hep-ph/9602393 26 Feb 1996 Contents Working Small angle Introduction T Ohl B Pietrzyk F Piccinini H Czy z M Dallavalle Event Comparisons ments Light Higher Sensitivity Brief characteristics Vacuum p olarization Exp erimental Group pairs Event selections Summary Multiple Comparison Dep endence on Wide Wide Reference event selections order photonic Bhabha Generators of and H Anlauf A Arbuzov M Bigi of event generators LEP event other small photon scattering Conveners J Field F Filthaut of Narrow Narrow observables of the programs calculati ons selection exp onentiated corrections energy and radiation B F L Ward contributions W P laczek E Remiddi M Skrzyp ek L Trentadue S Jadach and and to luminosity for small angle at for acollinearity versus Wide Narrow theory LEP and order by order F Jegerlehner E Kuraev G Montagna Bhabha Z W as and uncertainty H Burkhardt M Cacciari O Nicrosini cuts LEP Bhabha in calculations scattering acceptance Scattering luminosity measure M Ca o
First order technical precision
Beyond rst order physical precision
Asymmetric and very narrow event selections
Z and vacuum p olarizatio n included
The total theoretical error for small angle Bhabha scattering
Large angle Bhabha scattering
Physics
On Z p eak LEP
Far o Z p eak LEP
Short write up s of the programs
BHAGEN
BHAGENE
BHLUMI
BHWIDE
NLLBHA
SABSPV
UNIBAB
Conclusions and outlo ok
Introduction
The main goals of the Bhabha working group are to make an inventory of all the available
Monte Carlo MC event generators for small angle SABH and large angle LABH Bhabha
pro cesses at LEP and LEP and to improve our understanding of their theoretical uncertain
ties through systematic comparisons of the MC event generators developed indep endently
among themselves and with other non MC programs The presented activity is of course an
obvious continuation of the previous workshops on LEP physics In the b eginning of
the present workshop the theoretical uncertainty at LEP for the SABH pro cess was typically
estimated as and for the LABH pro cess was estimated at level at Z p eak and
on the wings of the Z resonance There were no estimates sp eci c to LEP
We shall concentrate on the comparison of all the presently available theoretical calculations
published and unpublished This will b e done for several kinds of event selection ES de ned
as a set of exp erimental cuts and apparatus acceptances starting from ES s unrealistic but
useful for some studies oriented towards the QED matrix element and ending on ES s very
close to the exp erimental ones
Let us add a few comments to clarify our priorities and to set the prop er p ersp ective for our
work In spite of the considerable e ort of several theoretical groups at present the theoretical
error on the small angle Bhabha cross section dominates the luminosity error at LEP This
inhibits from taking full advantage of the high exp erimental precision of the nal LEP data
for precision tests of the Standard Mo del As a consequence the reduction of the theoretical
error in the SABH pro cess at LEP is the biggest challenge and was the main ob jective of
our working group The precision requirements of LEP are lower than those of LEP The
total cross section of W pair pro duction will b e measured with to precision at b est
so it is su cient to keep the theoretical uncertainty of the SABH pro cess at the level
Furthermore at LEP the detectors and exp erimental techniques for measuring the SABH
pro cess are almost the same as for LEP Radiative corrections to the SABH cross section
dep end on the center of mass energy but smo othly moreover in the small angle regime the
center of mass energy is not so imp ortant from the p oint of view of the physics involved we are
always faced with a t channel photon exchange dominated pro cess hence improving the small
angle Bhabha generators for LEP is generally a su cient condition for improving them also
for LEP The only subtle p oint concerns the error estimate a error at LEP do es not
guarantee such a small error also at LEP so that an additional analysis has to b e p erformed
For the LABH pro cess the nal LEP data analysis requires a theoretical uncertainty of the
co des used to b e at the level The LABH pro cess at LEP is not of ma jor interest
and we think that a precision of the order of is enough Nevertheless the physics of the
LABH pro cess at LEP is signi cantly di erent from LEP di erent Feynman diagrams rise to
imp ortance so p erforming additional study for the LABH pro cess at LEP is a new nontrivial
1
Actually the main di erence is that due to machine background radiation the internal part of luminosity
detectors may b e obscured by sp ecial masks We shall discuss the impact of such mo di cation on the theoretical
errors This asp ect was brought to our attention by B Blo ch Devaux during our WG meeting in January
work
In view of the ab ove our strategy was to do all the work for the SABH and the LABH
pro cesses rst for LEP exp erimental conditions and to supplement it with all necessary
work discussion which would assure control of the precision at the level su cient for LEP
exp eriments This practically means that all the numerical comparisons were done for LEP
and rep eated for LEP or in rare cases a convincing argument was given that it is not nec
essary sometimes numerical results for LEP were obtained but are not shown in full form
b ecause they were trivially identical to those for LEP
We include in our rep ort two main parts one part on the SABH pro cess and a second one on
the LABH pro cess with the cases of LEP and LEP discussed in parallel These two pro cesses
are governed by di erent physics i e dominated by di erent Feynman diagrams Also the
theoretical precision requirements in calculating SABH and LABH cross sections are di erent
by a factor of ve ten These two parts are followed by a section including short descriptions
of all the involved Monte Carlo MC event generators or other co des and a nal section on
conclusions and outlo ok
Small angle Bhabha scattering
Small angle Bhabha SABH scattering is used at LEP and LEP to measure the accelerator
luminosity The LEP exp eriments have reached in a systematic uncertainty of
b etter than in selecting luminosity Bhabha events see Ref and Refs
On the theory side QED calculations have still an uncertainty larger than in
determining the Bhabha cross section in the detector acceptance which is caused mostly by
the non existence of a Monte Carlo program including complete O next to leading terms
Actually there exist O calculations with complete next to leading contributions which
claim a precision of the order of but they can not b e used in a straightforward way
b ecause they are not implemented in the Monte Carlo event generators The size of the O
contributions dep ends not only on the angular range covered by the detector and on the electron
energy cut o but also on crucial exp erimental asp ects such as the sensitivity to soft photons
or such as the electron cluster size This means that the main interest is in the theoretical
predictions for the Bhabha pro cess including as many higher order radiative corrections at it
is necessary to reach a precision of in a form of a Monte Carlo event generator
Monte Carlo event generators are very p owerful to ols b ecause they are able to provide a
theoretical prediction cross sections and any kind of distribution for arbitrary ES s However
event generators are di cult to construct and what is even more serious they are very di cult
to test one has to have at least two of them to compare with one another for a wide range of
2
The radiative LABH pro cess is an imp ortant background to other pro cesses like pair pro duction
+
W W ee new physics like SUSY pro cesses and so on but a detailed analysis of these items go es
b eyond the aims of the present study
ES s
For the SABH pro cess the task of comparing various Monte Carlo event generators was the
main goal of the Bhabha Working Group There were only a few comparisons of indep endently
developed Monte Carlo event generators for the SABH pro cess in the past A few examples can
b e found in Ref However we shall include in the comparisons results from non Monte Carlo
calculations as well They are usually limited to certain sp ecial primitive ES s Nevertheless
they provide additional valuable cross checks
What shall we learn from these comparisons The calculations from various Monte Carlo
event generators will of course di er The di erences have to b e understo o d In a certain class
of the comparisons the underlying QED matrix element will b e the same and in that case
the di erences will b e only due to numerical e ects The results from two or more computer
programs will di er due to rounding errors programming bugs numerical approximations The
di erence measures uncertainties of this kind and we say that we are determining the technical
precision of the tested programs One has to remember that the technical precision is dep endent
on the ES and it is therefore absolutely necessary to use several at least semi realistic quite
di erent examples of ES s In other cases we shall compare Monte Carlo event generators
which are based on di erent QED matrix elements In this case the di erence b etween results
will tell us typically ab out higher order e ects which are not included in some of these event
generators or which are approximated di erently in these programs In this situation we shall
talk ab out exploring the physical precision of the tested Monte Carlo event generators Needless
to say the physical precision is the main goal but one has to remember that without a technical
precision of at least a factor of two b etter than the physical precision it is p ointless to discuss
the physical precision at all
Before we come to the actual comparisons of the programs let us characterize various
contributions corrections to the SABH pro cess We shall also characterize brie y the various
Monte Carlo event generators and non Monte Carlo calculations involved in the comparisons
If we remember that the SABH pro cess was chosen for the luminosity measurement b ecause
it is calculable from rst principles within quantum electro dynamics QED then it is natural
to group corrections to the SABH pro cess into pure QED and non QED corrections The
latter ones are due to s channel Z exchange and the corrections induced by low energy strong
interactions QCD through vacuum p olarizatio n and light quark pair pro duction Among the
pure QED corrections we may distinguish photonic bremsstrahlung corrections related to
multiple photon emission and non photonic corrections for instance lepton pairs leptonic
vacuum p olarization multiperipheral diagrams Numerically the biggest ones are the photonic
corrections and the vacuum p olarizatio n correction They also contribute the most to the
physical precision Photonic corrections dominate completely the technical precision due to
the MC integration over the complicated multi bo dy phase space The QED non photonic
corrections are small but are di cult to calculate and quite uncertain technical precision
For all the comparisons of the event generators it is crucial esp ecially for SABH to under
stand the exp erimental ES In the main comparisons we shall compare al l the available event
generators for four types of ES s However the problem of the variation of the parameters in
the ES is so imp ortant that we include also a separate subsection on this sub ject in which for
a limited number of three event generators we p erform a detailed study of the dep endence of
the higher order corrections on all p ossible cut o s involved in the real exp eriment This will
allow us to see all our work in the prop er p ersp ective from the p oint of view of the exp erimental
analysis and will also give us clear hints on the dep endence of the higher order corrections on
the ne details of the ES This study will b e limited to the SABH pro cess
Sensitivity of LEP observables to luminosity
The imp ortance of the improvement of the theoretical luminosity error on the LEP results
is shown in Table The results of the lineshap e parameter ts made with the theoretical
luminosity error of and are given corresp onding to the reduction of error
achieved during this workshop A pro jection concerning a further reduction of the theoretical
luminosity error to is also given The results of the four LEP exp eriments used as input
to the ts as well as the tting pro cedure are describ ed in Ref From the ve parameter
is sensitive to the luminosity error The decreased error in this variable causes t only
h
a reduction of the errors of the derived parameters shown in the lower part of Table As
we see the ab ove improvement in the theoretical luminosity error in uences signi cantly not
only quantities like the number of light neutrino s N but also other LEP observables used
routinely in the tests of the Standard Mo del
theoretical luminosity error
m GeV
Z
GeV
Z
nb
h
R
l
l
A
FB
GeV
had
MeV
ll
nb
ll
had Z
ll Z
MeV
inv
inv ll
N
Table Line shap e and asymmetry parameters from parameter ts to the data of the four LEP
exp eriments made with a theoretical luminosity error of and In the lower part of
the Table also derived parameters are listed
At LEP the normalization of the total cross section for the WW pro duction pro cess enters
in a nontrivial way into tests of the W b oson coupling constants The precision requirements
for the total cross section is limited by statistics of the WW pro cess and a luminosity error
at the level is su cient see the chapter WW cross sections and distributions these
pro ceedings
Higher order photonic corrections at LEP and LEP
Canonical co e cients
mrad mrad
min min
LEP LEP LEP LEP
L O L
O
L O L
L O L
L O L
Table The canonical co e cients indicating the generic magnitude of various leading and subleading
contributions up to third order The big log L ln jtj m is calculated for mrad and
min
e
p
s M where the mrad and for two values of the center of mass energy at LEP
Z min
p
corresponding jtj s are and GeV and at LEP energy s GeV where the
min
corresponding jtj are and GeV resp ectively
For the SABH pro cess the smallness of the electron mass ruins the normal p erturbative
expansion order in the following sense for instance the O QED contributions can b e
expanded into O L O L and pure non log O The non log O corrections
are completely uninteresting while the O L corrections are as imp ortant as the O L
corrections Here L ln jtj m is the so called big log in the leading loga rithmic LL ap
e
proximation where t is the momentum transfer in the t channel of the order of GeV This
phenomenon is illustrated in Tab From this table it is clear that for a precision of the order
of for calorimetric ES s it is enough to include the O L O and O L For
a precision of the order of or b etter one has to add O L and O L These scale
co e cients have to b e kept in mind when discussing various QED calculations progra ms As
we shall see the higher order e ects seen in the numerical results presented in the next sections
generally conform to the ab ove scale co e cients
Table demonstrates also the scaling laws for various QED corrections b etween LEP
Z p eak and LEP energies If the angular range is kept the same then t channel transfer is
prop ortional to s E Actually at LEP exp eriments the luminosity measurement will
beam
rely more on the SABH pro cess at larger angles ab ove and this is why we also included in
the table another two columns for this angular range As we see photonic corrections do not
p
change very much due to the increase of s from Z p eak energy to LEP energy GeV and
due to going to twice larger angles Actually the change in canonical co e cients is negligible
One has only to pay attention to the O L corrections which in the worst case increase by
a factor however as we shall see they are under go o d control
One has to remember that as it was shown explicitly in ref the radiative corrections to
the SABH pro cess with the typical double tag detection are prop ortional to ln
max min
i e they are bigger for narrower angular acceptance and smaller for wider angular ac
ceptance This has to b e remembered b ecause at LEP in some exp eriments the angular range
might b e narrowed by placing masks in front of the SABH detectors in order to eliminate
machine background radiation We conclude that the change for narrower angular acceptance
is more dangerous from the p oint of view of the increase of the pure photonic corrections and
we shall address this problem with a separate numerical exercise
In ref it was also shown numerically using an O calculation that for the purp ose
of the SABH pro cess b elow we may neglect the real and virtual QED interference contri
butions b etween photon emission from the electron and p ositron lines the so called up down
interference In the numerical example in ref it was shown that for the angular range
the up down interference is b elow It is even smaller for smaller an
gles It means that it is negligible for all practical purp oses in the luminosity measurements
This phenomenon was also discussed in ref b eyond O in the framework of the eikonal
approximation
Light pairs and other small contributions
To calculate pair corrections to the SABH two approaches have b een used The rst one
is based on direct analysis of Feynman graphs and analytical extraction of graphs and terms
contributing to the SABH within the O accuracy Both leading and next to leading
terms are considered The other metho d uses the LL approximation to nd the dominant
pair contributions to SABH and to discard the negligible ones Having isolated the dominant
mechanism an actual MC program for this particular mechanism is constructed
