Page 1
arXiv:0912.0749v1 [hepph] 3 Dec 2009
EPJ manuscript No.
(will be inserted by the editor)
BIHEPTH2009005, BUHEPP0908,
CERNPHTH/2009201, DESY 09092,
FNT/T 2009/03, FreiburgPHENO09/07,
HEPTOOLS 09018, IEKPKA/200933,
LNF09/14(P), LPSC 09/157,
LPTORSAY0995, LTH 851, MZTH/0938,
PITHA09/14, PSIPR0914,
SFB/CPP0953, WUB/0907
Quest for precision in hadronic cross sections at low energy:
Monte Carlo tools vs. experimental data
Working Group on Radiative Corrections and Monte Carlo Generators for Low Energies
S. Actis38, A. Arbuzov9,43, G. Balossini32,33, P. Beltrame13, C. Bignamini32,33, R. Bonciani15, C. M. Carloni Calame35,
V. Cherepanov25,26, M. Czakon1, H. Czy˙ z19,44,47,48, A. Denig22, S. Eidelman25,26,45, G. V. Fedotovich25,26,43,
A. Ferroglia23, J. Gluza19, A. Grzeli´ nska8, M. Gunia19, A. Hafner22, F. Ignatov25, S. Jadach8, F. Jegerlehner3,19,41,
A. Kalinowski29, W. Kluge17, A. Korchin20, J. H.K¨ uhn18, E. A. Kuraev9, P. Lukin25, P. Mastrolia14,
G. Montagna32,33,42,48, S. E. M¨ uller22,44, F. Nguyen34,42, O. Nicrosini33, D. Nomura36,46, G. Pakhlova24,
G. Pancheri11, M. Passera28, A. Penin10, F. Piccinini33, W. P? laczek7, T. Przedzinski6, E. Remiddi4,5, T. Riemann41,
G. Rodrigo37, P. Roig27, O. Shekhovtsova11, C. P. Shen16, A. L. Sibidanov25, T. Teubner21,46, L. Trentadue30,31,
G. Venanzoni11,47,48, J. J. van der Bij12, P. Wang2, B. F. L. Ward39, Z. Was8,45, M. Worek40,19, and C. Z. Yuan2
1Institut f¨ ur Theoretische Physik E, RWTH Aachen Universit¨ at, D52056 Aachen, Germany
2Institue of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China
3Institut f¨ ur Physik HumboldtUniversit¨ at zu Berlin, D12489 Berlin, Germany
4Dipartimento di Fisica dell’Universit` a di Bologna, I40126 Bologna, Italy
5INFN, Sezione di Bologna, I40126 Bologna, Italy
6The Faculty of Physics, Astronomy and Applied Computer Science, Jagiellonian University, Reymonta 4, 30059 Cracow,
Poland
7Marian Smoluchowski Institute of Physics, Jagiellonian University, Reymonta 4, 30059 Cracow, Poland
8Institute of Nuclear Physics Polish Academy of Sciences, PL31342 Cracow, Poland
9Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980 Dubna, Russia
10Department of Physics, University of Alberta, Edmonton, AB T6G 2J1, Canada
11Laboratori Nazionali di Frascati dell’INFN, I00044 Frascati, Italy
12Physikalisches Institut, AlbertLudwigsUniversit¨ at Freiburg, D79104 Freiburg, Germany
13CERN, Physics Department, CH1211 Gen` eve, Switzerland
14CERN, Theory Department, CH1211 Gen` eve, Switzerland
15Laboratoire de Physique Subatomique et de Cosmologie, Universit´ e Joseph Fourier/CNRSIN2P3/INPG,
F38026 Grenoble, France
16University of Hawaii, Honolulu, Hawaii 96822, USA
17Institut f¨ ur Experimentelle Kernphysik, Universit¨ at Karlsruhe, D76021 Karlsruhe, Germany
18Institut f¨ ur Theoretische Teilchenphysik, Universit¨ at Karlsruhe, D76128 Karlsruhe, Germany.
19Institute of Physics, University of Silesia, PL40007 Katowice, Poland
20National Science Center “Kharkov Institute of Physics and Technology”, 61108 Kharkov, Ukraine
21Department of Mathematical Sciences, University of Liverpool, Liverpool L69 3BX, U.K.
22Institut f¨ ur Kernphysik, Johannes Gutenberg  Universit¨ at Mainz, D55128 Mainz, Germany
23Institut f¨ ur Physik (THEP), Johannes GutenbergUniversit¨ at, D55099 Mainz, Germany
24Institute for Theoretical and Experimental Physics, Moscow, Russia
25Budker Institute of Nuclear Physics, 630090 Novosibirsk, Russia
26Novosibirsk State University, 630090 Novosibirsk, Russia
27Laboratoire de Physique Th´ eorique (UMR 8627),Universit´ e de ParisSud XI, Bˆ atiment 210, 91405 Orsay Cedex, France
28INFN, Sezione di Padova, I35131 Padova, Italy
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2
29LLREcole Polytechnique, 91128 Palaiseau, France
30Dipartimento di Fisica, Universit` a di Parma, I43100 Parma, Italy
31INFN, Gruppo Collegato di Parma, I43100 Parma, Italy
32Dipartimento di Fisica Nucleare e Teorica, Universit` a di Pavia, I27100 Pavia, Italy
33INFN, Sezione di Pavia, I27100 Pavia, Italy
34Dipartimanto di Fisica dell’Universit` a “Roma Tre” and INFN Sezione di Roma Tre, I00146 Roma, Italy
35School of Physics and Astronomy, University of Southampton, Southampton SO17 1BJ, U.K.
