Beyond Higgs Couplings: Probing the Higgs with Angular Observables at Future $e^+ e^-$ Colliders

We study angular observables in the $e^+e^-\to Z H\to \ell^+ \ell^-\,b\bar{b}$ channel at future circular $e^+ e^-$ colliders such as CEPC and FCC-ee. Taking into account the impact of realistic cut acceptance and detector effects, we forecast the precision of six angular asymmetries at CEPC (FCC-ee) with center-of-mass energy $\sqrt{s} =$ 240 GeV and 5 (30) ${\rm ab}^{-1}$ integrated luminosity. We then determine the projected sensitivity to a range of operators relevant for the Higgs-strahlung process in the dimension-6 Higgs EFT. Our results show that angular observables provide complementary sensitivity to rate measurements when constraining various tensor structures arising from new physics. We further find that angular asymmetries provide a novel means of both probing BSM corrections to the $H Z \gamma$ coupling and constraining the"blind spot"in indirect limits on supersymmetric scalar top partners.


roperties has
become one of the highest priorities for current and future colliders.Highluminosity electron-positron colliders are particularly well suited to this end, promising a large sample of relatively clean Higgs production events and the ability to directly probe Higgs properties in a model-independent fashion.Such precision tests of Higgs couplings will provide a window into physics beyond the Standard Model (BSM) well above the weak scale.

Thus far much attention has focused on the potential of future e + e − colliders to probe deviations in Higg properties in terms of a re-scaling of Standard Model couplings [3][4][5], with sensitivity exceeding the percent level in some channels.However, in general deviations in Higgs properties may encode additional information, for example in the form of operators with different tensor structure in the Higgs Effective Field Theory (EFT).Disentangling contributions from these different operators provides a further handle on BSM physics by both increasing the effective reach of e + e − colliders and distinguishing different BSM scenarios in the event of deviations from the Standard Model.

The ZH production cross section provides a particularly sharp tool, as it allows a modelindependent measuremen of the Higgs-Z coupling in Higgsstrahlung events identified solely by Z recoils.The relatively large number of ZH production events at proposed e + e − colliders is expected to give sub-percent level precision in the measurement of the hZZ coupling, providing sensitivity to a range of BSM scenarios [6][7][8][9][10][11][12].But ZH production provides more than just a probe of rescalings of the hZZ coupling.The measurement of angular observables in ZH production provides sensitivity to a variety of tensor structures and therefore allows discrimination among a range of BSM scenarios.In this paper we investigate the potential for future circular e + e − colliders to discriminate between different BSM contributions to the ZH production cross section through the use of angular asymmetries.

Needless to say, there is a long history of studying angular distributions of Higgs production events at e + e − c lliders, both at LEP and for the planned ILC [13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30].In this work we build on these previous studies by forecasting sensitivity to a complete basis of dimension-6 operators in the Higgs EFT at proposed future e + e − circular colliders such as CEPC and FCC-ee, accounting for realistic cut acceptance and detector effects.We believe this is the first comprehensive study of angular observables at proposed e + e − colliders of its kind.

Our paper is organized as follows: In Section 2 we first review aspects of the Higgs EFT relevant for Higgsstrahlung identifying a complete set of dimension-6 operators relevant for characterizing deviations in e + e − → ZH.We construct a complete set of angular observables in e + e − → ZH, closely following [26], and demonstrate their sensitivity to various dimension-6 operators at √ s = 240 GeV.In Section 3 we forecast the sensitivity to these angular observables at proposed e + e − colliders, taking into account the effects of realistic cuts and detector effects.We then consider the sensitivity of proposed e + e − colliders to a range of BSM scenarios in Section 4, demonstrating the ability of angular observables to break degeneracies in rate measurements.We conclude in Section 5 and reserve some details of the angular observables and our statistical treatment for the Appendix.


Angular Observables

In this section we begin by reviewing aspects of the Higgs EFT relevant for the Higgsstrahlung pro

ss, and identify a c
mplete set of operators for characterizing various BSM contributions to e + e − → ZH.We then construct a set of independent angular asymmetries suitable for identifying contributions from different operators, following [26].


