LHC benchmark scenarios for the real Higgs singlet extension of the standard model
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Abstract
We present benchmark scenarios for searches for an additional Higgs state in the real Higgs singlet extension of the Standard Model in Run 2 of the LHC. The scenarios are selected such that they fulfill all relevant current theoretical and experimental constraints, but can potentially be discovered at the current LHC run. We take into account the results presented in earlier work and update the experimental constraints from relevant LHC Higgs searches and signal rate measurements. The benchmark scenarios are given separately for the lowmass and highmass region, i.e. the mass range where the additional Higgs state is lighter or heavier than the discovered Higgs state at around 125 GeV. They have also been presented in the framework of the LHC Higgs Cross Section Working Group.
Keywords
Higgs Boson Production Cross Section Vacuum Expectation Value Benchmark Point Standard Model Particle1 Introduction
In this work we consider the simplest extension of the SM Higgs sector, where an additional real scalar field is added, which is neutral under all quantum numbers of the SM gauge groups [7, 8] and acquires a vacuum expectation value (VEV). This model has been widely studied in the literature [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 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, 49, 50, 51, 52], also in the context of electroweak higher order corrections [53, 54] or offshell and interference effects [33, 34, 55, 56, 57, 58, 59]. Here, we present an update of the exploration of the model parameter space presented in Ref. [38], where we take the latest experimental constraints into account. As before, we consider masses of the second (nonstandard) Higgs boson in the whole mass range up to \(1\,\mathrm{TeV}\). This minimal setup can be interpreted as a limiting case for more generic BSM scenarios, e.g. models with additional gauge sectors [60] or additional matter content [61, 62]. Experimental searches for the model have been presented in [63, 64, 65, 66, 67, 68, 69, 70].
As in Ref. [38] we take the following theoretical and experimental constraints into account: bounds from perturbative unitarity and electroweak (EW) precision measurements, in particular focussing on higher order corrections to the W boson mass [32]; perturbativity, vacuum stability and correct minimization of the model up to a high energy scale using renormalization group (RG) evolved couplings; exclusion limits from Higgs searches at the LEP, Tevatron and LHC experiments via the public tool HiggsBounds [71, 72, 73, 74, 75], and compatibility of the model with the signal strength measurements of the discovered Higgs state using HiggsSignals [76] (cf. also Ref. [77]).
We separate the discussion of the parameter space into two different mass regions: (i) the highmass region, \(m_H \in [130, 1000]\,\mathrm{GeV}\), where the lighter Higgs boson h is interpreted as the discovered Higgs state; (ii) the lowmass region, \(m_h \in [1,120]\,\mathrm{GeV}\), where the heavier Higgs boson H is interpreted as the discovered Higgs state.
We find that the most severe constraints in the whole parameter space for the second Higgs mass \(m_H \lesssim 250~\mathrm{GeV}\) are mostly given by limits from collider searches for a SM Higgs boson as well as by the LHC Higgs boson signal strength measurements. For \(m_H\gtrsim 250\,\mathrm{GeV}\) limits from higher order contributions to the W boson mass prevail, followed by the requirement of perturbativity of the couplings.
For the remaining viable parameter space we present predictions for signal cross sections of the yet undiscovered second Higgs boson for the LHC at a CM energy of \(14\,\mathrm{TeV}\), discussing both the SM Higgs decay signatures and the novel HiggstoHiggs decay mode \(H\rightarrow hh\). For both the highmass and the lowmass regions we present a variety of benchmark scenarios. These are designed to render a maximal direct production rate for the collider signature of interest. Whenever kinematically accessible we give two different benchmark points for each mass, for which the HiggstoHiggs decay \(H\rightarrow hh\) is maximal or minimal, respectively.
The paper is organized as follows: In Sect. 2 we briefly review the model and the chosen parametrization. In Sect. 3 we review the constraints that are taken into account and in particular discuss the impact of the new constraints on the parameter space. In Sect. 4 we provide benchmark points and planes discussed above. We summarize and conclude in Sect. 5.
2 The model
In the following we briefly review the main features of the real Higgs singlet extension of the SM that are important for the benchmark choices. More details as regards the model can e.g. be found in Refs. [29, 32, 38, 54] and references therein.
2.1 Potential and couplings

