MC generator for the process e  e   e  e 

1. MC generator for the process ee  ee G.V.Fedotovich Budker Institute of Nuclear Physics Novosibirsk

2. Outline • Why is the accuracy for the cross sections with RC better than • 0.5 %required? • MC generator for the process ee  ee + (n) • (MCGPJ – Monte Carlo generator Photon Jets). • Comparison with BHWIDE code and CMD-2 data • MC generator for muon pairs production: ee   +(n). • Comparison with KKMC generator and CMD-2 data • Calculations of the cross sections ee   + (n) • Vacuum polarization effects at low energy range • Conclusion

3. CMD-2 detector at VEPP-2M 1 – vacuum chamber; 2 – drift chamber; 3 – Z-chamber; 4 – main solenoid; 5 – compensating solenoid; 6 – BGO calorimeter; 7 – CsI calorimeter; 8 – muon range system; 9 – yoke; 10 – quadrupoles ()  0.002rad, ()  0.001 rad,   

4. Nee L= ee  ee() Bhabha scattering events at large angles are preferable. Many reasons are. Luminosity measurement How does get number of Nee ee Select collinear events in tracking system Separate ee events by energy deposition in CsI calorimeter  Crude separation – number of events in red box More precise separation – unbinned fit of energy distribution ,  About 30 million Bhabha events at large angles were detected Systematic error of separation techniques is about 0.2% - 0.4%

5. ee  ee cross section calculation 1. Vacuum polarization by leptons and hadrons is included by each diagram 2. Matrix element due to one photon emission at large angle is treated With O() corrections exactly 3. Photon “jets” emission inside narrow cones along initial or final particles ( < 0) is described by SF function –D(z) 4. All enhanced terms proportional to [(ln(s/m²)]n comes from collinear regions are included in SF 5. Interference due to soft photons radiation by initial and final particles is taken into account 6. Non-leading contribution of the first order of  (so called K-factor) is Included 7. Theoretical accuracy estimation is about of 0.2%

6. ee  ee cross section calculation 2 + + 2 + + 2 + - “compensators” “Compensator” is required to remove from D(z) the part caused by emission of one photon at large angles

7. ee  ee cross section calculation “Shifted” Bhabha cross section for reaction e(z1) + e(z2)  e + e Initial electrons and positrons loose some fraction of energy z1 and z2 by emission photon jets in collinear regions Vacuum polarization effects for t- and s-channels are included in photon propagator Mandelshtam variables are defined in c.m.s.

8. ee  ee cross section calculation Why is the accuracy for the cross sections with RC better than 0.5% required ?  60 ppm 60 ppm  0.5   0.3 ppm, BNL (E821)  0.5 ppm New BNL (E969) is planned to improve the accuracy for this value at least by a factor of two and to reach 0.2 – 0.3 ppm Formulae for cross section with RC must have accuracy about of 0.2 – 0.3% or better

9. ee  ee cross section calculation Main contributions to cross section, under photons radiation, come from collinear region and they are proportional to ln(s/m²)  30 Four terms proportional to ln(s/m²) are taken into account in SF in all orders of . Remaining four terms can be interpreted as four “compensators”

10. ee  ee cross section calculation Expression for Bhabha scattering events at large angles includes:

11. ee  ee cross section calculation (cuts) – set of the kinematics selection criteria for collinear events

12. ee  ee cross section calculation Particular value of  has to be chosen for simulation. Soft photon approximation requires  to be small. But very small value of  could even produce unphysical negative weights in “master” formula. As a compromise between these two requirements the cutoff energy  was chosen about ten electron masses to optimize simulation efficiency ( / ~ 1%). • Since cross section depends very strongly on some • variables and to increase the generator simulation • efficiency main singularities have been isolated. • Photon energy - 1/ for initial and - 1/ for final particles • Photon angle - 1/(1  ²cos²) • Electron polar angle - 1/(1  cos(1)²

13. ee  ee cross section calculation • Selection criteria for collinear events : • 1. || < 0.25 rad, where  = 1 + 2 - , (Eph ~ 250 MeV) • 2. || < 0.15 rad, where  = |1 - 2| -  • 3. 1.1 < aver <  - 1.1, where aver = (1 - 2 + )/2 • 4. p > 90 MeV/c • 5. 0 = 1/sqrt(), it is close to 2° for VEPP-2M energy range • Tests have been performed for beams energy 900 MeV in c.m.s.

14. ee  ee cross section calculation Relative contribution of different parts to the total cross section: (default selection criteria are imposed)  ~ 50%, Born cross section with virtual and soft photon jets emission; 2 + 3 + 4 + 5 ~ 30%, one photon jet emission; 6 + 7 + 8 + 9 + 0 + 1 ~ 3%, two photon jets emission; 12 + 13 + 14 + 15 ~ 0.3%, three photon jets emission; 16~ 0.03%, four photon jets emission; 17 ~ 10%, one hard photon emission out of narrow cones.

15. ee  ee cross section calculation Cross section dependence with the auxiliary parameter . Cross section dependence with the auxiliary parameter 0. In both case cross section variations are inside ± 0.1%

16. ee  ee cross section calculation Code comparison with BHWIDE S.Jadach, W.Placzek, B.F.L.Ward hep-ph/9608412 Difference between cross sections calculated by MCGPJ and BHWIDE is inside corridor 0.1%. Events distribution as a function of missing energy electron pair for MCGPJ and BHWIDE code.

17. ee  ee cross section calculation Relative cross sections difference for MCGPJ code and “Berends” vs acollinearity polar angle. Relative cross sections difference for MCGPJ code and BHWIDE vs acollinearity polar angle.