36Theory Center, KEK, Tsukuba, Ibaraki 3050801, Japan
37Instituto de Fisica Corpuscular (IFIC), Centro mixto UVEG/CSIC, Edificio Institutos de Investigacion, Apartado de Correos
22085, E46071 Valencia, Espanya
38Paul Scherrer Institut, W¨ urenlingen and Villigen, CH5232 Villigen PSI, Switzerland
39Department of Physics, Baylor University, Waco, Texas 767987316, USA
40Fachbereich C, Bergische Universit¨ at Wuppertal, D42097 Wuppertal, Germany
41Deutsches ElektronenSynchrotron, DESY, D15738 Zeuthen, Germany
42Section 2 conveners
43Section 3 conveners
44Section 4 conveners
45Section 5 conveners
46Section 6 conveners
47Working group conveners
48Corresponding authors: henryk.czyz@us.edu.pl, guido.montagna@pv.infn.it, graziano.venanzoni@lnf.infn.it
Received: date / Revised version: date
Abstract. We present the achievements of the last years of the experimental and theoretical groups working
on hadronic cross section measurements at the low energy e+e−colliders in Beijing, Frascati, Ithaca,
Novosibirsk, Stanford and Tsukuba and on τ decays. We sketch the prospects in these fields for the
years to come. We emphasise the status and the precision of the Monte Carlo generators used to analyse
the hadronic cross section measurements obtained as well with energy scans as with radiative return, to
determine luminosities and τ decays. The radiative corrections fully or approximately implemented in the
various codes and the contribution of the vacuum polarisation are discussed.
PACS. 13.66.Bc Hadron production in e−e+interactions – 13.35.Dx Decays of taus – 12.10.Dm Unified
theories and models of strong and electroweak interactions – 13.40.Ks Electromagnetic corrections to
strong and weakinteraction processes – 29.20.c Accelerators
Page 3
2
Contents
1
2
3
4
5
6
7
Introduction . . . . . . . . . . . . . . . . . . . . . . .
Luminosity . . . . . . . . . . . . . . . . . . . . . . .
R measurement from energy scan . . . . . . . . . . .
Radiative return . . . . . . . . . . . . . . . . . . . .
Tau decays. . . . . . . . . . . . . . . . . . . . . . .
Vacuum polarisation . . . . . . . . . . . . . . . . . .
Summary . . . . . . . . . . . . . . . . . . . . . . . .
2
4
35
44
72
82
88
1 Introduction
The systematic comparison of Standard Model (SM) pre
dictions with precise experimental data served in the last
decades as an invaluable tool to test the theory at the
quantum level. It has also provided stringent constraints
on “new physics” scenarios. The (so far) remarkable agree
ment between the measurements of the electroweak ob
servables and their SM predictions is a striking experi
mental confirmation of the theory, even if there are a few
observables where the agreement is not so satisfactory.
On the other hand, the Higgs boson has not yet been ob
served, and there are clear phenomenological facts (dark
matter, matterantimatter asymmetry in the universe) as
well as strong theoretical arguments hinting at the pres
ence of physics beyond the SM. New colliders, like the
LHC or a future e+e−International Linear Collider (ILC),
will hopefully answer many questions, offering at the same
time great physics potential and a new challenge to pro
vide even more precise theoretical predictions.
Precision tests of the Standard Model require an ap
propriate inclusion of higher order effects and the knowl
edge of very precise input parameters. One of the basic
input parameters is the finestructure constant α , deter
mined from the anomalous magnetic moment of the elec
tron with an impressive accuracy of 0.37 parts per billion
(ppb) [1] relying on the validity of perturbative QED [2].
However, physics at nonzero squared momentum trans
fer q2is actually described by an effective electromagnetic
coupling α(q2) rather than by the lowenergy constant α
itself. The shift of the finestructure constant from the
Thomson limit to high energy involves low energy non
perturbative hadronic effects which spoil this precision.
In particular, the effective finestructure constant at the
scale MZ, α(M2
in basic EW radiative corrections of the SM. An important
example is the EW mixing parameter sin2θ, related to α,
Z) = α/[1 − ∆α(M2
Z)], plays a crucial role
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3
the Fermi coupling constant GF and MZ via the Sirlin
relation [3,4,5]
sin2θScos2θS=
πα
Z(1 − ∆rS),
√2GFM2
(1)
where the subscript S identifies the renormalisationscheme.
∆rSincorporates the universal correction ∆α(M2
contributions that depend quadratically on the top quark
mass mt[6] (which led to its indirect determination before
this quark was discovered), plus all remaining quantum ef
fects. In the SM, ∆rSdepends on various physical param
eters, including MH, the mass of the Higgs boson. As this
is the only relevant unknown parameter in the SM, impor
tant indirect bounds on this missing ingredient can be set
by comparing the calculated quantity in Eq. (1) with the
experimental value of sin2θS(e.g. the effective EW mixing
angle sin2θlept
eff
measured at LEP and SLC from the on
resonance asymmetries) once ∆α(M2
mental inputs like mtare provided. It is important to note
that an error of δ∆α(M2
electromagnetic coupling constant dominates the uncer
tainty of the theoretical prediction of sin2θlept
an error δ(sin2θlept
with the experimental value δ(sin2θlept
determined by LEPI and SLD [8,9]) and affecting the up
per bound for MH[8,9,10]. Moreover, as measurements of
the effective EW mixing angle at a future linear collider
may improve its precision by one order of magnitude, a
much smaller value of δ∆α(M2
low). It is therefore crucial to assess all viable options to
further reduce this uncertainty.
The shift ∆α(M2
∆αlep(M2
culable in perturbation theory and known up to three
loop accuracy: ∆αlep(M2
hadronic contribution ∆α(5)
(u, d, s, c, and b) can be computed from hadronic e+e−
annihilation data via the dispersion relation [12]
?αM2
Z), large
Z) and other experi
Z) = 35×10−5[7] in the effective
eff, inducing
eff) ∼ 12 × 10−5(which is comparable
eff)EXP= 16 × 10−5
Z) will be required (see be
Z) can be split in two parts: ∆α(M2
had(M2
Z) =
Z)+∆α(5)
Z). The leptonic contribution is cal
Z) = 3149.7686 × 10−5[11]. The
had(M2
Z) of the five light quarks
∆α(5)
had(M2
Z) = −
Z
3π
?
Re
?∞
m2
π
ds
R(s)
s(s − M2
Z− iǫ),
(2)
where R(s) = σ0
tal cross section for e+e−annihilation into any hadronic
states, with vacuum polarisation and initial state QED
corrections subtracted off. The current accuracy of this
dispersion integral is of the order of 1%, dominated by
the error of the hadronic cross section measurements in
the energy region below a few GeV [13,14,15,7,16,17,18,
19,20,21,22,23].
