Status of the Higgs singlet extension of the standard model after LHC run 1
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Abstract
We discuss the current status of theoretical and experimental constraints on the real Higgs singlet extension of the standard model. For the second neutral (nonstandard) Higgs boson we consider the full mass range from \(1~\mathrm{GeV}\) to \(1~\mathrm{TeV}\) accessible at past and current collider experiments. We separately discuss three scenarios, namely, the case where the second Higgs boson is lighter than, approximately equal to, or heavier than the discovered Higgs state at around \(125~\mathrm{GeV}\). We investigate the impact of constraints from perturbative unitarity, electroweak precision data with a special focus on higherorder contributions to the \(W\) boson mass, perturbativity of the couplings as well as vacuum stability. The latter two are tested up to a scale of \(\sim \) \(4 \times 10^{10}\,\mathrm{GeV}\) using renormalization group equations. Direct collider constraints from Higgs signal rate measurements at the LHC and \(95\,\%\) confidence level exclusion limits from Higgs searches at LEP, Tevatron, and LHC are included via the public codes HiggsSignals and HiggsBounds, respectively. We identify the strongest constraints in the different regions of parameter space. We comment on the collider phenomenology of the remaining viable parameter space and the prospects for a future discovery or exclusion at the LHC.
1 Introduction
In this work, we consider the simplest extension of the SM Higgs sector, where an additional real singlet field is added, which is neutral under all quantum numbers of the SM gauge group [19, 20] and acquires a vacuum expectation value (VEV). This model has been widely studied in the literature [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48]. Here, we present a complete exploration of the model parameter space in the light of the latest experimental and theoretical constraints. We consider masses of the second (nonstandard) Higgs boson in the whole mass range up to \(1\,\mathrm{TeV}\), thus extending and updating the findings of previous work [41]. This minimal setup can be interpreted as a limiting case for more generic BSM scenarios, e.g. models with additional gauge sectors [49] or additional matter content [50, 51].
In our analysis, we study the implications of various constraints: We take into account bounds from perturbative unitarity and electroweak (EW) precision measurements, in particular focussing on higherorder corrections to the \(W\) boson mass [43]. Furthermore, we study the impact of requiring perturbativity, vacuum stability, and a correct minimization of the model up to a high energy scale using renormalization group evolved couplings.^{1} We include the exclusion limits from Higgs searches at the LEP, Tevatron and LHC experiments via the public tool HiggsBounds [52, 53, 54, 55], and use the program HiggsSignals [56] (cf. also Ref. [57]) to test the compatibility of the model with the signal strength measurements of the discovered Higgs state.
 (i)

the high mass region, \(m_H \in [130, 1000]\,\mathrm{GeV}\), where the lighter Higgs boson \(h\) is interpreted as the discovered Higgs state;
 (ii)

the intermediate mass region, where both Higgs bosons \(h\) and \(H\) are located in the mass region \([120,130]\,\mathrm{GeV}\) and potentially contribute to the measured signal rates, and
 (iii)

the low mass region, \(m_h \in [1,120]\,\mathrm{GeV}\), where the heavier Higgs boson \(H\) is interpreted as the discovered Higgs state.
The paper is organized as follows: in Sect. 2 we briefly review the model and the chosen parametrization. In Sect. 3 we elaborate upon the various theoretical and experimental constraints and discuss their impact on the model parameter space. In Sect. 4 a scan of the full model parameter space is presented, in which all relevant constraints are combined. This is followed by a discussion of the collider phenomenology of the viable parameter space. We summarize and conclude in Sect. 5.
2 The model
2.1 Potential and couplings
The real Higgs singlet extension of the SM is described in detail in Refs. [19, 20, 41, 58]. Here, we only briefly review the theoretical setup as well as the main features relevant to the work presented here.

the suppression of the production cross section of the two Higgs states induced by the mixing, which is given by \(\sin ^2\alpha \,(\cos ^2\alpha )\) for the heavy (light) Higgs, respectively;

