The leptonic future of the Higgs

Journal of High Energy Physics, Sep 2017

Precision study of electroweak symmetry breaking strongly motivates the construction of a lepton collider with center-of-mass energy of at least 240 GeV. Besides Higgsstrahlung (e + e − → hZ), such a collider would measure weak boson pair production (e + e − → WW) with an astonishing precision. The weak-boson-fusion production process (\( {e}^{+}{e}^{-}\to \nu \overline{\nu}h \)) provides an increasingly powerful handle at higher center-of-mass energies. High energies also benefit the associated top-Higgs production (\( {e}^{+}{e}^{-}\to t\overline{t}h \)) that is crucial to constrain directly the top Yukawa coupling. The impact and complementarity of differential measurements, at different center-of-mass energies and for several beam polarization configurations, are studied in a global effective-field-theory framework. We define a global determinant parameter (GDP) which characterizes the overall strengthening of constraints independently of the choice of operator basis. The reach of the CEPC, CLIC, FCC-ee, and ILC designs is assessed.

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The leptonic future of the Higgs

HJE The leptonic future of the Higgs Gauthier Durieux 0 1 3 6 Christophe Grojean 0 1 2 3 4 6 Jiayin Gu 0 1 3 5 6 Kechen Wang 0 1 3 5 6 0 Chinese Academy of Sciences 1 Newtonstra e 15 , D-12489 Berlin , Germany 2 On leave from Institucio Catalana de Recerca i Estudis Avancats , 08010 Barcelona , Spain 3 Notkestra e 85 , D-22607 Hamburg , Germany 4 Institut fur Physik, Humboldt-Universitat zu Berlin 5 Center for Future High Energy Physics, Institute of High Energy Physics 6 19B YuquanLu , Beijing 100049 , China Precision study of electroweak symmetry breaking strongly motivates the construction of a lepton collider with center-of-mass energy of at least 240 GeV. Besides Higgsstrahlung (e+e ! hZ), such a collider would measure weak boson pair production ! W W ) with an astonishing precision. The weak-boson-fusion production process h) provides an increasingly powerful handle at higher center-of-mass energies. Beyond Standard Model; E ective Field Theories; Higgs Physics - ! 1 Introduction 2 3 4 5 Higgs production through weak boson fusion Higgs production in association with tops Weak-boson pair production Global t and determinant parameter Results Conclusions A E ective- eld-theory parameter de nitions B Measurement inputs C Additional gures D Numerical expressions for the observables E Numerical results of the global t design studies through global ts in the so-called kappa framework [5]. As new physics is being constrained to lie further and further above the electroweak scale, the description of its e ects at future lepton colliders seems to fall in a low-energy regime. E ective eld theories (EFTs) therefore look like prime exploration tools [6{11]. Given that the parity of an operator dimension is that of ( B L)=2 [12], all operators conserving baryon and lepton numbers are of even dimension: LEFT = LSM + X c(6) i 2 Oi i (6) + X c(8) j j 4 Oj (8) + (1.1) is a mass scale and ci(d) are the dimensionless coe cients of the Oi (d) operators of canonical dimension d. The standard-model e ective eld theory (SMEFT) allows for a systematic exploration of the theory space in direct vicinity of the standard model, encoding established symmetry principles. As a genuine quantum eld theory, its predictions are also perturbatively improvable. It therefore relies on much rmer theoretical bases than the kappa framework. While very helpful in illustrating the precision reach of Higgs measurements, the latter can in particular miss interactions of Lorentz structure di erent from that of the standard model, or correlations deriving from gauge invariance, notably between Higgs couplings to di erent gauge bosons. Many e ective- eld-theory studies have been performed, for Higgs measurements at LHC [13{17], electroweak (EW) precision observables at LEP [18{22], diboson measurements at both LEP [23] and LHC [24, 25], or the combination of measurements in several sectors [26, 27]. Among the studies performed in the context of future Higgs factories [28{ 36], many estimated constraints on individual dimension-six operators. A challenge related to the consistent use of the EFT framework is indeed the simultaneous inclusion of all operators up to a given dimension. It is required for this approach to retain its power and generality. As a result, various observables have to be combined to constrain e ciently all directions of the multidimensional space of e ective-operator coe cients. The rst few measurements included bring the more signi cant improvements by lifting large approximate degeneracies. Besides Higgsstrahlung production and decay rates in di erent channels, angular distributions contain additional valuable information [29, 31]. Our knowledge about di erential distributions could also be exploited more extensively through statistically optimal observables [37, 38]. Higgs production through weak-boson fusion provides complementary information of increasing relevance at higher center-of-mass energies. Diproduction in association with a pair of tops. Measurements at p rect constraints on the top Yukawa coupling can moreover only be obtained through Higgs s = 350 GeV and above can thus be very helpful. As the sensitivities to operator coe cients can vary with ps, these higher-energy runs would also constrain di erent directions of the parameter space and therefore resolve degeneracies. Beam polarization, more easily implemented at linear colliders, could be similarly helpful. Finally, the Higgs and anomalous triple gauge cou{ 2 { plings (aTGCs) are related in a gauge-invariant EFT, and a subset of operators relevant for Higgs physics can be e ciently bounded through diboson production e+e ! W W [17, 23]. We parametrize deviations from the standard-model in the processes enumerated above through dimension-six operators, in the so-called Higgs basis [39]. Translation to other bases is however straightforward. Our assumption of perfectly standard-model-like electroweak precision measurements is more easily implemented in that framework. No deviation in the gauge-boson couplings to fermions, or W mass is permitted. Given the poor sensitivity expected for the Yukawa couplings of lighter fermion, we only allow for modi cations of the ( avor-conserving) muon, tau, charm, bottom, and top ones. Neither CP-violating, nor fermion dipole operators are considered. The potential impact of these discussed in view of their respective design and run plan. The rest of this paper is organized as follows. In section 2, we lay down the EFT framework used. In section 3, we detail the observables included in our study. The results of the global ts are shown in section 4. The reach of the di erent colliders is summarized in gure 7. Our conclusions are drawn in section 5. Further details are provided in the appendix. We de ne our twelve e ective- eld-theory parameters and provide their expressions in the SILH' basis in appendix A. Additional information about the measurement inputs is provided in appendix B. Supplementary gures and results are available in appendix C. In appendix D, we provide numerical expressions for the observables used in terms of our twelve e ective- eld-theory parameters. Finally, the numerical results of the global ts are tabulated in appendix E. They could be used to set limits on speci c models while accounting for the correlations in the full twelve-dimensional parameter space. 2 E ective- eld-theory framework A global e ective- eld-theory treatment of any process requires to consider simultaneously all contributing operators appearing in a complete basis, up to a given dimension. Assuming baryon and lepton number conservations, we restrict ourselves to dimension-six operators. As mentioned in the introduction, we would like to model the following processes: Higgsstrahlung production: e+e ! hZ (rates and distributions), followed by Higgs decays in various channels, Higgs production through weak-boson fusion: e+e Higgs production in association with top quarks: e+e ! h, ! tth, weak-boson pair production: e+e ! W W (rate and distributions). Several combinations of operators a ecting these processes are however well constrained by other measurements. As discussed in section 3.4, electroweak precision observables could be constrained to a su cient level, although this remains to be established explicitly. { 3 { At leading order, CP-violating operators give no linear contribution to the Higgs rates but could manifest themselves in angular asymmetries [29, 31]. They could moreover be well constrained by dedicated searches. Under restrictive assumptions, indirect constraints arising from EDM experiments [40{42] for instance render Higgs CP-violating asymmetries inaccessible at future colliders [31], even thought some room may be left in the CP violating Yukawa of the charm and bottom quarks for which the direct and indirect bounds are not that restrictive [43]. It is also possible for CP violating Yukawa couplings of heavy avor leptons to evade the constraints from EDM experiments which could be probed in Higgs decays [44]. As a rst working hypothesis, we thus assume electroweak and CP-violating observables are perfectly constrained to be standard-model like. Throughout this paper, we only retain the interferences of e ective- eld-theory amplitudes with standard-model ones. The squares of amplitudes featuring a dimension-six operator insertion are discarded. They are formally of the same c2= 4 order as the interferences of dimension-eight operators with standard-model amplitudes. The relative importance of these two kinds of c2= 4 contributions