The dominant pair pro duction corrections enhanced by factors of L and L arise from
kinematical con gurations where one or b oth of the pro duced leptons is almost collinear with
the incoming or outgoing e These contributions have b een calculated analytically
The analytical calculation of the real hard pair pro duction cross section within
logarithmic accuracy takes into account the contributions of the collinear and semi collinear
kinematical regions All p ossible mechanisms for pair creation Singlet and Non Singlet as
well as the identity of the particles in the nal state are taken into account In the case of
3 +
Here we have taken into account only e e pair pro duction An estimate of the muon pair contribution gives
2 2 2 2
less than since ln Q m ln Q m Contributions of pion and tau lepton pairs give still smaller
corrections Therefore within the accuracy one may omit any pair pro duction contribution except the
p p
q q
p
* * * *
p p p
p
p
p p
p p
q q
------
q
q p p q q p p q
p q p q p p p p
------
- -
p p p
q
- -
p q p p
p q p q p q p q
Figure The Feynman diagrams giving logarithmically enhanced contributions in the kinematical region
where the created pair go es along the electron direction The signs represent the Fermi Dirac statistics of
the interchanged fermions
Channel e e cc uu dd ss total
nb
p
Table Double Tag cross sections for fermion pair production from multip eripheral graphs s
GeV mrad mrad For u d s quarks W GeV The uncorrected Born cross
e e
section is nb
B or n
SABH only a part of the total Feynman diagrams are relevant i e the scattering diagrams
shown in Fig
The analytical formulae for virtual soft hard and total pair pro duction contributions can b e
found in Numerical results for the pair contribution cross sections based on these
formulae can b e obtained by using the co de NLLBHA see b elow for a description of the co de
e
x Numerical The leading term can b e describ ed by the electron structure function D
e
results can b e found in Refs The contribution to SABH of the pro cess of pair
pro duction accompanied by photon emission when b oth pair and photons may b e real and
virtual has also b een analyzed and the relevant analytical formulae are given in
With the help of a Monte Carlo generator a dedicated study has b een done for the
contribution of the multiperipheral graphs Fig b eing for many kinematical setups the
dominant mechanism of pair pro duction The total cross sections for the pro duction of fermion
pairs as detailed in Table were obtained The total contribution from the multiperipheral
graphs is then estimated to b e with a relative error from MC statistics of
B or n
+
e e one
4
It can b e veri ed that the interference b etween the amplitudes describing the pro duction of
pairs moving in the electron direction and the p ositron one cancels This is known as up down interference
cancelation
z GeV GeV
min
LL
B or n
NN
LL
B or n
WW
LL
B or n
NW
Table LL Non Singlet ee pair correction to SABH SiCAL angular cuts WW NN
p
0
in Born units s GeV GeV for last two entries z s s
min min
This correction which still do es not take into account a further reduction factor of
coming from a cut on the acoplanarity angle of the detected e is thus negligible for
SABH
The LL calculation of photonic corrections to SABH of Ref has b een extended
to pair corrections in Analytical formulae for arbitrary asymmetric angular cuts for
b oth Singlet and Non Singlet corrections have b een given in These formulae based
on include b oth pairs and photons up to the exp onentiated second or third order The
semianalytical program BHPAIR based on this calculation has b een written Numerical
results for the LCAL type angular cuts have b een given in For the SiCAL type angular
cuts the Singlet contribution is negligible b elow and the Non Singlet contribution
B or n
up to third order with exp onentiation is calculated in Table also for the LEP energies
The strong dep endence of the result on angular cuts WW NN or NW may indicate signi cant
e ects due to more realistic ES s This can only b e analyzed with the MC simulation Such
a MC program has already b een constructed This program b eing an extension of the
BHLUMI MC co de is based on the extension of the YFS resummation of soft photons to
the resummation of infrared and collinear pairs cf Preliminary results show that a
calorimetric ES reduces further the pair correction of Table
To summarize numerical values of pair corrections as given in and Table
agree within for the NN and WW cuts The total contribution from pairs and
B or n
multiperipheral diagrams for the energy cut in the exp erimentaly interesting range x
c
is also at most With the help of a MC simulation of a realistic ES one
B or n
should b e able to control the pair contribution with an accuracy of or b etter A
B or n
similar conclusion is to b e exp ected also for the LEP energies
Vacuum p olarization
Vacuum p olarization contributes ab out and resp ectively to the e e cross section
in the angular region of the rst and second generation of the luminosity detectors at LEP
The leptonic part of this contribution is known with excellent precision The quark
5
Extending further the analysis of Ref with the help of the parton like picture together with appro
priate choices of structure functions and hard scattering cross sections one can calculate the other pair creation
mechanisms including the multiperipheral one as well as other leptonic backgrounds to SABH resulting from
the charge blindness of the detectors This analysis will app ear elsewhere
part however is more di cult since the quark masses are not unambiguously de ned and
p erturbative QCD cannot b e used for reliable calculations Therefore this part is
calculated using a disp ersion integral of R
had
e e hadr ons
R
had
e e
measured exp erimentally
jtj a b b a
rad GeV Ref Ref Ref
Table The hadronic part of the vacuum p olarization contribution to the small angle Bhabha scattering
as a function of the scattering angle and corresponding momentum transfer t In column and also
the ratio of the error to the value of the hadronic contribution is given in brackets The last column gives
the di erence b etween the results of Refs and
Recently several reevaluations of the hadronic contribution to the QED vacuum p olarizatio n
have b een p erformed mainly to determine the e ective QED coupling m and
Z
the anomalous magnetic moment g of the leptons At the same time the vacuum
p olarization contribution to the small angle Bhabha scattering has b een recalculated
Table compares the results of these two calculations of the hadronic contribution in the
angular region of small angle Bhabha scattering used at LEP for the luminosity measurements
They are in excellent agreement as is evident from the very small di erences listed in the last
column In brackets the error is given as a p ercentage of the total hadronic contribution We
see that the error of Ref varies b etween and of that of Ref in the angular
region presented here Numbers have b een obtained with the help of FORTRAN routines
HADR and REPI available from the authors Finally the values of the previously
used hadronic contribution from Ref are also shown
Fig from Ref shows the contribution of di erent energy regions of R to the value of
the hadronic contribution and its error while the Fig from Ref shows the uncertainty ρ safunction a as
narrow resonances e Ref
Burkhardt, Pietrzyk '95 Burkhardt, > 12.GeV 7 - 12 GeV 5 - 7. GeV
2.5 - 5 GeV nmgiueaducranyfo h e Ref the from uncertainty and magnitude in in uncertainty cteigcluainfo the from calculation scattering
1.05 - 2.5 GeV GeV ρ ftehdoi aumpoaiaincontribution olarization p vacuum hadronic the of t > 12.GeV 7 - 12 GeV ercent p in 5 - 7. GeV
2.5 - 5 GeV contribution in magnitude eaieuncertainty Relative eaiecnrbtosto contributions Relative luminosity measurement luminosity
1.05 - 2.5 GeV narrow resonances narrow Figure iue Figure ftemmnu rnfri h ml nl Bhabha small angle the in transfer momentum the of
hadr onic
rad Ref Ref Ref
total
Table The vacuum p olarization contribution to the small angle Bhabha scattering as a function of the
scattering angle The last column gives the ratio of the hadronic part to the total vacuum p olarization
contribution
Generation typical rad Ref Ref
rst
second
Table Summary of the uncertainty of the vacuum p olarization calculation for the rst and second
generation of the luminosity detectors of LEP according to Ref
of the hadronic vacuum p olarization contribution to the calculation of the small angle Bhabha
scattering as a function of the momentum transfer
The total vacuum p olarization contribution is obtained as sum of the leptonic contribution
and the hadronic one It is shown in Table The contribution of the vacuum p olarizatio n error
to the total error of the luminosity measurement is ab out twice the error given in the Table
The typical angular region of the rst and second generation of the LEP luminosity detectors
is and mrad resp ectively The contribution of the vacuum p olarization error to the
luminosity calculation for the LEP detector is given in Table
The vacuum p olarization correction and its uncertainty are smaller for the lower angles
covered by the second generation of luminosity detectors
In conclusion the error of the hadronic contribution of Ref makes a negligible contri
bution to the total error of the calculation of the small angle Bhabha scattering The error of
Ref is even smaller Thus the error of Ref can b e considered as a conservative one
Brief characteristics of the programs calculations
Here we will very brie y summarize the basic features of the co des involved in the SABH
comparisons The only aim of the following is to just settle the frame and not to give an
exhaustive description of the co des which can b e found in the original literature and or in the
dedicated write up s at the end of the present rep ort
BHAGEN It is a Monte Carlo integrator for b oth small and large angle Bhabha
scattering It is a structure function based program for all orders resummation including
complete photonic O and leading logarithmic O L corrections in all channels
BHLUMI Full scale Monte Carlo event generator for small angle Bhabha scattering It
includes multiphoton radiation in the framework of YFS exclusive exponentiation Its matrix
element includes complete O and O L The program provides the full event in terms
of particle avors and their four momenta with an arbitrary number of radiative photons
LUMLOG It is a Monte Carlo event generator for SABH part of BHLUMI see Photonic
corrections are treated at the leading logarithmic level at the strictly collinear and inclusive way
Structure functions exp onentiated up to O L are included and without exp onentiation up
to O L
NLLBHA It is the FORTRAN translation of a fully analytical up to O calculation
including all the next to leading corrections It is also able to provide O L photonic cor
rections and light pair corrections including simultaneous photon and light pair emission Not
an event generator
OLDBIS Classical Monte Carlo event generator for SABH from PETRA times the
mo dernized version is incorp orated in the BHLUMI set It includes photonic corrections
at the exact O
OLDBIS LUMLOG It is the well known tandem developed in order to take into account
higher order corrections LUMLOG on top of the exact O result OLDBIS The matching
b etween O and higher orders is realized in an additive form
SABSPV It is a new Monte Carlo integrator designed for small angle Bhabha scattering
It is based on a prop er matching of the exact O cross section for t channel photon exchange
and of the leading logarithmic results in the structure function approach The matching is
p erformed in a factorized form in order to preserve the classical limit
Exp erimental event selection and theory uncertainty in lumi
nosity measurements
In this section we discuss the interplay b etween exp erimental selection and higher order radia
tive corrections All numerical examples are for LEP at Z p eak energy The discussion of
the results is generally limited to LEP but using scaling rules from the introduction one
may easily extend it LEP In particular one has to remember that third order LL corrections
have the strongest energy dep endence and going from the Z p eak to the highest LEP energy
introduces in them a factor of almost two
In this subsection three di erent event generators are used i a generator based on a com
plete rst order calculation OLDBIS which has at most one photon radiated it includes
O and O L ii a generator based on a leading logari thmic third order exp onentiated
calculation LUMLOG it includes O L O L O L in strictly collinear ap
proximation the momenta of the nal state photons are added to the electrons iii a truly
multi photon generator based on an exp onentiated calculation BHLUMI it includes
complete O O L and O L while O L and O L are incomplete it generates
explicitly momenta of all photons ab ove an arbitrary user de ned energy threshold typi
cally a fraction k typically of the b eam energy The Bhabha cross section calculated
with BHLUMI will b e compared to the one calculated with the hybrid calculation consisting
of OLDBIS plus higher order contributions from LUMLOG LUMLOG The cross section
HO
di erences BHLUMI OLDBIS and BHLUMI OLDBIS LUMLOG are studied as
HO
a function of variations in the event selection parameters Note that BHLUMI OLDBIS is
dominated by O L O L and O L while BHLUMI OLDBIS LUMLOG is
HO
dominated by O L and O L
Only the QED t channel part of the generators is used with photon vacuum polarizatio n
switched o We use an improved version of the BHLUMI event generator as discussed in
Ref BHLUMI OLDBIS is used to estimate the higher order contributions We choose
BHLUMI b ecause the BHLUMI Monte Carlo distributions are in excellent agreement with
the data distributions for all LEP exp eriments A quantitative measurement of doubly
radiative events has shown consistency with the BHLUMI exp ectations and also with
OLDBIS LUMLOG exp ectations while OLDBIS alone fails to describ e this contribution
HO
as exp ected However although the MC di erential distributions agree with the data the
absolute scale of the integrated cross section remains uncertain since the bulk of the radiative
corrections are either virtual or involve soft MeV photons
In order to set the scale for the following numerical investigation let us remind the reader
that the LEP exp eriments have reached in a systematic exp erimental uncertainty in
the measuring the SABH luminosity cross section b etter than
Reference event selections
We de ne an imaginary detector consisting in a pair of cylindrical calorimeters covering the
region b etween and mm radially out from the b eam pip e centre and lo cated at mm
from the interaction p oint at opp osite sides of it The b eams are p ointlike and centered within
the b eam pip e The calorimeters are each divided into azimuthal segments sub divided into
radial pads A parton electron or photon dep osits all its energy in the pad it hits Photons
and electrons from Bhabha events that hit the detector within a region of radial pads and
azimuthal segments centered on the pad struck by the largest energy parton are combined
into a cluster The cluster energy is the pad energy sum Co ordinates of the cluster centroid
are the energy weighted average p olar co ordinates R summing over all pads in the cluster
Partons falling outside the principal cluster can originate secondary clusters with no overlap
Only one cluster the most energetic of all clusters is used Bhabha events are selected using
the cluster energy E and the radial co ordinate of the cluster centroid in b oth calorimeters
cluster
We then de ne a reference small angle selection for Bhabha events RSA selection The
radial acceptance edges for Bhabha events are set at pad b oundaries The Wide acceptance
b oundary extends up to two pads away from the detector inner and outer edges
mrad The Narrow acceptance b oundary extends up to six pads away from
the detector inner and outer edges mrad A similar angular range
is covered by the OPAL L ALEPH luminometers An event is selected when the
cluster co ordinates are within the Wide acceptance at one side side and within the narrow
acceptance at the opp osite side side Events must satisfy the criterion x x
with x E E Selection criteria are also applied on the acoplanarity rad and
cluster beam
the acollinearity mrad b etween the electron and p ositron clusters
Another selection is also considered similar to the previous one but extending over the
angular range covered by the DELPHI luminometer RLA reference large angle selection
The calorimeters are lo cated at mm from the interaction p oint and cover radially the region
b etween and cm A cluster is formed starting from the most energetic particle hitting
the calorimeter and considering all particles whose angular distance in radians from
the initial one satis es the two shower separation condition determined from the comparison
with the data The cluster energy is the sum over the energies of
all particles inside the cluster while the cluster co ordinates are given by the energy weighted
sum of their p olar co ordinates Bhabha events are selected by cutting on the minimum cluster
energy min x x on the acoplanarity and on the cluster radial co ordinate The radial
acceptance is de ned on the Narrow side by the condition mrad and on
the Wide side by the condition mrad
Comparison of exp onentiated and order by order calculations
First Order Calculation
The Bhabha cross section for the RSA and RLA selections has b een calculated with OLDBIS
The results are shown in gure for the RSA selection where the cross section is sub di