Table 1 (from Ref. [16]) shows that an uncertainty
δ∆α(5)
linear collider, requires the measurement of the hadronic
cross section with a precision of O(1%) from threshold up
to the Υ peak.
Like the effective finestructure constant at the scale
MZ, the SM determination of the anomalous magnetic mo
ment of the muon aµis presently limited by the evaluation
had(s)/(4πα2/3s) and σ0
had(s) is the to
had∼ 5×10−5, needed for precision physics at a future
δ∆α(5)
had×105
22
δ(sin2θlept
eff)×105
7.9
Request on R
Present
72.5δR/R ∼ 1% up to J/ψ
δR/R ∼ 1% up to Υ
had(first column)
5 1.8
Table 1.
and the errors induced by these uncertainties on the theoretical
SM prediction for sin2θlept
eff
(second column). The third column
indicates the corresponding requirements for the R measure
ment. From Ref. [16].
Values of the uncertainties δ∆α(5)
of the hadronic vacuum polarisation effects, which cannot
be computed perturbatively at low energies. However, us
ing analyticity and unitarity, it was shown long ago that
this term can be computed from hadronic e+e−annihila
tion data via the dispersion integral [24]:
?∞
α2
3π2
m2
π
aHLO
µ
=
1
4π3
m2
π
dsK(s)σ0(s)
=
?∞
dsK(s)R(s)/s.(3)
The kernel function K(s) decreases monotonically with
increasing s. This integral is similar to the one entering
the evaluation of the hadronic contribution ∆α(5)
in Eq. (2). Here, however, the weight function in the inte
grand gives a stronger weight to lowenergy data. A recent
compilation of e+e−data gives [25]:
had(M2
Z)
aHLO
µ
= (695.5 ± 4.1) × 10−10.(4)
Similar values are obtained by other groups [23,26,27,28].
By adding this contribution to the rest of the SM con
tributions, a recent update of the SM prediction of aµ,
which uses the hadronic lightbylight result from [29] gives
[25,30]: aSM
tween the experimental average[31], aexp
10−11and the SM prediction is then ∆aµ= aexp
+246(80)×10−11, i.e. 3.1 standard deviations (adding all
errors in quadrature). Slightly higher discrepancies are
obtained in Refs. [23,27,28]. As in the case of α(M2
the uncertainty of the theoretical evaluation of aSM
dominated by the hadronic contribution at low energies,
and a reduction of the uncertainty is necessary in order to
match the increased precision of the proposed muon g2
experiments at FNAL [32] and JPARC [33].
The precise determination of the hadronic cross sec
tions (accuracy ? 1%) requires an excellent control of
higher order effects like Radiative Corrections (RC) and
the nonperturbative hadronic contribution to the running
of α (i.e. the vacuum polarisation, VP) in Monte Carlo
(MC) programs used for the analysis of the data. Partic
ularly in the last years, the increasing precision reached
on the experimental side at the e+e−colliders (VEPP
2M, DAΦNE, BEPC, PEPII and KEKB) led to the de
velopment of dedicated high precision theoretical tools:
BabaYaga (and its successor BabaYaga@NLO) for the
µ = 116591834(49)× 10−11. The difference be
µ
= 116592080(63)×
µ − aSM
µ =
Z),
µ is still
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4
measurement of the luminosity, MCGPJ for the simula
tion of the exclusive QED channels, and PHOKHARA for
the simulation of the process with Initial State Radiation
(ISR) e+e−→ hadrons + γ, are examples of MC genera
tors which include NLO corrections with per mill accuracy.
In parallel to these efforts, welltested codes such as BH
WIDE (developed for LEP/SLC colliders) were adopted.
Theoretical accuracies of these generators were esti
mated, whenever possible, by evaluating missing higher
order contributions. From this point of view, the great
progress in the calculation of twoloop corrections to the
Bhabha scattering cross section was essential to establish
the high theoretical accuracy of the existing generators
for the luminosity measurement. However, usually only
analytical or semianalytical estimates of missing terms
exist which don’t take into account realistic experimental
cuts. In addition, MC event generators include different
parametrisations for the VP which affect the prediction
(and the precision) of the cross sections and also the RC
are usually implemented differently.
These arguments evidently imply the importance of
comparisons of MC generators with a common set of in
put parameters and experimental cuts. Such tuned com
parisons, which started in the LEP era, are a key step for
the validation of the generators, since they allow to check
that the details entering the complex structure of the gen
erators are under control and free of possible bugs. This
was the main motivation for the “Working Group on Ra
diative Corrections and Monte Carlo Generators for Low
Energies” (Radio MontecarLow), which was formed a few
years ago bringing together experts (theorists and experi
mentalists) working in the field of low energy e+e−physics
and partly also the τ community.
In addition to tuned comparisons, technical details of
the MC generators, recent progress (like new calculations)
and remaining open issues were also discussed in regular
meetings.
This report is a summary of all these efforts: it pro
vides a selfcontained and uptodate description of the
progress which occurred in the last years towards preci
sion hadronic physics at low energies, together with new
results like comparisons and estimates of high order effects
(e.g. of the pion pair correction to the Bhabha process) in
the presence of realistic experimental cuts.
The report is divided into five sections: Sections 2, 3
and 4 are devoted to the status of the MC tools for Lumi
nosity, the Rscan and Initial State Radiation (ISR).
Tau spectral functions of hadronic decays are also used
to estimate aHLO
µ
, since they can be related to e+e−anni
hilation cross section via isospin symmetry [34,35,36,37].
The substantial difference between the e+e− and τbased
determinations of aHLO
µ
, even if isospin violation correc
tions are taken into account, shows that further common
theoretical and experimental efforts are necessary to un
derstand this phenomenon. In Section 5 the experimental
status and MC tools for tau decays are discussed. The re
cent improvements of the generators TAUOLA and PHO
TOS are discussed and prospects for further developments
are sketched.
