Decays of the neutral Higgs bosons into SM fermions and gauge bosons in the \(\mathcal{CP}\)violating NMSSM
Abstract
The NexttoMinimal Supersymmetric Standard Model (NMSSM) offers a rich framework embedding physics beyond the Standard Model as well as consistent interpretations of the results about the Higgs signal detected at the LHC. We investigate the decays of neutral Higgs states into Standard Model (SM) fermions and gauge bosons. We perform full oneloop calculations of the decay widths and include leading higherorder QCD corrections. We first discuss the technical aspects of our approach, before confronting our predictions to those of existing public tools, performing a numerical analysis and discussing the remaining theoretical uncertainties. In particular, we find that the decay widths of doubletdominated heavy Higgs bosons into electroweak gauge bosons are dominated by the radiative corrections, so that the treelevel approximations that are often employed in phenomenological analyses fail. Finally, we focus on the phenomenological properties of a mostly singletlike state with a mass below the one at 125 GeV, a scenario that appears commonly within the NMSSM. In fact, the possible existence of a singletdominated state in the mass range around or just below 100 GeV would have interesting phenomenological implications. Such a scenario could provide an interpretation for both the \(2.3\,\sigma \) local excess observed at LEP in the \(e^+e^\rightarrow Z(H\rightarrow b\bar{b})\) searches at \(\mathord {\sim }\,98\) GeV and for the local excess in the diphoton searches recently reported by CMS in this mass range, while at the same time it would reduce the “Little Hierarchy” problem.
1 Introduction
The signal that was discovered in the Higgs searches at ATLAS and CMS at a mass of \(\mathord {\sim }\,125\) GeV [1, 2, 3] is, within the current theoretical and experimental uncertainties, compatible with the properties of the Higgs boson predicted within StandardModel (SM) of particle physics. No conclusive signs of physics beyond the SM have been reported so far. However, the measurements of Higgs signal strengths for the various channels leave considerable room for Beyond Standard Model (BSM) interpretations. Consequently, the investigation of the precise properties of the discovered Higgs boson will be one of the prime goals at the LHC and beyond. While the mass of the observed particle is already known with excellent accuracy [4, 5], significant improvements of the information about the couplings of the observed state are expected from the upcoming runs of the LHC [3, 6, 7, 8, 9] and even more so from the highprecision measurements at a future \(e^+e^\) collider [10, 11, 12, 13].
Motivated by the “Hierarchy Problem”, Supersymmetry (SUSY)inspired extensions of the SM play a prominent role in the investigations of possible new physics. As such, the Minimal Supersymmetric Standard Model (MSSM) [14, 15] or its singlet extension, the NexttoMSSM (NMSSM) [16, 17], have been the object of many studies in the last decades. Despite this attention, these models are not yet prepared for an era of precision tests as the uncertainties at the level of the Higgsmass calculation [18, 19, 20] are about one order of magnitude larger than the experimental uncertainty. At the level of the decays, the theoretical uncertainty arising from unknown higherorder corrections has been estimated for the case of the Higgs boson of the SM (where the Higgs mass is treated as a free input parameter) in Refs. [21, 22] and updated in Ref. [23]: depending on the channel and the Higgs mass, it typically falls in the range of \(\mathord {\sim }\,0.5\)–\(5\%\). To our knowledge, no similar analysis has been performed in SUSYinspired models, but one can expect the uncertainties from missing higherorder corrections to be larger in general – with many nuances depending on the characteristics of the Higgs state and the considered point in parameter space: we provide some discussion of this issue at the end of this paper. In addition, parametric uncertainties that are induced by the experimental errors of the input parameters should be taken into account as well. For the case of the SM decays those parametric uncertainties have been discussed in the references above. In the SUSY case the parametric uncertainties induced by the (known) SM input parameters can be determined in the same way as for the SM, while the dependence on unknown SUSY parameters can be utilized for setting constraints on those parameters. While still competitive today, the level of accuracy of the theoretical predictions of Higgsboson decays in SUSY models should soon become outclassed by the achieved experimental precision on the decays of the observed Higgs signal. Without comparable accuracy of the theoretical predictions, the impact of the exploitation of the precision data will be diminished – either in terms of further constraining the parameter space or of interpreting deviations from the SM results. Further efforts towards improving the theoretical accuracy are therefore necessary in order to enable a thorough investigation of the phenomenology of these models. Besides the decays of the SMlike state at 125 GeV of a SUSY model – where the goal is clearly to reach an accuracy that is comparable to the case of the SM – it is also of interest to obtain reliable and accurate predictions for the decays of the other Higgs bosons in the spectrum. The decays of the nonSMlike Higgs bosons can be affected by large higherorder corrections as a consequence of either large enhancement factors or a suppression of the lowestorder contribution. Confronting accurate predictions with the available search limits yields important constraints on the parameter space.
In this paper we present an evaluation of the decays of the neutral Higgs bosons of the \(\mathbb {Z}_3\)conserving NMSSM into SM particles. The extension of the MSSM by a gaugesinglet superfield was originally motivated by the ‘\(\mu \) problem’ [24], but also leads to a richer phenomenology in the Higgs sector (see e.g. the introduction of Ref. [25] for a recent summary of related activities). Several public tools provide an implementation of Higgs decays in the NMSSM: HDECAY [26, 27, 28], focusing on the SM and MSSM, was the object of various extensions to the \(\mathcal{CP}\)conserving or violating NMSSM for NMSSMTools [29, 30, 31, 32] and NMSSMCALC [33, 34]; SOFTSUSY [35, 36] recently released its own set of routines [37], which generally are confined to the leading order or leading QCD corrections; a oneloop evaluation of twobody decays in the \(\overline{\mathrm {DR}}\) \(\left( \overline{\mathrm {MS}}\right) \) scheme for generic models [38] has been recently presented for SPHENO [39, 40, 41, 42], which employs SARAH [43, 44, 45, 46]. Moreover, the nonpublic code SloopS [47] has been extended to the NMSSM [48, 49] and applied to the calculation of Higgs decays [49, 50].
The current work focusing on NMSSM Higgs decays is part of the effort for developing a version of FeynHiggs [18, 51, 52, 53, 54, 55, 56, 57] dedicated to the NMSSM [25, 58]. The general methodology relies on a Feynmandiagrammatic calculation of radiative corrections, which employs FeynArts [59, 60], FormCalc [61] and LoopTools [61]. The implementation of the renormalization scheme within the NMSSM [25] has been done in such a way that the result in the MSSM limit of the NMSSM exactly coincides with the MSSM result obtained from FeynHiggs without any further adjustments of parameters (cases where the NMSSM result is more complete than the current implementation of the MSSM result will be discussed below). Concerning the Higgs decays, our routines, in their current status, contain an evaluation of all the twobody decays of neutral and charged Higgs bosons. In the present paper, we wish to focus on decays of the neutral Higgs bosons into SM final states, where we have obtained results including higherorder contributions as detailed below as well as further refinements. The channels of the type HiggstoHiggs and HiggstoSUSY are currently only implemented at leading order.^{1}
For the evaluation of the decays of the neutral Higgs bosons of the NMSSM into SM final states we have followed the same general approach as for the implementation of the MSSM Higgs decays in FeynHiggs, which have been described in Refs. [63, 67]. At the level of the external Higgs fields, mixing effects are consistently taken into account as explained in Refs. [25, 68]. Full oneloop contributions are considered in the fermionic decay channels, supplemented by QCD higherorder corrections. For the bosonic decay modes generated at the radiative order, leading QCD corrections are taken into account. For decays into massive electroweak gauge bosons, the implementation in the MSSM is such that FeynHiggs first extracts the loopcorrected width that Prophecy4f [69, 70, 71] calculates in the SM for a Higgs boson at a given mass, and then rescales this result by the squared coupling of the MSSM Higgs boson to WW and ZZ, normalized to the SM value. We go beyond this approach and include full oneloop onshell results for these decay widths – however, the onshell kinematical factor of the treelevel contribution is replaced by its offshell counterpart, leading to a treelevel estimate below threshold.^{2} More generally, the refinements that we implement, e.g. the inclusion of higherorder corrections, often surpass the assumptions made in public codes dedicated to the NMSSM. However, we stress that we strictly confine ourselves to the freeparticle approximation for the final states. Accordingly, dedicated analyses would be needed for a proper treatment of decays close to threshold, since finitewidth effects need to be taken into account in this region, and effects of the interactions among the final states can be very large [72, 73, 74, 75, 76].
The case where the Higgs spectrum contains a singletdominated Higgs state with a mass below the one of the signal detected at 125 GeV is a particularly interesting scenario that commonly emerges in the NMSSM. Among the appealing features of this class of scenarios it should be mentioned that the somewhat high mass (in an MSSM context) of the SMlike Higgs state observed at the LHC could be understood more naturally as the result of the mixing with a lighter singlet state – see e.g. Ref. [77] for a list of references and an estimate of the possible uplift in mass. Furthermore, the existence of a mostly singletlike state in the range of \(\mathord {\lesssim }\,100\) GeV has been suggested [78, 79] as a possible explanation of the \(2.3\,\sigma \) local excess observed at LEP in the \(e^+e^\rightarrow Z(H\rightarrow b\bar{b})\) searches [80]. More recently, the CMS collaboration has reported a local excess in its diphoton Higgs searches for a mass in the vicinity of \(\mathord {\sim }\,96\) GeV [81]. This excess reaches the (local) \(2.9\,\sigma \) level in the Run II data and has already received attention from the particlephysics community [82, 83, 84, 85]. A similar excess (\(2.0\,\sigma \)) was already present in the Run I data in the same mass range. In an NMSSM context, the possibility of large diphoton signals is wellknown [86, 87, 88]. Here we show that it is in fact possible to describe simultaneously the LEP and the CMS excesses within the NMSSM. However, it should be kept in mind that the excesses that were observed at LEP and CMS at \(\mathord {\lesssim }\,100\) GeV could of course just be statistical fluctuations of the background and that their possible explanation in terms of an NMSSM Higgs state remains somewhat speculative. In particular, the diphoton excess observed at CMS would of course require confirmation from the ATLAS data as well [89].
In the following section, we discuss the technical aspects of our calculation, describing the conventions, assumptions and higherorder corrections that we address. Section 3 illustrates the workings of our decay routines in several scenarios of the NMSSM, and we perform comparisons with existing public tools. We also investigate the NMSSM scenario with a mostly singletlike state with mass close to 100 GeV. Furthermore we discuss the possible size of the remaining theoretical uncertainties from unknown higherorder corrections. The conclusions are summarized in Sect. 4.
2 Higgs decays to SM particles in the \(\mathcal{CP}\)violating NMSSM
In this section, we describe the technical aspects of our calculation of the Higgs decays. Our notation and the renormalization scheme that we employ for the \(\mathbb {Z}_3\)conserving NMSSM in the general case of complex parameters were presented in Sect. 2 of Ref. [25], and we refer the reader to this article for further details.
2.1 Decay amplitudes for a physical (onshell) Higgs state – Generalities

the Higgs selfenergies include full oneloop and leading \(\mathcal {O}{(\alpha _t\alpha _s,\alpha _t^2)}\) twoloop corrections (with twoloop effects obtained in the MSSM approximation via the public code FeynHiggs^{3});

the pole masses correspond to the zeroes of the determinant of the inversepropagator matrix;

the \((5 \times 5)\) matrix \(\mathbf {Z}^{ \text{ mix }}\) is obtained in terms of the solutions of the eigenvector equation for the effective mass matrix evaluated at the poles, and satisfying the appropriate normalization conditions (see Sect. 2.6 of Ref. [25]).
We stress that all public tools, with the exception of FeynHiggs, neglect the full effect of the transition to the physical Higgs states encoded within \(\mathbf {Z}^{ \text{ mix }}\), and instead employ the unitary approximation \(\mathbf {U}^0\) neglecting external momenta (which is in accordance with leadingorder or QCDimproved leadingorder predictions). We refer the reader to Refs. [25, 53, 68] for the details of the definition of \(\mathbf {U}^0\) or \(\mathbf {U}^m\) (another unitary approximation) as well as a discussion of their impact at the level of Higgs decay widths.

