. \(\phi _{\kappa }\simeq 0.11\). Additionally, the dominantly \(\mathcal{CP}\)odd singlet \(h_1\) then has a mass close to 100 GeV. As it acquires a doublet \(\mathcal{CP}\)even component via mixing, it could explain the LEP local excess in \(b\bar{b}\) final states [166]. The mostly \(\mathcal{CP}\)even singlet \(h_3\) (in green), with mass at \(\sim \)210 GeV plays no significant role. The masses of the heavier doubletlike fields \(h_4\) and \(h_5\) are approximately constant and close to \(M_{H^\pm }\).
In Fig. 4, we observe good agreement between our results (solid lines), computed as described in Sect. 2, and the predictions of NMSSMCALC (squares), although the corresponding masses are defined in different schemes and at different orders. For a more quantitative comparison, we turn to our “modified” scheme for the mass calculation. On the lefthand side of Fig. 5, we plot the deviation between the corresponding results and the predictions of NMSSMCALC for the three lightest Higgs states. We checked that the oneloop results are virtually identical, so that the differences between NMSSMCALC and our calculation are entirely controlled by twoloop effects. We observe typical deviations of order 0.5–1 GeV, which should be interpreted as the impact of \(\mathcal {O}{\left( \alpha _t\alpha _s\right) }\) corrections beyond the MSSMapproximation. As could be expected, the masses of the mostly singlet states (red and green lines) tend to exhibit the largest effect, though the mass predictions for the SMlike state may still differ by \(\sim \)0.5 GeV (for \(\phi _{\kappa }\simeq 0\)). The plot on the righthand side of Fig. 5 depicts the magnitude of \(\mathcal {O}{\left( \alpha _t^2\right) }\) effects, which is quantified in our “default” scheme. Here again, the typical impact on the masses is of order 1 GeV. Expectedly, the masses of the almost pure singlet states (red curve at \(\phi _{\kappa }\simeq 0\) or green curve) are insensitive to the corrections implemented in the MSSMapproximation. The mass of the mostly \(\mathcal{CP}\)odd singlet (red curve) is only affected when the corresponding state acquires a nonvanishing doublet component (\(\phi _{\kappa }\ne 0\)).
Subsequently, we present our results in another region of the parameter space: \(\lambda =0.2\), \(\kappa =0.6\), \(\tan \beta =25\), \(m_{H^{\pm }}=1\) TeV, \(\mu _{\text {eff}}=200\) GeV, \(A_{\kappa }=750\) GeV, the gaugino soft masses as well as the soft masses for the sfermions of first and second generations are chosen as before; for the third generation, the soft sfermion mass is set to 1.1 TeV; the trilinear soft terms are set to \(2\) TeV. With this choice of parameters, the singletlike \(\mathcal{CP}\)even state and the heavy \(\mathcal{CP}\)even and \(\mathcal{CP}\)odd doubletlike states receive comparable masses of the order of 1 TeV. This results in a sizable mixing for the corresponding fields \(h_2\), \(h_3\) and \(h_4\), which includes both singletdoublet admixture as well as \(\mathcal{CP}\)violation (for nonvanishing \(\phi _{\kappa }\)). The SMlike Higgs state has a mass close to \(\sim \)124 GeV on the whole range of \(\phi _{\kappa }\), which leads to good agreement with the Higgs properties measured at the LHC (as tested with HiggsSignals). The heaviest state \(h_5\) has a mass of \(\sim \)1.2 TeV, which we will not comment further below.
On the lefthand side of Fig. 7, we show our prediction for the mass of the lightest (SMlike) Higgs state (full curve). The mass delivered by NMSSMCALC is represented by the squares at about 118 GeV, which is substantially smaller than ours (by \(\approx \)6 GeV). If we mimic the settings of NMSSMCALC (our “modified result”, dotted curve), this discrepancy is considerably reduced. In fact, the difference between our full result and NMSSMCALC’s is largely driven by the \(\mathcal {O}{\left( \alpha _t^2\right) }\) twoloop contributions, missing in NMSSMCALC. Again, both results are virtually identical at the oneloop order.