Higgs EFT for Higgsstrahlung

Given the apparent parametric separation of scales between the Higgs boson and new physics

the Higgs EFT provides a usef
l framework for characterizing deviations in Higgs properties from their Standard Model (SM) predictions.There are many independent operators at a given order in power-counting, including 59 dimension-6 operators.The Lagrangian out to dimension 6 prior to electroweak symmetry breaking takes the form
L eff = L SM + 1 Λ 2 59 i=1 α i O i (2.1)
Only a subset of these operators contribute to the e + e − → ZH process, and of th nitions or equations of motion.

Here we will work in terms of a minimal operator basis given in [26]; for a comparable choice of basis, see [9].The relevan operators defining our operator basis are given in Table 1.Although there is no invariant meaning to a particular choice of basis, this basis is sufficient to characterize all dimension-6 contributions to e + e − → ZH in the sense that all other operators contributing to e + e − → ZH can be re-written in terms of this operator basis plus additional operators irrelevant to e + e − → ZH.After electroweak symmetry breaking, these dimension-6 operators give rise to a variety of interaction terms relevant for e + e − → ZH of the form
O Φ = (Φ † Φ) (Φ † Φ) O ΦW = (Φ † Φ)W I µν W Iµν O ΦD = (Φ † D µ Φ) * (Φ † D µ Φ) O ΦB = (Φ † Φ)B µν B µν O (1) Φ = (Φ † i ↔ D αβ V αβ .Here we again use the notation of [26] for clarity.

The couplings in this broken-phase effective Lagrangian may be straightforwardly expressed in terms of coefficients in the dimension-6 Higgs EFT.In this respect it is useful to consider the following linear combinations of (dimensionless) dimension-6 operator coeffi-cients:
α (1) ZZ = α Φ − 1 2 δ G F + 1 4 α ΦD , α ZZ = c 2 W α ΦW + s 2 W α ΦB + s W c W α ΦWB , α Z Z = c 2 W α Φ W + s 2 W α Φ se effective Lagrangian via
c (1) ZZ = m 2 Z ( √ 2G F ) 1/2 1 + α (1) ZZ , c(2)
ZZ = (
√ 2G F ) 1/2 α ZZ , c Z Z = ( √ 2G F ) 1/2 α Z Z ,(2.4
)
c AZ = coefficient c (1)
ZZ decomposes into the Standard Model prediction parameterize lus corrections given by new physics encoded in the Higgs EFT.The remaining coefficients are zero at tree-level in the Standard Model, and so the leading contributions are taken to be those from the dimension-6 operators.Note that the CP-even coeficients are generated at one loop in the Standard Model.

Similarly, for the hZ contact terms in Eq. (2.2) it is useful to form the linear combinations
α V Φ = α Φe + α (1) Φ + α (3) α A Φ = α Φe − α (1) Φ + α (3) Φ . (2.5)
which are simply related to the coefficients in Eq.

+0.
+0 th e peri ents up to ross sect on a

an
ul θ W and they remain at the sub-perc em m 2
0.4 α  ZZ α  ZZ (1) α  Φl V α  Φl A α  AZ δg V δg A α  Z Z ∼ α  A Z ∼ +0. +0. +0. +0. +0.02 +0. +0. +0.
+0.37 while the angular observables scale as
∂A θ1 ∂α i α=0 A θ 1 -0.6-0.4-0.2 0.0 0.2 0.4 α ZZ − 0.59 Φ − 0.44 AZ + 0 )A ( ) φ 0.0 36 − 0.004 α Z + 1 06 α V re r presen . 4 5.The . 2 & 3, but provi 1 0.2 α  ZZ α  ZZ (1) α  Φl V α  Φl α  Φ 6 +0 +0. +0. 0.11 ∂A (4) se.T e to al c oss ecti n is ial -lepton he a ymmetry are particularly sensitive symm try (3 φ are part cula ly s nsitive to hZ µν A µν , the vector contact pton coupling Z µ ¯ γ µ .The asymmetries A  It is worth pointing out that the choice of scales at which the couplings are evaluated has a significant impact on the central values of asymmetry observables.In particular, A

φ , A

φ , and A cθ 1 ,cθ 2 are proportional to the coupling combination g
V g A g 2 V +g 2 A 2
, which is quite sensitive to the value of sin 2 θ W . Evaluating the couplings at ∼ 240 GeV instead of the Zpole, the central values of asymmetry variables varies by O(10%).Although our choice of scales is adequate for forecasting sensitivity to asymmetry variables, a careful treatment of the Standard Model prediction f r these asymmetry observables will ultimately be required in order to extract useful information from angular observables at future e + e − colliders.


