**3. Left–Right Asymmetry** *A***LR**

A scheme to measure the *A*LR polarization asymmetry at the *Z* peak was suggested in [21]. It was shown that this observable can be used as for extraction of electroweak couplings as well as for a polarimeter calibration.

If we neglect the initial electron masses, the polarized cross-section can be rewritten in the following form:

$$
\sigma(P\_{\mathfrak{e}^-}, P\_{\mathfrak{e}^+}) \quad = \quad (1 - P\_{\mathfrak{e}^-} P\_{\mathfrak{e}^+}) [1 - P\_{\text{eff}} A\_{\text{LR}}] \sigma\_{0\prime} \tag{6}
$$

where *σ*<sup>0</sup> is the unpolarized cross-section.

The left–right asymmetry in the presence of partially polarized (|*P*eff| < 1) initial beams is defined as

$$A\_{\rm LR} = \frac{1}{P\_{\rm eff}} \frac{\sigma(-P\_{\rm eff}) - \sigma(P\_{\rm eff})}{\sigma(-P\_{\rm eff}) + \sigma(P\_{\rm eff})},\tag{7}$$

where *σ* is the cross-section with polarization *P*eff.

In the case of fully polarized initial particles (|*Pe*<sup>±</sup> | = 1) the definition (7) becomes:

$$A\_{\rm LR} = \frac{\sigma\_{L\_{\rm r}} - \sigma\_{R\_{\rm r}}}{\sigma\_{L\_{\rm r}} + \sigma\_{R\_{\rm r}}},\tag{8}$$

where *Le* and *Re* refer to the left and right helicity states of the incoming electron.

Equations (6) and (7) show that *A*LR does not depend on the degree of the initial beam polarization. This type of asymmetry is sensitive to weak interaction effects in the initial vertex. In the Born approximation at energies close to the *Z* resonance, it is directly related to the electron coupling:

$$A\_{\rm LR} \approx A\_{\rm c}.\tag{9}$$

The left–right asymmetry *A*LR as a function of the center-of-mass system (c.m.s.) energy in the ranges 20 <sup>≤</sup> <sup>√</sup>*<sup>s</sup>* <sup>≤</sup> 500 GeV (Left) and 70 <sup>≤</sup> <sup>√</sup>*<sup>s</sup>* <sup>≤</sup> 110 GeV (Right) is shown in Figure 1. We explore *A*LR in different approximations and the corresponding shifts Δ*A*LR between the Born level and 1-loop corrected approximations taking into account either pure QED, or weak, or complete EW effects: Δ*A*LR=*A*LR(1-loop corrected)-*A*LR(Born). The right figure shows the behavior of *A*LR near the *Z* resonance, and the value *Ae* at <sup>√</sup>*<sup>s</sup>* <sup>=</sup> *MZ* is indicated by a black dot (see (9)).

One can notice that although the total 1-loop EW corrections to the process cross-section are equal to the sum of the pure QED and weak ones, the corresponding shifts Δ*A*LR are not additive. That is because the asymmetry is defined as a ratio and the corrections affect both the numerator and denominator.

In Figure 2 we show *A*LR for the Born and weak 1-loop corrected levels of accuracy in different EW schemes and the corresponding shifts Δ*A*LR=*A*LR(weak, some EW scheme)-*A*LR(Born). We see that the effects due to weak corrections in different EW schemes behave in a similar way. Nevertheless the scheme dependence is visible within the expected precision of future measurements. The deviations between the results in different schemes can be treated as a contribution into the theoretical uncertainty due to missing higher order corrections.

**Figure 1.** (**Left**) The *A*LR asymmetry in the Born and 1-loop (weak, pure quantum electrodynamics (QED), and electroweak (EW)) approximations and Δ*A*LR vs. center-of-mass system (c.m.s.) energy in a wide range; (**Right**) the same for the *Z* peak region.

**Figure 2.** The *A*LR asymmetry at the Born level and with 1-loop weak radiative corrections (RCs); the corresponding shifts Δ*A*LR within *α*(0), *Gμ*, and *α*(*M*<sup>2</sup> *<sup>Z</sup>*) EW schemes vs. c.m.s. energy in the peak region.

The impact of 1-loop EW contributions to Δ*A*LR is of the order −0.1 in the resonance region, but at energies above <sup>√</sup>*<sup>s</sup>* <sup>=</sup> 200 GeV there are considerable cancellations between weak and QED effects so that the combined EW corrections becomes small (but still numerically important for high-precision measurements).

#### **Summary for** *A*LR

The left–right asymmetry *A*LR is almost insensitive to the details of particle detection since the corresponding experimental uncertainties tend to cancel out in the ratio (7). It (almost) does not depend on the final state fermion couplings in the vicinity of the *Z* boson peak and can be measured for any final state with a large gain in statistics. For this reasons it is appropriate for extraction of the sin<sup>2</sup> *ϑ*eff *<sup>W</sup>* value.

We observe that the values Δ*A*LR due to weak and pure QED 1-loop corrections are very significant at high energies in general, but in the resonance region impact of QED is small, while the weak contribution to Δ*A*LR reaches 0.07. Therefore, it is necessary to evaluate all possible radiative correction contributions to the weak parts of RCs carefully and thoroughly.