Section 6 discusses vacuum polarisation at low ener
gies, which is a key ingredient for the high precision de
termination of the hadronic cross section, focusing on the
description and comparison of available parametrisations.
Finally, Section 7 contains a brief summary of the report.
2 Luminosity
The present Section addresses the most important exper
imental and theoretical issues involved in the precision
determination of the luminosity at meson factories. The
luminosity is the key ingredient underlying all the mea
surements and studies of the physics processes discussed
in the other Sections. Particular emphasis is put on the
theoretical accuracy inherent to the event generators used
in the experimental analyses, in comparison with the most
advanced perturbative calculations and experimental pre
cision requirements. The effort done during the activity
of the working group to perform tuned comparisons be
tween the predictions of the most accurate programs is
described in detail. New calculations, leading to an up
date of the theoretical error associated with the predic
tion of the luminosity cross section, are also presented.
The aim of the Section is to provide a selfcontained and
uptodate description of the progress occurred during the
last few years towards highprecision luminosity monitor
ing at flavour factories, as well as of the still open issues
necessary for future advances.
The structure of the Section is as follows. After an in
troduction on the motivation for precision luminosity mea
surements at meson factories (Section 2.1), the leading
order (LO) cross sections of the two QED processes of
major interest, i.e. Bhabha scattering and photon pair
production, are presented in Section 2.2, together with
the formulae for the nexttoleadingorder (NLO) pho
tonic corrections to the above processes. The remarkable
progress on the calculation of nexttonextleadingorder
(NNLO) QED corrections to the Bhabha cross section, as
occurred in the last few years, is reviewed in Section 2.3.
In particular, this Section presents new exact results on
lepton and hadron pair corrections, taking into account
realistic event selection criteria. Section 2.4 is devoted
to the description of the theoretical methods used in the
Monte Carlo (MC) generators for the simulation of multi
ple photon radiation. The matching of such contributions
with NLO corrections is also described in Section 2.4. The
main features of the MC programs used by the experimen
tal collaborations are summarised in Section 2.5. Numer
ical results for the radiative corrections implemented into
the MC generators are shown in Section 2.6 for both the
Bhabha process and twophoton production. Tuned com
parisons between the predictions of the most precise gen
erators are presented and discussed in detail in Section 2.7,
considering the Bhabha process at different centreofmass
(c.m.) energies and with realistic experimental cuts. The
theoretical accuracy presently reached by the luminosity
tools is addressed in Section 2.8, where the most impor
tant sources of uncertainty are discussed quantitatively.
The estimate of the total error affecting the calculation of
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5
the Bhabha cross section is given, as the main conclusion
of the present work, in Section 2.9, updating and improv
ing the robustness of results available in the literature.
Some remaining open issues are discussed in Section 2.9
as well.
2.1 Motivation
The luminosity of a collider is the normalisation constant
between the event rate and the cross section of a given
process. For an accurate measurement of the cross section
of an electronpositron (e+e−) annihilation process, the
precise knowledge of the collider luminosity is mandatory.
The luminosity depends on three factors: beambeam
crossing frequency, beam currents and the beam overlap
area in the crossing region. However, the last quantity is
difficult to determine accurately from the collider optics.
Thus, experiments prefer to determine the luminosity by
the counting rate of well selected events whose cross sec
tion is known with good accuracy, using the formula [38]
?
where N is the number of events of the chosen reference
process, ǫ the experimental selection efficiency and σ the
theoretical cross section of the reference process. There
fore, the total luminosity error will be given by the sum in
quadrature of the fractional experimental and theoretical
uncertainties.
Since the advent of low luminosity e+e−colliders, a
great effort was devoted to obtain good precision in the
cross section of electromagnetic processes, extending the
pioneering work of the earlier days [12]. At the e+e−col
liders operating in the c.m. energy range 1 GeV <√s <
3 GeV, such as ACO at Orsay, VEPPII at Novosibirsk
and Adone at Frascati, the luminosity measurement was
based on Bhabha scattering [39,40] with finalstate elec
trons and positrons detected at small angles, or single and
double bremsstrahlung processes [41], thanks to their high
statistics. The electromagnetic cross sections scale as 1/s,
while elastic e+e−scattering has a steep dependence on
the polar angle, ∼ 1/θ3, thus providing a high rate for
small values of θ.
Also at highenergy, accelerators running in the ’90s
around the Z pole to perform precision tests of the Stan
dard Model (SM), such as LEP at CERN and SLC at
Stanford, the experiments used smallangle Bhabha scat
tering events as a luminosity monitoring process. Indeed,
for the very forward angular acceptances considered by
the LEP/SLC collaborations, the Bhabha process is dom
inated by the electromagnetic interaction and, therefore,
calculable, at least in principle, with very high accuracy.
At the end of the LEP and SLC operation, a total (ex
perimental plus theoretical) precision of one per mill (or
better) was achieved [42,43,44,45,46,47,48], thanks to the
work of different theoretical groups and the excellent per
formance of precision luminometers.
Ldt =N
ǫσ,
(5)
At current low and intermediateenergy highlumino
sity meson factories, the small polar angle region is diffi
cult to access due to the presence of the lowbeta inser
tions close to the beam crossing region, while wideangle
Bhabha scattering produces a large counting rate and can
be exploited for a precise measurement of the luminosity.
Therefore, also in this latter case of e±scattered at
large angles, e.g. larger than 55◦for the KLOE experi
ment [38] running at DAΦNE in Frascati, and larger than
40◦for the CLEOc experiment [49] running at CESR in
Cornell, the main advantages of Bhabha scattering are
preserved:
1. large statistics. For example at DAΦNE, a statistical
error δL/L ∼ 0.3% is reached in about two hours of
data taking, even at the lowest luminosities;
2. high accuracy for the calculated cross section;
3. clean event topology of the signal and small amount of
background.
In Eq. (5) the cross section is usually evaluated by
inserting event generators, which include radiative correc
tions at a high level of precision, into the MC code sim
ulating the detector response. The code has to be devel
oped to reproduce the detector performance (geometrical
acceptance, reconstruction efficiency and resolution of the
measured quantities) to a high level of confidence.
In most cases the major sources of the systematic er
rors of the luminosity measurement are differences of effi
ciencies and resolutions between data and MC.