the suppression of the Higgs decay modes to SM particles, which is realized if the competing decay mode \(H\rightarrow hh\) is kinematically accessible.
2.2 Model parameters
3 Theoretical and experimental constraints
 (1)
limits from perturbative unitarity,
 (2)
limits from EW precision data in form of the \(S,\,T,\,U\) parameters [59, 60, 61, 62] as well as the singlet–induced NLO corrections to the \(W\) boson mass as presented in Ref. [43],
 (3)
perturbativity of the couplings as well as the requirement on the potential to be bounded from below, Eqs. (4) and (5),
 (4)
limits from perturbativity of the couplings as well as vacuum stability up to a certain scale \(\mu _\text {run}\), where we chose \(\mu _\text {run}\,\sim \,4\times 10^{10}\,\mathrm{GeV}\) as benchmark point (these constraints will only be applied in the high mass region; see Sect. 3.3 for further discussion),
 (5)
upper cross section limits at \(95\,\%\) confidence level (CL) from null results in Higgs searches at the LEP, Tevatron and LHC experiments,
 (6)
consistency with the Higgs boson signal rates measured at the LHC experiments.
3.1 Perturbative unitarity
3.2 Perturbativity of the couplings
3.3 Renormalization group equation evolution of the couplings
First, the upper value of \(\tan \beta \) for fixed Higgs masses is determined by requiring perturbativity of \(\lambda _2\) as well as perturbative unitarity, cf. Sect. 3.1.
Second, the allowed range of the mixing angle \(\sin \alpha \) is determined by perturbativity of the couplings as well as the requirement of vacuum stability, especially when these are required at renormalization scales \(\mu _\text {run}\), which are significantly larger than the electroweak scale. Small mixings are excluded by the requirements of vacuum stability^{4} as well as minimization of the scalar potential. This corresponds to the fact that we enter an unstable vacuum for \(\mu _\text {run}\gtrsim \mu _\text {SM,bkdw}\) for \(\sin \alpha \,\sim \,0\).
In summary, the constraints from RGE evolution of the couplings pose the strongest bounds on the minimally allowed value \(\sin \alpha \) and the maximal value of \(\tan \beta \) in the high mass scenario. Note that, for lower \(m_H\), the \((\sin \alpha ,\,\tan \beta )\) parameter space is less constrained, as will be discussed in Sect. 4.1.
In the low mass scenario, i.e. where the heavier Higgs state is considered to be the discovered Higgs boson, none of the points in our scan fulfilled vacuum stability above the electroweak scale. This is due to the fact that for a relatively low \(m_h\), the value of \(\lambda _1\) at the electroweak scale is quite small, cf. Eq. (13). In the nondecoupled case, \(\sin \alpha \ne 1\), \(\lambda _1\) then receives negative contributions in the RG evolution toward higher scales, leading to \(\lambda _1(\mu _\text {run})\,\le \,0\) already at relatively low scales \(\mu _\text {run}\), corresponding to the breakdown of the electroweak vacuum. Hence, in the low mass scenario, the theory breaks down even earlier than in the SM case. In the analysis presented here, we will therefore refrain from taking the limits from RGE running into account in the low mass scenario. Then the theoretically maximally allowed value of \(\tan \beta \) is determined from perturbative unitarity and rises to quite large values, where we obtain \(\tan \beta _\text {max}\,\lesssim 50\), depending on the value of the light Higgs mass \(m_h\).
Further constraints on \(\tan \beta \) in the low mass scenario stem from the Higgs signal rate observables through the potential decay \(H\rightarrow hh\), as will be discussed in Sect. 3.6.
3.4 The \(W\) boson mass and electroweak oblique parameters \(S\), \(T\), \(U\)
Recently, the oneloop corrections to the \(W\) boson mass, \(m_W\), for this model have been calculated in Ref. [43]. In that analysis, \(m_W\) is required to agree within \(2\sigma \) with the experimental value \(m_W^\text {exp}= 80.385\,\pm \,0.015\,\mathrm{GeV}\) [69, 70, 71], leading to an allowed range for the purely singletinduced corrections of \(\Delta \,m_W^\text {sing}\,\in \,\left[ 5\,\mathrm{MeV};\,55\,\mathrm{MeV}\right] \). Theoretical uncertainties due to contributions at even higher orders have been estimated to be \(\mathcal {O}\left( 1\,\mathrm{MeV}\right) \). The oneloop corrections are independent^{5} of \(\tan \beta \) and give rise to additional constraints on \(\sin \alpha \), which in the high mass scenario turn out to be much more stringent [43] than the constraints obtained from the oblique parameters \(S\), \(T\), and \(U\) [59, 60, 61, 62].
In the low mass region, as discussed in Ref. [43], the NLO contributions within the Higgs singlet extension model even tend to decrease the current \(\sim \) \(20\,\mathrm{MeV}\) discrepancy between the theoretical value \(m_W\) in the SM [72] and the experimental measurement [69, 70, 71]. However, substantial reduction of the discrepancy only occurs if the light Higgs has a sizable doublet component. Hence, this possibility is strongly constrained by exclusion limits from LEP and/or LHC Higgs searches (depending on the light Higgs mass) as well as by the LHC Higgs signal rate measurements.
In the low mass region the electroweak oblique parameters pose nonnegligible constraints, as will be shown in Sect. 4.2. However, these constraints are again superseded once the Higgs signal strength as well as direct search limits from LEP are taken into account, cf. Sects. 3.5 and 3.6 respectively.
3.5 Exclusion limits from Higgs searches at LEP and LHC
Null results from Higgs searches at collider experiments limit the signal strength of the second, non SMlike Higgs boson. Recall that its signal strength is given by the SM Higgs signal rate scaled by \((\cos \alpha )^2\) in the low mass region and, in the absence of HiggstoHiggs decays, \((\sin \alpha )^2\) in the high mass region. Thus, the exclusion limits can easily be translated into lower or upper limits on the mixing angle \(\sin \alpha \), respectively.^{7}
We employ HiggsBounds4.2.0 [52, 53, 54, 55] to derive the exclusion limits from collider searches. The exclusion limits from the LHC experiments^{8} are usually given at the \(95\,\%~\mathrm {CL}\). For most of the LEP results we employ the \(\chi ^2\) extension [55] of the HiggsBounds package.^{9} The obtained \(\chi ^2\) value will later be added to the \(\chi ^2\) contribution from the Higgs signal rates, cf. Sect. 3.6, to construct a global likelihood.
In the presence of HiggstoHiggs decays, \(\mathrm {BR}(H \rightarrow hh)\ne 0\), additional constraints arise. In case of very low masses, \(m \,\lesssim \,3.5\,\mathrm{GeV}\), these stem from the CMS search in the \(H\rightarrow hh \rightarrow \mu ^+\mu ^\mu ^+\mu ^\) channel [94], and for large masses, \(m \in [260, 360]~\mathrm{GeV}\), from the CMS search for \(H\rightarrow hh\) with multileptons and photons in the final state [95]. These limits will be discussed separately in Sect. 4. Note that the limit from SM Higgs boson searches in the mass range \(m\gtrsim 250~\mathrm{GeV}\), as presented in Fig. 4b, will diminish in case of nonvanishing \(\mathrm {BR}(H \rightarrow hh)\) due to a suppression of the SM Higgs decay modes. We find in the full scan (see Sect. 4) that, in general, \(\mathrm {BR}(H \rightarrow hh)\) can be as large as \(\sim \) \(70\,\%\) in this model. Neglecting the correlation between \(\sin \alpha \) and \(\mathrm {BR}(H \rightarrow hh)\) for a moment, such large branching fractions could lead to a reduction of the upper limit on \(\sin \alpha \) obtained from SM Higgs searches by a factor of \(\sim \) \(1{/}\sqrt{1\text {BR}(H\rightarrow hh)}\,\lesssim \, 1.8\). However, once all other constraints (in particular from the NLO calculation of \(m_W\)) are taken into account, only \(\text {BR}({H\rightarrow hh})\) values of up to \(40\,\%\) are found; see Sect. 4.1, Fig. 11b. Moreover, in the mass region \(m_H \sim \) 270–360 \(\mathrm{GeV}\) where the largest values of \(\text {BR}(H\rightarrow hh)\) appear, the \(m_W\) constraint on \(\sin \alpha \) is typically stronger than the constraints from SM Higgs searches, even if \(\text {BR}(H\rightarrow hh) = 0\) is assumed in the latter. Therefore, given the present Higgs search exclusion limits, the signal rate reduction currently does not have a visible impact on the viable parameter space.^{10}
3.6 Higgs boson signal rates measured at the LHC
Higgs boson signal rate and mass observables from the LHC experiments, as implemented in HiggsSignals1.3.0 and used in this analysis. For the mass measurements we combined the systematic and statistical uncertainty in quadrature
Experiment  Channel  Obs. signal rate  Obs. mass (\(\mathrm{GeV}\)) 

