One-Loop Electroweak Radiative Corrections to Polarized Møller Scattering

This work is devoted to a theoretical description of polarized Møller scattering. Complete one-loop electroweak radiative corrections are calculated in the helicity amplitude approach with allowance for the exact dependence on the muon mass. Numerical results are presented for integrated unpolarized and polarized cross sections as well as angular differential distributions. Calculations are performed using ReneSANCe Monte Carlo generator and MCSANC Monte Carlo integrator.


Introduction
The next generation of electron colliders -the International Linear Collider(ILC) [1][2][3][4][5][6], the e + e − Future Circular Collider (FCCee) [7][8][9][10][11], the Compact Linear Collider (CLIC) [12][13][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 − γ 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 Bhabha and Compton-like processes is a good candidate for luminosity 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 tool is needed not only to estimate luminosity, i.e. polarized experiments CLIC [13], ILC [16], but also to study muon-muon polarized scattering at µTRISTAN [17].
Equal lepton scattering, e − e − → 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][20][21][22][23][24][25][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] and in the first time beyond the ultra-relativistic approximation in [31].
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++ [32], based on the helicity amplitudes and being a part of SHERPA; COMPHEP [33], using the traditional trace techniques to evaluate the matrix elements; GRACE [34,35] (with the packages BASES and SPRING), calculating matrix elements via helicity amplitude techniques; WHIZARD [36] 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 [37].
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 [38], polarized e + e − → ZH [39], s -channel [40], e + e − → γZ [41] and γγ → ZZ [42] are available in the ReneSANCe MC generator [43] 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 → 0, namely, the equal lepton scattering at the one-loop level with allowance for polarization.
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 next Section 3. Summary is drawn in Section 4.
Within the SANC system we calculate all processes using the on-mass-shell renormalization scheme in two gauges: the R ξ gauge and the unitary gauge as a cross-check.
We apply the helicity approach (HA) to all components of the one-loop cross sections: where σ Born is the Born cross section, σ virt is the contribution of virtual (loop) corrections, σ soft(hard) is the soft (hard) photon emission contribution (the hard photon energy E γ > ω).
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 at ϑ = 0, π.

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 (CA).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 CA is represented in a certain basis made of strings of Dirac matrices and/or external momenta (structures) contracted with polarization vectors of vector bosons, ǫ(k), if any.CA can be written in an explicit form using scalar form factors (FFs).All masses, kinematical factors and coupling constant and other parameter dependences are included into these FFs F i , but tensor structures with Lorenz indices made of strings of Dirac matrices are given by the basis.
The number of FFs is equal to the number of independent structures.Loop integrals are expressed in terms of standard scalar Passarino-Veltman functions A 0 , B 0 , C 0 , D 0 [44].We presented the CA for the 4f → 0 process in [45], 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 4 momenta incoming.Therefore, the derived universal scalar form factors for the amplitude of the process 4f → 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: where where m l is the final lepton mass and , the angle ϑ 3 is the c.m.s. the 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 H χ 1 χ 2 χ 3 χ 4 denote the signs of the fermion spin projections to corresponding momenta.Some basic definitions are c ± = 1 ± cos ϑ 3 , and the scattering angle ϑ 3 is related to the Mandelstam invariants t, u: 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 HAs in the t channel for this case is: Here χ Z (t) is the Z/γ propagator ratio: Note that tilded FFs absorb couplings, which leads to a compactification of formulas for the amplitude, while explicit expressions will be given for untilded quantities.The expressions for tilded FFs are Fgg = (I l F qd (s, t, u).We also use the coupling constants In order to get HAs for the Born level, one should set F gg,ll,lq,ql,qq = 1 and F ld,qd = 0.

Real photon emission corrections
The real corrections consist of soft and hard radiative contributions.They are calculated using the bremsstrahlung modules.The soft bremsstrahlung radiation has a Born-like kinematics, while the phase space of hard radiation has an extra particle -photon.The soft bremsstrahlung radiation has the following form: were with I = t, u.
In presenting the results we used our universal massive module for the hard photon bremsstrahlung for 4f → 0 [40] by appropriate unfolding it in the right channel.
3 Numerical results

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 [33] and WHIZARD [36,46,47] codes.
The results are calculated in the α(0) EW scheme with fixed 100% polarized initial states for √ s = 250 GeV and angular cuts | cos ϑ e | ≤ 0.9.For the hard bremsstrahlung cross sections, an additional cut on the photon energy E γ ≥ ω = 10 −4 √ s/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.

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 ϑ 3 .The channels e − (e − e − → e − e − (γ)) and µ + (µ + µ + → µ + µ + (γ)) of reaction (3) are considered below.

Integrated cross sections
CLIC would provide high-luminosity e − e − collisions covering a centre-of-mass energy range from 380 GeV to 3 TeV.They are three main c.m.s.energy stages at √ s CLIC : 380 GeV, 1.5 TeV and 3 TeV.The ILC offers many opportunities for measurements with collider energies from 90 GeV to 1 TeV.Three main c.m.s.energy stages can be distinguished -√ s ILC : 250 GeV, 500 GeV, and 1 TeV, with electron polarization of P e − = ±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.s.energy √ s ILC&CLIC and set (2) of the initial particle polarization degrees in the α(0) EW scheme.Under the µTRISTAN experimental conditions, the energy is assumed to be √ s µTRISTAN : 0.6, 1, 2 TeV and the polarization of both beams will reach P µ + = ±0.8 for the µ + -channel.Table 3 presents the same observables as in Table 2 in the conditions of the µTRISTAN experiment.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.s.energies √ s = 250−1000 GeV.At higher c.m.s.energies the polarization P e,µ = (±0.8,±0.8) increases the cross section as well, but the absolute value of the relative correction became larger than for the unpolarized one.

Differential cross sections
At Figures 1 and 2 the differential distributions for the LO and EW NLO cross sections (in pb) as well as the relative corrections (in %) are shown for the e − -and µ + -channels for the The differential distributions over cos θ 3 are symmetric for all c.m.s.energies and the maximum of the relative corrections is at the zero angle while the minimum is close to the | cos θ 3 | = 1.This is due to dominance of the Born contribution in the | cos θ 3 | ≈ 1 region due to a photon propagator 1/t(1/u).
It should be noted that while the integrated relative corrections for the c.m.s.energy √ s = 1000 GeV for the e − -and µ + -channels differ by 0.4% (see Tables 2,3) the differential difference is larger, being about 5-6 % at cos θ 3 = 0.

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 HAs to evaluate virtual and soft parts for Møller scattering.We used our previous module for HAs of the hard photon bremsstrahlung [40].We showed the results of interest for unpolarized FCCee and polarized ILC, CLIC, µTRISTAN experiments.
Our results show that it is necessary to include more then one-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.

Funding
This research was supported by the Russian Foundation for Basic Research, project N 20-02-00441.

Figure 1 :Figure 2 :
Figure 1: Fig. 1.The LO and NLO EW unpolarized cross sections (upper panel) and relative corrections (lower panel) of the e − -channel for the c.m.s.energy √ s = 250 GeV as a function of cos θ 3

Table 1 :
Table 1.The 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 P e − , P e −

Table 2 :
Table2.Integrated Born and one-loop cross section in pb and relative corrections in percent for e − -channel scattering for c.m.s.energy √ s ILC&CLIC and set (2) of the initial particle polarization degrees in the α(0) EW scheme P e − , P e −

Table 3 :
Table3.Integrated Born and one-loop cross section in pb and relative corrections in percent for the µ + -channel for c.m.s.energy √ s µTRISTAN and set (2) of the initial particle