vided into x bins separately for the narrow acceptance side and for the large acceptance side
x x A sample of events is used The total Bhabha cross section within the
N ar r ow W ide
g en
RSA acceptance is nb Displacing the generation minimum angle from
min
mrad as recommended in to mrad changes the accepted cross section by
nbarns No sizeable k E E dep endence is observed when varying k from to
beam
1.00 1.00
0:0498(2)
0:1050(3) 0:8251(8) 73:009(9) 1:0763(9) 7:053(3) 7:269(3) 41:278(7)
0.999 0.85
1:339(1) 0:00180(3)0:00393(5) 7:331(3)
0 0
x
N ar r ow
0.75 0.99
5
0:1774(3) 0:00175(3) 7:577(3)
0 0
< 10
0.65 0.90
0:0836(2)
1:415(1)
0 0 0
x
W ide
0.50 0.85
0.50 0.65 0.75 0.85 1 0.85 0.90 0.99 0.999 1
Figure OLDBIS Bhabha cross section nb in phase space bins for the RSA selection see text
Higher Order Leading Log Contribution
The cross section di erence LUMLOG LUMLOG all orders LUMLOG rst order is
HO
used to estimate the higher order leading loga rithmic contribution gure for the RSA se
lection In LUMLOG only the initial state radiation has an impact on the measured cluster
energies and angles b ecause the momenta of the nal state photons are combined together
with the electrons A sample of events is used There is a total higher order leading log
contribution of nb to the Bhabha cross section within the RSA acceptance the
higher order contribution is negative in the phase space region dominated by singly radiative
events it is p ositive in the non radiative Bhabha p eak and in the phase space region of hard
doubly radiative events
Exp onentiated Calculation
The Bhabha cross section in phase space bins for the RSA selection obtained with BHLUMI
is presented in gure A sample of events is used The total Bhabha cross section
accepted by the RSA selection is nb The accepted cross section changes by
g en
when decreasing the t minimum generated four momentum transfer squared value
min
as recommended in to half of it
Comparison of Exp onentiated and Order by Order Calculations
The BHLUMI and OLDBIS cross sections di er for the RSA selection by showing
that the estimated contribution to the accepted cross section from higher order radiative e ects
is very small This estimate is also in reasonable agreement with the LUMLOG exp ectation
HO
of
A similar study for the RLA selection results in a BHLUMI OLDBIS relative di erence
1.00 1.00
7
7(3) 10
:00905(4) :0142(8) 0:140(8) :2255(9) 1:729(2) 2:170(3) 6:626(6)
0.999 0.85
:04243(9) :054(1) :0640(1) :3970(3) :3794(3) 2:181(3)
x
N ar r ow
0.75 0.99
:0197(1) :0734(1) :4445(3) :4264(3) 1:852(3) :0623(6)
0.65 0.90
:000067(3)
:01522(6) :0888(1) :0852(1) :302(1)
x
W ide
0.50 0.85
0.50 0.65 0.75 0.85 1 0.85 0.90 0.99 0.999 1
Figure LUMLOG higher order contribution to the Bhabha cross section nb in phase space bins for
the RSA selection see text
1.00 1.00
0:0466(1)
0:1090(2) 0:7904(5) 73:199(6) 0:7945(7) 4:913(2) 4:632(2) 49:494(6)
0.999 0.85
0:0487(1) 1:2527(7) 0:0754(2) 0:4691(5) 0:4436(5) 4:682(2)
x
N ar r ow
0.75 0.99
0:1890(3) 0:0859(2) 0:5344(6) 0:5004(6) 5:304(2) 0:0750(8)
0.65 0.90
0:0765(2)
0:01724(9) 0:1033(2) 0:1006(2) 1:0476(8)
x
W ide
0.50 0.85
0.50 0.65 0.75 0.85 1 0.85 0.90 0.99 0.999 1
Figure BHLUMI Bhabha cross section nb in phase space bins for the RSA selection see text
of to b e compared with a LUMLOG exp ectation of
HO
Dep endence on energy and acollinearity cuts
The cross section relative di erence BHLUMI OLDBIS BHLUMI RSA where BHLUMI RSA
refers to the RSA selection is studied in table for several selection criteria on energy and
cut cut
is applied Through we mean that the energy cut min x x x acollinearity With x
min min
transverse momentum conservation energy and acollinearity cuts are strongly correlated in
events with initial state radiation The relative di erence BHLUMI OLDBIS is indicative
cut
of the higher order contribution which clearly app ears in table to b e huge for large x It
min
cut
b ecomes progressively smaller for smaller x It should b e stressed that the h o corrections
min
cut
and acollinearity are small at the p er mille level over a very broad region of x
min
A second estimate of the Bhabha cross section with higher order radiative corrections can
b e obtained with OLDBIS LUMLOG The three generator relative di erence BHLUMI
HO
OLDBIS LUMLOG BHLUMI RSA in table shows that the h o corrections in
HO
BHLUMI and in LUMLOG track each other very well giving con dence that the h o contri
butions are in fact small when they are estimated to b e so The unstable region is limited to
cut
The BHLUMI and OLDBIS LUMLOG Bhabha cross sections agree at very large x
HO
min
the level over an extremely broad range of energy and acollinearity cuts
The cross section di erences BHLUMI OLDBIS and BHLUMI OLDBIS LUMLOG
HO
for the RSA selection change by when the acoplanarity cut is not applied
For the RLA selection the cross section di erences BHLUMI OLDBIS and BHLUMI
OLDBIS LUMLOG normalized to the BHLUMI result are shown in table as a function
HO
cut
The higher order contribution to the Bhabha cross section for the RLA of the cut on x
min
cut
selection b oth in BHLUMI and in LUMLOG is very small over a broad range of x
min
BHLUMI OLDBIS LUMLOG
HO
BHLUMI OLDBIS
Acollinearity cut rad Acollinearity cut rad
cut cut
x no cut x no cut
min min
triang triang
Table Cross section di erences BHLUMI OLDBIS and BHLUMI OLDBIS LUMLOG normalized
HO
to the BHLUMI Bhabha cross section for the RSA selection The lab el triangular stands for the cut
x x
cut cut
x BHL OB x BHL OB LL
HO
min min
Table Cross section di erences BHLUMI OLDBIS and BHLUMI OLDBIS LUMLOG normalized
HO
to the BHLUMI Bhabha cross section for the RLA selection
Wide Wide Narrow Narrow versus Wide Narrow acceptance
In the reference selections RFA and RLA an asymmetric acceptance Wide on one side and
Narrow on the opp osite side is used All LEP exp eriments use an asymmetric acceptance for
the LEP luminosity measurement We study in table how the results change when using a
symmetric Wide Wide or Narrow Narrow The BHLUMI OLDBIS cross section di erence
b ecomes large for the Narrow Narrow acceptance A similar result is also obtained
using LUMLOG and then the BHLUMI OLDBIS LUMLOG di erence is small
HO HO
We thus conclude that the higher order contributions to the accepted Bhabha cross section as
estimated with BHLUMI or LUMLOG are largely reduced when using an asymmetric Wide
Narrow acceptance
WN WW NN
BHLUMI nb nb nb
OLDBIS nb nb nb
LUMLOG nb nb nb
HO
BHL OB BHL
BHL OB LL BHL
HO
Table Comparison of BHLUMI OLDBIS and LUMLOG Bhabha cross sections for Wide Narrow
HO
Wide Wide Narrow Narrow event selections All other cuts as in the RSA selection
Multiple photon radiation
A very relevant prop erty of exclusive exp onentiation is that there are many more multi photon
events than exp ected from p erturbation theory at a xed order in In a sample of BHLUMI
Bhabha events the events have up to eight photons with energy larger than k E MeV
beam
as shown in gure This may enhance the di erence b etween cross section calculations
p erformed with BHLUMI and with OLDBIS LUMLOG In the following we study the
HO
stability of the BHLUMI OLDBIS and BHLUMI OLDBIS LUMLOG ho di erences in
table and in table when varying those parameters in the exp erimental selection which are
sensitive to the presence of many photons
Lower Energy Photon Cut o
We de ne a K parameter in MeV expressing the sensitivity to soft photons the detector
c
is fully e cient for photons of energy larger than K An implicit K cut o is present in
c c
BHLUMI at K k E MeV for the cross sections calculations presented ab ove The
c beam
relative variation of the BHLUMI Bhabha cross section when varying K is rep orted in table
c
for the RSA selection and in table for the RLA selection The e ect is at most of
for the RSA acceptance in the extreme case of K MeV The relative changes
c
in the BHLUMI and OLDBIS cross sections are compared in gures and The large x
cut
region is very di erent most of the di erence has already disapp eared for x LUMLOG
min
remains una ected it has in the output only the electron and p ositron momenta with the
nal state photons combined with the electrons p ositrons Hence the e ect on the relative
cross section di erences BHLUMI OLDBIS and BHLUMI OLDBIS LUMLOG for
HO
the RSA selection is at most
Figure Distribution in numb er of emitted photons for a sample of unweighted BHLUMI events
The Removal ag is switched on in BHLUMI with K k E MeV
c beam
Cluster Size
The relative variation of the accepted Bhabha cross section with resp ect to the RSA selection
when changing the cluster size is studied in gure using BHLUMI generated events and using
OLDBIS generated events For large cluster sizes BHLUMI and OLDBIS track each other very
well and the BHLUMI OLDBIS relative di erence observed for the RSA selection remains
unchanged On the contrary for small cluster sizes the e ect of many photons in BHLUMI
generated events shows up strongly The LUMLOG result remains una ected Thus for the
K MeV
c
cut
x
min
triangular
Table Variation of the BHLUMI Bhabha cross section when changing the photon minimum detectable
energy K Normalization is with resp ect to the RSA selection with K k E MeV The lab el
c c beam
triangular stands for the cut x x
K MeV
c
cut
x
min
Table Variation of the BHLUMI Bhabha cross section when changing the photon minimum detectable
energy K from K MeV for the RLA selection
c c
1.00 1.00
:001(1)
:0013(9) :006(2) :012(3) 0:072(7) 0:111(8) 3:79(5) 9:45(8)
0.999 0.85
:004(2) 0:072(7) 0:50(2) 1:11(3) 3:65(5)
< 0:001
x
N ar r ow
0.75 0.99
:006(2) 0:009(2) 0:080(7) 0:54(1) 0:137(9)
0.65 0.90
:001(1)
0:010(3) 0:105(8) 0:106(8)
< 0:001
x
W ide
0.50 0.85
0.50 0.65 0.75 0.85 1 0.85 0.90 0.99 0.999 1
Figure Percentage variation of the BHLUMI Bhabha cross section when setting the photon minimum
detectable energy K to MeV see also gure instead of K MeV
c c
1.00
0:35(1) 7:80(5) 16:32(8)
< 0:001
0.999
7:85(5)
< 0:001 < 0:001
0
x
N ar r ow
0.99
0:39(1)
0 0 < 0:001
0.90
< 0:001
0 0 0
x
W ide
0.85
0.85 0.90 0.99 0.999 1
Figure Percentage variation of the OLDBIS Bhabha cross section when setting the photon minimum
detectable energy K to MeV see also gure instead of K MeV
c c
RSA selection we can exclude an e ect larger than on the BHLUMI OLDBIS and
on the BHLUMI OLDBIS LUMLOG cross section di erences
HO
Cluster Co ordinate
The energy weighting algorithm for extracting the cluster co ordinates couples the co ordinates
to the cluster size A di erent co ordinate reconstruction algorithm PADMAX is then used
we select the pad with the largest energy dep osit and use the momentum sum of the partons
which enter that pad to calculate an impact p oint in the pad the impact p oint so calculated
de nes the cluster co ordinates indep endent of the cluster dimensions The BHLUMI cross
section when changing from energy weighted co ordinates to PADMAX co ordinates in the
RSA selection changes by The OLDBIS cross section when changing from
energy weighted co ordinates to PADMAX co ordinates changes by The
LUMLOG result is una ected The e ect on the BHLUMI OLDBIS and on the BHLUMI
OLDBIS LUMLOG cross section di erences in the RSA selection when using the
HO
PADMAX co ordinates instead of the energy weighted co ordinates is
Summary
We have shown that there is a strong correlation b etween the magnitude of the O radiative
corrections to the Bhabha cross section and distinctive characteristics of the exp erimental
Bhabha event selection In particular we have shown that the Bhabha selections used by
the LEP exp eriments to measure the accelerator luminosity minimize the sensitivity to O
radiative corrections
P AD
BHLUMI
all : : : : :
: : : : : :
:
:
: : :
P AD OLDBIS
: : : : : all
: : : : : :
:
:
: : :
all
N
SEG
Figure Relative variation of the accepted Bhabha cross section with resp ect to the RSA selection when
changing cluster radial PAD s and azimuthal SEGments dimensions A cluster extends for pads
P AD
and N segments around the pad containing the largest energy dep osit A pad subtends a p olar angle of
SEG
ab out mrad a segment covers azimuthally an angle of degrees The RSA selection has
P AD
and N
SEG
The O contributions have b een estimated using BHLUMI OLDBIS and LUMLOG
HO
LUMLOG LUMLOG The cross section di erences BHLUMI OLDBIS
all or der s f ir st or der
and BHLUMI OLDBIS LUMLOG are very small at the p er mille level in a broad
HO
region of phase space around the exp erimental selections We have considered two angular
ranges mrad and mrad with a variety of energy and acollinearity
cuts The sensitivity to the p ossible presence of many photons predicted by exclusive exp o
nentiation the e ect of small or large cluster sizes and di erent ways of reconstructing the
cluster co ordinates have b een investigated Large cluster sizes rather soft energy cuts and a
Wide Narrow metho d are very e ective in minimizing the cross section di erences BHLUMI
OLDBIS and BHLUMI OLDBIS LUMLOG Vice versa these same e ects could b e
HO
used to enhance the sensitivity to the O radiative corrections in order to p erform measure
ments and test the theory predictions
Comparisons of event generators for small angle Bhabha scat
tering
In contrast to the previous section where we have seen results from many variants of ES s
with varying cut parameters but for only three types of QED calculations here we shall limit
ourselves to only four ES s two of which very close to realistic exp erimental situations
but we shall discuss al l the available theoretical calculations The outline of this section is
the following the actual comparisons will b e presented rst at the O level in order to
determine the basic technical precision and later for more advanced QED matrix elements
b eyond O in order to explore physical precision These comparisons will b e done rst for
LEP energy and later will b e also extended to LEP energies
BARE
Backward hemisphere Forward hemisphere
W
N
max
max
R
I
N
I
W
min
min
E
E
W W N N
WW 2 NN 2
i i
max max
min min
W N W 0 N
NW 2 2
s u s
min
max max
min min
Figure Geometry and acceptance of the simple non calorimetric ES BARE This ES restricts p olar
angles in the forward backward hemispheres and requires a certain minimum energy to b e detected
i
simultaneously in b oth hemispheres Photon momentum is not constrained at all The entire ducial
N W N W
rad and the narrow N range is range i e wide W range is
max max
min min
N W N W W W
where and This ES can b e symmetric Wide
max max max
min min min
Wide WW or Narrow Narrow NN or asymmetric Narrow Wide NW see the description in the gure
0 0
The energy cut s u s involves momenta of outgoing e s q q only
min
Event selections
One cannot talk ab out the cross section for the small angle Bhabha SABH pro cess without
de ning precisely all cuts or in other terms without sp ecifying the ES The most interesting
ES is that of the actual exp eriment LEP and LEP exp eriments employ in the measurement
of the small angle Bhabha scattering cross section a rich family of ES s They do however
CALO
Backward hemisphere Forward hemisphere
W
N
max
max
cl
cl
R
I N
W
min
I
min
cl
E
cl
E
cl cl
cl W W cl N N
WW 2 NN 2
max max
i min i min
cl cl
cl W W cl N N
NW 2 E z E 2 E
min
max max
min beam min
Figure Geometry and acceptance of the calorimetric ES CALO This ES restricts p olar angles in
i
the forward backward hemispheres and requires a certain minimum energy to b e detected simultaneously in
W W
b oth hemispheres The entire ducial range i e wide W range is rad
max
min
N N N W N W W
and the narrow N range is where and
max max max max
min min min
W
This ES can b e symmetric Wide Wide WW or Narrow Narrow NN or asymmetric Narrow
min
Wide NW see the description in the gure The energy cut involves the de nition of the cluster the
cl cl
i is identical to the angular p osition of the p ositron in the forward and cluster center
i i
the electron in the backward hemisphere The angular cone of radius rad around e is called
cluster The cone cluster in the plane is an elongated ellipsis due to smallness of theta The total
cl
energy registered in the cluster is denoted by E Note that for back to back con guration
i
have essential common features The most imp ortant is the double tag It means that e
and e are both detected with a certain minimum energy and minimum scattering angle in the
forward and backward direction close to the b eams The other imp ortant feature of the typical
exp erimental ES is that except for rare cases the photons and e cannot b e distinguished
only the combined energy and angle is registered It is said that the typical exp erimental ES
is calorimetric On the other hand for comparing theoretical calculations it is useful to deal
with simpli ed ES s in which only e are measured and the accompanying bremsstrahlung
photons e pairs are ignored The double tag is done on bare e Actually in order
to compare e ciently numerical results from the various programs we employed the family
of four ES s connecting in an almost continuous way the exp erimentally unrealistic but use
ful for theorists examples of ES s to exp erimentally realistic but di cult for some class of
theoretical calculations ones In order to compare theoretical results for SABH we use one
simple non calorimetric ES called BARE see Figs and three calorimetric ES s called
CALO CALO and SICAL with increasing degrees of sophistication They are de ned in
Figs and Fig The last one SICAL of Fig corresp onds very closely to the
ES of the real silicon detector of OPAL or ALEPH
CALO
Backward hemisphere Forward hemisphere
f
max
R
W
max
N
max
cl
cl
R
N
I W
min
min
I
f
cl
E
min
cl
E
cl cl
cl W W cl N N
WW 2 NN 2
max max
i min i min
cl cl
cl W W cl N N
NW 2 E z E 2 E
min
