Benchmark Planes for Higgs-to-Higgs Decays in the NMSSM

We present benchmark planes (or lines) with cross sections via gluon fusion for the processes $H\to h+H_{S}$, resonant Higgs pair and triple Higgs production, $A \to h+(A_S \to \gamma\gamma)$, $A \to Z+ h$ and $A \to Z+ H_{S}$ within the Next-to-Minimal Supersymmetric Standard Model. Moreover we propose new searches for $H\to h+(H_{S} \to t\bar{t})$, $A \to Z+ (H_{S}\to h+h)$ and $A \to Z+ (H_{S}\to t\bar{t})$ for which possible cross sections are given. These allow the experimental collaborations to verify in the future which search channels cover yet unexplored regions of the parameter space. Expressions for the dominant contributions to trilinear Higgs couplings and Higgs--Z couplings are discussed which allow to identify the dominant processes contributing to a given final state.

However, in models with extended Higgs sectors such as the NMSSM, heavy scalars can have sizeable branching fractions into two lighter scalars, or a Z boson and one scalar. For scalars with small direct production cross sections, such processes can be the only way to discover them. Corresponding searches have been performed in [11][12][13][14][15][16][17][18]. Finally heavy CP-even scalars can also be searched for in final states corresponding to resonant SM Higgs pair production, see [19][20][21][22][23][24][25][26][27][28]. For a recent review of boson pair production at the LHC see [29].
The benchmark points presented here are chosen such that constraints from the existing searches above are satisfied. In addition we impose constraints from B-physics, constraints from properties of the SM-like Higgs boson (a mass within 125 ± 2 GeV allowing for theoretical uncertainties, and couplings in the κ-framework satisfying combined limits of ATLAS and CMS [30][31][32]), and constraints from stability of the electroweak vacuum. Constraints from the anomalous magnetic moment of the muon are left aside as these concern the smuon/gaugino sector which is irrelevant here.
We also require that the lightest supersymmetric particle is neutral (the lightest neutralino), since it is stable and contributes necessarily to the relic density of the universe. We do not require that it accounts for all of the observed dark matter relic density as there may exist additional contributions from physics far above the weak scale. (For this reason we do not require the absence of a Landau singularity below the GUT scale but confine ourselves to λ < 0.7 in order to avoid a strong coupling regime close to the weak scale.) However, the stable lightest neutralino unavoidably contributes to dark matter direct detection experiments, and must satisfy corresponding constraints which are imposed on the benchmark points since the properties of the lightest neutralino (mass and annihilation rate typically via a CP-even or CP-odd scalar in the s-channel) depend on the same parameters as the NMSSM Higgs sector.
The above constraints are implemented in the code NMSSMTools 5.6.2 [52,53] (for more details see the website [54]) coupled to MicrOmegas [55] for the calculation of the dark matter relic density and direct detection cross sections.
Usually the production cross section of heavy Higgs states via gluon fusion dominates and is considered here, although vector boson fusion can be relevant in some particular regions of the parameter space [56]. For the calculation of the cross sections ggF → H/A (with M H/A ≥ 400 GeV) we start with the BSM Higgs production cross sections at √ s = 13 TeV (update in CERN Report4 2016) from the twiki web page [57]. These are multiplied by the reduced couplings squared of H/A. Thereby we capture most of the radiative QCD corrections in the form of K-factors; the remaining theoretical uncertainties are at most of O(10%).
In the next Section 2 we discuss masses and trilinear couplings (including H i -A j -Z) in the Higgs basis in the NMSSM confining ourselves to numerically dominant contributions. This allows to estimate which cascade decays are usually dominant. In Section 3 we present benchmark planes for various final states corresponding to H → h+H S in the space M H −M H S . For M H S we confine ourselves to the range M H S > 60 GeV: Otherwise the couplings of H S must be small enough in order to satisfy constraints from h → H S +H S leading to small allowed cross sections for its production via cascade decays, and its discovery seems more likely via decays of h. We also present a benchmark line with the largest possible cross sections for resonant SM-Higgs pair production H 2 → h + h as function of M H 2 (H 2 is a mixture of H S and H), cross sections for triple SM-Higgs production and for the yet unexplored process H → h+ (H S → tt).