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 Constraints
In this section we list all theoretical and experimental constraints that we take into account, and give an overview over the impact of these constraints on the parameter space. We refer the reader to Ref. [38] for details of the implementation of these constraints. With respect to Ref. [38] we update the experimental limits from LHC Higgs searches, leading to a change in the allowed parameter space especially in the lower mass range, \(m_H\,\in \,[130, 250 ] \,\mathrm{GeV}\). We also include constraints from the combined ATLAS and CMS Higgs signal strength [82], rendering a significantly stronger limit on the mixing angle. However, this limit is still not as strong as the constraint from the W boson mass measurement in most of the parameter space.
3.1 Theoretical constraints

vacuum stability and minimization of model up to a scale \(\mu _{\mathrm{run}}\,=\,4\,\times \,10^{10}\,\mathrm{GeV}\),

perturbative unitarity of the \(2\rightarrow 2\) Smatrix for \((W^+\,W^,ZZ,hh,hH,HH)\) initial and final states,

perturbativity of the couplings in the potential, \(\lambda _i\,\le \,4\,\pi \), up to a high energy scale, \(\mu _{\mathrm{run}}\,=\,4\,\times \,10^{10}\,\mathrm{GeV}\), employing oneloop renormalization group equations (RGEs) [83].
3.2 Experimental constraints

agreement with electroweak precision observables, employing the oblique parameters \(S,\,T,\,U\) [84, 85, 86, 87] and using the results from the global fit from the GFitter Group [88],

agreement with the observed W boson mass [89, 90, 91], \(M_W = 80.385 \pm 0.015~\mathrm{GeV}\), employing the NLO calculation presented in Ref. [32],
 agreement with limits from direct Higgs searches at LEP, Tevatron, and the LHC using HiggsBounds (version 4.3.1) [71, 72, 73, 74, 75]. With respect to the results presented in Ref. [38], limits from the following searches have been included here:

ATLAS search for \(H\rightarrow WW\) [92],

ATLAS search for \(H\rightarrow ZZ\) [70],

combination of ATLAS searches for \(H\rightarrow hh\rightarrow bb\tau \tau , \gamma \gamma WW^*, \gamma \gamma bb, bbbb\) [67],

CMS search for \(H\rightarrow VV~(V=W^\pm ,Z)\) [66],

CMS search for \(H\rightarrow hh\rightarrow 4\tau \), where H is the SMlike Higgs boson at \(125\,\mathrm{GeV}\) [93].

 agreement with the observed signal strengths of the 125 \(\mathrm GeV\) Higgs boson, using HiggsSignals (version 1.4.0) [76], and using the results from the ATLAS and CMS combination of the LHC Run 1 data, \(\mu \,=\, 1.09 \pm 0.11\) [82], leading tofor the heavy Higgs mass range \(m_H \gtrsim 150\,\mathrm{GeV}\) (highmass range, \(m_h\,\sim \,125\,\mathrm{GeV}\)), and$$\begin{aligned} \sin \alpha \,\le \,0.36 \end{aligned}$$(21)for the light Higgs mass range \(m_h \lesssim 100\,\mathrm{GeV}\) (lowmass range, \(m_H\,\sim \,125\,\mathrm{GeV}\)). In these mass regions potential signal overlap with the SMlike Higgs at \(125\,\mathrm{GeV}\) can be neglected. For Higgs masses in the range \([100, 150]\,\mathrm{GeV}\) we employ HiggsSignals using observables from the individual Higgs channels, which enables to approximately take into account a potential signal overlap [76], see also Ref. [38] for details.$$\begin{aligned} \sin \alpha \,\ge \,0.87 \end{aligned}$$(22)
3.3 Allowed parameter regions and sensitivity of the constraints
3.3.1 Highmass region
The importance of the different constraints on the mixing angle \(\sin \alpha \) in the highmass region, where \(m_h\,\sim \,125\,\mathrm{GeV}\), is summarized in Fig. 1. Recall that this angle is responsible for the global suppression of the production cross section with respect to the SM prediction at the same Higgs mass. We see that in the lower mass region, \(m_H\,\lesssim \,250\,\mathrm{GeV}\), the most important constraints stem from direct Higgs searches [66, 70, 94, 95, 96] and the combined Higgs signal strength [82], whereas for higher masses, \(m_H\,\in \,[250\,\mathrm{GeV};\;800\,\mathrm{GeV}]\), the W boson mass becomes the strongest constraint [32]. Requiring perturbativity of the couplings yields the upper limit on \(\sin \alpha \) for very heavy Higgs bosons, \(m_H\,\ge \,800\,\mathrm{GeV}\).
Range of \(m_H\,[\hbox {GeV}]\)  Search channel  Reference 