the Ward identity in \(h_i\rightarrow \gamma \gamma \) is not satisfied (see also Ref. [87]);

infrared (IR) divergences of the virtual corrections in \(h_i\rightarrow W^+W^\) do not cancel their counterparts in the bremsstrahlung process \(h_i\rightarrow W^+W^\gamma \) (see also Ref. [94]);

computing \(h_i\rightarrow f\bar{f}\) in an \(R_{\xi }\) gauge entails nonvanishing dependence of the amplitudes on the electroweak gaugefixing parameters \(\xi _Z\) and \(\xi _W\).
Numerical input in the oneloop corrections As usual, the numerical values of the input parameters need to reflect the adopted renormalization scheme, and the input parameters corresponding to different schemes differ from each other by shifts of the appropriate loop order (at the loop level there exists some freedom to use a numerical value of an input parameter that differs from the treelevel value by a oneloop shift, since the difference induced in this way is of higher order). Concerning the input values of the relevant light quark masses, we follow in our evaluation the choice of FeynHiggs and employ \(\overline{\text{ MS }}\) quark masses with threeloop QCD corrections evaluated at the scale of the mass of the decaying Higgs, \(m_q^{\overline{ \text{ MS }}}(M_{h_i})\), in the loop functions and the definition of the Yukawa couplings. In addition, the input value for the pole top mass is converted to \(m_t^{\overline{ \text{ MS }}}(m_t)\) using up to twoloop QCD and oneloop top Yukawa/electroweak corrections (corresponding to the higherorder corrections included in the Higgsboson mass calculation). Furthermore, the \(\tan \beta \)enhanced contributions are always included in the defining relation between the bottom Yukawa coupling and the bottom mass (and similarly for all other downtype quarks). Concerning the Higgs vev appearing in the relation between the Yukawa couplings and the fermion masses, we parametrize it in terms of \(\alpha (M_Z)\). Finally, the strong coupling constant employed in SUSYQCD diagrams is set to the scale of the supersymmetric particles entering the loop. We will comment on deviations from these settings if needed.
2.2 Higgs decays into SM fermions
Our calculation of the Higgs decay amplitudes into SM fermions closely follows the procedure outlined in the previous subsection. However, we include the QCD and QED corrections separately, making use of analytical formulae that are welldocumented in the literature [95, 96]. We also employ an effective description of the Higgs–\(b\bar{b}\) interactions in order to resum potentially large effects for large values of \(\tan \beta \). Below, we comment on these two issues and discuss further the derivation of the decay widths for this class of channels.
Here, \(Q_f\) is the electric charge of the fermion f, \(C_2(f)\) is equal to 4 / 3 for quarks and equal to 0 for leptons, \(M_{h_i}\) corresponds to the kinematic (pole) mass in the Higgs decay under consideration and the functions \(\Delta _{S,P}\) are explicated in e.g. Sect. 4 of Ref. [105]. In the limit of \(M_{h_i}\gg m_f\), both \(\Delta _{S,P}\) reduce to yy\(\left[ 3\log {\left( M_{h_i}/m_f\right) }+\frac{9}{4}\right] \). As noticed already in Ref. [95], the leading logarithm in the QCDcorrection factor can be absorbed by the introduction of a running \(\overline{\mathrm {MS}}\) fermion mass in the definition of the Yukawa coupling \(Y_f\). Therefore, it is motivated to factorize \(m_f^{\overline{ \text{ MS }}}(M_{h_i})\), with higher orders included in the definition of the QCD beta function.
The QCD (and QED) correction factors generally induce a sizable shift of the treelevel width of as much as \(\mathord {\sim }\,50\%\). While these effects were formally derived at the oneloop order, we apply them over the full amplitudes (without the QCD and QED corrections), i.e. we include the oneloop vertex amplitude without QCD / QED corrections Open image in new window and Open image in new window in the definitions of the couplings \(g_{h_jff}^{S,P}\) that are employed in Eq. (2.12) – we will use the notation \(g_{h_jff}^{S,P\,\text {1L}}\) below. The adopted factorization corresponds to a particular choice of the higherorder contributions beyond the ones that have been explicitly calculated.
The kinematic masses of the fermions are easily identified in the leptonic case. For decays into top quarks the ‘pole’ mass \(m_t\) is used, while for all other decays into quarks we employ the \(\overline{\text{ MS }}\) masses evaluated at the scale of the Higgs mass \(m_q^{\overline{ \text{ MS }}}(M_{h_i})\). We note that these kinematic masses have little impact on the decay widths, as long as the Higgs state is much heavier. In the NMSSM, however, singletlike Higgs states can be very light, in which case the choice of an \(\overline{\text{ MS }}\) mass is problematic. Yet, in this case the Higgs state is typically near threshold so that the freeparton approximation in the final state is not expected to be reliable. Our current code is not properly equipped to address decays directly at threshold independently of the issue of running kinematic masses. Improved descriptions of the hadronic decays of Higgs states close to the \(b\bar{b}\) threshold or in the chiral limit have been presented in e.g. Refs. [74, 75, 76, 106, 107, 108].
2.3 Decays into SM gauge bosons
Now we consider Higgs decays into the gauge bosons of the SM. Almost each of these channels requires a specific processing in order to include higherorder corrections consistently or to deal with offshell effects.
Decays into electroweak gauge bosons Higgs decays into onshell Ws and Zs can be easily included at the oneloop order in comparable fashion to the fermionic decays. However, the notion of WW or ZZ final states usually includes contributions from offshell gauge bosons as well, encompassing a wide range of fourfermion final states. Such offshell effects mostly impact the decays of Higgs bosons with a mass below the WW or ZZ thresholds. Instead of a full processing of the offshell decays at oneloop order, we pursue two distinct evaluations of the decay widths in these channels.
Our second approach consists in a oneloop calculation of the Higgs decay widths into onshell gauge bosons (see Ref. [94] for the MSSM case), including treelevel offshell effects. This evaluation is meant to address the case of heavy Higgs bosons at the full oneloop order. The restriction to onshell kinematics is justified above the threshold for electroweak gaugeboson production (offshell effects at the oneloop level could be included via a numerical integration over the squared momenta of the gauge bosons in the final state – see Refs. [109, 110] for a discussion in the MSSM). Our implementation largely follows the lines described in Sect. 2.1, with the noteworthy feature that contributions from Higgs–electroweak mixing Open image in new window vanish. In the case of the \(W^+W^\) final state, the QED IRdivergences are regularized with a photon mass and cancel with bremsstrahlung corrections: soft and hard bremsstrahlung are included according to Refs. [111, 112] (see also [94]). We stress that the exact cancellation of the IRdivergences is only achieved through the replacement of the \(h_iG^+G^\) coupling by the expression in terms of the kinematical Higgs mass, as discussed in Sect. 2.1. This fact had already been observed by Ref. [94]. In order to extend the validity of the calculation below threshold, we process the Bornorder term separately, applying an offshell kinematic integration over the squared external momentum of the gauge bosons – see e.g. Eq. (37) in Ref. [113]. Thus, this evaluation is performed at tree level below threshold and at full oneloop order (for the onshell case) above threshold. The vanishing onshell kinematical factor multiplying the contributions of oneloop order ensures the continuity of the prediction at threshold. Finally, we include the oneloop squared term in the calculation. Indeed, as we will discuss later on, the treelevel contribution vanishes for a decoupling doublet, meaning that the Higgs decays to WW / ZZ can be dominated by oneloop effects. To this end, the infrared divergences of twoloop order are regularized in an adhoc fashion – which appears compulsory as long as the twoloop order is incomplete – making use of the oneloop real radiation and estimating the logarithmic term in the imaginary part of the oneloop amplitude.
Radiative decays into gauge bosons Higgs decays into photon pairs, gluon pairs or \(\gamma Z\) appear at the oneloop level – i.e. Open image in new window for all these channels. We compute the oneloop order using the FeynArts model file, although the results are wellknown analytically in the literature – see e.g. Ref. [87] or Sect. III of Ref. [50] ([113] for the MSSM). The electromagnetic coupling in these channels is set to the value \(\alpha (0)\) corresponding to the Thomson limit.
The use of treelevel Higgs–Goldstone couplings together with loopcorrected kinematic Higgs masses \(M_{h_i}\) in our calculation would induce an effective violation of Ward identities by twoloop order terms in the amplitude: as explained in Sect. 2.1, we choose to restore the proper gauge structure by redefining the Higgs–Goldstone couplings in terms of the kinematic Higgs mass \(M_{h_i}\). Since our calculation is restricted to the leading – here, oneloop – order, the transition of the amplitude from treelevel to physical Higgs states is performed via \(\mathbf {U}^m\) or \(\mathbf {X}\) instead of \(\mathbf {Z}^{ \text{ mix }}\) in order to ensure the appropriate behavior in the decoupling limit.
Leading QCD corrections to the diphoton Higgs decays have received substantial attention in the literature. A frequently used approximation for this channel consists in multiplying the amplitudes driven by quark and squark loops by the factors \(\left[ 1\alpha _s(M_{h_i})/\pi \right] \) and \(\left[ 1+8\,\alpha _s(M_{h_i})/(3\,\pi )\right] \), respectively – see e.g. Ref. [114]. However, these simple factors are only valid in the limit of heavy quarks and squarks (compared to the mass of the decaying Higgs boson). More general analytical expressions can be found in e.g. Ref. [115]. In our calculation, we apply the correction factors \(\left[ 1+C^S{(\tau _q)}\,\alpha _s(M_{h_i})/\pi \right] \) and \(\left[ 1+C^P{(\tau _q)}\,\alpha _s(M_{h_i})/\pi \right] \) to the contributions of the quark q to the \(\mathcal{CP}\)even and the \(\mathcal{CP}\)odd \(h_i\gamma \gamma \) operators, respectively, and \(\Big [1+C{(\tau _{\tilde{Q}})}\,\alpha _s(M_{h_i})/\pi \Big ]\) to the contributions of the squark \(\tilde{Q}\) (to the \(\mathcal{CP}\)even operator). Here, \(\tau _X\) denotes the ratio \(\left[ 4\,m^2_X\right. \left. {(M_{h_i}/2)}/M^2_{h_i}\right] \). The coefficients \(C^{S,P}\) and C are extracted from Ref. [116] and Ref. [117]. In order to obtain a consistent inclusion of the \(\mathcal {O}{(\alpha _s)}\) corrections, the quark and squark masses \(m_X\) entering the oneloop amplitudes or the correction factors are chosen as defined in Eq. (5) of Ref. [116] and in Eq. (12) of Ref. [117] (rather than \(\overline{\mathrm {MS}}\) running masses).
The QCD corrections to the digluon decays include virtual corrections but also gluon and lightquark radiation. They are thus technically defined at the level of the squared amplitudes. In the limit of heavy quarks and squarks, the corrections are known beyond NLO – see the discussion in Ref. [113] for a list of references. The full dependence in mass was derived at NLO in Refs. [116, 117], for both quark and squark loops. In our implementation, we follow the prescriptions of Eqs. (51), (63) and (67) of Ref. [113] in the limit of light radiated quarks and heavy particles in the loop. For consistency, the masses of the particles in the oneloop amplitude are taken as pole masses. Effects beyond this approximation can be sizable, as evidenced by Fig. 20 of Ref. [116] and Fig. 12 of Ref. [117]. As the \(\mathcal{CP}\)even and \(\mathcal{CP}\)odd Higgs–gg operators do not interfere, it is straightforward to include both correction factors in the \(\mathcal{CP}\)violating case. Finally, we note that parts of the leading QCD corrections to \(h_i\rightarrow gg\) are induced by the real radiation of quark–antiquark pairs. In the case of the heavier quark flavors (top, bottom and possibly charm), the channels are experimentally welldistinguishable from gluonic decays. Therefore, the partial widths related to these corrections could be attached to the Higgs decays into quarks instead [118]. The resolution of this ambiguity would involve a dedicated experimental analysis of the kinematics of the gluon radiation in \(h_i\rightarrow gq\bar{q}\) (collinear or backtoback emission). In the following section, we choose to present our results for the \(h_i\rightarrow gg\) decay in the threeradiatedflavor approach, while the contributions from the heavier quark flavors are distributed among the \(h_i\rightarrow c\bar{c}\), \(b\bar{b}\), and \(t\bar{t}\) widths (provided the Higgs state is above threshold). These contributions to the fermionic Higgs decays are of \(\mathcal {O}{(\alpha _s^2)}\).
The QCD corrections to the quark loops of an SMHiggs decay into \(\gamma Z\) have been studied in Refs. [119, 120, 121], but we do not consider them here.
3 Numerical analysis
In this section, we present our results for the decay widths of the neutral Higgs bosons into SM particles in several scenarios and compare them with the predictions of existing codes. While a detailed estimate of the uncertainty associated to missing higherorder corrections goes beyond the scope of our analysis, we will provide some discussion at the end of this section, based on our observations and comparisons.
Throughout this section, the pole mass of the topquark is chosen as \(m_t = 173.2\) GeV. Moreover, all \(\overline{\mathrm {DR}}\) parameters are defined at the scale \(m_t\), and all stop parameters are treated as onshell parameters. Concerning 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 HiggsBounds 4.3.1 (and 5.1.1beta) [122, 123, 124, 125, 126, 127] and HiggsSignals1.3.1 (and 2.1.0beta) [127, 128, 129]. We refer the reader to the corresponding publications for a detailed list of experimental references. The input parameters employed in our scenarios are summarized in Table 1.
Input parameters for the scenarios considered in Sect. 3. The bilinear soft SUSYbreaking parameter of all the squarks of the third generation is denoted by \(m_{\tilde{Q}}\). Moreover, \(2\,M_1=M_2=M_3/5=500\) GeV and \(m_{\tilde{F}}=1.5\) TeV, where \(\tilde{F}\) represents any sfermion of the first two generations or sleptons of the third generation. Furthermore, \(A_t=A_t\exp {\left( \imath \,\phi _{A_t}\right) }\) and \(\kappa =\kappa \exp {\left( \imath \,\phi _\kappa \right) }\) in the conventions of Ref. [25]. We vary x in the interval [0, 1]
Sect. 3.1: comparison with FeynHiggs  

