LHC Benchmark Scenarios for the Real Higgs Singlet Extension of the Standard Model

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 low mass and high mass 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.


I. INTRODUCTION
The first run of the LHC at center-of-mass (CM) energies of 7 and 8 TeV has been completed in 2015. Its remarkable success is highlighted by the breakthrough discovery of a scalar boson in July 2012 and the measurements of its coupling properties, which thus far are well compatible with the interpretation in terms of the Higgs boson of the Standard Model (SM) Higgs mechanism [1][2][3][4][5]. The combination of the Higgs mass measurements performed by ATLAS and CMS yields [6] m H = 125.09 ± 0.21 (stat.) ± 0.11 (syst.) GeV. (1) If the discovered particle is indeed the Higgs boson of the SM, its mass measurement determines the last unknown ingredient of this model, as all other properties of the electroweak sector then follow directly from theory. In the coming years a thorough investigation of the Higgs boson's properties is needed in order to identify whether the SM Higgs sector is indeed complete, or instead, the structure of a more involved Higgs sector is realized. This includes detailed and accurate measurements of its coupling strengths and CP structure at the LHC and ultimately at future experimental facilities for Higgs boson precision studies.
Complementary to this, collider searches for additional Higgs bosons need to be continued over the full accessible mass range. The discovery of another Higgs boson would inevitably prove the existence of a non-minimal Higgs sector.
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 , 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 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 -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 high mass region, m H ∈ [130, 1000] GeV, where the lighter Higgs boson h is interpreted as the discovered Higgs state; (ii) the low mass region, m h ∈ [1, 120] 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 250 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 250 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 TeV, discussing both the SM Higgs decay signatures and the novel Higgs-to-Higgs decay mode H → hh. For both the high mass and low mass region 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 Higgs-to-Higgs decay H → hh is maximal or minimal, respectively.
The paper is organized as follows: In Section II we briefly review the model and the chosen parametrization. In Section III we review the constraints that are taken into account and in particular discuss the impact of the new constraints on the parameter space. In Section IV we provide benchmark points and planes discussed above. We summarize and conclude in Section V.

II. 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 about the model can e.g. be found in Refs. [29,32,38,54] and references therein.

A. Potential and couplings
The real Higgs singlet extension of the SM [7,8,78] contains a complex SU (2) L doublet, in the following denoted by Φ, and in additional a real scalar S which is a singlet under the SM gauge group. The most general renormalizable Lagrangian compatible with an additional Z 2 symmetry is then given by 3 II The model with the scalar potential The implicitly imposed Z 2 symmetry forbids all linear or cubic terms of the singlet field S in the potential. We assume that both Higgs fields Φ and S have a non-zero vacuum expectation value (VEV), denoted by v and x, respectively. In the unitary gauge, the Higgs fields are given by After diagonalization of the mass matrix we obtain the mass eigenstates h and H with mass eigenvalues given by and m 2 h ≤ m 2 H by convention. The gauge and mass eigenstates are related via the mixing where the mixing angle − π 2 ≤ α ≤ π 2 is given by It follows from Eq. (7) that the light (heavy) Higgs boson couplings to SM particles are suppressed by cos α (sin α). If kinematically allowed, the additional decay channel H → hh is present. Its partial decay width at leading order (LO) is given by [7,78] where the coupling strength µ of the H → hh decay reads 4 Next-to-leading order (NLO) corrections to the H → hh decay width for this model have been calculated recently in Ref. [54]. The branching ratios of the heavy Higgs mass eigenstate m H are then given by where Γ SM, H→SM is the partial decay width of the SM Higgs boson and H → SM represents any SM Higgs decay mode. The total width is then where Γ SM, tot denotes the total width of the SM Higgs boson with mass m H . The suppression by sin 2 α directly follows from the suppression of all SM-like couplings, cf. Eq. (7). For µ = 0, the decay H → hh vanishes and we recover the SM Higgs boson branching ratios.
For the collider phenomenology of the model two features are important: • the suppression of the production cross section of the two Higgs states induced by the mixing, which is given by sin 2 α (cos 2 α) 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 → hh is kinematically accessible.
For the high mass (low mass) scenario, i.e. the case where the light (heavy) Higgs boson is identified with the discovered Higgs state at ∼ 125 GeV, | sin α| = 0 (1) corresponds to the complete decoupling of the second Higgs boson and therefore the SM-like scenario.