130–145  H\(\rightarrow \)ZZ\(\rightarrow \)4l  [94] (CMS) 
145–158  H\(\rightarrow \)VV (V=W,Z)  [66] (CMS) 
158–163  SM comb.  [95] (CMS) 
163–170  H\(\rightarrow \)WW  [96] (CMS) 
170–176  SM comb.  [95] (CMS) 
176–211  H\(\rightarrow \)VV (V=W,Z)  [66] (CMS) 
211–225  H\(\rightarrow \)ZZ\(\rightarrow \)4l  [94] (CMS) 
225–445  H\(\rightarrow \)VV (V=W,Z)  [66] (CMS) 
445–776  H\(\rightarrow \)ZZ  [70] (ATLAS) 
776–1000  H\(\rightarrow \)VV (V=W,Z)  [66] (CMS) 
Note that these plots were obtained using a simple rescaling of production cross section of a SM Higgs boson of the same mass as given in Ref. [23], i.e. contributions due to interference with the additional scalar are not included. Tools which can handle these have been presented e.g. in Refs. [55, 56, 58, 59]. These studies, however, focus on effects on the lineshape of the heavy scalar boson after a possible discovery. Moreover, thus far, their calculations neglect additional higher order corrections, whereas these have been calculated to great precision for the SM Higgs boson and are included in Fig. 3 [23]. For the future, it would be desirable to perform a dedicated study of interference effects including higher order corrections for the benchmark points presented in this work in order to estimate their effects (and the systematic uncertainty introduced here by neglecting them).
3.3.2 Lowmass region
Limits on \(\sin \alpha \) and \(\tan \beta \) in the lowmass scenario for various light Higgs masses \(m_h\) and \(\tan \beta =1\). In the second column we give the lower limit on \(\sin \alpha \) stemming from exclusion limits from LEP or LHC Higgs searches (evaluated with HiggsBounds). If the lower limit on \(\sin \alpha \) obtained from the Higgs signal rates (evaluated with HiggsSignals) results in stricter limits, they are displayed in the third column. The fourth column displays the upper limit on \(\tan \beta \) that stems from perturbative unitarity in the complete decoupling case (\(\sin \alpha \,=\,1\)). 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. The table is taken from Ref. [38]
\(m_h~[\mathrm{GeV}]\)  \(\sin \alpha _{\mathrm{min}, {\mathrm{HB}}}\)  \(\sin \alpha _{\mathrm{min}, {\mathrm{HS}}}\)  \((\tan \beta )_{\mathrm{max}}\)  \((\tan \beta )_{\text {no}~H\rightarrow hh} \) 

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 
Maximally allowed cross section for light Higgs production in gluon fusion, \(\sigma _{gg}=\left( \cos ^2\alpha \right) _{\mathrm{max}}\times \sigma _{gg,{\mathrm{SM}}}\), at the LHC at CM energies of 8 and \(14~\mathrm{TeV}\) after all current constraints have been taken into account, corresponding to the mixing angles from Table 2. This is an updated version of Tab. V in Ref. [38]
\(m_h~[\mathrm{GeV}]\)  \(\sigma _{gg}^{8\,\mathrm{TeV}}[\mathrm{pb}]\)  \(\sigma _{gg}^{14\,\mathrm{TeV}}[\mathrm{pb}]\)  \(m_h~[\mathrm{GeV}]\)  \(\sigma _{gg}^{8\,\mathrm{TeV}}[\mathrm{pb}]\)  \(\sigma _{gg}^{14\,\mathrm{TeV}}[\mathrm{pb}]\) 

