1 INTRODUCTION

The next generation of electron colliders including the International Linear Collider (ILC) [1–6], the \({{e}^{ + }}{{e}^{ - }}\) Future Circular Collider (FCCee) [7–11], the Compact Linear Collider (CLIC) [12–14], and the Circular Electron Positron Collider (CEPC) [15] will allow an extensive program of experiments with unique opportunities for precision measurements. A major advantage to fulfill this goal is the universality of linear colliders, as they can operate in four \({{e}^{ + }}{{e}^{ - }}\), \({{e}^{ - }}{{e}^{ - }}\), \({{e}^{ - }}\gamma \), and γγ modes with strongly polarized electron and photon beams. An important feature of linear colliders is a high degree of polarization which can be obtained for electron beams.

The Møller scattering along with the Compton-like processes is a good candidate for beam polarization measurements and the background estimation in many searches for new physics beyond the Standard Model. At high energies for the polarized Møller scattering the most advanced Monte-Carlo (MC) tool is needed not only to estimate beam polarization, i.e., polarized experiments CLIC [13], ILC [16], but also   to study muon-muon polarized scattering at μ TRISTAN [17].

Equal lepton scattering, \({{e}^{ - }}{{e}^{ - }} \to {{e}^{ - }}{{e}^{ - }}\) was first calculated by C. Møller in 1932 [18]. There are a great number of theoretical works for description polarized case of this process [19–26]. In this series of papers, the calculations are given for the QED and electroweak (EW) one-loop corrections with taking into account the polarization.

A calculation of the radiative corrections (RCs) was performed for the unpolarized Møller scattering for the experiment [27] at one-loop level [28, 29], partly at two-loop level [30, 31], and in the first time beyond the ultra-relativistic approximation in [32].

However, all of the above-mentioned studies are not accompanied by the development of the Monte Carlo event generator which is the standard of the modern theoretical support of the high-precision experiments.

The following Monte Carlo generators currently exist, which take into account polarization at tree level: AMEGIC++ [33], based on the helicity amplitudes and being a part of SHERPA; COMPHEP [34], using the traditional trace techniques to evaluate the matrix elements; GRACE [35, 36] (with the packages BASES and SPRING), calculating matrix elements via helicity amplitude techniques; WHIZARD [37] a software system, intended for the effective calculation of scattering cross sections of many-particle and simulated events, where polarization is processed for both the initial and final states.

Theoretical support of experiments by the MERADGEN MC generator for polarized Møller scattering within QED theory is presented in [38].

In our previous works we estimated the theoretical uncertainty for the complete one-loop and leading higher-order EW corrections for \({{e}^{ + }}{{e}^{ - }}\) and γγ polarized beams. The implementations of polarized Bhabha scattering [39], polarized \({{e}^{ + }}{{e}^{ - }} \to ZH\) [40], s-channel [41], \({{e}^{ + }}{{e}^{ - }} \to \gamma Z\) [42] and \(\gamma \gamma \to ZZ\) [43] are available in the ReneSANCe MC generator [44] and the MCSANC integrator in the fully massive case and in total phase space.

This article is the next step in the series of SANC papers devoted to the implementation of one of the channels \(4f \to 0\), namely, the equal lepton scattering at the one-loop level with allowance for polarization.

The \(\alpha (0)\) EW scheme is used in the calculations. All the results are obtained for the center-of-mass (c.m.) energies from \(\sqrt s = 250\) GeV up to 3 TeV. The sensitivity to the initial polarization for the Born and hard photon bremsstrahlung cross sections was estimated for four beam polarization data sets:

$$\begin{gathered} ({{P}_{{{{e}^{ - }}}}},{{P}_{{{{e}^{ - }}}}}) \\ = (0,0),( - 1, - 1),( - 1, + 1),( + 1, - 1),( + 1, + 1). \\ \end{gathered} $$
(1)

The one-loop contributions were calculated for the following degrees of polarization:

$$({{P}_{{{{e}^{ - }}}}},{{P}_{{{{e}^{ - }}}}}) = (0,0),( \pm 0.8, \pm 0.8).$$
(2)

The statistical uncertainties were estimated using the SANC tools: ReneSANCe MC generator and MCSANC integrator.

This article consists of four sections. We describe the methodology of calculations of the polarized cross sections at the complete one-loop EW level within the helicity approach in Section 2. Numerical results and comparison are presented in the Section 3. Summary is drawn in Section 4.

2 ELECTROWEAK ONE-LOOP RADIATIVE CORRECTIONS

We consider the differential cross section for processes

$$\begin{gathered} {{l}^{ \pm }}({{p}_{1}},{{\chi }_{1}})\; + \;{{l}^{ \pm }}({{p}_{2}},{{\chi }_{2}}) \\ \to {{l}^{ \pm }}({{p}_{3}},{{\chi }_{3}})\; + \;{{l}^{ \pm }}({{p}_{4}},{{\chi }_{4}})\;\;( + \gamma ({{p}_{5}},{{\chi }_{5}})), \\ \end{gathered} $$
(3)

with \(l = e,\mu \) and arbitrary longitudinal polarization of initial particles (\(\chi \) corresponds to the helicity of the particles).

Within the SANC system we calculate all processes using the on-mass-shell renormalization scheme in two gauges: the \({{R}_{\xi }}\) gauge and the unitary gauge as a cross-check.

We apply the helicity approach to all components of the one-loop cross sections:

$${{\sigma }^{{{\text{one-loop}}}}} = {{\sigma }^{{{\text{Born}}}}} + {{\sigma }^{{{\text{virt}}}}}(\lambda ) + {{\sigma }^{{{\text{soft}}}}}(\lambda ,\omega ) + {{\sigma }^{{{\text{hard}}}}}(\omega ),$$
(4)

where \({{\sigma }^{{{\text{Born}}}}}\) is the Born cross section, \({{\sigma }^{{{\text{virt}}}}}\) is the contribution of virtual (loop) corrections, \({{\sigma }^{{{\text{soft(hard)}}}}}\) is the soft (hard) photon emission contribution (the hard photon energy \({{E}_{\gamma }} > \omega \)). The auxiliary parameters λ (“photon mass”) and ω are canceled after summation. The corresponding expressions for the Møller scattering cross section cannot be integrated over all angles because the integral diverges.

2.1 Born and Virtual Parts

To calculate the virtual part at the one-loop level using the procedure basement of SANC, we start with considering the covariant amplitude. The covariant one-loop amplitude corresponds to the result of the straightforward standard calculation of all diagrams contributing to a given process at the one-loop level. The covariant amplitude is represented in a certain basis made of strings of Dirac matrices and/or external momenta (structures) contracted with polarization vectors of vector bosons, \(\epsilon (k)\), if any.

The covariant amplitude can be written in an explicit form using scalar form factors. All masses, kinematical factors and coupling constant and other parameter dependences are included into these form factors \({{\mathcal{F}}_{i}}\), but tensor structures with Lorenz indices made of strings of Dirac matrices are given by the basis.

The number of form factors is equal to the number of independent structures.

Loop integrals are expressed in terms of standard scalar Passarino–Veltman functions \({{A}_{0}},{\kern 1pt} {{B}_{0}},{\kern 1pt} {{C}_{0}},{\kern 1pt} {{D}_{0}}\) [45]. We presented the covariant amplitude for the \(4f \to 0\) process in [46], where we considered it at the one-loop level of annihilation into a vacuum. Recall that in SANC we always calculate any one-loop process amplitude as annihilation into vacuum with all four-momenta incoming. Therefore, the derived universal scalar form factors for the amplitude of the process \(4f \to 0\) after an appropriate permutation of their arguments can be used for the description of the next-to-leading (NLO) corrections of this particular case unfolding into t and u channels.

The virtual (Born) cross section of processes (3) can be written as follows:

$$\frac{{d\sigma _{{{{\chi }_{1}}{{\chi }_{2}}}}^{{{\text{virt(Born)}}}}}}{{d\cos {{\vartheta }_{3}}}} = \pi {{\alpha }^{2}}\frac{{{{\beta }_{s}}}}{{2s}}{{\left| {\mathcal{H}_{{{{\chi }_{1}}{{\chi }_{2}}}}^{{{\text{virt(Born)}}}}} \right|}^{2}},$$
(5)

where

$${{\left| {\mathcal{H}_{{{{\chi }_{1}}{{\chi }_{2}}}}^{{{\text{virt(Born)}}}}} \right|}^{2}} = \sum\limits_{{{\chi }_{3}},{{\chi }_{4}}} {{\left| {\mathcal{H}_{{{{\chi }_{1}}{{\chi }_{2}}{{\chi }_{3}}{{\chi }_{4}}}}^{{{\text{virt(Born)}}}}} \right|}^{2}},$$
(6)

where \({{m}_{l}}\) is the final lepton mass and \({{\beta }_{s}} = \sqrt {1 - \frac{{4m_{l}^{2}}}{s}} \), the angle \({{\vartheta }_{3}}\) is the c.m. angle between \({{p}_{1}}\) and \({{p}_{3}}\).

