Higgs boson properties and supersymmetry: Constraints and sensitivity from the LHC to an e + e − collider

Higgs boson properties and supersymmetry: Constraints and sensitivity from the LHC to an e+e−collider A. Arbey,1,2 M. Battaglia ,2,3 A. Djouadi ,4,5 F. Mahmoudi,1,2 M. Mühlleitner,6and M. Spira 7 1Universit´e de Lyon, Universit´e Claude Bernard Lyon 1, CNRS/IN2P3, Institut de Physique des 2 Infinis de Lyon, UMR 5822, F-69622, Villeurbanne, France 2CERN, CH–1211 Geneva 23, Switzerland 3Santa Cruz Institute of Particle Physics, University of California, Santa Cruz, California 95064, USA 4CAFPE and Departamento de Física Teórica y del Cosmos, Universidad de Granada, E–18071 Granada, Spain 5NICPB, Rävala pst. 10, 10143 Tallinn, Estonia 6Institute for Theoretical Physics, Karlsruhe Institute of Technology, 76131 Karlsruhe, Germany 7Paul Scherrer Institut, CH–5232 Villigen PSI, Switzerland (Received 24 February 2022; accepted 14 August 2022; published 2 September 2022) The study of the Higgs boson properties offers compelling perspectives for testing the effects of physics beyond the Standard Model and has deep implications for the LHC program and future colliders. Accurate determinations of the Higgs boson properties can provide us with a distinctively precise picture of the Higgs sector, set tight bounds, and predict ranges for the values of new physics model parameters. In this paper, we discuss the constraints on supersymmetry that can be derived by a determination of the Higgs boson mass and couplings. We quantify these constraints by using scans of the 19-parameter space of the so-called phenomenological minimal supersymmetric Standard Model. The fraction of scan points that can be excluded by the Higgs measurements is studied for the coupling measurement accuracies obtained in LHC run 2 and expected for the HL-LHC program and eþe−colliders and contrasted with those derived from missing transverse energy searches at the LHC and from dark matter experiments. DOI: 10.1103/PhysRevD.106.055002 I. INTRODUCTION The discovery [1] of the Higgs boson [2] at the LHC has opened a vast program of studies of its fundamental properties [3], allowing new and intensive tests of the Standard Model (SM) of particle physics as well as indirect and tight constraints on models of new physics beyond it. In this context, supersymmetric models [4–6] were considered for a long time as the most interesting benchmarks for new physics. In these models, the particle spectrum is more than doubled as every SM particle has a partner of different spin and the Higgs sector is extended to contain more states than the sole SM–like Higgs boson that has been observed at the LHC. Although present LHC studies set stringent bounds on the masses of the new particles, to the extent where supersymmetry (SUSY) appears now to be less “natural” than initially thought, it is nevertheless still worthwhile to keep using and studying it as it remains among the best benchmarks for new physics searches and provides a rich laboratory for testing the SM. The determination of the Higgs boson mass and the measurement of its couplings to SM fermions and gauge bosons with sufficient accuracy have crucial implications for supersymmetry. Indeed, while in the SM the properties of the Higgs particle are fixed once its mass is determined, the contributions from the extended Higgs sector and those of the additional SUSY particles may shift the couplings of the SM–like neutral Higgs state and hence, its production rates and decay branching fractions. A precision study of the mass and the production and decay rates is thus essential for establishing the mechanism of electroweak symmetry breaking and of mass generation, for exploring the contributions of new physics models to the Higgs sector and for eventually setting constraints on their parameter spaces. These constraints need to be compared to those obtained from direct searches for the heavier Higgs bosons of the theory and for the SUSY particles in channels with missing transverse energy (MET), as the lightest SUSY particle that always appears at the end of the decay chains is stable and undetectable in the model’s most popular versions with R-parity conservation. The bounds from these searches by the LHC experiments are already significant and will extend to heavier and Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3. PHYSICAL REVIEW D 106, 055002 (2022) 2470-0010=2022=106(5)=055002(28) 055002-1 Published by the American Physical Society heavier SUSY particles, if no signal is observed in the next LHC run (run 3) and in the high–luminosity LHC (HLLHC) program. It is essential to assess the impact of the constraints derived from the Higgs property measurements, which will also improve in accuracy, on those scenarios that survive the tests of the SUSY direct searches. Conversely, it is important to understand how a given accuracy in the Higgs measurements may enable the reconstruction of the new physics model parameters, in the case where deviations from the SM predictions are observed. The experience gained with the current analyses from the LHC run 2 data provides us with firm guidelines for the evolution of the accuracy of the Higgs measurements and of the bounds from new particle searches with the larger datasets that are expected in the future. In defining the accuracy of the determination of the Higgs properties, systematic uncertainties, both theoretical and parametric, will play an important role and need to be properly accounted for. With results for most of the Higgs decay and production channels of interest now in hand and mass bounds set by a broad variety of SUSY searches, the time for a detailed assessment of the interplay between Higgs physics and SUSY at the LHC and beyond has come. After the LHC era, including the HL-LHC program, a new eþe−collider promises to deliver measurements that are inherently more precise and cover virtually all the Higgs decay channels. The importance of these data and the requirements for their accuracy, to be considered with the full set of LHC measurements and bounds already in hand, need to be precisely evaluated. Several analyses, some comprehensive and others more focused, that address this important issue have appeared quite recently [7]. In this paper, we attempt to answer these questions by considering two approaches. First, we study the relation between the Higgs coupling modifiers, κi, and the fundamental SUSY parameters. Then, we explore the sensitivity of the Higgs measurements to SUSY by quantifying the fraction of the scenarios excluded by the Higgs measurements, the constraints on its parameters, but also the sensitivity to their values in case deviations are observed. The study is conducted in the framework of the so-called phenomenological minimal supersymmetric extension of the Standard Model (pMSSM) [8]. The reduction of the viable pMSSM parameter space obtained by imposing the Higgs properties is compared to that derived from direct SUSY searches in the MET channels through the different stages of the LHC program as well as to the bounds derived from flavor physics and dark matter searches. II. STUDY OF HIGGS PROPERTIES IN THE PMSSM A. The pMSSM In the MSSM, two doublets of complex scalar fields of opposite hypercharge, Huand Hd, are required to break spontaneously the electroweak symmetry leading to the presence of five Higgs states, two CP-even Higgs bosons h and H, where the former is considered to be the lightest, a CP-odd Higgs state A, and two charged Higgs bosons H. Because of SUSY constraints, the tree-level masses of the various Higgs bosons and their couplings depend only on two input parameters generally taken to be the pseudoscalar Higgs mass MAand the ratio of the two vacuum expectation values tan β. However, many other MSSM parameters will enter the radiative corrections to the Higgs sector, which are known to play an extremely important role [9–15]. In principle, all soft SUSY–breaking parameters, which in general are of Oð100Þin addition to those of the SM, become relevant. Hence, in the most general MSSM, the analysis of the Higgs sector is tremendously complicated. A phenomenologically more viable MSSM framework, the pMSSM, that is easier to use in practice, can be defined by adopting the following three assumptions: first, all soft SUSY–breaking parameters are real, and there is no new source of CP violation, second, the matrices for the sfermion masses and for the trilinear couplings are all diagonal implying no flavor change at tree level, and third, the soft SUSY– breaking masses and trilinear couplings of