Angular Observables at CE − → ZH, we now develop projections for the sensitivity attainable at various proposed Higgs factories.

In particular, we study the reach in angular observables at two proposed future e + e − colliders: the Circular Electron-Positron Collider (CEPC) and the e + e − mode of the CERN Future Circular Collider (FCC-ee).Both of these colliders are designed to produce large numbers of e + e − → ZH events at the center-of-mass energy √ s = 240 GeV.With a proposed luminosity of

× 10 34 cm −2 s −1 per Interaction Poin
(IP), the integrated luminosity at CEPC will be 5 ab −1 over a running time of 10 years with 2 IPs [5,35].The machine parameters of FCC-ee [36] project that its luminosity can reach 6 × 10 34 cm −2 s −1 at √ s = 240 GeV, which is three times that of CEPC.In addition, there is a factor of 2 increase in luminosity on account of the projected 4 IPs at FCC-ee, bringing the total FCC-ee luminosity to six times that of CEPC.

Considering the same running time of 10 years, we therefore take the integrated luminosity at FCC-ee to be 30 ab −1 for the purpose of our projections.


Expected precision and statistical uncertainty

In Table 3 we list the theoretical expectations for all the relevant asymmetry observables assuming only Standard Model contributions, as well as the 1σ errors (σ A ) calculated from Eq. (B.4) for various integrated luminosity benchmarks.We consider only the process with Z → µ + µ − /e + e − and H → b b, which is almost entirely background-free.According to Section 3.3.3.1 in the pre-CDR of CEPC [5], the number of events after basic cuts in both µ + µ − and e + e − channels is 11067 + 11033 = 22100 for 5 ab −1 .We assume for simplicity that FCC-ee wil

conduct a very similar study on this channel, a
d consequently scale the statistics up directly to 30 ab −1 for FCC-ee.We emphasize that our study is very conservative by focusing only on the Z → µ + µ − /e + e − decays, which comprise only about 7% of all Z decays.A more comprehensive study could employ other visible Z boson decays to improve signal statistics, albeit at the cost of requiring a detailed background analysis.If all visible decays of the Z boson -about 80% of the total branching ratio -and additional decay channels of the Higgs beyond b b can be included, it is in principle possible to gain a factor of ten improvement in the statistics for our analysis.[5], the number of events after basic cuts is 22100 for 5 ab −1 .We use this number here and also scale it up with luminosity for 30 ab −1 and the full statistics scenario detailed in text.

Selection Therefore, although we do not perform such an analysis here, to demonstrate the full diagnostic power of asymmetry observables we consider a third benchmark with ten times larger statistics than the FCC-ee case with only the dilepton channel. 2 We refer to this as the "Full Statistics" (FS) benchmark with an eye towards the maximum sensitivity obtainable by using additional Z and H decay products.
A θ 1 A (1) φ A (2) φ A (3) φ A (4) φ A cθ 1 ,cθ 2

Simulation procedure and detector effects

At √ s = 240 GeV, the dominant Higgs production process will be e + e − → ZH.As discussed above, we choose the low-background, pr − b b to demonstrate the diagnostic power of angular observables and their complementarity to inclusive observables such as the rate measurement.

To understand the detailed effects of selection cuts on BSM contributions, we perform a numerical study on the signals using Madgraph5 [37] with dimension-six operator model file generated via FeynRules [38].The signal process is very clean.For our stud

we employ the event selection of CEPC pre
DR analysis [5] for this channel, including lepton angular acceptance of 10 • < θ µ < 170 • , lepton pair p T of 10 GeV < p T (µ + µ − ) < 90 GeV, lepton pair on-shell condition of 81 GeV < m µ + µ − < 101 GeV, recoil mass of 120 GeV < m recoil < 150 GeV and b-tagging. 3After applying these cuts, the background is negligi le and the dominant uncertainty is due to signal statistics.