Then we estimate the cross section as a function of eight helicity amplitudes. Helicity amplitudes depend on kinematic variables, coupling constants and seven scalar form factors. Helicity indices \({{\mathcal{H}}_{{{{\chi }_{1}}{{\chi }_{2}}{{\chi }_{3}}{{\chi }_{4}}}}}\) denote the signs of the fermion spin projections to corresponding momenta. Some basic definitions are \({{c}^{ \pm }} = 1 \pm \cos {{\vartheta }_{3}}\), and the scattering angle \({{\vartheta }_{3}}\) is related to the Mandelstam invariants \(t,u\):

$$t = 2m_{l}^{2} - \frac{s}{2}(1 + \beta _{l}^{2}\cos {{\vartheta }_{3}}),$$
(7)
$$u = 2m_{l}^{2} - \frac{s}{2}(1 - \beta _{l}^{2}\cos {{\vartheta }_{3}}).$$
(8)

The presence of the electron masses gives additional terms proportional to the factor \({{m}_{l}}\), which can be considered significant in calculations at low energy.

The set of the corresponding helicity approaches in the t channel for this case is:

$${{\mathcal{H}}_{{ \mp \mp \mp \mp }}} = \frac{1}{t}[({{c}^{ + }}k - 4m_{l}^{2}){{\tilde {F}}_{{gg}}} + {{\chi }_{Z}}(t)(({{c}^{ + }}k - 4m_{l}^{2}){{\tilde {F}}_{{qq}}}$$
$$ + \;4(k \mp \sqrt {{{\lambda }_{e}}} ){{\tilde {F}}_{{ll}}} + 2({{c}^{ + }}k - 4m_{l}^{2} \mp 2\sqrt {{{\lambda }_{e}}} ){{\tilde {F}}_{{lq}}}$$
$$ + \;2{{c}^{ - }}m_{l}^{2}k[{{\tilde {F}}_{{qd}}} + (k \mp \sqrt {{{\lambda }_{e}}} ){{\tilde {F}}_{{ld}}}])],$$
$${{\mathcal{H}}_{{ \mp - - \pm }}} = \sin {{\vartheta }_{3}}\frac{{\sqrt s {{m}_{l}}}}{t}[{{\tilde {F}}_{{gg}}} + {{\chi }_{Z}}(t)({{\tilde {F}}_{{qq}}} + 2{{\tilde {F}}_{{lq}}} - k{{\tilde {F}}_{{qd}}}$$
$$ - \;(k - \sqrt {{{\lambda }_{e}}} ){{\tilde {F}}_{{ld}}})],$$
$${{\mathcal{H}}_{{ \pm + + \mp }}} = - {{\mathcal{H}}_{{ \mp - - \pm }}}(\sqrt {{{\lambda }_{e}}} \to - \sqrt {{{\lambda }_{e}}} ),$$
$${{\mathcal{H}}_{{ \mp \mp \pm \pm }}} = - \frac{{m_{l}^{2}}}{t}[2{{c}^{ + }}{{\tilde {F}}_{{gg}}} + {{\chi }_{Z}}(t)(2{{c}^{ + }}({{\tilde {F}}_{{qq}}} + 2{{\tilde {F}}_{{lq}}})$$
$$ + \;8{{\tilde {F}}_{{ll}}} - s{{c}^{ + }}({{\tilde {F}}_{{ld}}} + {{\tilde {F}}_{{qd}}}))],$$
$${{\mathcal{H}}_{{ \mp \pm \mp \pm }}} = - \frac{{m_{l}^{2}}}{t}[2{{c}^{ + }}{{\tilde {F}}_{{gg}}} + {{\chi }_{Z}}(t)(2{{c}^{ + }}({{\tilde {F}}_{{qq}}} + 2{{\tilde {F}}_{{lq}}})$$
$$ - \;s{{c}^{ + }}({{\tilde {F}}_{{ld}}} + {{\tilde {F}}_{{qd}}}))],$$
$${{\mathcal{H}}_{{ - \mp \pm - }}} = - {{\mathcal{H}}_{{ + \mp \pm + }}},$$
$${{\mathcal{H}}_{{ + \mp \pm + }}} = \sin {{\vartheta }_{3}}\frac{{\sqrt s {{m}_{l}}}}{t}[{{\tilde {F}}_{{gg}}} + {{\chi }_{Z}}(t)({{\tilde {F}}_{{qq}}} + 2{{\tilde {F}}_{{lq}}}$$
$$ - \;2m_{l}^{2}({{\tilde {F}}_{{ld}}} + {{\tilde {F}}_{{qd}}}))],$$
$${{\mathcal{H}}_{{ \mp \pm \pm \mp }}} = \frac{{{{c}^{ - }}}}{t}[k{{\tilde {F}}_{{gg}}} + {{\chi }_{Z}}(t)(k({{\tilde {F}}_{{qq}}} + 2{{\tilde {F}}_{{lq}}})$$
$$ - \;2m_{l}^{2}[(k \pm \sqrt {{{\lambda }_{e}}} ){{\tilde {F}}_{{ld}}} + k{{\tilde {F}}_{{qd}}}])].$$