120  3.28  8.41  60  0.63  1.38 
110  3.24  8.17  50  0.45  0.96 
100  6.12  15.10  40  0.76  1.59 
90  6.82  16.47  30  1.60  3.09 
80  2.33  5.41  20  5.04  8.97 
70  1.72  3.91  10  18.44  29.74 
3.3.3 Intermediate mass region
4 Benchmark scenarios for LHC Run 2
The benchmark scenarios that are presented in this section are chosen such that they feature the maximally allowed production cross section at the LHC. We first present the benchmark scenarios for the highmass region, where the light Higgs plays the role of the discovered SMlike Higgs at \(125\,\mathrm{GeV}\), and then turn to the lowmass range, where the heavy Higgs state is the SMlike Higgs boson.^{3}
4.1 Highmass region
Benchmark points for mass ranges where the onshell decay \(H\rightarrow hh\) is kinematically forbidden. Maximal values of \(\tan \beta \) were calculated at the maximal mixing angle, and should be applied for consistency reasons
\(m_H [\mathrm{GeV}]\)  \(\sin \alpha _{{\mathrm{max}}}\)  \(\tan \beta _{{\mathrm{max}}}\)  \(m_H [\mathrm{GeV}]\)  \(\sin \alpha _{\mathrm{max}}\)  \(\tan \beta _{\mathrm{max}}\) 

130  0.42  1.79  195  0.28  1.22 
135  0.38  1.73  200  0.29  1.19 
140  0.36  1.69  210  0.28  1.14 
145  0.35  1.62  215  0.33  1.12 
150  0.34  1.57  220  0.34  1.10 
160  0.36  1.49  230  0.35  1.05 
180  0.30  1.32  235  0.34  1.03 
185  0.27  1.28  240  0.31  1.00 
190  0.29  1.26  245  0.28  0.98 
Maximal and minimal allowed branching ratios of the decay \(H\rightarrow hh\), taken at the maximally allowed value of \(\sin \alpha \). Note that minimal values for the \(\text {BR}(H\rightarrow hh)\) stem from \(\sin \alpha \,\le \,0\)
\(m_H [\mathrm{GeV}]\)  \(\sin \alpha _{\mathrm{max}}\)  \(BR^{H\rightarrow hh}_{\mathrm{min}}\)  \(BR^{H\rightarrow hh}_{\mathrm{max}}\)  \(m_H [\mathrm{GeV}]\)  \(\sin \alpha _{\mathrm{max}}\)  \(BR^{H\rightarrow hh}_{\mathrm{min}}\)  \(BR^{H\rightarrow hh}_{\mathrm{max}}\) 