B. Model parameters
At the Lagrangian level, the model has five free parameters, while the values of the additional parameters µ 2 , m 2 are fixed by the minimization conditions. A more intuitive basis, where the free model parameters are represented by physical (i.e. observable) quantities, is given by 1 The vacuum expectation value of the Higgs doublet Φ is given by the SM value v ∼ 246 GeV, and one of the Higgs masses is fixed to m h/H = 125.09 GeV, eliminating two of the five parameters. We are thus left with only three independent parameters, where the latter enters the collider phenomenology only through the heavy Higgs decay mode into the lighter Higgs, H → hh. Note that from a collider perspective, for cases where the decay mode H → hh is kinematically allowed, the input parameter tan β could be replaced by either the total width of the heavier state, Γ(H), the branching ratio BR (H → hh), or the partial decay width of this channel, Γ(H → hh), respectively, rendering the following viable parameter choices besides Eq. (17): If the insertion starts on the Lagrangian level (via e.g. FeynRules [79], SARAH [80,81] or similar), also the Lagrangian parameters as such can be used as input values, but then care must be taken to correctly translate these into the phenomenologically viable parameter regions.

III. 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 on 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 ∈ [130, 250] 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.

A. Theoretical Constraints
We consider the following theoretical constraints in the selection of the benchmark scenarios: • vacuum stability and minimization of model up to a scale µ run = 4 × 10 10 GeV, • perturbative unitarity of the 2 → 2 S-matrix for (W + W − , ZZ, hh, hH, HH) initial and final states, • perturbativity of the couplings in the potential, |λ i | ≤ 4 π, up to a high energy scale, µ run = 4 × 10 10 GeV, employing one-loop renormalization group equations (RGEs) [83].

B. Experimental Constraints
The following experimental constraints are taken into account at the 95% C.L.: • 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 ± 0.015 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: combination of ATLAS searches for H → hh → bbτ τ, γγW W * , γγbb, bbbb [67], • Agreement with the observed signal strengths of the 125 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, µ = 1.09 ± 0.11 [82], leading to | sin α| ≤ 0.36 (21) for the heavy Higgs mass range m H 150 GeV (high mass range, m h ∼ 125 GeV), and | sin α| ≥ 0.87 (22) for the light Higgs mass range m h 100 GeV (low-mass range, m H ∼ 125 GeV). In these mass regions potential signal overlap with the SM-like Higgs at 125 GeV can be neglected. For Higgs masses in the range [100, 150] 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.

High mass region
The importance of the different constraints on the mixing angle sin α in the high mass region, where m h ∼ 125 GeV, is summarized in Figure 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 250 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 ∈ [250 GeV; 800 GeV], the W boson mass becomes the strongest constraint [32]. Requiring perturbativity of the couplings yields the upper limit on | sin α| for very heavy Higgs bosons, m H ≥ 800 GeV.
The updated combined signal strength reduces the maximally allowed mixing angle from previously | sin α| 0.50 [38] Fig. 2. We see that the updated constraints yield stronger limits in particular for m H ≤ 250 GeV as well as for m H 400 GeV. We supplement this comparison by giving a detailed list in Tab. I of the LHC Higgs search channels that have been applied by HiggsBounds in the various mass regions. 2 The relatively strong constraints on the mixing angle lead to a significant suppression of the direct production rates of the heavy Higgs boson at LHC run 2. Fig. 3 shows the predicted production cross section at 14 TeV after all constraints have been taken into account. The production cross sections rapidly decrease with higher masses m H due to both the stronger constraints on the mixing angle (cf. Fig. 1) and a reduction of the available phase space for higher masses. The cross section for direct production in gluon fusion and successive decay into SM final states ranges from about 10 pb at lower masses to about 10 fb for masses around 800 GeV. Note that in order to obtain the predictions for a particular SM decay mode, H → XX, these numbers need to be multiplied by a factor of   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 line-shape 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).

Low mass region
In the low mass region, where the heavier Higgs state takes the role of the discovered Higgs boson, m H ∼ 125 GeV, the parameter space is extremely constrained by the Higgs signal strength and exclusion limits from LEP Higgs searches [89]. The updated experimental results do not change the limits presented in Ref. [38]. We review these limits in Tab. II. Note that in the low mass region the couplings of the heavy Higgs boson at 125 GeV become SM-like for | sin α| = 1.
Tab. III gives the direct production cross section in gluon fusion for the undiscovered light Higgs state at a 8 and 14 TeV LHC, respectively. Again, the production cross section stems from a simple rescaling of the corresponding cross section for a SM Higgs boson of that mass [23,98].  In the second column we give the lower limit on sin α stemming from exclusion limits from LEP or LHC Higgs searches (evaluated with HiggsBounds). If the lower limit on sin α 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 β that stems from perturbative unitarity in the complete decoupling case (| sin α| = 1). In the fifth column we give the tan β value for which Γ H→hh = 0 is obtained given the maximal mixing angle allowed by the Higgs exclusion limits (second column). At this tan β value, the | sin α| limit obtained from the Higgs signal rates (third column) is abrogated. The table is taken from Ref. [38].