the first and second sfermion generations are the same at the electroweak symmetry breaking scale. Making these assumptions will lead to only 22 input parameters in the pMSSM: (i) tan β: the ratio of the two vacuum expectation values (vevs) of the two Higgs doublet fields, which is expected to lie in the range 1≲tan β≲mt=mb; (ii) MA: the mass of the pseudoscalar Higgs boson that ranges from MZto the SUSY–breaking scale; (iii) μ: the Higgsino (supersymmetric) mass parameter (which can have both signs); (iv) M1,M2,M3: the bino, wino, and gluino mass parameters; (v) m˜ Q;m ˜ tR;m˜ bR;m˜ L;m˜τR: the third generation sfermion mass parameters; (vi) At,Ab,Aτ: the third generation trilinear couplings. (vii) m˜ q;m˜ uR;m˜ dR;m ˜ l;m˜ eR: the first and second generation sfermion mass parameters; (viii) Au,Ad,Ae: the first and second generation trilinear couplings; The first and second generation trilinear couplings Au,Ad,andAewill only play a minor role in general and can be ignored in most cases (and, if it is not the case, they can be equated to those of the third generation) so that at the end, one would have 19 basic parameters in practice. This gives the model more predictability and offers an adequate framework for extensive phenomenological studies. B. Higgs masses and couplings Let us now come back to the MSSM Higgs sector and discuss the Higgs masses and mixing angles. In the A. ARBEY et al. PHYS. REV. D 106, 055002 (2022) 055002-2 basis ðHd;HuÞ, the CP even Higgs mass matrix can be written as1 M2¼M2 Zc2 β−sβcβ −sβcβs2 βþM2 As2 β−sβcβ −sβcβc2 β þΔM2 11 ΔM2 12 ΔM2 12 ΔM2 22 ;ð1Þ where we use the short-hand notation sβ≡sin βetc. and introduce the radiative corrections through the general 2×2matrix ΔM2 ij. The masses of the neutral CP even h,Hbosons and the mixing angle αthat diagonalizes the two states can then be written as M2 h=H ¼1 2ðM2 AþM2 ZþΔM2 þ∓NÞ;ð2Þ tan α¼2ΔM2 12 −ðM2 AþM2 ZÞsβ ΔM2 −þðM2 Z−M2 AÞc2βþN;ð3Þ with ΔM2 ¼ΔM2 11 ΔM2 22; N¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi M4 AþM4 Z−2M2 AM2 Zc4βþC q; C¼4ΔM4 12 þðΔM2 −Þ2−2ðM2 A−M2 ZÞ ×ΔM2 −c2β−4ðM2 AþM2 ZÞΔM2 12s2β:ð4Þ The leading radiative corrections to the Higgs mass matrix of Eq. (1) are controlled by the top Yukawa coupling, λt¼ mt=v sin βwith v¼246 GeV, which appears with the second power accompanied by two additional powers of the top mass. We obtain a very simple analytical expression for the correction matrix ΔM2 ij at one-loop if only this contribution is taken into account [9], ΔM2 11 ∼ΔM2 12 ∼0; ΔM2 22 ∼ 3¯ m4 t 2π2v2sin2βlog M2 S ¯ m2 tþX2 t M2 S1−X2 t 12M2 S;ð5Þ where MSis the geometric average of the two stop masses MS¼ffiffiffiffiffiffiffiffiffiffiffiffiffi m˜ t1m˜ t2 pdefined to be the SUSY–breaking scale and Xtis the stop mixing parameter given by Xt¼At−μ=tan βand ¯ mtis the running MS top quark mass at the scale MSto account for the leading two-loop QCD corrections in a renormalization-group improved approach. Other SUSY parameters than Xtsuch as μand Ab, and, in general, the corrections controlled by the bottom Yukawa coupling λb¼mb=v cos βas well as the gaugino mass parameters M1;2;3, provide a small but non-negligible correction to ΔM2 ij and can also have an impact on the loop corrections [10–12,14]. At tree level, the lightest hboson mass is bounded by Mh≤MZjcos 2βj≤MZ≈91 GeV and is thus far from the measured value at the LHC, Mh¼125.09 0.24 GeV [3]. The radiative corrections have therefore to be rather large in order to attain this value. In the leading one-loop approximation above, the maximal value mass Mmax his given by M2 h⟶ MA≫MZM2 Zcos22βþΔM2 22s2 β;ð6Þ and is obtained for the following choice of SUSY parameters [16]: (i) a decoupling regime with heavy A states, MA∼OðTeVÞin order to minimize Higgs mixing; (ii) large values of the parameter tan β,tanβ≳10,inorder to maximize the tree-level contribution MZjcos 2βj; (iii) heavy stop squarks, i.e., large MSvalues to enhance the logarithmic contributions; (iv) a stop trilinear coupling of Xt¼ffiffiffi 6 pMS, the so-called maximal mixing scenario that maximizes the stop loops [16]. If the parameters are optimized as above, the maximal Mhvalue can then reach the level of the measured value Mh¼125 GeV for MS>1TeV. The basic feature of the hMSSM approach [17–19],that we will adopt in most cases for our effective Higgs coupling study, is that we can trade the radiative correction ΔM2 22 of Eq. (5) for the measured Higgs mass value Mh¼125 GeV. In this case, the MSSM Higgs sector with solely the dominant radiative corrections included, can be again described with only two unknown parameters, such as tan βand MAas it was the case at tree-level. The dominant radiative corrections involving the SUSY parameters are fixed by the value of Mh. This observation leads to a rather simple and accurate2parametrization of the MSSM Higgs sector and, more specifically, the heavier CP-even Higgs mass and the CP-even mixing angle can be expressed in terms of MA,Mh,andtanβas 1If the SUSY scale is very large, the evolution from this very high scale down to the electroweak scale could mix the quartic couplings of the MSSM Higgs sector in a nontrivial way such that the structure of the mass matrix at the low energy scale could be different from this expression. However, detailed studies in an effective two Higgs doublet model, that is renormalization group improved to resum the large logarithms involving the SUSY– breaking scale, suggest that this assumption is justified in most cases [15]. 2If the SUSY scale is not extremely high, the approach would need some refinements at high values of tan βand μto describe properly the Yukawa couplings to bottom quarks and τleptons as will be discussed later. In addition, there are subleading contributions to the Higgs mass matrix other than ΔM2 22, but these have been shown to be rather small as will also be discussed. HIGGS BOSON PROPERTIES AND SUPERSYMMETRY: …PHYS. REV. D 106, 055002 (2022) 055002-3 M2 H¼ðM2 AþM2 Z−M2 hÞðM2 Zc2 βþM2 As2 βÞ−M2 AM2 Zc2 2β M2 Zc2 βþM2 As2 β−M2 h ; α¼−arctanðM2 ZþM2 AÞcβsβ M2 Zc2 βþM2 As2 β−M2 h:ð7Þ The mass of the charged Higgs state MHis simply given by the tree-level relation, M2 H¼M2 AþM2 W;ð8Þ as the SUSY radiative corrections in this particular case are known to be small [20]. We can now discuss the production and decay rates of the MSSM Higgs bosons, restricting for the moment to the SM-like one h. C. hproduction and decays In many respects, we are fortunate enough as the mass value of Mh¼125 GeV of the SM–like hparticle allows us to produce the state in several redundant channels and to detect it in a variety of decay modes. First, many production processes have significant rates for a light SM-like Higgs boson. The by far dominant gluon fusion mechanism, gg →h that we will denote ggh, develops a large cross section for Mh¼125 GeV, σtot ggh ≈50 pb at ffiffiffi s p¼14 TeV. For such a Higgs mass, the subleading channels, i.e., the vector boson fusion (VBF) process, qq →hqq and the Higgs–strahlung (hV) mechanisms, q¯ q→hV with V¼W,Z, have cross sections at ffiffiffi s p¼14 TeV that are of the order of, respectively, σtot VBF ≈4pb and σtot hV ≈ 2.5pb when the two channels hZ and hW are combined. These rates would lead to large samples that would allow for a detailed study of the Higgs particle with the large amount of integrated luminosity, L≈3000 fb−1, that is expected to be collected at the high–luminosity option of the LHC (HL–LHC). Even the associated Higgs production with top quark pairs (tth), pp →t¯ th with a cross section of σtot tth ≈0.6pb and, to a much lesser extent, double Higgs production in the dominant gluon– fusion channel (gghh), gg →hh with a cross section of σtot gghh ≈50 fb could be probed with such a high luminosity. Second, for Mh¼125 GeV, the Higgs boson mainly decays into b¯ bpairs, h→b¯ b with a branching ratio of ≈60%, but the decays into massive gauge boson final states, h→WWand ZZ before allowing the gauge bosons to decay leptonically W→lνand Z→ll (l¼e,μ), are also significant with branching ratios of ≈20% and 2.5%, respectively. The leptonic decay channel, h→τþτ− is also of significance with a branching fraction of ≈5% as is the case for the h→gg (≈8%) and h→c¯ c(≈3%) decay modes (that are not detectable at the LHC to first approximation). The clean loop induced decay mode, h→γγ can be easily detected albeit its small branching ratio of 2×10−3. Even the rare h→μþμ−decay with a branching fraction of order 2×10−4has provided first evidence in the ATLAS [21] and CMS [22] latest searches, and the h→Zγ channel should be accessible at the HL-LHC. The branching fractions of the