max max
min beam min
Figure Geometry and acceptance of the calorimetric ES CALO This ES restricts p olar angles in
i
the forward backward hemispheres and requires a certain minimum energy to b e detected simultaneously
f
f
rad includes the wide in b oth hemispheres The entire ducial range
max
min
f
W W N N W W f
W range and the narrow N range where
max max max max
min min min
min
f f
N f N f
This ES can b e symmetric Wide Wide and
max max max
min
min min
WW or Narrow Narrow NN or asymmetric Narrow Wide NW see the description in the gure The
cl cl
energy cut involves the de nition of the cluster the cluster center i is identical to the
i i
angular p osition of the p ositron in the forward and electron in the backward hemisphere The angular
cl cl cl cl
plaquette where around e is called
i i i i
cl
Note that for back to back cluster The total energy registered in the cluster is denoted by E
i
con guration
First order technical precision
We start the numerical comparisons of the various theoretical calculations with the calibration
exercise in which we limit ourselves to strict O with Z exchange up down interference
and vacuum p olarization switched o i e we examine pure photonic corrections without up
down interferences We calculate the corresp onding total cross section for all our four ES s
p
at the LEP energy s GeV The purp ose of this exercise is to eliminate p ossible
trivial normalization problems in the core MC programs and in the testing programs which
implement our ES s Since O is unique and common the di erence of the results will
b e entirely due to numerical technical problems and following ref where the analogous
exercise of this type was done for the rst time we call it the technical precision of the
involved calculations progra ms The results are shown in Tab Since tables are hard to
read we always include a gure which contains exactly the same result in the pictorial way
In the gure one of the cross sections is used as a reference cross section and is subtracted
from the other ones It is plotted however on the horizontal line with its true statistical error
SICAL
Backward hemisphere Forward hemisphere
f
max
R
W
max
N
max
cl
cl
R
N
I W
min
min
I
f
cl
E
min
cl
E
cl cl
cl W W cl N N
WW 2 NN 2
max max
i min i min
cl cl
cl W W cl N N
NW 2 E z E 2 E
min
max max
min beam min
Figure Geometry and acceptance of the calorimetric ES SICAL This ES restricts p olar angles in
i
the forward backward hemispheres and requires a certain minimum energy to b e detected simultaneously
in b oth hemispheres No restrictions on azimuthal angles are there The entire ducial range
i
f
f W W
rad includes the wide W range and the narrow N range
max max
min
min
N N
exactly as depicted in the gure This ES can b e symmetric Wide Wide WW or Narrow
max
min
Narrow NN or asymmetric Narrow Wide NW The energy cut and cuts involve the de nition of the
cluster Eeach side detector consists of equal plaquetes A single plaquete registers the total energy
of electrons and photons The plaquete with the maximum energy together with its neighb orhoo d is
cl cl cl
called cluster The total energy registered in the cluster is E and its angular p osition is i
i i i
More precisely the angular p osition of a cluster is the average p osition of the centers of all plaquetes
weighted by their energies the de nitions of s are adjusted in such a way that for back to back
con guration The plaquetes of the cluster which spill over the angular range outside thick lines are also
used to determine the total energy and the average p osition of the cluster see backward hemisphere
Here Tab is visualized in Fig In this gure the cross sections from the Monte Carlo
OLDBIS an improved version of the MC program written originall y by Berends and Kleiss in
PETRA times now part of BHLUMI is used as a reference As we see all calculations agree
well within relative deviation The apparent discrepancy of the O SABSPV for
the SICAL ES is not statistically signi cant The cross section from the non Monte Carlo
type of calculation NLLBHA is available only for the simplest BARE As we have already
discussed the photonic radiative corrections for the SABH pro cess scale smo othly with energy
so we regard this test to b e valid for LEP energies within a factor two i e within
Beyond rst order physical precision
Having found go o d agreement of the various calculations at the rst order level we now reinstall
the photonic corrections b eyond rst order More precisely we keep again Z exchange up down
z OLDBIS nb SABSPV nb BHAGEN nb NNLBHA nb BHLUMI nb
min
a BARE
b CALO
c CALO
d SICAL
Table Monte Carlo results for the symmetric Wide Wide ES s BARE CALO CALO and SICAL
for the O matrix element Z exchange up down interference and vacuum p olarization are switched
p
o The center of mass energy is s GeV Not available x sections are set to zero
interference and vacuum p olarization switched o but compare numerical results which include
O L O L and O L contributions due to photon bremsstrahlung We do not include
pro duction of light fermion pairs unless stated otherwise The numerical results are shown in
Tab and Fig In the gure the cross section from the second order exp onentiated
Monte Carlo BHLUMI is used as a reference cross section The di erences b etween various
calculations now represent not only technical precision but also physical precision b ecause the
cross sections are calculated using di erent QED matrix elements
The results shown in Tab and Fig have remarkable prop erties For values of the
energy cut variable in the exp erimentally interesting range z the cross sec
min
tion from the programs BHLUMI and SABSPV agree throughout all the four ES s from the
unrealistic BARE to very realistic SICAL to within relative deviation This agree
ment is de nitely b etter than the di erence b etween BHLUMI and OLDBIS LUMLOG which
:0015 :0015
BARE CALO
REF REF
OLDBIS
O
REF
REF REF
:0010 :0010
O
SABSPV
O
?
BHAGEN
O
NNLBHA
O
BHLUMI
:0005 :0005
:0000 :0000
?
? ?
?
?
?
?
?
:0005 :0005
:0010 :0010
z z
min min
:0015 :0015
:25 :50 :75 1:00 :25 :50 :75 1:00
:0015 :0015
CALO SICAL
REF REF
REF REF
:0010 :0010
:0005 :0005
?
:0000 :0000
?
? ?
?
?
?
?
?
?
:0005 :0005
:0010 :0010
z z
min min
:0015 :0015
:25 :50 :75 1:00 :25 :50 :75 1:00
Figure Monte Carlo results for the symmetric Wide Wide ES s BARE CALO CALO and SICAL
for the O matrix element Z exchange up down interference and vacuum p olarization are switched
p
o The center of mass energy is s GeV In the plot the cross section from the program OLDBIS
part from BHLUMI a originally written by Berends and Kleiss is used as a reference cross section
in the last years was routinely used see Refs in order to estimate missing higher order
and subleading corrections Remarkably the OLDBIS LUMLOG results coincide extremely
well with BHAGEN Let us note that the OLDBIS LUMLOG matrix element do es not ex
z BHLUMI nb SABSPV nb BHAGEN nb OBI LMG nb NLLBHA nb
min
a BARE
b CALO
c CALO
d SICAL
Table Monte Carlo results for the symmetric Wide Wide ES s BARE CALO CALO and SICAL
for matrix elements b eyond rst order Z exchange up down interference and vacuum p olarization are
p
switched o The center of mass energy is s GeV Not available x sections are set to zero
p onentiate prop erly O L corrections i e they are wrong in the soft photon limit This may
explain why BHLUMI and SABSPV which do not have such problems agree b etter According
to the authors BHAGEN do es not su er of the same problem as it has the soft photon limit
prop erly treated by construction but some corrections are exp ected due to the approximate
treatment of two hard photon emission The result from NLLBHA is present only for unreal
istic BARE selection and for z it agrees to within with BHLUMI and
min
SABSPV It is an interesting result b ecause NLLBHA features complete O L corrections
while all the other programs have only incomplete O L contributions In Tab and Fig
the results of BHLUMI SABSPV and BHAGEN include exp onentiation and therefore they
include necessarily O L e ects incomplete We therefore compare them with a version
of NLLBHA which includes b esides O L also O L corrections All the ab ove results
will b e used as an input in our nal estimate of the total theoretical uncertainty of SABH cross
:002 :002
BARE CALO
REF REF
REF REF
:000 :000
?
?
?
?
?
?
?
:002 :002
BHLUMI
O
exp
REF
O
SABSPV
exp
O
?
? BHAGEN
exp
O
NNLBHA
NNL
O
OBI LUMG
exp
:004 :004
z z
min min
:25 :50 :75 1:00 :25 :50 :75 1:00
:002 :002
CALO SICAL
REF REF
REF REF
:000 :000
?
?
?
?
?
?
?
?
:002 :002
?
:004 :004
?
z z
min min
:25 :50 :75 1:00 :25 :50 :75 1:00
Figure Monte Carlo results for the symmetric Wide Wide ES s BARE CALO CALO and SICAL
for matrix elements b eyond rst order Z exchange up down interference and vacuum p olarization are
p
YFS
switched o The center of mass energy is s GeV In the plot the O cross section
exp
BHL
from BHLUMI a is used as a reference cross section
section for LEP LEP energies
Finally we present similar numerical comparisons of the calculations b eyond O at one
p
s GeV As b efore since the tables are hard to read we accompany LEP energy
z BHLUMI nb SABSPV nb BHAGEN nb OBI LUM nb
min
a CALO LEP
b SICAL LEP
:002 :002
CALO SICAL
REF REF
REF REF
:000 :000
?
?
?
?
?
?
:002 :002
BHLUMI
O
exp
REF
LEP
O
SABSPV
exp
O
? BHAGEN
exp
O
OBI LUMG
exp
:004 :004
?
z z
?
min min
:25 :50 :75 1:00 :25 :50 :75 1:00
p
Table In this table gure we show cross sections for LEP center of mass energy s GeV
Monte Carlo results are shown for various symmetric Wide Wide ES s and matrix elements b eyond rst
order Z exchange up down interference and vacuum p olarization are switched o Not available x sections
are set to zero In the plot the O cross section from BHLUMI a is used as a reference
exp
BHL
cross section
the table with a gure which shows the same numerical result in a pictorial way the caption
is common for the table and gure This way of presenting results in the form of the twin
table gure will b e used often in the following As b efore in the gure one of the cross sections
is used as a reference cross section and is subtracted from the other ones The main result is
shown in table gure Here results are shown for the symmetric Wide Wide variant of the
CALO and SICAL ES s As exp ected the di erence b etween the programs is almost the
same The higher order corrections at LEP are only slightly stronger This result was already
anticipated when analyzing scaling rules derived from Tab From the scaling rules we also
know that this result will b e essentially the same for the wider angular range The
practical message is that within the precision estimates derived from the numerical
exercises for the SABH process at LEP should be valid also for LEP
Precision requirements at LEP are less stringent In the gure we draw a LEP type b ox
which spans over and extends over the exp erimentally interesting range z
min
All programs come together within the ab ove range The ab ove limit will b e used
as an input in our nal estimate of the total theoretical uncertainty of the SABH cross section
for LEP energies This limit has obviously a large safety margin close to a factor of two
Asymmetric and very narrow event selections
The numerical comparisons shown in the previous section were done for pure technical reasons
less chances for programming errors in the testing programs for the symmetric Wide Wide
version of the ES As we know very well see the introduction the higher order contribu
tions are sensitive to the asymmetricity of the ES In order to avoid any danger due to the
ab ove simpli cation we have done another series of comparisons of the various calculations
for the symmetric Narrow Narrow and asymmetric Narrow Wide versions of the ES s CALO
which are de ned in Fig Let us remind the reader that the variation of the di erence
BHLUMI OLDBIS LUMLOG over the WW NN and NW selection was the cornerstone
HO
of the previous estimates of the size of uncontrolled higher order photonic corrections to
gether with technical precision We b elieve that CALO is close enough to our most realistic
ES SICAL and the results obtained for CALO are valid for SICAL Let us also recall that
the typical exp erimental ES is of the asymmetric Narrow Wide type The corresp onding results
are shown in table gure for the matrix elements in the O class we have checked that
for the O level the same programs agree b etter than but we omit the corresp onding
table plot due to lack of space
As we see in tables gures and for all the three types of the CALO ES WW NN
and NW BHLUMI and SABSPV stay within from one another for all the values of the
energy cut variable in the exp erimentally interesting range z This is a new
min
nontrivial result which will b e exploited to decrease the estimated error due to the higher order
photonic corrections from down to In a sense we replace the old estimate based on
BHLUMI OLDBIS LUMLOG with a new one based on BHLUMI SABSPV Hybrid
HO
Monte Carlo s OLDBIS LUMLOG and BHAGEN are o of ab out in the NN
HO
case but noticeably they are on the same ground as BHLUMI and SABSPV for the most
interesting NW case The ab ove exercise was done for the LEP energy and in view of the
results shown in table gure and our scaling rules see the introduction we do not foresee
any problem with extending its validity to LEP energies
As we already stressed in the introduction for the purp ose of LEP it is more imp ortant
however to check if the change of the narrowness i e the ratio to smaller
max min
z BHLUMI nb OBI LMG nb SABSPV nb BHAGEN nb
min
CALO Symmetric Narrow Narrow
CALO Asymmetric Narrow Wide
:002 :002
Narrow Narrow
Narrow Wide
REF REF
REF REF
:000 :000
?
?
?
?
?
?
?
?
:002 :002
BHLUMI
O
exp
REF
O
SABSPV
exp
?
O
? BHAGEN
exp
O
OBI LUMG
exp
:004 :004
z z
?
min min
:25 :50 :75 1:00 :25 :50 :75 1:00
Table In this table gure we show cross sections for various symmetric asymmetric versions of the
CALO ES for matrix elements b eyond rst order Z exchange up down interference and vacuum p olar
p
ization are switched o The center of mass energy is s GeV Not available x sections are set to
zero The wide range is de ned by and and the narrow range by
w f seg m w f seg m
and and rad
n f seg m n f seg m seg m f f f f
resp ectively
values do es not sp oil the agreement of the table gure As we have already indicated
at LEP the decrease of the narrowness may cause a signi cant increase in
max min
the photonic radiative corrections The relevant cross check is done in table gure It
represents the worst possible scenario at LEP The results are shown for the narrower version
of the CALO ES which we call CALO in the symmetric and asymmetric versions As we
see BHLUMI and SABSPV di er again for the ab ove ES by less than This result will
z BHLUMI nb OBI LMG nb SABSPV nb BHAGEN nb
min
CALO Symmetric Narrow Narrow LEP
CALO Asymmetric Narrow Wide LEP
:002 :002
Narrow Narrow
Narrow Wide
REF REF
REF REF
:000 :000
?
?
?
?
:002 :002
BHLUMI
O
exp
REF
LEP
O
SABSPV
exp
O
? BHAGEN
exp
?
?
?
O
OBI LUMG
exp
?
:004 :004
?
z z
min min
:25 :50 :75 1:00 :25 :50 :75 1:00
Table In this table gure we show cross sections for for the symmetric asymmetric CALO ES s the
narrower version of CALO for matrix elements b eyond rst order Z exchange up down interference
p
and vacuum p olarization are switched o The center of mass energy is s GeV Not available
x sections are set to zero The wide range is de ned by and and
w f seg m w f seg m
the narrow range by and
n f seg m n f seg m seg m f f f
and rad resp ectively
f
b e used for estimating theoretical uncertainty of the SABH pro cess at LEP Hybrid Monte
Carlo s OLDBIS LUMLOG and BHAGEN are o of ab out in the NN case but
HO
noticeably they are on the same ground as BHLUMI and SABSPV for the most interesting
NW case
z BHLUMI nb SABSPV nb BHAGEN VP Z Bhlumi
min
CALO Symmetric Wide Wide
CALO Symmetric Narrow Narrow
CALO Asymmetric Narrow Wide
Table Monte Carlo results for various symmetric asymmetric versions of the CALO ES for matrix
elements b eyond rst order Z exchange up down interference and vacuum p olarization are switched ON
p
The center of mass energy is s GeV Not available x sections are set to zero The wide range is
de ned by and and the narrow range by and
w f seg m w f seg m n f seg m
and rad resp ectively
n f seg m seg m f f f f
Z and vacuum p olarization included
In all the previous comparisons the small contributions from s channel Z exchange and s
channel photon exchange diagrams were switched o in order to enhance the p ossibili ty of
seeing more clearly the most imp ortant pure photonic higher order corrections In the following
part of numerical comparisons we restore in the calculations the contributions from these
s channel Z exchange and s channel photon exchange diagrams together with the e ect of
vacuum p olarization The comparison of various calculations is done for the semi realistic ES
CALO in the versions Wide Wide Narrow Narrow and Narrow Wide as de ned in Fig
The resulting cross sections are shown for a LEP energy in Tab and Fig Again
BHLUMI and SABSPV for values of the energy cut variable in the exp erimentally interesting
range z agree within for all the three versions of the ES WW NN and
min
WN BHAGEN is also in agreement in this case for all the three versions of the ES due to
a slightly bigger correction in these added contributions We do not exp ect that switching on
the small s channel Z exchange and s channel photon exchange corrections would change our
conclusions for LEP Vacuum p olarizatio n enters essentially only in the normalization of the
SABH cross section and Z contribution at LEP can b e safely neglected We therefore extend
the validity of the ab ove exercise to LEP
:002
Wide Wide
REF
REF
:000
?