Intermediate mass region
The intermediate mass region, where both Higgs bosons have masses between 120 GeV and 130 GeV, was originally discussed in Ref. [38]. In this mass region the observed Higgs signal at 125 GeV may be due to a signal overlap of both Higgs bosons, depending on the mass separation and the mass resolution of the experimental analysis. We show the allowed parameter space in the (m h , m H ) and (m h , sin α) plane from the updated fit in Fig. 4. The updated signal strength observables in HiggsSignals-1.4.0 yield only marginal improvements in the constrained parameter space, while the updated limits from direct Higgs searches are irrelevant in this mass region.

IV. 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 high mass region, where the light Higgs plays the role of the discovered SM-like Higgs at 125 GeV, and then turn to the low mass range, where the heavy Higgs state is the SM-like Higgs boson. 3

A. High mass region
We distinguish between two different search channels: • Higgs decays into SM particles: Maximizing the production cross section corresponds to maximizing the parameter [29] κ ≡ σ σ SM × BR(H → SM) = sin 4 α Γ SM,tot Γ tot .
In general, following Eq. (13) is maximized to obtain the largest possible signal yield. Figure 5 shows the allowed range of these two quantities, after all constraints have been taken into account. For the Higgs decay channel into SM particles, we see that searches from CMS pose important constraints for m H 400 GeV. For the Higgs-to-Higgs decay channel H → hh, on the other hand, both ATLAS [67] and CMS [100,101] searches are not yet sensitive enough to exclude points that are not already in conflict with other constraints.
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 → hh, respectively. While these maximal and minimal points define benchmark points, all BR(H → 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. Eq. (17)-(20)) is floating.
We furthermore distinguish scenarios for which the H → hh on-shell decay mode is kinematically allowed or forbidden. As we neglect all other triple and quartic Higgs selfcouplings apart from µ , and work in the on-shell approximation, tan β only influences the collider phenomenology for regions in parameter space where the decay H → hh is kinematically allowed, i.e. for heavy Higgs masses m H ≥ 2m h ≈ 250 GeV. For lower masses tan β 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 Tab. IV and V, respectively. Parameter ranges which are not explicitly listed can to a first approximation be linearly interpolated.
In addition, we also list exemplary benchmark points for this mass region in Tables VI and VII, where we additionally give the predictions for other relevant decay modes. Whenever kinematically accessible, we provide two benchmark points for every heavy Higgs mass, representing the maximal and minimal branching ratio for the H → hh decay, respectively. 4 The mixing angle is always chosen such that the production rate of the additional scalar is maximized.      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].

B. Low mass region
For the case that the heavier Higgs boson is taken to be the discovered SM-like Higgs boson with m H ∼ 125 GeV, | sin α| = 1 corresponds to the SM limit, and deviations from this value parametrize the new physics contributions. As in the high mass region, the following channels are interesting: • Direct production of the lighter Higgs state h and successive decay into SM particles, • Decay of the SM-like Higgs boson H into the lighter Higgs states, H → hh.
For the direct production of the light Higgs state smaller | sin α| values are of interest, as the cross section scales with cos 2 α. We provide the minimally allowed values for | sin α| in Tab. II. Tab. III lists the respective direct production cross sections at 8 and 14 TeV. These values can directly be used as benchmark scenarios for collider searches for direct light Higgs production.
For the second channel -the decay of the SM-like Higgs into two lighter Higgs stateswe list maximal branching ratios for the decay H → hh in Tab. VIII. As long as the decay H → hh is kinematically accessible, the maximal value of its branching ratio, BR(H → hh) 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 τ lepton pairs in the mass range m h ∈ [5,15] GeV has already been performed by the CMS experiment [93].
We present benchmark points for fixed masses in Tab. IX. Here, | sin α| 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 Tab. III. Again, we recommend to scan over tan β between the values of scenario a and b (thus defining a higher dimensional benchmark scenario) in order to obtain a range of possible branching ratios.     a and b). In scenario b we have tan β = − cot α. The | sin α| 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 Tab. II and III should be used. For BHM50 -BHM10, the production cross section for the SM like Higgs is σ(gg → H) = 49.66 pb. Reference production cross sections have been taken from the upcoming CERN Yellow Report 4 by the LHC Higgs Cross Section Working Group [104].
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 GeV ≤ m H ≤ 250 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 high mass and low mass region 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 TeV, and supplemented these with information about 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.