In the NMSSM, singlet-like pseudoscalars A S can have dominant branching fractions into γγ: tree level couplings to SM gauge bosons and fermions can vanish, but couplings to higgsino-like charginos ∼ λ remain. While decays into chargino pairs are kinematically forbidden, chargino loops induce a coupling of A S to photons making the diphoton channel the dominant decay mode. (The branching fraction into the loop induced Z + γ channel is about half as large unless kinematically suppressed, in which case the branching fractions into γγ can become 99%.) The production of A S can proceed via the production of the MSSM-like pseudoscalar A, and its decay A → A S + h. Benchmark points for this process will be given as well.
Furthermore we show benchmark planes for final states corresponding to A → H S + Z in the space M A − M H S . These benchmark points are the same as for H → h + H S which allows to compare the cross sections, and hence to estimate the corresponding relative sensitivities. Cross sections for the yet unexplored processes A → Z + (H S → h + h) and A → Z + (H S → tt) are also given. Finally we present a benchmark line with the largest possible cross sections for A → h + Z as function of M A . A summary is given in Section 4.

The Higgs Sector and Trilinear Higgs Couplings in the NMSSM
2 ). The mass matrices have to be diagonalized, and in the CP-conserving case one obtains three neutral CP-even scalars and two neutral CP-odd scalars (after elimination of the Goldstone boson). General expressions for these mass matrices including the dominant radiative corrections are given in [2]. A first approximation to the physical states is obtained in the so-called Higgs basis where singlet-doublet mixing is neglected and the CP-even doublets are rotated by the same angle as the CP-odd sector. Defining tan β = vu v d and using hats for the Higgs basis ( H SM is near to but not yet exactly equal to the physical SM Higgs boson h, and H is near to but not yet exactly equal to the physical MSSM-like Higgs boson H) one has To these the pure singlet states H S and A S have to be added. The tree level elements of the 2 × 2 mass matrix in the CP-odd sector in the basis A, A S are for the typical case s, The tree level elements of the 3 × 3 mass matrix in the CP-even sector in the basis H SM , H, H S are Hence singlet-doublet mixing is of O( v s , v A λ ) relative to the diagonal elements, but can still be large if the corresponding diagonal elements are close to each other.
Trilinear couplings are proportional to one of the vacuum expectation values v u , v d and s, or to one of the trilinear soft supersymmetry breaking terms A λ or A κ . The latter contributes only to the trilinear singlet Higgs couplings which play a negligible role for Higgs-to-Higgs decays since pure singlets have tiny production cross sections. 1 General expressions for the trilinear couplings can be found in [2], but it is instructive to compare the ones relevant for Higgs-to-Higgs decays for the typical case s, A λ ≫ v u , v d ≈ M Z . In the Higgs basis these are (neglecting If the fields in the Higgs basis are good approximations to the physical fields, relevant processes for searches for ggF → X → Y + h are ggF → H → H S + H SM (using the trilinear coupling a)) and ggF → A → A S + H SM (using the trilinear coupling e)); singlet-like scalars have small production cross sections. The production cross sections for H and A are similar, but the trilinear couplings a) are larger than the trilinear couplings e) for 2 sin β cos β < 1 and/or cancellations in (−2κs + A λ ) for κs, A λ > 0 as considered here. This explains why (for similar masses of H and A) the process ggF An exception is the final state h+γγ with γγ from a BSM scalar or pseudo-scalar. As stated in the introduction, the mostly singlet-like pseudoscalar A S can have a dominant branching fraction up to ∼ 99% into diphotons if Z + γ is kinematically suppressed (up to ∼ 66% otherwise). For maximal cross sections, radiative corrections from supersymmetric particles to the pseudoscalar mass matrix play a relevant role. In the next section we consider a benchmark plane for this process as well. (A corresponding decoupling of H S does not happen since in the scalar sector two mixing angles would have to vanish simultaneously, which would require λs → 0 leading to massless higgsinos.) At first sight the prospects for resonant SM Higgs pair production look dim: For H with the largest production cross section via gluon fusion the dominant trilinear coupling b) vanishes, whereas H S would not be produced via gluon fusion. However the scalar fields in the Higgs basis are not necessarily close to physical fields, and H and H S can strongly mix. Indeed we found that the cross sections for resonant SM Higgs pair production can be quite large (see the next section) in this case.
Next we turn to decays H → A + Z and A → H + Z. The relevant couplings are where C H i denote the H components of the physical states H i , and C A j the A components of the physical states A j .
Decays On the other hand searches for ggF → A → h + Z are also frequently performed. In the NMSSM, the possible cross sections (for a given mass M A ) are also discussed in the next Section.

