**6. Final-State Fermion Polarization** *Pf*

The polarization of a final-state fermion *Pf*<sup>=</sup>*μ*,*<sup>τ</sup>* can be expressed as the ratio between the difference of the cross sections for right and left handed final state helicities and their sum

$$P\_f = \frac{\sigma\_{\mathcal{R}\_f} - \sigma\_{\mathcal{L}\_f}}{\sigma\_{\mathcal{R}\_f} + \sigma\_{\mathcal{L}\_f}}.\tag{16}$$

In an experiment, it can be measured for the *τ*+*τ*<sup>−</sup> channel by reconstructing the *τ* polarization from the pion spectrum in the decay *τ* → *πν*. Details of the analysis of *P<sup>τ</sup>* measurements at LEP are described in [13]. Computer programs TAOLA [26] and KORALZ [27,28] were applied for this analysis. Estimated improvement for *Pτ* and *τ* decay products over LEP time in ILC in the GigaZ program was done in [5].

In the case for unpolarized beams in the vicinity of the *Z* peak, the expression for channel *<sup>e</sup>*+*e*<sup>−</sup> <sup>→</sup> *<sup>τ</sup>*+*τ*<sup>−</sup> is simplified to

$$P\_{\tau}(\cos \theta\_{\tau}) \approx -\frac{A\_{\tau} + \frac{2 \cos \theta\_{\tau}}{1 + \cos^{2} \theta\_{\tau}} A\_{\ell}}{1 + \frac{2 \cos \theta\_{\tau}}{1 + \cos^{2} \theta\_{\tau}} A\_{\ell} A\_{\tau}}.\tag{17}$$

From this observable, one can extract information on the couplings *Aτ* and *Ae*, simultaneously.

In Figure 8 (left) we show the distribution of *P<sup>τ</sup>* in the cosine of the scattering angle at the *Z* peak in the Born and 1-loop (weak, QED, and EW) approximations. The same conventions as in previous sections are applied for the shifts Δ*Pτ*. The shift due to pure QED RCs is approximately a constant close to zero. But one can see that this observable is very sensitive to the presence of weak-interaction corrections.

In the presence of initial beams polarization the expression depends on *P*eff:

$$P\_{\rm{\tau}}(\cos \theta) \approx -\frac{A\_{\rm{\tau}}(1 - A\_{\rm{\epsilon}}P\_{\rm{eff}}) + \frac{2\cos \theta\_{\rm{\tau}}}{(1 + \cos^{2} \theta\_{\rm{\tau}})}(A\_{\rm{\epsilon}} - P\_{\rm{eff}})}{(1 - A\_{\rm{\epsilon}}P\_{\rm{eff}}) + \frac{2\cos \theta\_{\rm{\tau}}}{(1 + \cos^{2} \theta\_{\rm{\tau}})}A\_{\rm{\tau}}(A\_{\rm{\epsilon}} - P\_{\rm{eff}})}.\tag{18}$$

which can be reduced to the short form neglecting the *AeA<sup>τ</sup>* and *AeP*eff terms:

$$P\_{\rm{T}}(\cos \theta\_{\rm{T}}) \approx -A\_{\rm{T}} - \frac{2 \cos \theta\_{\rm{T}}}{(1 + \cos^{2} \theta\_{\rm{T}})} (A\_{\rm{t}} - P\_{\rm{eff}}).\tag{19}$$

The influence of the initial particle polarization on *Pτ* at the *Z* peak is demonstrated in the Figure 8 (right). For comparison the unpolarized and two polarized cases (13) as functions of cos *ϑτ* are shown. It is seen that the behavior of *Pτ* depends on the polarization set choices very much, note that it even changes the sign for the *P*<sup>2</sup> case. The corresponding shifts Δ*P<sup>τ</sup>* also strongly depend on the initial beam polarization degrees and change the shape accordingly (note the maximum for *P*1).

In Figure 9 we show the dependence of *P<sup>τ</sup>* on the c.m.s. energy in the Born and 1-loop approximations (weak, QED, and EW). We see that at energies above the *Z* resonance, both weak and QED radiative corrections to *P<sup>τ</sup>* are large and considerable cancellations happen between their contributions. Note that theoretical uncertainties in weak and QED RCs are not correlated, so it is necessary to take into account higher-order effects to reduce the resulting uncertainty in the complete 1-loop result for *Pτ* at high energies.

**Figure 8.** (**Left**) The *Pτ* polarization in the Born and 1-loop (weak, pure QED, and EW) approximations as a function of cos *ϑτ* at <sup>√</sup>*<sup>s</sup>* <sup>=</sup> *MZ*. (**Right**) The *<sup>P</sup><sup>τ</sup>* polarization for unpolarized and polarized cases with (13) degrees of initial beam polarizations in the Born and EW 1-loop approximations vs. cosine of the final *τ* lepton scattering angle at the *Z* peak.