hbosons in our pMSSM scans in the b¯ b,WþW−and ZZ and τþτ−decay channels normalized to the SM predictions are shown in Fig. 1. Some of their correlations are given in Fig. 2. They have been obtained with the program HDECAY [23], used in this study to precisely evaluate the various Higgs partial decay widths and branching ratios and show a broad range of variation away from the SM predictions. D. The invisible Higgs decay width In many extensions of the SM, the light scalar Higgs boson can decay into pairs of non-SM particles. In the case of the MSSM, Higgs decays into squarks and gluinos are kinematically excluded as present experimental bounds constrain these particles to be much heavier than 1 2Mh. These decays are also kinematically closed for the charged sleptons, the two charginos, and the three heavier neutralinos as searches from the LEP experiment have set limits beyond 100 GeVon the masses of these particles. Invisible hboson decays are still possible in the context of a fully unconstrained MSSM in which the various soft A. ARBEY et al. PHYS. REV. D 106, 055002 (2022) 055002-4 SUSY-breaking parameters are unrelated, and they are of two types. The first kinematically still possible SUSY mode for the hboson is the decay into a pair of the lightest neutralino χ0 1 which, in most cases, is the lightest supersymmetric particle (LSP) and is stable if R-parity is conserved. This is largely allowed by present experimental constraints in particular in nonconstrained or nonunified models in which the soft SUSY-breaking gaugino mass parameters M1,M2,M3are not related and the relatively strong experimental bounds on the masses of the charginos, mχ 1≳100 GeV from LEP2, and the gluino, m˜ g≈M3≳1TeV from present LHC searches, do not affect the invisible neutralino, and one could have mχ0 1<1 2Mh. The partial width for the invisible Higgs decay into neutralinos is given by Γðh→χ0 1χ0 1Þ¼GFM2 WMh 2ffiffiffi 2 pπg2 hχ0 1χ0 1 β3 χ;ð9Þ with the neutralino velocity being βχ¼ð1−4m2 χ0 1 =M2 hÞ1=2 and the normalized Higgs-neutralino coupling given by ghχ0 1χ0 1¼ðZ12 −tan θWZ11Þðsin βZ14 −cos βZ13Þ; where Zis the 4×4matrix that diagonalizes the neutralino mass matrix [6]. The decay is important only for moderate and comparable values of the bino mass parameter M1and the Higgsino parameter μ: moderate M1≲Oð60 GeVÞ to have a light enough LSP as one has mχ0 1 ≈M1for M1≲jμj;M 2and comparable values jμj¼OðM1Þas the h boson prefers to couple to neutralinos, which are a mixture of gauginos and Higgsinos,3Z13;Z 14 ¼Oð1Þ. In this parameter range, the decay h→χ0 1χ0 1can be substantial if Mhis above the 2mχ0 1threshold; close to this value, the width is strongly suppressed by the β3 χvelocity factor. VV μ 0 0.5 1 1.5 2 bb μ 0 0.5 1 1.5 2 All pMSSM bb μ 0 0.5 1 1.5 2 ττ μ 0 0.5 1 1.5 2 All pMSSM VV μ 0 0.5 1 1.5 2 γγ μ 0 0.5 1 1.5 2 All pMSSM FIG. 2. Distributions of the correlations between pairs of the h decay branching fractions normalized to their SM predictions for pMSSM points. bb μ 0.8 0.9 1 1.1 1.2 μ1/N dN/d 4− 10 3− 10 2− 10 1− 10 1 All pMSSM VV μ 0.8 0.9 1 1.1 1.2 4− 10 3− 10 2− 10 1− 10 1 All pMSSM ττ μ 0.8 0.9 1 1.1 1.2 4− 10 3− 10 2− 10 1− 10 1 All pMSSM γγ μ 0.8 0.9 1 1.1 1.2 4− 10 3− 10 2− 10 1− 10 1 All pMSSM FIG. 1. Distributions of hdecay branching fractions normalised to their SM prediction, μ, for the b¯ b(top), WþW−and ZZ (second from top), τþτ−(bottom) and γγ (second from bottom) for pMSSM points. 3Note, however, that the LEP2 bound on the lightest chargino mass forces the parameters M2and μto be somewhat large, minðjμj;M2Þ≈mχ 1≳100 GeV. HIGGS BOSON PROPERTIES AND SUPERSYMMETRY: …PHYS. REV. D 106, 055002 (2022) 055002-5 The neutralino LSP with such a mass would have the relic density required by PLANCK results for ΩCDM, since it will annihilate efficiently through the exchange of the h boson [24]. However, in this case, the invisible branching fraction should be relatively small, BRðh→χ0 1χ0 1Þ≲ Oð10%Þas discussed, for instance, in Ref. [25]. Scenarios in which μand M1are small enough to lead to a light LSP with reasonable couplings to the hboson would lead to large rates for chargino-neutralino pair production at the LHC (in particular, a large amount of trilepton events from the process q¯ q→W→χ 1χ0 i→WZχ0 1χ0 1→ 3lþEmis T). However, because these states have rather compressed spectra, with the χ 1;χ0 imasses too close to that of the LSP, the missing energy is small and the process could escape observation at the LHC. Another, albeit less likely, possibility would be that the Higgs boson decays into a pair of sneutrinos which, if they are lighter than the chargino χ 1and the second neutralino χ0 2and the sleptons, would have as only possible decay the channel ˜ ν→νχ0 1with χ0 1the LSP and the decay is thus also invisible. As the experimental lower bound on the ˜ νmasses is rather low, m˜ν≳45 GeV from the invisible Zdecay width [26], there is a tight room 1 2MZ≲m˜ ν≲1 2Mh,forthe decay h→ ˜ ν˜ νto occur. However, as a result of SUð2ÞL invariance, the soft SUSY-breaking sneutrino and left-handed charged slepton mass parameters are related and to cope with the bound on slepton masses from LEP2 searches, m˜ l≳90 GeV, the mass m˜ νLshould be high enough. Nevertheless, a small room is still possible for relatively light sneutrino as a splitting between ˜ νand ˜ lLmasses can be generated by the D terms which, for small values of the common scalar mass ˜ mand for large tan βvalues for which they become maximal, govern the slepton masses (note that additional significant contributions to the D terms could arise beyond the MSSM). As these D terms tend to increase m˜ lLand decrease m˜ ν, sneutrino masses m˜ ν≲1 2Mhcan be obtained while keeping the m˜ l≳90 GeV experimental constraint still valid. The partial width for the decay mode, summing over the three possible sneutrinos, is given by Γðh→ ˜ ν˜ νÞ¼ 3GFM4 Z 8ffiffiffi 2 pπMh g2 h˜ ν˜ νβ˜ ν;β˜ ν¼1− 4m2 ˜ν M2 h1=2 ð10Þ The hboson coupling to sneutrinos, gh˜ν˜ν∝cos 2βin the decoupling limit, is also maximal at high tan βand is much larger than the bottom-quark Yukawa coupling, making the partial width huge. When the decay is not kinematically suppressed, it would dominate all other decays in contrast to current LHC observation. If the possibility of light sneutrinos is to hold, one thus needs to strongly suppress this decay channel and bring it at the few 10% level at most. Two possibilities are then at hand. A first one is that the sneutrino mass is close to the kinematical threshold m˜ν≈1 2M2 hso that the partial decay width in Eq. (10) is strongly suppressed by the sneutrino velocity β˜ ν. A second possibility would be to suppress the h˜ ν˜ ν couplings and hence, adopt a value of tan βthat is very close to unity for which cos 2β→0. Both options can coexist of course. Another possibility would be Higgs decays into the very light gravitinos and the next-to-LSP neutralino χ0 1in gauge mediated SUSY–breaking models, h→χ0 1 ˜ G, with a very long-lived next-to-LSP that decays outside the detector making the photon in the decay χ0 1→ ˜ Gγunobservable. However, for the h→χ0 1 ˜ Grate to be substantial, the scale for SUSY-breaking should be rather low, and this is already excluded in most realistic scenarios [27]. Nevertheless, one should keep this possibility in mind in more complex models. Concretely, all these cases can be parametrized by an invisible Higgs partial width, Δinv ¼Γinv h=Γtot h¼Γðh→invisibleÞ=ΓSMðh→allÞ:ð11Þ For scenarios where the decay channel is open, the rate into invisible particles is expected to be mostly at the few percent to the 10% level, and they cannot yet be excluded by the LHC data [28–31]. E. Direct corrections to the Higgs couplings Higgs decays into quark pairs are strongly affected by (SUSY-)QCD corrections, while (SUSY-)electroweak corrections are in general of moderate size [32,33]. The dominant part of the QCD corrections to the decays into bottom and charm quarks can be absorbed in the running Yukawa coupling if it is evaluated at the scale of the Higgs mass [34]. This requires the introduction of the running quark masses that are defined in the MS scheme in the program HDECAY [23].ThepureQCD corrections beyond the running-mass effects are included up to N4LO [34]. The SUSY-QCD corrections mediated by gluino-squark exchange are fully known up to NLO [33,35] and included in HDECAY. The dominant part of the SUSY-QCD and SUSY–electroweak corrections in the Higgs decays into bottom quarks can be approximated by the Δbterms that at one-loop order are given by [36–38] Δb¼ΔQCD bþΔEW;t bþΔEW;1 bþΔEW;2 b;ð12Þ with the individual