255  0.31  0.09  0.27  430  0.25  0.19  0.30 
260  0.34  0.11  0.33  470  0.24  0.19  0.28 
265  0.33  0.13  0.36  520  0.23  0.19  0.26 
280  0.32  0.17  0.40  590  0.22  0.19  0.25 
290  0.31  0.18  0.40  665  0.21  0.19  0.24 
305  0.30  0.20  0.40  770  0.20  0.19  0.23 
325  0.29  0.21  0.40  875  0.19  0.19  0.22 
345  0.28  0.22  0.39  920  0.18  0.19  0.22 
365  0.27  0.21  0.36  975  0.17  0.19  0.21 
395  0.26  0.20  0.32  1000  0.17  0.19  0.21 
 Higgs decays into SM particles: Maximizing the production cross section corresponds to maximizing the parameter [29]In general, following Eq. (13), Higgs decays into SM particles follow the hierarchy of the branching ratios of a SM Higgs of the same mass. This, together with the observation that the branching ratio for \(H\,\rightarrow \,hh\) is \(\mathcal {O}\left( 0.2 \right) \) in large parts of the parameter space, translates into the fact that for most of the highmass region the dominant decay mode is \(H\rightarrow WW\).$$\begin{aligned} \kappa \,\equiv \,\frac{\sigma }{\sigma _\text {SM}}\times \text {BR} (H\rightarrow \mathrm {SM})=\sin ^4\alpha \,\frac{\Gamma _\text {SM,tot}}{\Gamma _{\mathrm{tot}}}. \end{aligned}$$
 Higgs decays into two light Higgs bosons, \(H\rightarrow hh\): Here, the parameteris maximized to obtain the largest possible signal yield.$$\begin{aligned} \kappa '\,\equiv \,\frac{\sigma }{\sigma _\text {SM}} \times \text {BR}(H\rightarrow hh)=\sin ^2\alpha \,\frac{\Gamma _{H\rightarrow hh}}{\Gamma _{\mathrm{tot}}}, \end{aligned}$$
We quantify the benchmark scenarios for both signal channels in this regime by considering the maximally allowed mixing angle together with the maximal and minimal branching ratio for the decay \(H\rightarrow hh\), respectively. While these maximal and minimal points define benchmark points, all \(\text {BR}(H\rightarrow hh)\) values in between are in principle allowed. Therefore, an interpolation between the minimal and maximal values defines a higherdimensional benchmark scenario (benchmark slope or plane), where the additional third parameter (cf. Eqs. (17)–(20)) is floating.
We furthermore distinguish scenarios for which the \(H\rightarrow hh\) onshell decay mode is kinematically allowed or forbidden. As we neglect all other triple and quartic Higgs selfcouplings apart from \(\mu '\), and work in the onshell approximation, \(\tan \beta \) only influences the collider phenomenology for regions in parameter space where the decay \(H\rightarrow hh\) is kinematically allowed, i.e. for heavy Higgs masses \(m_H \ge 2m_h \approx 250\,\mathrm{GeV}\). For lower masses \(\tan \beta \) is irrelevant for the phenomenology considered here. However, to be consistent, we recommend to still keep the values within the respective parameter regions allowed by perturbativity and perturbative unitarity.
Benchmark scenarios for both cases are given in Tables 4 and 5, respectively. Parameter ranges which are not explicitly listed can to a first approximation be linearly interpolated.
Benchmark scenarios for the highmass region for fixed masses and \(\sin \alpha \), floating \(\tan \beta \) (between scenarios a and b). Reference production cross sections have been taken from the upcoming CERN Yellow Report 4 by the LHC Higgs Cross Section Working Group [104]
Benchmark scenarios for the real singlet  

