**4. Forward–Backward Asymmetry** *A***FB**

The forward–backward asymmetry is defined as

$$\begin{aligned} A\_{\rm FB} &= \frac{\sigma\_{\rm F} - \sigma\_{\rm B}}{\sigma\_{\rm F} + \sigma\_{\rm B}}, \\ \sigma\_{\rm F} &= \int\_{0}^{1} \frac{d\sigma}{d\cos\theta\_{f}} d\cos\theta\_{f}, \qquad \sigma\_{\rm B} = \int\_{-1}^{0} \frac{d\sigma}{d\cos\theta\_{f}} d\cos\theta\_{f}. \end{aligned} \tag{10}$$

where *ϑ<sup>f</sup>* is the angle between the momenta of the incoming electron and the outgoing negatively charged fermion. It can be measured in any *<sup>e</sup>*+*e*<sup>−</sup> <sup>→</sup> *<sup>f</sup>* ¯ *f* channels but for precision test the most convenient channels are *f* = *e*, *μ*. The channels with production of *τ* leptons, *b* or *c* quarks are very interesting as well.

At the Born level, this asymmetry is proportional to the product of initial and final state couplings and is caused by parity violation at both production and decay vertices:

$$A\_{\rm FB} \approx \frac{3}{4} A\_{\rm \epsilon} A\_{\rm f}. \tag{11}$$

In the case of partially polarized initial beams the condition (11) reduces to the following one

$$A\_{\rm FB} \approx \frac{3}{4} \frac{A\_{\rm \varepsilon} - P\_{\rm eff}}{1 - A\_{\rm \varepsilon} P\_{\rm eff}} A\_f. \tag{12}$$

In Figure 3 we show the behavior of the *A*FB asymmetry in the Born and 1-loop approximations (with weak, pure QED, or complete EW contributions) and the corresponding Δ*A*FB for c.m.s. energy range 20 <sup>≤</sup> <sup>√</sup>*<sup>s</sup>* <sup>≤</sup> 500 GeV in the left plot and for the *<sup>Z</sup>* peak region of c.m.s. energy 70 <sup>≤</sup> <sup>√</sup>*<sup>s</sup>* <sup>≤</sup> 110 GeV in the right one. As in the previous case of *A*LR, we indicate by a black dot the value *A*FB ≈ 3/4*AeA<sup>μ</sup>* at the resonance. We observe that the weak contribution to *A*FB is small and practically does not depend on energy. The shift Δ*A*FB changes the sign at the resonance and tends to a constant value (∼−0.3) above 200 GeV. The huge magnitude of the shift Δ*A*FB out of the *Z* resonance region is coming mainly from the pure QED corrections. In particular, above the peak the effect due to radiative return to the resonance is very important.

Figure 4 shows the dependence of *A*FB for different levels of accuracy (Born and 1-loop weak) on the EW scheme choice: either *α*(0), or G*μ*, or *α*(*M*<sup>2</sup> *<sup>Z</sup>*). The corresponding shifts Δ*A*FB between the Born and the 1-loop weak corrected approximations are shown in the lower plot.

**Figure 3.** (**Left**) The *A*FB asymmetry in the Born and 1-loop (weak, QED, EW) approximations and the corresponding shifts Δ*A*FB for a wide c.m.s. energy range; (**Right**) the same for the *Z* peak region.

**Figure 4.** The *A*FB asymmetry and Δ*A*FB in the Born and complete 1-loop EW approximations within the *α*(0), G*μ*, and *α*(*M*<sup>2</sup> *<sup>Z</sup>*) EW schemes vs. the c.m.s energy.

Below we investigate two sets of polarization degree *Pi* = (*Pe*<sup>−</sup> , *Pe*<sup>+</sup> ):

$$P\_1 = (-0.8, 0.3) \qquad \text{and} \qquad P\_2 = (0.8, -0.3). \tag{13}$$

In Figure 5 we compare the values of *A*FB asymmetry and the corresponding shifts due to EW corrections for the unpolarized case and two choices of polarized beams defined in the above equation. One can see that a combination of polarization degrees of initial particles can either increase or decrease the magnitude of the *A*FB asymmetry with respect to the unpolarized case.

There is an interesting idea [22] to use the *A*FB asymmetry at the FCC-ee in order to directly access the value of QED running coupling at *MZ*. This idea was supported in [23] where it was demonstrated that higher-order QED radiative corrections to *A*FB are under control. Our results show that higher-order effects due to weak interactions are not negligible in this observable; further studies are required.

At the Born level there are contributions suppressed by the small factor *m*<sup>2</sup> *<sup>f</sup>* /*s* with the fermion mass squared. It is interesting to note that in 1-loop radiative corrections there are contributions of the relative order *α* · *mf* / <sup>√</sup>*<sup>s</sup>* with the fermion mass to the first power [24], which are numerically relevant at high energies especially for the *b* quark channel.

**Figure 5.** The *A*FB asymmetry at the Born level (upper panel) and the corresponding Δ*A*FB in the 1-loop EW approximation (bottom panel) for unpolarized and polarized cases with degrees of beam polarizations *P*1,2 (13) vs. c.m.s. energy in the *Z* peak region. The constants *C*(*P*1,2) stand for the expression (12) with polarization degrees (13).

#### **Summary for** *A*FB

The weak 1-loop contribution Δ*A*FB is rather small for the whole energy range, see Figure 3. Nevertheless in this asymmetry the difference between the pure QED and the complete 1-loop approximations near the resonance is numerically important. The dependence on the EW scheme choice, see Figure 4, is small but still relevant for high-precision measurements. The dependence of this asymmetry on polarization is very significant.