Here \({{\chi }_{Z}}(t)\) is the \(Z{\text{/}}\gamma \) propagator ratio:

$${{\chi }_{Z}}(t) = \frac{1}{{4s_{W}^{2}c_{W}^{2}}}\frac{t}{{t - M_{z}^{2}}}.$$
(9)

Note that tilted form factors absorb couplings, which leads to a compactification of formulas for the amplitude, while explicit expressions will be given for untilted quantities.

The expressions for tiled form factors are:

$${{\tilde {F}}_{{gg}}} = (I_{l}^{{(3)}}{{)}^{2}}{{F}_{{gg}}}(s,t,u),$$
(10)
$${{\tilde {F}}_{{ll}}} = \delta _{l}^{2}{{F}_{{ll}}}(s,t,u),$$
$${{\tilde {F}}_{{qq}}} = I_{l}^{{(3)}}{{\delta }_{l}}{{F}_{{qq}}}(s,t,u),$$
$${{\tilde {F}}_{{lq}}} = {{\delta }_{l}}I_{l}^{{(3)}}{{F}_{{lq}}}(s,t,u),$$
$${{\tilde {F}}_{{ql}}} = {{\delta }_{l}}I_{l}^{{(3)}}{{F}_{{ql}}}(s,t,u),$$
$${{\tilde {F}}_{{ld}}} = (I_{l}^{{(3)}}{{)}^{2}}{{F}_{{ld}}}(s,t,u),$$
$${{\tilde {F}}_{{qd}}} = {{\delta }_{l}}I_{l}^{{(3)}}{{F}_{{qd}}}(s,t,u).$$

We also use the coupling constants

$$\begin{gathered} I_{l}^{{(3)}},\quad {{\sigma }_{l}} = {{v}_{l}} + {{a}_{l}},\quad {{\delta }_{l}} = {{v}_{l}} - {{a}_{l}}, \\ {{s}_{W}} = \frac{e}{g},\quad {{c}_{W}} = \frac{{{{M}_{W}}}}{{{{M}_{Z}}}}, \\ \end{gathered} $$

with \(l = e,\mu \).

In order to get helicity approaches for the Born level, one should set \({{F}_{{gg,ll,lq,ql,qq}}} = 1\) and \({{F}_{{ld.qd}}} = 0\).

2.2 Real Photon Emission Corrections

The real corrections consist of soft and hard radiative contributions. They are calculated using the bremsstrahlung modules. The soft bremsstrahlung has a Born-like kinematics, while the phase space of hard radiation has an extra particle, a photon.

The soft bremsstrahlung has the form

$${{\sigma }^{{{\text{soft}}}}} = - Q_{e}^{2}\frac{{2\alpha }}{\pi }{{\sigma }^{{{\text{Born}}}}}\left[ {(1 + \frac{{1 - 2m_{l}^{2}{\text{/}}s}}{{{{\beta }_{s}}}}\ln {{x}^{2}})\ln \left( {\frac{{4{{\omega }^{2}}}}{{{{\lambda }^{2}}}}} \right)} \right.$$
$$ + \;\frac{1}{{{{\beta }_{s}}}}\left[ { - \ln {{x}^{2}} + (1 - 2m_{l}^{2}{\text{/}}s)({\text{L}}{{{\text{i}}}_{2}}{{{(1 - {{x}^{2}})}}_{{\begin{subarray}{l} \\ \end{subarray}} }}} \right.$$
(11)
$$ - \;{\text{L}}{{{\text{i}}}_{2}}\left. {\left. {\left. {\left( {1 - \frac{1}{{{{x}^{2}}}}} \right)} \right)} \right] - F(t) - F(u)} \right],$$

