Measurement of the effective weak mixing angle at the CEPC

We present a study of the measurement of the effective weak mixing angle parameter ($\sin^2\theta^{\ell}_\text{eff}$) at the Circular Electron Positron Collider (CEPC). As a fundamental physics parameter, $\sin^2\theta^{\ell}_\text{eff}$ plays a key role not only in the global test of the standard model electroweak sector, but also in constraining the potential beyond standard model new physics at high energy frontier. CEPC proposes a two year running period around the Z boson mass pole at high instataneous luminosity, providing a large data sample with $4\times 10^{12}$ $Z$ candidates generated in total. It allows a high precision measurement of $\sin^2\theta^{\ell}_\text{eff}$ both in the lepton and quark final states, of which the uncertainty can be one order of magnitude lower than any previous measurement at the LEP, SLC, Tevatron and LHC. It will not only improve the overall precision of the $\sin^2\theta^\ell_\text{eff}$ experimental determination to be comparable to the preicision of the theoretical calculation with two-loop radiative corrections, but also provide direct comparisons between different final states. In this paper, we also study the measurement of $\sin^2\theta^{\ell}_\text{eff}$ at high mass region. With one month data taken, the precision of $\sin^2\theta^{\ell}_\text{eff}$ measured at 130 GeV from $b$ quark final state is 0.00010, which will be an important experimental observation on the energy-running effect of $\sin^2\theta^{\ell}_\text{eff}$.


INTRODUCTION
The weak mixing angle, θ W , is one of the fundamental parameters of the standard model (SM).It governs the relative strength of the axial-vector couplings to the vector couplings in the neutral-current interactions with Lagrangian where g f A and g f V are the axial-vector and vector couplings, defined as g f A = I f 3 and g f V = I f 3 − 2Q f • sin 2 θ W where I f 3 and Q f are the weak isospin component and the charge of the fermion f .Z µ describes the Z boson exchange.To include higher order electroweak radiative corrections, the effective weak mixing angles are defined as where κ f is a flavor-dependent effective scaling factor absorbing the higher order corrections [1].By doing this, sin 2 θ f eff can be directly measured from experimental observations, thus is very sensitive to both the precision validation of SM and the search for new physics beyond SM.
It is as custom to quote the leptonic effective weak mixing angle sin 2 θ ℓ eff , so that measurements at the LEP, SLC, Tevatron and LHC can be directly compared with each other.To do this, shifts between sin 2 θ ℓ eff and sin 2 θ q eff need to be calculated under the standard model assumptions.In this work, it is calculated using the zfitter package [2], which gives a shift of −0.0001 and −0.0002 for sin 2 θ u eff and sin 2 θ d eff with respect to sin 2 θ ℓ eff , respectively.For sin 2 θ b eff as a special case, the shift is +0.0014E.Such calculation has a high precision as long as the energy of the interaction is not approaching 160 GeV where the correction from box-diagrams becomes sizable [3].sin 2 θ ℓ eff has been measured in the past two decades using the f i fi → Z/γ * → f j fj productions.The results are shown in Figure 1.The most precise determinations of sin 2 θ ℓ eff come from the electron-positron colliders, which are 0.23221±0.00029from the combined LEP b quark production, and 0.23098±0.00026from the SLD [1].After that, a similar precision was achieved at the protonantiproton collider Tevatron, giving 0.23148±0.00033[4].sin 2 θ ℓ eff are also measured by ATLAS, CMS and LHCb collaborations [5][6][7].sin 2 θ ℓ eff was extracted from the forward-backward charge asymmetry (A F B , to be introduced in the following sections) in all those measurements above, except for the SLD result which also used the leftright polarization asymmetry.
Previous measurements achieved an relative precision at O(0.1%) with respect to the central value of sin 2 θ ℓ eff .It played an important role in the global fitting of the SM electroweak sector in the past years.However, the experimental precision, which is generally limited by the size of the data sample, is much worse than the precision of the theoretical calculations.At two-loop level, the uncertainty in sin 2 θ ℓ eff calculation is reduced to 0.00004 [8].It would be essential to improve the experimental precision on sin 2 θ ℓ eff to be comparable to the theoretical calculations.
Though a large data sample will be collected at the LHC in the next 10 years, it will be very difficult to have the uncertainty on sin 2 θ ℓ eff to be smaller than 0.00010 using the LHC data.At hadron colliders, the initial state fermions of the neutral-current interactions are quarks and antiquarks acting as partons in the hadrons.Their effective momentum are described by the parton distribution functions (PDFs), which extrapolate large uncertainties to the sin 2 θ ℓ eff extraction.In the recent LHC measurements, the PDF-induced uncertainties on sin 2 θ ℓ eff are larger than 0.00020 [5,6], and will become the most leading uncertainty in the future.Such uncertainty will not be naturally reduced as the LHC data is introduced in the PDF global analysis, due to a strong correlation between the PDF and the sin 2 θ ℓ eff in the LHC observations [9].The QCD-induced uncertainty is also larger than 0.00010 at the LHC [6], extrapolated via soft-gluon radiations in the initial state.In conclusion, it would be most likely to achieve a high precision determination on sin 2 θ ℓ eff at next generation electron-positron colliders, which are generally free from PDF and QCD, and have the capability to generate a large data sample.
In this paper, we study the measurement of sin 2 θ ℓ eff at the proposed Circular Electron Positron Collider (CEPC).CEPC is a powerful machine providing physics interactions with high energy electron-positron initial state [10].It is proposed to have a 2-year running plan around the Z boson mass pole.We focus on the precision of sin 2 θ ℓ eff in lepton and b quark final states, with both statistical uncertainty and potential experimental systematics taken into account.