can however not be determined without assuming a de nite power counting or referring to a speci c model. Nevertheless, thanks to the high precision to which most observables are measured at lepton colliders that collect large amount of integrated luminosity in clean environments, we generically expect the discarded terms to have small impact on our results. The percent-level measurement of an observable of schematic O OSM cE2 = 1 + O(1) 2 + O(1) cE2 2 2 e ective- eld-theory dependence (where E is a typical energy scale) will for instance constrain c E2= 2 at the percent level. The quadratic term then only induces a relative percent-level correction to this limit. In speci c cases, the interference of dimension-six operators with standard-model amplitudes can however su er accidental suppressions. This could invalidate the nave hierarchy above between linear and quadratic terms. Helicity selection rules [45] can for instance cause signi cant suppressions of the linear contribution compared to the quadratic one, at energies higher than electroweak mass scales. If the standard model and dimension-six operators give rise to amplitudes with electroweak bosons of di erent helicities, their interference is expected to scale as c m2V = 2. A measurement of O=OSM with precision x would still imply a limit of order x on cm2V = 2 at low energies but this bound would receive corrections scaling as xE4=m4V for increasing E. Given mV of order 100 GeV, only measurements of 10 2, 10 3, 10 4, 10 5 and 10 6 precisions at least are roughly expected to be dominated by linear e ective- eld-theory contributions at 250, 500, 1000, 1400 and 3000 GeV energies, respectively. We will comment further on accidental suppressions and on their possible impact on our results in section 4. Light fermion dipole operators also have interferences with standard-model amplitudes that su er drastic mass suppressions. As a consequence, their dominant e ects arise at the c2= 4 level. We however leave the study of this family of operators for future work. Under the above assumptions, together with avor universality, it was shown that there are 10 independent combinations of operators that contribute to Higgs (excluding { 4 { its self coupling) and TGC measurements [13, 14, 16, 23].1 We however lift the avor universality requirement and treat separately the top, charm, bottom, tau, and muon Yukawa couplings. No avor violation is allowed and we refer to refs. [46{48] for studies of the possible means to probe the light-fermion Yukawas at present and future experiments. In total, 12 degrees of freedom are thus considered. While all non-redundant basis are equivalent, we nd the Higgs basis [39] particularly convenient. It is de ned in the broken electroweak phase and therefore closely related to experimental observables. Distinguishing the operators contributing to electroweak precision measurements from the ones of Higgs and TGC measurements is also straightforward in this basis. The parameters we use are: cZ ; cZZ ; cZ ; c ; cZ ; cgg ; yt ; yc ; yb ; y ; y ; Z : (2.1) Their exact de nitions as well as a correspondence map to the SILH' basis of gaugeinvariant dimension-six operators can be found in appendix A. The numerical expressions of the various observables we use as functions of these parameters are given in appendix D. Compared to the widely-used kappa framework, an important feature of this e ective eld theory is the appearance of Higgs couplings with Lorentz structures di ering from SM ones. In addition to cZ hZ Z which modi es an existing SM coupling, the cZZ hZ interactions are for instance also generated by gaugeinvariant dimension-six operators. The e+e ! hZ rate, at a given center-of-mass energy and for a xed beam polarization, depends on one combination of these parameters. Runs at various energies, with di erent beam polarizations, as well as additional measurements are therefore crucial to constrain all other orthogonal directions. Measurements at higher center-of-mass energies have an enhanced sensitivity to cZZ and cZ . Angular asymmetries in e+e ! hZ, weak-boson-fusion production rate, weak-boson pair production, or the h ! ZZ and h ! W W decay is crucial too. The cZ decays, each play a role. The measurement of the h ! Z coupling which contributes to the Higgsstrahlung process otherwise remains loosely constrained and weakens the whole t. The treatment of the h ! gg, , and Z decays requires some special attention. Given that they are loop-level generated in the standard model, one may wish to include their loop-level dependence in e ective parameters like yt, yb, cW which rescale standardmodel interactions, or cZZ , cZ , etc. which do not. Complete e ective- eld-theory results at that order are however not currently available for the above processes (see ref. [49] for the treatment of h ! ). The computation of next-to-leading-order e ective- eld-theory contributions to processes that are not loop-level generated in the standard model would also be needed to ensure a consistent global treatment. Misleading results can otherwise be obtained. Let us illustrate this point with the dependence of the h ! partial width on c and yt, at tree- and loop-level, respectively. The Higgsstrahlung, weak-boson fusion, and weak-boson pair production processes also depend at tree level on c and receive loop corrections proportional to yt. A combination of these two parameters similar to the one entering in the h ! partial width may moreover be expected. Including the 1Refs. [26, 32, 35] additionally set lepton and down-type Yukawa couplings equal while ref. [17] focuses on third-generation fermions instead of assuming avor universality. { 5 { HJEP09(217)4 dependence of this partial width on yt, but not that of the e+e and e+e ! W W cross sections, one would arti cially render their constraints orthogonal. Tight bounds on yt would then be obtained. Consistently including all one-loop dependences on these parameters might however still leave a combination of c and yt at least nearly unconstrained. To avoid such a pitfall, we choose not to include any loop-level dependence on e ective- eld-theory parameters in the h ! direct constraints on the top Yukawa coupling (from the LHC or from e+e and Z partial widths. Once included, we however checked that including the whole loop dependence of the h ! branching fraction has only marginal e ects on our results.2 For our purpose, it is on the ! tth) are contrary safe to account for the loop-level yt and yb dependences of the h ! gg partial width. It remains to be examined whether the loop-level dependence on yt in processes measured at lepton collider, below the tth threshold, could serve to improve on the highluminosity LHC constraints. A similar question, asked for the trilinear Higgs coupling [51] ! hZ, e+e ! h, Absorbing also, for convenience, a standard-model normalization factor into barred could be further investigated. e ective parameters, we thus obtain: and gg SM gg c SM ' 1 2c ; Z Z SM ' 1 2cZ ; ' 1 + 2cgeg ' 1 + 2 cgg + 2:10 yt We will sometimes display results in terms of the cgeg parameter that is directly probed by the h ! gg branching fraction. It is particularly informative to do so when cgg and yt are only poorly constrained individually. Measurement of the h ! ZZ rate relies on its fermionic decay products and has some sensitivity on c and cZ , in addition to cZ , cZZ and cZ . Higgs decays to o -shell photons can indeed produce the same nal state. Each fermionic decay channel actually has a somewhat di erent sensitivity which depends strongly on the invariant mass of fermion pairs. Loosened cuts would provide increased sensitivities to c and cZ [52].3 For simplicity, we however neglect the contributions of those two e ective- eld-theory parameters to h ! ZZ . Standard invariant mass cuts together with the constraints on c and cZ arising from the direct measurements of h ! Z and h ! to limit the impact of this approximation on our results. decays should be su cient The standard-model e ective eld theory we use speci cally assumes the absence of new states below the electroweak scale. It does therefore not account for possible invisible decays 2We used the numerical expressions derived from the results of ref. [49] in the appendix of ref. [50]. 3See also ref. [53] for a recent EFT study of the Higgs decay into four charged leptons exploiting both the rates and kinematic distributions. { 6 { (2.2) (2.3) (2.4) of the Higgs. The corresponding branching fraction would nevertheless be signi cantly constrained at future lepton colliders. An integrated luminosity of 5 ab 1 collected at 240 GeV would for instance bound (hZ) BR(h ! inv) to be smaller than 0:28% of (hZ) at the 95% CL [1]. Other exotic Higgs decays not modeled in a SMEFT framework would also be constrained very well at future lepton colliders [54]. We do therefore not expect an e ective eld theory modi ed to include such decays to lead to results widely di erent from the ones we obtain. 