18. ee  ee cross section calculation Events distribution with acollinearity azimuthal angle. Solid line – MCGPJ, dashed line - BHWIDE . Events distribution with acollinearity polar angle. Solid line – MCGPJ, dashed line – BHWIDE.

19. ee  ee cross section calculation Events number versus angle . Solid line – simulation (MCGPJ), histogram – experiment. All data upper 1040 MeV are collected on this plot. Events number versus angle . Solid line – simulation (MCGPJ), histogram – experiment. All data upper 1040 MeV are collected on this plot.

20. ee  ee cross section calculation Relative contribution of photon jets with respect to cross section with one hard photon (Berends), in %.

21. ee  ee cross section calculation Contribution of vacuum polarization to Bhabha cross section as a function of cms energy, in %

22. ee  ee cross section calculation Two dimensional plot of simulated events. Left – MCGPJ code, right – “Berends” About 1% events have total energy E1+E2 < 600 MeV This differences strongly depend on the cut for transverse momentum p

23. ee  ee cross section calculation Relative cross sections difference as a function of cut for transverse momentum applied to both particles.

24. ee  ee cross section calculation Crucial point is the estimate of theoretical accuracy of this approach. To quantify the error independent comparison was performed with generator based on “Berends”, where first order corrections in  are treated exactly. It was found the rel. difference is less than 0.2% for  = 0.25 rad. Short summarize: radiation two and more photons in collinear regions contributes to cross section for amount 0.2% only. Since the accuracy of this contribution is known better than 100% therefore theoretical accuracy of the cross section with RC certainly is better than 0.2% for our “soft” selection criteria. Unaccounted higher order corrections are estimated to be at the level 0.2%. 1. Weak interactions contribute less than 0.1% for 2E < 3 GeV. 2. A part of the second order next-to-leading radiative corrections proportional to ²ln(s/m²) ~ 10-4 are fortunately small with respect to 0.1% and were omitted. Among these contributions are effects due to double hard photons emission – one inside narrow cones and one at large angle.

25. ee  ee cross section calculation 3. Soft and virtual photon emission simultaneously with one hard photon emission and so on. If to assume that a coefficient before these terms will be of order of ten their contribution can not exceed 0.1%. 4. Fourth source of uncertainty is hadronic vacuum polarization contribution. Systematic error of hadronic cross section in 1% leds to leptonic cross sections changes at the level of 0.04%. 5. Fifth source of uncertainty about of 0.1% is connected with the models which are used to describe energy dependence of the hadronic cross section. 6. The last source of uncertainty is mainly driven by collinear approximation – several terms proportional to )0² and (0² were omitted. Indeed photons inside jets have angular distribution. Numerical estimations show that a contribution of these factors is about of 0.1%. Considering the uncertainties sources as independent the total systematic error of the cross section with RC is smaller than 0.2%.

26. A little fantasy • Measurement of the cross section of the process • ee   is a unique source to direct extract • vacuum polarization effects for future precise • dispersion calculations !!! • Special detector is needed to aim on this process. • 2. QED approach is enough to calculate this cross section • with ISR and FSR at the accuracy 0.1% level or better. • 3. Process ee    is a good instrument for luminosity. • No vac. polar. effects are needed to calculate this cross section. • It is not necessary to measure exclusive hadronic cross sections – MAIN CRUCIAL ARGUMENT !!!

27. Cross section ee   calculation MC generator to simulate muon pairs production in the reaction e(z1) + e(z2)   +. Initial particles lose some energy by photon jets radiation in collinear regions. “Shifted” Born cross section modified by vacuum polarization effects in photon propagator.

28. 2  2  +  2  “compensator”  Vacuum polarization by leptons and hadrons is included by each diagram. ee   cross section calculation Precise matrix element for one photon radiation out of narrow cones with respect to the momentum directions of initial particles ( > 0, E >E). + +  Soft, virtual and one hard photon radiation for FSR + interference. Photon jets emission along initial particles (inside narrow cones,  < 0) . Theoretical accuracy estimation is about  0.2%.

29. ee   cross section calculation “Master” formula for moun pairs production consists of: photon jets radiation inside narrow cones along beam axes, two “compensators”, cross section with one photon emission outside collinear region.

30. ee   cross section calculation Dependence of muon cross section with auxiliary parameters  and 0 . Relative cross section deviations do not exceed ± 0.1%, when  and 0 change their values more than 4 order of magnitude.

31. Code comparison with KKMC (0.1%) Vacuum polarization effects switch off in both generators Difference with KKMC with FSR, % Difference with KKMC without FSR, %  0.17 %  0.06 %

32. ee   cross section calculation Contribution of FSR, %. CMD-2 selection criteria were used.

33. ee   cross section calculation Relative difference between cross sections calculated by MCGPJ and KKMC code at VEPP-2M energy range. Vacuum polarization effects switched off in both generators. Number of selected muon pairs to electrons ones divided on the ratio of theoretical cross sections. Average deviation is: –2% ± 1.4%st± 0.7%syst.

34. Conclusions • MC generator (MCGPJ) for processes ee  ee (n), • ee  (n)with precise RC is done. • Accuracy of cross sections calculation for our selection criteria is estimated to be about 0.2% or better. • We have good agreement between many kinematics distributions produced by MCGPJ generator and BHWIDE, KKMC and BABAYAGA. • Comparison with CMD-2 data was done: • Bhabha scattering events - ,  distributions (2E > 1040 MeV), • 2. Muon cross section – deviation from QED prediction is about • –2% ± 1.4% ± 0.7% - best result for low energy region. • Process ee  2 (n)is a good instrument for luminosity. • Process ee  (n)is a unique tool for vac. pol. measurement.