Exploring Heavy Higgs Bosons at a 100 TeV Hadron Collider within the Semi-Constrained NMSSM

In this study, we explore the detectability of heavy Higgs bosons in the $pp \to b\bar{b}H/A \to b\bar{b}t\bar{t}$ channel at a 100 TeV hadron collider within the semi-constrained Next-to-Minimal Supersymmetric Standard Model (NMSSM). We calculate their production cross sections and decay branching ratios, comparing these with simulation results from existing reference. We focus on the heavy, doublet-dominated CP-even Higgs $H$ and CP-odd Higgs $A$, with mass limits set below 10 TeV to ensure detectability. We find that at a collider with 3 ab$^{-1}$ of integrated luminosity, the potential for detecting heavy Higgs bosons varies significantly with their mass and $\tan\beta$. Heavy Higgs bosons below 2 TeV are within the testable range, while those heavier than 7 TeV fall below the exclusion and discovery thresholds, rendering them undetectable. For masses between 2 and 7 TeV, heavy Higgs bosons with $\tan\beta$ less than 20 can be detected, whereas those with $\tan\beta$ greater than 20 are beyond the current discovery or exclusion capabilities.


I. INTRODUCTION
It has been widely acknowledged for an extended period that the Standard Model (SM) of particle physics, despite its remarkable success in explaining a vast array of phenomena, fails to provide a complete description of the fundamental aspects of the universe.Consequently, the search for new physics beyond the SM (BSM) has become a crucial direction in modern physics research.
The exploration of new physics phenomena, especially the detection of heavy Higgs bosons, requires colliders with higher energy.Future high-energy colliders, designed to exceed the capabilities of current facilities, will be able to examine these heavy particles.Notably, the Future hadron-hadron Circular Collider (FCC-hh) at CERN [47] and the Super-pp-Collider (SppC) [48,49] in China are among the most ambitious projects in this direction.Both initiatives aim to construct a 50-100 TeV pp collider [50], promising a significant leap in the energy frontier and potentially uncovering phenomena beyond the SM.Moreover, the concept of a multi-TeV muon collider presents an innovative approach to high-energy physics experiments [51,52].Extensive research has been conducted on the detection of heavy Higgs bosons at future colliders.In particular, the studies described in Ref. [53] examined the pp → b bH/A → b bτ τ and pp → b bH/A → b bt t channels at a 100 TeV pp collider, and proposed pushing the exclusion limits for heavy Higgs searches up to M H ∼ 10 TeV, with exceptions in regions of low tan β.Furthermore, the analysis in Ref. [13] explored the pp → H/A → χ0 1 χ∓ 2 process, revealing the 4ℓ + E / signal at a 100 TeV hadron collider, demonstrating its ability to probe new supersymmetric model sectors.In addition, Ref. [54] explored the potential of a multi-TeV muon collider to discover heavy Higgs bosons within Two Higgs Doublet Models (2HDMs) and assess the discriminative power among different 2HDM types.
In the current study, we extend the investigation initiated in our previous work on heavy Higgs bosons within the framework of the semi-constrained NMSSM (scN-MSSM) [55].The NMSSM incorporates an additional singlet superfield to the MSSM, thereby enriching the Higgs and neutralino sectors.Our analysis focuses on the computational evaluation of production cross sections and decay branching ratios for these heavy Higgs bosons.Through these calculations, we aim to provide a comprehensive understanding of the behavior and detectability of heavy Higgs bosons within the scNMSSM.Furthermore, we delve into the exploration of the discovery potential of these heavy Higgs bosons through the pp → b bH/A → b bt t channel at a future 100 TeV col-arXiv:2112.15570v2[hep-ph] 18 Apr 2024 lider.The selection of a 100 TeV collider is driven by its exceptional ability to achieve the high energy levels required for producing such massive particles, thus opening up new avenues for their discovery.