Main features  Real singlet extension, with two vevs and no hidden sector interaction with heavy Higgs H and light Higgs h 
Fixed parameters  \(M_h = 125.1\) GeV or \(M_H = 125.1\) GeV 
Irrelevant parameters  \(\tan \beta \) whenever channel \(H\rightarrow hh\) kinematically not accessible 
Additional comments  Predictions at LO, factorized production and decay; a, b signify maximal and minimal \(\text {BR}(H\rightarrow hh)\); for b, \(\sin \alpha <0\); any values for \(\tan \beta \) between scenario a and b are allowed 
Production cross sections at 14 TeV [pb] and branching fractions  
BHM300 a, b  
Spectrum  \(M_H=300\) GeV, \(\sin \alpha = {0.31}, \,\tan \beta ~(a)= {0.79}, \,\tan \beta ~(b)= {0.79} \) 
\(\sigma (gg \rightarrow h)\)  44.91 
\(\sigma (gg \rightarrow H)\)  1.09 
BR(\(H\rightarrow hh\))  0.41 (a), 0.17 (b) 
BR(\(H \rightarrow W W\))  0.41 (a), 0.57 (b) 
BR(\(H \rightarrow Z Z\))  0.18 (a), 0.25 (b) 
BHM400 a, b  
Spectrum  \(M_H=400\) GeV, \(\sin \alpha =0.26,\,\tan \beta ~(a)=0.58,\,\tan \beta ~(b)=0.59\) 
\(\sigma (gg \rightarrow h)\)  46.32 
\(\sigma (gg \rightarrow H)\)  0.76 
BR(\(H\rightarrow hh\))  0.32 (a), 0.20 (b) 
BR(\(H \rightarrow W W\))  0.40 (a), 0.47 (b) 
BR(\(H \rightarrow Z Z\))  0.18 (a), 0.22 (b) 
BR(\(H \rightarrow t \bar{t}\))  0.10 (a), 0.12 (b) 
BHM500 a, b  
Spectrum  \(M_H=500\) GeV, \(\sin \alpha =0.24,\,\tan \beta ~(a)=0.44,\,\tan \beta ~(b)=0.46\) 
\(\sigma (gg \rightarrow h)\)  46.82 
\(\sigma (gg \rightarrow H)\)  0.31 
BR(\(H\rightarrow hh\))  0.26 (a), 0.19 (b) 
BR(\(H \rightarrow W W\))  0.41 (a), 0.44 (b) 
BR(\(H \rightarrow Z Z\))  0.19 (a), 0.21 (b) 
BR(\(H \rightarrow t \bar{t}\))  0.14 (a), 0.16 (b) 
Benchmark scenarios for the highmass region for fixed masses and \(\sin \alpha \), floating \(\tan \beta \) (between scenarios a and b). Reference production cross sections have been taken from the upcoming CERN Yellow Report 4 by the LHC Higgs Cross Section Working Group [104]
Production cross sections at 14 TeV [pb] and branching fractions (continued)  

BHM600 a, b  
Spectrum  \(M_H=600\) GeV, \(\sin \alpha =0.22,\,\tan \beta ~(a) =0.37, \,\tan \beta ~(b)=0.38 \) 
\(\sigma (gg \rightarrow h)\)  47.28 
\(\sigma (gg \rightarrow H)\)  0.12 
BR(\(H\rightarrow hh\))  0.25 (a), 0.19 (b) 
BR(\(H \rightarrow W W\))  0.41 (a), 0.45 (b) 
BR(\(H \rightarrow Z Z\))  0.21 (a), 0.22 (b) 
BR(\(H \rightarrow t \bar{t}\))  0.13 (a), 0.14 (b) 
BHM700 a, b  
Spectrum  \(M_H=700\) GeV, \(\sin \alpha =0.21,\,\tan \beta ~(a)=0.31,\,\tan \beta ~(b)=0.32\) 
\(\sigma (gg \rightarrow h)\)  47.49 
\(\sigma (gg \rightarrow H)\)  0.050 
BR(\(H\rightarrow hh\))  0.24 (a), 0.19 (b) 
BR(\(H \rightarrow W W\))  0.44 (a), 0.47 (b) 
BR(\(H \rightarrow Z Z\))  0.22 (a), 0.23 (b) 
BR(\(H \rightarrow t \bar{t}\))  0.10 (a), 0.11 (b) 
BHM800 a, b  
Spectrum  \(M_H=800\) GeV, \(\sin \alpha =0.2,\,\tan \beta ~(a)=0.25, \tan \beta ~(b)=0.27 \) 
\(\sigma (gg \rightarrow h)\)  47.69 
\(\sigma (gg \rightarrow H)\)  0.022 
BR(\(H\rightarrow hh\))  0.23 (a), 0.19 (b) 
BR(\(H \rightarrow W W\))  0.46 (a), 0.48 (b) 
BR(\(H \rightarrow Z Z\))  0.23 (a), 0.24 (b) 
BR(\(H \rightarrow t \bar{t}\))  0.08 (a), 0.09 (b) 
BHM200  
Spectrum  \(M_H=200\) GeV, \(\sin \alpha =0.29,\,\tan \beta = 1.19 \) 
\(\sigma (gg \rightarrow h)\)  45.50 
\(\sigma (gg \rightarrow H)\)  1.74 
BR(\(H \rightarrow \text {SM}\))  As for a SM Higgs boson with mass of \(200\,\mathrm{GeV}\) 
Maximal branching ratios for \(H\rightarrow hh\). This BR can always be zero for the choice \(\tan \beta \,=\,\cot \,\alpha \)
\(m_h [\mathrm{GeV}]\)  \(\sin \alpha \)  \(BR^{H\rightarrow hh}_{\mathrm{max}}\) 

