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 , D12489 Berlin , Germany
2 On leave from Institucio Catalana de Recerca i Estudis Avancats , 08010 Barcelona , Spain
3 Notkestra e 85 , D22607 Hamburg , Germany
4 Institut fur Physik, HumboldtUniversitat 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 centerofmass 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 weakbosonfusion production process h) provides an increasingly powerful handle at higher centerofmass 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
Weakboson pair production Global t and determinant parameter
Results
Conclusions
A E ective eldtheory 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 socalled 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 lowenergy
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 standardmodel 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 eldtheory 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 dimensionsix 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 ectiveoperator 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 weakboson fusion provides
complementary information of increasing relevance at higher centerofmass 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 higherenergy 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 gaugeinvariant 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 standardmodel in the processes enumerated above
through dimensionsix operators, in the socalled Higgs basis [39]. Translation to other
bases is however straightforward. Our assumption of perfectly standardmodellike
electroweak precision measurements is more easily implemented in that framework. No
deviation in the gaugeboson 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 ( avorconserving) muon, tau, charm, bottom, and top ones. Neither
CPviolating, 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 eldtheory 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 eldtheory 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 twelvedimensional parameter space.
2
E ective eldtheory framework
A global e ective eldtheory 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 dimensionsix 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 weakboson fusion: e+e
Higgs production in association with top quarks: e+e
!
h,
! tth,
weakboson 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, CPviolating 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 CPviolating 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 CPviolating
observables are perfectly constrained to be standardmodel like.
Throughout this paper, we only retain the interferences of e ective eldtheory
amplitudes with standardmodel ones. The squares of amplitudes featuring a dimensionsix
operator insertion are discarded. They are formally of the same c2= 4 order as the
interferences of dimensioneight operators with standardmodel 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 percentlevel measurement of an
observable of schematic
O
OSM
cE2
= 1 + O(1)
2 + O(1)
cE2 2
2
e ective eldtheory 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
percentlevel correction to this limit. In speci c cases, the interference of dimensionsix
operators with standardmodel 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 dimensionsix 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 eldtheory 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 standardmodel 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 lightfermion Yukawas at present and future experiments.
In total, 12 degrees of freedom are thus considered.
While all nonredundant 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 dimensionsix 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 widelyused 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 dimensionsix operators. The e+e
! hZ rate, at a given centerofmass 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
centerofmass energies have an enhanced sensitivity to cZZ and cZ . Angular
asymmetries in e+e
! hZ, weakbosonfusion production rate, weakboson 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 looplevel generated in the standard model, one may wish to include their
looplevel dependence in e ective parameters like yt, yb, cW which rescale
standardmodel interactions, or cZZ , cZ , etc. which do not. Complete e ective eldtheory results
at that order are however not currently available for the above processes (see ref. [49] for
the treatment of h !
). The computation of nexttoleadingorder e ective eldtheory
contributions to processes that are not looplevel 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 looplevel, respectively. The Higgsstrahlung, weakboson fusion,
and weakboson 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 downtype Yukawa couplings equal while ref. [17] focuses
on thirdgeneration 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 oneloop
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 looplevel
dependence on e ective eldtheory 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 looplevel yt and yb dependences of the h ! gg partial
width. It remains to be examined whether the looplevel 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 standardmodel 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 eldtheory 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 standardmodel 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 centerofmass 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 FCCee 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 centerofmass 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 centerofmass energies of about 450 GeV
where weakboson 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 centerofmass 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 FCCee 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 weakboson 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 schannel 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 righthanded 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 centerofmass 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 righthanded 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 eldtheory 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 CPviolating 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 eldtheory parameters. The corresponding Higgsbasis expressions
are provided in appendix D.
{ 9 {
e−
e
+
W −
W +
ν
ν¯
h
!
e−
e
+
Z/γ
Z
h
ν¯
h process: weakboson 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 weakboson 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 weakboson fusion cross section
grows with energy, so that it is better measured at a centerofmass 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 weakboson fusion (see gure 3) and has a rate about six times larger
at a centerofmass energy of 240 GeV (without beam polarization). At this centerofmass
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 weakboson fusion contribution is
therefore di cult. For the CEPC and FCCee 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 fourfermion 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 weakboson 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 weakbosonfusion 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 weakboson fusion contribution. Both processes reach equal rates at a
centerofmass energy close to 350 GeV (without beam polarization). At this and higher energies,
we thus assume that their distinct recoilmass 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 centerofmass 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 schannel 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 fourpoint Zhtt
interaction depicted on gure 4 (bottomright), nor fourfermion 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
Weakboson pair production
The diagrams contributing to the e+e
! W W process, at leading order, are depicted in
gure 5. The schannel 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 schannel diagram on the
couplings. Considering CPeven dimensionsix 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 FCCee 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 standardmodel values.