3 To the best of our knowledge, the most updated run plans of each machine are the following: According to its preCDR, the CEPC would collect 5 ab 1 of integrated luminosity at 240 GeV. Recently, the reference circumference of its tunnel has been xed to 100 km [55]. A run at 350 GeV could therefore be envisioned. The luminosity to expect at that center-of-mass energy however depends on the machine design and is currently unknown. To study the impact of the measurements at 350 GeV, we take a conservative benchmark value of 200 fb 1 and explore a larger range in section 4. The CDR of the FCC-ee project is expected by the year 2018 [56] and will supersede the TLEP white paper [2] that still contains the most recent results on Higgs physics. The latter document, we rely on, assumes that 10 ab 1 of data would be collected at 240 GeV and 2:6 ab 1 at 350 GeV. Recent ILC documents suggest that, with a luminosity upgrade, it could collect 2 ab 1 at 250 GeV, 200 fb 1 at 350 GeV, and 4 ab 1 at 500 GeV [57, 58]. This signi cantly extends the plans presented in its TDR [3]. The updated estimations are adopted in our study. The ILC could also run with longitudinally polarized beams. We follow refs. [3, 58] and assume that a maximum polarization of 80% ( 30%) can be achieved for the incoming electron (positron). While collecting 1 ab 1 of integrated luminosity at a center-of-mass energy of 1 TeV, with P (e ; e+) = ( 0:8; +0:2) polarization, is also considered in the TDR [3], we follow refs. [57, 58] and do not take such a run into account. Nevertheless, results including the 1 TeV measurements of precision quoted in ref. [59] are shown in appendix C. Recent ref. [4] proposed that CLIC would collect 100 fb 1 at the top threshold, 500 fb 1 at 380 GeV, 1:5 ab 1 at 1:5 TeV, and 3 ab 1 at 3 TeV. The more speci c study of Higgs measurements of ref. [60] however assumed 500 fb 1 at 350 GeV, 1:5 ab 1 at 1:4 TeV and 2 ab 1 at 3 TeV. We follow the latter plan in order to make use of its estimations. While the implementation of beam polarization is also likely at CLIC, we follow again ref. [60] and assume unpolarized beams. In the rest of this section, we summarize the important aspects of each of the measurements we take into account. We detail the assumptions made in the many cases where necessary information is not provided in the literature. The numerical inputs we use are given in appendix B. { 7 { production modes at lepton colliders below center-of-mass energies of about 450 GeV where weak-boson fusion takes over. Its cross section is maximized around 250 GeV but bremsstrahlung makes it more advantageous for circular colliders to run at 240 GeV. At this energy, an integrated luminosity of 5 ab 1 would yield about 1:06 250 GeV, 2 ab 1 of data collected with P (e ; e+) = ( 0:8; +0:3) beam polarization would contain approximatively 6:4 105 Higgses. The latter polarization con guration maximizes the e+e e+e ! hZ cross section. The recoil mass of the Z gives access to the inclusive ! hZ rate independently of the exclusive Higgs decay channels measurements. The Higgsstrahlung process can also be measured at higher center-of-mass energies. Despite the smaller cross sections, this allows to probe di erent combinations of EFT parameters and is thus helpful for resolving (approximate) degeneracies among them. The estimated measurement precisions at each collider and at di erent energies are shown in table 2, 3 and 4 of appendix B, where further details are also provided. A few important comments are in order. As mentioned in section 2, the measurement of the rare h ! Z decay, while not very constraining for the SM hZ coupling, could be hZ ing c very important to resolve the degeneracies of EFT parameters in the production processes. Therefore, while the estimation of this measurement is not available for the FCC-ee and ILC, we scale the precision estimated for the CEPC, assuming the dominance of statistical uncertainties. Some care must also be taken to avoid potential double counting between the e+e ; h ! bb process and the weak-boson fusion e+e ! h; h ! bb, which yield the same nal state. This is further discussed in section 3.2 and appendix B. Note also that the interferences between s-channel Z and photon amplitudes are accidentally suppressed by a factor of 1 4 sin2 W ' 0:06 in the total unpolarized cross section. This factor arises from the sum of the left- and right-handed couplings of the electron to e e the Z, 2sW cW ( 1 + 2s2W ) and 2sW cW (2s2W ), respectively. Beam polarization thus significantly a ects the sensitivity of the Higgsstrahlung rate to operators contributing to the vertex.4 Numerical expressions in the Higgs basis are provided in eq. (D.1). Introducde ned in eq. (A.3) and contributing for an o -shell photon however renders this e ect more transparent. For P (e ; e+) = (0; 0), ( 0:8; +0:3), (+0:8; 0:3) polarization 4We thank Michael Peskin for helping us understand this interesting phenomena. { 8 { HJEP09(217)4 h Z ℓ− b ℓ+ θ1 z Note the two polar angles are respectively de ned in the center-of-mass and Z restframes. con gurations at p s = 250 GeV, we for instance obtain: HJEP09(217)4 hZ SM hZ 250 GeV 0 (0; 0) 1 (+0:8; 0:3) ' 1 + 2 cZ + 1:6 cZZ + 3:5 cZ 1 C C 1 C C : (3.1) An increase in the sensitivity magnitude of more than an order of magnitude is brought by beam polarization. Reversing the polarization also ips the sign of the cZ and c prefactors, given the opposite signs of the left- and right-handed couplings of the Z to electrons. Angular asymmetries. Three angles and two invariant masses fully characterize the di erential distribution of the e+e ! hZ ! hf f process (see gure 2). It naturally provides information complementary to that of the total rate alone. The e ective- eldtheory contributions to the angular distributions have been thoroughly studied in ref. [29]. At tree level and linear order in the e ective- eld-theory parameters, they can all be captured through the following asymmetries: A 1 = A A A A (1) = (2) = (3) = (4) = Ac 1;c 2 = = f 1; 2; g and the sgn function gives the sign of its argument. Among these (2) are sensitive to CP-violating parameters (or absorptive parts (4) depend on the same combination of operator coe cients. In the absence of CP violation, the angular observables therefore provide three independent constraints on e ective- eld-theory parameters. The corresponding Higgs-basis expressions are provided in appendix D. { 9 { e− e + W − W + ν ν¯ h ! e− e + Z/γ Z h ν¯ h process: weak-boson fusion (left), and e+e hZ; Z ! (right). ! HJEP09(217)4 A phenomenological study of these angular asymmetries at circular e+e colliders has been performed in ref. [31]. In particular, it was shown that the uncertainties on their determination is statistics dominated for leptonic Z decays. The absolute statistical uncertainty (one standard deviation) on each asymmetry A measured with N events is given by [31] A = r 1 A 2 N p 1 N : Following ref. [31], we use only the events with Higgs decays to bottom quarks (e+e hZ ; Z ! `+` ; h ! bb) which has negligible backgrounds. Reference [31] refers to a preliminary version of the CEPC preCDR which suggests the signal selection e ciency of this channel at 240 GeV is around 54%. For simplicity, we assume a universal e ciency of 60% for the event selection of this channel at all energies for the angular asymmetry analysis. For the CEPC, with 5 ab 1 collected at 240 GeV, this constitutes a subsample of approximately 2:7 104 Higgsstrahlung events. For the ILC, the e ects of beam polarizations on the asymmetries is taken into account. No systematic uncertainty is included. We however expect statistical uncertainties to be dominant given the fairly rare but clean Z decay to leptons. 3.2 Higgs production through weak boson fusion The Higgs couplings to W , Z bosons, and photons are related by SU(2)L gauge invariance. As such, the measurement of the weak-boson fusion process, rst considered in e+e colliders in ref. [61], is complementary to that of the Higgsstrahlung process. So, a combination of the two measurements can e ciently resolve the degeneracy among the EFT parameters that contribute to the production processes. The weak-boson fusion cross section grows with energy, so that it is better measured at a center-of-mass energy of 350 GeV or above. Nevertheless, the measurement at 240 GeV can still provide important information, especially if runs at higher energies are not performed. Importantly, Higgsstrahlung with Z decay to neutrinos (e+e the same nal state as weak-boson fusion (see gure 3) and has a rate about six times larger at a center-of-mass energy of 240 GeV (without beam polarization). At this center-of-mass energy the missing mass distributions for both processes moreover peak at similar energies (see gure 3.16 on page 75 of ref. [1]). Isolating the weak-boson fusion contribution is therefore di cult. For the CEPC and FCC-ee at 240 GeV, we therefore consider an inclusive (3.3) ! Z/γ ¯ t Z/γ t t ¯ t h ¯ t h HJEP09(217)4 e− e + e− e + Z/γ Z/γ h t ¯ t t ¯ t ! tth process. In the SM, the dominant contribution are the ones involving the top Yukawa coupling. Other EFT contributions (including that of four-fermion operators, not depicted) should be well constrained by other measurements. e+e e+e for which the precision on the e+e h rate measurement is reported in the literature. ! h sample to which the two processes contribute, and only use the h ! bb channel We neglect the contributions of the weak-boson fusion in the other Higgs decay channels of . For the ILC, ref. [59] states that a 2 t of the recoil mass distribution is used to separate the weak-boson-fusion and the Higgsstrahlung processes. We thus consider that the precision on (e+e ! h) BR(h ! bb) quoted in ref. [58] applies directly to the weak-boson fusion contribution. Both processes reach equal rates at a centerof-mass energy close to 350 GeV (without beam polarization). At this and higher energies, we thus assume that their distinct recoil-mass distributions are su cient to e ciently separate them. More details on the treatment of this measurement can be found in appendix B. 3.3 Higgs production in association with tops The e+e ! tth production of a Higgs boson in association with top quarks (see gure 4) requires a large center-of-mass energy which is only achieved at a linear collider. A 10% p precision on (tth) BR(h ! bb) could be achieved with 4 ab 1 of ILC data collected at s = 500 GeV (scaled from 28% of the 500 fb 1 result in ref. [58]). At CLIC, 1:5 ab 1 of 1:4 TeV data should yield an 8:4% precision [60]. In the SM, the dominant contributions to this process involve a top Yukawa coupling. The radiation of a Higgs from the s-channel Z boson is comparatively negligible [3]. In the e ective eld theory, we only include modi cations of the top Yukawa coupling. Other contributions should be su ciently constrained by the measurement of top pair production and other processes. Neither the four-point Zhtt interaction depicted on gure 4 (bottom-right), nor four-fermion operator contributions are thus accounted for here. This channel could also be used to establish the CP properties of the Higgs boson [62], which we simply assumed to be a 0+ state throughout our analysis. 3.4 Weak-boson pair production The diagrams contributing to the e+e ! W W process, at leading order, are depicted in gure 5. The s-channel diagrams with an intermediate Z or photon involve triple gauge Z/γ W − W + e + ν W − W + left with an intermediate Z or photon involves a triple gauge coupling. ! W W . The s-channel diagram on the couplings. Considering CP-even dimension-six operators only, the aTGCs are traditionally parameterized using g1;Z , and Z [63, 64], de ned in eq. (A.5). Among them, g1;Z and are generated by e ective operators that also contribute to Higgs observables. As pointed out in ref. [23], this leads to an interesting interplay between Higgs and TGC measurements. Triple gauge couplings have been measured thoroughly at LEP2 [65]. Various studies of future lepton colliders' reach have also been carried out [66{71]. At future circular colliders, most of the W pairs are likely to be produced at 240 GeV, as a byproduct of the Higgs measurement run which requires large luminosities. At this energy, the e+e W W cross section is approximately two orders of magnitude larger than that of e+e hZ. With 5 ab 1, the CEPC would thus produce about 9 107 e+e ! ! ! W W events, thereby improving signi cantly our knowledge of TGCs. A run at 350 GeV, probing a di erent combination, could bring further improvement on the constraints. Longitudinal beam polarization is also very helpful in probing the aTGCs. With 500 fb 1 collected at 500 GeV and equally shared between four P (e ; e+) = ( 80%; 30%) beam polarization con gurations, the ILC could constrain the three TGCs to the 10 4 level [68]. Note the runs with ++ and polarizations are mostly meant to provide a simultaneous and su ciently accurate polarization magnitude measurement. Comparable results can be expected for more realistic repartitions of the luminosities [69]. For the CEPC and FCC-ee prospects, we follow ref. [71] which exploited kinematic distributions in the e+e ! W W ! 4f process. Five angles can be reconstructed in each such event: the polar angle between the incoming e and the outgoing W , and two angles specifying the kinematics of each W decay products. When both W s decay leptonically, the W mass constraints allow to fully reconstruct the kinematics up to a fourfold ambiguity at most. Here, we make the optimistic assumption that the correct solution is always found. In the hadronic W decays, one can not discriminate between the quark and antiquark. The angular distributions of the W decay products are thus folded. We divide the di erential distributions of each angle into 20 bins (10 in folded distributions). Uncorrelated Poisson distributions are assumed in each bin and their 2 are summed over. The total 2 is constructed by summing over the 2 of all the angular distributions of all decay channels. The statistical correlation between angular distributions is neglected. Given the huge statistics that would be collected, and although they were neglected in ref. [71], the systematic uncertainties could play an important role. Theoretical uncertainties could also become limiting. At the moment, there is however no dedicated experimental study of TGC measurements at future circular colliders. We therefore introduce a benchmark systematic uncertainty of 1% in each bin of the di erential distributions. This guess is probably too conservative compared to few 10 4 systematic uncertainties on the g1;Z , , and Z TGC parameters recently estimated by the ILC collaboration [72]. We therefore examine the impact of variation of this value in section 4 and also provide constraints obtained by assuming no deviation on the TGC from their standard-model values. For the prospects of the full ILC program, we use the one-sigma statistical uncertainties obtained in ref. [68] ( g1;Z = 6:1 these numbers to higher luminosities, as systematic uncertainties are likely to become important. The current estimates by the ILC collaboration for systematics uncertainties are of a few 10 4 [72]. When focusing on the 250 and 350 GeV runs of the ILC, we use the strategy described above for the CEPC instead. As a dedicated experimental study of TGC measurements at CLIC is also missing,5 we assume a precision similar to the ILC one can be reached there. It should be noted, however, that the 1.4 and 3 TeV runs at CLIC could potentially provide even stronger constraints on the aTGCs due to the increase of sensitivities with energy [35]. Another important issue raised by the signi cant improvement in the e+e ! W W ! 4f measurement precision concerns the uncertainty on electroweak precision observables. In the extraction of the constraints on aTGCs, one usually makes the TGC dominance assumption and neglects the impact of new physics on all other parameters. At LEP, this was justi ed given the better precision of Z-pole and W -mass measurements compared to that of W pair production. In this work, we also assume that runs at lower energies will give us su cient control on such e ects. Exploiting diboson data could also be an alternative if runs at lower energies are not performed. Further investigations are required in this direction. The W mass can be measured very well at a Higgs factory by reconstructing the W decay products in the e+e a ect the di erential distributions of e+e ! W W process. To leading order, the aTGCs ! W W , but not the W invariant mass. The two measurements are thus approximatively independent. A precision of 3 MeV could be achieved at the CEPC with this method [1]. A dedicated threshold scan at center-of-mass energies of 160{170 GeV could also be performed. As such, it is reasonable to assume the W mass will be su ciently well constrained at future e+e colliders. The corrections to gauge-boson propagators and fermion gauge couplings could however have a non-negligible impact on the determination of triple gauge couplings, especially without a future Z factory to improve their constraints.6 While the CEPC and FCC-ee could perform a run at 5For CLIC at 3 TeV and an integrated luminosity of 1 ab 1, ref. [67] bases itself on ref. [66] which derived individual constraints and quotes = 0:9 , Z = 1:3 10 4 constraints (we thank Philipp Rolo for pointing out this reference). These results are however insu cient to serve as input for our global analysis. A phenomenological study for CLIC based on total e+e in ref. [35]. The results in section 3.2 and eq. (4.2) there imply individual constraints rescaled for 1 ab 1 ! W W rates only was also performed that are less than a factor of two better than that of ref. [67]. 6See also ref. [25] for a recent discussion on this topic in the context of LHC measurements. the Z pole, the interest of such a Z-pole run at the ILC and CLIC is still under investigation. Notably, the ILC precision on aTGCs quoted above already surpasses the precision obtained at LEP on the electroweak observables. A global t including Higgs, TGC and the Z-pole measurements would be instructive but is beyond the scope of this paper. Global t and determinant parameter Our total 2 can be rewritten as the sum of that of the measurements described previously in this section: where7 2 tot = 2 hZ= h; rates + 2 hZ; asymmetries + 2W W ; The i are the signal strengths (rates normalized to SM predictions) of the rate measurements, summed over (hZ), (hZ) BR and ( h) BR. The corresponding one-sigma uncertainties are listed in table 2, 3 and 4 of appendix B, for the di erent colliders. Ai are the asymmetries of eq. (3.2), and Ai their uncertainties, given in eq. (3.3). For the e+e ! W W measurements at CEPC and FCC-ee, the 2 is summed over all W -boson where ni is the number of events in that bin. For ILC and CLIC, the 2 decay channels, over the ve angular distributions, and over all their bins. A systematic uncertainty i sys is included in each bin. Unless otherwise speci ed, we take i sys=ni = 1% W W is directly reconstructed from one-sigma bounds and the correlation matrix of aTGCs from ref. [68] (shown in table 7 of appendix B). Finally, the 2 is summed over runs with di erent energies and beam polarizations (if applicable). As we only retained the linear dependence of all observables in terms of e ectiveoperator coe cients, our 2 are quadratic functions: 2 = X(c ij c0)i ij2 (c c0)j ; where ij 2 ( ci ij cj ) 1 ; (3.8) where ci=1; ::: 12 denotes the 12 parameters of eq. (2.1) and c0 are the corresponding central values, which are vanishing by construction in our study. The uncertainties ci and the correlation matrix can thus be obtained from It should also be noted that the measured Higgs decay width reported in the corresponding documents of the colliders is a quantity derived (with certain assumptions) from several measurements which are already included in the t. We therefore do not include it in our t as an additional independent measurement. (3.4) (3.5) (3.6) (3.7) c2 Δχ2=1 is proportional to the square root of the determinant of the covariance matrix, pdet 2. In n dimensions, the nth root of this quantity or global determinant