MC generator for the process
ee  ee
G.V.Fedotovich
Budker Institute of Nuclear Physics
Novosibirsk
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
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,   
Luminosity measurement
Bhabha scattering events at large angles are preferable. Many reasons are.
L=
Nee
ee  ee()
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%
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%
ee  ee cross section calculation
2
+
+
2
Σ
+
+
2
Σ
+
-
“compensators”
4
“Compensator” is required to remove from D(z) the part caused by
emission of one photon at large angles
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.
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
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”
ee  ee cross section calculation
Expression for Bhabha scattering events at large angles includes:
ee  ee cross section calculation
(cuts) – set of the kinematics selection criteria for collinear events
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.
1. Photon energy - 1/ for initial and - 1/ for
final particles
2. Photon angle - 1/(1  ²cos²)
3. Electron polar angle - 1/(1  cos(1)²
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.
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.
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%
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.
ee  ee cross section calculation
Relative cross sections difference for
MCGPJ code and BHWIDE vs
acollinearity polar angle.
Relative cross sections difference for
MCGPJ code and “Berends” vs
acollinearity polar angle.
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.
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.
ee  ee cross section calculation
Relative contribution of photon jets with respect to
cross section with one hard photon (Berends), in %.
ee  ee cross section calculation
Contribution of vacuum polarization to Bhabha cross
section as a function of cms energy, in %
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
ee  ee cross section calculation
Relative cross sections difference as a function of cut for transverse
momentum applied to both particles.
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.
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%.
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 !!!
1. 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.
4. It is not necessary to measure exclusive hadronic cross
sections – MAIN CRUCIAL ARGUMENT !!!
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.
ee   cross section calculation
Vacuum polarization by leptons and hadrons is included by each diagram.
Precise matrix element for one photon radiation out of narrow cones with
respect to the momentum directions of initial particles ( > 0, E > E).


2
+


2
+


+
2

“compensator”
2
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%.
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.
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.
Code comparison with KKMC (0.1%)
Vacuum polarization effects switch off in both generators
Difference with KKMC with
FSR, %
  0.17 %
Difference with KKMC without
FSR, %
  0.06 %
ee   cross section calculation
Contribution of FSR, %. CMD-2 selection criteria were used.
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.
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:
1. 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.