**Figure 9.** (**Left**) The *Pτ* polarization in the Born and 1-loop (weak, pure QED, and EW) approximations and Δ*P<sup>τ</sup>* vs. c.m.s. energy in a wide range; (**Right**) the same for the *Z* peak region. The black dot indicates the value *Pτ* at the *Z* resonance.

In Figure 10 we show *P<sup>τ</sup>* in the Born and 1-loop EW approximations for different sets of beam polarization degrees in a narrow bin around the *Z* resonance. The beam polarizations sets *P*<sup>1</sup> and *P*<sup>2</sup> are defined in Equation (13). One can see that the energy dependence of *P<sup>τ</sup>* is strongly affected by a beam polarization choice outside the *Z* peak region. The same concerns the size of radiative corrections to *Pτ*, which are represented on the lower plot.

#### **Summary for** *Pτ*

The *Pτ* asymmetry is very sensitive to weak-interaction corrections and to the polarization degrees of the initial beams. Near the *Z* resonance the value of theoretical uncertainty of *Pτ* is determined by the interplay of uncertainties of rather large contributions pure QED and weak radiative corrections.

**Figure 10.** The *Pτ* polarization for (13) degrees of the initial beam polarizations in the Born and 1-loop EW approximations vs. c.m.s. energy in the *Z* peak region.

#### **7. Conclusions**

New opportunities of the future *e*+*e*<sup>−</sup> colliders: GigaZ options and new energy scale up to several TeV require modern tools for high-precision theoretical calculations of observables. We investigated *<sup>A</sup>*LR, *<sup>A</sup>*FB and *<sup>A</sup>*LRFB for *<sup>e</sup>*+*e*<sup>−</sup> <sup>→</sup> *<sup>μ</sup>*+*μ*<sup>−</sup> channel and polarization *<sup>P</sup><sup>τ</sup>* for the final state in *<sup>e</sup>*+*e*<sup>−</sup> <sup>→</sup> *<sup>τ</sup>*+*τ*<sup>−</sup> channel on the *Z* resonance and in the high energy region up to 500 GeV by using MCSANC. We evaluated the resulting shifts of asymmetries at the Born and EW levels of accuracy in different EW schemes. The numerical results presented above for pure QED, weak, and complete EW radiative corrections show an interplay between the weak and QED contributions to asymmetries. This fact indicates the necessity to consider those contributions always in combined way.

Asymmetries in *e*+*e*<sup>−</sup> annihilation processes provide a powerful tool for investigation of symmetries between three fermion generations. By studying all available asymmetries, one can extract parameters of weak interactions in the neutral current for all three charged leptons. So, by comparing the parameters it will be possible to verify the lepton universality hypothesis at a new level of precision.

Hypothetical extra neutral *Z* vector bosons [29] can contribute to the processes of *e*+*e*<sup>−</sup> annihilation. For example, effects of Kaluza–Klein excited vector bosons in the gauge Higgs unification on *e*+*e*<sup>−</sup> annihilation cross sections were considered in [30,31]. Since the new bosons can have couplings to left and right fermions being different from the SM ones, the asymmetries (especially with polarized beams) can help a lot in search for such *Z*bosons.

At the FCC-ee we have experimental precision tag in the sin<sup>2</sup> *ϑ*eff *<sup>W</sup>* measurement of the order of <sup>5</sup> <sup>×</sup> <sup>10</sup>−6, which means more than a thirty-fold improvement with respect to the current precision of 1.6 <sup>×</sup> <sup>10</sup>−4. This is due to a factor of several hundred improvement on statistical errors and because of a considerable improvement in particle identification and vertexing. In order to provide theoretical predictions for the considered asymmetries with sufficiently small uncertainties which would not spoil the precision of the future experiments besides the complete 1-loop EW radiative corrections presented here we need:


Challenges in calculations of higher order QED effects for FCC-ee were discussed in Ref. [32]. The complete two-loop electroweak corrections in the vicinity of the *Z* boson peak have been presented in [33]. More details on challenges for high-precision theoretical calculations for future *e*+*e*<sup>−</sup> colliders can be found in [34,35].

**Author Contributions:** Conceptualization, A.A., S.B. and L.K.; methodology, A.A., S.B. and L.K.; software, A.A., S.B. and L.K.; validation, A.A., S.B. and L.K.; formal analysis, A.A., S.B. and L.K.; investigation, A.A., S.B. and L.K.; resources, A.A., S.B. and L.K.; data curation, A.A., S.B. and L.K.; writing—original draft preparation, A.A., S.B. and L.K.; writing—review and editing, A.A., S.B. and L.K. The authors claim to have contributed equally and significantly in this paper. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by RFBR grant 20-02-00441.

**Acknowledgments:** The authors are grateful to Ya. Dydyshka, R. Sadykov, V. Yermolchyk, and A. Sapronov for fruitful discussions and numerical cross checks, and to A. Kalinovskaya for the help with preparation of the manuscript.

**Conflicts of Interest:** The authors declare no conflict of interest.

## **Abbreviations**

The following abbreviations are used in this manuscript:

