HL-LHC and ILC sensitivities in the hunt for heavy Higgs bosons

The prediction of additional Higgs bosons is one of the key features of physics beyond the Standard Model (SM) that gives rise to an extended Higgs sector. We assess the sensitivity of the Large Hadron Collider (LHC) in the high luminosity (HL) run alone and in combination with a possible future International Linear Collider (ILC) to probe heavy neutral Higgs bosons. We employ the Minimal Supersymmetric Standard Model (MSSM) as a framework and assume the light CP-even MSSM Higgs boson to be the Higgs boson observed at 125 GeV. We discuss the constraints on the MSSM parameter space arising from the precision measurements of the rates of the detected signal at 125 GeV and from direct searches for new heavy Higgs bosons in the $\tau^+\tau^-$, $b\bar{b}$ and di-Higgs ($hh$) final states. A new benchmark scenario for heavy Higgs searches in the $b\bar{b}$ channel is proposed in this context. For the future Higgs rate measurements at the HL-LHC and ILC two different scenarios are investigated, namely the case where the future rate measurements agree with the SM prediction and the case where the rates agree with the predictions of possible realizations of the MSSM Higgs sector in nature.


Introduction
The Large Hadron Collider (LHC) continues to measure the properties of the discovered Higgs boson [1,2] at 125 GeV with increasing precision. So far, given the current experimental and theoretical uncertainties, the measurements are in good agreement with the SM predictions [3][4][5][6]. Nevertheless, in certain parameter regions, models with extended Higgs sectors often feature a SM-like Higgs boson that is compatible with the experimental data. Such models, often motivated by theoretical arguments or observational puzzles (such as dark matter), are therefore equally viable in the light of the Higgs observation.
The MSSM [7][8][9] is one of the best studied models with an extended Higgs sector. In contrast to the case of the SM, the MSSM contains two Higgs doublet fields. This results in five physical Higgs bosons instead of the single Higgs boson in the SM. These are (in the CPconserving case, which is assumed throughout this paper) the light and heavy CP-even Higgs bosons, h and H, the CP-odd Higgs boson, A, and the charged Higgs bosons, H ± . At tree-level the Higgs sector is determined by the ratio of the two vacuum expectation values, tan β, and the mass of the CP-odd Higgs boson, M A . In addition, the MSSM predicts two scalar partners for all fermions as well as fermionic partners for all bosons, which for the electroweak gauge bosons and Higgs bosons are the so-called neutralinos and charginos, or electroweakinos.
In order to facilitate collider searches for the additional MSSM Higgs bosons a set of new benchmark scenarios for MSSM Higgs boson searches at the LHC has been proposed recently [10,11]. The scenarios are compatible -at least over wide portions of their parameter space -with the most recent LHC results for the properties of the Higgs boson at 125 GeV [3][4][5][6] and the bounds on masses and cross sections of new SUSY particles, as well as with global fits of the phenomenological MSSM, see e.g. [12][13][14]. Each scenario contains one CP-even scalar with a mass around 125 GeV which is identified with the observed Higgs boson. Nevertheless, the scenarios differ significantly in their predictions for the light Higgs-boson phenomenology (within the allowed uncertainties), in particular in the non-decoupling regime, as well as in the phenomenology of the additional, so far undetected Higgs bosons. In many cases, such differences occur due to the presence of additional light particles like, for example, the above-mentioned electroweakinos.
The search for the additional Higgs bosons will continue during LHC Run 3 and subsequently at the high luminosity (HL)-LHC [15]. At possible future e + e − colliders, such as the International Linear Collider (ILC), the Compact Linear Collider (CLIC), the Future Circular Collider (FCC)-ee or the Circular Electron Positron Collider (CEPC), the search for extended Higgs sectors can be performed either directly, via the search for new Higgs bosons, or indirectly, via the precise measurements of the properties of the Higgs boson at 125 GeV. The new benchmark scenarios, due to their distinct phenomenology of the MSSM Higgs sector, can be employed to assess the reach of current and future colliders.
In this paper we present the HL-LHC and ILC 1 sensitivities to heavy Higgs bosons in the M 125 h scenario [10], in which all supersymmetric particles are heavy, leaving essentially a Two-Higgs-Doublet Model (THDM)-like Higgs sector at the electroweak scale, and the M 125 h (χ) scenario [10], which features light neutralinos and charginos. For the low tan β region we furthermore study the analogous scenarios M 125 h,EFT and M 125 h,EFT (χ), respectively, which employ a dedicated effective field theory (EFT) calculation and a varying SUSY mass scale [11]. In addition, we define in this work a new benchmark scenario in which the decays of the Higgs bosons to pairs of bottom quarks are enhanced. A selection of our results has already been summarized in the report of the HL/HE-LHC Higgs working group Ref. [15]. This paper is organized as follows: In Section 2 we briefly review the MSSM benchmark models used in this study and introduce the new benchmark scenario with enhanced Higgsbottom-quark couplings. In Section 3 we discuss the experimental input and the projections both for the direct and indirect searches at the HL-LHC and two ILC running scenarios. We present the exclusion reach of HL-LHC heavy Higgs boson searches and the indirect sensitivity from HL-LHC Higgs rate measurements in Section 4. In Section 5 we investigate constraints on BSM physics that can be obtained from the HL-LHC results in combination with precision Higgs rate measurements at the ILC. In this context we analyze both the case where the future rate measurements agree with the SM prediction and the case where the rates agree with the predictions of possible realizations of the MSSM Higgs sector in nature. We conclude in Section 6.

MSSM benchmark models
We perform our study of the physics potential of the HL-LHC and the ILC in the exploration of heavy neutral Higgs bosons in the framework of five MSSM benchmark scenarios. This allows us to compare the sensitivity of direct searches for heavy neutral Higgs bosons with the indirect sensitivity from precision measurements of the light Higgs boson signal rates. The benchmark scenarios chosen here feature quite different predictions for the considered search channels and for the rates of the Higgs boson at 125 GeV, thus providing a certain variety in the phenomenology of the two approaches and for the different collider scenarios.
Four of the benchmark scenarios chosen here were proposed in Refs. [10,11]. We shall furthermore define one new scenario in this work in which the sensitivity of searches for heavy Higgs boson decays to bottom quarks is enhanced (see below). All considered scenarios are defined as two-dimensional planes in M A and tan β, the two parameters that fix the MSSM Higgs-boson sector at tree-level. We emphasize that the five benchmark scenarios considered in this work feature the decoupling limit [16], i.e. the light CP-even Higgs boson acquires SM-like tree-level couplings if M A becomes large, M A M Z . In scenarios where loop corrections to the Higgs-boson mass matrix lead to an accidental alignment without decoupling [16][17][18][19][20], lower bounds on M A from the measurements of the rates of the Higgs boson at 125 GeV become much weaker [10,19]. While scenarios featuring at least one additional Higgs boson that is lighter than the one at 125 GeV are tightly constrained within the MSSM [10] and not considered in the present paper, such a situation can occur generically in singlet extensions of the Higgs sector, see e.g. Refs. [21][22][23][24][25][26][27].
The theory predictions for the Higgs-boson masses, couplings and branching ratios are obtained from FeynHiggs (see below for more details on specific versions employed in this work) [28][29][30][31][32][33][34][35][36][37]. For Higgs production via gluon fusion we make use of SusHi (version 1.7.0) [38][39][40][41][42][43][44][45][46][47][48][49]. For Higgs production via bottom-quark annihilation "matched predictions" are employed [50][51][52][53]. The M 125 h scenario [10] is characterized by relatively heavy superparticles, such that the Higgs phenomenology at the LHC resembles that of a THDM with MSSM-inspired Higgs couplings. The light Higgs boson has SM-like couplings, and in the region M A 1.9 TeV the heavy Higgs bosons can only decay into SM particles (including the light SM-like Higgs boson). For larger values of M A decays into electroweakinos (EWinos) are kinematically open. In contrast, the M 125 h (χ) scenario [10] features light EWinos via the choice of relatively small values of the bino, wino and Higgsino mass parameters, M 1 , M 2 and µ, respectively. This leads to sizable decay rates of the heavy Higgs bosons H and A into charginos and neutralinos throughout the parameter plane, thus diminishing the event yield of the τ + τ − and bb final state signatures that are used to search for the additional Higgs bosons at the LHC. Furthermore, the branching ratio of the light Higgs boson h into a pair of photons is enhanced for small values of tan β due to loop corrections involving light charginos.
In these two scenarios all other SUSY masses are fixed around the TeV scale. Consequently, in both scenarios, the light Higgs-boson mass becomes too small (within experimental and theoretical uncertainties, as taken over from Ref. [10], where a theory uncertainty of ±3 GeV was assumed [30]), M h 122 GeV, for tan β 6, rendering this parameter region phenomenologically inconsistent with the observed state at 125 GeV. For these two standard scenarios we used FeynHiggs version 2.14.3. More details and a discussion of the current constraints from Higgs-boson searches and rate measurements in these two scenarios can be found in Ref. [10].
Low-tan β (EFT) scenarios: M 125 h,EFT and M 125 h,EFT (χ) The M 125 h,EFT and M 125 h,EFT (χ) scenarios [11] have similar phenomenological properties as the standard scenarios described above, but are in agreement with the observed Higgs-boson mass value at lower values of tan β. A light Higgs-boson mass M h 125 GeV is achieved by tuning the scalar top masses to very large values for every point in the parameter plane. Accordingly, for these scenarios we employ the THDM as effective field theory (EFT) below the SUSY scale to calculate the higher-order corrections to the Higgs-boson masses and self-energies [35] [11]. For these two EFT scenarios we used an extended version of FeynHiggs 2.14.3, which includes the THDM as EFT below the SUSY scale [35]. This feature was officially released (alongside other changes) in version 2.16.0.

Large H/A → bb scenario(s)
In the MSSM the relation between the experimentally measured bottom-quark mass, m b , and the bottom-quark Yukawa coupling, h b , is affected by loop corrections which are enhanced for large tan β. These corrections are summarized in the parameter ∆ b ∝ µ tan β [54][55][56][57][58][59] according to Consequently, large negative values of µ enhance the bottom-quark Yukawa coupling and thus lead to enhanced rates of Higgs-boson decays to bottom quarks as well as of Higgs boson production in association with a bottom-quark pair. This enhancement is particularly strong for the heavy non-SM Higgs bosons H and A, while the SM-like light Higgs boson h is only affected to a much lesser extent, as these corrections are suppressed in the decoupling limit. 2 The Higgsino mass parameter µ can therefore via ∆ b lead to a change in the relative strengths of the Higgs boson couplings to bottom quarks and to τ -leptons. This, in turn, changes the relative sensitivity of heavy Higgs boson searches in bb and τ + τ − final states. In order to investigate this change in sensitivity we define one new benchmark scenario, which we denote by M 125 h (µ = −2 TeV) or the short-hand notation M 125,µ− h . 3 The SUSY input parameters are chosen to be All parameters are defined in the on-shell scheme. For the SM input parameters, we follow the recommendations of the LHC-HXSWG [61].
Apart from µ, all parameters are chosen as in the M 125 h scenario, in which µ = +1 TeV. In addition to this default "large H/A → bb scenario", we also investigate two other choices of µ (µ = −1 TeV and µ = −3 TeV). For the scenarios with negative µ we employ FeynHiggs version 2.14.4 which includes additional two-loop ∆ b corrections [62,63] in the Higgs-boson decays.
In Fig. 1 (left) we show the current LHC constraints at 95% confidence level (C.L.) from heavy Higgs searches in the bb and τ + τ − final states as well as from Higgs signal rate measurements for the standard M 125   CMS pp → H/A → τ + τ − search [65] with 35.9 fb −1 of data, both at a center-of-mass energy of 13 TeV, using HiggsBounds [66][67][68][69][70][71]. The indirect constraints from Higgs rate measurements are evaluated with HiggsSignals (version 2.3.0) [72,73] by means of a negative log-likelihood ratio (LLR) test with the SM as alternative hypothesis, and approximating the likelihood with a χ 2 function. This test uses Run-1 [3] and recent Run-2 results up to around 80 fb −1 from ATLAS [74] and CMS [75][76][77][78][79][80][81][82][83][84]. Fig. 1 (left) clearly illustrates that pp → H/A → bb searches become more sensitive for scenarios with large negative µ values due to the enhancement of the bottom-quark Yukawa coupling, as the excluded regions probe lower values of tan β for larger negative µ values. 4 It is noteworthy that the exclusion limit from pp → H/A → τ + τ − searches does not vary significantly with µ. 5 This is because the pp → H/A → bb signal rate profits from an enhancement in the production (in the gg → bbH/A production mode) and in the decay branching ratio (BR) of the H/A → bb decay, while the pp → H/A → τ + τ − signal rate only gains from the enhancement in the production rate whereas in combination with the decay rate BR(H/A → τ + τ − ) a large compensation of ∆ b effects occurs [85]. Still, we observe that heavy Higgs-boson searches in the bb final state cover significantly less parameter space in the benchmark plane than searches in the τ + τ − final state, regardless of the choice of µ.
We can furthermore see in Fig. 1(left) that, currently, the indirect exclusion derived from 125 GeV Higgs signal rate measurements is stronger than the H/A → bb (but not H/A → τ + τ − ) search limits for all choices of µ. At larger tan β values these indirect constraints dominantly originate from deviations of the light Higgs-boson Yukawa coupling to bottom quarks. By approaching the decoupling limit, i.e. with increasing M A , the allowed parameter space opens towards larger tan β values. However, at around (M A , tan β) ∼ (1.75 TeV, 28) for µ = −2 TeV, and at around (M A , tan β) ∼ (1.3 TeV, 19) for µ = −3 TeV, and extending to larger M A values, the exclusion from the 125 GeV Higgs measurements becomes a constant upper limit on tan β in this scenario. This is caused by the light Higgs-boson mass dropping below 122 GeV (see the discussion above) for larger values of tan β for these two choices of µ. Similarly, the light Higgs-boson mass drops below 122 GeV for tan β ≤ 7. This is illustrated in Fig. 1 (right), which shows contours for the lowest acceptable value (taking into account the theoretical uncertainty) of the light-Higgs boson mass, M h = 122 GeV, for the different choices of µ.

Experimental input for HL-LHC and ILC projections
We assess the reach of direct LHC searches in the τ + τ − final state by applying the modelindependent 95% CL limit projections for 6 ab −1 by the CMS experiment [15,86], serving as a proxy for a future ATLAS and CMS search combination using the full HL-LHC data. We implemented these limits -presented as one-dimensional (marginalized) cross section limits on either the gluon fusion or bb-associated production mode -into HiggsBounds [66][67][68][69][70][71] to obtain the projected 95% C.L. exclusion in our scenarios.
For LHC searches for heavy Higgs bosons decaying into bb or di-Higgs (hh) final states, unfortunately, no experimental HL-LHC projection has been performed by ATLAS and CMS. 6 In order to estimate the future sensitivity of these searches, we approximate the future 95% C.L. limit, σ 95%CL HL−LHC , by rescaling the current expected limit, σ 95%CL current , based on the current integrated luminosity, L current , at √ s = 13 TeV, by the expected increase of statistics to the future combined ATLAS and CMS integrated luminosity, L HL−LHC = 6 ab −1 , at a slightly higher center-of-mass energy of 14 TeV, 7 6 See Ref. [87] for a theorists' analysis of the HL-LHC prospects of H → hh searches for various final states. 7 In our naive projection of the heavy Higgs-boson searches in bb and hh final states we neglect effects related to the slightly higher center-of-mass energy √ s = 14 TeV in the HL-LHC runs. As the cross sections for the background processes increase with respect to the current run, the resulting upper cross section limit at 14 TeV will be larger than our estimated σ 95%CL HL−LHC . This degradation however depends strongly on the experimental background estimation methods and the specific process. Furthermore, it will most likely be overcome by the simultaneous increase in the signal cross section for 14 TeV, thus leading overall to a net gain in sensitivity in particular at large Higgs boson masses. In that sense, our projection may be regarded as a conservative estimate.
Obviously, this estimate has to be taken with a grain of salt, as we assumed the full uncertainty to improve like a statistical uncertainty. Our projection of heavy Higgs boson searches in the bb final state is based on the current ATLAS search with L current = 28.7 fb −1 [64], while we use the current CMS H → hh analysis [88] based on L current = 36 fb −1 (which combined several final states) for the projection of searches in the di-Higgs final state. The current limit of these searches is strongly dominated by statistical uncertainties, which motivates our naive projection approach. Only more sophisticated projection studies by ATLAS and/or CMS would give a more reliable picture.
We do not take into account direct searches in the A → Zh final states, as in the MSSM they are expected to be less relevant than the H → hh searches [11]. We also do not assess the HL-LHC reach of searches for heavy Higgs bosons in the tt final state, as there are no official projections available and since the limit is strongly influenced by interference effects between signal and the SM background, as discussed in Refs. [89][90][91][92][93][94][95], and is therefore rather modeldependent. It is clear that these searches will be able to constrain low values of tan β [96,97], where the heavy Higgs bosons predominantly decay into di-top final states. Thus, in particular in the EFT-scenarios they will have a significant impact at low values of tan β, see for comparison Fig. 6 in Ref. [97] for the current hMSSM [98][99][100][101][102] limits obtained with an integrated luminosity of 36 fb −1 . Similarly, also for the charged Higgs-boson searches at the HL-LHC no official projections in benchmark planes are available. 8 Consequently, we also leave charged Higgs sensitivity studies for future work.
We estimate the indirect reach through Higgs rate measurements by using the detailed HL-LHC signal strength projections for the individual Higgs production times decay modes, including the corresponding correlation matrix, as evaluated by the ATLAS and CMS collaborations (see Tab. 35 in Ref. [15]). These projections assume the evolution of systematic uncertainties as estimated by the HL-LHC Working Group 2 (see Section 1.1.3 in Ref. [15] and references therein for details) and are referred to by future scenario S2 or 'YR18' systematic uncertainties. We furthermore take cross-correlations of theoretical rate uncertainties between future ATLAS and CMS measurements into account, assuming that theoretical and parametric uncertainties are halved with respect to current estimates [61] (as prescribed in S2). All projections of Higgs-boson rate measurements are implemented into HiggsSignals, which we use to evaluate the projected reach in the MSSM parameter space.
For the indirect reach of Higgs signal rate measurements at the ILC we consider two future scenarios: First, the inital stage scenario at √ s = 250 GeV with 2 ab −1 of data (denoted ILC250), and second, the ILC program including a second run at 350 GeV with 0.2 ab −1 of data, and a third run at √ s = 500 GeV with 4 ab −1 of data (for brevity we denote this future scenario by ILC500). All ILC runs assume −80% polarization of electrons and +30% polarization of positrons. The projected precisions of various Higgs-boson signal channels are listed in Tab. 21 and 22 of Ref. [104]. We implemented these future anticipated measurements in HiggsSignals. For the Higgs branching ratios, we assume that theoretical and parametric uncertainties are halved (as done for the HL-LHC, see above), while we assume a theoretical uncertainty on the e + e − → Zh (ννh) cross section of 0.5% (1%) [105]. The projections for both future scenarios ILC250 and ILC500 are combined in our analysis with the projected HL-LHC Higgs rate measurements.
For illustration, we shall furthermore compare for specific MSSM parameter points the predictions of Higgs coupling scale factors [61], κ i , with the anticipated precision of the future κ determination at the HL-LHC and ILC, taken from Tabs. 4 and 5 of Ref. [104]. Note that these projections of the future κ i determination are based on identical experimental input as in our study, but have been evaluated in a Bayesian statistical analysis instead of a log-likelihood ratio (LLR) test as employed here.

Sensitivity of the HL-LHC
We start with a discussion of the future HL-LHC sensitivity via rate measurements and direct searches for heavy MSSM Higgs bosons. In the subsequent figures we present in the (M A , tan β) planes of the considered MSSM benchmark scenarios the projected direct and indirect 95% C.L. sensitivity of a future combination of ATLAS and CMS data at the HL-LHC. For comparison, we also include the current sensitivity of the corresponding searches and measurements, as presented in part in Refs. [10,11]. In the next section we shall address the improvements of the indirect reach obtained by ILC measurements.
Our projections in the M 125 h scenario are presented in Fig. 2. Corresponding planes for the M 125 h (χ) and M 125 h,EFT scenario are given in Figs. 3 and 4, respectively. In the figures we show the current limit (magenta dotted line) for the indirect reach of the LHC in the benchmark scenarios, as evaluated in Ref. [10,11], as well as the expected limit from current direct BSM Higgs searches by ATLAS [106] (red dashed line) 9 and CMS [65] (green dashed line) in the τ + τ − final state, using ∼ 36 fb −1 of data from Run 2 at 13 TeV. The dashed black curve and blue filled region indicate the HL-LHC reach via direct heavy Higgs searches in the τ + τ − channel with 6 ab −1 of data (with the dark blue regions indicating the 1 and 2σ experimental uncertainty). The future HL-LHC sensitivity via combined ATLAS and CMS Higgs rate measurements is shown as black dotted contours, accompanied with a hatching of the prospectively excluded region.
Within the M 125 h scenario, displayed in Fig. 2, the indirect constraints at the HL-LHC have the sensitivity for a prospective exclusion limit that is given by a nearly vertical band extending to M A values of around 900 GeV and by a nearly horizontal band with tan β values of up to 6. The nearly vertical band arises from the measurements of the Higgs signal strengths, while the nearly horizontal band is due to the prediction for the mass of the SM-like Higgs boson. In the nearly horizontal band this prediction is below 122 GeV for the parameters of the M 125 h scenario, such that the interpretation of the observed Higgs signal in terms of the light CP-even MSSM Higgs boson h is incompatible in this specific benchmark scenario with the experimental mass value for the adopted theory uncertainty of 3 GeV. Fig. 2 shows that in this scenario the indirect sensitivity of the Higgs rate measurements at the HL-LHC is not  scenario, assuming YR18 systematic uncertainties (scenario S2 in Ref. [15]). The dashed black curve and blue filled region indicate the expected HL-LHC reach via direct heavy Higgs searches in the τ + τ − channel with 6 ab −1 of data (with the dark blue regions indicating the 1 and 2σ uncertainty), whereas the red and green dashed lines show the expected limit from current searches in this channel by ATLAS [106] and CMS [65], respectively. The current and future HL-LHC sensitivity via combined ATLAS and CMS Higgs rate measurements is shown as magenta and black dotted contours, respectively (the latter being accompanied with a hatching of the prospectively excluded region). sufficient to probe parameter regions that are not covered by the direct Higgs searches (black dashed line). Those direct searches in the τ + τ − final state will probe the parameter space up to M A 2.5 TeV for the highest displayed tan β values of tan β ∼ 50. At tan β = 20 the reach extends up to M A 2000 GeV. The change in the curvature of the black dashed line around M A ∼ 1.9 TeV can be understood from the fact that for larger values of M A decays of H and A into electroweakinos open, thus diminishing the event yield of the τ + τ − final state. The kink in the exclusion boundary at M A ∼ 800 GeV is caused by a transition of the main production channel from gluon fusion (low tan β values) to bottom quark associated production (high tan β values). 10 In this scenario the prospective combined sensitivity from direct and indirect searches in the absence of a signal would yield a lower bound on M A of about M A 1200 GeV. In order to correctly interpret this result, the following should be taken into account. As explained above, this bound is not a consequence of prospective Higgs signal strength measurements at  the HL-LHC, but it is rather driven by the direct Higgs search reach in combination with the Higgs-mass prediction. Since by definition for this benchmark scenario all parameters except M A and tan β are set to fixed values, the adopted theoretical uncertainty of the Higgs-mass prediction has a major impact on the resulting bound. For a smaller theoretical uncertainty the allowed region in this scenario would be shifted to larger tan β values, so that the lower bound on M A would rise to values above 2 TeV. On the other hand, in scenarios where the prediction for the mass of the light Higgs boson is compatible with the measured Higgs-boson mass also for low tan β values, the indirect constraints on M A from the rate measurements can exceed the sensitivity from the direct searches (see the discussion below).
The picture is somewhat different in the M 125 h (χ) scenario. Here the large branching ratio of the heavy neutral Higgs boson decaying to charginos and neutralinos already at lower values of M A leads to a strongly reduced direct reach of H/A → τ + τ − searches. The kink in the exclusion boundary at M A ∼ 600 GeV is as in For values of tan β ≤ 12.5 the enhancement of the h → γγ partial width is so large that the HL-LHC has the potential to exclude this parameter region via the precise measurements of the h → γγ rates, which have an anticipated precision of 2.6% [15]. The combination of direct and indirect bounds yields a prospective lower limit of M A 1250 GeV in the M 125 h (χ) scenario. This bound is less sensitive to the theoretical uncertainties of the Higgs-mass prediction than for the M 125 h scenario but instead depends on the mass scale of the charginos that has been chosen in the M 125 h (χ) scenario. In fact, the ongoing searches for electroweakinos in the mass-compressed region put the M 125 h (χ) scenario under some tension as the parameters of the electroweakino sector are fixed to rather low values of M 1 = 160 GeV and M 2 = µ = 180 GeV. However, one can increase the mass parameters of the electroweakino sector by, for instance, +100 GeV without a major impact on the sensitivity reach of the τ + τ − searches. This can be understood from the fact that the heavy Higgs bosons, as long as the decay modes are kinematically open, still show dominant branching ratios into electroweakinos at low and moderate values of tan β. On the other hand, increasing the electroweakino masses significantly lowers the effect of the chargino loop contributions on the h → γγ partial decay width. We will investigate this effect in more detail below in the context of the M 125 h,EFT (χ) scenario (see Fig. 5). The M 125 h,EFT scenario in Fig. 4 is only shown up to tan β = 10, as this reflects the main feature of this scenario, which is to provide access to the low tan β region. Since this scenario allows a light Higgs-boson mass of 125 GeV even for tan β values as low as tan β ∼ 1, the indirect reach through Higgs rate measurements is now almost vertical. The horizontal band yielding a lower bound on tan β in the M 125 h scenario from the compatibility of the Higgs-boson mass prediction with 125 GeV is not present in the M 125 h,EFT scenario. Since the M 125 h,EFT scenario does not feature light charginos, also the lower bound on tan β arising from precise measurements of the h → γγ rates in the M 125 h (χ) scenario does not occur in Fig. 4. Again the indirect reach in M A is similar to the scenarios discussed beforehand, driven by the behavior of the couplings of the light CP-even Higgs boson when approaching the decoupling limit, which is mostly a function of M A and hardly dependent on tan β. The projected sensitivity at the HL-LHC in this scenario therefore corresponds to an almost vertical expected exclusion region for CP-odd Higgs boson masses M A 1000 GeV. The direct search reach for heavy neutral Higgs bosons is similar to the M 125 h scenario for the displayed tan β region. However, due to the fact that the low tan β region is not excluded by the light Higgs-boson mass predictions, the decay H → hh can cover some additional parameter space, up to M A ∼ 1000 GeV for tan β = 1. At low tan β searches for heavy neutral Higgs bosons in the di-top final state would be of relevance as well, but are not further discussed here (see Sec.  largest coverage for tan β values up to tan β ∼ 5.5, while for higher values of tan β the direct searches for heavy Higgs bosons in the τ + τ − final state have the best prospects. In order to cover the low-tan β region, further experimental sensitivity studies for direct searches for H/A → tt, H → hh and A → Zh decays as well as heavy Higgs boson decays into electroweakinos are of interest (see Refs. [87,95] for recent theorists' projections of H/A → tt and H → hh, and Ref. [15] for experimental projections in different scenarios). The searches for decays to electroweakinos are of particular importance in both the M 125 h (χ) and the M 125 h,EFT (χ) scenario, see also Ref. [10,11,108].
We now turn to the second EFT scenario, M 125 h,EFT (χ), with a light EWino spectrum. As for the case of the M 125 h (χ) scenario discussed above, the HL-LHC measurement of the di-photon Higgs-boson signal rate has the potential to set a lower bound on tan β for the chosen values of the chargino masses. In fact, restricting ourselves to the tan β range between 1 and 10 that was originally proposed for this scenario, the entire (M A , tan β) plane of the M 125 h,EFT (χ) scenario can be probed by the HL-LHC measurement of the di-photon Higgs-boson signal rate. Accordingly, this parameter plane could be excluded at the HL-LHC if no deviation from the SM prediction is observed. Therefore, instead of displaying the (M A , tan β) plane, we instead investigate the reach of the HL-LHC in the (M 2 , µ) parameter plane, where M 2 is the soft-breaking wino mass parameter and µ the Higgs mixing parameter. This is shown in Fig. 5, where we highlight the prospective 2σ excluded region, assuming HL-LHC Higgs signal rate measurements that agree with the SM expectation. The results are shown for three different values of tan β = 2.5, 5, 10 and fixed M A = 3 TeV. As can be seen in Fig. 5, the reach in the chargino mass parameters M 2 and µ increases with decreasing tan β, caused by a larger mixing of the charginos with decreasing tan β, which directly impacts the h → γγ partial decay width. Similarly, the largest values of the light chargino mass, Mχ± 1 , can be probed if M 2 ≈ µ, as in this case the chargino mixing is large, and in turn, the Higgs boson coupling to charginos is maximized. For instance, for tan β = 2.5 (5) and M 2 ≈ µ, light chargino masses up to ∼ 255 (190) GeV can be probed at the 2 σ level (in this case, the heavier chargino mass is ∼ 410 (320) GeV). In contrast, in case of a larger hierarchy, M 2 µ or M 2 µ, the smaller of the two mass parameters has to be rather low in order to be able to probe the electroweakino sector via the di-photon signal strength measurements. The nominal values of M 2 and µ that were chosen in the definition of the M 125 h,EFT (χ) scenario, marked by an orange star in Fig. 5, could be probed for tan β 12.5, which is in agreement with the findings in the M 125 h (χ) scenario, see Fig. 3. We emphasize that this indirect probe for electroweakinos via their loop contributions to the h → γγ partial decay width is complementary to the direct searches for electroweakinos at the HL-LHC [109].
95% C.L. sensitivity at HL-LHC (BSM Higgs searches) (left) can be found here for the HL-LHC projections as well: Searches for heavy Higgs bosons in the τ + τ − final state cover a larger area in the (M A , tan β) parameter plane than those in the bb final state, and the H/A → bb search sensitivity shows a strong dependence on the size and sign of µ while there is only a moderate impact on the searches in the τ + τ − final state. On the other hand, Fig. 6 shows that the anticipated reach of heavy Higgs boson searches in the bb final state is competitive with the indirect reach of the anticipated Higgs-boson rate measurements. Except for µ = −3 TeV the direct searches in the bb final state yield a stronger expected exclusion in the high-M A region than the Higgs-boson rate measurements. The flat regions towards large values of M A in the upper bounds on tan β for µ = −2 TeV and µ = −3 TeV are again caused by the fact that the prediction for the light Higgs-boson mass is below 122 GeV in this region (see Fig. 1 (right)), and the same applies to the lower limit in tan β (which is almost identical for all values of µ). However, for M A 2 TeV in the scenario with µ = −2 TeV and for M A 1.5 TeV in the scenario with µ = −3 TeV the Higgs rate measurements provide sensitivity for a non-trivial upper bound on tan β.