Fig. 1  Fig. 3  Fig. 4  Fig. 5  
#1  \(\lambda \)  \(1\cdot 10^{5}\)  \(1\cdot 10^{5}\)  \(1\cdot 10^{5}\)  \(1\cdot 10^{5}\)  
#2  \(\kappa \)  \(1\cdot 10^{5}\)  \(1\cdot 10^{5}\)  \(1\cdot 10^{5}\)  \(1\cdot 10^{5}\)  
#3  \(\phi _{\kappa }\)  0  0  0  0  
#4  \(\tan \beta \)  10  10  \(1+39 \cdot x\)  10  
#5  \(\mu _{ \text{ eff }}\) (GeV)  250  250  250  250  
#6  \(m_{H^{\pm }}\) (TeV)  1  \(0.15+1.85\cdot x\)  1  0.5  
#7  \(A_{\kappa }\) (GeV)  \(100\)  \(100\)  \(100\)  \(100\)  
#8  \(m_{\tilde{Q}}\) (TeV)  \(0.7+1.3\cdot x\)  1.5  1.5  1.5  
#9  \(A_{t}\) (TeV)  \(1.4+1.6\cdot x\)  2.3  2.3  2.5  
#10  \(\phi _{A_{t}}\)  0  0  0  \(\pi (2\cdot x1)\)  
#11  \(A_{b}\) (TeV)  \(1.4+1.6\cdot x\)  2.3  2.3  2.5  
Sect. 3.2: comparison with NMSSMCALC  Sect. 3.3: singlet Higgs at \(\mathord {\lesssim }\,100\) GeV  
Fig. 6  Fig. 8  Fig. 9  Fig. 11  Fig. 13  Fig. 15  Fig. 17  
#1  0.3  0.3  0.3  0.2  0.1  0.6  0.7 
#2  0.4  0.4  0.4  0.6  0.15  0.035  0.1 
#3  0  0  0  \(\pi (2\cdot x1)\)  0  0  \(\frac{\pi }{8}(2\cdot x1)\) 
#4  10  10  \(1+39 \cdot x\)  25  12  2  2 
#5  250  250  250  200  140  \(397+15\cdot x\)  500 
#6  1  \(0.15+1.85\cdot x\)  1  1  1.4  1  1.175 
#7  \(100\)  \(100\)  \(100\)  \(750\)  \(830+150\cdot x\)  \(325\)  \(70\) 
#8  \(0.7+1.3\cdot x\)  1.5  1.5  1.5  1.5  1  0.5 
#9  \(1.4+2.6\cdot x\)  3  3  2.5  2.5  0  0.1 
#10  0  0  0  \(\pi \)  \(\pi \)  0  0 
#11  \(1.4+2.6\cdot x\)  3  3  \(2.5\)  0.5  0  0.1 
3.1 Comparison with FeynHiggs in the MSSMlimit
The MSSM limit of the NMSSM is obtained at vanishingly small values of \(\lambda \) and \(\kappa \): the singlet superfield then decouples from the MSSM sector but the \(\mu _{\text {eff}}\) term remains relevant as long as \(\kappa \sim \lambda \). It is then possible to compare our results for the Higgs decays to the corresponding predictions of FeynHiggs 2.13.0. The settings of FeynHiggs are thus adjusted in order to match the level of higherorder contributions and renormalization conditions of our NMSSM mass calculation: the corresponding FeynHiggs input flags read FHSetFlags[4,0,0,3,0,2,0,0,1,1]. We will denote the MSSM(like) Higgs bosons as h and H for the \(\mathcal{CP}\)even states, and A for the \(\mathcal{CP}\)odd one.
In Fig. 1, we show the variations of the Higgs decay widths into \(b\bar{b}\) and \(t\bar{t}\) for the doublet states (when kinematically allowed). The solid lines correspond to our predictions in several approximations: at the ‘tree level’ with Yukawa couplings defined in terms of running \(\overline{\mathrm {MS}}\) quark masses at the scale of the physical Higgs mass (black); including QCD and QED corrections (but without SQCD contributions) as well as the transition to the physical Higgs state via \(\mathbf {Z}^{ \text{ mix }}\) (blue); replacing also the Higgs couplings to the downtype quarks by their effective form as expressed in Eq. (2.9) (green); at full one loop, with higherorder improvements as described above (red). The green dotted line is similar to the solid green line, up to the replacement of \(\mathbf {Z}^{ \text{ mix }}\) by its unitary approximation \(\mathbf {U}^m\). The purple diamonds are obtained with FeynHiggs. As the Higgs masses vary little over the range of the scan, the modification of the decay widths is essentially driven by radiative effects: this explains the relatively flat behavior of the treelevel results – at least in the case of the heavy states; for the light state, the mass and decay widths vary somewhat more. The same remark applies to the QCD/QEDcorrected widths with \(\mathbf {Z}^{ \text{ mix }}\), although the \(H\rightarrow t\bar{t}\) decay width displays a more pronounced variation due to the \(h^0\)–\(H^0\) mixing at the loop level. The oneloop corrections to the \(b\bar{b}\) decay width of the SMlike state shift this quantity upwards by \(\mathord {\sim }\,20\%\). The bulk of this effect, beyond the QCD corrections included in the running b quark mass, is already contained within the QCD/QEDcorrected width. For the heavy doublet states, however, QCD and QED corrections (beyond the effect encoded within the running Yukawa coupling) do not lead to a significant improvement of the prediction of the decay widths, as the ‘treelevel’ result often appears closer to the full oneloop widths than the blue curve – for the \(b\bar{b}\) final state, this deviation is only partially explained by the radiative corrections that can be resummed within the effective Higgs couplings to the bottom quark (solid green curve).
In the case of the \(t\bar{t}\) final state, the solid green curve and the blue curve are essentially identical as the effective couplings to the downtype quarks only play a secondary part. The difference between the solid and dotted green lines originates from the treatment regarding the transition matrix employed for the description of the external Higgs leg – \(\mathbf {Z}^{ \text{ mix }}\) or the approximation \(\mathbf {U}^m\): the consequences for the predicted width reach \(\mathcal {O}(5\%)\). We note that, with the exception of FeynHiggs, public tools usually neglect the effects associated with the momentum dependence of the Higgs selfenergies (only properly encoded within \(\mathbf {Z}^{ \text{ mix }}\)). On the other hand, the estimate using \(\mathbf {U}^m\) is somewhat closer to the full oneloop result (using \(\mathbf {Z}^{ \text{ mix }}\)), which means that, as long as one restricts to an ‘improved treelevel approximation’, the choice of a unitary transition matrix might provide slightly more reliable results than the same level of approximation with \(\mathbf {Z}^{ \text{ mix }}\). As we mentioned earlier, the \(\mathord {\sim }\,10\%\) difference between the red and green curves in the case of the heavy doublets hints at sizable electroweak effects of oneloop order. This can be understood in terms of large electroweak Sudakov logarithms for the heavy Higgs bosons. The impact of the sfermion spectrum on the decay widths into SM fermions consists in a suppression in the presence of light stops and sbottoms. This effect is of order \(10\%\) over the considered range of squark masses. Concerning the comparison with FeynHiggs, we observe a very good agreement of the predicted oneloop widths for the \(\mathcal{CP}\)even states: this is expected since we essentially apply the same processing of the parameters. However, a small discrepancy in the \(h\rightarrow b\bar{b}\) width is noticeable: it is related to our inclusion of the contribution from \(h\rightarrow g(g^*\rightarrow b\bar{b})\) to the decay width. Subtracting this contribution (dashed red curve), we recover the FeynHiggs prediction. This effect is negligible for the other Higgs states.^{6} Furthermore, for the \(\mathcal{CP}\)odd state, we find a deviation of a few percent in the \(t\bar{t}\) channel: this difference is due to FeynHiggs employing \(\mathcal{CP}\)even QCD/QEDcorrection factors instead of the \(\mathcal{CP}\)odd ones. The agreement is restored if we adopt the same approximation (dashed red curve). No such discrepancy appears for the \(b\bar{b}\) final state, as the \(\mathcal{CP}\)even and \(\mathcal{CP}\)odd QCD/QEDcorrection factors converge in the limit of very light fermions (as compared to the mass of the Higgs state). Furthermore, the processing of the effective Higgs couplings to downtype quarks differs between FeynHiggs and our implementation, leading to small numerical effects (below \(1\%\)) for light SUSY spectra: we evaluate \(\Delta _b\) at the scale defined by the arithmetic mean of the SUSY masses involved, while FeynHiggs employs a geometric mean (‘modified \(Y_b^{\text {eff}}\)’). Another (numerically minor) difference with FeynHiggs is the slightly different treatment of the Goldstone–Higgs couplings regarding the restoration of gauge invariance of the result – see Sect. 2.1.
We choose to discuss the diphoton and digluon decay widths in the MSSMlimit in a scenario where \(\tan \beta \) scans the range [1, 40] – \(m_{H^{\pm }}\) is set to 1 TeV again; the details of the input are available in Table 1. The mass of the SMlike Higgs state is of order 100 GeV at \(\tan \beta =1\), but settles in the interval [123.5, 126.5] when \(\tan \beta \gtrsim 7\). Correspondingly, the low\(\tan \beta \) limit is disfavored by HiggsSignals in this scenario. The heavy doublet states have a mass of about 996–997 GeV. The endpoints at large \(\tan \beta \sim 40\) are excluded by HiggsBounds due to constraints on heavyHiggs searches in the \(\tau \tau \) channel [130].
We display the Higgs decay widths into gg and \(\gamma \gamma \) in Fig. 4. The transition to the physical Higgs states is performed using various approximations: \(\mathbf {U}^0\) (black curves), \(\mathbf {U}^m\) (blue curves) and \(\mathbf {X}\) (red curves). These three descriptions agree rather well. The predictions for the digluon final state employ the QCD corrections for three radiated quark flavors – as radiated \(c\bar{c}\), \(b\bar{b}\) or \(t\bar{t}\) are regarded as contributions to the fermionic Higgs decays. Nevertheless, in order to compare with FeynHiggs, we also show the results in the fiveradiatedflavor approach (solid green curves) and cutting off universal QCD corrections beyond NLO (dashed green curves). The resulting deviation from the predictions of FeynHiggs (purple diamonds) is resolved when replacing the pole quark masses by \(\overline{\mathrm {MS}}\) masses in the oneloop amplitude (dotted green curves). The appropriate choice for the considered QCDcorrection factor is that of pole masses in the loop, however. The difference between the widths depicted by the black/blue/red lines and by the solid green lines is of order \(20\%\): this enhancement is due to the larger number of radiated flavors. Universal QCD corrections beyond NLO (difference between the solid and the dashed green curves) represent almost \(15\%\) of the width. In the case of the diphoton widths, the results of FeynHiggs (purple diamonds) should be compared to our predictions employing \(\mathbf {U}^m\) (blue curves): a deviation of somewhat less than \(10\%\) is noticeable for the heavy states. We checked that this discrepancy can be interpreted in terms of the heavyquark/squark approximation that FeynHiggs employs for the NLO QCD corrections as well as the use of \(\overline{\mathrm {MS}}\) running masses (instead of the running masses defined in Eq. (5) of Ref. [116]): simplifying our processing of the widths to this approximation (dashed blue curves) yields a very good agreement with the results of FeynHiggs.