The remainder of this manuscript is organized as follows.In Sec.II, we provide a brief overview of the scN-MSSM, outlining its fundamental aspects and theoretical significance.In Sec.III, we present a detailed account of our computational methodology, followed by a comprehensive discussion of the results obtained from our analysis.In Sec.IV, we conclude the paper by summarizing the main results and their implications for future research in this area.

II. THE SEMI-CONSTRAINED NMSSM
The NMSSM extends the MSSM by introducing an additional singlet superfield, denoted as Ŝ, where the superpotential of the Z 3 -symmetric NMSSM is defined as: where W MSSM | µ=0 is the superpotential of the MSSM without the µ-term, λ and κ are coupling constants, Ĥu and Ĥd are the doublet Higgs superfields, and Ŝ is the added singlet superfield.After electroweak symmetry breaking, the singlet scalar's vacuum expectation value (VEV), denoted v s , dynamically generates the massive µ-term [56,57] µ ≡ λv s .
Concurrently, the scalar components H u and H d also attain VEVs, labeled v u and v d , respectively.This leads to the introduction of a new parameter tan β, defined as where the sum of their squares is The NMSSM introduces specific soft SUSY breaking terms, distinct from those in the MSSM, as given by: where L soft MSSM | µ=0 denotes the MSSM's soft SUSY breaking terms with the µ parameter set to zero.The symbols H u and H d refer to the scalar components of the Higgs doublets, A λ and A κ represent the trilinear coupling constants with mass dimension, and m S is the mass of the singlet scalar field.
In the scNMSSM, the Higgs sector is allowed to deviate from universality at the Grand Unified Theory (GUT) scale, a characteristic also known as the NMSSM with non-universal Higgs masses.Specifically, the soft masses for the Higgs fields, m 2 Hu , m 2 H d , and m 2 S , can differ from M 2 0 + µ 2 .Furthermore, the trilinear coupling constants A λ and A κ may vary independently from A 0 .Consequently, the parameter space of the scNMSSM is defined by nine parameters: λ, κ, tan β, µ, A λ , A κ , A 0 , M 1/2 , M 0 . (5) Here, M 1/2 and M 0 represent the universal sfermion mass and the universal gaugino mass, respectively, while A 0 denotes the universal trilinear coupling constant in the sfermion sector.The Higgs sector within the NMSSM is predicted to contain three CP-even Higgs bosons, two CP-odd Higgs bosons, and a pair of charged Higgs bosons.For convenience, the scalar components of the superfields H u , H d , and S are often rotated so that they can be represented as where ε = 0 1 −1 0 , and S 1 , S 2 , and S 3 create the CPeven basis, while P 1 and P 2 establish the CP-odd one.H 2 is identified as the SM-like Higgs, H 1 represents a new Higgs doublet field, and H 3 introduces a new singlet field.
The CP-even Higgs mass matrix M 2 S in the basis (S 1 , S 2 , S 3 ) is given by [21,58]: where M A is defined as: The CP-odd Higgs mass matrix M 2 P in the basis (P 1 , P 2 ) is given by: Three CP-even mass eigenstates h 1 , h 2 , and h 3 (m h1 < m h2 < m h3 ) are derived from the mixture of (S 1 , S 2 , S 3 ), and two CP-odd mass eigenstates a 1 and a 2 (m a1 < m a2 ) are derived from (P 1 , P 2 ).This can be represented as: where S ij and P ij are the mixing matrices that diagonalize the mass matrices M 2 S and M 2 P , respectively.Among the three CP-even Higgs bosons (h i , where i = 1, 2, 3), the 125 GeV SM-like Higgs could be either h 1 or h 2 , both of which are predominantly doublet-dominated scalars.The remaining CP-even Higgs bosons include another doublet-dominated and a singlet-dominated scalar.For the two CP-odd Higgs bosons (a i , where i = 1, 2), one is doublet-dominated, and the other is singlet-dominated.The singlet-dominated Higgs boson rarely couples to fermions because the singlet S interacts only with the Higgs sector.This property makes it difficult to detect at the LHC.In contrast, the doublet-dominated Higgs boson couples to fermions, which facilitates its detection.Our study focuses only on the heavy, doublet-dominated CP-even Higgs H and CP-odd Higgs A, because of their detectability.The couplings to up/down-type fermions of these heavy Higgs bosons, H and A, are defined as follows: The reduced couplings of these heavy Higgs bosons, H and A, are defined as follows: Furthermore, it is observed that when tan β is significantly larger than 1 (tan β ≫ 1), M 2 S,11 closely approximates M 2 A .In addition, the Higgs bosons H and A become degenerate, meaning they have the same mass and exhibit identical couplings to quarks.