contributions, A. ARBEY et al. PHYS. REV. D 106, 055002 (2022) 055002-6 ΔQCD b¼2 3 αs πm˜gμtanβIðm2 ˜ b1 ;m2 ˜ b2 ;m2 ˜ gÞ; ΔEW;t b¼λ2 t ð4πÞ2AtμtanβIðm2 ˜ t1;m2 ˜ t2;μ2Þ; ΔEW;1 b¼−α1 12πM1μtanβf1 3Iðm2 ˜ b1 ;m2 ˜ b2 ;M2 1Þ þc2 b 2þs2 bIðm2 ˜ b1 ;M2 1;μ2Þ þs2 b 2þc2 bIðm2 ˜ b2 ;M2 1;μ2Þ; ΔEW;2 b¼−α2 4πM2μtanβc2 tIðm2 ˜ t1;M2 2;μ2Þþs2 tIðm2 ˜ t2;M2 2;μ2Þ; þc2 b 2Iðm2 ˜ b1 ;M2 2;μ2Þþs2 b 2Iðm2 ˜ b2 ;M2 2;μ2Þ;ð13Þ where αsdenotes the strong coupling, λt¼ffiffiffi 2 pmt=ðvsinβÞ the top Yukawa coupling, α1¼g02=4πand α2¼g2=4πthe electroweak gauge couplings. The masses m˜ g;m˜ b1;2,and m˜ t1;2are the gluino, sbottom, and stop masses. The terms s=ct;b ¼sin =cos θt;b are related to the stop/sbottom mixing angles θt;b. The generic function Iis defined as Iða; b; cÞ¼ab log a bþbc log b cþca log c a ða−bÞðb−cÞða−cÞ:ð14Þ The corresponding Δτterm for the tau-lepton couplings acquires contributions from the EW gauge couplings α1 and α2only. These are given by [36] Δτ¼ΔEW;1 τþΔEW;2 τ;ð15Þ with the individual contributions, ΔEW;1 τ¼α1 4πM1μtan βIðm2 ˜τ1;m 2 ˜τ2;M2 1Þþc2 τ 2−s2 τ ×Iðm2 ˜τ1;M 2 1;μ2Þþs2 τ 2−c2 τIðm2 ˜τ2;M2 1;μ2Þ; ΔEW;2 τ¼−α2 4πM2μtan βIðm2 ˜ ντ;M 2 2;μ2Þ þc2 τ 2Iðm2 ˜τ1;M 2 2;μ2Þþs2 τ 2Iðm2 ˜τ2;M2 2;μ2Þ;ð16Þ where s=cτ¼sin =cos θτis related to the ˜τmixing angle θτ and m˜τ1;2;m˜ ντdenote the stau and tau sneutrino masses, respectively. These Δfðf¼b; τÞterms modify the effective bottom and τYukawa couplings ˜ gϕ fðϕ¼h; H; AÞof the two neutral CP-even states h,Hand the CP-odd state A, as follows [37,38]4: ˜ gh f¼gh f 1þΔf1− Δf tan αtan β; ˜ gH f¼gH f 1þΔf½1þΔf tan α tan β; ˜ gA f¼gA f 1þΔf1− Δf tan2β;ð17Þ in terms of the original Yukawa couplings gϕ f, gh u¼cos α sin β;g h d¼−sin α cos β; gH u¼sin α sin β;g H d¼cos α cos β; gA u¼cot β;g A d¼tan β;ð18Þ for upand down-type fermions, where αis the mixing angle of the neutral CP-even Higgs states. It has been shown that the effective bottom Yukawa couplings absorb the bulk of the SUSY-QCD and -EW corrections to most of the production and decay processes mediated by these couplings up to a remainder of a few percent, while the full SUSY–QCD corrections can reach about 100%. The twoloop QCD corrections to Δbhave been calculated and shown to add a moderate correction of about 10% [39]. They are included in HDECAY [23] thus increasing the reliability of the predictions even in cases of large Δbterms. Similarly to the isospin down-type fermions, direct corrections also affect the ht¯ tcouplings. Contrary to the terms discussed so far, these are suppressed by tan βand hence, can be important only at very low tan β, tan β≈1. The corresponding Δtcorrection is given by Δt¼μcot β2αs 3πm˜gIðm2 ˜ t1;m 2 ˜ t2;m 2 ˜ gÞ þλ2 b ð4πÞ2AbIðm2 ˜ b1 ;m 2 ˜ b2 ;μ2Þ;ð19Þ and hence, only the first term would contribute since λb¼ ffiffiffi 2 pmb=ðvcos βÞis small as there is no relative enhancement by tan βvalues. However, for the same reason, these corrections turn out to be small in total as they are not enhanced by tan βfactors at all. For the decays into photon pairs, the additional SUSYloop contributions mediated by chargino, sfermion, and charged Higgs loops are taken into account. In this way, a significant dependence on the SUSY parameters is induced that can lead to sizeable modifications of the corresponding partial width from the SM expression, if SUSY particles are relatively light. The QCD and EW corrections are small for the light scalar Higgs decay into photons [40,41]. Similar features also apply to the less relevant Higgs decays into Zγ [42] and, for the EW corrections, into gg [43]. In the latter 4Analogous corrections emerge for the muon and strange Yukawa couplings, but they do not play a role in our analysis. HIGGS BOSON PROPERTIES AND SUPERSYMMETRY: …PHYS. REV. D 106, 055002 (2022) 055002-7 case of gluonic decays, the QCD corrections are large [44], however, and are included in HDECAY [23]. The QCD and EW corrections to the H→Zγdecay width are not included in HDECAY. Within the MSSM there are novel Higgs decays as, e.g., the heavy scalar Higgs decay into a pair of two light CPeven Higgs bosons H→hh that plays a role for small values of tan βbelow the t¯ tthreshold [45]. Higher order corrections to this decay mode are only taken into account for the effective trilinear Higgs coupling, obtained in the framework of the RG-improved effective potential, while process-dependent corrections are not included [46]. Finally, the MSSM Higgs bosons can also decay into SUSY particles where the partial decay widths into final states with charginos and neutralinos may be significant for the heavier Higgs particles [47]. The possibility of the light scalar Higgs decay into a pair of neutralinos is reduced to very small and exceptional regions of the MSSM parameter space and is thus not of relevance for our study as discussed in Sec. II D. Higgs decays into sfermion pairs may only play a role for the heavy Higgs bosons but not for the light scalar h[47] in the context of the present LHC bounds on the sfermion masses (except for sneutrinos as discussed before). The MSSM Higgs sector is implemented in HDECAY [23] within the renormalization group-improved effective potential approach, including the dominant topand bottom-Yukawa coupling induced two-loop corrections. The residual uncertainties of this approach on the light scalar Higgs mass amount to about 3–5 GeV, depending on the MSSM scenario. The same perturbative level is also extended to the trilinear and quartic Higgs self-couplings by modifying the official version of the subh subroutine of Refs. [10] accordingly. The present state-of-the-art calculations for the Higgs masses include partial three-loop corrections that reduce the uncertainty on the light scalar Higgs mass to a level of 1–2GeV[48]. These latter contributions are not included in HDECAY. F. Coupling modifiers and effective Higgs couplings The deviations of Higgs production cross sections and decay branching fractions from the SM predictions due to new physics contributions can be analyzed in terms of coupling-modifier terms, κX, in the context of the so-called κ-framework formalism [49]. The factors κXfor a particle X are defined as κX¼gMSSM hXX =gSM HSMXX;ð20Þ so that the hcross sections and partial decay widths to the particle type X, normalized to the ones of the SM Higgs particle HSM, scale as κ2 X. It must be stressed that, under these assumptions, only the Higgs boson couplings of processes existing in the SM are modified by new physics and possible modifications to the kinematics of the production and decay processes are not considered. This is justified in the context of the pMSSM with the lightest neutralino χ0 1being the LSP, provided that Mχ0 1>1 2Mhso that decays to any SUSY particle are kinematically forbidden. The relations between the coupling modifiers κXand the SUSY parameters can also be discussed in the context of the so-called hMSSM approach. This has been shown to be a good approximation to the full MSSM, since its basic underlying assumption, that the radiative corrections to the Higgs mass matrix of Eq. (1) are dominated by the ΔM2 22 entry, is verified in most cases.5 The MSSM couplings normalized to their SM-like values, c0 X¼gMSSM hXX =gSM hXX, correspond to the κXmodifiers when all direct radiative corrections are neglected. These values for the couplings of the light hstate to third generation t,bfermions and V¼W=Z gauge bosons, including the radiative corrections entering in the MSSM Higgs masses and mixing only, are given by c0 V¼sinðβ−αÞ;c 0 t¼gh u;c 0 b¼gh d;ð21Þ with the couplings gh uand gh dof Eq. (18) and the angle α given by Eq. (7). In the decoupling regime with a heavy pseudoscalar Higgs boson, MA≫MZ, the mixing angle α gets close to α≈β−π 2making the hcouplings to fermions and massive gauge bosons SM-like, namely c0 V;c 0 t;c 0 b→1. In this limit, the heavier CP-even and the charged Higgs states become almost degenerate in mass with the CP-odd Aboson, MH≈MH≈MA≫Mh, while the couplings of the neutral Hand Astates become similar. In particular, there are no more Hcouplings to the weak bosons as is the case for the state Aby virtue of CP invariance, gHVV ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1−ðc0 VÞ2 q¼cosðβ−αÞ ⟶ MA≫MZ0≡gAVV ð22Þ In fact, the magnitude of the coupling gHVV is a very good measure of the decoupling limit in which the hcouplings are SM–like. Performing an expansion in terms of the inverse pseudoscalar Higgs mass, one obtains in the approach to this limit, gHVV ⟶ MA≫MZχ≡1 2 M2 Z M2 A sin 4β− 1 2 ΔM2 22 M2 A sin 2β;ð23Þ 5In Ref. [17], the impact of the subleading corrections ΔM2 11 and