SANC Monte Carlo programs for small-angle Bhabha scattering

Luminosity monitoring at $e^+e^-$ colliders is investigated using SANC Monte Carlo event generator ReneSANCe and integrator MCSANC for simulation of Bhabha scattering at low angles. Results are presented for center-of-mass energies of the Z boson resonance and 240 GeV for the conditions of typical luminosity detectors. It is shown that taking into account bremsstrahlung events with extremely low electron scattering angles is relevant to match the precision tags of the future electron-positron colliders.


Introduction
Luminosity monitoring is the standard task for all collider experiments.One of the traditional processes for high-precision luminosity measurements at electron-positron colliders is Small-Angle Bhabha Scattering (SABS).This process has a clean detector signature and very large cross section which sharply increases at small scattering angles.From the theoretical point of view it is almost a pure QED process and thus can be described very accurately within perturbative quantum field theory.SABS occupies a special place in the physics programme of future e + e − colliders like FCCee [1] and CEPC [2].Given the extremely large expected statistics, the luminosity measurement with precision 10 −4 or better is necessary.The theoretical accuracy for SABS calculations must be significantly better than this target precision in order not to spoil the resulting uncertainty.
The most advanced codes for theoretical estimation of luminosity with the help of SABS are BabaYaga [3][4][5][6][7], BHLUMI [8].The Monte Carlo (MC) generator BHLUMI is a pure QED tool and its theoretical uncertainty is estimated to be about 0.037%, see Table 2 in [9].In that paper the future prospects of theoretical precision 1 × 10 −4 was presented for luminosity measurement at the future colliders at the Z peak.
The new release of BabaYaga [10] is accounting for the various sources of radiative corrections, i.e.QED, (electro)weak and higher order effects.This generator is mainly intended for large angle Bhabha scattering, with theoretical errors of about 0.1%.
In this paper we present a study of SABS based on MCSANC integrator [11] and ReneSANCe generator [12].The process of polarized Bhabha scattering (see Fig. 1) e + (p 1 ) + e − (p 2 ) → e − (p 3 ) + e + (p 4 ) + (γ(p 5 )) was calculated at the complete one-loop electroweak level [13].In addition here we consider the higher order corrections by δρ parameter, which are necessary to meet the high-precision requirements of the future e + e − experiments.The details of Bhabha scattering implementation into MC ReneSANCe are described in [12].
The aim of the present paper is to report on the study of the Bhabha scattering cross section at arbitrarily small or even vanishingly small scattering angles.The contribution from electron scattering at very small angles introduces additional, potentially sizeable, effect in the theoretical interpretation of the measured SABS cross section value.We provide the advanced assessment of SABS events with scattering angles under 10 mrad.Earlier, this kinematic region could be described by BHAGEN-1PH [14], however the calculations were limited to the contribution of hard photon Bremsstrahlung.The outline of the paper is as follows.In Section 2 we show the comparisons with the results of alternative MC code in the conditions and setup of the CERN Workshop [15].In Section 3 we give numerical results for the integral cross sections and angular event distributions of experimental interest in SABS.We also discuss different sources of radiative corrections and study the effect from the minimum cut-off on electron scattering angle.

Cross-check with the 1996 LEP Workshop
To verify the technical precision of our codes we produced the tuned comparison with results presented in the proceedings of the CERN Workshop [15] devoted to event generators for Bhabha scattering at LEP for the non-calorimetric event selection called BARE1 and the calorimetric one called CALO1.All numbers are produced within the setup of this workshop for the O(α) matrix element without contribution of the Z exchange, up-down interference and vacuum polarization and with various values of the energy-cut z min = s ′ /s, where s ′ is the collision energy after initial state radiation (ISR).Table 1 shows a good agreement within numerical precision.

Numerical results
All results were obtained in the α(0) electroweak scheme using the set of input parameters listed in Table 2.
In addition, the following conditions were taken into account: • electrons were allowed to scatter by any angle, down to zero, • luminosity acceptance was assumed 30 mrad < θ < 174.5 mrad.
To demonstrate ReneSANCe capabilities, we generated 100 million events for the Bhabha cross section for the two c.m.s.energies √ s = 91.18GeV and 240 GeV, where each arm of the luminometer registered an energy shower from an electron or photon.We do not apply any restrictions on the minimum scattering angle of an electron, i.e. the electron can scatter down to zero.
We used two different setup for event selection (ES) called ES-BARE (non-calorimetric) and ES-CALO (calorimetric).In the ES-BARE case we define the SABS cross section by choosing events where each arm of the calorimeter is hit by an electron or positron.These electrons or positrons must have an energy of at least half the energy of the beam (E beam ).For the ES-CALO setup we consider a calorimetric detector that can't distinguish electrons from photons.In other words, the cross section is determined by events in which each arm of the calorimeter is hit by either a photon or an electron carring at least half of the beam energy.

