**5. Left–Right Forward–Backward Asymmetry** *A***LRFB**

In order to measure the weak couplings of the final state fermions, it was suggested to analyze the so-called left–right forward–backward asymmetry [25]:

$$A\_{\rm LRFB} = \frac{(\sigma\_{\rm L\_{\rm \varepsilon}} - \sigma\_{\rm R\_{\rm \varepsilon}})\_F - (\sigma\_{\rm L\_{\rm \varepsilon}} - \sigma\_{\rm R\_{\rm \varepsilon}})\_B}{(\sigma\_{\rm L\_{\rm \varepsilon}} + \sigma\_{\rm R\_{\rm \varepsilon}})\_F + (\sigma\_{\rm L\_{\rm \varepsilon}} + \sigma\_{\rm R\_{\rm \varepsilon}})\_B},\tag{14}$$

where *σ<sup>L</sup>* and *σ<sup>R</sup>* are the cross sections with left and right handed helicities of the initial electrons.

From the definition (14) it follows that *A*LRFB partially inherits the properties of the *A*LR and, in particular, does not depend on the degree of the initial beam polarizations.

In the case of unpolarized beams on the *Z* resonance peak, the Born-level asymmetry is

$$A\_{\rm LRFB} \approx \frac{3}{4} A\_f. \tag{15}$$

In Figure 6 we present the predictions for the *A*LRFB asymmetry in several approximations, namely at the Born level and with 1-loop weak, pure QED, and complete EW contributions.

**Figure 6.** (**Left**) The *A*LRFB asymmetry in the Born and 1-loop (weak, QED, EW) approximations and Δ*A*LRFB for c.m.s. energy range; (**Right**) the same for the *Z* peak region.

Next, we repeat the study of the *A*LRFB asymmetry behavior in different EW schemes. We have illustrated the energy dependence of the *A*LRFB asymmetry in *α*(0), G*μ*, and *α*(*M*<sup>2</sup> *<sup>Z</sup>*) schemes and the corresponding Δ*A*LRFB in Figure 7. The impact of weak corrections on *A*LRFB is large. For example, the Born-level value of *A*LRFB at the *Z* peak is about 0.17, while accounting for the weak RCs contribution reduces the asymmetry value down to ∼0.11.

**Figure 7.** The *A*LRFB asymmetry in the Born and 1-loop EW approximations and Δ*A*LRFB within *α*(0), G*μ*, and *α*(*M*<sup>2</sup> *<sup>Z</sup>*) EW schemes vs. c.m.s. energy in the *Z* peak region.

#### **Summary for** *A*LRFB

We would like to emphasize that the above Formula (15) appears to be a rather rough approximation since radiative corrections shift the observable value of *A*LRFB quite a lot. Apparently the *A*LRFB asymmetry is more affected by weak corrections than *A*LR. The shifts Δ*A*LRFB only slightly depend on an EW scheme choice. The *A*LRFB asymmetry at the *Z* boson peak depends on the final lepton coupling that could be used to measure the *μ* and *τ* weak couplings and their difference from the initial lepton (electron) one.