Constraints on BSM physics from the prospective rate measurements at the HL-LHC and the ILC
We now extend our investigations to the situation where the results from the HL-LHC are combined with prospective high-precision measurements of the Higgs signal rates at a future linear e + e − collider (for definiteness, we focus on the ILC as the currently most advanced project for which the anticipated precision levels are based on full detector simulations). We do not take into account in this context the capabilities of an e + e − linear collider for detecting new light states, like the light electroweakinos occurring in the benchmark scenarios discussed above and the possible production of additional light Higgs bosons [110][111][112][113][114]. The latter possibility is of less relevance in the benchmark scenarios that we discuss in the present paper, but within the MSSM context can occur in scenarios with (approximate) alignment without decoupling [10]. Light additional Higgs bosons that are compatible with present experimental constraints can occur as a generic feature in extended Higgs sectors with an additional singlet, see e.g. Refs. [21][22][23][24].
In the following we first discuss the indirect constraints on M A that could be inferred in the absence of a deviation from the SM predictions, i.e. under the assumption that the measured rates exactly agree with the SM prediction. It should be obvious from the discussion of the HL-LHC sensitivities in the previous section, where in the considered benchmark scenarios the prospective constraints on M A are already quite far in the decoupling region of the MSSM, that an improved precision of the detected Higgs-boson rates will only have a moderate effect in such a scenario. This is due to the fact that the dependence of the Higgs-boson rates on M A is essentially flat in this region, with only very small deviations from the SM value. It should be noted that a precision measurement of the Higgs-boson rates would have a much higher impact for the case, for instance, where the Higgs boson at 125 GeV would have a non-vanishing decay mode into BSM particles.
In a second line of analysis we investigate scenarios that would correspond to the situation where a particular MSSM parameter point was realized in nature. We show in this context how on the basis of the Higgs rate measurements alone such a scenario could be distinguished from the SM case and how well the parameters M A and tan β could be indirectly constrained.

Impact of the rate measurements for the case where the SM is realized
In Fig. 7 we present the indirect 2σ expected exclusion regions through precision Higgs-boson rate measurements that are obtained under the assumption that the measured rates exactly agree with the SM predictions. The results are given in the scenario this parameter space is largely covered by the indirect constraint from confronting the prediction for the mass of the light Higgs boson with the measured value (see also Fig. 2), but as discussed above this kind of constraint is highly sensitive to the details of the considered benchmark scenario.
The result for the M 125 h (χ) scenario (upper right panel of Fig. 7) shows that there is additional sensitivity in the low-tan β region in comparison to the case of the M 125 h scenario. As already noted in the discussion of Fig. 3 It should be noted that the displayed results in the scenarios with negative µ, see Figs. 6 and 7, can also be interpreted in the following way: the Higgs-rate measurements in the scenarios with negative µ, as a consequence of their dependence on potentially large ∆ b corrections, provide sensitivity to set an upper bound on |µ| (assuming that µ < 0 holds) depending on M A and tan β. As an example, measuring SM Higgs rates at the HL-LHC would exclude the point M A = 1300 GeV and tan β = 20 for µ = −3 TeV but not for µ = −2 TeV. While this sensitivity is restricted to a certain range of M A (see the discussion of Fig. 6) and relies on the assumption µ < 0, it could nevertheless be of interest in the context of future direct searches for supersymmetric particles. We come back to the discussion of the influence of the bottom-quark Yukawa coupling enhancement on the Higgs rates in the next subsection, however at a lower value of tan β, where the effect is not as pronounced, see Fig. 10 below.
As stressed above, the analysis within specific benchmark scenarios, where besides M A and tan β all other SUSY parameters are fixed to specific values by definition, and the assumption used in this section that the detected rates exactly agree with the SM predictions, cannot demonstrate the full potential of the precision measurements of the Higgs boson rates at the HL-LHC and the ILC. In particular, assuming that the underlying nature of the probed physics scenario is the MSSM implies very important correlations between the couplings, the production and decay rates of the Higgs boson at 125 GeV. Thus, a precise measurement of only few observables is already sufficient to determine the decoupling behavior if the underlying structure of the physics scenario is assumed to be known. Instead, if no such assumption is made the full breadth of precision measurements at the HL-LHC and the ILC will be crucial in order to either determine the nature of observed patterns of deviations from the SM or to set constraints on wide classes of possible extensions or alternatives to the SM. Specifically, the model-independent measurement of the total Higgs production cross section of e + e − → Zh at the ILC (which allows a model-independent determination of the Higgs branching ratios and provides a robust method for obtaining the total width with high precision) has no direct impact in our benchmark scenarios, while it is crucial for probing models that both allow for additional Higgs boson decay mode(s) as well as a compensating enhancement of the production rates [115].
Furthermore, the assumption that all future HL-LHC and ILC measurements of Higgs production and decay modes will yield exact agreement with the SM predictions is of course not realistic. Even in the absence of any contribution of new physics one would still expect that the measured values are scattered around the SM values according to their statistical uncertainty.
In order to answer the question whether the actually observed pattern of measurements with a certain amount of data hints at a non-zero deviation from the SM or not, the full set of observables accessible at the HL-LHC and the ILC measured with the highest possible accuracy will be instrumental.

Impact of the rate measurements for the case where a particular MSSM scenario is realized
We now take a different perspective and assume that future precision Higgs-boson rate measurements reveal deviations from the SM prediction, which are compatible with MSSM predictions for the Higgs-boson rates. Specifically, we assume that a certain parameter point within the considered MSSM benchmark scenario is realized and set the central values of all projected HL-LHC and ILC measurements to the predicted rates at this parameter point (while keeping the same relative uncertainty as for the SM case). We then again analyze how well the MSSM parameter space can be indirectly constrained from precision measurements of the rates of the Higgs boson at 125 GeV. 11 We first assume the realization of MSSM parameter points at a moderate value of tan β that are expected to elude a direct 5σ discovery in heavy Higgs boson searches. The chosen points lie close to the border of the 2σ sensitivity of HL-LHC direct searches in the τ + τ − channel. Furthermore we briefly discuss the case that a relatively large value of tan β could be realized in nature. In light of the existing search limits this would mean that the associated value of M A has to be significantly higher, and correspondingly the impact of such a scenario on the rates of the Higgs boson at 125 GeV is expected to be rather small.
We start with the M 125 h scenario in Fig. 8, where we assume in the left panels (M A , tan β) = (700 GeV, 8) and in the right panels (M A , tan β) = (1000 GeV, 8) being realized in nature. In fact, both points will be probed at the 2σ level by heavy Higgs searches in the τ + τ − final state at the HL-LHC, see Fig. 2, which would thus give complementary information. Here we want to focus on the indirect constraints on the parameters that can be obtained from the precision rate measurements of the Higgs boson at 125 GeV. In the upper panels of Fig. 8 we show the parameter regions which are preferred at the 2σ level by the HL-LHC alone (faint red), and in combination with the ILC250 measurements (medium red) and the ILC500 measurements (dark red). The lower panels of Fig. 8 show contours for R V h bb , i.e. the Higgs boson signal rate for pp → V h (V = W ± , Z) production, followed by the decay h → bb, and normalized to its SM prediction (at the same Higgs boson mass). In the considered parameter space R V h bb is very similar to the SM-normalized rates of the ILC processes e + e − → Zh, h → bb and e + e − → ννh, h → bb, as the gg → Zh contribution to R V h bb is subdominant, and the rates of the other production processes contributing to R V h bb scale uniformly with the squared hV V coupling. This signal process is the most sensitive search channel for the bb final state, with an anticipated HL-LHC (ATLAS and CMS combined) precision for the signal strength of around 4.6% in the pp → Zh production mode [15].
As already discussed above, the Higgs rate measurements have almost no sensitivity on tan β in the M 125 The bottom panels in Fig. 8 show that in the 2σ-allowed region, the deviations of R V h bb are 2% for the case where M A = 1000 GeV has been assumed and 4% for the case with M A = 700 GeV. Accordingly, the future HL-LHC measurements of the pp → Zh, h → bb rate have only a limited sensitivity for constraining the parameter space in the considered scenarios. On the other hand, the inclusive rate measurements in the h → γγ, h → V V (V = W ± , Z) and h → τ + τ − channels at the HL-LHC have sensitivity to modifications of the hbb coupling  , 8), the Higgs-to-diphoton rate is suppressed by 7% and 3% with respect to the SM prediction, respectively, as a result of a slightly enhanced bottom-quark Yukawa coupling and its impact on the total Higgs width. The combination of the measurements of the Higgs signal rates at the HL-LHC in various channels involving the product of the production cross sections and decay branching ratios will therefore provide sensitive information on possible deviations from the SM, while it will be non-trivial to disentangle the source of the deviations. Concerning  [104] to Higgs rate measurements at the HL-LHC, where the theoretical assumption κ V ≤ 1 is employed, and including prospective measurements at ILC250 and ILC500 (but without imposing an assumption on κ V ). the prospective rate measurements at the ILC, the most precise Higgs rate measurements will be performed in the e + e − → Zh, h → bb channel during the run at 250 GeV and in the e + e − → ννh, h → bb channel in the 500 GeV run [104], each with a precision at the sub-percent level. The ILC measurements will therefore complement the information obtainable at the HL-LHC with high-precision input on observables that cannot be well exploited at the LHC. The ILC will furthermore provide model-independent measurements of individual branching ratios. This kind of information will be crucial in order to determine the source of possible deviations without invoking model assumptions. In order to investigate the underlying nature of detected deviations from the SM, the indirect constraints that we have discussed here should of course be applied in the context of the information that is available from the direct searches for additional Higgs bosons (see in particular  Fig. 9. The displayed plots, which we denote as "Wäscheleinen-plots" (washing line plots) in the following, show the predicted light Higgs couplings (normalized to the SM prediction) at the assumed MSSM points in the κ framework [61], along with the anticipated 1σ precision of a rather general κ determination [104] from prospective Higgs rate measurements at the HL-LHC and the ILC. 12 It should be noted that the coupling determination at the HL-LHC is based on the theoretical assumption κ V ≤ 1, while no such assumption is needed for the coupling determinations at the ILC. The plots in Fig. 9 show that for the assumed parameter points sizable deviations from the SM only occur for the bottom-quark and tau-lepton Yukawa couplings, represented by κ b and κ τ , respectively. At the HL-LHC the precision of the κ b and κ τ determination is at the 2% and 1.5% level, respectively. One can see that in particular for the assumed point (M A , tan β) = (1000 GeV, 8) the ILC accuracy will be crucial in order to experimentally establish a significant deviation of the rates of the Higgs boson at 125 GeV from the SM predictions. Moreover, the ILC measurements will provide a crucial consistency test of the correlations between couplings, for instance, between κ b and κ τ , and will therefore help to further discriminate between different BSM interpretations.
The prospects for the indirect constraints from the Higgs rate measurements at the HL-LHC and ILC for the assumed parameter points (M A , tan β) = (700 GeV 1400 GeV that restricts the displayed values of tan β to tan β 27 (this feature is not visible in the left panels due to the smaller displayed range of M A ). This cutoff is not a consequence of the rate measurements but arises in this benchmark scenario because of the incompatibility of the light Higgs mass with the observed value, as discussed above (see the right panel of Fig. 1). The impact of the Higgs rate measurements for constraining tan β can be seen in this case for M A 1400 GeV. For the assumed parameter point (M A , tan β) = (1000 GeV, 8) an indirect upper limit on M A from the Higgs rate measurements can only be obtained using the ILC measurements. Here the measurements from later ILC stages (ILC500) will sharpen the upper limits on M A obtained from using the measurements at the initial ILC run (ILC250) by around 400 GeV, independently of the tan β value. For both assumed parameter points the incorporation of the Higgs rate measurements at the ILC leads to a large reduction of the allowed parameter space in the (M A , tan β) plane.
The pattern of the deviations in R V h bb in the preferred regions of the (M A , tan β) plane (lower panels of Fig. 10) is similar to the case of Fig. 8. Thus, also in this case the incorporation of the ILC measurements would lead to a much larger set of observables showing sizable deviations from the SM. This kind of information will be crucial to determine the underlying nature of the detected deviations. As discussed above, those investigations should of course be based on both the direct information from searches and the indirect constraints. For the M 125,µ− h scenario large parts of the parameter region that would be preferred by the prospective Higgs rate measurements are within the 2σ reach of heavy Higgs searches in the τ + τ − and possibly even bb final states at the HL-LHC, see Fig. 6. A robust excess in these searches would provide clues for the mass scale of the heavy Higgs bosons, M A . The 125 GeV Higgs rate measurements could then, together with first potential measurements of the strength of such a heavy Higgs boson signal, allow one to put new physics interpretations under scrutiny and, within the considered scenario, lead to strongly improved constraints on the model parameters.  , 8) and different values of µ the prospects for κ b , where for comparison also the corresponding prediction in the THDM-II (see text) is indicated (dotted line), see text for details. The Higgs couplings in the κ framework predicted in the displayed scenarios are compared with the anticipated 1σ precision from Higgs rate measurements, where at the HL-LHC the theoretical assumption κ V ≤ 1 is employed, while for the results including prospective measurements at ILC250 and ILC500 no assumption on κ V is employed.
In Fig. 11 Fig. 11 indicates the prediction for the THDM type II, disregarding any additional SUSY effect beyond the mixing of the CP-neutral Higgs bosons (obtained approximatively by averaging between the MSSM prediction for µ = −1 TeV (∆ b -enhanced) and the MSSM prediction for µ = +1 TeV (∆ b -suppressed) in the M 125 h scenario). The comparison of the κ b value for the THDM-II with the variation of κ b and κ τ in the MSSM shows that for M A 1 TeV the prospects for distinguishing between the MSSM and a THDM-II via the differences in the higher-order contributions to κ b and κ τ do not appear to be promising. We note this point since parameter points that have been used elsewhere in the literature seem to indicate that such a distinction could be possible. 13 We now turn to the M 125 h,EFT (χ) scenario where we consider as possible realizations the points (M A , tan β) = (700 GeV, 3) and (M A , tan β) = (1000 GeV, 3). In comparison to the previously discussed points in the M 125 h and M 125,µ− h scenarios (as well as its variations with different values of µ), these parameter points are more difficult to probe directly by experimental searches for heavy Higgs bosons, since at such low values of tan β the pp → H/A → τ + τ − rate does not contain a large enhancement factor. Other direct heavy Higgs searches, in particular in the di-top final state and in electroweakino final states as suggested in Refs. [10,11,108] could improve the sensitivity. Moreover, as discussed above, LHC searches for direct production of mass-compressed electroweakinos have the potential to probe a scenario with such low values of tan β in the electroweakino sector and can thus provide complementary information [109].