ATLAS  \( h\rightarrow WW\rightarrow \ell \nu \ell \nu \) [96]  \( 1.08\begin{array}{c} + 0.22\\  0.20 \end{array}\)  – 
ATLAS  \( h\rightarrow ZZ\rightarrow 4\ell \) [5]  \( 1.44\begin{array}{c} + 0.40\\  0.33 \end{array}\)  \(124.51 \pm 0.52\) 
ATLAS  \( h\rightarrow \gamma \gamma \) [4]  \( 1.17\begin{array}{c} + 0.27\\  0.27 \end{array}\)  \(125.98 \pm 0.50\) 
ATLAS  \( h\rightarrow \tau \tau \) [97]  \( 1.42\begin{array}{c} + 0.43\\  0.37 \end{array}\)  – 
ATLAS  \( Vh\rightarrow V(b\bar{b})\) [6]  \( 0.51\begin{array}{c} + 0.40\\  0.37 \end{array}\)  – 
CMS  \( h\rightarrow WW\rightarrow \ell \nu \ell \nu \) [10]  \( 0.72\begin{array}{c} + 0.20\\  0.18 \end{array}\)  – 
CMS  \( h\rightarrow ZZ\rightarrow 4\ell \) [9]  \( 0.93\begin{array}{c} + 0.29\\  0.25 \end{array}\)  \(125.63 \pm 0.45 \) 
CMS  \( h\rightarrow \gamma \gamma \) [7]  \( 1.14\begin{array}{c} + 0.26\\  0.23 \end{array}\)  \(124.70 \pm 0.34\) 
CMS  \( h\rightarrow \tau \tau \) [8]  \( 0.78\begin{array}{c} + 0.27\\  0.27 \end{array}\)  – 
CMS  \( Vh\rightarrow V(b\bar{b})\) [8]  \( 1.00\begin{array}{c} + 0.50\\  0.50 \end{array}\)  – 
When both Higgs masses are fixed, the fit depends on two free parameters, namely \(\sin \alpha \) and \(\tan \beta \). The latter can only influence the signal rates of the Higgs boson \(H\) if the additional decay mode \(H\rightarrow hh\) is accessible. The branching fraction \(\mathrm {BR}(H\rightarrow hh)\) then leads to a decrease of all other decay modes and hence to a reduction of the predictions for the measured signal rates, cf. Eq. (16). The sensitivity to \(\tan \beta \) via the signal rate measurements is thus only given if the heavier Higgs state is interpreted as the discovered particle, \(m_H \sim 125\,\mathrm{GeV}\) (low mass region), and the second Higgs state is sufficiently light, \(m_h \lesssim 62\,\mathrm{GeV}\). If the \(H\rightarrow hh\) decay is not kinematically accessible, or in the case where the light Higgs is considered as the discovered Higgs state at \(\sim \) \(125\,\mathrm{GeV}\), there are no relevant experimental constraints on \(\tan \beta \).
In Fig. 5b we show a zoom of the (\(\sin \alpha \), \(\tan \beta \)) plane, focusing on the low\(\mathrm {BR}(H\rightarrow hh)\) valley and \(\sin \alpha \) values close to \(1\). We furthermore indicate the parameter regions which are allowed at the \(1\), \(2\), and \(3\sigma \) level by the Higgs signal rate measurements by the gray contour lines. The maximally values of \(\mathrm {BR}(H\rightarrow hh) \approx 26\,\%\) allowed by the Higgs signal rate measurements at \(95\,\%~\mathrm {CL}\) are found for \(\sin \alpha \) very close to \(1\) and large \(\tan \beta \) values, i.e. in the vicinity of case (ii) discussed above. In the given example with \(m_h =50~\mathrm{GeV}\), the \(95\,\%~\mathrm {CL}\) exclusion from LEP searches, as discussed in Sect. 3.5 (cf. Fig. 4a), imposes \(\sin \alpha \lesssim 0.985\) and is indicated in Fig. 5b by the green, dashed line.
We now want to draw the attention to the intermediate mass range, where both mass eigenstates can contribute to the signal strength measurements at the LHC. If the masses of the two Higgs bosons are well separated, the signal yields measured in the LHC Higgs analyses can be assumed to be solely due to the one Higgs boson lying in the vicinity of the signal, \(m\sim 125\,\mathrm{GeV}\). However, in analyses with a poor mass resolution, as is typically the case in search analyses for the decay modes \(H\rightarrow W^+W^\), \(H\rightarrow \tau ^+\tau ^\), and \(VH\rightarrow b\bar{b}\), the signal contamination from the second Higgs boson needs to be taken into account if its mass is not too far away from \(125~\mathrm{GeV}\). While a proper treatment of this case can only be done by the experimental analyses, HiggsSignals employs a Higgs boson assignment procedure to approximately account for this situation [56]. Based on the experimental mass resolution of the analysis and the difference between the predicted mass and the mass position where the measurement has been performed, HiggsSignals decides whether the signal rates of multiple Higgs states need to be combined. Hence, superpositions of the two Higgs signal rates considered here are possible if the second Higgs mass lies within \(100~\mathrm{GeV}\lesssim m \lesssim 150~\mathrm{GeV}\).
In the case of nearly massdegenerate Higgs bosons, \(m_h \approx m_H = 125.14~\mathrm{GeV}\), the sensitivity on the mixing angle \(\sin \alpha \) significantly decreases, as the signal rates of the two Higgs states are always superimposed. There remains a slight dependence of the total signal rate on the Higgs masses, though, since the production cross sections and branching ratios are mass dependent. Moreover, depending on the actual mass splitting and mixing angle, potential effects may possibly be seen in the invariant mass distributions of the highresolution LHC channels \(pp \rightarrow H\rightarrow \gamma \gamma \) [7] and \(pp\rightarrow H\rightarrow ZZ^*\rightarrow 4\ell \), at a future linear collider like the ILC [18, 42] or eventually a muon collider [42, 98]. However, the sensitivity on \(\sin \alpha \) completely vanishes in the case of exact mass degeneracy, \(m_h = m_H\), such that the singletextended SM becomes indistinguishable from the SM.
The weak \(\Delta \chi ^2\) dependence on \(m\) outside of the massdegenerate region, i.e. for \(m \gtrsim 128~\mathrm{GeV}\) and \(m\lesssim 122~\mathrm{GeV}\), is caused by the superposition of the signal rates of both Higgs bosons in some of the \(H\rightarrow W^+W^, \tau ^+\tau ^\), and \(b\bar{b}\) channels, as discussed above. These structures depend on the details of the implementation within HiggsSignals, in particular on the assumed experimental resolution for each analysis. For Higgs masses \(m\) below \(100~\mathrm{GeV}\) and beyond around \(152~\mathrm{GeV}\) the \(\sin \alpha \) limit from the signal rates is independent^{11} of \(m\).
We see that for Higgs masses \(m\) in the range between \(\sim \) \(100\) and \(150\,\mathrm{GeV}\), the constraints from the Higgs signal rates are more restrictive than the exclusion limits from Higgs searches at LEP and LHC. For lower Higgs masses, \(m < 100~\mathrm{GeV}\), the LEP limits (cf. Fig. 4a) generally yield stronger constraints on the parameter space. For higher Higgs masses, \(m \in [150,500]~\mathrm{GeV}\), the direct LHC limits (cf. Fig. 4b) are slightly stronger than the constraints from the signal rates, however, this picture reverses again for Higgs masses beyond \(500~\mathrm{GeV}\), where direct heavy Higgs searches become less sensitive.
4 Results of the full parameter scan
4.1 High mass region
In this section, we explore the parameter space of the high mass region, \(m \in [130, 1000]\,\mathrm{GeV}\). In general, for masses \(m \ge 600~\mathrm{GeV}\), our results agree with those presented in Ref. [41]. However, we obtain stronger bounds on the maximally allowed value of \(\sin \alpha \) due to the constraints from the NLO calculation of \(m_W\) [43], which has not been available for the previous analysis [41]. As has been discussed in Sect. 3.4, Fig. 3, the constraints from \(m_W\) are much more stringent than those obtained from the oblique parameters \(S\), \(T\), and \(U\) in the high mass region.
We compile all previously discussed constraints on the maximal mixing angle in Fig. 8. Furthermore, the (onedimensional) allowed regions in \(\sin \alpha \) and \(\tan \beta \) are given in Table 2 for fixed values of \(m\).^{12} Here, the allowed range of \(\sin \alpha \) is evaluated for fixed \(\tan \beta = 0.15\) and we explicitly specify the relevant constraint that provides in the upper limit on \(\sin \alpha \). We find the following generic features: for Higgs masses \(m \gtrsim \) 200–300 \(\mathrm{GeV}\), the \(W\) boson mass NLO calculation provides the upper limit on \(\sin \alpha \), at lower masses the LHC constraints at \(95\,\%~\mathrm {CL}\) from direct Higgs searches and the signal rate measurements are most relevant. The purely theorybased limits from perturbativity of \(\lambda _1\) only become important for \(m\gtrsim 800\,\mathrm{GeV}\). Furthermore, in the whole mass range, the minimal value of \(\sin \alpha \) and the maximal value of \(\tan \beta \) are determined by vacuum stability and perturbativity of the couplings.

