**Andrej Arbuzov 1,2,\*,†, Serge Bondarenko 1,† and Lidia Kalinovskaya 3,†**


Received: 14 June 2020; Accepted: 6 July 2020; Published: 7 July 2020

**Abstract:** Processes of electron–positron annihilation into a pair of fermions were considered. Forward–backward and left–right asymmetries were studied, taking into account polarization of initial and final particles. Complete 1-loop electroweak radiative corrections were included. A wide energy range including the *Z* boson peak and higher energies relevant for future *e*+*e*<sup>−</sup> colliders was covered. Sensitivity of observable asymmetries to the electroweak mixing angle and fermion weak coupling was discussed.

**Keywords:** high energy physics; electron–positron annihilation; forward–backward asymmetry; left–right asymmetry

**PACS:** 12.15.-y; 12.15.Lk; 13.66.Jn

## **1. Introduction**

Symmetries play a key role in the construction of physical theories. In fact, they allow us to describe a huge variety of observables by means of compact formulae. We believe that the success of theoretical models based on symmetry principles is due to the presence of the corresponding properties in Nature. The Standard Model (SM) is the most successful physical theory ever. Its predictions are in excellent agreement with practically all experimental results in particle physics. The renormalizability of the model allows us to preserve unitarity and provide finite verifiable results. Both phenomenological achievements and nice theoretical features of the SM are mainly due to the extended usage of symmetries in its construction. The model is based on several symmetries of different type, including the Lorentz (Poincaré) symmetry, the gauge *SU*(3)*<sup>C</sup>* × *SU*(2)*<sup>L</sup>* × *U*(1)*<sup>Y</sup>* symmetries, the CPT symmetry, the spontaneously broken global *SU*(2)*<sup>L</sup>* × *SU*(2)*<sup>R</sup>* symmetry in the Higgs sector, etc. Some symmetries of the model are exact (or seem to be exact within the present precision) while others are spontaneously or explicitly broken. In particular, the nature of the symmetry among the three generations of fermions is one of the most serious puzzles in the SM and verification of the lepton universality hypothesis is on the task list of modern experiments.

Despite the great successes of the SM, we can hardly believe that it is the true fundamental theory of Nature. Most likely, it is an effective model with a limited applicability domain. The search for the upper energy limit of the SM applicability is the actual task at all high-energy colliders experiments. Up to now, all direct attempts to find elementary particles and interactions beyond the Standard Model have failed. The accent of experimental studies has shifted towards accurate verification of the SM features. Deep investigation of the SM symmetries is an important tool in this line of research.

Asymmetries form a special class of experimental observables. First of all, they explicitly access the breaking of a certain symmetry in Nature. Second, they are usually constructed as a ratio of observed quantities, in which the bulk of experimental and theoretical systematic uncertainties is canceled out. So the asymmetries provide independent additional information on particle interactions. They are especially sensitive to non-standard weak interactions including contributions of right currents and new intermediate *Z*vector bosons, see e.g., [1].

The physical programs of future (super) high-energy electron–positron colliders such as CLIC [2], ILC [3–5], FCC-ee [6], and CEPC [7] necessarily include accurate tests of the SM. Studies of polarization effects and asymmetries will be important to probe of the fundamental properties of Higgs boson(s) and, in particular, in the process of annihilation into top quarks [8–10]. The future colliders plan to start operation in the so-called GigaZ mode at the *Z* peak and improve upon the LEP both in statistical and systematical uncertainties in tests of the SM [11] by at least one order of magnitude. Among these collider projects, the FCC-ee one has the most advanced program of high-precision measurements of SM processes at the *Z* peak. Such tests have been performed at LEP and SLC and they have confirmed the validity of the SM at the electroweak (EW) energy scale of about 100 GeV [12,13]. During the LEP era, extensive experimental and theoretical studies of asymmetries made an important contribution to the overall verification of the SM, see review [14] and references therein. The new precision level of future experiments motivates us to revisit the asymmetries and scrutinize the effects of radiative corrections (RCs) to them. In the analysis of LEP data, semi-analytic computer codes like ZFITTER [15] and TOPAZ0 [16] were extensively used. The forthcoming new generation of experiment requires more advanced programs, primarily Monte Carlo event generators.

The article is organized as follows. The next section contains preliminary remarks and the general notations. Section 3 is devoted to the left–right asymmetry. The forward–backward asymmetry is considered in Section 4. Discussion of the left–right forward–backward asymmetry is presented in Section 5. In Section 6, we provide results related to the final state fermion polarization. Section 7 contains a discussion and conclusions.

#### **2. Preliminaries and Notations**

In the recent paper [17] by the SANC group, high-precision theoretical predictions for the process *<sup>e</sup>*+*e*<sup>−</sup> <sup>→</sup> *<sup>l</sup>* +*l* <sup>−</sup> (*l* = *μ* or *τ*) were presented. With the help of computer system SANC [18], we calculated the complete 1-loop electroweak radiative corrections to these processes, taking into account possible longitudinal polarization of the initial beams. The calculations were performed within the helicity amplitude formalism, taking into account the initial and final state fermion masses. So, the SANC system provides a solid framework to access asymmetries in *e*+*e*<sup>−</sup> annihilation processes and to study various relevant effects. In particular, the system allows us to separate effects due to quantum electrodynamics (QED) and weak radiative corrections.