For the samples satisfying the above theoretical and experimental constraints, we observe the following properties: • The squarks of the first two generations are heavier than 2.2 TeV, with the lightest squark, t1 , exceeding 1 TeV.
• The third-generation sleptons can be as light as approximately 170 GeV.
• The glugino mass exceeds 2 TeV.Consequently, given the universal gaugino mass condition at the GUT scale, the bino and wino masses are more than 340 GeV and 620 GeV, respectively.
• The mass range for the lightest neutralino varies from 4 GeV to 4 TeV, typically dominated by bino and singlino compositions, with some higgsino admixture.
• In the Higgs sector, we categorize the samples into two types: h 1 is the 125 GeV SM-like Higgs.In this study, we focus on doublet-dominant heavy Higgs due to the difficulty of detecting singlet-dominant Higgs.There are two types to consider.In the first type, h 1 resembles the SM-like Higgs, with either h 2 or h 3 being doublet-dominant, and the same holds for a 1 and a 2 , where one of them is doublet-dominant.We label the heavy CP-even and CP-odd doublet-dominant Higgs as H and A, respectively.In the second type, h 2 acts as the SM-like Higgs, with h 3 and a 2 typically being doublet-dominant, also denoted as H and A. Thus, H and A represent the heavy CP-even and CP-odd doublet-dominant Higgs in subsequent discussions.For the heavy Higgs H and A, which have masses ranging from 0.6 TeV to 10 TeV, we calculate their production cross sections and decay branching ratios.We also compare the pp → b bH → b bt t signal with simulation results found in Ref. [53].
In Fig. 1, we show the mass and reduced coupling of the heavy doublet-dominated Higgs bosons H and A in the scNMSSM.The following observations can be made from these figures: • In the left panel, the surviving samples are plotted on the m A versus m H plane, with colors indicating M A .It is observed that H and A have nearly identical masses, approximately equal to the parameter M A .This similarity arises because, according to Eq. 16, P 1 is the CP-odd doublet-dominated Higgs, and since A is also denoted as the CP-odd doubletdominated Higgs, it follows that m A ≈ M A .Furthermore, S 1 is the CP-even doublet-dominated Higgs, labeled here as H. From Eq. 9, when tan β ≫ 1, it is derived that m H ≈ M A .And the mass of H and A is between 0.6 TeV and 10 TeV.
• In the middle panel, the surviving samples are displayed on the plane of reduced coupling with up-type fermions for A versus H, with the colors indicating 1/ tan β.It is evident that for most samples, C Auu and C Huu are approximately equal to 1/ tan β.This approximation arises because the doublet components of H and A are neither exactly equal nor exactly equal to 1. Furthermore, the values of the reduced couplings C Huu and C Auu range from 0 to 0.8.
• In the right panel, the surviving samples are plotted on the plane of reduced coupling with down-type fermions for A versus H, with colors representing tan β.The results are similar to those in the middle panel; for most samples, C Add and C Hdd approximate tan β.This approximation is also due to the doublet components of H and A not being exactly equal or exactly equal to 1. Additionally, the values of the reduced couplings C Hdd and C Add vary from 0 to 50.