light gray points include all scan points which are not further classified,

dark gray points fulfill constraints from perturbative unitarity, perturbativity of the couplings, RGE running and the \(W\) boson mass, as discussed in Sects. 3.1–3.4,

blue points additionally pass the \(95\,\%~\mathrm {CL}\) exclusion limits from Higgs searches,

red/yellow points fulfill all criteria above and furthermore lie within a \(1/\,2\,\sigma \) regime favored by the Higgs signal rate observables.
We now turn to the discussion of the collider phenomenology of the high mass region. Experimentally, the model can be probed by searches for a SMlike Higgs boson with a reduced signal rate and total decay widths, or by direct searches for the HiggstoHiggs decay mode \(H\rightarrow hh\), where \(h\) is the light Higgs boson at around \(125~\mathrm{GeV}\).
We show the allowed values of the branching ratio \(\text {BR} (H\rightarrow hh)\), given by Eq. (16), in Fig. 11. In Fig. 11a we show the dependence on \(\sin \alpha \) exemplarily for fixed Higgs masses \(m_H\), whereas the full \(m_H\) dependence is displayed in Fig. 11b, using the same color code as above. We observe that the maximal values of \(\text {BR} (H\rightarrow hh)\) are \(\sim \) \(40\,\%\), reached for large, positive \(\sin \alpha \) values [41], and low Higgs masses \(m \sim 300~\mathrm{GeV}\). At higher Higgs masses the branching ratio \(\text {BR} (H\rightarrow hh)\) is around \(20\,\%\) or slightly higher.
In general, the total width of the heavy Higgs boson is of high interest for collider searches. In the SM, the width of the SM Higgs boson rapidly rises with its mass. In Ref. [41] it was shown that in the singletextended SM the total width of the heavy resonance, \(\Gamma _\text {tot}\left( m_H \right) \), is highly suppressed due to the small mixing angle required. The same behavior is observed here. We show the ratio \(\Gamma _\text {tot} / m_H\), as well as the suppression of the width, \(\Gamma _\text {tot}/\Gamma _\text {SM}\), in Fig. 14. We see that the total width of the heavy Higgs only amounts to up to \(\sim \)20–25 % at lower masses \(m_H\lesssim 200~\mathrm{GeV}\), while it is even further suppressed to below 5–15 % of the SM Higgs width for masses \(m_H > 300~\mathrm{GeV}\). At \(m_H = 1000~\mathrm{GeV}\), the total width is still below \(25~\mathrm{GeV}\). In comparison to SM Higgs boson of the same mass, the total width of these resonances is therefore highly suppressed, which promises to enhance the validity of a narrow width approximation in this mass range.^{13}
4.2 Low mass region
Limits on \(\sin \alpha \) and \(\tan \beta \) in the low mass scenario for various light Higgs masses \(m_h\). The limits on \(\sin \alpha \) have been determined at \(\tan \beta =1\). The lower limit on \(\sin \alpha \) stemming from exclusion limits from LEP or LHC Higgs searches evaluated with HiggsBounds is given in the second column. If the lower limit on \(\sin \alpha \) obtained from the test against the Higgs signal rates using HiggsSignals results in stricter limits, we display them in the third column. The upper limit on \(\tan \beta \) in the fourth column stems from perturbative unitarity for the complete decoupling case (\(\sin \alpha = 1\)), cf. Fig. 1. In the fifth column we give the \(\tan \beta \) value for which \(\Gamma _{H\rightarrow hh}=0\) is obtained, given the maximal mixing angle allowed by the Higgs exclusion limits (second column). At this \(\tan \beta \) value, the \(\sin \alpha \) limit obtained from the Higgs signal rates (third column) is abrogated
\(m_h~(\mathrm{GeV})\)  \(\sin \alpha _\text {min, {HB}}\)  \(\sin \alpha _\text {min, {HS}}\)  \((\tan \beta )_\text {max}\)  \((\tan \beta )_{\text {no}~H\rightarrow hh} \) 