Table 4 shows the cut flow for the 6 angular observables at each stage of event selection.As shown in the line 3 of this table, the acceptance of the polar angle can greatly change some anglular observables.This suggests, among other things, that the power of angular observables might be improved at future e + e − colliders by enlarging the detector acceptance for charged leptons beyond current projections.

The signal selection cuts also shift the central value of the observable asymmetries, which likewise effects their diagnostic power in the event of a BSM contribution.In Fig. 6 we show the asymmetry observable and cross section values resulting from our analytical calculation; our simulation before cuts; and our simulation after cuts, in each case following the input parameter choices discussed in Section 2. To demonstrate the impact of realistic cuts on different values of the dimension-6 operator coefficients, we choose to plot these predictions as a function of ĉΦB , the coefficient one of the higher dimensional op rators only constrained by precision Higgs measurements.The operator O ΦB affects all CP-even observables σ, A θ 1 , A
φ , A(3)
φ and A cθ 1 ,cθ 2 , and is a reasonable proxy for the impact of cuts on various BSM contributions to the angular asymmetries.

We can see in Fig. 6 that our simulation before cuts agrees well with our analytical calculation, and the cuts alter the asymmetries consistently for different values of ĉΦB .The central values of the asymmetry variables for different values of the coefficients ĉΦB before and after cuts are shown in blue and red lines.Unsurprisingly, the asymmetries change as a result of the cuts, ranging from sub-percent level for most asymmetry variables to around 40% (reduction) for A (4) φ .Cruciall of the asymmetries as a function of the Wilson coefficient do not change much before and after cuts.This allows us greatly sim lify our forecasting, as it demonstrates that realistic cuts do not alter how asymmetry observables respond to new operators in the regime of interest.

To study the impact of realistic cuts and detector acceptance on asymmetry observables we compare our Madgraph5 analysis result with the data sample from the CEPC Pre-CDR study ground for the SM expectations.Their sample data is generated using Whizard [39,40] for the process of e + e − → HZ, with H → b b at √ s = 240 GeV.A fast simulation of detector effects using the energy resolution and acceptance in polar angle of the CEPC detector is applied as well.Our study with the Madgraph5 sample and with the CEPC preCDR sample φ (right panel) as a function of the coefficient of operator O ΦB .The blue and red lines indicate the simu ated results at parton level before and after cuts.The green and yellow band is projected precision for corresponding observable with CEPC 5 ab −1 , assuming measured values follow SM.The numbers are the slope of the asymmetry observable with respect to coefficients ĉΦB at parton level, after cuts and in theory, respectively.agree well in the values of the asymmetries under selection cuts, after taking into account the difference caused by different SM input parameters.With the initial state radiation the initial state e + e − and the intermediate Higgs boson and Z boson are no longer coplanar, causing ambiguities in the definition of asymmetries.We compare the asymmetry values obtained in the lab frame and the e + e − → ZH collision center of mass frame.We find the difference are negligibly small for the precisions of symmetries at CEPC.

Following our simulation study and the validation of CEPC pre-CDR data sample, we judiciously apply a universal 5% penalty factor for the sensitivities of asymmetry observables to Wilson coefficients under onsideration.This penalty factor is to account for detector effects and other systematics.

In Fig. 6 we also show the one-and two-sigma bands (green and yellow) indicating the constraint that can be placed at CEPC assuming a Standard Model-like central value.Here we see that the angular variable A θ 1 places an order-of-magnitude stronger bound on O ΦB compared to A (4) φ .This is unsurprising given our discussion in Section 2, but also owes in part to the different precision attainable in the two asymmetries after cuts.We reserve a more complete discussion of the constraints on Wilson coefficients for Section 4.

Apart from the systematic impact of cuts and acceptance discussed above, there are several other sources of uncertainty to consider in a realistic treatment of angular asymmetries at future colliders:

• There re instrumental uncertainties related to uncertainty in the integrated luminosity which affect the cross section measurement, but this cancels for asymmetry observables by construction.

• Instrumental uncertainty from beam energy resolution affects the cross section measure-ment, but this can be calibrated well by other processes, and its effect on asymmetry observables is very small due to th weak dependence on center-of-mass energy exhibited in [26].

• Instrumental uncertainty from initial-state-radiation4 affects the reconstruction of the scattering plane, but we have verified that the size of this effect is small at the level of our Whizard simulation and is well within the size of our conservatively-assigned overall uncertainty.To verify this in detail, we use the signal events for the same process from the CEPC Pre-CDR study ground with initial-state-radiation, reconstruct the angular observables ignoring such effects (i.e., assuming lab frame and c.m. frame are identical) and compare the values of these observable with the true asymmetry using the radiated photon information.Uncertainties resulting from these effects are numerically sub-dominant.

• Instrumental uncertainty from particle reconstruction energy resolution is also small because the leptons can be measured very precisely and our asymmetry observables rely only on the lepton momentum.This may be ome a more significant uncertainty if asymmetries are constructed from non-leptonic decay products, or if more information from the decay of the Higgs is employed.

• Theoretical uncertainties such as uncertainties on input parameters and electroweak corrections are very important for the precision measurement of angular observables.We consider them to be factorizable from oth r systematics and note that these uncertainties can be significantly improved in the near future.Current estimates place uncertainties due to NLO electroweak corrections at the (sub)percent level.While they may be somewhat higher in angular observables, substantial improvements may be realized by the advent of future Higgs factories.We note that some angular observables such as A

φ and A cθ 1 ,cθ 2 are highly sensitive to the values of input parameters, and the central values of Standard Model predictions can be altered by as large as a factor of three by different choices of SM input parameters

his makes cle
r the need for a careful future treatment of Standard Model predictions for these observables.


Applications

Given our estimates for the sensitivity attainable in angular observables at future e + e − colliders, we now consider the implications of this sensitivity for a variety of BSM physics scenarios.Broadly speaking, angular observables both improve the overall reach for BSM physics and constrain linear combinations o Wilson coefficients in the dimension-6 HEFT orthogonal to those constrained by coupling measurements alone.

For the purposes of forecasting, we assume the experimental results are SM-like and obtain the expected constraints on new physics using a simple χ 2 fit.For the sake of con-creteness, we focus on the channel e + e − → ZH → + − b b at CEPC with √ s = 240 GeV and 5 ab −1 integrated luminosity, although we also forecast sensitivity for several scenarios at FCC-ee.As we have justified in Section 3, statistical uncertainties dominate for the angular observables in this channel.As such, in this section we neglect systematic uncertainties and consider only statistical uncertainties based on a sample size of 22100 events (the expected number of events collected by CEPC after selection cuts with 5 ab −1 integrated luminosity).To compensate the omission of systematics, we judiciously apply a universal 5% penalty factor for the sensitivities of asymmetry observables to Wilson coefficients, as mentioned in Section 3.For the uncertainty in the cross section, we adopt the values in the preCDR [5], which are 0.9% for the µ + In what follows we will also neglect theory uncertainties in the Standa d Model predictions for the total cross section and for angular observables.Recent forecasts for precision measurements of the associated production cross section suggest that uncertainties on the order of O(0.5%) are realistic.For angular observables the situation is somewhat less clear, but we expect substantial progress in the study of NLO corrections to angular variables in anticipation of a future e + e − collider.

The χ 2 from the rate measurements (χ 2 rate ), the angular measurements χ 2 angles ) and the cominbation of all measurements (χ 2 total ) are defined as
5 χ 2 rate = (X NP − X SM ) 2 σ 2 X ,(4.1)χ 2 angles = i (A i NP − A i SM ) e; X NP and A i NP are the predictions of new physics, which can be written as functions of the Wilson coefficients as e.g. in Eq. (2.18)-(2.20);and the A i are summed over A θ 1 , A
φ , A(1)φ , A(2)φ , A(3)
φ and A cθ 1 ,cθ 2 .Here σ X and σ A i are the 1σ un observables, respectively.We have also neglected the correlations among (the experimental measurements of) the observables, which we expect to be small.