?
?
?
:002
All Included
?
BHLUMI
O
exp
REF
O
? BHAGEN
exp
O
SABSPV
exp
:004
z
min
:25 :50 :75 1:00
:002 :002
Narrow Narrow
Wide Narrow
REF REF
REF REF
?
?
?
?
:000 :000
?
?
?
?
?
:002 :002
?
:004 :004
z z
min min
:25 :50 :75 1:00 :25 :50 :75 1:00
Figure Monte Carlo results for various symmetric asymmetric versions of the CALO ES for matrix
elements b eyond rst order Z exchange up down interference and vacuum p olarization are switched ON
p
The center of mass energy is s GeV Not available x sections are set to zero In the plot the
YFS
O cross section from BHLUMI x is used as a reference cross section
exp
BHL
The total theoretical error for small angle Bhabha scattering
In this section we present some supplementary numerical material concerning higher order
corrections from MC and non MC programs and we summarize on the total theoretical error
z BHLUMI alf e BHLUMI alf e BHLUMI alf NLLBHA alf NLLBHA alf NLLBHA alf p
min
a BARE
b SICAL
BARE SICAL
:002 :002
REF REF
2 2
BHLUMI BHLUMI
O O
exp exp
REF REF
REF REF
3 3
O O
BHLUMI BHLUMI
exp exp
2 2
O O
BHLUMI BHLUMI
:001 :001
2
2
2
?
?
?
:000 :000
2
?
2
:001 :001
?
3
O
+
NLLBHA
N LLp
3
? O
NLLBHA
N LL
2
2
O
NLLBHA
N LL
:002 :002
z z
min min
:25 :50 :75 1:00 :00 :25 :50 :75
p
Table In this table gure we show cross sections for LEP center of mass energy s GeV
Results from BHLUMI and NLLBHA for the symmetric Wide Wide ES s BARE and SICAL are shown Not
available x sections are set to zero In the table column BHLUMI alf e represents O BHLUMI
exp
a col BHLUMI alf shows O BHLUMI without exp onentiation col BHLUMI alf e shows
missing O in BHLUMI a as calulated with the new unpublished version of LUMLOG col
LL
NLLBHA alf shows O result from NLLBHA including NLL corrections col NLLBHA alf is the
previous plus O and col NLLBHA alf p is the previous plus light pair corrections In the plot the
LL
O cross section from BHLUMI a is used as a reference cross section except for missing
exp
REF
O for which we show
LL RE F
for the SABH pro cess at LEP and LEP
Let us discuss again the size of the O L and O L corrections In the next ta
LEP LEP
Type of correction error Ref Present Present
a Missing photonic O L
a Missing photonic O L
c Vacuum p olarization
d Light pairs
e Z exchange
Total
Table Summary of the total physical technical theoretical uncertainty for a typical calorimetric
detector For LEP the ab ove estimate is valid for the angular range within and for LEP it covers
energies up to GeV and angular range within and see the text for further comments
ble gure we address this question showing once again some results from Tab Fig
and adding some new numerical results from the BHLUMI event generator and the semianalyt
ical program NLLBHA for the unrealistic ES BARE and the realistic ES SICAL symmetric
WW variants First let us recall that in Tab Fig the O L e ects were included
through exp onentiation in all calculations but in most cases they were incomplete In the case
of BHLUMI the recent version of LUMLOG is able to answer the question how big is the
missing O L in BHLUMI a In table gure we see black dots that it is b elow
for b oth BARE and SICAL ES s According to our scaling rules we conclude that
it is b elow at LEP Hence from the practical p oint of view O L in BHLUMI
a is complete In table gure we also include for the unrealistic BARE ES numerical
results from NLLBHA stars which includes complete O L and O L corrections
LL
The di erence b etween BHLUMI crosses and NLLBHA stars should b e in principle due to
O L and technical precision b ecause O L should cancel completely As we see the
ab ove di erence is within the one p er mil b ox but for stronger cuts z it grows
min
slightly b eyond Luckily enough we may push the ab ove exercise in the interesting direc
tion we have also in table gure the results from BHLUMI circles and NLLBHA b oxes
in which exp onentiation and O L was removed completely As we see these results agree
LL
b etter even for strong energy cut z Actually this result di erence b etween b oxes
min
and circles represents an interesting quantity missing O L in BHLUMI The ab ove result
suggests that it is rather small b elow One has to keep in mind that if the ab ove is
true then the former di erence with O L crosses and stars is a puzzle and needs to
LL
b e examined further In any case the fact that all the four ab ove results from BHLUMI and
NLLBHA are within the one p er mil b ox is interesting encouraging and reinforcing our nal
conclusion that photonic corrections are under control within For the present time the
ab ove interesting comparison is limited to BARE ES For SICAL and BARE ES s we see
that the di erence b etween BHLUMI with and without exp onentiation is quite sizeable
and from that we conclude that the inclusive Yennie Frautschi Suura exp onentiation in BH
6 3 3
The new LUMLOG includes nal state radiation in addition to the initial up to O L It was
LL
discussed in the Bhabha Working Group and will b e included in the next release of BHLUMI
LUMI is necessary and instrumental for getting go o d control over the O L corrections
LL
even if they are not complete in the matrix element As a matter of fact all the other MC co des
involved in the present study include exp onentiation and so are on a rm ground from this
p oint of view In table gure we show also results from NLLBHA including pair pro duction
in addition to the O L and O L corrections plus marks in the plot The di erence
b etween pluses and stars represents the net e ect of the light fermion pair pro duction For the
BARE ES with z in the exp erimentally interesting range it is at most We exp ect
min
this e ect to b e ab out a factor of two smaller for calorimetric ES s
The total theoretical error for the SABH pro cess at LEP LEP is summarized in table
The errors in the table are understo o d to b e with resp ect to the cross section calculated for any
typical asymmetric ES for the LEP exp eriment in the angular range with resp ect to
the cross section calculated using BHLUMI a In the case of LEP the estimate extends
to the angular range and to the case of the angular range ab out twice narrower than
usual see the discussion of the numerical results in the previous sections The entries include
combined technical and physical precision In this table entry a for Missing O L is based
mainly on the agreement b etween BHLUMI and SABSPV as seen in tables and
It should b e stressed that we rely on the agreement b etween BHLUMI and SABSPV for al l
the three types of ES Wide Wide Wide Narrow and Narrow Narrow The agreement b etween
BHLUMI and SABSPV is now b etter than the one b etween BHLUMI and OLDBIS LUMLOG
used in the previous b est error estimate of Ref Noticeably alb eit the agreement b etween
BHLUMI on the one side and BHAGEN OLDBIS LUMLOG on the other side is not
always b elow for all the ES s considered it is at least for the exp erimentally most interest
ing NW case This fact is a further reinforcement of the present theoretical error estimate for
the SABH pro cess in the NW case and it is a suggestion for the exp erimentalists to continue
to choose the NW ES s The fact that for the unrealistic ES BARE the di erence b etween
BHLUMI and NLLBHA see g is also within con rms this evaluation Entry b is
based on table gure In entry c the new improved uncertainty of the vacuum p olariza
tion is taken from Tab We take the biggest of the results from refs The light pair
pro duction uncertainty entry c is based on new estimates rep orted during the workshop see
Ref and Ref see also table gure In tab we quote for LEP the
present error due to light fermion pairs contribution to b e This is based on all the refer
ences quoted ab ove and on the discussion during the WG meetings The previous estimate
in Ref is therefore con rmed and improved slightly This is under the assumption that the
pair e ect is corrected for at least in the LL approximation If the e ect is not corrected for
then we recommend to use for LEP as an estimate for the missing pair e ect
for LEP The material presented at the workshop suggests that the nal uncertainty of the
light pair contribution will b e at the level of In entry e the reduced uncertainty of
the Z exchange contribution is based on Ref work done during this Workshop
The improvement of the theoretical luminosity error from down to is basically
7
Pro duction of the light pairs is not included in the standard version of BHLUMI It is implemented only in
the testing unpublished version
due to successful comparisons of the programs BHLUMI and SABSPV for a wide range WW
NN and NW of exp erimentally realistic ES s SICAL and also due to an encouraging al
though limited to the unrealistic ES BARE comparison of un exp onentiated BHLUMI and
NLLBHA in table gure Furthermore the agreement of BHLUMI SABSPV BHAGEN
and OLDBIS LUMLOG within that same error in the NW ES recommends safely this
choice in the exp erimentally relevant cases At last the analysis describ ed in subsection
shows that the actual Bhabha selections used by the LEP exp eriments to measure the acceler
ator luminosity minimize the sensitivity to O radiative corrections thus putting the ab ove
conclusions on an even rmer ground We would like to stress very strongly that the ab ove
new estimate of the total luminosity error is based on new results which although pretty
stable numerically are generally still quite fresh and they are unpublished We exp ect these new
results to b e published in journals shortly after the workshop together with the corresp onding
computer programs
The total theoretical error for the SABH pro cess at LEP is also summarized in Tab
We assume that the cross section is calculated for any typical asymmetric ES for LEP
exp eriment in the typical angular range or The error estimate covers also
the worst case scenario of the sup er narrow angular range see the example of ES CALO
in table gure In entry a the estimate of the total photonic uncertainty is based again
up on the agreement b etween BHLUMI and SABSPV on all the variants of ES s considered
and reinforced by the fact that BHAGEN OLDBIS LUMLOG are on the same ground as
BHLUMI and SABSPV in the exp erimentally more interesting NW case see tables gures
and Note that sometimes in the case of other angular range and higher energies
the scaling laws from the introduction were used instead of direct calculation to extend
the actual numerical results to these situations see the comments accompanying the relevant
tables plots We do not see much danger in this b ecause usually the large safety margin close
to a factor of two was present Entry b is pro duced out of LEP result using the scaling
rule The vacuum p olarization for LEP case in the Tab is taken from Tab at the
jtj GeV corresp onding to LEP energy and the angle of mrad
min
Type of correction error Error estimate
a Missing O L O L
b Technical precision photonic
c Vacuum p olarization
d Light fermion pairs
e Z contribution
Total
Table Future projection of the total physical technical theoretical uncertainty for a typical calori
metric detector within the angular range at LEP energies
Finally in view of all the work reviewed during the workshop we are also able to estimate
the precision which will b e attained in the next step It is shown schematically in table
At the time when Monte Carlo programs will include the matrix element from O L the
uncertainty due to higher order corrections will b e negligible The dominant contribution will
b e of technical origin and we think that as we have seen from O comparisons it can
b e reduced to provided we can successfully tune two indep endent Monte Carlo event
generators at that precision level for the same or very similar O matrix elements The
vacuum p olarization is now taken according to Ref and from the discussions during the
workshop meetings it was obvious that a further reduction of the uncertainty due to pairs and
Z exchange is also p ossible The corresp onding work is in progress
Large angle Bhabha scattering
In the present section the LABH pro cess is considered b oth at LEP and LEP The aim of
the study rather than up dating the conclusions of Ref concerning the theoretical accuracy
of the LABH pro cess at LEP is twofold on the one hand the comparison b etween the semi
analytical b enchmarks and the Monte Carlo co des used by the LEP collab oratio ns on the other
one the study of the LABH pro cess at LEP accompanied by the development of dedicated
software
Physics
The main physics interest of Bhabha scattering measurements at large angles say
around the Z resonance is a precise test of the electroweak sector of the Standard Mo del In this
angular region more than of the cross section is due to resonant s channel Z exchange For
p
s M the interference contributions b etween s channel Z exchange and the other diagrams
Z
either vanish or are completely irrelevant and the s channel photon exchange contribution is
small of the Z exchange cross section The only other relevant contribution is
t channel photon exchange For electroweak analyzes one thus subtracts the t channel and
s t interference contributions from the large angle exp erimental data typically calculated
using the ALIBABA semi analytical SA program After correcting for the e ects of real
and virtual photon radiation using the analytical programs MIBA TOPAZ or
p
peak
may b e extracted For ZFITTER the Z exchange cross section s M
Z
Z Z Z
where
peak
e
Z
M
Z Z
For the other charged lepton pair decay mo des of the Z the quantity in Eqn
e
is replaced by resp ectively while for hadronic q q decays it is replaced by
e e e had
Thus the electronic width of the Z which app ears in the cross section for all decay mo des of
e
peak
only the Z is measured directly and with improved sensitivity b ecause in this case
Z e
in large angle Bhabha scattering It is worth noting however that in principle the so called
t channel subtraction is not unavoidable Actually the program TOPAZ could b e
used to t directly the data for large angle Bhabha scattering without relying up on t channel
subtracted data
The resulting sensitivity of the backward forward charge asymmetry in large angle Bhabha
top H I GGS
scattering to the imp ortant electroweak parameters and
p
G c
W
top
m
t
s
W
p
M G M
H
W
H I GGS
ln
M
W
is similar to that of the other dilepton channels The ab ove formulae of course
indicate only the leading dep endence of the one lo op corrections on the masses of the top quark
and the Higgs b oson Actually at the nowadays precision level a complete electroweak library
is mandatory
At the Z p eak the purely QED corrections to the large angle Bhabha cross section are for
typical exp erimental cuts O O These corrections are much larger
than those in small angle Bhabha scattering when typical wide narrow cuts are used
O O Thus theoretical errors on QED radiatively corrected cross sections
are exp ected to b e considerably larger in large angle than in small angle Bhabha scattering
This is indeed found to b e the case in the comparisons b etween di erent co des shown b elow
In the energy regime of LEP the Z b oson e ects on the large angle Bhabha cross section
are much smaller than at LEP Actually b efore entering the details of the comparisons
it is worth noting that large angle Bhabha scattering shows very di erent physical features
dep ending on the energy regime at which it is considered As can b e seen from Fig around
the Z p eak the cross section is largely dominated by Z b oson annihilatio n whereas already
some GeV o resonance the cross section is largely dominated by t channel photon exchange
From this p oint of view large angle Bhabha scattering at LEP is much more similar to small
angle Bhabha scattering than to large angle Bhabha scattering at LEP Hence at LEP the
large angle Bhabha cross section cannot b e a useful to ol for precise tests of the electroweak
sector of the Standard Mo del but rather for general QED tests
The state of the art of large angle Bhabha scattering up to now can b e found in Ref In
that pap er an extensive comparison b etween two semi analytical co des namely ALIBABA
and TOPAZ is shown On the other hand although extensive in the sense that cross
sections and asymmetries are considered that comparison is in some sense limited actually it
involves only semi analytical co des on very simple academic ES s only at the Z p eak
In view of the ab ove considerations the tasks of the present Working Group as far as
large angle Bhabha scattering is concerned are the following ones
involving in the comparisons also the Monte Carlo MC co des to day available and used
by the LEP collab orations
8
See for a discussion of pseudo observables for precision calculations at the Z p eak 1.5 γ γ 1.25 t t 1
0.75
0.5 ZsZs 0.25 0 -0.25 γ γ s t -0.5 Ζ γ s t -0.75 80 100 120 140 160 180 200
ECMS (GeV)
Figure The relative contributions to the integrated cross section at the Born level The individual
contributions are from top to b ottom on the right hand side of the plot t t Z t Z t s s
Z s Z s Z s s Z s Z t s Z t Z s t t Z t and s t
considering also more realistic alb eit simple ES s
providing results also for the LEP energy range eventually developing dedicated soft
ware
The ES s considered in the present study are the following ones
BARE This ES for the sake of simplicity is de ned exactly as in namely
max
and E GeV for b oth electron and
min
acoll
p ositron
CALO This ES is de ned as ab ove but with E GeV for the nal fermion energy