On the righthand side of Fig. 7, we turn to the heavier states \(h_2\), \(h_3\) and \(h_4\) of this scenario. Our default results (full curves) are compatible with the predictions of NMSSMCALC (squares). The discrepancies are of order 1–3 GeV only, which should be considered from both the perspective of the different renormalization scheme of the electric coupling e and of the different twoloop contributions. Actually, the mass predictions match almost exactly when comparing NMSSMCALC with our modified scheme (dotted curves). The corresponding deviations are shown on the lefthand side of Fig. 8 and fall in the range of 100 MeV. In this precise case, the difference between our results and NMSSMCALC is essentially driven by the resummation of large\(\tan \beta \) effects in the bquark Yukawa coupling. On the righthand side of Fig. 8, we quantify the associated massshift and find an impact of a few GeV.
Finally, we turn to the \(\mathbf {U}^0\) matrix elements for \(h_2\), \(h_3\) and \(h_4\) in Fig. 9. There, we observe sizable deviations between our default result (solid curves) and NMSSMCALC (squares), which, however, have no deepreason to agree in view of the diverging options. If we keep in mind that the main difference between our full scheme and NMSSMCALC is controlled by the large\(\tan \beta \) corrections to \(Y_b\) in this precise scenario, it is not surprising to observe large shifts, as mixing angles are indeed very sensitive to small deviations in the massmatrix for states that are very close in mass. These differences largely vanish when we identify the output of NMSSMCALC with our modified results (dotted lines), which is better equipped for comparisons with this code.
3.4 The matrix \(\mathbf {Z}^\mathrm{{mix}}\) and the \(h_i\rightarrow \tau ^+\tau ^\) decays
The matrix \(\mathbf {Z}^\mathrm{{mix}}\) is not an observable quantity in itself. It is a renormalizationscheme dependent object relating the treelevel mass states of the Higgs sector to the physical Higgs fields. For onshell renormalized fields \(\mathbf {Z}^\mathrm{{mix}}\) is trivial. In any other renormalization scheme, however, it is mandatory to include this transition to the physical fields for a proper description of external legs in Feynman diagrams at higher orders.
A remarkable aspect of \(\mathbf {Z}^\mathrm{{mix}}\) is that the eigenvectors that it contains do not preserve unitarity with respect to the treelevel fields. Instead, they satisfy the normalization condition given in Eq. (2.37). This is a feature that the approximations \(\mathbf {U}^0\) and \(\mathbf {U}^m\) are unable to capture (by construction). In a first step, we will show that the norms in \(\mathbf {Z}^\mathrm{{mix}}\) can differ from 1 by a few percent in the scheme that we have described in Sect. 2. Beyond the normalization of the fields, \(\mathbf {U}^0\) and \(\mathbf {U}^m\) also differ from \(\mathbf {Z}^\mathrm{{mix}}\) in that they diagonalize the massmatrix away from the poles of the propagator.
However, as we wrote above, \(\mathbf {Z}^{\mathrm{mix}}\) is a schemedependent object and we should not pay excessive attention to its actual structure. In order to characterize its role in an observable quantity, we will consider the \(h_i\rightarrow \tau ^+\tau ^\) decays at the oneloop level. We have chosen this particular channel as it is one of the main fermionic Higgs decays and proves technically simple to implement in a predictive way. Moreover, oneloop corrections are of purely electroweak nature—QCD contributions occur only at threeloop order and beyond—so that radiative corrections are expected to be moderate. This allows for a clean appreciation—free of large higherorder uncertainties—of the impact of the wavefunction normalization matrix \(\mathbf {Z}^\mathrm{{mix}}\). Radiative corrections^{7} are computed with our model file, except for the QED contributions, which are included according to the prescriptions of Refs. [170, 171]. There, \(\mathbf {Z}^\mathrm{{mix}}\) intervenes in the decay amplitudes of the physical fields according to Eq. (2.32) (we drop the superscript ‘phys’ throughout this section). We will show that the substitution of \(\mathbf {Z}^\mathrm{{mix}}\) by the approximations \(\mathbf {U}^0\) and \(\mathbf {U}^m\) may lead to sizable deviations in certain regions of the NMSSM parameter space. This result confirms the outcome of similar studies in the MSSM [109].