parameter (GDP 2pndet 2) provides an average of constraints strengths. GDP ratios measure improvements in global constraint strengths independently of an e ective- eld-theory operator basis. ( n2 = ( n2 + 1) p 2pndet 2 Global determinant parameter (GDP). We introduce a metric, dubbed global determinant parameter, for assessing the overall strength of constraints. In a global analysis featuring n degrees of freedom, it is de ned as the determinant of the covariance matrix raised to the 1=2n power, GDP . In a multivariate Gaussian problem, the square root of the determinant is proportional to the volume of the one-sigma ellipsoid det 2) and therefore measures the allowed parameter space size (see gure 6). Its nth root is the geometric average of the half lengths of the ellipsoid axes and can thus serve as an average constraint strength. Interestingly, the ellipsoid volume transforms linearly under rescalings of the t parameters. So, ratios of GDPs do not depend on parameters' normalization. They are obviously also invariant under rotations in the multidimensional parameters space. Such ratios are thus independent on the choice of e ective-operator basis used to describe the same underlying physics. We therefore judge these quantities especially convenient to measure the improvement in global constraints brought by di erent run scenarios of future lepton colliders. It is however to be noted that the GDP measure weights equally all directions in the e ective- eld-theory parameter space, so that it is on its own certainly not accounting for the fact some directions are privileged by speci c power countings or models. 4 Results We rst discuss in this section the precision reach of the whole program of each collider before examining, in subsequent subsections, the impact of di erent measurements, centerof-mass energies, systematic uncertainties, and beam polarization. The CEPC is then taken as an illustrative example (except when studying polarization) and the corresponding gures for the FCC-ee and ILC are provided in appendix C. 7Note that we have used the symbol to denote both cross sections and standard deviations. What we mean in each case should be clear from the context. 10-1 n o i s i rce10-2 p 10-3 10-4 * * * precision reach of the 12-parameter fit in Higgs basis * * **** - -* -* * -**-* * * -*-*-*-* ----λZ GDP eters. All results but the light-shaded columns include the 14 TeV LHC (with 3000 fb 1) and LEP measurements. LHC constraints also include measurements carried out at 8 TeV. Note that, without run above the tth threshold, circular colliders alone do not constrain the cgg and yt e ective- eldtheory parameter individually. The combination with LHC measurements however resolves this at direction. The horizontal blue lines on each column correspond to the constraints obtained when one single parameter is kept at the time, assuming all others vanish. The red stars correspond to the constraints assuming vanishing aTGCs. The GDPs of future lepton colliders are shown on the right panel. See main text for comparisons with the LHC GDPs. We show in gure 7 the one-sigma precision reach at various future lepton colliders on our e ective- eld-theory parameters. These projections are compared to the reach of the Higgs measurements at the 14 TeV LHC with 300 fb 1 and 3000 fb 1 of integrated luminosity, combined with the diboson production measurement at LEP. The estimated reach of Higgs measurements at the high-luminosity LHC derives from projection by the ATLAS collaboration [73] which collected information from various other sources. Information about the composition of each channel are extracted from refs. [74{78]. Theory uncertainties on these LHC measurements are not included in our estimations. In LHC results, we also assume the charm Yukawa to be SM-like as ref. [73] does not provide estimations on the h ! cc branching fraction precision reach. The constraints from the diboson measurements at LEP are obtained from ref. [23]. We do not include the LHC constraints arising from diboson production, as issues related to the validity of the e ective- eld-theory [24, 79] and of the TGC dominance assumption [25] need to be simultaneously considered. A dedicated study of the reach of the high-luminosity LHC on these processes should be carried out. The constraints set at future lepton colliders are however expected to be much more stringent. Compared with LHC and LEP, future lepton colliders would improve the measurements of e ective- eld-theory parameters by roughly one order of magnitude. A combination with the LHC measurements provides a marginal improvement for most of the parameters. For c , cZ and y , the improvements are more signi cant, as the small rates and clean signals make the LHC reaches comparable to that of lepton colliders. It should be noted that the measurements of the h ! gg branching fraction only constrain a linear combination of cgg and yt. These two parameters are thus only constrained independently by lepton colliders when tth production is measured. Therefore, the combination with LHC measurements is required for CEPC and FCC-ee to constrain cgg and yt. The resulting bounds on yt are then even substantially better than that set by the LHC alone. The twelve-parameter GDPs for the combination of future lepton collider, LHC 3000 fb 1 and LEP measurements are displayed on the right panel of gure 7. Corresponding numerical values are 0.0077, 0.0054, 0.0049, 0.0058 for CEPC, FCC-ee, ILC and CLIC, respectively. Varying prospective constraints on the charm Yukawa measurement complicate the comparison with the high-luminosity LHC. The ATLAS collaboration estibranching fraction could be constrained to be smaller than 15 times its standard model value with 3 ab 1 at 14 TeV [80]. Such a constraint would translate into a one-sigma precision reach on yc of order one. To broadly cover the range spent by other studies [81{85], we vary the expected precision reach on yc in the 0:01 10 range. The combination of LHC 300 fb 1 (3000 fb 1) and LEP measurements only then leads to GDPs in the 0:065 0:069) interval, one order of magnitude worst than when future lepton collider measurements are included. On the other hand, with yc set to zero, the eleven-parameter GDP for the combination of LHC 300 fb 1 (3000 fb 1 ) and LEP measurements only is of 0:078 (0:044). In comparison, when future lepton collider measurements are also included, the corresponding eleven-parameter GDP are 0.0073, 0.0053, 0.0046, 0.0052 for CEPC, FCC-ee, ILC and CLIC, respectively. Let us also comment further on the impact of having discarded the quadratic dependence on dimension-six operator coe cients. As stressed in section 2, no signi cant e ect is expected given the good precision achieved at future lepton colliders in the measurement of most observable. Note that even the branching ratios for rare Higgs decays like h ! Z are su ciently well constrained for quadratic contributions to be subleading. Only cases in which accidental suppressions of the standard-model interference with e ective- eld-theory amplitudes require a case-by-case discussion. We identify two such cases. First, helicity selection rules are known to suppress the ratio of linear and quadratic dependences on the Z aTGC at high energies. Reproducing the analysis made at 250 GeV for a center-of-mass energy of 500 GeV and 500 fb 1 shared between two beam polarization con gurations, with and without quadratic aTGC contributions, we obtained di erences in the derived limits of 10% at most. The linear approximation thus seems to be reasonably accurate in that case also checked that quadratic contributions would be subleading at p and no strong quadratic aTGC dependence should a ect the bounds derived in ref. [68]. We s = 3 TeV, provided the whole di erential information is included. The non-interference between standard-model and dimension-six operator indeed does not hold when the azimuthal angles of the W decay products are not integrated over. Secondly, as noted in section 3.1, the interference between the s-channel photon and Z amplitudes in the unpolarized Higgsstrahlung cross section su ers from an accidental numerical suppression. Moreover, at high energies, the Higgsstrahlung cross section goes down and so does the accuracy with which it can be measured. Therefore, one can expect the quadratic dependence on the operator modifying the HZ vertex with an o -shell photon to be important in that speci c case. Although we use unpolarized cross section measurements to determine CLIC reach on e ective- eld-theory 0.10 precision reach at CEPC with different sets of measurements ve columns exploit 5 ab 1 of 240 GeV data while the last column also includes 200 fb 1 at 350 GeV. Only Higgs rate measurements (e+e ! hZ= h) are included in the rst ! column. One single measurement is excluded at the time in the three subsequent columns: e+e W W in the second, e+e h in the third, and the angular asymmetries of e+e the fourth. Note that Z is left unconstrained by Higgs data. All measurements at 240 GeV are included to obtain the constraints in the fth column. A run at 350 GeV is also included in the last, sixth, column. The dark shades correspond to the constraints obtained when one single parameter ! ! hZ in is kept at the time, assuming all other vanish. parameters to match experimental studies, beam polarization would actually be available at CLIC and we checked explicitly that the quadratic e ective- eld-theory contributions become unimportant once measurements with polarized beams are performed. Impact of the various measurements. We examine, in gure 8, the impact of di erent measurements. The one-sigma precision are displayed with one or more measurements removed from the global t, using