FORWARD-BACKWARD CHARGE ASYMMETRY
The forward-backward charge asymmetry A F B of the f i fi → Z/γ * → f j fj process is an ideal experimental observable to probe the electroweak interaction with a high precision.It is defined as where N F and N B are the numbers of the forward and backward events, judged by the scattering angle θ ij formed by the directions of the initial state negative charged electron beam and the final state fermion.
Events with cos θ ij > 0 are classified as forward (F ) and those with cos θ ij < 0 as backward (B).At the CEPC, the initial state fermions are electrons and positrons, while the final state fermions can be leptons and quarks.
The asymmetry arises from the interference between vector and axial vector coupling terms, and precisely governed by sin 2 θ ℓ eff .The value of A F B changes with the center-of-mass energy √ s. Figure 2 shows the A F B spectrum as a function of √ s for different productions.The predictions are calculated using the effective born approximation package zfitter corresponding to next-tonext-to-leading order (NNLO) radiative corrections [2].A F B is very sensitive to sin 2 θ ℓ eff around the Z mass pole.In the low mass region, the asymmetry is approaching zero as √ s goes lower due to the rising contribution of the photon exchange.In the high mass region, the asymmetry has roughly a constant value dominated by the interference between the γ and Z boson exchanges.Therefore, the sensitivity of off-pole A F B to sin 2 θ ℓ eff is significantly reduced.The sensitivity, defined as is given as a function of √ s in Figure 3 for b quark and lepton productions as an example.Predictions are calculated using zfitter as well.In the following sections, we estimate the uncertainty on sin 2 θ ℓ eff based on the sensitivity S of A F B to sin 2 θ ℓ eff .The observed asymmetry, denoted as A obs F B , could be biased due to the imperfect detector performance.Three major contributions of the potential experimental systematics are discussed in this work, which come from the √ s determination, the charge measurement and the inefficiency of particle reconstruction and selection.These systematics are generally small.At lepton colliders, √ s of an event can be precisely controlled by the beam energy, instead of reconstructed from the final state particles measured in the detector.According to the CEPC conceptual design report, the uncertainty of the electron and positron beam energy can be controlled around 100 KeV [11], extrapolating a relative uncertainty on sin 2 θ ℓ eff much smaller than 0.01%.The inefficiency of particle reconstruction and selection could have a large effect especially in quark productions.However, A F B is defined as a relative asymmetry where total cross section perfectly cancelled.Therefore, the limited efficiency only enlarges the statistical uncertainty of the A F B observation and sin 2 θ ℓ eff extraction and causes no systematics, as long as there is no difference between the forward and backward event efficiencies.With CEPC's large data sample, the statistical uncertainty will be negligibal anyway.
A more complicated case is the charge misidentification of the final state particles, which could cause both systematic extrapolation and statistical uncertainty increase.The forward and backward categories are classified according to the charge of the final state particles.If an event has a probability of f to be wrongly classified between forward and backward due to the mis-identification of charge, the observed A F B will be diluted from the original A F B , written as: When f = 50%, there will be no observed asymmetry.Such dilution causes reduction in the A F B to sin 2 θ ℓ eff sensitivity, appearing as an enlarged statistical uncertainty.For the selected e + e − → Z/γ * → f f events, f can be determined from the following relationship where N ss is the number of selected events with same charge sign of the final state fermions, while N total is the total number of selected events.ω is the probability to mis-identify the charge of a single fermion, and we ).The precision of the f determination is dominated by the statistics.Considering the large data sample at the CEPC, f could be precisely determined by this data-driven method and causes very small systematics.Besides, the final state fermions are usually required to have opposite charge in order to suppress the mis-identification of the forward and backward categories.
According to the definition of A F B in Eq. 4 and assuming that the value of A F B around Z pole is close to zero, the statistical uncertainty on the observed asymmetry A obs F B is approximately written as where N is the number of the selected events.Taking the above effects into consideration, the statistical uncertainty of sin 2 θ ℓ eff measured from A F B can be expressed as: where ϵ is the overall efficiency of detecting and selecting an e + e − → Z/γ * → ℓ + ℓ − event.The term ϵ • (1 − 2f ) 2 is defined as the tagging power parameter.For lepton final states, the overall efficiency is very close to 100%, and f is negligibly small.Therefore the tagging power is almost 100% [10].For b quark final state, it is much more complicated.It is difficult to determine the b quark charge by measuring the final state jet.To have a better charge measurement, only a small part of the b quark production events, where b quarks decay to leptons or Kaons, could be used in the sin 2 θ ℓ eff measurement.Therefore, the tagging power parameter for b quark productions needs to be optimized between the overall efficiency and the charge mis-identification probability.According to the CEPC simulation study [10,12], with a selection of b quark with 98% purity, the optimized tagging power for b quark production is 0.088.