Different radiative correction contributions
To present the main sources of theoretical uncertainties of the one-loop cross section σ 1−loop , we divide it into gauge-invariant subsets.When evaluating the contribution cross section at the Born level (leading order, LO) σ Born , both photon and Z-boson exchanges are taken into account.In order to quantify the impact of different contributions, we divide them into gauge-invariant subsets: QED one-loop corrections σ QED , the vacuum polarization contribution σ VP , and the pure weak contribution σ weak as the difference between the complete one-loop electroweak correction and the pure QED part σ QED of it.The leading higher-order (ho) corrections we denoted as σ ho .
In Table 3 we show the results of the various radiative contributions to the total cross section for the Z-pole and √ s = 240 GeV and evaluate corresponding relative corrections as δ = σ contr./σ Born .The leading higher-order EW corrections δ ho to SABS are included in our calculations through the ∆α and ∆ρ parameters.A detailed description of our implementation of this contribution was presented in [16].At two-loop level the above corrections consist of the EW at O(G 2 µ ) and the mixed EW⊗QCD at O(G µ α s ) parts.For SABS the bulk of the considered higher-order effects is due to running α.

SABS, analysis of events for ES-BARE and ES-CALO setups
Another possible bias to the luminosity measurement arises from events where an electron is scattered at a very small angle and escape detection.Such events can be accepted by a luminometer due to energetic photons radiated at angles large enough to be detected in the detector.This effect would lead to a bias in luminosity measurements if the data is analyzed with an MC tool which uses a minimum scattering angle cut-off.
We use MCSANC integrator to compare results from ES-BARE setup (ignoring photons) with ES-CALO setup in which calorimeter can be hit by either an electron or a photon.The presence of a high-energy photon provides a natural regularisation of divergence at zero electron scattering angles.Althoug electrons are allowed to be scattered by zero angle, the number of such events is small because of the requirement to have an energetic photon within the acceptance of the calorimeter.We define the luminosity acceptance in the range of 30 mrad to 10 degrees (174.5 mrad), which is typical for LEP detectors as well as for future e + e − colliders like FCCee, CEPC and ILC.
This effect is presented in Table 4.We observe that the ES-CALO cross-section at √ s = 91.18 and 240 GeV is 3% larger than ES-BARE Bhabha cross-section, when both beam particles must hit the luminometer.The largest part of the difference is due to the events with collinear photon or due to the events in which electron is scattered by an angle larger than the luminosity acceptance, while hard ISR photon hits the luminometer.Such effect can not introduce any experimental bias because the electron can be detected by large-angle calorimeters and the process can be simulated by any Bhabha generator.
Additionally, it was found that approximately 1.4 permille of the total cross section for both energies is represented by events with electron scattering angles below the given luminometer acceptance angle 30 mrad.The size of this effect ∆ QED (ϑ < 0.030) can be derived from Table 4 as the difference between δ QED 3 and δ QED 2 .Note that only the technical uncertainty of numerical integration are shown in the Table, estimates of the corresponding theoretical uncertainties will be presented elsewhere.