In Fig. 12  These M A ranges are very similar to those found in the M 125 h scenario for the two assumed points at the same M A values. In particular, for the assumed point with M A = 1 TeV the prospective accuracy of the signal strength measurements at the HL-LHC is not sufficient to place an indirect upper bound on M A at the 2σ level. As in the previous scenarios, the ILC measurements would be essential in this case to obtain an upper bound on M A . In contrast to the scenarios discussed above, however, for the assumed value of tan β = 3 in this scenario with light charginos the Higgs rate measurements would not only provide a high sensitivity for a distinction from the SM but would also allow one to constrain tan β to a narrow range This high sensitivity to tan β is caused by the fact that the chargino contributions to the h → γγ decay rate strongly depend on the chargino mixing, which in turn depends on tan β. The Higgs rate measurements would complement the information from the direct searches for new particles, which in such a scenario could yield a significant excess or even a signal from the production of light electroweakinos. The lower panels of Fig. 12 display two SM-normalized Higgs rates that are of particular importance at the HL-LHC in this case: The inclusive rate for pp → h → V V (V = W ± , Z), denoted R h V V , and the inclusive rate for pp → h → γγ, denoted R h γγ . The displayed results for R h γγ indicate that the di-photon rate is strongly influenced by loop contributions of charginos, which become large at small tan β values. In contrast, the V V rate follows the basic trend that its decoupling behavior is mostly governed by M A , see also the rate R V h bb in the discussion of the M 125 h scenario. The pattern of how the decoupling is approached, however, shows that for low tan β values slightly larger deviations from the SM value occur for the same value of M A than for larger tan β. The interplay of the constraints from in particular these two rate measurements at the HL-LHC leads to the displayed shape of the allowed region that extends to the highest values of M A (going beyond the plotted range in the right plot) in the vicinity of tan β ∼ 3. It should be noted that for the assumed point at M A = 1 TeV (right plot) the enhancement of R h γγ compared to the SM value is even somewhat larger (10%) than for the assumed point at M A = 700 GeV (∼ 6.5%, left plot). This is because at lower M A values the enhancement of the h → bb decay is stronger, which in turn suppresses the h → γγ decay rate via its contribution to the total decay width. For the considered scenario the impact of the Higgs rate measurements at the ILC would mainly be a significant improvement of the indirect constraints on M A .
In Fig. 13 we show contour lines of equal M SUSY in the same parameter space as considered in Fig. 12. Superimposed (as dotted lines) are the expected 2σ-allowed parameter regions shown previously in Fig. 12 for the same MSSM points that we assume to be realized. M SUSY denotes the scale of all scalar fermion soft-SUSY breaking masses. As explained in Sec. Those indirect constraints could of course be significantly improved with the results of the direct searches for additional Higgs bosons and electroweakinos, which in the considered scenario would have good prospects for a significant excess or even a discovery.   Fig. 9, and here the ILC measurements will be crucial to achieve a significant discrimination with respect to the SM prediction.
We now turn to the discussion of the case that a relatively large value of tan β could be realized in nature. For this purpose we choose a heavy Higgs-boson mass of M A = 1.75 TeV. In the M 125 h and M 125 h (χ) scenarios the tan β value is chosen to be tan β = 50, close to the expected exclusion bound of the current pp → H/A → τ + τ − analysis [65]. For the M 125,µ− h scenario we fix tan β = 25, close to the current indirect exclusion from Higgs rate measurements. The chosen value of M A = 1.75 TeV is a "best-case" scenario if the MSSM with a large value of tan β is realized, in the sense that it would certainly lead to a discovery of heavy Higgs bosons at the HL-LHC (see our discussion above of the projections in the different benchmark scenarios) and possibly even already in the near future.
For definiteness, we quote here the 13 TeV signal rates of the processes pp → H/A → τ + τ − and pp → H/A → bb, whose production is completely dominated by bottom-quark associated Higgs production at these parameter points. They are given by It should be noted that for additional Higgs bosons in the mass range considered here the possibility that they could have a sizable branching fraction into other BSM particles should be taken seriously. For instance, for the considered parameter point in the M 125 h (χ) scenario, the decay rates of H and A to electroweakinos amount to around 32%. Therefore, in this scenario not only the decays of the heavy Higgs bosons to third generation leptons and quarks but also the decays into electroweakinos would be promising channels for their detection.
We now turn to the effects in the rates of the Higgs boson at 125 GeV that would arise for the assumed parameter points with M A = 1.75 TeV and large values of tan β. The predicted light Higgs boson couplings, presented in terms of κ scale factors, for the three discussed MSSM scenarios are shown as Wäscheleinen-plots in Fig. 15, in comparison with the precision of the prospective κ determination at the HL-LHC and the ILC. As expected for such a large value of M A , corresponding to a scenario that is quite far in the decoupling region, we find that even for the relatively large chosen values of tan β the coupling deviations from the SM predictions are very small. Deviations at this level will be extremely challenging to resolve experimentally. The ultimate precision of ILC500 seems necessary in order to experimentally establish a significant pattern of deviations. The best chances in this context would occur if a relatively large negative value of µ was realized, as exemplified in the M 125,µ− h scenario in Fig. 15, where an enhancement of the bottom-quark (and perhaps also tau-lepton) Yukawa coupling with respect to the SM prediction could clearly be established at the ILC. In this case the pattern and size of deviations is driven by genuine SUSY effects, i.e. beyond the THDM type II.
We summarize this discussion of MSSM points assumed to be realized in nature by giving the χ 2 values of the SM hypothesis in these assumed scenarios for the prospective accuracies of the rate measurements at the HL-LHC and the ILC in Tab. 1. These χ 2 values can be interpreted either directly as the goodness-of-fit of the SM, i.e. how well it fits the set of observations, or they can be used to perform a likelihood-ratio test that quantifies the significance of the tension between the SM prediction and the MSSM interpretation (with the assumed parameter point   . The predicted Higgs couplings in the κ framework are compared with the anticipated 1σ precision from Higgs rate measurements, where at the HL-LHC the theoretical assumption κ V ≤ 1 is employed, while for the results including prospective measurements at ILC250 and ILC500 no assumption on κ V is employed. used as null hypothesis). Here we want to focus on the latter. As the future measurements will naturally feature statistical fluctuations, we rather refer to the χ 2 of the SM hypothesis as ∆χ 2 SM ≡ χ 2 SM − χ 2 MSSM , where in our projection study with idealized measurements we have χ 2 MSSM = 0 for the considered realized MSSM parameter point. In this likelihood ratio test between two simple hypotheses, with no adjustable model parameters, the levels ∆χ 2 = 4 and 9 correspond to a 2σ and 3σ tension, respectively, between the SM hypothesis and the MSSM hypothesis. It should be noted that this level of sensitivity does not allow one to exclude the SM hypothesis on grounds of the measurements alone, but instead only allows one to discriminate between two models. As these tensions are inferred indirectly from the signal rates of the  SM-like Higgs boson, the combination with the information from direct collider searches, both regarding limits and possible hints for signals, will be crucial to firmly establish an observed pattern of BSM physics and to narrow down its possible nature.
From Tab. 1 we see that in the M 125 h and the M 125,µ− h scenario the HL-LHC will reveal a significant tension between the SM and the MSSM interpretation only if a realization with M A = 700 GeV is assumed. For the larger value of M A = 1000 GeV, the HL-LHC can only discriminate the SM at the 2σ level from the MSSM hypothesis, while the ILC measurements would be crucial to clearly establish a deviation from the SM. The situation is somewhat different in the M 125 h,EFT (χ) scenario at low values of tan β, where the large enhancement of h → γγ from loop contributions of light charginos leads to a very strong tension with the SM, already with its precise determination at the HL-LHC. In fact, such a strong deviation in the h → γγ rate would clearly exclude the SM as null hypothesis, based solely on the goodnessof-fit, with very high significance. For the scenarios with M A = 1.75 TeV shown in Tab. 1 the deviations in the properties of the Higgs boson at 125 GeV are so small that the HL-LHC accuracy for the Higgs rate measurements will yield only small shifts in χ 2 between the SM and the MSSM. A significant distinction between the SM hypothesis and the MSSM hypothesis, with ∆χ 2 > 4, can only be achieved for the M 125,µ− with the present LHC reach and the projected HL-LHC reach for direct searches for additional heavy Higgs bosons. For the direct searches we have taken into account present results and projected sensitivities in the τ + τ − , bb and di-Higgs (hh) final states. Concerning the projections of the rate measurements of the detected Higgs boson at 125 GeV at a future e + e − Higgs factory, we have considered for definiteness the case of the ILC since it is the currently most advanced project, and its projected accuracies are based on full detector simulations. We have performed our analysis for the specific example of the MSSM, based on benchmark scenarios that were defined in Refs. [10,11] and a new benchmark scenario, called the M 125,µ− h scenario, which we have defined in the present paper. The M 125,µ− h scenario is characterized by a relatively large negative value of the parameter µ, namely µ = −2 TeV, leading to an enhancement of the Higgs-boson couplings to bottom quarks via SUSY loop contributions. This new benchmark scenario should be well suited in particular for the presentation of experimental results for the search for heavy Higgs bosons in the H/A → bb decay channel.