Figure 5 displays the Higgs decay widths into \(b\bar{b}\) and WW. For the \(b\bar{b}\) final state, we consider the treelevel widths (corresponding to an \(\overline{\mathrm {MS}}\) running Yukawa coupling; black lines), incorporate QCD/QED corrections and transform to the physical Higgs state using \(\mathbf {Z}^{ \text{ mix }}\) (blue), subtract and redistribute the SUSY corrections to the bottomquark mass in the definition of the Higgsbottom couplings (green) and finally evaluate the widths at full oneloop order (red, including twoloop pieces as described in Sect. 2). The predictions of FeynHiggs (purple diamonds) are in good agreement with our full oneloop results. While the inclusion of the corrections associated to QED/QCD effects and the definition of effective Higgs couplings to \(b\bar{b}\) notably improve the treelevel width of the SMlike state, as compared to the oneloop results – from a discrepancy of \(\mathord {\sim }\,20\%\) to less than \(\mathord {\sim }\,4\%\) – the performance of these ‘leading’ corrections is less convincing in the case of the heavy doublet states: the deviations with respect to the full oneloop results remain of order 5–\(10\%\).
Turning to the WW final state, we expectedly recover the results of FeynHiggs in the approximation with the rescaled SM widths of Prophecy4f (black lines). The impact of the oneloop corrections (red curves) on the width of the mostly \(\mathcal{CP}\)even heavy doublet state \(h_2\) are rather mild, which we should put in perspective with the fact that this state is comparatively light. However, for the mostly \(\mathcal{CP}\)odd state \(h_3\), the treelevel approximations sizably underestimate the oneloop widths.
To summarize, in this comparison with FeynHiggs in the MSSM limit of the NMSSM, we were able to quantitatively recover the widths predicted by FeynHiggs and interpret the origin of the differences with our results. In the case of the Higgs decays into electroweak gauge bosons, our oneloop approach goes beyond the current approach of FeynHiggs and shows that the treelevel approximation for the SUSY contributions – even though rescaled from the Prophecy4f SM widths – leads to significant deviations for the heavy doublet states. In the case of the digluon decay, universal QCD corrections beyond NLO as well as the number of radiated quark flavors have a sizable impact on the widths. Finally, we observed that accounting for the mass dependence in the QCD corrections to the diphoton widths has a mild effect on the decay of the heavy states. It is planned to include all the refinements that go beyond the current status of FeynHiggs and that we have employed here into the predictions of the MSSM Higgs decays of a future update of FeynHiggs.
3.2 Comparison with NMSSMCALC
We will compare our estimates for the Higgs decay widths to the predictions of NMSSMCALC 2.1 [33, 34]. In order to minimize the impact of the Higgsmass calculation, we bypass the massevaluation routines of NMSSMCALC– we refer the reader to Refs. [20, 25] for a comparison of the Higgsmass predictions – and directly inject our Higgs spectrum (twoloop masses and \(\mathbf {U}^m\) mixing matrix) in the SLHA [131, 132] file serving as an interface between the massevaluation and the decayevaluation routines. We proceed similarly with the squark masses and mixing angles. The subroutine of NMSSMCALC that evaluates the Higgs decays is based on a generalization of HDECAY6.1 [26, 27]. The corresponding widths include leading NLO effects (e.g. QCD/QED corrections, effective Higgs couplings to the bottom quark, etc.) but do not represent a complete oneloop order evaluation. Furthermore, internal parameters, renormalization scales or RGE runnings are not strictly identical to our choices. For instance, the decay widths predicted by NMSSMCALC are systematically normalized to \(G_F\), while we employ other parametrizations in terms of \(M_W\), \(M_Z\) and e.g. \(\alpha (M_Z)\): the corresponding treelevel contributions numerically differ by a few percents. In the case of NMSSMCALC, such effects are of oneloop electroweak order, hence beyond the considered approximation. In our calculation, the oneloop electroweak order is consistently implemented with respect to our renormalization scheme. Other small numerical differences appear at the level of e.g. running quark masses. Therefore, the numerical comparison between the two sets of results is expected to show a certain level of deviations.
Finally, we consider a \(\mathcal{CP}\)violating scenario with the parameters \(\lambda =0.2\), \(\kappa =0.6\), \(\tan \beta =25\) and \(A_{\kappa }=750\) GeV. We scan over \(\phi _{\kappa }\in [\pi ,\pi ]\), which is phenomenologically realistic in the sense that EDMs in principle allow for large variations of this phase [133, 134]; however, scenarios with large \(\mathcal{CP}\)violating mixing of the Higgs states – as the one we consider – tend to be constrained. The Higgs spectrum consists of an SMlike state with a mass of \(\mathord {\sim }\,124.5\) GeV, a triplet of \(\mathcal{CP}\)even/\(\mathcal{CP}\)odd doublet and \(\mathcal{CP}\)even singlet states near \(\mathord {\sim }\,1\) TeV with large and fluctuating mixing depending on \(\phi _{\kappa }\), as well as a mostly \(\mathcal{CP}\)odd singlet at \(\mathord {\sim }\,1.2\) TeV. The masses are depicted in Fig. 10. The Higgs decays into \(b\bar{b}\) and WW are plotted in Fig. 11. For the \(b\bar{b}\) final state we find a relatively good agreement between the predictions of NMSSMCALC (purple diamonds) and our predictions employing the same approximation (shown in green: transition via \(\mathbf {U}^m\), QCD/QED corrections and effective SUSYcorrected Higgs–\(b\bar{b}\) couplings). A sizable discrepancy with our full oneloop result appears for \(\phi _{\kappa }\simeq 2.6\) in the case of \(h_{2,3}\): this difference originates in large Higgsmixing effects encoded within \(\mathbf {Z}^{ \text{ mix }}\) that are not captured by the \(\mathbf {U}^m\) approximation (see Sect. 3.3 and Sect. 3.4 of Ref. [25] for a detailed discussion). In the case of the WW channel, deviations can be large when the treelevel approximation fails to capture the leading contribution to the decay width (as e.g. for \(h_{2,3,4}\)): the predictions of NMSSMCALC are similar to our rescaled widths from Prophecy4f.
We have thus observed a qualitative agreement of our results with the decay widths predicted by NMSSMCALC, when employing the same approximations as this code. Our calculation goes beyond the approach of NMSSMCALC/HDECAY at the level of the Higgs decays into SM fermions or into WW / ZZ, since we consider the full oneloop order. Sizable differences may thus appear, e.g. in the case of the decays into electroweak gauge bosons. Concerning the decays into gluon or photon pairs, the two codes are considering the decay widths at the same order, and the results are similar.
3.3 A singletdominated state at \(\mathord {\lesssim }\,100\) GeV and possible explanation of slight excesses in the CMS and LEP data
The possible presence of a singletdominated state with mass in the ballpark of \(\mathord {\sim }\,100\) GeV is a longstanding phenomenological trademark of the NMSSM – see e.g. Refs. [78, 79]. One motivation for such a scenario is for instance the \(2.3\,\sigma \) local excess observed in Higgs searches at LEP in the \(e^+e^\rightarrow Z(H\rightarrow b\bar{b})\) channel [80], which would be consistent with a scalar mass of \(\mathord {\sim }\,98\) GeV (but with a rather coarse mass resolution). It would correspond to a signal strength with respect to the SM at the level of \(\mathord {\sim }\,10\%\). A natural candidate to explain this excess consists in a mostly singletlike Higgs with a doublet component of about \(10\%\) (mixing squared). Interestingly, recent LHC Run II results [81] for CMS Higgs searches in the diphoton final state show a local excess of \(\mathord {\sim }\,3\,\sigma \) in the vicinity of \(\mathord {\sim }\,96\) GeV, while a similar upward fluctuation of \(2\,\sigma \) had been observed in the CMS Run I data at a comparable mass. A hypothetical signal of this kind would amount to \(60\% \pm 20\%\) of that of an SM Higgs boson at the same mass. In the NMSSM, relatively large Higgs branching fractions into \(\gamma \gamma \) are possible due to the threestate mixing, in particular when the effective Higgs coupling to \(b\bar{b}\) becomes small – see e.g. Refs. [86, 87]. However, it then appears more difficult to interpret the LEP excess simultaneously.^{7} Below, we consider the decays of a light singletlike Higgs in this regime employing our calculation, to show that it is indeed possible to describe both ‘excesses’ simultaneously (without exploiting all possibilities within the NMSSM to describe these effects).
The Higgs properties for a few example points in Sect. 3.3 are shown. The input parameters are provided in Table 1. The Higgs width into \(xx'\) is denoted by \(\Gamma _{xx'}\), and the width normalized to the SM width at the same mass is represented by \(\widehat{\Gamma }_{xx'}\). The symbol \(\widehat{\text {BR}}_{xx'}\) represents the Higgs branching ratio into \(xx'\), normalized to the SM branching ratio at the same mass. The mass values and the partial decay widths are given in GeV. The variation in the values of \(M_{h_1}\) is an artifact of the scans performed for the three scenarios
Fig. 13  Fig. 15  Fig. 17  