ΔM2 12 has been proven to be small via a scan of the MSSM parameter space (in particular, the parameters μ;A t;A b;M1; M2;M3, and MS) in which the full radiative corrections to the Higgs sector up to two loops are implemented. Several other independent analyses [19,50] have reached the same conclusions. A. ARBEY et al. PHYS. REV. D 106, 055002 (2022) 055002-8 where the first term is due to the tree-level contribution and the second one to the dominant contribution of the radiative corrections. A look at the tree-level component shows that for both large tan βand tan βvalues close to unity, the decoupling limit is reached more quickly as the expansion parameter involves the factor sin 4βwhich, in the two limiting cases, behaves as sin 4β¼4tan βð1−tan2βÞ ð1þtan2βÞ2 →−4=tan βfor tan β≫1 1−tan2βfor tan β∼1 →0ð24Þ and the gHVV coupling is doubly suppressed by both M2 Z=M2 Aand tan βterms in these limits. In the radiatively generated component, the one-loop correction ΔM2 22 of Eq. (5) involves a term that goes like 1=sin2β, which makes it proportional to −ΔM2 22=M2 A× cot βand thus, vanishes at high tan βvalues. This leads to the well-known fact that the decoupling limit gHVV →0is reached very quickly in this case, in fact as soon as MA≳150–200 GeV. Instead, for tan β≈1, this radiatively generated component is maximal. However, when both the tree-level and radiative components are included, the largest departure of the coupling gHVV from zero for a fixed MAvalue occurs when sin 4β≈−1. This corresponds to an angle β¼3π=8 and hence to the value tan β≈2.4. Similarly to the coupling to gauge bosons, one can write the couplings of the hstate to isospin 1 2and −1 2fermions in the approach to the decoupling limit as c0 t¼sinðβ−αÞþcot βcosðβ−αÞ⟶ MA≫MZ1þχcot β→1 c0 b¼sinðβ−αÞ−tan βcosðβ−αÞ⟶ MA≫MZ1−χtan β→1; ð25Þ where the expansion parameter χ∝1=M2 Ais the same as the one given in Eq. (23). In the approach to the decoupling limit MA≫MZ, the hcouplings to bottom (top) quarks have an additional tan β(cot β) factor. Hence, at high tan β values, the lightest Higgs couplings to bottom quarks can be different from the SM–one even at high MAvalues, contrary to the hcouplings to gauge bosons and top quarks. At low tan βvalues, significant deviations from the decoupling can be observed for all hcouplings even for a relatively heavy Astate. The couplings to bb and ττ as a function of MAfrom our pMSSM scans are compared to the scaling of Eq. (25) with MAin Fig. 3. The simple picture of the hMSSM is altered by additional direct radiative corrections. These contributions can be effectively mapped into few parameters and these can, in principle, be isolated experimentally. The most important direct correction occurs in the case of b-quarks where additional one-loop vertex terms modify the tree-level hb¯ b coupling outside the decoupling regime. They grow as μtan βand are thus very large at high tan β. The dominant component comes from the SUSY-QCD corrections in which sbottoms and gluinos are exchanged in the loops, but an important electroweak contribution comes from stop and chargino loops in addition. In the case of the hb¯ bcoupling, this correction appears only outside the decoupling regime and results in an effective correction term, κb¼˜ gh b¼c0 b×1−Δbcot αcot β 1þΔbð26Þ where the resummation of Eq. (17) has been included. In the decoupling regime, tan αapproaches −1=tan βwhen [GeV] A M 500 1000 1500 2000 b κ 1 1.2 1.4 1.6 1.8 2 [GeV] A M 500 1000 1500 2000 τ κ 1 1.2 1.4 1.6 1.8 2 FIG. 3. The variation of the h0b¯ b(upper) and ττ (lower panel) coupling modifiers as a function of MA. The histograms are obtained with HDECAY for the valid pMSSM scan points. The lines show the scaling of Eq. (25) with MAfor MS¼1000 GeV, Xt¼1000 GeV, and tan β¼2(upper line) and 20 (lower line). The color scale ranges from dark to light according to the increasing fraction of scan points in each bins. HIGGS BOSON PROPERTIES AND SUPERSYMMETRY: …PHYS. REV. D 106, 055002 (2022) 055002-9 taken from those estimated by ATLAS for the analyses listed in Table IV. The 95% CL exclusion of each SUSY point in presence of background only is determined using the CLs method [105]. Results from the present run 2 analyses are projected to the integrated luminosities reachable after run 3 and HLLHC operation by rescaling the signal and background event yields. Accepted pMSSM scan points yielding a rate of SUSY events incompatible with the background-only hypothesis using the CLs method for a given integrated luminosity are considered as “excluded”at the corresponding stage of the LHC program. The other pMSSM points are considered as “not excluded”. The fraction of accepted pMSSM points not excluded by the present searches and beyond the sensitivity of run 3 and the HL-LHC are shown in Fig. 6for the H=A search channels as a function of MAand in Fig. 7for the jet=lþ MET searches as a function of the ˜ t1,˜ g, and ˜ χ0 1masses. In particular, the MAvalues at which more than 90% of the accepted pMSSM points are excluded by the H=A searches at 95% CL are ∼500 GeV for the present run 2 ATLAS data, 1000 GeV for the run 3, and 1400 GeV for the HL-LHC statistics. The fraction of pMSSM points excluded by the H=A →ττ and the jets and leptons þMET searches are summarized in Table V. The jet=lþMET searches have already excluded more than 90% of the accepted pMSSM points up to a gluino mass of 1400 GeV, a lighter scalar top mass of 400 GeV, and a lightest chargino mass of 200 GeV, with an expected sensitivity extending to 2000, 1200, and 500 GeV, respectively, by the end of the HL-LHC program. C. Higgs measurement accuracy at the LHC and e+e−colliders The measurements of the Higgs properties provide at least two sets of constraints on the MSSM. First, the prediction of the Mhvalue in SUSY provides a constraint on the parameters when the measured value is imposed. Given the measured value of 125 GeV, the MSSM parameters need to be properly chosen to reach this mass, for example, by selecting the so-called “maximal mixing”solutions. While imposing the constraint 122 GeV <M h<128 GeV keeps [GeV] 0 A M 500 1000 1500 2000 2500 Fract. of non-Excluded pMSSM Points 2− 10 1− 10 1 -1 LHC 140 fb-1 LHC 400 fb -1 HL-LHC 4000 fb FIG. 6. Fraction of accepted pMSSM points not excluded at 95% CL by the present H=A searches and beyond the expected sensitivity of run 3 and HL-LHC as a function of MA. [GeV] stop 1 M 0 1000 2000 3000 4000 5000 6000 Fract. of non-Rejected pMSSM Points 1− 10 1 LHC 140 fb-1 LHC 400 fb-1 [GeV] gluino M 01000 2000 3000 4000 5000 6000 Fract. of non-Rejected pMSSM Points 1− 10 1 LHC 140 fb-1 LHC 400 fb-1 [GeV] 1 ± χ M 0 500 1000 1500 2000 Fract. of non-Excluded pMSSM Points 2− 10 1− 10 1 -1 LHC 140 fb-1 LHC 400 fb -1 HL-LHC 4000 fb FIG. 7. Fraction of accepted pMSSM points not excluded at 95% CL by the present jet=lþMET searches and beyond the expected sensitivity of run 3 and HL-LHC as a function of the scalar top (top panel), gluino (center panel), and lightest chargino (bottom panel) masses. A. ARBEY et al. PHYS. REV. D 106, 055002 (2022) 055002-16 only ∼10% of the accepted pMSSM points, the Mhvalue per se does not provide any discrimination between SM and SUSY Higgs. Then, the Higgs decay yields in the accessible final states provide us with an opportunity to further constrain the MSSM parameter space and possibly tell a SUSY Higgs from the SM boson. In this study, we consider the values of the coupling modifiers, κX, obtained by ATLAS combining the present analyses of run 2 data with luminosities up to 139 fb−1 [119] given in Table VI and the HL-LHC program [120] given in Table VII. The accuracies for the determination of the Higgs couplings at future eþe−Higgs factories are taken from the compilation in [121] and given in Table VII. For each accepted pMSSM point, the values of the coupling modifiers, κX, for the accessible channels are computed using HDECAY. The compatibility of the pMSSM with the LHC measurements is determined by computing the χ2value as χ2¼xTM−1x, where xis the vector, of size N, of the differences between the values of the coupling modifiers predicted for the pMSSM point xpMSSM and those measured by the experiment xexp (or the SM predictions for the future results) and Mis the N×Ncovariance matrix of the measurement, including correlations, reported by the experiment [119]. This expression extends the χ2definition to a set of Ncorrelated variables. The compatibility of each pMSSM point is determined at a given CL from the χ2 probability Probðχ2;NÞ, where the number of degrees of freedom corresponds to the number of observables, N.For