\(120\)  0.410  0.918  8.4  – 
\(110\)  0.819  \({0.932}\)  9.3  – 
\(100\)  0.852  \({0.891}\)  10.1  – 
\(90\)  0.901  –  11.2  – 
\(80\)  0.974  –  12.6  – 
\(70\)  0.985  –  14.4  – 
\(60\)  0.978  \({0.996}\)  16.8  0.21 
\(50\)  0.981  \({0.998}\)  20.2  0.20 
\(40\)  0.984  \({0.998}\)  25.2  0.18 
\(30\)  0.988  0.998  33.6  0.16 
\(20\)  0.993  0.998  50.4  0.12 
\(10\)  0.997  0.998  100.8  0.08 

Light gray points which fail theoretical constraints.

Dark gray points which are excluded by LHC Higgs searches.

Blue points allowed by LHC Higgs searches, but excluded by \(>\) \(95\,\%~\mathrm {CL}\) by LEP searches.

Dark green points consistent with LEP constraints within \(2\sigma \).

Light green points consistent with LEP constraints within \(1\sigma \).

Yellow points favored within \(2\sigma \) in the global fit (HiggsSignals \(\chi ^2\) \(+\) LEP \(\chi ^2\)).

red points favored within \(1\sigma \) in the global fit (HiggsSignals \(\chi ^2\) \(+\) LEP \(\chi ^2\)).
In Fig. 16b, we observe a drastic change in the distribution of allowed parameter points when going to Higgs masses \(m_h < m_H/2 \approx 62~\mathrm{GeV}\), where the decay mode \(H\rightarrow hh\) becomes kinematically accessible. As discussed earlier in Sect. 3.6, cf. Fig. 5, the decay \(H\rightarrow hh\) easily becomes the dominant decay mode if \(\tan \beta \gtrsim 1\), unless the mixing angle is very close to \(\sin \alpha  = 1\). Hence, for \(m_h < m_H/2\), most of the allowed points are found for small values of \(\tan \beta \), since the Higgs signal rates favor small values of \(\mathrm {BR}(H\rightarrow hh)\). At larger Higgs masses, \(m_h > m_H /2\), the favored points are equally distributed over the entire \(\tan \beta \) range allowed by perturbative unitarity.
It is interesting to investigate the allowed range of the \(H\rightarrow hh\) signal rate in dependence of the light Higgs mass. This is shown in Fig. 17a, where the signal rate is normalized to the SM Higgs boson production. Note that, due to the LEP constraints, the favored points feature a mixing angle \(\sin \alpha  \approx 1\) and thus the displayed signal rate closely resembles \(\mathrm {BR}(H\rightarrow hh)\). We see that the maximally allowed \(H\rightarrow hh\) signal rate is about \(22\,\%\) and is roughly independent on the light Higgs mass.^{14} This upper limit solely stems from the observed signal rates of the SM–like Higgs boson at \(\sim \) \(125~\mathrm{GeV}\). These constraints therefore also limit the total width of the heavy Higgs at \(125.14\,\mathrm{GeV}\) to values \(\sim \)3–5 \(\mathrm MeV\), being in the vicinity of the SM total width of \(\sim \)4.1 \(\mathrm MeV\).
We now discuss the case of very low Higgs masses, \(m_h\lesssim 4~\mathrm{GeV}\). Here, the LEP constraints stem from the decaymode independent analysis of \(e^+e^ \rightarrow Zh\) by OPAL [93], yielding a slightly weaker limit on the mixing angle, \(\sin \alpha  \gtrsim 0.965\), than at larger masses, cf. Fig. 4a. In the mass region \(m_h \in [1, 3]~\mathrm{GeV}\), the branching fraction for the light Higgs decay \(h\rightarrow \mu \mu \) amounts between 3–6 %, thus allowing to search for the signature \((pp)\rightarrow H\rightarrow hh \rightarrow \mu ^+\mu ^ \mu ^+\mu ^\) at the LHC. We show the predicted signal rate for this signature^{15} for the LHC at a centerofmass energy of \(8~\mathrm{TeV}\) in Fig. 17b. A search for this signature has been performed by CMS [94], yielding the observed upper limit^{16} displayed as magenta line in the figure. As can be seen, the CMS limit provides competitive constraints in this parameter region, excluding a sizable amount of the parameter region favored by the global fit. Future LHC searches for the \(4\mu \) signature therefore have a good discovery potential in this mass region. Other final states, composed of \(\tau \) leptons, strange or charm quarks, could be exploited at a future linear \(e^+e^\) collider like the ILC.
A very light Higgs boson \(h\) with mass values up to the \(b\bar{b}\) threshold can also be probed at \(B\)factories in the radiative decay \(\Upsilon \rightarrow h \gamma \) [103], with successive decay of the light Higgs boson to \(\tau \)–lepton, muon or hadron pairs. Here, we provide a rough estimate of the present constraints.
Constraints from \(B\)factories on a light Higgs boson with mass \(m_h\). The second to fifth column list the current experimental \(90\,\%~\mathrm {CL}\) upper bounds on the decay rate of \(\Upsilon (1s) \rightarrow h \gamma \) and successive Higgs decay (specified in the second title row). The inferred upper limit on the rescaling factor of the bottom Yukawa coupling in given in the sixth column, and—if possible—the lower limit on the singletdoublet mixing angle \(\sin \alpha \) is given in the last column. We indicate the most relevant constraint for the model (yielding the listed limits on the model parameters) by bold numbers
\(m_h~[\mathrm{GeV}]\)  \(90\,\%~\mathrm {CL}\) upper limit on \(\mathrm {BR}(\Upsilon (1s)\rightarrow h\gamma , h\rightarrow \cdots )\),  \(g_b^2\) (upper limit)  \(\sin \alpha \) (lower limit)  