CEPC as an example. Since the degeneracy between cgg and yt can not be resolved with measurements at 240 and 350 GeV, we display the constraint on cgeg , de ned in eq. (2.3). The rst ve columns use the measurements at 240 GeV (5 ab 1) only. The rst column on the left shows the results from rate measurements in Higgs processes (e+e ! hZ= one single measurement is excluded at the time: e+e and the angular asymmetries of e+e h) only. To obtain the second, third, and fourth columns, ! W W (2nd), e+e ! h (3rd), ! hZ (4th), respectively. The fth column expresses the constraints deriving from all measurements at 240 GeV. In the last column, 200 fb 1 of data at 350 GeV is also included. The dark shades nally display the constraints deriving when one single e ective- eld-theory parameter is kept at a time. su cient to constrain simultaneously all parameters to a satisfactory degree. They leave poorly constrained some directions of the multidimensional parameter space, thereby weakening the whole t. As already stressed, in such a global treatment, the combination of several observables is capital to e ectively bound all parameter combinations. The global strength of constraints is dramatically improved by the rst few measurement which resolve approximate degeneracies. Once a su cient number of constraints is imposed, the exclusion of one single observable does not dramatically a ect the overall precision. The individual constraints (obtained by switching on one parameter at a time), on the other hand, receive little improvement from the additional measurements | a clear demonstration that global constraints are driven by approximate degeneracies. A marginal improvement of the constraints obtained for a given run would be obtained by including a set of observables even more complete than the one we use. Impact of a 350 GeV run at circular colliders. As already visible in gure 8, a 350 GeV run signi cantly improves the strength of the constraints set by circular colliders. An important question for their design is the optimal amount of luminosity to gather at that energy, in view of the physics performance and the budget cost. In addition to the top mass and electroweak coupling measurements, the improvement on the precision of Higgs coupling could be considered too. This is addressed in gure 9 which shows the reach of the CEPC for increasing amounts of integrated luminosity collected at 350 GeV, from 0 to 2 ab 1. It is clear that a run at this energy is able to lift further approximate degeneracy among e ective- eld-theory parameters. A GDP reduction of about 17% is obtained with only 200 fb 1, and reaches about 34% with 2 ab 1 . The improvements on the c , cZ , and y e ective parameters are less signi cant. The Higgs decay channels which provide the dominant constraints on these parameters su er from small rates. These constraints are thus mainly statistics limited and approximate degeneracies play a secondary role. It should be noted that the estimations for Higgs measurements at 350 GeV for various luminosities are obtained by scaling from the ones in table 2, assuming statistical uncertainties dominate. This assumption may cease to be valid for large integrated luminosities. Impact of beam polarization at linear colliders. The possibility of longitudinal beam polarization constitutes a distinct advantage for linear colliders. Implementing it at circular colliders may be di cult (especially at high center-of-mass energies) and not economically feasible [2]. Dividing the total luminosity into multiple runs of di erent polarization con gurations e ectively provides several independent observables and helps constraining di erent direction of the e ective-theory parameter space. In gure 10, we examine what subdivision of the total ILC luminosity at 250 GeV would optimize the nal precision reach. We follow the ILC TDR [3] and assume that the ILC could achieve a maximum beam polarization of 80% for electrons and 30% for positrons. Ref. [58] proposes a run plan with four polarization con gurations sgnfP (e ; e+)g = ( ; +), (+; ), ( ; ), (+; +) and corresponding luminosity fractions of 67:5%, 22:5%, 5%, and 5%, respectively. The ( ; ) and (+; +) polarizations could serve to probe exotic new physics, like electron dipole or Yukawa operators. They however suppress the rate of Higgs and gauge boson production and are thus not very helpful for the precision study of these processes. For simplicity, we will thus only consider the ( ; +) and (+; ) polarizations. Uncertainty estimates are often only provided for an entire run in the P (e ; e+) = ( 0:8; +0:3) con guration. Scaling OW W = g2jHj2W a W a; OBB = g02jHj2B B OHW = ig(D H)y a(D H)W a OHB = ig0(D H)y(D H)B OGG = gs2jHj2GA GA; Oyu = yujHj2QLH~ uR Oyd = ydjHj2QLHdR Oye = yejHj2LLHeR O3W = 31! g abcW a W b W c TGC measurements, assuming there is no correction to the Z-pole and W mass measurements and no dipole interaction. We only consider the avor-conserving component of Oyu , Oyd and Oye contributing to the top, charm, bottom, tau, and muon Yukawa couplings. The aTGCs in this basis are given by g1;Z = = Z = c 2 HW ; W HW 3W ; HB ; which are obtained from the general results in ref. [21]. Finally, the expression of our e ective- eld-theory parameters in terms of the operators in table 1 are: cZ = cZZ = 1 2 cH ; 4 g2 + g02 cZ c cZ cgg = yf = 2 g2 16 g2 2 g2 16 1 2 ( HB g2 GG ; cH cyf : ( HW + t2W HB) ; ( W W + BB) ; t2W HB + 4 c2W W W + 4 t2W s2W BB) ; HW + 8 c2W W W 8 s2W BB) ; It should be noted that eq. (A.11) is only valid under the assumptions made in this paper. More general basis translations from the Higss basis to the SILH' basis (and others) are provided in ref. [39]. B Measurement inputs We provide here additional details about the input measurements used in our study, including the Higgs production rates (e+e (A.10) (A.11) in e+e ! hZ and TGC measurements from e+e ! W W . The estimated one-sigma precisions of the Higgs rate measurements are respectively displayed in table 2 for the CEPC and FCC-ee, in table 3 for ILC and, in table 4 for CLIC. When provided, the are respectively extracted from ref. [86] for the CEPC (which updates the preCDR [1]), ref. [2] for the FCC-ee, ref. [58] for the ILC and ref. [60] for CLIC. For CLIC, we also include the estimations for (hZ) BR(h ! bb) at 1.4 and 3 TeV from ref. [35]. While these measurements su er from smaller cross sections, they nevertheless signi cantly improve the constraints on cZZ and cZ due to the huge sensitivities at high energies.8 We also found the ZZ fusion measurements at CLIC (with (e+e h) BR(h ! bb) measured to a precision of 1:8% (2.3%) at 1.4 TeV (3 TeV) [60]) to have a negligible impact in our analysis.9 The numbers highlighted in green are obtained by scaling with luminosity when dedicated estimates are not available. For the ILC, the estimations of signal strengths are summarized in ref. [58] (table 13) but only for benchmark run scenarios with smaller luminosities. These are scaled up to the current run plan. For the 350 GeV run of CEPC and FCC-ee, relative precision are rescaled from the 350 GeV ILC ones.10 The precision of (hZ) BR(h ! Z ) is not provided for the FCC-ee and ILC. We thus scale it from the CEPC estimation. While a statistical precision of 2.2% is reported in ref. [2] for the ( at FCC-ee 240 GeV, it is not clear what assumptions on the e+e h) BR(h ! bb) measurement are made in obtaining this estimation. Therefore, we scale it with luminosity from the CEPC one. The di erence between unpolarized and polarized cross sections are taken into account in these rescalings. Given the moderate statistics in most of the relevant channels, it is reasonable to assume their precision is statistics limited. Nevertheless, it is important for these estimations to be updated by experimental groups in the future. The constraints from angular observables in e+e described in section 3.1, making use of the channels e+e ! hZ are obtained with the method ! hZ ; Z ! `+` ; h ! bb, cc, gg. They are included for all the e+e colliders at all energies except for the 1.4 TeV and 3 TeV runs of CLIC. The constraints on aTGCs derived from the e+e ! W W measurements are obtained using the method described in section 3.4, for the CEPC and FCC-ee. In particular, 1% systematic uncertainties are assumed in each bin with the di erential distribution of each measured angle divided in 20 bins (10 bins if the angle is folded ). The results, including the correlation matrices, are shown in table 5 and table 6, which are fed into the global t. For ILC, the constraints are shown in table 7, taken from ref. [68], which assumes 500 fb 1 data at 500 GeV and four P (e ; e+) = ( 0:8; 0:3) beam polarization con gurations. For CLIC, we simply use the ILC results. While the measurement inputs of LHC and LEP measurements are too lengthy to be reported in this paper, here we simply list the results from the global ts in terms of one sigma constraints and the correlation matrix, which can be used to reconstruct the chi8We thank Tevong You for pointing this out. 9It is nevertheless possible to further optimize the precision reach of the cross section measurements of ZZ fusion using judicious kinematic cuts, as pointed out in ref. [87]. For simplicity, we do not perform such optimizations in our study. 