the present results of Ref. [119], the fitted values are used as central values xexp, while for the projected performance at run 3 and HL-LHC the xexp are taken to be the SM values and the correlation matrix is assumed to be the same as that of the current combination [119]. At an eþe−collider, the eþe−→Zh →llXprocess is available for a model-independent determination of the production cross section from the recoil mass of the ll system and the branching fraction can be directly measured [122]. The accuracy in the determination of the Higgs decay branching fractions and couplings have been studied in great details first for an eþe−linear collider at ffiffiffi s p¼ 250 −500 GeV [123,124] and, more recently, extended to higher center-of-mass energies. Results obtained first with fast simulation have been confirmed by subsequent studies based on detailed GEANT -4 full simulation and reconstruction with the inclusion of (at least some of) the beam-induced backgrounds. Detector R&D has also greatly progressed towards a validation of the unprecedented response performances assumed in the ILC studies. An important outcome of the studies of eþe−collisions at energies above 500 GeV has been the indications of significant improvements in the determination of the Higgs properties through the eþe−→WWνν →hνν fusion process, whose cross section increases ∝log s M2 h and exceeds the peak value of the eþe−→hZ Higgs-strahlung cross section for ffiffiffi s penergies ≥480 GeV. TABLE V. Fractions of accepted pMSSM points excluded by the LHC H=A →ττ,ZZ,t¯ t, and the jet=lþMET searches at three stages of the LHC program. LHC LHC HL-LHC 140 fb−1400 fb−14ab−1 H=A →ττ,ZZ,t¯ t0.10 0.12 0.19 þj=lsþMET 0.47 0.52 0.65 TABLE VII. Assumed accuracies on the hboson coupling modifiers κifor future collider projects (from Ref. [121]). ILC ILC ILC FCC-ee FCC-ee Channel HL-LHC 250 GeV 500 GeV 1 TeV 240 GeV 365 GeV κW0.017 0.0180 0.0029 0.0024 0.013 0.0043 κZ0.015 0.0029 0.0023 0.0022 0.0020 0.0017 κt0.033  0.0690 0.016   κb0.036 0.0180 0.0058 0.0048 0.0130 0.0067 κc 0.025 0.0130 0.0090 0.018 0.013 κτ0.019 0.0190 0.0070 0.0057 0.0140 0.0073 κγ0.019 0.0670 0.034 0.019 0.047 0.039 κg0.023 0.0230 0.0097 0.0066 0.0170 0.0100 TABLE VI. Best fit values for the Higgs boson coupling modifiers κXfrom the combination of the ATLAS measurements, with effective photon and gluon couplings under the assumption that the SM decay channel saturates the Higgs decay width [119]. 13 TeV Channel 25–79.8fb−1 κW1.06 0.06 κZ0.99 0.06 κt0.92 0.10 κb0.87 0.11 κτ0.92 0.07 κγ1.04 0.06 κg0.92þ0.07 −0.06 HIGGS BOSON PROPERTIES AND SUPERSYMMETRY: …PHYS. REV. D 106, 055002 (2022) 055002-17 V. RESULTS In this section, we analyze the potential of measurements of the Higgs boson effective couplings at the LHC and future colliders, discussed in the previous sections, to exclude, or identify, MSSM solutions in the parameter space of the pMSSM not excluded by direct searches for heavy Higgs bosons and for SUSY particles in channels with missing ETsignatures. The analysis uses as inputs the coupling modifier terms, κX, obtained in the context of the so-called κframework, discussed in Sec. III,by assuming the central values and uncertainties summarized in Sec. IV C. First, we compare the distributions of these coupling modifiers predicted for the valid pMSSM points in our scans to those obtained for the points not excluded by the direct searches in run 2 and to the current measurements (see Fig. 8). It is also instructive to compare the coupling modifier values, κX, for the pMSSM points to those obtained so far for the LHC data by correlating particles as shown in Fig. 9. In all these comparisons, the current experimental accuracy is close to the full spread of the pMSSM predictions. In this sense, the statement that the properties of the observed Higgs boson are SM-like could also be rephrased by saying that they are also MSSM-like. However, a fraction of the pMSSM solutions is found to be incompatible with these measurements, and more will be tested at the HL-LHC and later by a possible eþe−Higgs factory. It is interesting to understand the Higgs coupling properties of the points that are preferentially discarded by direct searches. Comparing the κXdistributions for all the valid pMSSM points to those for the points not excluded by the direct LHC searches for heavy Higgs bosons and those for SUSY particles in missing ETchannels, it is evident that these preferentially excluded pMSSM points are located on 1/N dN/dk 5− 10 4− 10 3− 10 2− 10 1− 10 1 10 All pMSSM pMSSM not Excl. LHC (140 fb-1) ATLAS CMS b κ 0.9 1 1.1 1.2 1.3 1.4 1.5 post Run2/All 0 0.5 1 1/N dN/dk 5− 10 4− 10 3− 10 2− 10 1− 10 1 10 All pMSSM pMSSM not Excl. LHC (140 fb-1) ATLAS CMS τ κ 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 post Run2/All 0 0.5 1 1/N dN/dk 5− 10 4− 10 3− 10 2− 10 1− 10 1 10 All pMSSM pMSSM not Excl. LHC (140 fb-1) ATLAS CMS t κ 0.85 0.9 0.95 1 1.05 1.1 post Run2/All 0 0.5 1 1/N dN/dk 5− 10 4− 10 3− 10 2− 10 1− 10 1 10 All pMSSM pMSSM not Excl. LHC (140 fb-1) ATLAS CMS g κ 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 post Run2/All 0 0.5 1 FIG. 8. hHiggs boson coupling modifiers, κX,tobquarks (upper left), τleptons (upper right), top quarks (lower left) and gluons (lower right) for all valid pMSSM points and those not excluded by the LHC run 2 searches compared to the present measurements by the ATLAS [119] and CMS [125] experiments. The lower panels show the fractions of nonexcluded pMSSM points as a function of κX. A. ARBEY et al. PHYS. REV. D 106, 055002 (2022) 055002-18 the tails of kidistributions at values away from the SM predictions, corresponding to κX¼1. The large reduction of pMSSM points at κvalues significantly above the SM expectations for the bquarks and τleptons induced by the SUSY searches is mostly due to the constraint on MAset by the searches in the H=A →τþτ−channel. These effects are discussed in quantitative terms in the next section. This is done by studying the fractions of valid and accepted pMSSM points that can be excluded by the Higgs coupling measurements with the accuracies expected for the next steps of the LHC program and for future colliders in relation to their observability at the LHC in new particle searches. In addition, the constraints that can be derived from these measurements on the relevant MSSM parameters are discussed. A. Constraints on the pMSSM The compatibility of the Higgs couplings with the predictions for the SM or for specific MSSM scenarios defines a way to study the sensitivity for identifying SUSY and discriminate between different scenarios as a function of the collider data accuracy in the light Higgs couplings. Since increasing the integrated luminosity increases the precision of the Higgs property measurements as well as the mass bounds from direct searches, if no signal is observed, this study estimates also the complementarity between the direct (jets=lsþMET and heavy Higgs bosons) and indirect (Higgs properties) probes of new physics in the context of SUSY. In particular, we study how the indirect sensitivity evolves with increasing the accuracy of the Higgs measurements and determine the expected impact of the improved accuracy from an eþe− Higgs factory, given the current reach of the SUSY searches by the LHC experiments. The test is performed by computing the χ2probability of the Higgs observables for each pMSSM point with respect to the actual measurements or the SM hypothesis and measuring the fraction of accepted pMSSM points incompatible with the SM at the 90% CL. This is the fraction of V κ 0 0.5 1 1.5 2 b κ 0 0.5 1 1.5 2 ATLAS W 95% C.L. ATLAS Z 95% C.L. All pMSSM pMSSM not Excl. LHC (140 fb-1) t κ 0 0.5 1 1.5 2 b κ 0 0.5 1 1.5 2 ATLAS 95% C.L. All pMSSM pMSSM not Excl. LHC (140 fb-1) γ κ 0 0.5 1 1.5 2 b κ 0 0.5 1 1.5 2 ATLAS 95% C.L. All pMSSM pMSSM not Excl. LHC (140 fb-1) τ κ 0 0.5 1 1.5 2 b κ 0 0.5 1 1.5 2 ATLAS 95% C.L. All pMSSM pMSSM not Excl. LHC (140 fb-1) FIG. 9. Correlations of the hHiggs boson κcoupling modifiers comparing the valid pMSSM points, those not excluded by the LHC run 2 searches and the 95% CL contours of the current measurements by the ATLAS experiment [119]. HIGGS BOSON PROPERTIES AND SUPERSYMMETRY: …PHYS. REV. D 106, 055002 (2022) 055002-19 pMSSM points that can be tested and excluded, assuming that the measured Higgs couplings coincide exactly with those predicted by the SM. The fraction of points within the range of the scans performed here (see Table II) excluded by the SUSY direct searches (see Table IV) are summarized in Table V. Considering the valid and accepted