\(h \rightarrow \mu ^+\mu ^\) [111]  \(h\rightarrow \tau ^+\tau ^\) [109]  \(h\rightarrow gg\) [112]  \( h\rightarrow s\bar{s}\) [112]  
\(1.0\)  \(\sim \) \(4\times 10^{6}\)  –  \(\sim \) \(\mathbf {5 \times 10^{6}}\)  –  \(\sim \) 0.25  \(\sim \) \(0.87\) 
\(2.0\)  \(\sim \) \(\mathbf {5\times 10^{6}}\)  –  \(\sim \) \(1 \times 10^{4}\)  \(\sim \) \(5 \times 10^{5}\)  \(\sim \)1.16  – 
\(3.0\)  \(\sim \) \(\mathbf {6\times 10^{6}}\)  –  \(\sim \) \(2 \times 10^{4}\)  \(\sim \) \(8 \times 10^{5}\)  \(\sim \)7.82  – 
\(4.0\)  \(\sim \) \(8\times 10^{6}\)  \(\mathbf {1.2 \times 10^{5}}\)  \(\sim \) \(4 \times 10^{4}\)  \(\sim \) \(3 \times 10^{4}\)  2.06  – 
\(5.0\)  \(\sim \) \(8\times 10^{6}\)  \(\mathbf {9.1 \times 10^{6}}\)  \(\sim \) \(3 \times 10^{4}\)  \(\sim \) \(3 \times 10^{4}\)  0.68  \(0.57\) 
\(6.0\)  \(\sim \) \(1\times 10^{5}\)  \(\mathbf {2.3 \times 10^{5}}\)  \(\sim \) \(5 \times 10^{5}\)  \(\sim \) \(8 \times 10^{5}\)  1.59  – 
\(7.0\)  \(\sim \) \(1\times 10^{5}\)  \(\mathbf {1.6 \times 10^{5}}\)  \(\sim \) \(3 \times 10^{4}\)  \(\sim \) \(1 \times 10^{4}\)  1.33  – 
\(8.0\)  \(\sim \) \(2\times 10^{5}\)  \(\mathbf {3.2 \times 10^{5}}\)  \(\sim \) \(1 \times 10^{2}\)  \(\sim \) \(4 \times 10^{5}\)  4.45  – 
Maximally allowed cross sections, \(\sigma _{gg}=\left( \cos ^2\alpha \right) _\text {max}\times \sigma _{gg,\text {SM}}\), for direct light Higgs production at the LHC at CM energies of \(8\) and \(14~\mathrm{TeV}\) after all current constraints have been taken into account. The SM Higgs production cross sections have been taken from Refs. [35, 121]
\(m_h~(\mathrm{GeV})\)  \(\sigma _{gg}^{8\,\mathrm{TeV}}\,(\mathrm{pb})\)  \(\sigma _{gg}^{14\,\mathrm{TeV}}\,(\mathrm{pb})\) 

\(120\)  3.28  8.40 
\(110\)  3.24  8.12 
\(100\)  6.12  14.96 
\(90\)  6.82  16.26 
\(80\)  2.33  5.41 
\(70\)  2.97  6.73 
\(60\)  0.63  1.38 
\(50\)  0.45  0.96 
\(40\)  0.74  1.50 
4.3 Intermediate mass region
For the intermediate mass region, which contains the special case of massdegenerate Higgs states, we treat both Higgs masses as free parameters in the fit, \(m_h, m_H \in [120,130]\,\mathrm{GeV}\). Note that the following discussion is based on a few simplifying assumptions about overlapping Higgs signals in the experimental analyses. It should be clear that a precise investigation of the near massdegenerate Higgs scenario can only be performed by analyzing the LHC data directly and is thus restricted to be done by the experimental collaborations (see e.g. Ref. [7] for such an analysis). Nevertheless, we want to point out this interesting possibility here and encourage the LHC experiments for further investigations.
Limits on \(\sin \alpha \) and \(\tan \beta \) in the intermediate mass scenario. We fix one Higgs mass at \(125.14~\mathrm{GeV}\) and vary the mass of the other Higgs state, \(m\). The limit on \(\sin \alpha \) that stems from LHC Higgs searches evaluated with HiggsBounds is given in the second column (if available). The limit on \(\sin \alpha \) obtained from the test against the Higgs signal rates with HiggsSignals is given in the third column. Note that, depending on the mass hierarchy, we have either an upper or lower limit on \(\sin \alpha \), indicated by the “\(<\)” and “\(>\)”, respectively. The upper limit on \(\tan \beta \) is given in the fourth column and always stems from perturbative unitarity; see also Fig. 1. Note that we do not impose constraints from perturbativity and vacuum stability at a high energy scale via RGE evolution of the couplings here
\(m~(\mathrm{GeV})\)  \(\sin \alpha _\text {HB}\)  \(\sin \alpha _\text {HS}\)  \((\tan \beta )_\text {max}\) 