where

$$F(I) = \frac{{1 - 2m_{l}^{2}{\text{/}}I}}{{{{\beta }_{I}}}}\left[ {\ln {{y}_{I}}\ln \left( {\frac{{4{{\omega }^{2}}}}{{{{\lambda }^{2}}}}} \right)} \right. + {\text{L}}{{{\text{i}}}_{2}}\left( {1 - \frac{{{{y}_{I}}x}}{{{{z}_{I}}}}} \right)$$
$$ - \;{\text{L}}{{{\text{i}}}_{2}}\left( {1 - \frac{x}{{{{z}_{I}}}}} \right) + {\text{L}}{{{\text{i}}}_{2}}\left( {1 - \frac{{{{y}_{I}}}}{{{{z}_{I}}x}}} \right) - {\text{L}}{{{\text{i}}}_{2}}\left. {\left( {1 - \frac{1}{{{{z}_{I}}x}}} \right)} \right],$$
$${{\beta }_{I}} = \sqrt {1 - \frac{{4m_{l}^{2}}}{I}} ,\quad x = \frac{{\sqrt s }}{{{{m}_{l}}}}\frac{{1 + {{\beta }_{s}}}}{2},$$
$${{y}_{I}} = 1 - \frac{I}{{m_{l}^{2}}}\frac{{1 + {{\beta }_{I}}}}{2},\quad {{z}_{I}} = \frac{{{{m}_{l}}}}{{\sqrt s }}\left( {1 + {{y}_{I}}} \right),$$

with \(I = t,u\).

In presenting the results we used our universal massive module for the hard photon bremsstrahlung for \(4f\gamma \to 0\) [41] by appropriate unfolding it in the right channel.

3 NUMERICAL RESULTS

3.1 Tree Level

In this section calculated polarized cross sections at tree level for the Born and hard photon bremsstrahlung are compared with the results of the CalcHEP [34] and WHIZARD [37, 47, 48] codes.

The results are calculated in the \(\alpha (0)\) EW scheme with fixed 100% polarized initial states for \(\sqrt s = \) 250 GeV and angular cuts \({\text{|}}\cos {{\vartheta }_{e}}{\text{|}} \leqslant 0.9\). For the hard bremsstrahlung cross sections, an additional cut on the photon energy \({{E}_{\gamma }} \geqslant \omega {{ = 10}^{{ - 4}}}\sqrt s {\text{/}}2\) is applied.

The Born results agree in all digits for all codes, and therefore the table is omitted. The hard bremsstrahlung results are shown in Table 1. Very good agreement within statistical errors with the above-mentioned codes is found.

Table 1. Tuned triple comparison between SANC (the second column), WHIZARD (the third column), and CalcHEP (the fourth column) and SANC results for the hard bremsstrahlung cross section
Full size table

3.2 One-Loop Level

In this Section we show the study of the complete one-loop EW RCs and polarization effects for Møller scattering in high-energy regions. Numerical estimates are presented for the total (integrated) cross sections (σ, pb) and relative corrections (δ, %) as well as for the differential distribution as function of the scattering angle \(\cos {{\vartheta }_{3}}\). The channels \({{e}^{ - }}\) (\({{e}^{ - }}{{e}^{ - }} \to {{e}^{ - }}{{e}^{ - }}(\gamma \))) and \({{\mu }^{ + }}\) (\({{\mu }^{ + }}{{\mu }^{ + }} \to {{\mu }^{ + }}{{\mu }^{ + }}(\gamma \))) of reaction (3) are considered below.

3.2.1. Integrated cross sections. CLIC would provide high-luminosity \({{e}^{ - }}{{e}^{ - }}\) collisions covering a center-of-mass energy range from 0.38 to 3 TeV. They are three main c.m. energy stages at \({{\sqrt s }_{{{\text{CLIC}}}}}\): 0.38, 1.5, and 3 TeV.

The ILC offers many opportunities for measurements with collider energies from 90 GeV to 1 TeV. Three main c.m. energy stages can be distinguished: \({{\sqrt s }_{{{\text{ILC}}}}}\) = 0.25, 0.5, and 1 TeV, with electron polarization of \({{P}_{{{{e}^{ - }}}}} = \pm 0.8\).

Table 2 presents the integrated Born and one-loop cross section in pb and relative corrections in percent for the \({{e}^{ - }}\)-channel for c.m. energy \({{\sqrt s }_{{{\text{ILC }\& \text{CLIC}}}}}\) and set (2) of the initial particle polarization degrees in the \(\alpha (0)\) EW scheme.