In the case of KLOE, the largest experimental error
of the luminosity measurement is due to a different polar
angle resolution between data and MC which is observed
at the edges of the accepted interval for Bhabha scatter
ing events. Fig. 1 shows a comparison between large angle
Bhabha KLOE data and MC, at left for the polar angle
and at right for the acollinearity ζ = θe+ + θe− − 180◦.
One observes a very good agreement between data and
MC, but also differences (of about 0.3 %) at the sharp
interval edges. The analysis cut, ζ < 9◦, applied to the
acollinearity distribution is very far from the bulk of the
distribution and does not introduce noteworthy system
atic errors. Also in the CLEOc luminosity measurement
with Bhabha scattering events, the detector modelling is
the main source of experimental error. In particular, un
certainties include those due to finding and reconstruc
tion of the electron shower, in part due to the nature of
the electron shower, as well as the steep e±polar angle
distribution.
The luminosity measured with Bhabha scattering events
is often checked by using other QED processes, such as
e+e−→ µ+µ−or e+e−→ γγ. In KLOE, the luminos
ity measured with e+e−→ γγ events differs by 0.3%
from the one determined from Bhabha events. In CLEOc,
e+e−→ µ+µ−events are also used, and the luminosity
determined from γγ (µ+µ−) is found to be 2.1% (0.6%)
larger than that from Bhabha events. Fig. 2 shows the
CLEOc data for the polar angle distributions of all three
processes, compared with the corresponding MC predic
tions. The three QED processes are also used by the BaBar
Page 7
6
θ (degrees)
1/N dN/dθ (degrees)1
0
0.005
0.01
0.015
0.02
0.025
5060 7080 90100110120 130
ζ (degrees)
1/N dN/dζ (0.2 degrees)1
10
3
10
2
10
1
012345678910
Fig. 1. Comparison between largeangle Bhabha KLOE data (points) and MC (histogram) distributions for the e±polar angle
θ (left) and for the acollinearity, ζ = θe+ + θe− − 180◦ (right), where the flight direction of the e±is given by the position of
clusters in the calorimeter. In each case, MC and data histograms are normalised to unity. From [38].
experiment at the PEPII collider, Stanford, yielding a lu
minosity determination with an error of about 1% [50].
Largeangle Bhabha scattering is the normalisation pro
cess adopted by the CMD2 and SND collaborations at
VEPP2M, Novosibirsk, while both BES at BEPC in Bei
jing and Belle at KEKB in Tsukuba measure luminos
ity using the processes e+e−→ e+e−and e+e−→ γγ
with the finalstate particles detected at wide polar angles
and an experimental accuracy of a few per cent. However,
BESIII aims at reaching an error of a few per mill in their
luminosity measurement in the near future [51].
The need of precision, namely better than 1%, and pos
sibly redundant measurements of the collider luminosity is
of utmost importance to perform accurate measurements
of the e+e−→ hadrons cross sections, which are the key
ingredient for evaluating the hadronic contribution to the
running of the electromagnetic coupling constant α and
the muon anomaly g − 2.
2.2 LO cross sections and NLO corrections
As remarkedin Section 2.1, the processes of interest for the
luminosity measurement at meson factories are Bhabha
scattering and electronpositronannihilation into two pho
tons and muon pairs. Here we present the LO formulae
for the cross section of the processes e+e−→ e+e−and
e+e−→ γγ, as well as the QED corrections to their cross
sections in the NLO approximation of perturbation the
ory. The reaction e+e−→ µ+µ−is discussed in Section
3.
2.2.1 LO cross sections
For the Bhabha scattering process
e−(p−) + e+(p+) → e−(p′
at Born level with simple onephoton exchange (see Fig. 3)
the differential cross section reads
?3 + c2
where
−) + e+(p′
+) (6)
dσBhabha
0
dΩ−
=α2
4s1 − c
?2
+ O
?m2
e
s
?
,(7)
s = (p−+ p+)2,c = cosθ−. (8)
The angle θ−is defined between the initial and final elec
tron threemomenta, dΩ−= dφ−dcosθ−, and φ− is the
azimuthal angle of the outgoing electron. The small mass
correction terms suppressed by the ratio m2
ligible for the energy range and the angular acceptances
which are of interest here.
At meson factories the Bhabha scattering cross sec
tion is largely dominated by tchannel photon exchange,
followed by st interference and schannel annihilation.
Furthermore, Zboson exchange contributions and other
electroweak effects are suppressed at least by a factor
s/M2
Z. In particular, for largeangle Bhabha scattering
with a c.m. energy√s = 1 GeV the Z boson contribu
tion amounts to about −1 × 10−5. For√s = 3 GeV it
amounts to −1 × 10−4and −1 × 10−3for√s = 10 GeV.
So only at B factories the electroweak effects should be
taken into account at tree level, when aiming at a per mill
precision level.
The LO differential cross section of the twophoton
annihilation channel (see Fig. 4)
e/s are neg
e+(p+) + e−(p−) → γ(q1) + γ(q2)
Page 8
7
Fig. 2. Distributions of CLEOc√s = 3.774 GeV data (cir
cles) and MC simulations (histograms) for the polar angle of
the positive lepton (upper two plots) in e+e−and µ+µ−events,
and for the mean value of cosθγ of the two photons in γγ
events (lower panel). MC histograms are normalised to the
number of data events. From [49].
γ
e−
e+
e−
e+
γ
e−
e+
e−
e+
Fig. 3. LO Feynman diagrams for the Bhabha process in QED,
corresponding to schannel annihilation and tchannel scatter
ing.
can be obtained by a crossing relation from the Compton
scattering cross section computed by Brown and Feyn
man [52]. It reads
dσγγ
dΩ1
0
=α2
s
?1 + c2
1
1 − c2
1
?
+ O
?m2
e
s
?
,(9)
where dΩ1denotes the differential solid angle of the first
photon. It is assumed that both final photons are regis
tered in a detector and that their polar angles with respect
e−
γ
e+
γ
e−
γ
e+
γ
Fig. 4. LO Feynman diagrams for the process e+e−→ γγ.
to the initial beam directions are not small (θ1,2≫ me/E,
where E is the beam energy).