The formulation in Eq. (4.1-4.3) is also applicable to the discrimination of different ew physics cases, for which one could assume the measured values of the observables are given by some benchmark new physics scenario.The changes in the expected precisions are negligible, unless the NP predictions are dramatically different from the SM ones (in which case the effective theory description will break down).Since there is no additional information, we will focus on forecasting in the scenario that the experimental results are SM-like.

The rest of this section is organized as follows: In Section 4.1, we present the constrai ts on the Wilson coefficients in the Higgs effective Lagrangian with a model-independent approach.In Section 4.2, we discuss in detail the constraints on the HZγ coupling, for which angular observables provide sensitivity comparable to that of direct measurement at e + e − colliders.In Section 4.3, we demonstrate how the constraints from angular observables can be applied to specific models of new physics, using light stops as an example.


Constraining Wilson coefficients

In this section we present the model-independent constra

ts on the Wilson coefficients in
he Higgs effective Lagrangian, Eq. (2.2), parameterized the 9 Wilson coefficients in Eq. (2.25), which are
α ZZ , α(1)ZZ , α V Φ , α A Φ , α AZ , δg V , δg A , α Z Z , α A Z .
Treating the 9 coeffici e rate and the six asymmetry observables, less than the number of unknowns.Therefore, one cannot obtain independent constraints on the Wilson coefficients without making further assumptions.However, with a reduced set of coefficients the angular observables can break the degeneracy of the rate measurement, which by itself could only constrain one linear combination of the Wilson coefficients.To illustrate this point, we focus on two coefficients at a time while setting the rest to zero.One of the coefficients is always chosen to be α

ZZ , which parameterizes a modification of the SM HZ µ Z µ interaction and is most strongly const ained by the rate measurement.The angular observables, being normalized to the total cross section, are independent of α ZZ by construction.In Fig. 7, we show the expected constraints in the two-dimensional parameter space consisting of α
(1)
ZZ and one of the remaining coefficients, assuming SM-like measurements.The total combined constraints from both rate and angular measurements are shown along with the ones from the rate measurements or the (combined) angular observables alone.From Fig. 7, it is clear that the rate measurement alone only constrains a linear combination of the two coefficients.The inclusion of angular observables typically allows considerable discrimination between Wilson coefficients.In particular, in Fig. 7 it is apparent that angular observables have appreciable discriminating power for the coefficients α V Φ , α AZ , δg V , α Z Z , and α A Z .Note that the angular obervables A
φ , A(1)
φ depend only on CP-odd operators, which are zero in the Standard Model and thus entirely dominated ons from the dimension-6 EFT.The remaining angular observables depend only on CP-even operators, which generally accumulate radiative contributions in the Standard Model.

In principle, precision e + e − colliders could be used to set appreciable bounds on CP-odd operators via measure ents of the observables
A (1) φ , A(2)
φ .However, these operators are much more strongly constrained by bounds on the electron EDM, as they t one-loop order through Barr-Zee type diagrams [41].In [42] the corresponding bounds on O ΦB , O ΦW , and O ΦW B were computed using |d e |/e < 1.05 × 10 −27 .Recently the ACME experiment improved this bound to |d e |/e < 8.7 × 10 −29 [43].Accounting for the improved limit, for Λ = 1 TeV, this corresponds to bounds of α ΦB < 1.4 × 10 −4 , α ΦW < 3.6 × 10 −5 , and α ΦW B < 6.7 × 10 −5 when the operators are considered independently.For α i = 1, this equivalently corresponds to bounds on the scales of the dimension-6 operators O ΦB , O ΦW , and O ΦW B of Λ > 83, 167, and 122 TeV.Thus we conclude that potential limits on A (1)
φ , A(2)
φ at e + e − colliders, while appreciable, are far from competitive with existing bounds coming from the el le sensitivity to the CP-odd observables is not competitive with EDM experiments, sensitivity to α V Φ , α AZ , and δ V provides meaningful improvement over bounds from rate measurements alone.We will discuss the particular utility of the α AZ constraint in the next subsection.

Of course, the combination of angular observables is not universally useful, and in some cases (namely α ZZ , α A Φ and δ A ), the inclusion of angular observables provides little additional information relative to the rate measurement.It should be noted that for these cases we have chosen quite large plot ranges for the sake of illustration; in reality, for large Wilson coefficients the effective theory description would break down.