The focus of this article is on the description and assessment of the asymmetry family: the left–right asymmetry *A*LR, the forward–backward asymmetry *A*FB, the left–right forward–backward asymmetry *A*LRFB, and the final state fermion polarization *P<sup>τ</sup>* in collisions of high-energy polarized or unpolarized *e*+*e*<sup>−</sup> beams. The main aim was to verify the effect of radiative corrections on the extraction of the SM parameters from the asymmetries and to analyze the corresponding theoretical uncertainty.

We performed calculations for polarized initial and final state particles. Beam polarizations play an important role:


the electroweak mixing angle) by an order of magnitude, through studies of the left–right asymmetry [1].

Numerical illustrations for each asymmetry are given in two energy domains: the wide center-of-mass energy range 20 <sup>≤</sup> <sup>√</sup>*<sup>s</sup>* <sup>≤</sup> 500 GeV and the narrow one around the *<sup>Z</sup>* resonance (70 <sup>≤</sup> <sup>√</sup>*<sup>s</sup>* <sup>≤</sup> 100 GeV), where a peculiar behavior of observables can be seen. All results were produced with the help of the *e*+*e*<sup>−</sup> branch [19] of the MCSANC Monte Carlo integrator [20].

Let us introduce the notation. First of all, we define quantities *Af* (*f* = *e*, *μ*, *τ*) which are often used for description of asymmetries at the *Z* peak:

$$A\_f \equiv 2\frac{g\_{V\_f}g\_{A\_f}}{g\_{V\_f}^2 + g\_{A\_f}^2} = \frac{1 - (g\_{R\_f}/g\_{L\_f})^2}{1 + (g\_{R\_f}^2/g\_{L\_f}^2)^2} \tag{1}$$

where the vector and axial-vector coupling constants of the weak neutral current of the fermion *f* with the electromagnetic charge *q <sup>f</sup>* (in the units of the positron charge *e*) are

$$\mathbf{g}\_{V\_f} \equiv I\_f^3 - 2q\_f \sin^2 \theta\_{\mathcal{W}\_f} \qquad \mathbf{g}\_{A\_f} \equiv I\_f^3. \tag{2}$$

The corresponding left and right fermion couplings are

$$\lg\_{L\_f} \equiv I\_f^3 - q\_f \sin^2 \theta\_W, \qquad \lg\_{R\_f} \equiv -q\_f \sin^2 \theta\_W. \tag{3}$$

The neutral current couplings *gLf* and *gRf* quantify the strength of the interaction between the *Z* boson and the given chiral states of the fermion.

We claim that there are sizable corrections to all observable asymmetries due to radiative corrections which affect simple Born-level analytic formulae relating the asymmetries with electroweak parameters. It is especially interesting to consider the behavior of asymmetries in different EW schemes: *α*(0), *α*(*M*<sup>2</sup> *<sup>Z</sup>*), and *Gμ*, see their definitions below. We also will compare the results in the Born and 1-loop approximation. The latter means inclusion of 1-loop radiative corrections of one of the following types: pure QED photonic RCs (marked as "QED"), weak RCs (marked as "weak"), and the complete 1-loop electroweak RCs (marked as "EW"):

$$
\sigma\_{\rm EW} = \sigma\_{\rm Born} + \sigma\_{\rm QED} + \sigma\_{\rm weak}.
$$

The weak part in our notation includes 1-loop self-energy corrections to photon and *Z* boson propagators. In our notation, higher-order effects due to interference of pure QED and weak contributions are a part of *σ*weak.

The cross section of a generic annihilation process of longitudinally polarized *e*<sup>+</sup> and *e*<sup>−</sup> with polarization degrees *Pe*<sup>+</sup> and *Pe*<sup>−</sup> can be expressed as follows:

$$\begin{aligned} \sigma(P\_{\varepsilon^{-}}, P\_{\varepsilon^{+}}) &= (1 + P\_{\varepsilon^{-}}) (1 + P\_{\varepsilon^{+}}) \sigma\_{\text{RR}} + (1 - P\_{\varepsilon^{-}}) (1 + P\_{\varepsilon^{+}}) \sigma\_{\text{LR}} \\ &+ (1 + P\_{\varepsilon^{-}}) (1 - P\_{\varepsilon^{+}}) \sigma\_{\text{RL}} + (1 - P\_{\varepsilon^{-}}) (1 - P\_{\varepsilon^{+}}) \sigma\_{\text{LL}} \end{aligned} \tag{4}$$

Here *σab* = ∑*ij*(*k*)|H*abij*(*k*)| <sup>2</sup> are the 2 <sup>→</sup> <sup>2</sup>(3) helicity amplitudes of the reaction, (*ab* <sup>=</sup> *RR*, *RL*, *LR*, *LL*) with right-handed *R*="+" or left-handed *L*="−" initial particles.

It is convenient to combine the electron *Pe*<sup>−</sup> and positron *Pe*<sup>+</sup> polarizations into the effective quantity

$$P\_{\rm eff} = \frac{P\_{\rm c^-} - P\_{\rm c^+}}{1 - P\_{\rm c^-} P\_{\rm c^+}}.\tag{5}$$

In the case when only the electron beam is polarized, the effective polarization coincides with the electron one.

To investigate theoretical uncertainties, we use the following three EW schemes:


Results of fixed-order perturbative calculations in these schemes differ due to missing higher-order effects. In what follows, numerical calculations are performed in the *α*(0) EW scheme if another choice is not explicitly indicated.