In Fig. 2 we show the decay properties of the CPeven doublet-dominated heavy Higgs H in the scN-MSSM, with colors representing tan β.Since both heavy Higgs A and H are doublet-dominated, their couplings to fermions show very little difference.The only difference is that Br(A → V V / Ṽ Ṽ ) = 0.However, this difference is minimal due to the small coefficient C HV V .As a consequence, the decay properties of the CP-odd doubletdominated heavy Higgs A are very similar to those of H; therefore, the plot for A is omitted.The following observations can be drawn from these plots: • The dominant branching ratios consistently arise from the decays to t t, b b and SUSY particles, with these branching ratios reaching values close to 1.
• In the upper left panel, the branching ratio of H to t t is inversely proportional to tan β; that is, the smaller the tan β, the larger the branching ratio Br(H → t t).This relationship is due to C Huu being directly proportional to 1/ tan β.Consequently, when tan β < 10, the branching ratio Br(H → t t) exceeds 0.2.As tan β approaches 1, the branching ratio Br(H → t t) tends towards 1.
• In the upper middle and right panels, the branching ratios of H to b b and τ + τ − are proportional to tan β; specifically, the larger the tan β, the higher the branching ratios Br(H → b b) and Br(H → τ + τ − ).This proportionality is due to C Hdd being directly proportional to tan β.Additionally, when tan β exceeds 40, the maximum branching ratio Br(H → b b) can reach up to 0.8, while the maximum branching ratio Br(H → τ + τ − ) can only reach up to 0.2.Furthermore, the branching ratio Br(H → τ + τ − ) is generally lower than Br(H → b b) due to the lower mass of τ compared to b, as the coupling strength of Higgs with fermions is proportional to their mass.
• In the lower left panel, the branching ratios of H to light Higgs bosons are relatively small, reaching a maximum of approximately 0.6.Additionally, for samples where tan β exceeds 40, the maximum branching ratio Br(H → light Higgs) is only 0.1.
• In the lower right panel, the maximum branching ratios of H to SUSY particles can approach 0.8.Additionally, it is observed that when tan β ranges from 10 to 30, the branching ratios Br(H → SUSY) can approach this maximum value of 0.8, remaining above 0.5.When tan β is less than 10, the heavy Higgs H predominantly decays into t t; conversely, when tan β exceeds 30, it primarily decays into b b.
We calculate the cross sections for the process pp → b bH in the SM with m H ranging from 0.5 to 10 TeV at √ s = 100 TeV using MG5 aMC v2.6.7 [93,94].The calculated cross section for m H or m A in our samples is multiplied by the square of the reduced coupling C Hbb and the branching ratio Br(H/A → t t).Since the masses and various couplings of the heavy Higgs H and A are very similar, along with nearly identical branching ratios Br(H → t t) and Br(A → t t), and similar reduced couplings C Hbb and C Abb , the heavy Higgs bosons H and A are considered degenerate in the detection channel pp → b bH/A → b bt t.Therefore, the cross section for the pp → b bH/A → b bt t channel is twice that of the individual H or A channels.Production rates for our samples in the pp → b bH/A → b bt t channel are presented in the left panel of Fig. 3, where colors indicate tan β.The red and green curves represent the model-independent exclusion and discovery reaches, respectively, with an integrated luminosity of 3 ab −1 at 100 TeV, as depicted in Fig. 9 of Ref. [53].In calculating the SM cross section, we employed both the four-flavor scheme (4FS) and five-flavor scheme (5FS) cross sections [95], and combined them using the formula from Ref. [96]: where In Fig. 3 we show the cross section for the pp → b bH/A → b bt t channel of the CP-even doublet-dominated heavy Higgs H/A in the scNMSSM, with colors representing tan β.Since the heavy Higgs A and H are considered degenerate, the cross section σ(pp The following observations can be drawn from these plots: • In the left panel, 1/ tan 2 β/Γ tot (H) appears to be directly proportional to Br(H → t t).This is because the decay diagram for Br(H → t t) includes a coupling vertex C Huu , and when calculating the decay cross-section, a C 2 Huu term is introduced.Thus, Br(H → t t) is proportional to 1/ tan 2 β.Additionally, the branching ratio Br(H → t t) is inversely proportional to the total decay width Γ tot (H), which is represented as: • In the middle panel, the total decay width Γ tot (H) of the heavy Higgs increases exponentially with tan β.Additionally, when tan β remains constant, Γ tot (H) increases with the mass of the heavy Higgs m H .