60  0.9996  0.259 
50  0.9999  0.259 
40  0.9999  0.259 
30  0.9999  0.259 
20  0.9998  0.259 
10  0.9999  0.259 
Lowmass benchmark scenarios for the HiggstoHiggs decay signature for fixed masses and \(\sin \alpha \), floating \(\tan \beta \) (between scenarios a and b). In scenario b we have \(\tan \beta \,=\,\cot \,\alpha \). The \(\sin \alpha \) values have been optimized for scenario a, which in turn leads to a suppression of direct production for the lighter state. For direct production of the lighter scalar, the parameters in Tables 2 and 3 should be used. For BHM50  BHM10, the production cross section for the SM like Higgs is \(\sigma (gg\,\rightarrow \,H)= {49.66}\, \mathrm{pb}\). Reference production cross sections have been taken from the upcoming CERN Yellow Report 4 by the LHC Higgs Cross Section Working Group [104]
BHM60 a, b  
Spectrum  \(M_h=60\) GeV, \(\sin \alpha =0.9997,\, \tan \beta ~(a) =3.48,\,\tan \beta ~(b)=0.025 \) 
\(\sigma (gg \rightarrow h)\)  0.10 
\(\sigma (gg \rightarrow H)\)  49.65 
BR(\(H\rightarrow hh\))  0.26 (a), 0 (b) 
BR(\(H \rightarrow \text {SM}\))  Rescaled by 0.74 (a), as in SM (b) 
BHM50 a, b  
Spectrum  \(M_h=50\) GeV, \(\sin \alpha =0.9998,\, \tan \beta ~(a)=3.25,\,\tan \beta ~(b)=0.020\) 
\(\sigma (gg \rightarrow h)\)  0.098 
BR(\(H\rightarrow hh\))  0.26 (a), 0 (b) 
BR(\(H \rightarrow \text {SM}\))  Rescaled by 0.74 (a), as in SM (b) 
BHM40 a, b  
Spectrum  \(M_h=40\) GeV, \(\sin \alpha =0.9998,\, \tan \beta ~(a)=3.13, \tan \beta ~(b)=0.020 \) 
\(\sigma (gg \rightarrow h)\)  0.16 
BR(\(H\rightarrow hh\))  0.26 (a), 0 (b) 
BR(\(H \rightarrow \text {SM}\))  Rescaled by 0.74 (a), as in SM (b) 
BHM30 a, b  
Spectrum  \(M_h=30\) GeV, \(\sin \alpha =0.9998,\, \tan \beta ~(a)=3.16, \tan \beta ~(b)=0.020 \) 
\(\sigma (gg \rightarrow h)\)  0.31 
BR(\(H\rightarrow hh\))  0.26 (a), 0 (b) 
BR(\(H \rightarrow \text {SM}\))  Rescaled by 0.74 (a), as in SM (b) 
BHM20 a, b  
Spectrum  \(M_h=20\) GeV, \(\sin \alpha =0.9998,\, \tan \beta ~(a)=3.23, \tan \beta ~(b)=0.020 \) 
\(\sigma (gg \rightarrow h)\)  0.90 
BR(\(H\rightarrow hh\))  0.26 (a), 0 (b) 
BR(\(H \rightarrow \text {SM}\))  Rescaled by 0.74 (a), as in SM (b) 
BHM10 a, b  
Spectrum  \(M_h=10\) GeV, \(\sin \alpha =0.9998,\, \tan \beta ~(a)=3.29, \tan \beta ~(b)=0.020 \) 
\(\sigma (gg \rightarrow h)\)  2.98 
BR(\(H\rightarrow hh\))  0.26 (a), 0 (b) 
BR(\(H \rightarrow \text {SM}\))  Rescaled by 0.74 (a), as in SM (b) 
4.2 Lowmass region