We turn to the following NMSSM input: the parameters are chosen as in Fig. 4, except for \(M_{H^{\pm }}=1.2\) TeV and \(A_{\kappa }=100\) GeV. We have plotted the masses of the three lighter Higgs fields as a function of \(\phi _\kappa \) in the plot on the lefthand side of Fig. 10. For vanishing \(\phi _{\kappa }\) the lightest Higgs state \(h_1\) is SMlike and we checked with HiggsBounds and HiggsSignals that this point is consistent with the experimental data. The dominantly \(\mathcal{CP}\)odd, singletlike state \(h_2\) is only slightly heavier than the state \(h_1\) in this case. For increasing values of \(\phi _{\kappa }\) the mixing of the states \(h_1\) and \(h_2\) tends to lower the mass of the SMlike state \(h_1\), which eventually becomes too light to accommodate the experimental data. The dominantly \(\mathcal{CP}\)even, singletlike state \(h_3\) has a near constant mass of \(\sim \)210 GeV for all depicted values of \(\phi _\kappa \). The two heavier, \(\mathcal{CP}\)even and \(\mathcal{CP}\)odd doubletlike states have masses close to \(\sim \)1.2 TeV.
The results for the squared norms \(Z_i^2\) of the eigenvectors—see Eq. (2.38)—in this scenario are shown in the plot on the righthand side of Fig. 10. We observe a departure from the value 1—which would correspond to a unitary transition, as modeled by the approximations \(\mathbf {U}^0\) and \(\mathbf {U}^m\)—by a few percent. The local extrema at \(\phi _{\kappa }\simeq 0\) for \(Z_1^2\) (red curve) and \(Z_2^2\) (blue curve) are associated to the sudden disappearance of the mixing between the light \(\mathcal{CP}\)odd singlet and the SMlike states at \(\phi _\kappa = 0\) (\(\mathcal{CP}\)conserving limit). The discontinuities of \(Z_2^2\) and \(Z_3^2\) (green curve) at \(\phi _{\kappa }\simeq \pm 0.5\) and \(\pm 0.7\) correspond to the crossing of decay thresholds (\(h_2\rightarrow W^+W^\), \(h_2\rightarrow 2\,Z\), \(h_{3}\rightarrow h_1\,h_2\)). These “spikes” are associated to the singularities of the first derivatives of the loop functions involved in the determination of \(\mathbf {Z}^{\mathrm{mix}}\)—the apparent singularities actually come with a finite height due to the imaginary parts of the poles. A proper description of these threshold regions would require that the interactions among the daughter particles (of the decays at threshold) are properly taken into account, which would result in e.g. interactions between the Higgs state and boundstates or swaves of the daughter particles. This, however, goes beyond the scope of the present work.