Within the M 125 h benchmark scenario, in which all supersymmetric particles have masses above the TeV scale such that the model at the electroweak scale resembles a THDM with MSSM relations in the Higgs sector, we find that the rate measurements at the HL-LHC could set a lower limit of about M A 900 GeV in the absence of any deviation of the measured results from the SM predictions. In this scenario the indirect sensitivity from the rate measurements at the HL-LHC is not sufficient to access allowed parts of the parameter space that would not be covered by the direct searches in the τ + τ − channel. This is in contrast to the M 125 h,EFT scenario, where the region that is compatible with a prediction of about 125 GeV for the mass of the SM-like Higgs boson extends to lower values of tan β around tan β ≥ 1 through a very heavy sfermion spectrum. At such low values of tan β the indirect HL-LHC sensitivity via the rate measurements of the state at 125 GeV surpasses the reach of the standard search for heavy Higgs bosons in the di-tau final state. We have emphasized in this context that the search channel H → hh has the potential to cover parts of the parameter space in the low-tan β region, and it is also of interest to exploit H/A → tt and A → Zh searches.
In addition to the M 125 h and M 125 h,EFT scenarios, where all SUSY particles are relatively heavy, we have also considered the specific case of very light electroweakinos, as defined in the M 125 h (χ) and M 125 h,EFT (χ) scenarios. These scenarios would constitute a particularly favorable case for the HL-LHC, since the light charginos would give rise to large loop contributions to the h → γγ rate (and could also be probed by direct searches for electroweakinos). In the context of a scenario of this kind, the Higgs rate measurements at the HL-LHC (assuming that they agree with the SM prediction) could be used to rule out a certain part of the plane of the two parameters M 2 and µ, which together with tan β fix the chargino sector at tree-level. For instance, at tan β = 2.5 the HL-LHC could indirectly probe mass values of up to ∼ 255 GeV at the 2σ level. We have emphasized in the context of this example of a scenario where some of the SUSY particles are relatively light that the impact of direct searches for heavy Higgs bosons at the LHC can be further strengthened by supplementing the searches in the τ + τ − and bb final states with dedicated searches for the decays of H and A to BSM particles, for instance charginos, neutralinos and sleptons.
We then extended our investigations to the situation where the results from the HL-LHC are combined with prospective high-precision measurements of the Higgs signal rates at a future e + e − machine. Specifically, we considered the combination of the HL-LHC results with the results that could be achieved at the first ILC stage at 250 GeV and 2 ab −1 of data, and furthermore also the additional combination with the results achievable at the ILC at 500 GeV with 4 ab −1 . In this context we did not take into account the capabilities of the ILC for detecting new light states, and we also did not attempt a combination with the prospective results from direct searches at the HL-LHC, but only commented on the possible interplay between the indirect information from the rate measurements and the results of the direct searches.
We first discussed the case where the future rate measurements at the HL-LHC and the ILC exactly agree with the SM predictions. This corresponds to an MSSM scenario that is far in the decoupling limit, where the sensitivity to variations in M A is small. Accordingly, the additional measurements from ILC250 and from ILC500 strengthen the indirect lower bound on M A from the HL-LHC measurements by only a moderate amount of about +100 GeV each in this scenario. It should be noted that in a realistic future analysis not all measurements would uniformly yield a push into the decoupling region, but even in case of the absence of any source of BSM physics statistical fluctuations could mimick hints for new physics. Both the gain in accuracy and the much larger set of high-precision observables that would be available from the combination of HL-LHC and ILC measurements would be instrumental to correctly interpret the observed pattern of the measurements. The analysis of scenarios with a relatively large negative value of µ revealed the interesting feature that the Higgs rate measurements in such a case have the potential to set an upper bound on tan β and / or the absolute value of µ.
We furthermore investigated the question to what extent measurements at the HL-LHC and the ILC can establish significant deviations in the properties of the Higgs boson at 125 GeV from the SM predictions via the Higgs rate measurements if a certain parameter point of the MSSM is realized in nature. In this context we also addressed the issues of how well the parameters M A and tan β can be indirectly constrained and to what extent the SM can be discriminated from the MSSM. This information, obtained on the basis of just the rate measurements of the state at 125 GeV, would be important in order to complement it with the (possibly inconclusive) information from direct searches for new physics. For this analysis we assumed that a particular parameter point of the (M A , tan β) plane of the considered benchmark scenario is realized and we used the corresponding predictions as hypothetical results of the rate measurements performed at the HL-LHC and the ILC. Specifically we considered the M A values of 700 GeV, 1 TeV and 1.75 TeV with different settings for the accompanying tan β value.
For the optimistic case where a value of M A = 700 GeV was realized nature, the accuracy of the rate measurememts at the HL-LHC would allow one to obtain an indirect upper bound on M A of about 1 TeV at the 2σ level, which would further be strengthened with the precision measurements at the ILC. In contrast, for the case where the true value would be M A = 1 TeV the prospective accuracy of the signal strength measurements at the HL-LHC would not be sufficient to place such an indirect upper bound on M A . This would be of relevance not only for discriminating between the SM and effects of new physics but also set a target for the direct searches for additional heavy Higgs bosons. The incorporation of the precision measurements at the ILC would not only have a strong impact regarding the distinction between the SM and the MSSM. Moreover, the much larger set of high-precision observables available from the combination of the HL-LHC with the ILC in comparison to the case of just the HL-LHC would also be crucial for disentangling the underlying nature of observed deviations from the SM. For the benchmark scenarios with light charginos the induced large loop contributions to the h → γγ rate would make it possible, in addition to the constraints on M A , to set tight indirect constraints on tan β. This would be possible via the rate measurements at the HL-LHC if a low value of tan β was realized in nature (we considered the example of tan β = 3). For the case of 1.75 TeV we considered the large values of tan β = 50 and tan β = 25, where for the former value this parameter region will be accessible via the H/A → τ + τ − searches at the LHC in the near future. As expected for such a scenario that is quite far in the decoupling region, the deviations in the Higgs rates from the SM predictions are very small. A significant discrepancy from the SM could only be established with the ultimate precision of ILC500 for the M 125,µ− h scenario.
Besides the analysis of indirect constraints in the (M A , tan β) plane we also displayed the sensitivities in the different scenarios via "Wäscheleinen-plots" (washing line plots) showing the predicted light Higgs couplings (normalized to the SM prediction) at the assumed MSSM points in the κ framework in comparison with the anticipated precision of the κ determination from the prospective Higgs rate measurements at the HL-LHC and the ILC. We summarized the capabilities of the HL-LHC and the two ILC realizations to discriminate between the SM and the MSSM by providing the goodnees-of-fit of the SM assuming the MSSM realizations discussed in this paper.
It should be noted that the analyses in this paper have been carried out within specific MSSM benchmark scenarios, where besides M A and tan β all other SUSY parameters are fixed to specific values by definition. This setting implies large correlations between the different Higgs rates, so that precise measurements of just a few observables already have a large impact on constraining the parameter space. As a consequence, these analyses cannot demonstrate the full potential of the entire set of high-precision observables that will be available by combining the information from the HL-LHC and the ILC. Instead, more realistically if no assumption is made on the underlying structure of the physics scenario that is realized in nature, the full breadth of precision measurements at the HL-LHC and the ILC will be crucial in order to either determine the nature of observed patterns of deviations from the SM or to set constraints on wide classes of possible extensions or alternatives to the SM. For the scenarios considered in this paper this would mean that the correlations arising from assuming the MSSM Higgs sector with the considered parameter settings could actually be tested, and results for the Higgs couplings could be obtained in a model-independent way. through the project FPA2016-78022-P, in part by the "Spanish Red Consolider MultiDark" FPA2017-90566-REDC, and in part by the AEI through the grant IFT Centro de Excelencia Severo Ochoa SEV-2016-0597.