\(h_1\)  \(h_2\)  \(h_1\)  \(h_2\)  \(h_1\)  \(h_2\)  
\(M_{h_i}\)  95.4  125.1  95.0  125.5  95.1  127.0 
\(\Gamma _{b\bar{b}}\)  \(1.9\cdot 10^{7}\)  \(2.4\cdot 10^{3}\)  \(1.4\cdot 10^{4}\)  \(2.1\cdot 10^{3}\)  \(4.0\cdot 10^{4}\)  \(1.6\cdot 10^{3}\) 
\(\Gamma _{\tau \tau }\)  \(3.8\cdot 10^{9}\)  \(2.7\cdot 10^{4}\)  \(1.5\cdot 10^{5}\)  \(2.4\cdot 10^{4}\)  \(4.3\cdot 10^{5}\)  \(1.8\cdot 10^{4}\) 
\(\Gamma _{c\bar{c}}\)  \(2.2\cdot 10^{6}\)  \(1.3\cdot 10^{4}\)  \(1.8\cdot 10^{5}\)  \(1.2\cdot 10^{4}\)  \(2.3\cdot 10^{5}\)  \(1.1\cdot 10^{4}\) 
\(\Gamma _{WW}\)  \(1.2\cdot 10^{7}\)  \(8.8\cdot 10^{4}\)  \(8.4\cdot 10^{6}\)  \(7.9\cdot 10^{4}\)  \(1.2\cdot 10^{6}\)  \(7.9\cdot 10^{4}\) 
\(\Gamma _{ZZ}\)  \(1.4\cdot 10^{8}\)  \(1.1\cdot 10^{4}\)  \(1.0\cdot 10^{7}\)  \(9.7\cdot 10^{5}\)  \(1.5\cdot 10^{7}\)  \(9.9\cdot 10^{5}\) 
\(\Gamma _{gg}\)  \(2.6\cdot 10^{6}\)  \(2.5\cdot 10^{4}\)  \(2.2\cdot 10^{5}\)  \(2.2\cdot 10^{4}\)  \(2.8\cdot 10^{5}\)  \(2.1\cdot 10^{4}\) 
\(\Gamma _{\gamma \gamma }\)  \(9.0\cdot 10^{8}\)  \(9.0\cdot 10^{6}\)  \(6.3\cdot 10^{7}\)  \(7.6\cdot 10^{6}\)  \(7.2\cdot 10^{7}\)  \(7.1\cdot 10^{6}\) 
\(\widehat{\Gamma }_{ZZ}\)  0.02  0.15  0.22  
\(\widehat{\Gamma }_{gg}\)  0.02  0.18  0.23  
\(\widehat{\text {BR}}_{b\bar{b}}\)  0.04  0.88  1.0  
\(\widehat{\text {BR}}_{\gamma \gamma }\)  11.8  2.2  1.0  
\(\xi _b\)  \(8.4\cdot 10^{4}\)  0.13  0.22  
\(\xi _{\gamma }\)  0.26  0.41  0.23 
We now turn to another scenario in the low\(\tan \beta \) regime with large \(\lambda \): the chosen input is provided in column ‘Fig. 15’ of Table 1. We vary \(\mu _{ \text{ eff }}\) in a narrow interval. The masses of the two lightest Higgs states are shown in Fig. 14. In the lower range of values for \(\mu _{ \text{ eff }}\), the mixing between the light singlet at \(\mathord {\sim }\,101\) GeV and the SMlike state at \(\mathord {\sim }\,121\) GeV almost vanishes. These points are in tension with the measured properties of the SMlike state, as tested with HiggsSignals, because of the relatively low mass of the SMlike state. However, with growing \(\mu _{ \text{ eff }}\), the mixing between the two light \(\mathcal{CP}\)even states increases, eventually pushing the singlet mass down to \(\mathord {\sim }\,90\) GeV and the mass of the SMlike state up to \(\mathord {\sim }\,128\) GeV. Consistency with the experimental results obtained on the observed state at 125 GeV is achieved for a mass of the SMlike state that is compatible with the LHC discovery within experimental and theoretical uncertainties. The \(\mathcal{CP}\)odd singlet has a mass of \(\mathord {\sim }\,150\) GeV, while the heavy doublet states are at \(\mathord {\sim }\,1\) TeV in this scenario. The decay properties of \(h_1\) are documented in Fig. 15, as well as in the column ‘Fig. 15’ of Table 2 (for a specific point). The branching ratio into \(b\bar{b}\) changes very significantly between the treelevel and the oneloop approach: again, the point of vanishing \(H_d^0\) component in \(h_1\) is shifted in parameter space from \(M_{h_1}\sim 98\) GeV to \(M_{h_1}\sim 101\) GeV. On the other hand, the \(H_u^0\) component in \(h_1\) vanishes at \(M_{h_1}\sim 101.5\) GeV, leading to a suppression of all the decay widths into gauge bosons at this mass. The magnitude of the estimated \(e^+e^\rightarrow Z(h_1\rightarrow b\bar{b})\) signal reaches \(\mathord {\sim }\,13\%\) of that of an SM Higgs at \(M_{h_1}\sim 95\) GeV, while \(pp\rightarrow h_1\rightarrow \gamma \gamma \) corresponds to more than \(40\%\) of an SM signal in the same mass range. In this example, \(\text {BR}[h_1\rightarrow \gamma \gamma ]\) (or \(\text {BR}[h_1\rightarrow gg]\)) is only moderately enhanced with respect to the SM branching fraction due to an \(H_u^0\)dominated doublet composition of \(h_1\), while \(\text {BR}[h_1\rightarrow b\bar{b}]\) remains dominant, albeit slightly suppressed. This scenario would thus simultaneously address the LEP and the CMS excesses in a phenomenologically consistent manner.
Finally, we consider a \(\mathcal{CP}\)violating scenario, still in the large\(\lambda \), low\(\tan \beta \) regime. We scan over \(\phi _{\kappa }\in \left[ \frac{\pi }{8},\frac{\pi }{8}\right] \). The lightest Higgs state is dominantly \(\mathcal{CP}\)odd, with a mass of \(\mathord {\sim }\,100\) GeV in the \(\mathcal{CP}\)conserving limit. The secondlightest Higgs is SMlike, with a mass of \(\mathord {\sim }\,120\) GeV for \(\phi _{\kappa }=0\). With increasing \(\phi _{\kappa }\), these states mix and the masses draw apart, reaching \(\mathord {\sim }\,60\) GeV and \(\mathord {\sim }\,150\) GeV at \(\phi _{\kappa }\sim \frac{\pi }{8}\), which can be seen in Fig. 16. Correspondingly, appropriate Higgs properties, as tested with HiggsBounds and HiggsSignals, are obtained for \(\phi _{\kappa }\simeq 0.1\)–0.2. The \(\mathcal{CP}\)even singlet and the heavy doublet states have masses of the order of 210 GeV and 1180 GeV, respectively. In Fig. 17, we show the branching ratios of the lightest Higgs state as a function of its mass. Contrarily to the previous cases, \(\text {BR}[h_1\rightarrow b\bar{b}]\) is nearly constant over the whole range; the treelevel and oneloop results agree reasonably well with each other. The branching ratios into gauge bosons show an abrupt decrease near \(M_{h_1}\sim 102\) GeV: this corresponds to the \(\mathcal{CP}\)conserving limit of our scenario – in which case \(h_1\) is a pure \(\mathcal{CP}\)odd state. The estimated signals in the \(e^+e^\rightarrow Z(h_1\rightarrow b\bar{b})\) and \(pp\rightarrow h_1\rightarrow \gamma \gamma \) channels reach \(\mathord {\sim }\,20\)–\(25\%\) of their SM counterparts at \(M_{h_1}\sim 95\) GeV. As can be seen in Table 2, the decays of \(h_1\) for this point (\(\phi _{\kappa }\sim 0.14\)) approximately stay in SMlike proportions. The \(b\bar{b}\) and \(\gamma \gamma \) signals would thus remain comparable in magnitude. In particular, a diphoton signal as large as \(60\%\) of the one for an SMHiggs boson could not be accommodated in this configuration, as the LEP limits on \(h_1\rightarrow b\bar{b}\) indirectly constrain the diphoton rate. Yet, it is remarkable that a mostly \(\mathcal{CP}\)odd Higgs state would trigger sizable signals at both LEP and the LHC.
3.4 Discussion concerning the remaining theoretical uncertainties
Below, we provide a summary of the main sources of theoretical uncertainties from unknown higherorder corrections applying to our calculation of the NMSSM Higgs decays. We do not discuss here the parametric theoretical uncertainties arising from the experimental errors of the input parameters. For the experimentally known SMtype parameters the induced uncertainties can be determined in the same way as for the SM case (see e.g. Ref. [21]). The dependence on the unknown SUSY parameters, on the other hand, is usually not treated as a theoretical uncertainty but rather exploited for setting indirect constraints on those parameters.
Higgs decays into quarks (\(\mathbf h _\mathbf i \rightarrow \mathbf q \bar{\mathbf{q }}\), \(\mathbf q =\mathbf c, b, t \)) In our evaluation, these decays have been implemented at full oneloop order, i.e. at QCD, electroweak and SUSY nexttoleading order (NLO). In addition, leading QCD logarithmic effects have been resummed within the parametrization of the Yukawa couplings in terms of a running quark mass at the scale of the Higgs mass. The Higgs propagatortype corrections determining the mass of the considered Higgs particle as well as the wave function normalization at the external Higgs leg of the process contain full oneloop and dominant twoloop contributions.