pMSSM points and following the fractions of those that can be excluded by the Higgs measurements at the LHC, we observe that these fractions range from 1% to 2% and 5% to 8% for the run 2 and HL-LHC datasets, respectively. If we restrict ourselves to considering the points that can be excluded only by the Higgs coupling measurements and are not excluded by the direct SUSY searches (heavy Higgs bosons and scalar quarks, leptons, and gauginos) conducted on the same data sets, these fractions become approximately 1% and 2% (see Table VIII. Despite the increasing accuracy of the Higgs measurements, the fraction of pMSSM points that can be excluded by the Higgs coupling measurements during the LHC program remains small. However, their increase indicates that the improvement of the sensitivity obtained by the Higgs measurements with higher accuracy beats the estimated increase in sensitivity of the direct searches moving from run 2 to the HL-LHC. The fractions of points excluded by the Higgs couplings at the LHC and at future eþe−colliders are presented in Table IX, in relation to the current sensitivity of the direct searches at LHC run 2. The reduction of sensitivity of the Higgs coupling measurements observed for the accepted points, fulfilling the flavor physics constraints, and for the points not excluded by the direct searches comes almost entirely from the exclusion of the low to moderate MA scenarios in both cases. The exclusion of the pMSSM at moderate values of MApushes the hboson into the decoupling regime. The deviations of the Higgs couplings due to the Δbeffect vanishing in the decoupling regime, the constraints from missing ETsearches on the mass of scalar quarks and gauginos have only a minor effect. By improving the accuracy of these Higgs coupling measurements to the percent, or subpercent, level, as expected at future eþe− Higgs factories, the ILC and the FCC-ee, can test up to about 10%–12% of the pMSSM points not excluded by the current LHC direct SUSY searches and flavor physics data (see Table IX), provided they can operate at a large enough energy to perform a full study of the Higgs profile. B. Extracting MSSM parameters from the Higgs profile By studying the Higgs couplings as a function of the MSSM parameters for the accepted pMSSM points, we observe four main groups of parameters to which the Higgs couplings are sensitive. In general, these are: MA,M˜ g; ˜ b;˜ t;˜τ, μtan β, and M˜ χ0 1, but this sensitivity is strongly reduced when only the pMSSM points compatible with the LHC run 2 searches are considered. The small fractions of pMSSM points viable after the run 2 searches that can be excluded by future precision measurements of the Higgs couplings, discussed in the previous section, highlight the decoupling properties of the lightest MSSM Higgs boson, once the value of MAis constrained to sufficiently large values. In this regime, the TABLE VIII. Fraction of pMSSM points excluded at 90% CL by the Higgs coupling measurements with the LHC run 2 and the estimated HL-LHC accuracies. The first lines of each block gives the fractions for all valid and accepted pMSSM points, the following row the fractions of the pMSSM points excluded only by Higgs couplings and not by the searches for H=A heavy Higgs bosons and those in the jets and/or leptons þMET channels with the run 2 and the HL-LHC sensitivity. pMSSM LHC HL-LHC Selection 140 fb−13ab−1 Valid excluded by Higgs couplings 0.023 0.077 Excluded only by Higgs couplings 0.015 0.024 Accepted excluded by Higgs couplings 0.012 0.047 Excluded only by Higgs couplings 0.006 0.019 TABLE IX. Fraction of pMSSM points excluded at 90% CL by the Higgs coupling measurements assuming SM central values and the accuracies estimated for the HL-LHC and the eþe−ILC and FCC-ee Higgs factory proposals. The fractions excluded by the current LHC run-2 Higgs coupling results are also given in the first column. The first lines of each block gives the fractions of all valid (neutralino LSP þ122 <M h<128 GeV) and accepted (valid þflavor þΩχh2) pMSSM points, respectively, followed by the fractions for the points not excluded by the direct searches for H=A heavy Higgs bosons and also by those in the jets and/or leptons þ MET channels with the run 2 sensitivity. pMSSM LHC HL-LHC ILC ILC ILC FCC-ee FCC-ee Selection 140 fb−13ab−1250 GeV 500 GeV 1 TeV 240 GeV 365 GeV Valid 0.023 0.077 0.083 0.166 0.191 0.102 0.156 H=A (LHC run-2) 0.017 0.061 0.068 0.115 0.133 0.078 0.108 H=A and j=lþMET (LHC run-2) 0.016 0.059 0.066 0.114 0.130 0.076 0.106 Accepted 0.012 0.048 0.054 0.137 0.161 0.072 0.125 H=A (LHC run-2) 0.012 0.047 0.053 0.107 0.125 0.067 0.098 H=A &j=lþMET (LHC run-2) 0.011 0.046 0.051 0.106 0.123 0.065 0.097 A. ARBEY et al. PHYS. REV. D 106, 055002 (2022) 055002-20 modifications of the Higgs coupling to b¯ bas a function of M˜g; ˜ b;˜ t;˜τand μtan βthrough the Δbterm are suppressed. Among the pMSSM parameters inducing effects on the Higgs couplings, discussed in Secs. II and III, only M˜ χ0 1, through invisible decays if Mχ<0.5Mh, and MAremain viable. 1. M˜ χ0 1 If Mχ<0.5Mhthe hcan decay to neutralino pairs, as discussed in Sec. II D. The fraction of accepted pMSSM points compatible with the Higgs couplings is shown as a function of M2μtan β, relevant to the determination of the Higgs invisible decay rate, in Fig. 10. Limits on the invisible decay rate can be used to constrain the value of the neutralino LSP mass. As we have already pointed out, the hχχ coupling also controls the χpscattering cross section thus introducing a correlation between the rate of h→χχ, the scattering cross section and the neutralino relic density, Ω˜χ0 1. A large rate for invisible Higgs decays implies a large χpscattering cross section, because they are both due to an enhanced hχχ coupling. This is shown in Fig. 11 visualizing the predicted spinindependent χNscattering cross section on nucleons in the portion of our pMSSM points with low M˜ χ0 1values highlighting the points with sizeable h→χχ branching fractions. Bounds on the χscattering cross section place nontrivial constraints on the Higgs invisible rate as shown in the upper panel in Fig. 11. These bounds are relaxed but not invalidated even when the predicted scattering cross section is rescaled by the ratio Ωχ=ΩCDM for points having neutralino relic density significantly lower that the current PLANCK result for ΩCDM (see the lower panel in Fig. 11). In particular, the XENON-1T upper limit on the scattering cross section on nucleons for neutralino masses below Mh=2from the 1.0 ton-yr exposure [101] removes almost all of the MSSM solutions having BRðh→χχÞabove 0.01 and provides a competitive constraint compared to the direct current upper bound on Higgs invisible decays at 0.11 [28–31] within the MSSM with the lightest neutralino as the only source of dark matter. This reduces the invisible Higgs rate likely below the sensitivity at the LHC, determined either directly through ] 2 [TeVβ tan μ 2 M 600−400−200−0 200 400 600 Fract. of non-Excluded pMSSM Points 0 0.2 0.4 0.6 0.8 1 Higgs Couplings 95% Exclusion -1 LHC 140 fb HL LHC ILC 1000 FIG. 10. Fraction of accepted pMSSM points not excluded at the 95% of CL by the Higgs couplings as a function of M2μtan β for the present run 2 ATLAS results (dark grey) and the expected HL-LHC (medium grey) and ILC-1000 (light grey) accuracies, assuming SM central values. FIG. 11. Predicted spin-independent χNscattering cross section on nucleons, as a function of M˜ χ0 1values with highlighted pMSSM points with sizeable h→χχ branching fractions. Points with sizeable h→χχ branching fractions are shown in color, the darker shade indicating those with branching fractions exceeding the ATLAS upper bounds on invisible Higgs boson decays [31]. The line represents the upper bound from the XENON 1T data. The lower panel has the χNscattering cross section values rescaled by Ωχ=ΩCDM. HIGGS BOSON PROPERTIES AND SUPERSYMMETRY: …PHYS. REV. D 106, 055002 (2022) 055002-21 ZH and VBF production or indirectly through the sum of the Higgs rates, and makes it unique to the eþe−collider program. In our pMSSM scenario with Mχ¼58.3GeV, the fit to the Higgs branching fractions makes it possible to indirectly reconstruct the neutralino mass with better than 10% relative statistical accuracy. 