Benchmark Planes and Lines
A significant excess in final states corresponding to H 3 → h + H 2 would imply the simultaneous discovery of two new bosons beyond the Higgs sector of the Standard model, which may correspond to the CP-even scalars H and H S in the NMSSM. (We recall that the physical states H and H S are generally mixtures of the weak eigenstates.) Based on an integrated luminosity of up to 140 fb −1 , corresponding searches have been performed by CMS in the channel H 3 → (h → τ τ ) + (H S → bb) for mass ranges 240 GeV < M H 3 < 3000 GeV and 60 GeV < M H S < 2800 GeV in [11], and in the channel H 3 → (h → bb) + (H S → bb) for mass ranges 900 GeV < M H 3 < 4000 GeV and 60 GeV < M H S < 600 GeV in [12].
We have prepared a plane of viable benchmark points for ggF → H → (H S → bb) + h covering the mass ranges 400 GeV < M H < 2000 GeV, and 60 GeV < M H S < 800 GeV (or M H S < M H −200 GeV). Generally, masses of H and H S are given with a precision of ±0.5 GeV, except for M H S = 60 GeV which means 60 ≤ M H S ≤ 60.5 GeV such that constraints from CMS in [11], valid for 60 ≤ M H S , are satisfied. 2 Details of the NMSSM-specific parameters, masses, branching fractions and more for each point can be obtained in SLHA format from the authors.
For given values of M H and M H S , the remaining parameters are chosen such that the cross sections for ggF → H → (H S → bb) + h are relatively large, sometimes just below the upper limits from present constraints from the LHC (and from B-physics and dark matter direct detection), see below. The cross sections for decays of h such as h → τ τ and h → γγ are closely related to the ones for h → bb since the branching fractions of h satisfy the combined constraints from ATLAS and CMS [30][31][32]. Still, deviations of ∼ 10% from the Standard Model values are possible within these constraints, and sometimes realized within the NMSSM. Therefore we show in Table 1 the cross sections for ggF → H → (H S → bb) + (h → XX) separately for XX = bb, τ τ, γγ for all benchmark points. Points indicated by (1) in the second column in Table 1 (and later in Table 4) have cross sections ggF → H → (H S → bb)+(h → τ τ ) at the boundary of the region excluded by CMS in [11]. (Points indicated by (2) or (3) have cross sections for resonant Higgs pair production or for ggF → A → Z + (H S → bb) just below the boundary of the region excluded by corresponding searches, see below.) For illustration we show in Fig. 1 [14].
H S has additional interesting decay modes other than H S → bb. For instance, the branching fraction into τ τ is always smaller by a factor 0.1 − 0.14, depending on M H S . This allows to estimate the cross sections for H S → τ τ from the ones into bb.
For M H S > 250 GeV, the branching fraction BR(H S → h + h) becomes sizeable (up to ∼ 20%), and the cascade H → H S + h leads to triple Higgs production. Furthermore, for M H S > 350 GeV, the branching fraction BR(H S → tt) becomes dominant. Since both processes are of interest, we added the cross sections for ggF → H → (H S → h + h) + h and for ggF → H → (H S → tt) + h (without branching fractions of h which are within ∼ 10% of the Standard Model values) in Table 1. The branching fractions for H S → h + h can become small for accidential cancellations within the corresponding trilinear coupling c) in (2.6); of potential interest are the cases where these cross sections are relatively large.
In principle the processes ggF → A → A S + h can lead to identical signatures as the considered processes ggF → H → H S + h. However, we found in Section 2 that the considered processes have larger cross sections and are thus more promising for potential discoveries (or exclusions). As discussed in the introduction and in Section 2 the final state from A S → γγ is an exception. In Table 2 Table 2.