For the prospects of the full ILC program, we use the onesigma 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 Zpole 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 centerofmass
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
gaugeboson propagators and fermion gauge couplings could however have a nonnegligible
impact on the determination of triple gauge couplings, especially without a future Z
factory to improve their constraints.6 While the CEPC and FCCee 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 Zpole 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 Zpole 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 onesigma
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 FCCee, 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 onesigma 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 eldtheory 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 onesigma 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 ectiveoperator 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 eldtheory 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,
centerofmass 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 FCCee 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.
101
n
o
i
s
i
rce102
p
103
104
*
*
*
precision reach of the 12parameter fit in Higgs basis
*
*
****

*
*
*
***
*
*
****
λZ
GDP
eters. All results but the lightshaded 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 onesigma precision reach at various future lepton colliders on
our e ective eldtheory 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 highluminosity 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 SMlike 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 eldtheory [24, 79] and
of the TGC dominance assumption [25] need to be simultaneously considered. A dedicated
study of the reach of the highluminosity 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 eldtheory 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 FCCee to constrain cgg and yt. The resulting bounds on yt are
then even substantially better than that set by the LHC alone.
The twelveparameter 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, FCCee, ILC and
CLIC, respectively. Varying prospective constraints on the charm Yukawa measurement
complicate the comparison with the highluminosity 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 onesigma 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 elevenparameter 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 elevenparameter GDP are 0.0073, 0.0053,
0.0046, 0.0052 for CEPC, FCCee, ILC and CLIC, respectively.
Let us also comment further on the impact of having discarded the quadratic
dependence on dimensionsix 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 standardmodel interference with e ective eldtheory
amplitudes require a casebycase 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 centerofmass
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 noninterference between standardmodel
and dimensionsix 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 schannel 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 eldtheory
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 eldtheory 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 onesigma 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 eldtheory 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 eldtheory 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 centerofmass 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 ectivetheory 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 Zpole and W
mass measurements
and no dipole interaction. We only consider the
avorconserving 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 eldtheory 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 onesigma
precisions of the Higgs rate measurements are respectively displayed in table 2 for the CEPC
and FCCee, 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
FCCee, 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 FCCee, relative precision
are rescaled from the 350 GeV ILC ones.10 The precision of (hZ)
BR(h ! Z ) is not
provided for the FCCee 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 FCCee 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 FCCee. 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
FCCee 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 FCCee.
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
FCCee
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 centerofmass 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 chisquare 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
thirdgeneration couplings. Since we explicitly assume the future results are SMlike, 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
101
o
i
s
i
104
1
101
n
o
i
s
i
c
re102
p
103
104

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 12parameter 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 HLLHC
blue line: individual constraints
CEPC
ILC
CLIC
FCCee 240GeV (10/ab) + 350GeV (2.6/ab)
yt
c
y
c
c
yb
c
yτ
c
y
μ
here for the FCCee 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 chisquare 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
FCCee 240GeV (10/ab) only
FCCee 240GeV (10/ab) + 350GeV (500/fb)
FCCee 240GeV (10/ab) + 350GeV (1/ab)
precision reach at FCCee 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 FCCee with different sets of measurements
FCCee 240GeV (10/ab), Higgs measurements (e+e→ hZ / ννh), rates only
FCCee 240GeV (10/ab), Higgs measurements only (e+e→ WW not included)
FCCee 240GeV (10/ab), e+e→ ννh not included
FCCee 240GeV (10/ab), angular asymmetries of e+e→ hZ not included
FCCee 240GeV (10/ab), all measurements included
FCCee 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 schannel 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 oneloop standardmodel 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 FCCee 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 FCCee 240GeV (10/ab) assuming different systematics for e+e→WW
FCCee 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 oneloop 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 oneloop 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 chisquared can be reconstructed using eq. (3.8),
which can be used to constrain any particular model that satis es the assumptions of the
12parameter 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 dimension6
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, FCCee
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 (CCBY 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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