We now turn to the decay widths \(\Gamma {(h_i \rightarrow \tau ^+\tau ^)}\) in the scenario of Fig. 10. The widths are displayed in the left column of Fig. 11 in the exhaustive description of the Higgs external leg (i.e. employing \(\mathbf {Z}^{\mathrm{mix}}\); solid curves), in the \(\mathbf {U}^m\) approximation (dashed lines) and in the \(\mathbf {U}^0\) approximation (dotted lines), for the five Higgsmass eigenstates. We observe a sharp variation close to \(\phi _{\kappa }=0\) for the decays of \(h_1\) and \(h_2\), both in the full and approximate descriptions. It is associated to the mixing that develops between the SMlike state \(h_1\) and the dominantly \(\mathcal{CP}\)odd, singletlike state \(h_2\): this effect transfers part of the doublet component of \(h_1\) to \(h_2\), so that the second state acquires a nonvanishing coupling to SM fermions at the expense of the first. The sum of the decay widths for both these states remains approximately constant in the vicinity of \(\phi _{\kappa }=0\). The width \(\Gamma (h_3\rightarrow \tau ^+\tau ^)\) appears to be an order of magnitude smaller than the corresponding widths for \(h_1\) and \(h_2\), an effect that is associated to the dominantly \(\mathcal{CP}\)even, singletlike nature of \(h_3\). Still, \(\Gamma (h_3\rightarrow \tau ^+\tau ^)\) nearly doubles in the considered interval of \(\phi _{\kappa }\), while the mass of \(h_3\) is fairly stable: we can understand this fact in terms of the acquisition of a larger doublet component, which is channeled by the increased proximity of the masses of \(h_2\) and \(h_3\). The widths of \(h_4\) and \(h_5\) are essentially constant with only small relative changes. The general \(\phi _\kappa \)dependency of the approximated and the full results are very similar. Yet, a systematic shift can be observed, especially in the case of \(\mathbf {U}^0\). This is consistent with the findings of similar studies in the context of the MSSM [109].
On the righthand side of Fig. 11, we show the difference between the full and the approximate results, \(\Delta {\Gamma } = \Gamma  \Gamma _{\text {appr}}\), normalized to the more accurate one obtained with \(\mathbf {Z}^{\mathrm{mix}}\). When \(\mathbf {Z}^{\mathrm{mix}}\) is approximated by \(\mathbf {U}^0\) (dotted lines), the typical discrepancy averages 4%, although the deviation reaches beyond 10% in the case of the decays of the two lightest Higgs states in the vicinity of, but not at, \(\phi _{\kappa }\simeq 0\). We stress that this interval close to \(\phi _{\kappa }=0\) corresponds to the phenomenologically relevant region from the perspective of the measured Higgs properties. The approximation of \(\mathbf {Z}^{\mathrm{mix}}\) by \(\mathbf {U}^m\) tends to provide better estimates of the full result, though deviations reach up to \(\sim \)7%. For both approximations the largest deviations from the more complete result employing \(\mathbf {Z}^{\mathrm{mix}}\) are intimately related to the proximity in mass of the SMlike and dominantly \(\mathcal{CP}\)odd, singletlike states: as the approximations capture the dependence on the external momentum either partially (\(\mathbf {U}^m\)) or not at all (\(\mathbf {U}^0\)), the gap between the diagonal elements of the Higgsmass matrix, hence the mixing between the two states, is not quantified properly. While this precise configuration might appear somewhat anecdotal, we wish to point out the popularity of NMSSM scenarios with a sizable singletdoublet mixing. Dismissing this extreme case, the approximations of \(\mathbf {Z}^{\mathrm{mix}}\) by \(\mathbf {U}^0\), and to a lesser extent by \(\mathbf {U}^m\), still generate errors of the order of a few percent at the level of the decay widths. In view of the precision of the measurements achievable at the LHC [3, 172, 173, 174, 175], such discrepancies may appear of secondary importance. In the long run, however, if the Higgs couplings are studied more closely, e.g. at a linear collider [176, 177, 178, 179], one would have to try and keep such sources of error to a minimum.
4 Conclusions
In this paper, we have discussed the renormalization of the NMSSM Higgs sector, including complex parameters. Radiative contributions to the Higgs selfenergies have been included up to the leading twoloop MSSMlike effects of \(\mathcal {O}{\left( \alpha _t\alpha _s+\alpha _t^2\right) }\). Beyond the calculation of onshell Higgs masses in this scheme, we were interested in determining the transition matrix \(\mathbf {Z}^{\mathrm{mix}}\) between the mass and treelevel states. The latter play an essential role in the proper description of external Higgs legs in physical processes at the radiative level.