• In the right panel, the cross section σ(pp → b bH/A → b bt t) decreases rapidly as the mass of the heavy Higgs m H increases.The cross section for pp → b bH → b bt t can be approximated as follows: This decline is because σ(pp → b bH → b bt t) is proportional to σ(pp → b bH SM ), and the production cross section σ(pp → b bH SM ) diminishes with an increase in mass.It can also be observed that samples with smaller tan β values have larger cross sections σ(pp → b bH → b bt t).This is because σ(pp → b bH → b bt t) is inversely proportional to the total decay width Γ tot (H), and Γ tot (H) exponentially increases as tan β increases.
• In the right panel, the regions above the green and red curves indicate where the samples can be covered by 2 σ and 5 σ, respectively, with an integrated luminosity of 3 ab −1 at 100 TeV.This implies that through the pp → b bH/A → b bt t channel in the scNMSSM, samples with a heavy Higgs mass m H < 2 TeV can be tested at at the 100 TeV collider with 3 ab −1 of integrated luminosity.For samples with the heavy Higgs mass m H > 7 TeV, they fall below the exclusion and discovery curves, thus they cannot be discovered or excluded.Samples with the heavy Higgs mass in the range of 2−7 TeV and tan β < 20 can be tested at the 100 TeV collider with 3 ab −1 of integrated luminosity.
In Table I, we present four benchmark samples detailing the Higgs sector, where σ(X) represents the cross section σ(pp → b bX → b bt t).The heavy Higgs bosons H and A, corresponding to h 3 and a 2 respectively, are doublet-dominated, while S 2 33 and P 2 22 indicate the singlet components in H and A. We find that H and A have minimal singlet components, suggesting that h 2 and a 1 are primarily singlet-dominated.Due to their weak coupling to fermions, these singlet-dominated bosons, h 2 and a 1 , are difficult to detect at colliders.

IV. CONCLUSION
In this study, we explore the potential for detecting heavy Higgs bosons in the pp → b bH/A → b bt t channel at a 100 TeV hadron collider within the semiconstrained NMSSM.First, we scan the relevant parameter space with the NMSSMTools package, which includes theoretical constraints such as vacuum stability and Landau poles, as well as experimental constraints like Higgs data, B physics, sparticle searches, dark matter relic density, and direct detection experiments.We observe that singlet-dominated Higgs bosons S are difficult to detect due to their limited interactions outside the Higgs sector.Therefore, our analysis primarily focuses on the more detectable heavy, doublet-dominated CP-even Higgs H and CP-odd Higgs A, limiting their masses to below 10 TeV to remain detectable.The presence of a CP-even Higgs (h 1 or h 2 ) resembling the 125 GeV SM-like Higgs does not affect these findings.Since that the heavy Higgs H and A are nearly identical in mass and couplings, the cross section for the combined channel pp → b bH/A → b bt t is effectively double that of the single H channel.
We calculated their decay branching ratios and production rates, and compared these with the simulation results cited in Ref. [53].Finally, we draw the following conclusions about the heavy Higgs bosons A and H, with masses ranging from 0.6 to 10 TeV, in the semiconstrained NMSSM: • When the heavy Higgs bosons are doubletdominated, their reduced couplings with up-type fermions, C Huu and C Auu , are approximately equal • The branching ratio Br(H → t t) is proportional to 1/ tan 2 β and inversely proportional to the total decay width Γ tot (H).Furthermore, the total decay width of the heavy Higgs, Γ tot (H), increases exponentially with tan β.