Direct production of the lighter Higgs state h and successive decay into SM particles,

Decay of the SMlike Higgs boson H into the lighter Higgs states, \(H\rightarrow hh\).
For the second channel—the decay of the SMlike Higgs into two lighter Higgs states—we list maximal branching ratios for the decay \(H\rightarrow hh\) in Table 8. As long as the decay \(H\rightarrow hh\) is kinematically accessible, the maximal value of its branching ratio, \(\text {BR}(H\rightarrow hh) \simeq 0.259\), is not dependent on the light Higgs mass. The lighter Higgs bosons then decay further according to the branching ratios of a SM Higgs of the respective mass. A first experimental search of this signature with the light Higgs boson decaying into \(\tau \) lepton pairs in the mass range \(m_h \in [5, 15]\,\mathrm{GeV}\) has already been performed by the CMS experiment [93].
We present benchmark points for fixed masses in Table 9. Here, \(\sin \alpha \) values closer to unity are needed in order to obtain maximal branching ratios for this channel, which in turn leads to the reduction of direct production for the lighter state by almost an order of magnitude with respect to the values presented in Table 3. Again, we recommend to scan over \(\tan \beta \) between the values of scenario a and b (thus defining a higherdimensional benchmark scenario) in order to obtain a range of possible branching ratios.
5 Conclusions
In this paper we have revisited and updated the constraints on the parameter space of the real scalar singlet extension of the SM. In comparison with the previous results presented in Ref. [38], the most important improvements have been made in the constraints from new results in LHC searches for a heavy Higgs boson decaying into vector boson final states, as well as from the ATLAS and CMS combination of the signal strength of the discovered Higgs state. We found that these modify our previous findings in the mass range \(130\,\mathrm{GeV}\le m_H\,\le \,250\,\mathrm{GeV}\), where now the direct Higgs searches as well as the ATLAS and CMS signal strength combination render the strongest constraints on the parameter space.
Based on these updated results, we have provided benchmark scenarios for both the highmass and the lowmass regions for upcoming LHC searches. Hereby, we pursued the philosophy of selecting those points which feature a maximal discovery potential in a dedicated collider search of the corresponding signature. We provided predictions of production cross sections for the LHC at 14 \(\mathrm TeV\), and supplemented these with information as regards the branching fractions of the relevant decay modes. We encourage the experimental collaborations to make use of these benchmark scenarios in the current and upcoming LHC runs.
Footnotes
 1.
Note that even if the \(Z_2\) symmetry is not imposed, the parameters of the model relevant for the collider phenomenology considered here can always be chosen in terms of the masses, a mixing angle, and an additional parameter determining the \(H\rightarrow hh\) decay channel.
 2.
HiggsBounds selects the most sensitive channel by comparing the expected exclusion limits first. In a second step, the predicted signal strength is confronted with the observed exclusion limit only of this selected channel. This welldefined statistical procedure allows one to systematically test the model against a plethora of Higgs search limits without diluting the \(95~\%\) C.L. of the individual limits.
 3.
See also Ref. [99] for recent benchmark point suggestions within the complex singlet model.
 4.
Electroweak corrections to the decay \(H\rightarrow hh\) have been presented for some of these benchmark points in Ref. [54].
Notes
Acknowledgments
We thank S. Dawson, C. Englert, M. Gouzevitch, S. Heinemeyer, I. Lewis, A. Nikitenko, M. Sampaio, R. Santos, M. Slawinska, and D. Stoeckinger for useful discussions, as well as G. Chalons, D. LopezVal, and G.M. Pruna for fruitful collaboration on earlier related work. TS is supported in parts by the U.S. Department of Energy grant number DESC0010107 and a FeodorLynen research fellowship sponsored by the Alexander von Humboldt foundation.
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