The number of the selected events N depends on the luminosity of the proposed running plan, and the cross section of each channel of the Z boson decay.The latest CEPC studies give a proposal of 2-year running period around the Z boson mass pole, with 50 ab −1 integrated luminosity per year.In another word, CEPC can provides 1.7 × 10 11 Z boson events every month.Considering the branching ratio [8] and Eq. 7, the expected statistical uncertainty of sin 2 θ ℓ eff measured from lepton final state (ee+µµ) is δ sin 2 θ ℓ eff (ℓ) = 5 × 10 −6 using 1 month data.For b quark productions, the uncertainty is δ sin 2 θ ℓ eff (b) = 4 × 10 −6 using 1 month data.This uncertainty can be further reduced using the proposed 2-year data sample.However, it would be more useful to run at different collision energy point off-pole rather than simply collecting data at the very peak of the Z mass line shape.When changing the collision energy in this study, the cross section of the Z boson production is altered according to its mass line shape.The drop of the instantaneous luminosity is estimated approximately as the third power of the increase of the collision energy [11].The expected statistical uncertainties of sin 2 θ ℓ eff with 1 month data collection at different collision energy points are summarized in Table I  As we can see, both lepton final state and b quark final state can provide precise determination on sin 2 θ ℓ eff at the Z boson mass pole.For the measurement of the energy running effect, b quark production has higher precision because the sensitivity S off-pole drops much slower than the lepton final state cases.To make a conservative estimation, the precision on the sin 2 θ ℓ eff determination, considering both the statistical uncertainty and the experimental systematics at the CEPC, can be 0.00001 in both lepton and b quark productions with 1 month data collection.The sin 2 θ ℓ eff can be measured as a function of collision energy up to 130 GeV, with a precision of around 0.0001 from the b quark productions.
Due to the contribution of the t-channel and the s-t interference in the electron final state, the uncertainty of the theoretical calculation in the electron final state can be very large, giving 0.00085 on the weak mixing angle according to Ref. [1].However, such uncertainty only affects the dielectron events.For other channels, the residual theoretical uncertainties can be much smaller, giving 0.00006 on the weak mixing angle [1].Therefore, the best precision of the sin 2 θ ℓ eff determination relies on the muon channel.The uncertainty on the calculations of the e + e − → f f process is much larger than the statistical uncertainty and the experimental systmatics, thus will be the major source limiting the final precision of the experimental measurement of sin 2 θ ℓ eff .However, it will still be much better than the expected precision at the hadron colliders.
Aside from electon, muon and b quark channel A F B measurement, other channels such as c quark can also be utilized to extract sin 2 θ ℓ eff .With the predicted sensitivity S f f listed in table II, one can estimate the precision of sin 2 θ ℓ eff from A F B measurement using Eq. 7, after the performance research for different final state particles.For instance, a recent CEPC simulation study [13] used leading particle and weighted jet charge combined information to give a higher performance of heavy flavor jet charge measurement, and the bare tagging power [17] of b/c quark was determined.The tagging power of b quark final state is doubled, therefore the estimated δ sin 2 θ ℓ eff with b quark final state is 2.5 × 10 −6 (with 1 month data collection at Z pole).However, for the c quark circumstance, due to the low purity of c flavor tagging, the estimation will need detailed simulation study of flavor tagging using Z → q q samples.Tau lepton is the only final state fermion we've known whose polarizarion(P τ ) can be measured at an unpolarized leptonic collider [1] with where σ r/l is the cross section for producing right/lefthanded final state tau leptons.And P τ is connected to sin 2 θ ℓ eff by where is the asymmetry parameter.This property was utilized by LEP to perform rather independent measurement for sin 2 θ ℓ eff .Compared to the whole lepton channel, the statistics of tau channel is small, and the efficiency and purity of tau reconstruction are low.But with a very high sensitivity of P τ to τ − Z vector coupling constant, it can give a high precision extraction of sin 2 θ ℓ eff .Measurement of P τ is based on the fact that τ has a short lifetime and that the kinematic spectrum of its decay production is different when tau has different helicity (shown in Figure 4 and Figure 5).We use pythia8 program [14] to generate e + e − → τ + τ − events, and then use tauola interface [15] to decay the taus.
Using two templates with helicity = +1 and −1, respectively, we can perform fit to the pseudo-data, whose P τ is a given number.Due to limited computing resources, only 2 × 10 8 Z → τ τ events are generated for pseudo-data and each template.The extrapolation of the fitting results shows that the statistical uncertainty on sin 2 θ ℓ eff with 1 month data collection at Z pole is 2.15 × 10 −6 using P τ measurement.Experimentally, the τ lepton is reconstructed from its daughter particles in the decay, thus relies on the precision of the measurement of the particle energy and the background control.It makes the detector systematics extrapolate more significantly to the weak mixing angle than in other channels, According to the stydy of LEP [1], the systematics in the τ measurement is at an order of O(10 −4 ) on the weak mixing angle.Given that the statistical uncertainty at the CEPC would be much smaller, the total uncertaintt in the τ channel measurement will be dominated by the systematics.All the are genarated using pythia8 genarator and tauola interface.