First, we should assess the magnitude of the missing QCD NNLO (twoloop) effects. We stress that there should be no large logarithms associated to these corrections, since these are already resummed through the choice of running parameters and the renormalization scale. For the remaining QCD pieces, we can directly consider the situation in the SM. In the case of the light quarks, the QCD contributions of higher order have been evaluated and amount to \(\mathord {\sim }\,4\%\) at \(m_{H}=120\) GeV (see e.g. Refs. [136, 137]). For the top quark, the uncertainty due to missing QCD NNLO effects was estimated to \(5\%\) [21].

Concerning the electroweak corrections, Fig. 1 suggests that the oneloop contribution is small – at the percent level – for an SMlike Higgs, which is consistent with earlier estimates in the SM [21]. For the heavy Higgs states, Fig. 1 indicates a larger impact of such effects – at the level of \(\mathord {\sim }\,10\%\) in the considered scenario. Assuming that the electroweak NNLO corrections are comparable to the squared oneloop effects, our estimate for pure electroweak higher orders in decays of heavy Higgs states reaches the percent level. In fact, for multiTeV Higgs bosons, the electroweak Sudakov logarithms may require a resummation. Furthermore, mixed electroweak–QCD contributions are expected to be larger than the pure electroweak NNLO corrections, adding a few more percent to the uncertainty budget. For light Higgs states, the electroweak effects are much smaller since the Sudakov logarithms remain of comparatively modest size.

Finally, the variations with the squark masses in Fig. 1 for the heavy doublet states show that the oneloop SUSY effects could amount to 5–\(10\%\) for a subTeV stop/sbottom spectrum. In such a case, the twoloop SUSY and the mixed QCD/electroweak–SUSY corrections may reach the percent level. On the other hand, for very heavy squark spectra, we expect to recover an effective singletextended TwoHiggsDoublet model (an effective SM if the heavy doublet and singlet states also decouple) at low energy. However, all the parameters of this lowenergy effective field theory implicitly depend on the SUSY radiative effects, since unsuppressed logarithms of SUSY origin generate terms of dimension \(\mathord {\le }\,4\) – e.g. in the Higgs potential or the Higgs couplings to SM fermions. On the other hand, the explicit dependence of the Higgs decay widths on SUSY higherorder corrections is suppressed for a large SUSY scale. In this case, the uncertainty from SUSY corrections reduces to a parametric effect, that of the matching between the NMSSM and the lowenergy Lagrangian – e.g. in the SMlimit, the uncertainty on the mass prediction for the SMlike Higgs continues to depend on SUSY logarithms and would indirectly impact the uncertainty on the decay widths.
Higgs decays into leptons Here, QCD corrections appear only at twoloop order in the Higgs propagatortype corrections as well as in the counterterms of the electroweak parameters and only from threeloop order onwards in the genuine vertex corrections. Thus, the theory uncertainty is expected to be substantially smaller than in the case of quark final states. For an SMlike Higgs, associated uncertainties were estimated to be below the percent level [23]. For heavy Higgs states, however, electroweak oneloop corrections are enhanced by Sudakov logarithms and reach the \(\mathord {\sim }\,10\%\) level for Higgs masses of the order of 1 TeV, so that the twoloop effects could amount to a few percent. In addition, light staus may generate a sizable contribution of SUSY origin, where the unknown corrections are of twoloop electroweak order.

In the SM, the uncertainty of Prophecy4f in the evaluation of these channels was assessed at the subpercent level below 500 GeV, but up to \(\mathord {\sim }\,15\%\) at 1 TeV [21]. For an SMlike Higgs, our Fig. 3 shows that the oneloop electroweak corrections are somewhat below \(10\%\), making plausible a subpercent uncertainty on the results employing Prophecy4f. On the other hand, the assumption that the decay widths for an NMSSM Higgs boson can be obtained through a simple rescaling of the result for the width in the SM by treelevel couplings, is in itself a source of uncertainties. We expect this approximation to be accurate only in the limit of a decoupling SMlike composition of the NMSSM Higgs boson. If these SMlike characteristics are altered through radiative corrections of SUSY origins or NMSSMHiggs mixing effects – both of which may still reach the level of several percent in a phenomenologically realistic setup – the uncertainty on the rescaling procedure for the decay widths should be of corresponding magnitude.

In the case of heavier states, Fig. 3 and Fig. 8 indicate that the previous procedure is unreliable in the mass range \(\mathord {\gtrsim }\,500\) GeV. In particular, for heavy doublets in the decoupling limit, radiative corrections dominate over the – then vanishing – treelevel amplitude, shifting the widths by orders of magnitude. In such a case, our oneloop calculation captures only the leading order and one can expect sizable contributions at the twoloop level: as we already mentioned in discussing Fig. 3, shifting the quark masses between pole and \(\overline{\mathrm {MS}}\) values – two legitimate choices at the oneloop order that differ in the treatment of QCD twoloop contributions – results in modifications of the widths of order \(\mathord {\sim }\,50\%\). On the other hand, one expects the decays of a decoupling heavy doublet into electroweak gauge bosons to remain a subdominant channel, so that a less accurate prediction may be tolerable. It should be noted, however, that the magnitude of the corresponding widths is sizably enhanced by the effects of oneloop order, which may be of interest regarding their phenomenological impact.

In the SM, the uncertainty on a Higgs decay into \(\gamma \gamma \) was estimated at the level of \(1\%\) in Ref. [21]: however, the corresponding calculation includes both QCD NLO and electroweak NLO corrections. In our case, only QCD NLO corrections (with full mass dependence) are taken into account. The comparison with NMSSMCALC in Fig. 9 provides us with a lower bound on the magnitude of electroweak NLO and QCD NNLO effects: both evaluations are at the same order but differ by a few percent. The uncertainty on the SUSY contribution should be considered separately, as light charginos or sfermions could have a sizable impact. In any case, we expect the accuracy of our calculation to perform at the level of \(\mathord {\gtrsim }\,4\%\) (the typical size of the deviations in Fig. 9).

In the case of the Higgs decays into gluons, for the SM prediction – including QCD corrections with full mass dependence and electroweak twoloop effects – an uncertainty of \(3\%\) from QCD effects and \(1\%\) from electroweak effects was estimated in Ref. [21]. In our case, the QCD corrections are only included in the heavyloop approximation, and NLO electroweak contributions have not been considered. Consequently, the uncertainty budget should settle above the corresponding estimate for the SM quoted above. In the case of heavy Higgs bosons, the squark spectrum could have a significant impact on the QCD twoloop corrections, as exemplified in Fig. 5 of Ref. [117].

For \(h_i\rightarrow \gamma Z\), QCD corrections are not available so far, so that the uncertainty should be above the \(\mathord {\sim }\,5\%\) estimated in the SM [21].

The mixing in the Higgs sector plays a central role in the determination of the decay widths. Following the treatment in FeynHiggs, we have considered \(\mathbf {Z}^{ \text{ mix }}\) in all our oneloop evaluations, as prescribed by the LSZ reduction. Most public codes consider a unitary approximation in the limit of the effective scalar potential (\(\mathbf {U}^0\) in our notation). The analysis of Ref. [25] and our discussion on Fig. 1 – employing \(\mathbf {U}^m\), a more reliable unitary approximation than \(\mathbf {U}^0\) – indicate that the different choices of mixing matrices may affect the Higgs decays by a few percent (and far more in contrived cases). However, even the use of \(\mathbf {Z}^{ \text{ mix }}\) is of course subject to uncertainties from unknown higherorder corrections. While the Higgs propagatortype corrections determining the mass of the considered Higgs boson and the wave function normalization contain corrections up to the twoloop order, the corresponding prediction for the mass of the SMlike Higgs still has an uncertainty at the level of about \(2\%\), depending on the SUSY spectrum.

In this paper, we confined ourselves to the evaluation of the Higgs decay widths into SM particles and did not consider the branching ratios. For the latter an implementation at the full oneloop order of many other twobody decays, relevant in particular for the heavy Higgs states, would be desirable, which goes beyond the scope of the present analysis. Furthermore, in order to consider the Higgs branching ratios at the oneloop order, we would have to consider threebody widths at the tree level, for instance \(h_i\rightarrow b\bar{b}Z\), since these are formally of the same magnitude as the oneloop effects for twobody decays. In addition, these threebody decays – typically real radiation of electroweak and Higgs bosons – exhibit Sudakov logarithms that would require resummation in the limit of heavy Higgs states.