In addition to the constraints in two-dimensional parameter spaces, we provide in Table 5 the constraints on individual Wi son coefficients with the assumption that all other coefficients are zero.Table 5 shows the 1σ uncertainties for each Wilson coefficient (setting others to zero) from the rate measurements only, the angular observables measurements only, and the combination of the two.We use "∞" to denote coefficients for which no constraint can be derived within our procedure.In particular, the angular observables are insensitive to α (1) ZZ by construction, while the rate measurements are independent of the CP-odd operators at leading order in the Wilson coefficients.As discussed in Section 3, with the same running time FCC-ee is able to deliver a sample size 6 times larger than that of CEPC.It is also reasonable to expect that statistical uncertainties dominate for the e + e − → ZH → + − b b process at FCC-ee as they do at CEPC.Furthermore, the inclusion of additional decay modes of H and Z would increase the statistics and could potentially significantly increase the constraining power.While the reaches of other channels would require further study, to illustrate their potential usefulness we perform a sample size of 22100, 132600 spectively.All contours correspond to 68% CL.The green dot at (0, 0) indicates the SM prediction.
α ZZ α (1) ZZ α V Φ α A Φ α AZ δg V δg A α Z Z α A

The HZγ coupling

As we have seen, angular observables in e + e − → ZH provide an additional means of probing α AZ , thereby constraining anomalous contributions to the hZγ coupling.To date, much attention has been devoted to constraining the hZγ coupling in decays h → Zγ (which can be observed at the LHC) and in the production mode e + e − → γh (which can be observed at future e + e − colliders).However, the contribution of the hZγ to e + e − → ZH via an intermediate photon provides a complementary probe at e + e − colliders that is not sensitive to the potentially complicated backgrounds faced by e + e − → γh.This is perhaps not surprising.In general, one expects angular observables to provide a powerful handle on small deviations in the hZγ coupling, insofar as BSM contributions appear at O(α AZ ) via interference with the tree-level SM process

+ e − → ZH, where
s in e − e − → γh they arise either directly at O(α 2 AZ ) or at O(α AZ ) via interference with the loop-level SM process e + e − → γh. 6he limits on hZγ that may be obtained by precision measurement of both the rate and angular distributions of e + e − → ZH (→ + − b b) is shown in Table 5, amounting to −0.0070 ≤ α AZ ≤ 0.0070 at CEPC.By comparison, direct measurement of the e + e − → γh process at CEPC (scaling up the result of [44] to 5 ab −1 ) leads to a projected bound of −0.008 ≤ α AZ ≤ 0.003.While this is somewhat better than the projected bound from e + e − → ZH, the analysis in [44] includes only the backgrounds from hard processes e + e − → γb b.In general one expects additional backgrounds from beamstrahlung that are more difficult to characterize and may further complicate the direct measurement.At the very least, it is clear that angular observables provide a surprising and competitive avenue for probing the hZγ coupling at future e + e − colliders.


Stops

As a final example of the discriminating power of angular observables, we consider a concrete weakly-coupled model that may be constrained with precision measurements at e + e − colliders: scalar top partners (stops) in supersymmetric extensions of the Standard Model.For simplicity, we will consider stops with degenerate stop soft masses m 2 t = m 2 Q3 = m 2 tR plus mixing terms of the form X t = A t − µ cot β.The mass scale of the effective operators is Λ = m t.Wilson coefficients for this scenario were computed in [45], while the constraint on the stop parameter space due to rate measurements at e + e − colliders was determined in [9] 7 .

Here we include the additional sensitivity contributed by angular observables by translating the results of [9,45] into our preferred basis of Wilson coefficients and applying the results of the previous section.In Fig. 9 we show the sensitivity provided by rate measurements and the inclusion of angular observables in the plane of the two stop mass eigenvalues M 1 and M 2 , which are functions of m 2 t and X t given by the stop mass mixing matrix.For definiteness we have set tan β = 10, while the results are insensitive to tan β as long as tan β few.Expected constraints in the (M 1 , M 2 ) plane for different collider scenarios, assumi g SM-like results.M 1 and M 2 are the two mass eigenvalues of the left-and right-handed stops.The three scenarios, CEPC, FCC-ee and FCC-ee FS are described in Section 4.1.We set tan β = 10.The blue contours show the constraints from the rate measurements only and the red contours show the total combined constraints from the measurements of rate and the angular observables .The solid (dotted) lines corresponds to 68%(95%) CL.The region in the upper-right part of each plot is allowed by projected coupling measurements.