\(130\)  \(<\) \(0.806\)  \(<\) \(0.370\)  7.76 
\(129\)  \(<\) \(0.881\)  \(<\) \(0.373\)  7.81 
\(128\)  \(<\) \(0.988\)  \(<\) \(0.377\)  7.88 
\(127\)  –  \(<\) \(0.381\)  7.94 
\(126\)  –  \(<\) \(0.552\)  8.00 
\(125\)  –  –  8.07 
\(124\)  –  \(>\) \(0.793\)  8.13 
\(123\)  –  \(>\) \(0.864\)  8.20 
\(122\)  –  \(>\) \(0.904 \)  8.26 
\(121\)  –  \(>\) \(0.913 \)  8.34 
\(120\)  \(>\) \(0.410\)  \(>\) \(0.918\)  8.41 
The upper limits on \(\tan \beta \) listed in Table 6 correspond to the perturbative unitarity bound (cf. Fig. 1). Similarly as in the low mass region, we do not impose constraints from perturbativity and vacuum stability at a high energy scale here. If these were additionally required, \(\tan \beta \) would be limited to values \(\lesssim \) \(1.86\) for \(m \ge 125.14~\mathrm{GeV}\). For lower Higgs masses \(m\) no valid points would be found. It should be noted, however, that the collider phenomenology does not depend on \(\tan \beta \) in the intermediate mass region, since HiggstoHiggs decays are kinematically not accessible.
Figure 18 also shows the correlations of \(\tan \beta \) with the mixing angle \(\sin \alpha \), Fig. 18b, and the Higgs masses, Fig. 18e, f. As stated earlier, \(\tan \beta \) does not influence the collider phenomenology in the intermediate mass range, thus we find allowed parameter points in the full \(\tan \beta \) range up to the maximal value given by perturbative unitarity.
A direct search for the second Higgs boson in the intermediate mass region at the LHC seems challenging. Even if the mass splitting between the two Higgs states is large enough to be resolved by the experimental analyses, we expect the second resonance to be much smaller than the established signal. Nevertheless we would like to encourage the LHC experiments to perform dedicated resonance searches, in particular in the mass region slightly above the current signal, \(m_H \sim \) (125.5–126.5) \(\mathrm{GeV}\), since in this case larger values of the mixing angle are still allowed while an improvement of the vacuum stability at the high scale may be obtained. More promising prospects to resolve the near massdegenerate Higgs scenario have future experimental facilities like the ILC [18, 42] or a muon collider [42, 98], where the latter provides excellent opportunities to measure the mass and the total width of the discovered Higgs boson via a lineshape scan.
5 Conclusions
In this work, we have investigated the theoretical and experimental limits on the parameter space of a real singlet extension of the SM Higgs sector, considering mass values of the second Higgs boson ranging from \(1~\mathrm{GeV}\) to \(1~\mathrm{TeV}\), i.e. within the accessible mass range of past, current and future collider experiments. This study complements a previous work [41], which was restricted to \(m_H\in [600\,\mathrm{GeV},1\,\mathrm{TeV}]\) and, moreover, did not include constraints from direct Higgs collider searches. In the present work, either the heavy or the light Higgs state can take the role of the discovered SMlike Higgs boson at \(125~\mathrm{GeV}\). We found that up to Higgs masses \(m \lesssim 300~\mathrm{GeV}\), exclusion limits from direct Higgs collider searches at LEP and the LHC, as well as the requirement of consistency with the measured SMlike Higgs signal rates pose quite strong constraints. At higher Higgs masses, strong limits stem from electroweak precision observables, in particular from the \(W\) boson mass calculated at NLO, as well as from requiring perturbativity of the couplings and vacuum stability. The latter two are tested both at the electroweak scale and at a high scale \(\mu \sim 4 \times 10^{10}~\mathrm{GeV}\) using the \(\beta \)functions of the theory (see e.g. Ref. [41] and references therein).
We performed an exhaustive scan in the three model parameters—specified by the Higgs mixing angle, the second Higgs mass and the ratio of the Higgs VEVs—and provided a detailed discussion of the viable parameter space and the relative importance of the various constraints. We translated these results into predictions for collider observables for the second yet undiscovered Higgs boson, which are currently investigated by the LHC experiments. In particular, we focused on the global rescaling factor \(\kappa \) for the SM Higgs decay modes, the signal rate for the HiggstoHiggs decay signature \(H\rightarrow hh\) as well as the total width \(\Gamma \) of the new scalar. A typical feature of the model is that the total width of the new scalar is quite suppressed with respect to the SM Higgs boson at such masses. At very light Higgs boson masses below \(10~\mathrm{GeV}\) we found that new results from LHC searches for the signature \(H\rightarrow h h \rightarrow 4\mu \) are complementary to LEP Higgs searches and thus probe an unexplored parameter region. Also future \(B\)factories should be able to probe these parameter regions through the decay \(\Upsilon \rightarrow h \gamma \).
We furthermore investigated the intermediate mass region, where both Higgs masses are between \(120\) and \(130~\mathrm{GeV}\), and discussed some of the experimental challenges in probing this scenario. Dedicated LHC searches for an additional resonance in the invariant mass spectra of the \(H\rightarrow \gamma \gamma \) (see Ref. [7] for a CMS analysis) and \(H\rightarrow ZZ^* \rightarrow 4\ell \) channel in the vicinity of the discovered Higgs boson as well as future precision experiments at the ILC or a muon collider may shed more light onto this case.
The discovery of additional Higgs states is one of the main goals of the upcoming runs of the LHC. In this model, two distinct and complementary signatures of the second Higgs state arise. Firstly, the \(H\rightarrow hh\) decay signature, where the best sensitivity for the LHC is obtained for heavy Higgs masses between \(250~\mathrm{GeV}\) and roughly \(500~\mathrm{GeV}\). These signatures have been recently explored by ATLAS and CMS [99, 100, 122] but the analyses are not yet sensitive to constrain the parameter space. Secondly, Higgs searches designed for a SM Higgs boson are sensitive probes of the parameter space. We strongly encourage the experimental collaborations to continue these searches in the full accessible mass range. However, some of the features of the second Higgs state discussed in this work, such as the strong reduction of the total width, should be taken into account in upcoming analyses. Finally, we hope that the predictions of LHC signal cross sections at a CM energy of \(14~\mathrm{TeV}\) will be found useful for designing some interesting benchmark points for the experimental analyses of this model.
Footnotes
 1.