min
which is the electron p ositron energy if there are no photons nearby whereas it is the
electron p ositron plus photon energy if the photon is within a cone of semi ap erture
from the electron p ositron
For all the cases considered the input parameters are M GeV m GeV
Z t
m GeV and M The predictions by ALIBABA are taken from Ref
H s Z
Let us now brie y summarize the features of the co des involved in the study Here only
the general features will b e highlighted for more details the reader is referred to the original
literature or to the write ups presented at the end of this section
ALIBABA It is a semi analytical co de implementing exact O QED and weak cor
rections The higher order QED corrections consist of leading log O corrections plus soft
photon exp onentiation Moreover the weak O corrections are folded with the leading log
structure functions The matching b etween the exact O QED matrix element and the
higher order corrections is p erformed in additive form The electroweak library is not up to
date Nonetheless the co de has to b e considered as a b enchmark
BHAGEN It is a Monte Carlo integrator for b oth small and large angle Bhabha scat
tering The value for the cross section is obtained from the event generator BHAGEN
a structure function based program for all orders resummation including complete photonic
O and leading logarithmic O L corrections in all channels and all relevant electroweak
corrections according to BHM WOH basic formulae from Ref The approximations intro
duced with the collinear kinematics of initial and nal radiation and in its angular distribution
are eliminated for the one hard photon emission by substitution with the exact calculation
BHAGENE It is a Monte Carlo event generator for large angle Bhabha scattering
and muon pair pro duction The program includes one lo op and the most imp ortant two
lo op electroweak as well as QED radiative corrections The O QED correction uses the
exact matrix element Higher order QED corrections are included in an improved soft photon
approximation with exp onentiation of initial state radiation Up to three hard nal state
photons are generated Events are generated in the full nal state phase space including explicit
mass e ects in the region of collinear mass singularities The minimum scattering angle for
p ercent level cross section accuracy is Extensive use is made in the program of one and
two dimensional lo ok up tables for fast exible and e cient Monte Carlo generation The
program was designed for the Z p eak region but may also b e used at LEP energies
BHWIDE It is a new Monte Carlo event generator for large angle Bhabha scattering
at LEP SLC and LEP It includes multiphoton radiation in the framework of O YFS
exp onentiation The O virtual b oth weak and QED corrections are in the current version
taken from ALIBABA The program provides the full event in terms of particle avors and
their four momenta with an arbitrary number of radiative photons In many asp ects it is
similar to the program BHLUMI for small angle Bhabha scattering and can b e considered as
its extension to large angles It has b een checked that for the pure QED pro cess BHWIDE at
O no exp onentiation agrees with the MC program OLDBIS within a statistical accuracy
of
SABSPV It is a new Monte Carlo integrator originall y designed for small angle Bhabha
scattering but adapted to the treatment of large angle Bhabha scattering at the LEP energy
range It is based on a prop er matching of the O corrected cross section for t channel
photon exchange and of the leading logarithmic results in the structure function approach
The matching is p erformed in a factorized form in order to preserve the classical limit At
present the e ect of up down interference in the t t contribution is not taken into
account and all the other contributions are corrected at the leading logarithmic level Due to
the present approximations the theoretical accuracy of the co de is of the order of as far
as large angle Bhabha scattering at LEP is concerned
TOPAZ It is a semi analytical co de developed for precision physics at LEP It
includes the state of the art concerning weak and QCD corrections according to Ref As
far as QED corrections are concerned they are exactly treated at O for s channel pro cesses
leptonic and hadronic at the leading logarithmic level for pure t channel and s t contributions
in the Bhabha scattering case On top of this higher order QED corrections are taken into
account in the structure functions approach in a factorized form in order to preserve the clas
sical limit A particular e ort has b een p erformed in order to implement as much analytically
as p ossible the exp erimental cuts typically applied by the LEP collab orati ons
UNIBAB It is a full Monte Carlo event generator that was originally designed for large
angle Bhabha scattering at LEP and SLC energies The QED corrections are implemented in a
fully factorized form by assuming s channel dominance and using photon shower algorithms for
initial and nal state radiation and therefore exp onentiation of soft photons and resummation
of the logarithms from multiple emission of hard collinear photons is automatic QED initial
nal interference corrections are not yet implemented The electroweak corrections are based
dep endence and on a library also used by ALIBABA but up dated to include the leading m
t
higher order QCD corrections to the Z width
On Z p eak LEP
The situation of the comparisons for LEP is summarized in Figs BARE and CALO
and corresp onding tables Conventionally the reference cross section with resp ect to which the
relative deviations are computed is taken from TOPAZ It has to b e stressed that this choice
has no particular meaning at all
Let us b egin with commenting the situation of Fig i e for the BARE ES As far as
the comparison b etween the two semi analytical co des ALIBABA and TOPAZ is concerned
the agreement is b etter than at the Z p eak energy p oints n and corresp onding to
the smallest exp erimental error which is of the order of statistical and systematic
and deteriorates on the wings where on the other hand the exp erimental error is larger for
instance at p eak GeV the exp erimental error is of the order of statistical and
systematic Note that the worst situation is for maximum acollinearity cut of ab ove the
Z p eak where the co des di er from one another of ab out this di erence is due to higher
order QED e ects as p ointed out in Ref factorized versus additive formulation As far
as the Monte Carlo co des BHAGENE and BHWIDE are concerned their agreement with the
semi analytical co des at p eak is within few p er mil whereas o p eak BHWIDE is within
and BHAGENE can deviate up to
The situation for the more realistic case the CALO ES Fig is generally b etter from
the p oint of view of the SA MC comparisons Note that ALIBABA is no more involved since
No E TOPAZ BHWIDE BHAGENE ALIBABA BHAGEN
CM
o
a BARE acol
max
0
b BARE acol
max
o o
BARE acol BARE acol
max max
BHWIDE BHWIDE
? BHAGENE ? BHAGENE
:02 :02
REF REF
ALIBABA ALIBABA
REF REF
2 2
BHAGEN BHAGEN
?
TOPAZ TOPAZ
REF REF
:01 :01
? ?
?
2
?
?
?
:00 :00
?
2
2
2
2
2
?
2
2
:01 :01
2
2
2
2
2 2
?
2
2
?
?
?
:02 :02
No of Energy p oint No of Energy p oint
?
?
1: 2: 3: 4: 5: 6: 7: 8: 1: 2: 3: 4: 5: 6: 7: 8:
o o
Figure Monte Carlo results for the BARE ES for two values and of acollinearity cut Center
of mass energies in GeV close to Z p eak In the plots the cross section from TOPAZ is used as
REF
a reference cross section Cross sections in nb
it cannot manage calorimetric measurements whereas UNIBAB app ears it is slow for very
small minimum fermion energy and therefore it did not contribute to the BARE case On
p eak the agreement b etween the co des is at the few p er mil level o p eak BHWIDE is within
No E TOPAZ BHWIDE BHAGENE UNIBAB BHAGEN
CM
o
a CALO acol
max
o
b CALO acol
max
o o
CALO acol CALO acol
max max
BHWIDE BHWIDE
? BHAGENE ? BHAGENE
:02 :02
REF REF
UNIBAB UNIBAB
REF REF
?
2 2
?
BHAGEN BHAGEN
?
TOPAZ TOPAZ
REF REF
:01 :01
?
?
?
?
?
2
?
:00 :00
2
2
2 2
?
2
2
?
:01 :01
2
2
2
?
2
2
?2
?
2
2
?
2
?
:02 :02
No of Energy p oint No of Energy p oint
1: 2: 3: 4: 5: 6: 7: 8: 1: 2: 3: 4: 5: 6: 7: 8:
o o
Figure Monte Carlo results for the CALO ES for two values and of acollinearity cut Center
of mass energies in GeV close to Z p eak In the plots the cross section from TOPAZ is used as
REF
a reference cross section Cross sections in nb
from TOPAZ whereas UNIBAB deviates up to b elow p eak and BHAGENE can
di er from TOPAZ by ab out
BHAGEN is within everywhere for b oth the BARE and CALO ES s around the
Z p eak The agreement is b etter few GeV ab ove and b elow the Z resonance However the
implementation of the complete weak and QCD library is very recent and still under tests
Far o Z p eak LEP
No BHWIDE TOPAZ BHAGENE UNIBAB SABSPV BHAGEN
o
a CALO acol
max
o
b CALO acol
max
o o
CALO acol CALO acol
max max
:10 :10
TOPAZ TOPAZ
? BHAGENE ? BHAGENE
REF REF
UNIBAB UNIBAB
REF REF
2 2
BHAGEN BHAGEN
SABSPV SABSPV
:05 :05
BHWIDE BHWIDE
REF REF
2
2
2
2
2
2
:00 :00
?
?
?
?
?
?
:05 :05
No of Energy p oint No of Energy p oint
:10 :10
1: 2: 3: 1: 2: 3:
o o
Figure Monte Carlo results for the CALO ES for two values and of acollinearity cut Center
of mass energies close to W pair production threshold E GeV GeV GeV In
CM
the plots the cross section from BHWIDE is used as a reference cross section Cross sections in pb
REF
The situation of the comparisons for LEP is shown in Fig CALO and corresp onding table
Conventionally the reference cross section with resp ect to which the relative deviations are
computed is taken from BHWIDE It has to b e stressed again that this choice has no particular
meaning at all Note that TOPAZ has b een developed in the Z dominance approximation and
UNIBAB do es not include initial nal interference e ects so that their results are at the leading
logarithmic level in the LEP energy range A new entry app ears namely SABSPV which has
b een conceived for small angle Bhabha scattering and further improved for large angle Bhabha
at LEP
BHAGEN BHWIDE and SABSPV stay within from one another More precisely
SABSPV is steadily around ab ove BHWIDE and b elow BHAGEN BHAGENE
for b oth the acollinearity cuts considered can deviate from the reference cross section up to
TOPAZ and UNIBAB show deviations from the reference cross section up to ab ove and
b elow resp ectively which dep end on the acollinearity cut and can b e presumably traced
back to the approximations intrinsic in these Z p eak designed co des Anyway the deviations of
the two co des from the reference cross section are consistent with what can b e exp ected from
leading logarithmic results
Short write up s of the programs
The aim of the following short write up s is to provide quick reference for the reader on basic
prop erties of all event generators used in the numerical comparisons throughout this article The
intention was that details are given only on new and or unpublished features of the programs
including bugs while other features are describ ed in general terms with help of references to
published works
BHAGEN
AUTHORS
M Ca o INFN and Dipartimento di Fisica dell Universit a Bologna Italy
caffo bo infn it
H Czy z University of Silesia Katowice Poland INFN Universit a Bologna Italy
czyz bo infn it
E Remiddi INFN and Dipartimento di Fisica dell Universit a Bologna Italy
remiddi bo infn it
GENERAL DESCRIPTION
BHAGEN is a collection of three programs to calculate the cross section for Bhabha scattering
for small and large scattering angles at LEP and LEP energies In its present form the
integrated cross section for a given selection of cuts is calculated as
H H
BHAGEN BH FO BHAGEN PH
BHAGEN is the integrated cross section obtained with the Monte Carlo event generator
BHAGEN a structure function based program for all orders resummation includ
ing complete photonic O and leading logarithmic O L corrections in all channels and
all relevant electroweak corrections according to BHM WOH basic formulae from Approx
imations are introduced with the collinear kinematics of initial and nal radiation and in its
angular distribution
H
BH FO is the integrated cross section of O for one hard photon emission obtained
with the Monte Carlo event generator BH FO the O expansion of BHAGEN
H
BHAGEN PH is the integrated cross section obtained with the one hard photon com
plete matrix element and exact kinematics implemented in the Monte Carlo event generator
BHAGEN PH
H H
The subtraction of BH FO and its substitution with BHAGEN PH is to eliminate
the error in the contribution coming from the one hard photon emission
FEATURES OF THE PROGRAM
The three programs provide cross sections which are summed as in Eq or used to obtain
other quantities such as forward backward asymmetry Due to the mentioned substitution
pro cedure the event generator feature of the constituent programs can not b e pro ted and the
use is simply that of a Monte Carlo integrator
At small angle we estimate the accuracy in the cross section evaluation which comes from the
uncontrolled higher orders terms O L and O L and from the incertitude in O L
s t interference to amount comprehensively to a The error due to approximate two
hard photon contribution strongly dep endent on the imp osed cuts is estimated on the basis
of the correction required for the one hard photon contribution times s to account
for the increase in p erturbative order All included we estimate at small angle an accuracy of
the order of for lo ose cuts z and of for sharp cuts z for b oth
min min
LEP and LEP energies
At large angle we estimate the O L s t interference accuracy up to dep ending on
cuts at LEP energy but much smaller at LEP The error coming from the approximate
treatment of two hard photon emission is estimated as ab ove and is smaller for more stringent
acollinearity cut All included we estimate an accuracy of the order of for b oth LEP and
LEP energies
HOW DOES THE CODE WORK
The three programs run separately They provide initializatio n and ducial volume de nition
according to input parameters then starts the generation of events according to some variables
which smo oth the cross section b ehavior Rejection is p erformed through the routine TRIGGER
where the sp ecial cuts can b e implemented The programs stop when the requested number of
accepted events is reached or alternatively when the requested accuracy is obtained
INPUT CARD
The following data have to b e provided in input mass of the Z mass of the top quark
mass of the Higgs value of M value of the b eam energy E the minimum energy
S Z beam
Z
for leptons E larger than GeV minimum and maximum angle for the scattered electron
min
p ositron with the initial electron p ositron direction maximum acollinearity allowed b etween
nal electron and p ositron number of accepted events to b e pro duced numbers to initialize the
random number generator One may also switch on or o pairs pro duction the channels to
b e considered and the recording of the events For O programs one has also to sp ecify the
minimum and maximum energy allowed for the photon For the input of BHAGEN PH one
has to give also the maximum acoplanarity and minimum angles of the emitted photon with
initial and nal fermion directions if one wants to exclude the contributions with the collinear
photons
DESCRIPTION OF THE OUTPUT
Each program return the input parameters and the values of the cross section obtained with
weighted and unweighted events with the relative statistical variance one standard deviation
Of course due to the e ciency the weighted cross section is usually much more precise than
the unweighted one The total integrated cross section is then calculated according to Eq
AVAILABILITY
On request to the authors and to b e p osted on WWW at http www bo infn it
BHAGENE
AUTHORS
J Field D epartement de Physique Nucl et Corpusculaire Univ de Gen eve
jfield cernvm cern ch
T Riemann DESY Platanenallee D Zeuthen
riemann ifh de
GENERAL DESCRIPTION
BHAGENE is a Monte Carlo event generator for muon pairs at all angles or for Bhabha
scattering in the large angle region The program which is intended for use at
or ab ove the Z p eak region contains all tree level diagrams with complete one lo op and the
leading two lo op virtual corrections The running is included with the correct scale in all
amplitudes The O QED correction uses the exact l l matrix elements Higher order
QED corrections are included in an improved soft photon approximation with exp onentiation
of initial state radiation Events with up to three hard photons are generated in the full
kinematically allowed phase space including explicit mass e ects for near collinear photon
I F
radiation If n n are the resp ective numbers of initial and nal state photons the di erent
I F
nal state top ologies generated are n n Initial Final
state interference e ects are taken into account only to O The photon energies are describ ed
p
i
by scaled variables y E s For y y typically y a Born top ology
i i
event is generated The corresp onding cross section contains all virtual V corrections and is
integrated over the phase space of all soft S photons with y y Exp onentiation of initial
i
state radiation is implemented by mo difying the O partial cross sections and interference
terms in such a way that the derivative of the VS cross section with resp ect to y is recovered in