The features of the exclusion provided by rate measurements were discussed extensively in [9].The most noteworthy feature of the rate measurements is the so-called "blind spot" along the line M 2 = M 1 + √ 2m t where the shift in the hZZ rate is zero.Such blind spots arise more generally in stop corrections to various Higgs properties such as hgg, hγγ couplings and precision

ectroweak ob
ervables.Each blind spot corresponds to a zero in physical linear combinations of Wilson coefficients.While the exact zeroes in observables arise in different places in the M 1 − M 2 plane, they are collected around the line in which the coupling of the lightest stop mass eigenstate to the Higgs goes to zero.In general, the addition of angular observables does not lead to immense improvements over the rate measurement in generic regions of parameter space.This is not surprising, since the relevant Wilson coefficients well-constrained by angular observables are generated at one loop and thus are small for all values of the stop masses.However it is apparent in Fig. 9 that the addition of angular observables provides significantly improved sensitivity in the blind spot of the hZZ rate measurement.This is simply because the Wilson coefficients contributing to angular observables are suppressed but nonzero along the line where the ZH cross section shift is zero, and so provide complementary sensitivity at small M 1 , M 2 provided sufficient statistics.This demonstrates the value of angular observables even in the case of BSM scenarios that are generally well-constrained by rate measurements.


Conclusions

Future e + e − provide unprecedented opportunities to explore the Higgs sector.The large sample size of clean Higgs events may be used to constrain no only deviations in Higgs couplings, but also non-standard tensor structures arising from BSM physics.While the former are readily probed by rate measurements, the latter may be effectively probed using appropriately-constructed angular asymmetries.In this work we have initiated the study of angular observables at future e + e − colliders such as CEPC and FCC-ee.We have taken particular care to account for the impact of realistic cut acceptance and detector effects on angular asymmetries.

Our primary result is a forecast of the precision with which angular asymmetries may be measured at future e + e − colliders.We have translated this forecast into projected sensitivity to a range of operators in the dimension-6 EFT, where angular measurements provide complementary sensitivity to rate measurements.Among other things, we have found that angular asymmetries provide a novel means of probing BSM corrections to the hZγ coupling beyond direct measurement of e + e − → hγ.We also apply our results to a complete model of BSM physics, namely scalar top partners in supersymmetric extensions of the Standard Mo

l, where angular observ
bles help to constrain the well-known "blind spot" in rate measurements.

There are a wide range of interesting future directions.In this work we have focused on ZH events with Z → + − and h → b b in order to obtain a relatively pure

ound contamination.Of course, there will be far more even
s involving alternate decays of the Z and Higgs which, while not backgroundfree, could add considerable discriminating power.It would be useful to conduct a realistic study of these additional channels to determine the maximum possible sensitivity of angular asymmetries.Although we have taken care to account for the impact of cut acceptance and detector effects on angular asymmetries, our work has neglected the possible impact l prediction for angular asymmetries.A detailed estimate of current and projected theory uncertainties in the Standard Model prediction for Higgsstrahlung differential distributions would be broadly useful to future studies.More generally, this work serves as a starting point for investigating the full set of Higgs properties accessible at future e + e − colliders.

Combining Eq. (B.2) and Eq.(B.3), we obtain
σ A = 1 − Ā2 N . (B.4)
In the ideal case with only statistical uncertainty and no background, no systematic error, and with perfect resolution, for the asymmetry observables considered in this paper, Ā is given by the theoretical predictions in Eq. (2. 19-2.20).The SM expectations and the corresponding uncertainties are listed in Table 3.If detector effects are included, Eq. (B.4) still applies, but Ā has to be modified accordingly to take count of that.In practice, as long as Ā