10A statistical precision of 0.6% is reported in ref. [2] for the ( h) BR(h ! bb) measurement at FCC-ee 350 GeV, which is in good agreement with our estimation from scaling (0:71%). [240 GeV, 5 ab 1] [350 GeV, 200 fb 1] [240 GeV, 10 ab 1] [350 GeV, 2:6 ab 1] 0:21%F | | | | | | | | | Zh h ! h ! h ! Z Zh Zh 2.4% BR h ! h ! h ! h ! Z Zh | 0.25% tth | | | | | | | | | h | tth | | | | | | | | | h | 0.3% tth | | | | | | | | | available estimations from refs. [1, 2, 86], while the missing ones (highlighted in green) are obtained from scaling with luminosity. See appendix B for more details. For (e+e ! h), the precisions marked with a diamond } are normalized to the cross section of the inclusive channel which includes both the W W fusion and e+e , while the unmarked precisions are normalized to the W W fusion process only. For the CEPC, the precision of the (hZ) (marked by a star F) reduces to 0.24% if one excludes the contribution from e+e BR(h ! bb) measurement ; h ! bb to avoid double counting with e+e h; h ! bb. The corresponding information is not available for the FCC-ee. 500 GeV runs, all numbers are scaled from ref. [58] (table 13), except for (hZ) which is scaled from the CEPC estimation. A beam polarization of P (e ; e+) = ( 0:8; +0:3) is assumed. The 1 TeV run is only included in gure 17 of appendix C, while the estimations are BR(h ! Z ) taken from ref. [59] which assumes a polarization of P (e ; e+) = ( 0:8; +0:2). | 2.6% 26% 17% 37% 33% 77% 275% | ILC HJEP09(217)4 production h ! h ! h ! h ! Z Zh 1.6% 1.9% 14.3% | | | | | | g1;Z Z g1;Z Z g1;Z 1 g1;Z 1 h | Z -0.93 -0.40 1 FCC-ee Z -0.93 -0.40 1 tth | | | | | | | | | nd the inclusion of the ZZ fusion (e+e unpolarized beams and considers only statistical uncertainties. In addition, we also include the estimations for (hZ) BR(h ! bb) at high energies in ref. [35], which are 3.3% (6.8%) at 1.4 TeV ! e+e h) measurements to have little impact in our analysis. 240 GeV(5 ab 1) 240 GeV(5 ab 1)+350 GeV(200 fb 1) uncertainty correlation matrix uncertainty correlation matrix ! W W measurement at CEPC using the methods described in section 3.4. Both the results from the 240 GeV run alone and the ones from the combination of the 240 GeV and 350 GeV runs are shown. 240 GeV(10 ab 1) 240 GeV(10 ab 1)+350 GeV(2:6 ab 1) uncertainty correlation matrix uncertainty correlation matrix | 0.3% g1;Z 1 -0.51 1 g1;Z 1 -0.61 1 Z -0.89 0.12 1 Z -0.88 0.19 1 uncertainty 1 in ref. [68], assuming 500 fb 1 of data equally shared between four P (e ; e+) = ( 0:8; 0:3) beam polarization con gurations at 500 GeV. We use the same results for CLIC. No scaling with statistics or center-of-mass energy is performed, given that systematic uncertainties may become important. ! W W measurements at ILC HJEP09(217)4 cZ cZZ cZ cZ cgg yu yd ye Z 0.17 0.42 0.19 uncertainty correlation matrix cZ 1 cZZ -0.04 1 cZ -0.21 -0.96 1 -0.76 0.37 -0.17 1 cZ -0.15 0.19 -0.10 0.20 1 cgg 8 TeV Higgs measurements and LEP e+e ! W W measurements. Flavor universality is imposed. To transform it into our framework we simply take yu ! yt, d ! yb, ye ! y . For consistency we also set the central values to zero. square. The chi-square can then be combined with the ones of the future e+e colliders to reproduce the results in gure 7 and table 11. In table 8, we list the current constraints in ref. [23], obtained from the LHC 8 TeV Higgs measurements and LEP e+e measurements. While ref. [23] explicitly assumes avor universality for the Yukawa cou! W W plings, it is a good approximation to simply assume the constraints given there apply to third-generation couplings. Since we explicitly assume the future results are SM-like, for consistency, we also set the central values of current results to zero when combining them with the future collider results. In table 9 and 10, we list the results for the 14 TeV LHC with 300 fb 1 and 3000 fb 1 luminosity, derived from projection by the ATLAS collaboration [73] which collected information from various other sources, while the information about the composition of each channel are extracted from refs. [74{78]. While yc is set to zero in obtaining these results (due to the fact that ref. [73] did not provide estimations for the decay h ! cc), it is not set to zero when the future e+e colliders. However, this has little impact on the results of the combined ts. 2 is combined with the ones from correlation matrix cZ 1 cZZ -0.029 1 cZ -0.037 -0.996 1 -0.61 -0.73 1 cZ 1 cZZ cZ -0.045 -0.998 1 c -0.54 0.81 -0.78 1 cZ cZZ cZ cZ cgg yb y cZ cZZ cZ cZ cgg yb y e+e C uncertainty uncertainty cZ -0.22 -0.44 cgg cgg -0.19 -0.20 -0.099 1 yt LHC 14 TeV Higgs measurements with 300 fb 1 data, using the ATLAS projection with no theory error [73]. yc to set to zero since ref. [73] did not provide estimations for the decay h ! cc. LHC 14 TeV Higgs measurements (3000 fb 1) correlation matrix obtaining these results, it is not set to zero when the . Note that while yc is set to zero in 2 is combined with the ones from future colliders. This has little impact on the results of the combined ts. Additional gures Here we provide additional results of the global ts. In our study, conservative estimates have been made for the measurements of the diboson process (e+e ! W W ) which often end up being systematics dominated. To give a sense of the impact of these systematic uncertainties we show, in gure 13, global t results in which aTGCs are assumed to be perfectly constrained. Figure 14 reproduces the results in gure 7 in the basis de ned by eq. (A.8) and table 1. The analogues to the gures presented in the main text for the CEPC, gure 8{12, are given 0.88 -0.21 -0.69 -0.41 yb -0.73 -0.85 -0.54 -0.094 y -0.62 -0.73 -0.39 0.13 -0.15 -0.55 -0.68 -0.41 -0.20 y -0.68 -0.81 -0.42 0.26 -0.30 10-1 o i s i 10-4 1 10-1 n o i s i c re10-2 p 10-3 10-4 - -δcZ -cZZ - - precision reach in Higgs basis, assuming zero aTGCs cgg δyt δyb δyμ Z = 0, both Z and cZ are eliminated, while the relation e2c + (g2 g02)cZ (g2 + g02)cZZ = 0 is imposed among cZZ , c and cZ . Note that the individual constraints are basis dependent. We use the above relation to eliminate cZ , hence its individual constraints are not shown. precision reach of the 12-parameter fit in the SILH' basis LHC 300/fb Higgs + LEP e+e-→WW LHC 3000/fb Higgs + LEP e+e-→WW light shade: e+e- collider only solid shade: combined with HL-LHC blue line: individual constraints CEPC ILC CLIC FCC-ee 240GeV (10/ab) + 350GeV (2.6/ab) yt c y c c yb c yτ c y μ here for the FCC-ee and ILC in gure 15{21. In particular, gure 16 shows the precision reach for ILC with di erent scenarios including runs at 250 GeV, 350 GeV and 500 GeV, while gure 17 further shows the potential improvement with the inclusion of a 1 TeV run. D Numerical expressions for the observables We express some of the important observables as numerical functions of the parameters in eq. (2.1), which is fed into the chi-square in eqs. (3.5){(3.7). The SM input parameters we use in our analytical expressions are GF = 1:1663787 10 5 GeV 2 , mZ = 91:1876 GeV, 0.015 0.005 0.000 o i s i rep0.010 0.005 0.000 FCC-ee 240GeV (10/ab) only FCC-ee 240GeV (10/ab) + 350GeV (500/fb) FCC-ee 240GeV (10/ab) + 350GeV (1/ab) precision reach at FCC-ee with different luminosities at 350 GeV dark shade: individual fit assuming all other 10 parameters are zero δcZ cZZ cZ□ cgefgf δyb δyτ δyμ/10 λZ precision reach at ILC with different run scenarios ILC 250GeV(2/ab, 2 polarizations) ILC 250GeV(2/ab, 1 polarization) + 350GeV(200/fb) ILC 250GeV(2/ab, 2 polarizations) + 350GeV(200/fb) ILC 250GeV(2/ab, 1 polarization) + 350GeV(200/fb) + 500GeV(4/ab) ILC 250GeV(2/ab, 2 polarizations) + 350GeV(200/fb) + 500GeV(4/ab) Light shades for columns 2&3: e+e-→WW measurements at 350GeV not included GDP GDP δcZ cZZ cZ□ cgefgf δyc δyb δyτ δyμ/10 λZ to the ILC 250 GeV run with 2 ab 1 luminosity which is divided into two runs with polarizations P (e ; e+) = ( 0:8; +0:3) and (+0:8; 0:3), and fractions 0:7 and 0:3, respectively (see gure 10). The 2nd and 3rd columns include ILC 250 GeV (2 ab 1 ) and 350 GeV (200 fb 1). For the 2nd column, only the ( 0:8; +0:3) polarization is used for the 240 GeV run, while for the 3rd column the 240 GeV run is divided in the same way as for the 1st column. The results of the ILC full run (2 ab 1 at 250 GeV, 200 fb 1 at 350 GeV and 4 ab 1 at 500 GeV) are shown in the 4th and 5th columns, while single polarization (two polarizations) at 250 GeV has been assumed for the 4th (5th) column, analogous to the 2nd and 3rd columns. P (e ; e+) = ( 0:8; +0:3) is assumed for the 350 GeV and 500 GeV runs. We found that dividing the runs at 350 GeV and 500 GeV into multiple polarization does not improve the results. For the ILC full program, we still show the constraint of cgeg instead of cgg and yt in order to compare with other scenarios. For the full program only the 500 GeV TGC results are used for consistency with the main results in gure 7. GDP 0.006 precision reach at ILC with different run scenarios at 1 TeV 0.008 1TeV run: only e+e-→ννh and e+ e- → tth are included ILC 250GeV(2/ab) + 350GeV(200/fb) + 500GeV(4/ab) ILC 250GeV(2/ab) + 350GeV(200/fb) + 500GeV(4/ab) + 1TeV(1/ab) ILC 250GeV(2/ab) + 350GeV(200/fb) + 500GeV(4/ab) + 1TeV(2.5/ab) rp0.004 0.002 0.000 o i e r ics0.04 0.02 0.00 δcZ cZZ cZ□ cγγ/10 cZγ/10 cgg/10 δyt/10 δyc λZ column corresponds to the ILC full run considered in our study with 2 ab 1 at 250 GeV, 200 fb 1 at 350 GeV and 4 ab 1 at 500 GeV and a xed polarization of P (e ; e+) = ( 0:8; +0:3). For the 2nd (3rd) column, an additional run at 1 TeV with an integrated luminosity of 1 ab 1 (2:5 ab 1 and polarization P (e ; e+) = ( 0:8; +0:2) is also included. For the 1 TeV run, the estimated measurement precisions in ref. [59] are used. Only the measurements of the e+e ! tth processes are included at 1 TeV, as the ones for e+e ! hZ and e+e ! W W are not ! h and provided. In particular, the precision of (tth) BR(h ! bb) at 1 TeV is estimated to be 6.0% with 1 ab 1 data and 3.8% with 2:5 ab 1 data, which signi cantly improves the precision at 500 GeV. As such, the constraints on both cgg and yt are greatly improved. It should be noted that the W W processes are more sensitive to