the y limit For example in the s channel photon exchange contribution with initial state
radiation
0 0 0
d s s u t u t
s s init
ln
0
d dy s y s m s
l
e
0 0 0 0
exp onentiation is carried out by the replacement u t u t f u t u t
i
e
where f C y Events with hard photons are generated according to distributions
V
where the soft photon eikonal factors are corrected by the Grib ov Lipatov kernels The
relative probabilities of di erent top ologies of nal state photons are chosen according to the
Poisson distribution P njN where n n and
I F
N ln y N ln y
e f
A short description of program together with comparisons with other muon pair and wide angle
Bhabha co des has b een published A long write up is also available
OIMZ Z mass GeV
OIMT Top quark mass GeV
OIMH Higgs b oson mass GeV
OMAS M
s Z
IOCH e e
IOEXP exp onentiated O
OW collision energy GeV
in the ODLR frame OCTC lower cos
l
in the ODLR frame OCTC lower cos
l
IOXI initial random number
Table Variables of the lab elled common ICOM OCTC OCTC are used in setting up the LUT of the
lepton scattering angles To allow for the e ects of the Lorentz b o ost the angular range should b e chosen
somewhat wider than that de ned by the cuts in the LAB system
FEATURES OF THE PROGRAM
The execution of the program has three distinct phases initialisati on generation and termina
tion In the initialisa tio n phase all relevant electroweak quantities are calculated from the input
parameters M M M and Also a number of lo ok up tables for quantities such as the
Z t H s
lepton scattering angle and photon energies are created for use in the subsequent generation
phase This pro cess is relatively time consuming so the user should not b e surprised if there
is some delay b etween the execution of the program and the start of event generation In the
generation phase events with unit weight are generated by the weight throwing technique The
corresp onding vectors are stored in common C VEC The user may apply arbitary cuts and
pro duce weighted histograms in subroutine FUSER Histograms of unit weight events may b e
pro duced in subroutine FHIST In the nal termination phase the input parameters are printed
CUT
out together with the exact cross section and its error together with all histograms and
plots
HOW TO USE THE PROGRAM
The program has a very short main program containing de nitions of the most imp ortant
NPAR weak lo op corrections ON OFF
NPAR parameterisations of had vac p ol
NPAR two loop m correction
s
t
NPAR weak b ox diagrams ON OFF
NPAR two loop terms m ON OFF
t
XPAR initial lepton charge D
XPAR nal lepton charge D
XPAR nal lepton colour D
XPAR nal lepton mass GeV
Z
XPAR QCD correction to non b quarks
q
Z
XPAR QCD correction to
b
P
YMA maximum value of E E D
beam
YMI minimum value of E E D
beam
WTMAX maximum value of the event weight D
Table Control parameters de ned in SUBROUTINE BHAGENE Default values are underlined or
given in parentheses
input parameters which are stored in the lab elled common blo ck ICOM These variables are
describ ed in Table The execution of the program has three distinct phases i Initialisati on
ii Generation of a single unit weight event iii Termination Each of these phases is entered
via a call to subroutine BHAGENE in the main program
CALL BHAGENE MODE CTP CTP CTM CTM CTAC EP EM
MODE is set to for the initialisa tion generation and termination phases resp ectively
The other parameters of BHAGENE de ne the kinematical cuts to b e applied to the generated
events
CTP minimum value of cos
l
CTP maximum value of cos
l
CTM minimum value of cos
l
CTM maximum value of cos
l
CTAC maximum value of cos
col
EP minimum energy of l GeV
EM minimum energy of l GeV
All these cuts are applied in the lab oratory incoming e e centre of mass system The
angle is the collinearity angle b etween the l and the l CATC for a back to back
col
con guration In the calls of BHAGENE with MODE only this parameter need b e
sp eci ed Other initialisa tion parameters of interest to users are de ned in BHAGENE itself
A list of the most imp ortant of these can b e found in Table
AVAILABILITY
From Compure Physics Communications Program Library see http www cs qub ac uk cpc
for more details
BHLUMI
AUTHORS
S Jadach Institute of Nuclear Physics Krak ow ul Kawiory a
jadach cernvm cern ch
E Richter W as Institute of Computer Science Jagellonian University Krak ow
erichter cernvm cern ch
B F L Ward Department of Physics and Astron University of Tennessee and SLAC
bflw slac stanford edu
Z W as CERN and Institute of Nuclear Physics Krak ow ul Kawiory a
wasm cernvm cern ch
GENERAL DESCRIPTION
The program is a multiphoton Monte Carlo event generator for low angle Bhabha providing four
momenta of outgoing electron p ositron and photons The rst O version was describ ed
YFS
in ref The actual version a includes several types of the matrix elements The most
pr ag
matrix elements M E is based on the Yennie Frautschi Suura YFS imp ortant O
YFS
exp onentiation This M E includes exactly the photonic rst order and second order leading
pr ag
M E the other higher order and subleading contributions log corrections In the O
YFS
are included in the approximate form The detailed description exists for the version in
ref For the di erences b etween the versions and the user has to consult ref
the README le in the distribution directory and comments in the main program of the
demonstration deck The only di erence b etween versions and a is correction to
an imp ortant bug a In order to correct it one has to replace v in eq in Ref with
v ln ln ln
p q
We also provide patch to correct this in the sorce co de of the versions see AVAILABILITY
b elow This correction can a ect the result of the program typicaly up to for
some event selections
FEATURES OF THE PROGRAM
BHLUMI consists in fact of the three separate event generators BHLUM OLDBIS and
LUMLOG where OLDBIS is an improved version of the OLDBAB written by Berends and
Kleiss at PETRA times and LUMLOG is an event generator with the inclusive many photons
emission strictly collinear to momenta of incomming outcogoing fermions M E of OLDBIS is
limited to O and M E of LUMLOG includes exp onentiated and non exp onentiated electron
structure functions up to O BHLUM includes four types of the exp onentited M E
LL
YFS YFS
O O and four types of the non exp onentited M E O O where
A B A B
A B A B
the cases A and B corresp ond to two kinds of ansatz employed for mo deling the O L second
order NNL contribution The BHLUM program includes vacuum p olarization s chanel and
Z exchange contributions see ref in the approximation suitable for the low angle b elow
rad scattering The BHLUM do es no include so called up down interferences However
OLDBIS do es include them so it can b e used to check how small they are
HOW DOES THE CODE WORK
The program ia a full scale Monte Carlo event generator A single CALL BHLUMI pro duces one
event i e the list of the nal state four momenta of electron p ositron and photons enco ded
in the common blo ck Dep ending on switch in the input parameters the program provides
event with the variable weight WTMOD or with constant weight WTMOD In the constant
YFS
weight mo de the calculation is done for M E of the O type In the variable weight mo de
B
WTMOD corresp onds to the ab ove M E but the user has also acces to all six types of the M E
listed ab ove and even more and may p erform in a single run calculation for various types of
the M E The choice of one of three sub generators BHLUM OLDBIS or LUMLOG is decided
through one of the input parameters Program requires initializatio n b efore pro ducing rst
MC event There are many input parameters The most imp ortant ones de ne the minimum
and maximum angle t chanel transfer For weighted events it is p ossible to cover the angular
range down to zero angle but the program is realy designed for double tag acceptance It is
p ossible to stop and restart the program from the next event in the series The distribution
directory incudes example demonstrating how to do it
DESCRIPTION OF THE OUTPUT
Program prints certain control output The basic output of the program is the series of the
Monte Carlo events and the user decides by himself which events are accepted or rejected ac
cording to his favourite selection criteria The total cross section in nanobarns can b e calculated
for arbitrary cuts in a standard way
X
W
I
N
Accepted E v ents
where the sum of the weights variable or constant is over all accepted events N is total
number of generated events and is a reference normalization cross section in nanobarns
provided by the program at the end of the MC generation In the analogous standard way one
may obtain any arbitrary distribution prop erly normalized
AVAILABILITY
The program is p osted on WWW at http hpjmiady ifj edu pl in the form of tar gz le
together with all relevant pap ers and do cumentation in p ostscript The version a which was
used to pro duce all numerical results in this workshop consists of the version describ ed in
ref and of the error patch p osted in the same lo cation http hpjmiady ifj edu pl After
workshop the equivalent version will b e released The new version of BHLUMI will also
contain new version of LUMLOG with the nal state bremsstrahlung which was used in in the
table gure and improved version of the BHLUMI matrix element without exp onentiation
which was used in this table gure
BHWIDE
AUTHORS
S Jadach Institute of Nuclear Physics Krak ow ul Kawiory a
jadach cernvm cern ch
W P laczek Dept of Physics and Astron Univ of Tennessee
placzek hephp phys utk edu
B F L Ward Dept of Physics and Astron Univ of Tennessee and SLAC
bflw slac stanford edu
GENERAL DESCRIPTION
The program evaluates the large wide angle Bhabha cross section at LEP SLC and LEP
energies The theoretical formulation is based on O YFS exp onentiation with O virtual
b oth weak and QED corrections taken from ref as formulated in the program AL
IBABA The YFS exp onentiation is realized via Monte Carlo metho ds based on BHLUMI type
Monte Carlo algorithm which is explained in refs Thus we achieved an event by event
realization of our calculation in which arbitrary detector cuts are p ossible and in which infrared
singularities are canceled to all orders in A detailed description of our work can b e found in
ref
FEATURES OF THE PROGRAM
The co de is a full edged Monte Carlo event generator so that the nal particle four momenta
for the entire e e n nal state are available for each event which may b e generated as
a weighted or unweighted event as the user nds more or less convenient accordingly Thus
it is trivial to imp ose arbitrary detector cuts on the events If the user wishes he she may
also use the original BABAMC type of pure weak corrections there is a simple switch
which accomplishes this The exp ected accuracy of the program when all tests are nished
is anticipated at in the Z region and in the LEP regime
HOW DOES THE CODE WORK
The co de works entirely analogous to the MC event generator BHLUMI describ ed in
ref A crude distribution consisting of the primitive Born level distribution and the most
dominant part of the YFS form factors which can b e integrated analytically is used to gener
ate a background p opulation of events The weight for these events is then computed by the
standard rejection techniques involving the ratio of the complete distribution and the crude dis
tribution As the user wishes these weights may either b e used directly with the events which
have the four momenta of all nal state particles available or they may b e accepted rejected
against a constant maximal weight WTMAX to pro duce unweighted events via again standard
MC metho ds Standard nal statistics of the run are provided such as statistical error analysis
total cross section etc
DESCRIPTION OF THE OUTPUT
Program prints certain control output The basic output of the program is the series of the
Monte Carlo events The total cross section in nanobarns can b e calculated for arbitrary cuts
in the same standard way as for BHLUMI i e user may imp osed arbitrary exp erimental cuts
by rejection
AVAILABILITY
The program can b e obtained via e mail from the authors It will b e p osted so on on WWW at
http enigma phys utk edu as well as on anonymous ftp at enigma phys utk edu in the
form of tar gz le together with all relevant pap ers and do cumentation in p ostscript It will
also b e available via anonymous ftp at enigma phys utk edu in the directory pub BHWIDE
NLLBHA
AUTHORS
A B Arbuzov Joint Institute for Nuclear Research Dubna Russia
arbuzov thsun jinr dubna su
E A Kuraev Joint Institute for Nuclear Research Dubna Russia
kuraev theor jinrc dubna su
L Trentadue CERN TH Division Universit a di Parma INFN Sezione di Milano
trenta vxcern cern ch
GENERAL DESCRIPTION
NLLBHA is a semi analytical program for calculations of radiative QED and electroweak correc
tions to the small angl e Bhabha scattering at high energies It takes into account complete
relevant at small angles rst order QED and electroweak corrections the leading and next
to leading QED corrections to O and the leading logarithmic contributions to O The
corrections due to photon emission as well as the ones due to pair pro duction are included The
theoretical uncertainty of the calculations is less then
FEATURES OF THE PROGRAM
NLLBHA integrates numerically analytical formulae For the Born cross section an ex
pansion for small scattering angles is used The contributions due to real particle emission
are integrated over symmetrical detector ap ertures For the case of asymmetrical detectors
narrow wide case leading logarithmic contributions are calculated next to leading are es
timated to b e equal or less the ones in the narrow narrow case Cuts on the nal particles
energies are p ossible Calorimetric set up as well as other sp ecial exp erimental conditions are
not implemented
HOW DOES THE CODE WORK
The co de consists of the main part and of a series of subroutines which calculate separately
radiative correction RC contributions from di erent Feynman diagrams and con gurations
In the main part the ags the parameters and the constants are de ned Using the ags one
can de ne with their help the event selection BARE symmetric or asymmetric are p ossible
only the order of corrections switch on or o di erent contributions like Z b oson exchange
vacuum p olarizatio n and light pair pro duction Then the user have to set the parameters the
b eam energy the angular range the energy cut The electroweak parameters are calculated
with the help of the DIZET package
DESCRIPTION OF THE OUTPUT
At rdt the co de prints the information ab out the chosen set up vacuum p olarization on o
Z b oson contribution on o Then the co de prints the b eam energy the angular range and
the electroweak parameters After calculations it prints for each value of x energy cut the
c
Born and the radiatively corrected to di erent orders and approximations cross sections in
nb It also prints a line with the values of the di erent corrections in p ercent with resp ect to
the Born cross section The normalizations and de nitions used do directly corresp ond to the
ones given in where also the origin of all RC contributions can b e found
AVAILABILITY
The co de is available up on request from the authors
SABSPV
AUTHORS
M Cacciari DESY Hamburg Germany
cacciari desy de
G Montagna University of Pavia Italy
montagna pavia pv infn it
O Nicrosini CERN TH Division Permanent address INFN Pavia Italy
nicrosini vxcern cern ch nicrosini pavia pv infn it
F Piccinini INFN Pavia Italy
piccinini pavia pv infn it
GENERAL DESCRIPTION
SABSPV evaluates small angle Bhabha cross sections in the angular region used for lumi
nosity measurement at LEP and large angle Bhabha cross sections at LEP The theoretical
formulation is based on a suitable matching b etween an exact xed order calculation and the
resummation of leading log radiative e ects provided by the structure function techniques
1
The matching b etween the all orders leading log cross section given by the convolution
LL
of structure functions with kernel Born cross sections and the O one is realized according
to the following general recip e the order content of the leading log cross section is extracted
by employing the O expansions of the structure functions thereby yielding Denoting
LL
S V
by k the cross section including virtual corrections plus soft photons of energy up to
H
E k E and by k the radiative O cross section the fully corrected cross section can
nally b e written as
1
S V H
k k
A
LL LL
Equation is in the additive form A factorized form can also b e supplied It has the same
O content but also leads to the so called classical limit according to which the cross section
must vanish in the absence of photonic radiation It reads
S V H
k k
1
NL LL
H H
C C
F
NL LL NL
H
b eing the Born cross section C contains the non log part of the O cross section rep
NL
resented by
NL
In order to b e exible with resp ect to the di erent kinds of exp erimental cuts and triggering
conditions it makes use of a multi dimensional Monte Carlo integration with imp ortance sam
pling A detailed description of the formalism adopted and the physical ideas b ehind it can b e
found refs and references therein
FEATURES OF THE PROGRAM
The co de is a Monte Carlo integrator for weighted events At every step two kinds of events
are fully accessible
A two bo dy events they include tree level events and radiative events in the collinear