CONCLUSION
We present an estimation of the precision on sin 2 θ ℓ eff determination at the CEPC in the lepton final state and b quark final state.With a high instantaneous luminosity, the statistical uncertainty can be reduced to be negligible.The experimental sysmtematics are also negligible in general since A F B is defined as a relative asymmetry so that the systematics cancel out.The dominant uncertainty will come from the theoretical calculation on the e − e + → f f process.As a result, the precision of sin 2 θ ℓ eff can be improved to O(10 −5 ), from the current precision of O(10 −4 ) at the LEP, SLC and Tevatron.Due to a large model uncertainty from QCD calculations and PDF modeling, such precision is difficult to be achieved using the LHC data in the future.This precision will, for the first time, be comparable to the precision of the theoretical calculation of sin 2 θ ℓ eff with radiative corrections at two-loop level, meaning that the precision of the SM electroweak global fit can be significantly improved.One thing worth emphasizing is that the high precision measurement of sin 2 θ ℓ eff at the CEPC is essential to the QCD studies at the LHC.As discussed in the introduction section, the observation of proton structure and electroweak symmetry breaking is highly correlated in pp(q q) → Z/γ * → ℓ + ℓ − events.In Ref. [9], it is proved that the single Z boson production can provide unique information of the relative difference between quarks and antiquarks.However, it is not available yet in the PDF global fitting due to a large uncertainty induced by the experimental determination of sin 2 θ ℓ eff .Using the CEPC's electron-positron interaction, the measured sin 2 θ ℓ eff can be used as high precision input in the PDF global fitting, fixing the electroweak calculations in predicting the single Z boson production cross sections.
One last thing to be emphasized is that our analysis uses high purity b b sample to exclude the contamination of other quark flavors.With a properly designed working point for jet flavor tagging, we can also use other quark flavors to measure sin

FIG. 4 :
FIG. 4: The relative contributions of different decay modes in the τ final state.

1 FIG. 5 :
FIG. 5: Kinematic spectrum of different tau decay modes.The red solid line and blue dashed line represent the kinematic spectrum of taus with helicity = +1 and −1, respectively.

TABLE I :
The expected statistical uncertainties on sin 2 θ ℓ eff .Results are estimated according to 1 month data collection.

TABLE II :
Sensitivity S of different final state particles.

TABLE III :
2 θ ℓ eff .Future development of detector optimization and advanced reconstruction algorithm, especially those based on machine learning, for example those has been used recently at CMS experiment [16], could also boost the relevant performances.√s(GeV ) σµ(mb) σ d (mb) σu(mb) σs(mb) σc(mb) σ b (mb) Cross section of process e + e − → f f calculated using the zfitter package.Values of the fundamental parameters are set as mZ = 91.1875GeV,mt = 173.2GeV,mH = 125GeV, αs = 0.118 and mW = 80.38GeV.