Angular distributions
In the following, we illustrate the numerical results by the example of several angular distributions obtained with the MC generator ReneSANCe.We consider the distribution of electron scattering angles between the outgoing electron and the incoming electron as well as the distribution by the angle at which the photon was emitted.
We present angular distributions of two types: a) distribution of events by scattering angle of the Bremsstrahlung photon ϑ 15 = ϑ γ , i.e., the angle between particle p 1 (initial positron) and particle p 5 (photon), b) distribution of events by positron scattering angle ϑ 14 , i.e., the angle between particle p 1 and particle p 4 .√ s = 91.18GeV.The axes show the relative fraction of events in the given bin.The sum of all events is normalized to 1.0 and the numbers in the frames show the fractions of events within the range of a given plot.As can be seen from the plots, the event yield vanishes when lepton scattering angle approaches zero.Sharp edges at 1.7 and 10 degrees correspond to acceptance of luminometer.Events with leptons scattered beyond the luminometer acceptance correspond to detection of energetic photons.In Figure 3 we show the distributions of electrons scattered at the angle less than the acceptance of the luminosity calorimeter.The peak at nearly zero electron scattering angles is due to the terms proportional to m 2 e /t 2 (here t is the square of the electron momentum transferred), which are present in the differential cross section of the radiative Bhabha process, see, e.g., ref. [17].The total fraction of the events within this angular range (0, 30) mrad is about 1.3 permille.For the angular range 0-10 mrad the relative event yield is 0.65 • 10 −4 .Therefore the MC generator cut-off on electron (and positron) scattering angles somewhat less than 10 mrad would be safe if the experimental systematic error on luminosity measurement is expected at  OPAL experiment at LEP has partially taken into account the effect of very low angle scattering of electrons by generating events with 18.9 mrad minimum angular cut-off [18], which is considerably lower than the experimental acceptance domain.The contribution of scattering by smaller angles was estimated by extrapolation to be less than 2 • 10 −5 and was neglected.But the simple extrapolation could underestimate the neglected contribution because of the peak at extremely small angles which is seen in Fig. 5. Our calculations show that the neglected contribution amounts to about 2.3 • 10 −4 at 91 GeV collision energy.This is still within the total theoretical uncertainty of 5.4 • 10 −4 assumed in [18] 1 .

Conclusions
In this way, we applied the Monte Carlo MCSANC integrator and ReneSANCe generator for description of small-angle Bhabha scattering.We verified that the results of the two programs are consistent with each other within statistical errors.At the level of one-loop QED radiative corrections their results agree also with the ones of the BHLUMI event generator [8].We took into account also the leading effects due to higher-order electroweak corrections and vacuum polarization.We examined SABS as a possible process to monitor the luminosity at future e + e − experiments aiming at the 10 −4 level of uncertainty.Here we limited ourselves to considering only perturbative effects, whereas in a realistic situation other effects must be taken into account, for example, beamstrahlung and the final size of the beams [19,20] must be taken into account.
The unique feature of the SANC tools allows one to generate radiative Bhabha events with electron scattering angles down to zero.This allowed us to take into account events in which one arm of the luminosity calorimeter is fired by an energetic ISR photon, while an electron is scattered by very small angle and escapes detection.Based on calculations of both MCSANC integrator and ReneSANCe generator we observe a contribution of 1.3-1.4permille from events with scattering angles less than 30 mrad, both at Z pole and at 240 GeV.This effect represent a significant bias given the high experimental precision expected at the future colliders.The bias can influence in particular the measurements of the total luminosity, the effective

1 Fig. 1 .
Fig.1.The s and t channels of Bhabha processes at lowest order.

3 ZFigure 2
Figure 2 presents the angular distributions of type a) on the left side and of type b) on the right side for c.m.s.√ s = 91.18GeV.The axes show the relative fraction of events in the given bin.The sum of all events is normalized to 1.0 and the numbers in the frames show the fractions of events within the range of a given plot.As can be seen from the plots, the event yield vanishes when lepton scattering angle approaches zero.Sharp edges at 1.7 and 10 degrees correspond to acceptance of luminometer.Events with leptons scattered beyond the luminometer acceptance correspond to detection of energetic photons.
Photon scattering angle[degree]Fraction of the number of events

Fig. 2 .
Fig. 2. Angular distributions of type a) on the left side and type b) on the right side for √ s = 91.18GeV.

Table 1 .
Comparison of BARE1 and CALO1 for the O(α) matrix element.Z exchange, up-down interference and vacuum polarization are switched off.The center of mass energy is √ s = 92.3GeV.The results are shown with various values of the energy-cut z min = s ′ /s.

Table 3 .
The results of the various radiative contributions to the total cross section for the Z-pole and

Table 4 .
Born cross sections and relative corrections for √ s = 91.18GeV and 240 GeV.Here δ QED 1 = δ(ES-BARE) is the QED correction for ES-BARE setup, δ QED 2 = δ(ES-CALO, ϑ > 0.030) is the QED correction for the ES-CALO setup with electron scattering angles larger than the minimum luminosity acceptance, and δ QED 3 = δ(ES-CALO) is the QED correction for ES-CALO setup with arbitrary electron scattering angles.