• The cross section σ(pp → b bH/A → b bt t) decreases rapidly as the mass of the heavy Higgs (m H ) increases, and it is inversely proportional to the total decay width Γ tot (H).Consequently, this cross section also decreases exponentially with increasing tan β.
• For the pp → b bH/A → b bt t channel at a 100 TeV collider with 3 ab −1 of integrated luminosity in the semi-constrained NMSSM: -Heavy Higgs bosons with a mass m H < 2 TeV can be tested.
-Heavy Higgs bosons with a mass m H > 7 TeV fall below the exclusion and discovery thresholds, and therefore cannot be discovered or excluded.
-For heavy Higgs masses in the range of 2-7 TeV, those with tan β ≲ 20 can be tested, while those with tan β ≳ 20 cannot be discovered or excluded.

h 2 and h 3
are heavy CP-even Higgs bosons, while a 1 and a 2 are heavy CP-odd Higgs bosons.h 2 is the 125 GeV SM-like Higgs.The light CP-even Higgs h 1 and the light CP-odd Higgs a 1 are typically singlet-dominant.The heavy CP-even Higgs h 3 and the light CP-odd Higgs a 2 are typically doublet-dominant.

FIG. 1 .
FIG. 1. Surviving samples are shown in the planes of mA versus mH (left), reduced coupling CAuu versus CHuu (middle), and reduced coupling C Add versus C Hdd (right).From left to the right, the colors represent MA, 1/ tan β, and tan β respectively.Samples with larger values of tan β are plotted on top of those with smaller values.

FIG. 2 .
FIG. 2. Surviving samples in the planes of branching ratios versus mH , with colors representing tan β.The branching ratios pertain to the decays of the heavy CP-even Higgs H into t t, b b, τ + τ − , all possible lighter Higgs bosons, and all possible SUSY particles, respectively.Samples with larger values of tan β are plotted on top of those with smaller values.

FIG. 3 .
FIG. 3. Surviving samples are shown in the planes of 1/ tan 2 β/Γtot(H) versus Br(H → t t) (left), heavy Higgs total decay width Γtot(H) versus heavy Higgs mass mH (middle), and cross section of pp → b bH → b bt t versus heavy Higgs mass mH (right), where the colors of the samples indicate tan β.The red and green curves represent the model-independent exclusion and discovery ranges, respectively, for the pp → b bH/A → b bt t channel, with an integrated luminosity of 3 ab −1 at 100 TeV, as taken from Fig.9 in Ref. [53].Samples with larger values of tan β are plotted on top of those with smaller values.

TABLE I .
Four Benchmark Points for Surviving Samples, where σ(X) denotes the cross section σ(pp → b bX → b bt t).Here, H and A represent the doublet-dominated heavy Higgs bosons, while S 2 33 and P 2 22 indicate the singlet components in H and A respectively. to 1/ tan β.This relationship causes the branching ratio of H/A to t t to be inversely proportional to tan β; specifically, a smaller tan β results in a larger branching ratio Br(H → t t).Conversely, the reduced couplings with down-type fermions, C Hdd and C Add , approximate to tan β, leading to branching ratios of H to b b and τ + τ − that are directly proportional to tan β.•When tan β is less than 10, the heavy Higgs H predominantly decays into t t, with the branching ratio Br(H → t t) reaching up to 1.When tan β exceeds 30, it primarily decays into b b, with the branching ratio Br(H → b b) up to 0.8 and Br(H → τ + τ − ) also reaching up to 0.2.For tan β values between 10 and 30, the branching ratio Br(H → SU SY ) is dominant, reaching up to 0.8.