Another interesting process within the NMSSM is resonant (SM) Higgs pair production. The role of the "resonance" can be played by H or by H S . The most constraining limits on SM Higgs pair production ggF → H → h + h originate from the combination of bbbb, bbτ τ and bbγγ final states by ATLAS in [27] for M H = 250 − 3000 GeV. In fact, for M H ≤ 650 GeV the cross sections in the NMSSM could be larger than the limits obtained by ATLAS in [27], hence these limits constrain the parameter space of the NMSSM. For M H S ≤ 120 GeV, these limits imply lower bounds on M H (depending on M H S ) slightly above 400 GeV. Among the selected benchmark points in Table 1 (and later in Table 4), points indicated by (2) in the second column (for M H near 400 GeV) have cross sections ggF → H → h + h just below the boundary of the region excluded by ATLAS in [27]. As discussed in Section 2, particularly large cross sections for resonant SM Higgs pair production can be found if H and H S strongly mix. Then the notation H 2 , H 3 is more appropriate, and the largest cross sections are found for ggF → H 2 → h + h. In Table 3 we show possible cross sections in the NMSSM (for points which differ from the benchmark points in Table 1) for ggF → H 2 → h + h for M H 2 > 700 GeV up to M H 2 = 1200 GeV; these are still below the limits from CMS in [28]. For illustration we show in Fig. 3 the allowed cross sections for resonant Higgs pair production as function of M H from Table 3.
Finally we turn to cascade decays into a Z boson. As discussed in Section 2, the largest cross sections in the NMSSM correspond to the processes ggF → A → H S + Z; cross sections for ggF → H → A S + Z are substantially smaller for equivalent masses of H/A and A S /H S . Corresponding searches have been performed by CMS in [13] after ∼ 36 fb −1 for the mass ranges 120 GeV < M A < 1000 GeV and 30 GeV < M H S < 780 GeV, and by ATLAS in [14] after ∼ 139 fb −1 for 230 GeV < M A < 800 GeV and 130 GeV < M H S < 700 GeV. (  For M H 2 < 700 GeV these correspond to the limits from ATLAS in [27]. adopted the notation to the interpretation within the NMSSM.) The latter search excludes some regions in the parameter space of the NMSSM; the benchmark points shown here satisfy these constraints. Points indicated by (3) in the second column of Tables 1 and 4 have cross sections ggF → A → Z + (H S → bb) at the boundary of the region excluded in [14]. In the Table 4 we show the cross sections for ggF → A → Z + (H S → XX) for various final states XX = bb, τ τ, γγ for the same benchmark points as in Table 1 which allows to compare the sensitivities in the various search channels. (For some points, the branching fraction for H S → γγ can be particularly small due to cancellations among different loop contributions.) Also shown in Table 4 are cross sections for the yet unexplored processes ggF → A → Z + (H S → h + h) and ggF → A → Z + (H S → tt) which can possibly be within reach.
Searches for A → h + Z have been performed by CMS in [15,16] after ∼ 36 fb −1 , and by ATLAS in [17] after ∼ 139 fb −1 . In the NMSSM these cross sections are dominated by the production of the MSSM-like pseudo-scalar. In the Table 5 we show the largest possible cross sections in the NMSSM for 400 < M A < 2000 GeV (for points different from the previous benchmark points) which are well below the present limits obtained by CMS and ATLAS.

Summary
Searches for Higgs-to-Higgs and Higgs-to-Higgs+Z cascade decays at the LHC allow to explore extended Higgs sectors beyond the SM. In the present paper we have presented various benchmark planes and lines which show which cross sections are possible in which final states within the NMSSM, subject to present phenomenological and theoretical constraints. Some of the available searches by ATLAS and CMS already touch the parameter space of the NMSSM, and our tables allow to estimate which future searches can be promising not only using available data, but also after the upgrade of the LHC to High Luminosity after a suitable rescaling. The proposed search channels H → h + (H S → tt), A → Z + (H S → h + h) and A → Z + (H S → tt) are new and have not been considered before.

Appendix
Tables in different format (csv) as well as SLHA files for the benchmark points are available from the authors upon request.
Points indicated by (1) in the second column have cross sections ggF → H → (H S → bb)+(h → τ τ ) at the boundary of the region excluded by CMS in [11]. Points indicated by (2) in the second column have cross sections ggF → H → h + h at the boundary of the region excluded by ATLAS in [27]. Points indicated by (3) in the second column have cross sections ggF → A → Z + (H S → bb) at the boundary of the region excluded by ATLAS in [14].  Table 1 continued  Table 1 continued  Table 1 continued Points indicated by (1) in the second column have cross sections ggF → H 3 → (H S → bb)+(h → τ τ ) at the boundary of the region excluded by CMS in [11]. Points indicated by (2) in the second column have cross sections ggF → H 3 → h + h at the boundary of the region excluded by ATLAS in [27]. Points indicated by (3) in the second column have cross sections ggF → A 2 → Z + (H S → bb) at the boundary of the region excluded by ATLAS in [14].