Table 2. Integrated Born and one-loop cross section in pb and relative corrections in percent for \({{e}^{ - }}\)-channel scattering for c.m. energy \({{\sqrt s }_{{{\text{ILC }\& \text{ CLIC}}}}}\) and set (2) of the initial particle polarization degrees in the \(\alpha (0)\) EW scheme
Full size table

Under the \(\mu \)TRISTAN experimental conditions, the energy is assumed to be \({{\sqrt s }_{{\mu {\text{TRISTAN}}}}}\): 0.6, 1, 2 TeV and the polarization of both beams will reach \({{P}_{{{{\mu }^{ + }}}}} = \pm 0.8\) for the \({{\mu }^{ + }}\)-channel. Table 3 presents the same observables as in Table 2 in the conditions of the \(\mu \)TRISTAN experiment.

Table 3. Integrated Born and one-loop cross section in pb and relative corrections in percent for the \({{\mu }^{ + }}\)-channel for c.m. energy \({{\sqrt s }_{{\mu {\text{TRISTAN}}}}}\) and set (2) of the initial particle polarization degrees in the \(\alpha (0)\) EW scheme
Full size table

As it seen in Tables 2 and 3, the use of the polarized beams significantly increases the cross section. At the same time the RCs increase at \({{P}_{{{{e}^{ - }}}}} = (0.8,0.8)\) and reduce at \({{P}_{{{{e}^{ + }}}}} = ( - 0.8, - 0.8)\) comparing to the unpolarized case in the region of c.m. energies \(\sqrt s = 250{\kern 1pt} - {\kern 1pt} 1000\) GeV. At higher c.m. energies the polarization \({{P}_{{e,\mu }}} = ( \pm 0.8, \pm 0.8)\) increases the cross section as well, but the absolute value of the relative correction became larger than for the unpolarized one.

3.2.2. Differential cross sections. Figures 1 and 2 show the differential distributions for the LO and EW NLO cross sections (in pb) as well as the relative corrections (in %) for the \({{e}^{ - }}\)- and \({{\mu }^{ + }}\)-channels for the c.m. energies \(\sqrt s = 250\), 1000 GeV as a function of \(\cos {{\theta }_{3}}\).

Fig. 1.
figure 1

(Color online) LO and NLO EW unpolarized cross sections (upper panel) and relative corrections (lower panel) of the \({{e}^{ - }}\)-channel for the c.m. energy \(\sqrt s = 250\) GeV versus \(\cos {{\theta }_{3}}\).

Full size image
Fig. 2.
figure 2

(Color online) LO and NLO EW unpolarized cross sections (upper panel) and relative corrections (lower panel) of the \({{e}^{ - }}\)- and \({{\mu }^{ + }}\)-channels for the c.m. energy \(\sqrt s = 1000\) GeV versus \(\cos {{\theta }_{3}}\).

Full size image

The differential distributions over \(\cos {{\theta }_{3}}\) are symmetric for all c.m. energies and the maximum of the relative corrections is at the zero angle while the minimum is close to the \({\text{|}}\cos {{\theta }_{3}}{\text{|}} = 1\). This is due to dominance of the Born contribution in the \({\text{|}}\cos {{\theta }_{3}}{\text{|}} \approx 1\) region due to a photon propagator \(1{\text{/}}t\;(1{\text{/}}u)\).

It should be noted that while the integrated relative corrections for the c.m. energy \(\sqrt s = 1000\) GeV for the \({{e}^{ - }}\)- and \({{\mu }^{ + }}\)-channels differ by 0.4% (see Tables 2, 3) the differential difference is larger, being about 5–6% at \(\cos {{\theta }_{3}} = 0\).

4 SUMMARY

We computed the NLO contributions RCs due to QED and purely weak corrections and implement them into a fully differential Monte Carlo event generator ReneSANCe and MCSANC integrator.

We presented explicit expressions for helicity approaches to evaluate virtual and soft parts for Møller scattering. We used our previous module for helicity approaches of the hard photon bremsstrahlung [41]. We showed the results of interest for unpolarized FCCee and polarized ILC, CLIC, μTRISTAN experiments.

Since the measurement of the beam polarization is expected at the level of 1% [19, 49], it is necessary to include more than one-loop EW RCs (leading logarithms QED and EW 2-loop corrections) to ensure the required level of the theoretical support.

The established SANC framework allows us to investigate the one-loop and higher-order corrections for any polarization, estimate the contribution of the selected helicity state, and take into account mass effects.