some of the EFT parameters at ! hZ and e+e ! reach of the global t. higher energies. The inclusion of their measurements could potentially further improve the overall precision reach at FCC-ee with different sets of measurements FCC-ee 240GeV (10/ab), Higgs measurements (e+e-→ hZ / ννh), rates only FCC-ee 240GeV (10/ab), Higgs measurements only (e+e-→ WW not included) FCC-ee 240GeV (10/ab), e+e-→ ννh not included FCC-ee 240GeV (10/ab), angular asymmetries of e+e-→ hZ not included FCC-ee 240GeV (10/ab), all measurements included FCC-ee 240GeV (10/ab) + 350GeV (2.6/ab) dark shade: individual fit assuming all other 10 parameters are zero (0.41) (0.19) δcZ cZ□ cgefgf δyc δyb δyτ δyμ/10 λZ p0.02 0.01 0.00 precision reach at ILC with different sets of measurements ILC 250GeV (2/ab, 2 polarizations), Higgs measurements (e+e-→ hZ / ννh), rates only ILC 250GeV (2/ab, 2 polarizations), Higgs measurements only (e+e-→ WW not included) ILC 250GeV (2/ab, 2 polarizations), e+eILC 250GeV (2/ab, 2 polarizations), angular asymmetries of e+e-→ hZ not included ILC 250GeV (2/ab, 2 polarizations), all measurements included ILC 250GeV (2/ab, 2 polarizations) + 350GeV (200/fb) dark shade: individual fit assuming all other 10 parameters are zero HJEP09(217)4 δcZ cZZ cZ□ cgefgf δyb δyμ/10 λZ 250 GeV run is divided into two runs with polarizations P (e ; e+) = ( 0:8; +0:3) and (+0:8; 0:3), and fractions 0:7 and 0:3, respectively (see gure 10). em(m2Z ) = 1=127:940 and mh = 125:09 GeV. For the rate of e+e ments with the following energies and polarizations P (e ; e+) are used, ! hZ, the measureB B B B B B B B B B B 0240GeV unpolarized1 BB250GeV ( 0:8;+0:3)CC BB250GeV (+0:8; 0:3)CC hZ BB350GeV unpolarizedCC SM hZ BB350GeV ( 0:8;+0:3)CC BB500GeV ( 0:8;+0:3)CC BB 1:4TeV unpolarized CC 3TeV unpolarized C C C C C C C C C A CC'1+2 cZ +BBB 2:8 CCCCcZZ +BBB 7:5 CCCCcZ B B B B B 0 1:8 1 BB 5:6 CC B B B B 2:9CC C C B B B B B 11 C BB 21 CC B C A 52 C C C C C C C B B B 0 3:7 1 BB 9:8 CC C B B B B 3:2CC B B B B B 20 C BB 41 CC B C A 526 C C C C C C C C B B B +BB 0 B B B B B B B B B B B B B B B B 0:0481 0:73 CC BB 0:79 CC C C C C C C C C 0:11 CCCc 1:5 C 3:3 CC 1:9 CC C C C A 8:8 0 B B B B B B B B B B B B B B B B B B B B B 0:0871 1:3 CC 1:5 C C C C C C C 8:1 CC 5:5 CC C C C C C A 26 +BB 0:24 CCCCcZ : 3:3 C (D.1) As noted in section 3.1, the interferences between s-channel Z and photon amplitudes are accidentally suppressed in the unpolarized total cross section. On the contrary, they have a signi cant impact when polarized beams are used, ipping for instance the sign of the cZZ prefactor as polarization is reversed at p s = 250 GeV.11 The relevant expressions for 11For simplicity, the one-loop standard-model contributions to the hZ vertex are not included in the expressions above. They have a relatively large impact on the numerical prefactors of the c coe cients which are accidentally suppressed in the unpolarized cross section, at 240 GeV in particular. Given that this measurement has little sensitivity to these coe cients, such contributions do however not a ect the results of our global analysis. Note that the c parameter, directly related to the hZ vertex, is written in terms of cZZ , cZ , c and cZ using eq. (A.3). precision reach of aTGCs at FCC-ee 240GeV (10/ab) solid shade: combined with Higgs measurement assuming the following systematics in each bin of the differetial distrubtions of e+e-→WW: 0% 0.5% 1% 2% 5% 10% Higgs measurements only HJEP09(217)4 0% 0.5% 1% 2% Higgs measurements only (0.040) δg1,Z λZ precision reach at FCC-ee 240GeV (10/ab) assuming different systematics for e+e-→WW FCC-ee 240GeV (10/ab), all measurements included, assuming the following systematics in each bin of the differetial distrubtions of e+e-→WW: o i is0.008 c s i e rp0.015 0.010 0.005 0.000 the WW fusion process are 0 B B B B B 240GeV B B BB250GeVCC B B350GeVCC BB 1TeV BB 1:4TeV CC 3TeV 1 C C C C C C C C C C A 0 B B B B B B B B B B B B B B B B 0:25 0:27CC 0:40C 0:76CC 0:86C 1:1 A 1 C C C C C C C C C C C C δcZ cZ□ cgefgf δyb δyτ δyμ/10 λZ W W !h B W W !h BBB500GeVCCC ' 1+2 cZ +BB C B SM B B 0:53CCcZZ +BB 1:5 CCcZ +BB0:075CCc +BB 0:20 CCcZ ; 0 B B B B B B B B B B B B B B B B B B 0:68 0:72CC 1:1 CCC C 1 C C C C C 2:2 CC C C C 2:5 CC 3:4 A 0 B B B B B B B B B 0:035 BB0:037CC BB0:056C C C BB 0:12 CC B BB 0:14 CC C C C C C C C C 0 B B B B B B B B B 0:090 BB0:097CC BBB 0:14 CCC C BB 0:32 CC BB 0:37 CC C C C C C C C is0.008 c p0.006 0.004 0.002 e rp0.015 0.010 0.005 0.000 0% 0.5% 1% 2% 5% 10% Higgs measurements only 0% 0.5% 1% 2% Higgs measurements only (0.042) precision reach of aTGCs at ILC 250GeV (2/ab, 2 polarizations) 0.014 solid shade: combined with Higgs measurement assuming the following systematics in each bin of the differetial distrubtions of e+e-→WW: δg1,Z λZ precision reach at ILC 250GeV (2/ab, 2 polarizations) assuming different systematics for e+e-→WW 0.035 ILC 250GeV (2/ab, 2 polarizations), all measurements included, assuming the following systematics in each bin of the differetial distrubtions of e+e-→WW: δcZ cZ□ cgefgf δyc δyb δyτ δyμ/10 λZ with polarizations P (e ; e+) = ( 0:8; +0:3) and (+0:8; 0:3), and fractions 0:7 and 0:3, respectively (see gure 10). which are obtained from MadGraph5 [88] with the BSMC model [89, 90] as functions of cW , cW W and cW and then transformed into the basis in eq. (2.1) with eq. (A.3). The default input parameters are used for these numerical computations. They apply to any polarizations since only the initial states with helicities H(e ; e+) = ( ; +) contribute to this process. For the e+e ! tth process, we only consider the dominate NP contribution which is from the modi cation of the top Yukawa, yt. It is therefore straight forward to write down the rate of the tth process as of fermions are cc cc bb bb SM ' 1 + 2 yb ; tth tth SM ' 1 + 2 yt : For Higgs decays, we make use of the results in ref. [16]. The Decay widths to a pair SM ' 1 + 2 y ; SM ' 1 + 2 y : (D.4) The decay width to W W ZZ (with 4f nal states) are given by where we assume there is no NP correction to the gauge couplings of fermions. As stated in section 2, we do not consider contribution from o -shell photons that gives the same nal states as ZZ , as they can be relatively easily removed by kinematic cuts. The decay of Higgs to gg, and Z are generated at one-loop level in the SM. The decay widths are given by12 leading EFT contribution could either be at tree level (which are generated in the UV theory by new particles in the loop) or come at loop level by modifying the couplings in the SM loops. As mentioned in section 2, we follow ref. [16] and include both the tree level EFT contribution (cgg) and the one-loop contribution (from only keeping the tree level EFT contribution (c yt and yb) for h ! gg, while and cZ ) for h ! and h ! Z . The and from gg gg SM ' 1 + 241 cgg + 2:10 yt cZ 8:3 5:9 10 2 10 2 2 2 StoMt = tot X i SMi BriSM : The branching ratio can be derived from the total decay width, which can be obtained In practice, one only needs to include the BSM e ects of the main channels in the calculation of the total width. Finally, the physical observables in the form of BR can be constructed from the above information. impact on the global t results. 12The choices of the bottom mass value would change the numerical values in eq. (D.7), but has little (D.3) (D.5) (D.6) (D.7) (D.8) (D.9) precision (one standard deviation) cZZ cZ cgg yc y Z CEPC For each collider, the LHC 3000 fb 1 (including 8 TeV results) + LEP measurements are also combined in the total 2 . E Numerical results of the global t We hereby list the numerical results of the global t for the future e+e colliders. The one standard deviation constraints on each of the 12 parameters in eq. (2.1) are listed in table 11, and the corresponding correlation matrices are shown in table 12{15. For each collider, the LHC 3000 fb 1 (including 8 TeV results) + LEP measurements are also combined in the total 2, so that the results represent the \best reach" for each scenario. With this information, the corresponding chi-squared can be reconstructed using eq. (3.8), which can be used to constrain any particular model that satis es the assumptions of the 12-parameter framework, where the 12 parameters in EFT are functions of a usually much smaller set of model parameters. To minimize the numerical uncertainties, three signi cant gures are provided for the one standard deviation constraints, which is likely more than su cient for the level of precision of our estimations. For easy mapping to dimension-6 operators and new physics models, we also switch back to the original de nitions of c , cZ and cgg (instead of c , cZ and cgg). 1 cZZ 1 -0.37 -0.39 cZ -0.69 1 0.083 correlation matrix, CEPC cZ -0.15 -0.18 cgg -0.028 -0.17 -0.014 1 -0.011 -0.99 1 -0.19 -0.11 -0.44 -0.11 -0.25 -0.33 -0.11 -0.18 -0.049 -0.069 -0.063 -0.040 -0.0076 -0.044 -0.021 -0.16 -0.21 -0.72 -0.016 -0.11 -0.011 cZZ cZ cgg yc y y Z cZ cZ cZ cgg yc yb y Z cZ 1 -0.49 1 cZ -0.88 1 -0.055 0.13 -0.077 1 correlation matrix, FCC-ee cZ -0.22 -0.20 cgg -0.0025 -0.0018 0.13 -0.0038 -0.99 1 0.19 -0.24 0.13 -0.10 -0.15 0.26 -0.44 -0.16 -0.27 0.26 -0.33 0.15 -0.13 -0.20 0.013 -0.027 -0.034 Z -0.81 -0.0070 -0.054 -0.083 -0.0014 -0.0083 0.0027 -0.073 0.0050 -0.0054 cZ 1 cZZ 1 cZ 0.039 -0.020 1 cZ -0.26 -0.024 1 cgg -0.0049 1 cZ 1 1 -0.0065 -0.14 0.089 -0.46 0.051 cZ 1 0.022 1 correlation matrix, CLIC cZ -0.17 cgg -0.0014 -0.041 -0.012 1 cZ cZZ cZ cgg yc y Z cZ cZ cgg yc y y Z This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. 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Gauthier Durieux, Christophe Grojean, Jiayin Gu, Kechen Wang. The leptonic future of the Higgs, Journal of High Energy Physics, 2017, 14, DOI: 10.1007/JHEP09(2017)014