2.2.2 NLO corrections
The complete set of NLO radiative corrections, emerging
at O(α) of perturbation theory, to Bhabha scattering and
twophoton annihilation can be split into gaugeinvariant
subsets: QED corrections, due to emission of real photons
off the charged leptons and exchange of virtual photons
between them, and purely weak contributions arising from
the electroweak sector of the SM.
The complete O(α) QED corrections to Bhabha scat
tering are known since a long time [53,54]. The first com
plete NLO prediction in the electroweak SM was per
formed in [55], followed by [56] and several others. At
NNLO, the leading virtual weak corrections from the top
quark were derived first in [57] and are available in the
fitting programs ZFITTER [58,59] and TOPAZ0 [60,61,
62], extensively used by the experimentalists for the ex
traction of the electroweak parameters at LEP/SLC. The
weak NNLO corrections in the SM are also known for the
ρparameter [63,64,65,66,67,68,69,70,71,72,73,74,75,76,
77,78,79] and the weak mixing angle [80,81,82,83,84,85],
as well as corrections from Sudakov logarithms [86,87,88,
89,90,91,92,93]. Both NLO and NNLO weak effects are
negligible at low energies and are not implemented yet in
numerical packages for Bhabha scattering at meson facto
ries. In pure QED, the situation is considerably different
due to the remarkableprogress made on NNLO corrections
in recent years, as emphasised and discussed in detail in
Section 2.3.
As usual, the photonic corrections can be split into
two parts according to their kinematics. The first part
preserves the Bornlike kinematics and contains the ef
fects due to oneloop amplitudes (virtual corrections) and
single softphoton emission. Examples of Feynman dia
grams giving rise to such corrections are represented in
Fig. 5. The energy of a soft photon is assumed not to ex
ceed an energy ∆E, where E is the beam energy and the
auxiliary parameter ∆ ≪ 1 should be chosen in such a
way that the validity of the softphoton approximation is
guaranteed. The second contribution is due to hard pho
ton emission, i.e. to single bremsstrahlung with photon
energy above ∆E and corresponds to the radiative pro
cess e+e−→ e+e−γ.
Page 9
8
Following [94,95], the soft plus virtual (SV) correction
can be cast into the form
?
−8α
π
dσBhabha
B+S+V
dΩ−
=dσBhabha
dΩ−
0
1 +2α
π(L − 1)
?
2ln∆ +3
?
2
?
ln(ctgθ
2)ln∆ +α
πKBhabha
SV
,(10)
where the factor KBhabha
SV
is given by
KBhabha
SV
= −1 − 2Li2(sin2θ
1
(3 + c2)2
+3c + 21)ln2(sinθ
2) + 2Li2(cos2θ
2)
+
?π2
3(2c4− 3c3− 15c) + 2(2c4− 3c3+ 9c2
2) − 4(c4+ c2− 2c)ln2(cosθ
−4(c3+ 4c2+ 5c + 6)ln2(tgθ
−5)ln(cosθ
2)
2) + 2(c3− 3c2+ 7c
2) + 2(3c3+ 9c2+ 5c + 31)ln(sinθ
2)
?
,(11)
and depends on the scattering angle, due to the contribu
tion from initialfinalstate interference and box diagrams
(see Fig. 6). It is worth noticing that the SV correction
contains a leading logarithmic (LL) part enhanced by the
collinear logarithm L = ln(s/m2
rections there is also a numerically important effect due
to vacuum polarisation in the photon propagator. Its con
tribution is omitted in Eq. (11) but can be taken into ac
count in the standard way by insertion of the resummed
vacuum polarisation operators in the photon propagators
of the Bornlevel Bhabha amplitudes.
The differential cross section of the single hard brems
strahlung process
e). Among the virtual cor
e+(p+) + e−(p−) → e+(p′
for scattering angles up to corrections of order me/E reads
+) + e−(p′
−) + γ(k)
dσBhabha
hard
=
α3
2π2sRe¯ eγdΓe¯ eγ,
+d3p′
ε′
(12)
dΓe¯ eγ=d3p′
−d3k
−k0
m2
(χ′
?s
?s1
+ε′
δ(4)(p++ p−− p′
?s
+t1
s+ 1
?2
+− p′
−− k),
Re¯ eγ=WT
4
−
e
+)2
t+t
s+ 1
?2
?2
−
m2
(χ′
e
−)2
t1
−m2
χ2
e
+
?s1
t
+
t
s1
+ 1
?2
−m2
χ2
e
−
t1
+t1
s1
+ 1,
where
W =
s
χ+χ−
+
s1
+χ′
1) + tt1(t2+ t2
χ′
−
−
t1
+χ+
χ′
−
t
χ′
−χ−
+
u
χ′
+χ−
1)
+
u1
−χ+,
χ′
T =ss1(s2+ s2
1) + uu1(u2+ u2
ss1tt1
,
Fig. 5. Examples of Feynman diagrams for real and virtual
NLO QED initialstate corrections to the schannel contribu
tion of the Bhabha process.
and the invariants are defined as
s1= 2p′
u = −2p−p′
NLO QED radiative corrections to the twophoton an
nihilation channel were obtained in [96,97,98,99], while
weak corrections were computed in [100].
In the oneloop approximation the part of the differ
ential cross section with the Bornlike kinematics reads
?
π
??
SV=π2
32(1 + c2
1)
?
c1= cosθ1,θ1= ?
In addition, the threephoton production process
−p′
+,t = −2p−p′
u1= −2p+p′
−,
−,
t1= −2p+p′
χ±= kp±,
+,
χ′
+,
±= kp′
±.
dσγγ
B+S+V
dΩ1
=dσγγ
dΩ1
0
1 +α
?
(L − 1)
?
2ln∆ +3
2
?
+Kγγ
SV
,
Kγγ
+
1 − c2
1
??
1 +3
?
2
1 + c1
1 − c1
ln21 − c1
?
ln1 − c1
2
+1 +1 − c1
1 + c1
+1
2
1 + c1
1 − c1
q1p−.