At decay thresholds, the approximation of free particles in the final state is not sufficient, and a more accurate treatment would require the evaluation of finalstate interactions. Several cases have been discussed in e.g. Refs. [75, 76, 138].
4 Conclusions
In this paper, we have presented our evaluation of neutral Higgs decay widths into SM final states in the (\(\mathcal{CP}\)conserving or \(\mathcal{CP}\)violating) NMSSM. Full oneloop corrections have been included for all the considered channels, as well as higherorder QCD corrections to the decays that are generated at the radiative level. The inclusion of oneloop contributions to the decays into SM fermions or electroweak gauge bosons goes beyond the usual approximation amounting to a QCD/QEDcorrected tree level. In addition, QCD corrections to the digluon and diphoton decay widths have been carefully processed, including the mass dependence in the \(\gamma \gamma \) case and corrections beyond NLO in the gg case. In its current form, this stateoftheart implementation of the neutral Higgs decays into SM particles is available as a Mathematica package, but should also be integrated into the FeynHiggs code in the near future.
In order to illustrate this calculation of the Higgs decay widths, we have presented our results in several regimes of the parameter space of the NMSSM. In the MSSM limit, we were able to recover the predictions of FeynHiggs and trace the origins of deviations from our new results. In particular, we emphasized the relevance of oneloop contributions in the decays of heavy doublet states into electroweak gauge bosons, for which the usual estimates based on the treelevel Higgs–gauge couplings are not appropriate. Minor effects in the treatment of QCD and QED corrections have also been noted. Beyond the MSSM limit, we have compared our decay widths to the output of NMSSMCALC. We observed a qualitative agreement wherever this could be expected. We also gave an account of the various sources of theoretical uncertainties from higherorder corrections and discussed the achieved accuracy of our predictions.
As a phenomenological application, we investigated in particular the case of a mostly singletlike state with mass in the vicinity of \(\mathord {\lesssim }\,100\) GeV. The decays of such a state can be notably affected by suppressed couplings to down or uptype quarks which can occur in certain parameter regions as a consequence of the mixing between the different Higgs states. In particular, an additional Higgs boson \(h_i\) of this kind could manifest itself via signatures in the channels \(e^+e^ \rightarrow Z(h_i\rightarrow b\bar{b})\) and/or \(pp \rightarrow h_i \rightarrow \gamma \gamma \). The presence of such a light Higgs boson could thus explain the slight deviations from the SM predictions reported by LEP and CMS in those channels.
Footnotes
 1.
Within the MSSM (including complex parameters) the HiggstoHiggs decays are implemented into FeynHiggs at the full oneloop level [62, 63], and HiggstoSUSY decays have been calculated at the full oneloop level in Refs. [64, 65] (see also Ref. [66]). Moreover, the Z factors (as implemented in FeynHiggs) include corrections beyond one loop, corresponding to the Higgsboson mass calculation.
 2.
As the onshell kinematical factor multiplying the contributions of oneloop order vanishes at threshold, the predicted width remains a continuous function of the Higgs mass.
 3.
 4.
We denote the imaginary unit by \(\imath \).
 5.
 6.
The option for a similar processing of the \(h\rightarrow g(g^*\rightarrow q\bar{q})\) contributions will be integrated in an upcoming version of FeynHiggs.
 7.
For instance, the authors of Ref. [135] came to a negative answer when considering the Run I ‘excess’ together with Dark Matter constraints in a specific region of the parameter space.
 8.
While this work was in its finalizing stages, preliminary Run II results from ATLAS with 80 fb\(^{1}\) in the \(\gamma \gamma \) searches below 125 GeV were released [89]. No significant excess above the SM expectation was observed in the mass range [65, 110] GeV, while the limit on crosssection times branching ratio – after taking into account the acceptance corresponding to the fiducial crosssection used by ATLAS – still allows for the ‘excess’ seen by CMS.
Notes
Acknowledgements
We thank T. Hahn, M. Mühlleitner, P. Slavich, M. Spira and D. Stöckinger for fruitful discussions. The work of F. D. and S. H. was supported in part by the MEINCOP (Spain) under contract FPA201678022P, in part by the “Spanish Agencia Estatal de Investigación” (AEI) and the EU “Fondo Europeo de Desarrollo Regional” (FEDER) through the project FPA201678022P, and in part by the AEI through the grant IFT Centro de Excelencia Severo Ochoa SEV20160597. In addition, the work of S. H. is supported in part by the “Spanish Red Consolider MultiDark” FPA201790566REDC. During different stages of the project, S. P. acknowledges support by the Collaborative Research Center SFB676 of the DFG, “Particles, Strings and the early Universe” and by the ANR grant “HiggsAutomator” (ANR15CE310002). G. W. acknowledges support by the Collaborative Research Center SFB676 of the DFG, “Particles, Strings and the early Universe”.
References
 1.ATLAS Collaboration, Phys. Lett. B 716, 1 (2012). arXiv:1207.7214
 2.CMS Collaboration, Phys. Lett. B 716, 30 (2012). arXiv:1207.7235
 3.ATLAS Collaboration, CMS Collaboration, JHEP 08, 045 (2016), arXiv:1606.02266
 4.ATLAS Collaboration, CMS Collaboration, Phys. Rev. Lett. 114, 191803 (2015), arXiv:1503.07589
 5.CMS Collaboration, JHEP 11, 047 (2017). arXiv:1706.09936
 6.CMS Collaboration, (2015), CDS 2055167Google Scholar
 7.ATLAS Collaboration, (2015), CDS 2055248Google Scholar
 8.ATLAS Collaboration, M. Testa, Prospects on Higgs Physics at the HLLHC for ATLAS, 2017, ATLAS Higgs HLLHCGoogle Scholar
 9.CMS Collaboration, M. Cepeda, HIGGS @ HLLHC, 2017, CMS Higgs HLLHCGoogle Scholar
 10.H. Baer et al., (2013). arXiv:1306.6352
 11.K. Fujii et al., (2015). arXiv:1506.05992
 12.K. Fujii et al., (2017). arXiv:1702.05333
 13.G. MoortgatPick et al., Eur. Phys. J. C75, 371 (2015). arXiv:1504.01726 ADSGoogle Scholar
 14.H. Nilles, Phys. Rep. 110, 1 (1984)ADSGoogle Scholar
 15.H. Haber, G. Kane, Phys. Rep. 117, 75 (1985)ADSGoogle Scholar
 16.U. Ellwanger, C. Hugonie, A. Teixeira, Phys. Rep. 496, 1 (2010). arXiv:0910.1785 ADSMathSciNetGoogle Scholar
 17.M. Maniatis, Int. J. Mod. Phys. A 25, 3505 (2010). arXiv:0906.0777 ADSMathSciNetGoogle Scholar
 18.G. Degrassi, S. Heinemeyer, W. Hollik, P. Slavich, G. Weiglein, Eur. Phys. J. C 28, 133 (2003). arXiv:hepph/0212020 ADSGoogle Scholar
 19.F. Staub, P. Athron, U. Ellwanger, R. Gröber, M. Mühlleitner, P. Slavich, A. Voigt, Comput. Phys. Commun. 202, 113 (2016). arXiv:1507.05093 ADSMathSciNetGoogle Scholar
 20.P. Drechsel, R. Gröber, S. Heinemeyer, M. Mühlleitner, H. Rzehak, G. Weiglein, Eur. Phys. J. C 77, 366 (2017). arXiv:1612.07681 ADSGoogle Scholar
 21.A. Denner, S. Heinemeyer, I. Puljak, D. Rebuzzi, M. Spira, Eur. Phys. J. C 71, 1753 (2011). arXiv:1107.5909 ADSGoogle Scholar
 22.LHC Higgs Cross Section Working Group, CDS CERN2013004 (2013), arXiv:1307.1347
 23.LHC Higgs Cross Section Working Group, CERN Yellow Reports:Monographs 2017002(2016). arXiv:1610.07922
 24.J. Kim, H. Nilles, Phys. Lett. B 138, 150 (1984)ADSGoogle Scholar
 25.F. Domingo, P. Drechsel, S. Paßehr, Eur. Phys. J. C 77, 562 (2017). arXiv:1706.00437 ADSGoogle Scholar
 26.A. Djouadi, J. Kalinowski, M. Spira, Comput. Phys. Commun. 108, 56 (1998). arXiv:hepph/9704448 ADSGoogle Scholar
 27.Tools and Monte Carlo Working Group, (2010). arXiv:1003.1643
 28.A. Djouadi, J. Kalinowski, M. Mühlleitner, M. Spira, (2018). arXiv:1801.09506
 29.U. Ellwanger, J. Gunion, C. Hugonie, JHEP 02, 066 (2005). arXiv:hepph/0406215 ADSGoogle Scholar
 30.U. Ellwanger, C. Hugonie, Comput. Phys. Commun. 175, 290 (2006). arXiv:hepph/0508022 ADSGoogle Scholar
 31.F. Domingo, JHEP 06, 052 (2015). arXiv:1503.07087 ADSMathSciNetGoogle Scholar
 32.
 33.J. Baglio, R. Gröber, M. Mühlleitner, D. Nhung, H. Rzehak, M. Spira, J. Streicher, K. Walz, Comput. Phys. Commun. 185, 3372 (2014). arXiv:1312.4788 ADSGoogle Scholar
 34.
 35.B. Allanach, Comput. Phys. Commun. 143, 305 (2002). arXiv:hepph/0104145 ADSGoogle Scholar
 36.B. Allanach, P. Athron, L. Tunstall, A. Voigt, A. Williams, Comput. Phys. Commun. 185, 2322 (2014). arXiv:1311.7659 ADSGoogle Scholar
 37.B. Allanach, T. Cridge, Comput. Phys. Commun. 220, 417 (2017). arXiv:1703.09717 ADSGoogle Scholar
 38.M. Goodsell, S. Liebler, F. Staub, Eur. Phys. J. C 77, 758 (2017). arXiv:1703.09237 ADSGoogle Scholar
 39.W. Porod, Comput. Phys. Commun. 153, 275 (2003). arXiv:hepph/0301101 ADSGoogle Scholar
 40.W. Porod, F. Staub, Comput. Phys. Commun. 183, 2458 (2012). arXiv:1104.1573 ADSGoogle Scholar
 41.M. Goodsell, K. Nickel, F. Staub, Eur. Phys. J. C 75, 32 (2015). arXiv:1411.0675 ADSGoogle Scholar
 42.
 43.F. Staub, Comput. Phys. Commun. 181, 1077 (2010). arXiv:0909.2863 ADSGoogle Scholar
 44.F. Staub, Comput. Phys. Commun. 182, 808 (2011). arXiv:1002.0840 ADSGoogle Scholar
 45.F. Staub, Comput. Phys. Commun. 184, 1792 (2013). arXiv:1207.0906 ADSGoogle Scholar
 46.F. Staub, Comput. Phys. Commun. 185, 1773 (2014). arXiv:1309.7223 ADSGoogle Scholar