2. MA The sensitivity to the pseudoscalar Higgs boson mass, MA, is the main MSSM benchmark for the light Higgs coupling measurements. The scaling of the coupling deviations with M2 Z=M2 A(see Sec. II F) offers us with an opportunity to discriminate a MSSM hfrom the SM H boson and to infer the mass of the pseudoscalar state from the precision measurements of the properties of the lighter state. The fraction of accepted pMSSM points excluded by the hcoupling determination for different accuracies is shown as a function of MAin Fig. 12. We have verified that our scans contain points corresponding to the so-called “alignment scenario”, where small MAvalues correspond to SM-like Higgs couplings [126]. However these points in our scans are removed by flavor data and direct searches at the LHC. The MAvalues at which more than 90% (95%) of the accepted pMSSM points are excluded by the Higgs couplings at 95% CL are 300 (275) GeV for the present run 2 ATLAS data, 675 (525) GeV, and 1075 (925) GeV for the expected HL-LHC and ILC-1000 accuracies assuming SM central values. This picture changes when only the pMSSM points not excluded by the jet=lþMET searches on the LHC run 2 data are considered. These searches remove a significant fraction of points with relatively light pseudoscalars, as discussed in Sec. IV B. These points see different contributions to the Higgs couplings, from MA as well as from the lighter SUSY particles, resulting in a spread of the values of the couplings. Once these points are largely removed, the dependence of the Higgs couplings on MAbecomes dominant, and the bounds on MA are improved. The MAvalues at which more than 90% (95%) of the pMSSM points not excluded by the SUSY searches as listed in Table IV are excluded by the Higgs couplings at 95% CL are 350 (325) GeV for the present run 2 ATLAS data, 750 (725) GeV and 1225 (1165) GeV for the expected HL-LHC and ILC-1000 accuracies assuming again SM central values. By the end of the LHC program, if no deviation from the SM prediction is observed, the Higgs couplings will probe MAup to 750 GeV. Owing to its improved accuracy, an eþe−Higgs factory is expected to extend this indirect sensitivity up to heavy Higgs boson masses of ≃1200 GeV (see Fig. 13). This reach is comparable to that of the direct H=A searches at the HL-LHC, discussed in Sec. IV B (see Fig. 6). As expected, the errors on the reconstructed mass increase with the MAvalue. Further, the accurate determination of the hcouplings can be used to estimate MA, in the case the measured values significantly deviate from the SM predictions. We evaluate the accuracy of this estimate as a function of the value of MAby considering a set of benchmark scenarios, not excluded by run 2 data, summarized in Table X. The h couplings and branching fractions are computed for these benchmarks and compared to the value for our accepted [GeV] 0 A M 500 1000 1500 2000 2500 Fract. of non-Rejected pMSSM Points 0 0.2 0.4 0.6 0.8 1 Higgs Couplings 95% Exclusion -1 LHC 140 fb HL LHC ILC 1000 FIG. 12. Fraction of accepted pMSSM points not excluded at the 95% of CL by the Higgs couplings as a function of the MA mass for the present run 2 ATLAS results (dark grey) and the expected HL-LHC (medium grey) and ILC-1000 (light grey) accuracies, assuming SM central values. [GeV] A Benchmark 9 M 0500 1000 1500 2000 2500 3000 1/N dN/dM 0 0.02 0.04 0.06 0.08 0.1 [GeV] A Benchmark 11 M 0500 1000 1500 2000 2500 3000 1/N dN/dM 0 0.01 0.02 0.03 0.04 0.05 0.06 [GeV] A Benchmark 13 M 0500 1000 1500 2000 2500 3000 1/N dN/dM 0 0.01 0.02 0.03 0.04 0.05 FIG. 13. Probability density function for MAcorresponding to the ILC 1000 accuracy obtained for the benchmark points 9, 11, and 13. A. ARBEY et al. PHYS. REV. D 106, 055002 (2022) 055002-22 pMSSM points in the scan. Each point is assigned a weight defined as the χ2probability of the point with the scenario tested, where the χ2is computed using the Higgs coupling modifiers with the accuracies for the LHC and the eþe− Higgs factories given in Table VII. The values of MAof these weighted points give distributions (some of which are shown in Fig. 13) from which the central value and uncertainty of the MAparameters are extracted. The central values are computed as the average in an interval integrating 68% of the entries around the most probable value. The uncertainties are obtained by determining the interval of parameter values around the central value, which integrates 68% of the weighted entries allowing for asymmetric ranges. The relation between the generated and reconstructed values of MAfor the chosen benchmark points is shown in Fig. 14. VI. CONCLUSIONS The study of the Higgs boson properties offers compelling perspectives for testing the effects of physics beyond the Standard Model at the LHC and at future colliders. This is particularly the case in the context of supersymmetric theories and its minimal version, the MSSM. The Higgs couplings to SM particles, both at tree level and through loops, are sensitive to new physics effects and can be used to discriminate the MSSM hfrom the SM H. In this study, we have reviewed the SUSY corrections to the couplings and decay rates of the SM-like Higgs boson and their dependence on the MSSM parameters. The constraints and predictivity of the Higgs measurements are applied directly on the relevant supersymmetric parameters using scans of the pMSSM parameter space and contrasted with those derived from direct searches for new particles at the LHC. Theoretical and parametric uncertainties in Higgs production and decay also need to be considered alongside the experimental accuracies. These are revisited and discussed in detail. The sources of these corrections can be classified in three main categories: the pseudoscalar Higgs mass MA, the invisible decays h→ ˜ χ0˜ χ0,˜ ν˜ ν, and the SUSY– QCD corrections generating the Δb,Δt, and Δτterms through scalar quarks and gluino or scalar tau and gauginoHiggsino contributions. The values of the lightest neutralino mass as well as the Higgsino and gaugino mass parameters, μand M1,M2, that can generate significant rates of invisible decays are already constrained by dark matter direct detection data, even more (GeV) A Generated M 500 1000 1500 (GeV) A Reconstructed M 500 1000 1500 2000 FIG. 14. Reconstructed most probable value vs generated value of MAfor the study points. The deviation of the reconstructed most probable value from the diagonal at large values of MAis due to the loss of sensitivity on the lower side of the MA probability density distribution. TABLE X. pMSSM scenarios adopted in the study of the extraction of MA. Scenario MA(GeV) tan βμ(GeV) M˜ t1(GeV) M˜τ1(GeV) M˜ χ0 1(GeV) M˜χ 1(GeV) M˜ g(GeV) 1 285 6.4 −12.24023 1546 3.3 14.9 5204 2 434 5.6 −549 1493 744 562 564 5059 3 472 6.6 1994 1652 1407 315 1459 390 4 510 5.5 −181 3749 1999 185 186 4585 5 554 6.5 2351 4613 2294 737 2012 4379 6 606 6.3 369 3688 1510 380 381 3571 7 649 5.7 −1411 1886 1178 444 445 2961 8 704 5.2 480 4924 865 170 493 3436 9 747 4.5 −3596 3483 3080 1072 1073 2324 10 798 5.8 −3302 2712 2258 1329 3284 2085 11 844 9.1 −1679 1902 2045 1695 1696 4502 12 893 7.1 −368 5038 2095 379 381 2889 13 947 7.7 −4268 2941 2276 364 3677 441 14 1002 5.4 716 3037 827 732 733 3234 15 1095 12.0 1330 3068 4060 1351 1352 5722 HIGGS BOSON PROPERTIES AND SUPERSYMMETRY: …PHYS. REV. D 106, 055002 (2022) 055002-23 severely than by the results of the current LHC invisible Higgs decay searches. For the range of MAvalues not yet probed by the ATLAS and CMS data or excluded by flavor data, the Δbcontribution to the light hboson coupling to b¯ bis largely reduced, by compensation of direct and indirect contributions, and the remaining one-loop SUSY corrections are tiny. The effects of the Δtand Δτcorrections are in general small. This reduces the sensitivity of Higgs couplings and decay rates to MSSM parameters other than MAas the main MSSM parameter that can be probed in the continuation of the LHC program and at future eþe−colliders. This study has shown that Higgs coupling measurements with the accuracies obtained on the LHC run 2 data and those expected for the HL-LHC and future eþe−colliders can exclude ∼2%, 8%, and 20%, respectively, of the accepted pMSSM points in our scans, but only ∼1%, 5%, and 12% of the points that are not yet excluded by flavor data and by the LHC heavy Higgs direct searches, while direct SUSY searches have only a mild impact. The indirect sensitivity to MAin the pMSSM through the Higgs coupling measurements will evolve from ∼450 GeV for the run 2 data to ∼800 GeV at HL-LHC and ∼1400 GeV at future eþe−colliders. Within this range, future eþe− colliders of sufficient energy can indirectly determine MAto a relative accuracy ranging from ≃8% to 40% for MAvalues from 700 GeV to 1.1 TeV, from the deviations of the measured lightest hcouplings with respect to their SM expectations. 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