2
+ (c1→ −c1)
?
,
(13)
e+(p+) + e−(p−) → γ(q1) + γ(q2) + γ(q3)
must be included. Its cross section is given by
dσe+e−→3γ=
α3
8π2sR3γdΓ3γ,
3+ (χ′
χ1χ2χ′
2
+(cyclic permutations),
dΓ3γ=d3q1d3q2d3q3
q0
(14)
R3γ= sχ2
3)2
1χ′
− 2m2
e
?
χ2
1+ χ2
χ1χ2(χ′
2
3)2+(χ′
1)2+ (χ′
χ′
2)2
1χ′
2χ2
3
?
1q0
2q0
3
δ(4)(p++ p−− q1− q2− q3),
where
χi= qip−,χ′
i= qip+,i = 1,2,3.
The process has to be treated as a radiative correction
to the twophoton production. The energy of the third
photon should exceed the softphoton energy threshold
∆E. In practice, the tree photon contribution, as well as
the radiative Bhabha process e+e−→ e+e−γ, should be
simulated with the help of a MC event generator in order
to take into account the proper experimental criteria of a
given event selection.
Page 10
9
Fig. 6. Feynman diagrams for the NLO QED box corrections
to the schannel contribution of the Bhabha process.
In addition to the corrections discussed above, also
the effect of vacuum polarisation, due to the insertion of
fermion loops inside the photon propagators, must be in
cluded in the precise calculation of the Bhabha scattering
cross section. Its theoretical treatment, which faces the
nontrivial problem of the nonperturbative contribution
due to hadrons, is addressed in detail in Section 6. How
ever, numerical results for such a correction are presented
in Section 2.6 and Section 2.8.
10
100
1000
10000
100000
σ (nb)
LO e+e−
NLO e+e−
LO γγ
NLO γγ
16
14
12
10
σ(LO)
8
6
4
0246810
σ(NLO)−σ(LO)
(%)
√s (GeV)
e+e−
γγ
Fig. 7. Cross sections of the processes e+e−→ e+e−and
e+e−→ γγ in LO and NLO approximation as a function of
the c.m. energy at meson factories (upper panel). In the lower
panel, the relative contribution due to the NLO QED correc
tions (in per cent) to the two processes is shown.
In Fig. 7 the cross sections of the Bhabha and two
photon production processes in LO and NLO approxima
tion are shown as a function of the c.m. energy between
√s ≃ 2mπand√s ≃ 10 GeV (upper panel). The results
were obtained imposing the following cuts for the Bhabha
process:
θmin
±
Emin
±
= 45◦,
= 0.3√s,
θmax
±
ξmax= 10◦,
= 135◦,
(15)
where θmin,max
the minimum energy thresholds for the detection of the
finalstate electron/positron and ξmax is the maximum
±
are the angular acceptance cuts, Emin
±
are
e+e−acollinearity. For the photon pair production pro
cesses we used correspondingly:
θmin
γ
Emin
γ
= 45◦,
= 0.3√s,
θmax
γ
ξmax= 10◦,
= 135◦,
(16)
where, as in Eq. (15), θmin,max
cuts, Emin
γ
is the minimum energy threshold for the de
tection of at least two photons and ξmaxis the maximum
acollinearity between the most energetic and nexttomost
energetic photon.
The cross sections display the typical 1/s QED be
haviour. The relative effect of NLO corrections is shown
in the lower panel. It can be seen that the NLO corrections
are largely negative and increase with increasing c.m. en
ergy, because of the growing importance of the collinear
logarithm L = ln(s/m2
are about one half of those to Bhabha scattering, because
of the absence of finalstate radiation effects in photon
pair production.
γ
are the angular acceptance
e). The corrections to e+e−→ γγ
2.3 NNLO corrections to the Bhabha scattering cross
section
Beyond the NLO corrections discussed in the previous Sec
tion, in recent years a significant effort was devoted to the
calculation of the perturbative corrections to the Bhabha
process at NNLO in QED.
The calculation of the full NNLO corrections to the
Bhabha scattering cross section requires three types of in
gredients: i) the twoloop matrix elements for the e+e−→
e+e−process; ii) the oneloop matrix elements for the
e+e−→ e+e−γ process, both in the case in which the ad
ditional photon is soft or hard; iii) the treelevel matrix
elements for e+e−→ e+e−γγ, with two soft or two hard
photons, or one soft and one hard photon. Also the pro
cess e+e−→ e+e−e+e−, with one of the two e+e−pairs
remaining undetected, contributes to the Bhabha signa
ture at NNLO. Depending on the kinematics, other final
states like, e.g., e+e−µ+µ−or those with hadrons are also
possible.
The advent of new calculational techniques and a deeper
understanding of the IR structure of unbroken gauge the
ories, such as QED or QCD, made the calculation of the
complete set of twoloop QED corrections possible. The
history of this calculation will be presented in Section 2.3.1.
Some remarks on the oneloop matrix elements with
three particles in the final state are in order now. The di
agrams involving the emission of a soft photon are known
and they were included in the calculations of the twoloop
matrix elements, in order to remove the IR soft diver
gences. However, although the contributions due to a hard
collinear photon are taken into account in logarithmic ac
curacy by the MC generators, a full calculation of the di
agrams involving a hard photon in a general phasespace
configuration is still missing. In Section 2.3.2, we shall
Page 11
10
comment on the possible strategies which can be adopted
in order to calculate these corrections.1
As a general comment, it must be noticed that the
fixedorder corrections calculated up to NNLO are taken
into account at the LL, and, partially, nexttoleading
log (NLL) level in the most precise MC generators, which
include, as will be discussed in Section 2.4 and Section
2.5, the logarithmically enhanced contributions of soft and
collinear photons at all orders in perturbation theory.
Concerning the tree level graphs with four particles
in the final state, the production of a soft e+e−pair was
considered in the literature by the authors of [102] by fol
lowing the evaluation of pair production [103,104] within
the calculation of the O(α2L) singlelogarithmic accurate
smallangle Bhabha cross section [43], and it is included
in the twoloop calculation (see Section 2.3.1). New re
sults on lepton and hadron pair corrections, which are at
present approximately included in the available Bhabha
codes, are presented in Section 2.3.3.