Our predictions for the Higgs masses have been compared to the calculations of existing tools in several NMSSM scenarios. In the MSSM limit of the model, we have recovered an excellent agreement with FeynHiggs. For nonvanishing \(\lambda \) and \(\kappa \), we first compared our Higgsmass prediction with the findings of a previous extension of FeynHiggs to the NMSSM in the case of real parameters, and found nearly identical results. Second, we compared our calculation in the case of complex parameters with NMSSMCalc and found values of the Higgs masses which are compatible, although small differences emerge as a result of different processing of the twoloop pieces, both for low and high \(\tan \beta \).
Finally, we investigated the impact of the transition matrix \(\mathbf {Z}^{\mathrm{mix}}\) on the \(h_i\rightarrow \tau ^+\tau ^\) width in a scenario with low \(\tan \beta \) and large \(\lambda \), where the SMlike Higgs state may have a sizable mixing with the \(\mathcal{CP}\)odd singlet. We compared the full oneloop calculation of the width—i.e. including \(\mathbf {Z}^{\mathrm{mix}}\)—with the popular approximations \(\mathbf {U}^0\) and \(\mathbf {U}^m\)—which are determined for fixed, unphysical momenta. We found typical deviations at the percent level, although larger effects can develop in the presence of almostdegenerate states, especially in the \(\mathbf {U}^0\) approximation. Such precision effects will matter when the measurement of fermionic Higgs couplings reaches comparable accuracy.
In its current form, our masscomputing tool is contained within a Mathematica package. In time, our routines should be incorporated in an extension of FeynHiggs to the NMSSM.
Footnotes
 1.
Besides Higgs–G mixing, we neglect the kinetic Higgs–Z and Higgs–photon mixing, since they are subleading effects of two and threeloop order, respectively.
 2.
 3.
Note that the system is nonlinear due to the momentum dependence of \(\hat{\varvec{\Sigma }}_{hG}\). However, \(\left\{ \mathcal {M}_{h_i}^2,(\mathbf {Z}^{\mathrm{mix}})_{i}\right\} \) is a genuine eigenstate of \(\mathbf {D}_{hG}  \hat{\varvec{\Sigma }}_{hG}{(\mathcal {M}_{h_i}^2)}\).
 4.
We remind the reader that both in [113] and in our calculation \(M_W^2\) and \(M_Z^2\) are chosen as independent, onshell parameters. Therefore, the corresponding terms in the Higgs massmatrix are not affected by the differences in the renormalization/reparametrization discussed here.
 5.
Note that it is somewhat more involved to compare our results quantitatively with RGEbased tools, as the input requires a conversion to the appropriate scheme (usually \(\overline{\mathrm {DR}}\)) and a running to the correct input scale [167]. For this reason, we shall confine our discussion to comparisons with NMSSMCALC, which shares closer characteristics with our approach. A similar comparison for real parameters has been presented in [168].
 6.
We thank K. Walz for providing a modified version of NMSSMCALC for this feature.
 7.
Details on the calculation of the decays at the oneloop level will be presented in a future publication.
Notes
Acknowledgements
We thank S. Heinemeyer for helpful comments on the manuscript. The work of F. Domingo is supported in part by CICYT (Grant FPA 201340715P), in part by the MEINCOP Spain under contract FPA201678022P, and in part by the Spanish “Agencia Estatal de Investigación” (AEI) and the EU “Fondo Europeo de Desarrollo Regional” (FEDER) through the project FPA201678645P. The work of P. Drechsel and S. Paßehr is supported by the Collaborative Research Center SFB676 of the DFG, “Particles, Strings and the early Universe.”
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