approximation in this last case information concerning the equivalent photons lost in the
b eam pip e is available
B three b o dy events they include the radiative events e e e e b eyond the collinear
approximation
No explicit photons b eyond O are generated on the generated events every kind of cuts can
b e imp osed O corrections are available for the t t contribution see for instance
for the soft plus virtual corrections and for the hard bremsstrahlung contribution
all the other channels are treated in the leading logarithmic approximation This theoretical
framework do es exploit the fact that the t t channel is by far the most dominant one It
is therefore su cient to evaluate exact order corrections for this channel only The other
channels which at the Born level contribute at the level of one p er cent in the small angle
region and of some p er cents in the large angle region at LEP energies can b e evaluated in the
leading log approximation Higher order corrections are implemented in the structure function
formalism The overall accuracy of the predictions p erformed by the co de is generically of
the order of in the small angle regime and of the order of in the large angle regime at
LEP energies
HOW DOES THE CODE WORK
The co de generates random integration variables within the ducial cuts supplied via the
input card see b elow These values are passed to the kinematics subroutines which construct
the full quadrimomenta for electron p ositron and photon The quadrimomenta are then fed
to a trigger routine which either accepts or rejects the event according to the cuts sp eci ed in
it by the user The control is then returned to the main integration routine which generates
weighted events accumulates the cross section result for each single contribution and comp ose
them as describ ed in eqs and Once in a given number of events the integrations
9
Actually in the present version of the program the up down interference contribution is neglected This is
of no practical relevance for the small angle cross section whereas it introduces an error of the order of some
p er mil in the large angle cross section at LEP
results and the related error estimates are evaluated and written to the output le The error is
also compared to the accuracy limit required and the run stops when the latter is reached The
program can b e restarted from its own output le by sp ecifying the same physical inputs
and either a larger number of events or a higher accuracy
INPUT CARD
The following data card has to b e provided via standard input
D EBEAM
D D D T MIN T MAX E MIN
D D D T MIN T MAX E MIN
D D D D D CALOINPUT
ISIM
ICALO
D D SABSPV OUT EVTS ACCLIM IRESTART OUTFILE
These parameters have the following meaning
D the electron and p ositron b eam energy EBEAM
D D D the electron minimum and maximum scattering angle in radians
and the minimum electron energy in GeV T MIN T MAX E MIN These cuts are to b e
interpreted as ducial cuts within which the events are generated b efore going through the
triggering routine
the same for the p ositron T MIN T MAX E MIN
D D D D D inputs that may b e required by the cutting routines
for the triggers These values are stored in the vector CALOINPUT via the common blo ck
COMMON CALOS
ag for symmetric cuts ISIM The user has to sp ecify if the exp erimental cuts asked
for are or not symmetric for electron p ositron exchange If they are choosing saves
computing time
ag for choosing the triggering routine ICALO
D D SABSPV OUT these are inputs related to the Monte Carlo integration
and to the management of the output Namely the total number of events EVTS the relative
accuracy limit aimed at ACCLIM the restarting ag IRESTART if the program tries to restart
execution from the indicated output le if it reinitializes it and the output le name
OUTFILE
DESCRIPTION OF THE OUTPUT
The output le OUTFILE contains a description of the inputs provided to the co de the results
of the Monte Carlo integrations for the various contributions and the nal results with their
standard statistical error Moreover informations concerning the random number generator and
the cumulants that can b e used to restart the program from where it stopp ed are provided
AVAILABILITY
The co de is available up on request to one of the authors
UNIBAB
AUTHORS
H Anlauf TH Darmstadt Universit at Siegen Germany
anlauf crunch ikp physik th darmstadt de
T Ohl TH Darmstadt Germany
ohl crunch ikp physik th darmstadt de
GENERAL DESCRIPTION
UNIBAB is a Monte Carlo event generator designed for large angle Bhabha scattering at LEP and
SLC energies In its original incarnation it was a simple QED dresser describing only
multiphoton initial state radiation thus fo cusing on the exp onentiation of soft photons and the
n
n
resummation to all orders of the leading logarithmic corrections of the form ln s m
e
The rst published version UNIBAB version contains improvements in the exclusive
photon shower algorithm used for the description of initial sta te radiation and many enhance
ments such as nal state radiation using a similar photon shower algorithm An electroweak
library based on ALIBABA was added Initial and nal state corrections are implemented
in a fully factorized form Version of the program features the inclusion of longitudinal
b eam p olarization During this workshop the current version was developed which uses an
implementation of the nal state photon shower based on the exact lowest order matrix element
for the pro cess Z f f Also the electroweak library has b een up dated slightly to include
the leading m dep endence and higher order QCD corrections to the Z width as discussed in
t
detail in
FEATURES OF THE PROGRAM
The event generator UNIBAB calculates the QED radiative corrections through a photon shower
algorithm The actual implementation is based on an iterative numerical solution of an Altarelli
Parisi type evolution equation for the electron structure function The e ective matrix element
for photon emission from the initial state assumes a factorized form of the radiative matrix
element Therefore it is exact for collinear emission It also allows to generate nite transverse
momenta of the radiated photons For nal state radiation the algorithm employs an iterated
form of the rst order matrix element for Z f f which gives a reasonable description
of exclusive distributions that are sensitive to the details of the approximations used for the
multiphoton matrix element such as acollinearity distributions on the Z p eak
UNIBAB generates only unweighted events It is implicitly assumed that all scales in the hard
subpro cess are of the same order of magnitude and the program do es not yet include initial
nal interference thus the program is generally limited to the large angle region Numerically
the e ects from initial na l interference are su ciently small in the vicinity of the Z p eak For
details see the long write up
HOW DOES THE CODE WORK
UNIBAB consists of two layers an external layer with a very simple user interface that allows easy
interactive and batch control of the program and an internal layer with a low level interface
to the internal routines It is however recommended to use the high level interface which
automatically takes care of parameter dep endencies and prop erly reinitializes the Monte Carlo
when a physics parameter is mo di ed
In order to run the program one has to sp ecify several steering parameters that are internally
translated into Monte Carlo parameters The actual physical cuts have to b e implemented in
an external analyzer The essential steering parameters are
ctsmin ctsmax cuts on cos where is the scattering angle in the b o osted subsystem
after taking initial state radiation into account
ecut minimum energy of the nal state fermions
acocut maximum acollinearity of the outgoing e e pair
An interactive run may lo ok like
set ebeam Beam energy in GeV
set mass z Z mass
set mass t top quark mass
set mass h Higgs mass
set ctsmin
set ctsmax
set ecut
set acocut acollinearity cut in degrees
init
generate
close
quit
Additional switches control the inclusion or omission of certain contributions like weak b ox
diagrams or t channel diagrams For more details please consult the manual
DESCRIPTION OF THE OUTPUT
UNIBAB stores the generated events and all supplementary information for analysis cross sec
tion Monte Carlo error in the prop osed standard hepevt common blo ck and must b e
read from there by a suitable analyzer A simple yet very exible to ol for implementing a the
orist s detector is given by HEPAWK which easily allows to obtain arbitrary distributions
from the generated events
AVAILABILITY
The current version of UNIBAB may b e downloaded via anonymous ftp from
ftp crunch ikp physik th darmstadt de pub anlauf unibab
along with up to date do cumentation At the time of this writing and for historical reasons
the program source and accompanying les are still distributed in the CERN patchy format
Platform dep endent Fortran source les will b e made available up on request For the sample
test run UNIBAB has also to b e linked with the analyzer HEPAWK A more mo dern auto
con guring and self contained version of the Monte Carlo generator will b e made available in
a future release after the end of the workshop
Conclusions and outlo ok
In this WG the rst systematic comparison of all the existing Monte Carlo event generators
for the Bhabha pro cess at LEP and LEP has b een p erformed This is one of our main
achievements The other one is that as a result of these comparisons the theoretical error
of the small angle Bhabha pro cess is now reduced from to for typical LEP
exp erimental ES s at the angular range of In parallel an estimate of the theoretical
error of the small angle Bhabha pro cess at LEP has also b een xed at for all p ossible
exp erimental situations The theoretical precision of the small angle Bhabha scattering should
b e still improved by a factor of two at LEP in order to match the exp erimental precision
From the analysis p erformed we conclude that a theoretical error of the order of is
reasonably feasible at LEP and the present study o ers a solid ground for the next step in
this direction
As far as the large angle Bhabha pro cess is concerned the main result of this WG is that
now we have comparisons not only among the semi analytical b enchmarks ALIBABA and
TOPAZ but also among Monte Carlo event generators and on the Monte Carlo co des versus
semianalytical programs In spite of the fact that the comparisons involving Monte Carlo s
do not change the conclusions of the previous LEP WG on the theoretical precision of large
angle Bhabha at LEP see they give information ab out the p erformances of the Monte
Carlo event generators themselves In particular except for some programs which have to b e
improved either on the QED libraries or on the pure weak ones the situation at LEP is
generally under control with resp ect to the present exp erimental accuracy b oth on and o Z
p eak As far as LEP is concerned a general agreement of the order of has b een achieved
There is certainly ro om for further improvements on this item but for practical purp oses the
situation can b e considered satisfactory
References
Part I Electroweak Physics in Reports of the working group on precision calculations
for the Z resonance edited by D Bardin W Hollik and G Passarino CERN Geneva
CERN Yellow Rep ort
Part I I I Small Angle Bhabha Scattering in Reports of the working group on precision
calculations for the Z resonance edited by D Bardin W Hollik and G Passarino CERN
Geneva CERN Yellow Rep ort
B Pietrzyk in Tennessee International Symposium on Radiative Corrections Status and
Outlook edited by B F L Ward World Scienti c Singap ore Gatlinburg Ten
nessee USA June
LEP Electroweak Working Group A Combination of Preliminary LEP Electroweak Results
from the Summer Conferences CERN rep ort LEPEWWG
LEP Collab oratio ns Collab ora tion notes ALEPH PHYSICS DEL
PHI PHYS L Note OPAL Technical Note TN August
S Jadach E Richter W as B F L Ward and Z W as Phys Lett B
CERN preprint CERN TH
A B Arbuzov E A Kuraev N P Merenkov and L Trentadue preprint JINR Dubna
E unpublished
A B Arbuzov E A Kuraev N P Merenkov and L Trentadue JETPh
F Teubert IFAE Barcelona private communication
The LEP Collab orati ons ALEPH DELPHI L OPAL and the LEP Electroweak Working
Group preprint CERN TH unpublished
S Jadach E Richter W as B F L Ward and Z W as Phys Lett B
A Arbuzov et al in Tennessee International Symposium on Radiative Corrections Status
and Outlook edited by B F L Ward World Scienti c Singap ore Gatlinburg
Tennessee USA June
N P Merenkov Sov J Nucl Phys
N P Merenkov Sov J Nucl Phys
A B Arbuzov E A Kuraev N P Merenkov and L Trentadue preprint CERN TH
unpublished
L N Lipatov Sov J Nucl Phys
G Altarelli and G Parisi Nucl Phys B
E A Kuraev and V S Fadin Sov J Nucl Phys
E A Kuraev and V S Fadin preprint INP Novosibirsk unpublished
O Nicrosini and L Trentadue Phys Lett B
E A Kuraev N P Merenkov and V S Fadin Sov J Nucl Phys
M Skrzyp ek Acta Phys Pol B
A Arbuzov et al in preparation
F A Berends P H Daverveldt and R Kleiss Nucl Phys B
F A Berends P H Daverveldt and R Kleiss Comput Phys Comm
J H Field presented in the meeting of the Bhabha WG unpublished
S Jadach E Richter W as B F L Ward and Z W as Phys Lett B
S Jadach M Skrzyp ek and B F L Ward Phys Rev D
S Jadach M Skrzyp ek and B F L Ward presented in the meeting of the Bhabha WG
and to app ear
S Jadach E Richter W as B F L Ward and Z W as Comput Phys Commun
S Jadach M Skrzyp ek and B F L Ward Phys Rev D
W Beenakker and B Pietrzyk Phys Lett B
S Eidelman and F Jegerlehner Z Phys C
H Burkhardt and B Pietrzyk Phys Lett B
H Burkhardt in Electroweak interactions and uni ed theories edited by J Tran Than
Van Editions Fronti eres Gif sur Yvette in print
H Burkhardt F Jegerlehner G Penso and C Verzegnassi Z Phys C
R B Nevzorov A V Novikov and M Vysotsky JETP Lett
B Geshkenbein and V Morgunov preprint HEPPH unpublished
N V Krasnikov Mo d Phys Lett A
M L Swartz preprint SLAC PUB revised November unpublished
M L Swartz preprint SLAC PUB September unpublished
K Ab el and F J Yndurain preprint FTUAM September unpublished
M Ca o H Czy z and E Remiddi BHAGEN short write up these Pro ceedings
S Jadach et al BHLUMI Monte Carlo to b e submitted to Comput Phys Commun
available from WWW lo cation http hp jmiady ifj edu pl a patch correcting bug a can
b e also found there unpublished
F A Berends and R Kleiss Nucl Phys B
M Cacciari G Montagna O Nicrosini and F Piccinini Comput Phys Commun
preprint CERN TH
M Acciarri et al Z Phys C
I C Bro ck et al Luminosity Measurement in L L Collab ora tio n in preparation
B Blo ch Devaux et al The SiCal luminosity analysis ALEPH internal note
ALEPH PHYSIC
D Buskulic et al Z Phys C
OPAL Collab ora tio n OPAL Physics Note PN
J Hart in Rencontres de Physique de la Vallee D Aoste edited by EDITOR PUB
LISHER NAME PUBLISHER TOWN p PAGE
G M Dallavalle in The Standard Model and Just Beyond th S Miniato Topical Seminar
edited by F L Navarria and P Pelfer World Scienti c Singap ore San Miniato
Italy June
D Bederede et al Nucl Instr and Meth A
B E Anderson et al IEEE Trans on Nucl Sci
M Merk in Electroweak interactions and uni ed theories edited by J Tran Than Van
Editions Fronti eres Gif sur Yvette
AUTHOR in EPS HEP edited by EDITOR PUBLISHER PUBLISHER
TOWN Brussels July August in print
S Jadach E Richter W as B F L Ward and Z W as Phys Lett B
S Jadach W P laczek and B Ward Phys Lett B CERN preprint TH
April
W Beenakker F A Berends and S C van der Marck Nucl Phys B
M Martinez and R Miquel Z Phys C
P Comas and R Martinez Z Phys C
G Montagna et al Comput Phys Commun
G Montagna O Nicrosini G Passarino and F Piccinini preprint CERN TH
to app ear in Comput Phys Commun unpublished
D Bardin et al ZFITTER An Analytical program for fermion pair production in e e
annihilation preprint CERN TH unpublished
D Bardin et al Nucl Phys B
J H Field Phys Lett B
J H Field and T Riemann BHAGENE a Monte Carlo Event Generator for Lepton
Pair Production and Wide Angle Bhabha Scattering in e e Col lisions near the Z Peak
rep ort UGVA DPNC DESY to b e published in Comp Phys
Commun
S Jadach W P laczek and B F L Ward preprint The University of Tennessee UTHEP
unpublished
H Anlauf et al Comput Phys Commun
G Montagna et al Nucl Phys B
M Ca o et al Bhabha Scattering in Z Physics at LEP edited by G Altarelli R
Kleiss and C Verzegnassi CERN Geneva p CERN Yellow Rep ort
M Ca o H Czy z and E Remiddi Nuovo Cim A
M Ca o H Czy z and E Remiddi Int J Mo d Phys
M Ca o H Czy z and E Remiddi Phys Lett B
M Ca o H Czy z and E Remiddi program BHAGEN in preparation
M Ca o H Czy z and E Remiddi program BHAGEN PH in preparation
D Bardinand W Hollik and T Riemann Z Phys C
A Djouadi and C Verzegnassi Phys Lett B
A Djouadi Nuovo Cim A
D Bardin and A Chizhov preprint INR Dubna E unpublished
M B ohm A Denner and W Hollik Nucl Phys B and references therein
F A Berends R Kleiss and W Hollik Nucl Phys B
V N Grib ov and L N Lipatov Sov J Nucl Phys
S Jadach and B F L Ward Phys Rev D
W Beenakker F A Berends and S C van der Marck Nucl Phys B
D Bardin et al Comp Phys Commun
M Greco G Montagna O Nicrosini and F Piccinini Phys Lett B and
references therein
G Montagna O Nicrosini and F Piccinini Comput Phys Commun
erratum ibid
H D Dahmen P Manakos T Mannel and T Ohl preprint SI IKDA un
published
H D Dahmen P Manakos T Mannel and T Ohl Z Phys C
T Ohl Comp Phys Commun
T Ohl rep ort IKDA hep ex unpublished