 47.
 48.G. Bélanger, V. Bizouard, F. Boudjema, G. Chalons, Phys. Rev. D 93, 115031 (2016). arXiv:1602.05495 ADSGoogle Scholar
 49.G. Bélanger, V. Bizouard, F. Boudjema, G. Chalons, Phys. Rev. D 96, 015040 (2017). arXiv:1705.02209 ADSGoogle Scholar
 50.G. Bélanger, V. Bizouard, G. Chalons, Phys. Rev. D 89, 095023 (2014). arXiv:1402.3522 ADSGoogle Scholar
 51.S. Heinemeyer, W. Hollik, G. Weiglein, Eur. Phys. J. C 9, 343 (1999). arXiv:hepph/9812472 ADSGoogle Scholar
 52.S. Heinemeyer, W. Hollik, G. Weiglein, Comput. Phys. Commun. 124, 76 (2000). arXiv:hepph/9812320 ADSGoogle Scholar
 53.M. Frank, T. Hahn, S. Heinemeyer, W. Hollik, H. Rzehak, G. Weiglein, JHEP 0702, 047 (2007). arXiv:hepph/0611326 ADSGoogle Scholar
 54.T. Hahn, S. Heinemeyer, W. Hollik, H. Rzehak, G. Weiglein, Phys. Rev. Lett. 112, 141801 (2014). arXiv:1312.4937 ADSGoogle Scholar
 55.H. Bahl, W. Hollik, Eur. Phys. J. C 76, 499 (2016). arXiv:1608.01880 ADSGoogle Scholar
 56.H. Bahl, S. Heinemeyer, W. Hollik, G. Weiglein, Eur. Phys. J. C 78, 57 (2018). arXiv:1706.00346 ADSGoogle Scholar
 57.See: http://feynhiggs.de/
 58.P. Drechsel, L. Galeta, S. Heinemeyer, G. Weiglein, Eur. Phys. J. C 77, 42 (2017). arXiv:1601.08100 ADSGoogle Scholar
 59.J. Küblbeck, M. Böhm, A. Denner, Comput. Phys. Commun. 60, 165 (1990)ADSGoogle Scholar
 60.T. Hahn, Comput. Phys. Commun. 140, 418 (2001). arXiv:hepph/0012260 ADSGoogle Scholar
 61.T. Hahn, M. PérezVictoria, Comput. Phys. Commun. 118, 153 (1999). arXiv:hepph/9807565 ADSGoogle Scholar
 62.K. Williams, G. Weiglein, Phys. Lett. B 660, 217 (2008). arXiv:0710.5320 ADSGoogle Scholar
 63.K. Williams, H. Rzehak, G. Weiglein, Eur. Phys. J. C 71, 1669 (2011). arXiv:1103.1335 ADSGoogle Scholar
 64.S. Heinemeyer, C. Schappacher, Eur. Phys. J. C 75, 198 (2015). arXiv:1410.2787 ADSGoogle Scholar
 65.S. Heinemeyer, C. Schappacher, Eur. Phys. J. C 75, 230 (2015). arXiv:1503.02996 ADSGoogle Scholar
 66.A. Fowler, G. Weiglein, JHEP 01, 108 (2010). arXiv:0909.5165 ADSGoogle Scholar
 67.S. Heinemeyer, W. Hollik, G. Weiglein, Eur. Phys. J. C 16, 139 (2000). arXiv:hepph/0003022, [,2367(2000)]ADSGoogle Scholar
 68.E. Fuchs, G. Weiglein, JHEP 09, 079 (2017). arXiv:1610.06193 ADSGoogle Scholar
 69.A. Bredenstein, A. Denner, S. Dittmaier, M. Weber, Phys. Rev. D 74, 013004 (2006). arXiv:hepph/0604011 ADSGoogle Scholar
 70.A. Bredenstein, A. Denner, S. Dittmaier, M. Weber, Nucl. Phys. Proc. Suppl. 160, 131 (2006). arXiv:hepph/0607060 ADSGoogle Scholar
 71.A. Bredenstein, A. Denner, S. Dittmaier, M. Weber, JHEP 02, 080 (2007). arXiv:hepph/0611234 ADSGoogle Scholar
 72.I. Bigi, V. Fadin, V. Khoze, Nucl. Phys. B 377, 461 (1992)ADSGoogle Scholar
 73.K. Melnikov, M. Spira, O. Yakovlev, Z. Phys. C 64, 401 (1994). arXiv:hepph/9405301 ADSGoogle Scholar
 74.M. Drees, K. Hikasa, Phys. Rev. D 41, 1547 (1990)ADSGoogle Scholar
 75.F. Domingo, U. Ellwanger, JHEP 06, 067 (2011). arXiv:1105.1722 ADSGoogle Scholar
 76.F. Domingo, JHEP 03, 052 (2017). arXiv:1612.06538 ADSGoogle Scholar
 77.F. Domingo, G. Weiglein, JHEP 04, 095 (2016). arXiv:1509.07283 ADSGoogle Scholar
 78.R. Dermisek, J. Gunion, Phys. Rev. D 77, 015013 (2008). arXiv:0709.2269 ADSGoogle Scholar
 79.G. Bélanger, U. Ellwanger, J. Gunion, Y. Jiang, S. Kraml, J. Schwarz, JHEP 01, 069 (2013). arXiv:1210.1976 ADSGoogle Scholar
 80.OPAL, DELPHI, LEP WG for Higgs boson searches, ALEPH, L3, Phys. Lett. B565, 61 (2003). arXiv:hepex/0306033
 81.CMS Collaboration, (2017). CDS 2285326Google Scholar
 82.A. Mariotti, D. Redigolo, F. Sala, K. Tobioka, (2017). arXiv:1710.01743
 83.A. Crivellin, J. Heeck, D. Müller, Phys. Rev. D 97, 035008 (2018). arXiv:1710.04663 ADSGoogle Scholar
 84.P. Fox, N. Weiner, (2017). arXiv:1710.07649
 85.G. Cacciapaglia, G. Ferretti, T. Flacke, H. Serodio, (2017). arXiv:1710.11142
 86.U. Ellwanger, Phys. Lett. B 698, 293 (2011). arXiv:1012.1201 ADSGoogle Scholar
 87.R. Benbrik, M. GomezBock, S. Heinemeyer, O. Stål, G. Weiglein, L. Zeune, Eur. Phys. J. C 72, 2171 (2012). arXiv:1207.1096 ADSGoogle Scholar
 88.U. Ellwanger, M. RodríguezVázquez, JHEP 02, 096 (2016). arXiv:1512.04281 ADSGoogle Scholar
 89.ATLAS collaboration, (2018). CDS 2628760Google Scholar
 90.S. Paßehr, G. Weiglein, Eur. Phys. J. C 78, 222 (2018). arXiv:1705.07909 ADSGoogle Scholar
 91.S. Borowka, S. Paßehr, G. Weiglein, Eur. Phys. J. C 78, 576 (2018). arXiv:1802.09886 ADSGoogle Scholar
 92.W. Hollik, E. Kraus, M. Roth, C. Rupp, K. Sibold, D. Stückinger, Nucl. Phys. B 639, 3 (2002). arXiv:hepph/0204350 ADSGoogle Scholar
 93.N. Baro, F. Boudjema, A. Semenov, Phys. Rev. D 78, 115003 (2008). arXiv:0807.4668 ADSGoogle Scholar
 94.P. González, S. Palmer, M. Wiebusch, K. Williams, Eur. Phys. J. C 73, 2367 (2013). arXiv:1211.3079 ADSGoogle Scholar
 95.E. Braaten, J. Leveille, Phys. Rev. D 22, 715 (1980)ADSGoogle Scholar
 96.M. Drees, K. Hikasa, Phys. Lett. B 240, 455 (1990). [Erratum: Phys. Lett. B262, 497 (1991)]ADSGoogle Scholar
 97.T. Banks, Nucl. Phys. B 303, 172 (1988)ADSGoogle Scholar
 98.L. Hall, R. Rattazzi, U. Sarid, Phys. Rev. D 50, 7048 (1994). arXiv:hepph/9306309 ADSGoogle Scholar
 99.R. Hempfling, Phys. Rev. D 49, 6168 (1994)ADSGoogle Scholar
 100.M. Carena, M. Olechowski, S. Pokorski, C. Wagner, Nucl. Phys. B 426, 269 (1994). arXiv:hepph/9402253 ADSGoogle Scholar
 101.M. Carena, D. Garcia, U. Nierste, C. Wagner, Nucl. Phys. B 577, 88 (2000). arXiv:hepph/9912516 ADSGoogle Scholar
 102.H. Eberl, K. Hidaka, S. Kraml, W. Majerotto, Y. Yamada, Phys. Rev. D 62, 055006 (2000). arXiv:hepph/9912463 ADSGoogle Scholar
 103.D. Noth, M. Spira, Phys. Rev. Lett. 101, 181801 (2008). arXiv:0808.0087 ADSGoogle Scholar
 104.D. Noth, M. Spira, JHEP 06, 084 (2011). arXiv:1001.1935 ADSGoogle Scholar
 105.A. Dabelstein, Nucl. Phys. B 456, 25 (1995). arXiv:hepph/9503443 ADSGoogle Scholar
 106.E. Fullana, M. SanchisLozano, Phys. Lett. B 653, 67 (2007). arXiv:hepph/0702190 ADSGoogle Scholar
 107.D. McKeen, Phys. Rev. D 79, 015007 (2009). arXiv:0809.4787 ADSGoogle Scholar
 108.M. Dolan, F. Kahlhoefer, C. McCabe, K. SchmidtHoberg, JHEP 03, 171 (2015). arXiv:1412.5174, [Erratum: JHEP 07, 103 (2015)]Google Scholar
 109.W. Hollik, J. Zhang, (2010). arXiv:1011.6537
 110.W. Hollik, J. Zhang, Phys. Rev. D 84, 055022 (2011). arXiv:1109.4781 ADSGoogle Scholar
 111.B. Kniehl, Nucl. Phys. B 357, 439 (1991)ADSGoogle Scholar
 112.B. Kniehl, Phys. Rep. 240, 211 (1994)ADSGoogle Scholar
 113.M. Spira, Prog. Part. Nucl. Phys. 95, 98 (2017). arXiv:1612.07651 ADSGoogle Scholar
 114.J. Lee, A. Pilaftsis, M. Carena, S. Choi, M. Drees, J. Ellis, C. Wagner, Comput. Phys. Commun. 156, 283 (2004). arXiv:hepph/0307377 ADSGoogle Scholar
 115.U. Aglietti, R. Bonciani, G. Degrassi, A. Vicini, JHEP 01, 021 (2007). arXiv:hepph/0611266 ADSGoogle Scholar
 116.M. Spira, A. Djouadi, D. Graudenz, P. Zerwas, Nucl. Phys. B 453, 17 (1995). arXiv:hepph/9504378 ADSGoogle Scholar
 117.M. Mühlleitner, M. Spira, Nucl. Phys. B 790, 1 (2008). arXiv:hepph/0612254 ADSGoogle Scholar
 118.A. Djouadi, M. Spira, P. Zerwas, Z. Phys. C 70, 427 (1996). arXiv:hepph/9511344 Google Scholar
 119.A. Djouadi, M. Spira, P. Zerwas, Phys. Lett. B 276, 350 (1992)ADSGoogle Scholar
 120.R. Bonciani, V. Del Duca, H. Frellesvig, J. Henn, F. Moriello, V. Smirnov, JHEP 08, 108 (2015). arXiv:1505.00567 ADSGoogle Scholar
 121.T. Gehrmann, S. Guns, D. Kara, JHEP 09, 038 (2015). arXiv:1505.00561 Google Scholar
 122.P. Bechtle, O. Brein, S. Heinemeyer, G. Weiglein, K. Williams, Comput. Phys. Commun. 181, 138 (2010). arXiv:0811.4169 ADSGoogle Scholar
 123.P. Bechtle, O. Brein, S. Heinemeyer, G. Weiglein, K. Williams, Comput. Phys. Commun. 182, 2605 (2011). arXiv:1102.1898 ADSGoogle Scholar
 124.P. Bechtle, O. Brein, S. Heinemeyer, O. Stål, T. Stefaniak, G. Weiglein, K. Williams, PoS 156, 024 (2012). arXiv:1301.2345 ADSGoogle Scholar
 125.P. Bechtle, O. Brein, S. Heinemeyer, O. Stål, T. Stefaniak, G. Weiglein, K. Williams, Eur. Phys. J. C 74, 2693 (2014). arXiv:1311.0055 ADSGoogle Scholar
 126.P. Bechtle, S. Heinemeyer, O. Stål, T. Stefaniak, G. Weiglein, Eur. Phys. J. C 75, 421 (2015). arXiv:1507.06706 ADSGoogle Scholar
 127.
 128.P. Bechtle, S. Heinemeyer, O. Stål, T. Stefaniak, G. Weiglein, Eur. Phys. J. C 74, 2711 (2014). arXiv:1305.1933 ADSGoogle Scholar
 129.P. Bechtle, S. Heinemeyer, O. Stål, T. Stefaniak, G. Weiglein, JHEP 11, 039 (2014). arXiv:1403.1582 ADSGoogle Scholar
 130.CMS Collaboration, (2015). CDS 2041463Google Scholar
 131.P. Skands et al., JHEP 07, 036 (2004). arXiv:hepph/0311123 ADSGoogle Scholar
 132.SUSY Les Houches Accord 2, Comput. Phys. Commun. 180, 8 (2009). arXiv:0801.0045
 133.K. Cheung, T. Hou, J. Lee, E. Senaha, Phys. Rev. D 84, 015002 (2011). arXiv:1102.5679 ADSGoogle Scholar
 134.S. King, M. Mühlleitner, R. Nevzorov, K. Walz, Nucl. Phys. B 901, 526 (2015). arXiv:1508.03255 ADSGoogle Scholar
 135.J. Cao, X. Guo, Y. He, P. Wu, Y. Zhang, Phys. Rev. D 95, 116001 (2017). arXiv:1612.08522 ADSGoogle Scholar
 136.P. Baikov, K. Chetyrkin, J. Kühn, Phys. Rev. Lett. 96, 012003 (2006). arXiv:hepph/0511063 ADSGoogle Scholar
 137.A. Kataev, JETP Lett. 66, 327 (1997). arXiv:hepph/9708292 ADSGoogle Scholar
 138.U. Haisch, J.F. Kamenik, A. Malinauskas, M. Spira, JHEP 03, 178 (2018). arXiv:1802.02156 Google Scholar
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