The value of this high energy scale is chosen to be larger than the energy scale where the running SM Higgs quartic coupling turns negative. This will be made more precise in Sect. 3.
 2.
In fact, all Higgs selfcouplings depend on \(\tan \beta \). However, in the factorized leadingorder description of production and decay followed here, and as long as no experimental data exists which constrains the Higgs boson selfcouplings, only the \(Hhh\) coupling needs to be considered.
 3.
As has been discussed in e.g. Ref. [67], the scale where \(\lambda _1= 0\) in the decoupling case strongly depends on the initial input parameters. However, as we are only interested in the difference of the running in the case of a nondecoupled singlet component with respect to the standard model, we do not need to determine this scale to the utmost precision. For a more thorough discussion of the behavior of the RGEresulting constraints in case of varying input parameters, see e.g. Ref. [41].
 4.
For the requirement of vacuum stability, we found that in some cases the coupling strengths vary very mildly over large variations of the RGE running scale. In these regions the inclusion of higherorder corrections in the spirit of Ref. [67] seems indispensable. Therefore, all lower limits on the mixing angle originating from RGE constraints need to be viewed in this perspective. In fact, such higherorder contributions to the scalarextended RGEs have recently been presented in Ref. [68]. However, the authors did not specifically investigate the higherorder effects on parameter points which exhibit small variations over large energy scales at NLO.
 5.
In the electroweak gauge sector, \(\tan \beta \) only enters at the 2loop level when the Higgs mass sector is renormalized in the onshell scheme.
 6.
 7.
Here we neglect the possible influence of interference effects in the production of the light and heavy Higgs boson and its successive decay. Recent studies [44, 75, 76, 77, 78, 79, 80, 81, 82] have shown that interference and finite width effect can lead to sizable deviations in the invariant mass spectra of prominent LHC search channels such as \(gg\rightarrow H\rightarrow ZZ^{*} \rightarrow 4\ell \) in the high mass region and thus should be taken into account in accurate experimental studies of the singletextended SM at the LHC. However, the inclusion of these effects is beyond the scope of the work presented here.
 8.
HiggsBounds also contains limits from the Tevatron experiments. In the singletextended SM, however, these limits are entirely superseded by LHC results.
 9.
The LEP \(\chi ^2\) information is available for Higgs masses \(\ge \) \(4~\mathrm{GeV}\). For lower masses, we take the conventional \(95\,\%~\mathrm {CL}\) output from HiggsBounds.
 10.
Note, however, that this may change in future with significantly improved exclusion limits from SM Higgs searches.
 11.
This statement is only true if the Higgs state at \(\sim \) \(125\,\mathrm{GeV}\) does not decay to the lighter Higgs. As discussed above, at low light Higgs masses \(m_h < m_H/2\), the branching ratio \(\mathrm {BR}(H\rightarrow hh)\) can reduce the signal rates of the heavy Higgs decaying to SM particles.
 12.Note that the upper limit on \(\sin \alpha \) from the Higgs signal rates is based on a twodimensional \(\Delta \chi ^2\) profile (for floating \(m_h\)) in Fig. 8, whereas in Table 2 the onedimensional \(\Delta \chi ^2\) profile (for fixed \(m_h\)) is used. This leads to small differences in the obtained limit.Table 2
Allowed ranges for \(\sin \alpha \) and \(\tan \beta \) in the high mass region for fixed Higgs masses \(m\). The allowed interval of \(\sin \alpha \) was determined at \(\tan \beta =0.15\). The \(95\,\%~\mathrm {CL}\) limits on \(\sin \alpha \) from the Higgs signal rates are derived from onedimensional fits and taken at \(\Delta \chi ^2 = 4\). The lower limit on \(\sin \alpha \) always stems from vacuum stability, and the upper limit on \(\tan \beta \) always from perturbativity of \(\lambda _2\), evaluated at \(\sin \alpha = 0.1\). The source of the most stringent upper limit on \(\sin \alpha \) is named in the third column. We fixed \(m_h={125.14}~\mathrm{GeV}\), and the stability and perturbativity were tested at a scale of \(\sim \) \(4\times 10^{10}\,\mathrm{GeV}\)
\(m~(\mathrm{GeV})\)
\(\sin \alpha \)
Source upper limit
\((\tan \beta )_\text {max}\)
\(1000\)
\([0.018, 0.17]\)
\(\lambda _1\) perturbativity
0.23
\(900\)
\([0.022, 0.19]\)
\(\lambda _1\) perturbativity
0.26
\(800\)
\([0.027, 0.21]\)
\(m_W\) at NLO
0.29
\(700\)
\([0.031, 0.21]\)
\(m_W\) at NLO
0.33
\(600\)
\([{0.038}, {0.23}]\)
\(m_W\) at NLO
0.39
\(500\)
\([{0.046}, {0.24}]\)
\(m_W\) at NLO
0.47
\(400\)
\([{0.055}, {0.27}]\)
\(m_W\) at NLO
0.59
\(300\)
\([0.067, {0.31}]\)
\(m_W\) at NLO
0.78
\(200\)
\([{0.090}, {0.43}]\)
\(m_W\) at NLO
1.17
\(180\)
\([0.10, {0.46}]\)
Signal rates
1.30
\(160\)
\([{0.11}, {0.46}]\)
Signal rates
1.46
\(140\)
\([{0.16}, 0.31]\)
Signal rates
1.67
 13.See e.g. Ref. [75] for the discussion of finite width effects for SMlike Higgs bosons in the mass range \(m_{h}\gtrsim 200~\mathrm{GeV}\).
 14.The reason why the density of allowed points still depends strongly on \(m_h\) is that regions which are strongly constrained by LEP searches require a large finetuning of \(\sin \alpha \) to render allowed points.
 15.
We consider here only the Higgs production via gluon–gluon fusion.
 16.
This exclusion limit is not provided with HiggsBounds4.2.0, because the expected limit from the CMS analysis is not publicly available.
 17.
We thank M. Grazzini for providing us with the production cross sections for \(m_h <80\,\mathrm{GeV}\).
 18.
Testing overlapping signals of multiple Higgs bosons against mass measurements by employing a mass average calculation is the default procedure in HiggsSignals since version 1.3.0.
 19.
This mass dependence is neglected per default in HiggsSignals since additional complications arise if theoretical mass uncertainties are present. This is, however, not the case here, since we use the Higgs masses directly as input parameters. The evaluation of the signal strength according to Eq. (36) can be activated in HiggsSignals by setting normalize_rates_to_ reference_position=.True. in the file usefulbits_ HS.f90.
Notes
Acknowledgments
We thank Klaus Desch and Sven Heinemeyer for inspiring remarks and for motivating us to perform this study. We furthermore acknowledge helpful discussions with Philip Bechtle, Howie Haber, Antonio Morais, Marco Sampaio, Rui Santos and Martin Wiebusch. The code for testing perturbative unitarity has been adapted from Ref. [41]. TS is supported in part by US Department of Energy Grant No. DEFG0204ER41286, and in part by a FeodorLynen research fellowship sponsored by the Alexander von Humboldt Foundation.
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