2.3.1 Virtual corrections for the e+e−→ e+e−process
The calculation of the virtual twoloop QED corrections to
the Bhabha scattering differential cross section was carried
out in the last 10 years. This calculation was made possible
by an improvement of the techniques employed in the eval
uation of multiloop Feynman diagrams. An essential tool
used to manage the calculation is the Laporta algorithm
[105,106,107,108], which enables one to reduce a generic
combination of dimensionallyregularised scalar integrals
to a combination of a small set of independent integrals
called the “Master Integrals” (MIs) of the problem under
consideration. The calculation of the MIs is then pursued
by means of a variety of methods. Particularly important
are the differential equations method [109,110,111,112,
113,114,115] and the MellinBarnes techniques [116,117,
118,119,120,121,122,123,124,125]. Both methods proved
to be very useful in the evaluation of virtual corrections
to Bhabha scattering because they are especially effective
in problems with a small number of different kinematic
parameters. They both allow one to obtain an analytic ex
pression for the integrals, which must be written in terms
of a suitable functional basis. A basis which was exten
sively employed in the calculation of multiloop Feynman
diagrams of the type discussed here is represented by the
Harmonic Polylogarithms [126,127,128,129,130,131,132,
133,134] and their generalisations. Another fundamental
achievement which enabled one to complete the calcula
tion of the QED twoloop corrections was an improved
understanding of the IR structure of QED. In particular,
the relation between the collinear logarithms in which the
electron mass me plays the role of a natural cutoff and
the corresponding poles in the dimensionally regularised
massless theory was extensively investigated in [135,136,
137,138].
1As emphasised in Section 2.8 and Section 2.9, the complete
calculation of this class of corrections became available [101]
during the completion of the present work.
The first complete diagrammatic calculation of the two
loop QED virtual corrections to Bhabha scattering can
be found in [139]. However, this result was obtained in
the fully massless approximation (me = 0) by employ
ing dimensional regularisation (DR) to regulate both soft
and collinear divergences. Today, the complete set of two
loop corrections to Bhabha scattering in pure QED have
been evaluated using me as a collinear regulator, as re
quired in order to include these fixedorder calculations in
available Monte Carlo event generators. The Feynman di
agrams involved in the calculation can be divided in three
gaugeindependent sets: i) diagrams without fermion loops
(“photonic” diagrams), ii) diagrams involving a closed
electron loop, and iii) diagrams involving a closed loop
of hadrons or a fermion heavier than the electron. Some
of the diagrams belonging to the aforementioned sets are
shown in Figs. 8–11. These three sets are discussed in more
detail below.
Photonic corrections
A large part of the NNLO photonic corrections can be
evaluated in a closed analytic form, retaining the full de
pendence on me [140], by using the Laporta algorithm
for the reduction of the Feynman diagrams to a combina
tion of MIs, and then the differential equations method for
their analytic evaluation. With this technique it is possi
ble to calculate, for instance, the NNLO corrections to the
form factors [141,142,143,144]. However, a calculation of
the twoloop photonic boxes retaining the full dependence
on meseems to be beyond the reach of this method. This
is due to the fact that the number of MIs belonging to
the same topology is, in some cases, large. Therefore, one
must solve analytically large systems of firstorder ordi
nary linear differential equations; this is not possible in
general. Alternatively, in order to calculate the different
MIs involved, one could use the MellinBarnes techniques,
as shown in [122,123,144,145,146,147], or a combination
of both methods. The calculation is very complicated and
a full result is not available yet.2However, the full depen
dence on meis not phenomenologically relevant. In fact,
the physical problem exhibits a well defined mass hierar
chy. The mass of the electron is always very small com
pared to the other kinematic invariants and can be safely
neglected everywhere, with the exception of the terms in
which it acts as a collinear regulator. The ratio of the pho
tonic NNLO corrections to the Born cross section is given
by
dσ(2,PH)
dσ(Born)=
?α
π
?2
2
?
i=0
δ(PH,i)(Le)i+ O
?m2
e
s,m2
e
t
?
, (17)
where Le= ln(s/m2
infrared logarithms and are functions of the scattering an
gle θ. The approximation given by Eq. (17) is sufficient
e) and the coefficients δ(PH,i)contain
2For the planar double box diagrams, all the MIs are known
[145] for small me, while the MIs for the nonplanar double
box diagrams are not completed.
Page 12
11
Fig. 8.
“photonic” NNLO corrections to the Bhabha scattering differ
ential cross section. The additional photons in the final state
are soft.
Some of the diagrams belonging to the class of the
for a phenomenological description of the process.3The
coefficients of the double and single collinear logarithm
in Eq. (17), δ(PH,2)and δ(PH,1), were obtained in [148,
149]. However, the precision required for luminosity mea
surements at e+e−colliders demands the calculation of
the nonlogarithmic coefficient, δ(PH,0). The latter was ob
tained in [135,136] by reconstructing the differential cross
section in the s ≫ m2
ally regularised massless approximation [139]. The main
idea of the method developed in [135,136] is outlined be
low: As far as the leading term in the small electron mass
expansion is considered, the difference between the mas
sive and the dimensionally regularised massless Bhabha
scattering can be viewed as a difference between two reg
ularisation schemes for the infrared divergences. With the
known massless twoloop result at hand, the calculation
of the massive one is reduced to constructing the infrared
matching term which relates the two above mentioned reg
ularisation schemes. To perform the matching an auxiliary
amplitude is constructed, which has the same structure of
the infrared singularities but is sufficiently simple to be
evaluated at least at the leading order in the small mass
expansion. The particular form of the auxiliary amplitude
is dictated by the general theory of infrared singularities
in QED and involves the exponent of the oneloop correc
tion as well as the twoloop corrections to the logarithm
of the electron form factor. The difference between the
full and the auxiliary amplitudes is infrared finite. It can
be evaluated by using dimensional regularisation for each
amplitude and then taking the limit of four spacetime
dimensions. The infrared divergences, which induce the
asymptotic dependence of the virtual corrections on the
electron and photon masses, are absorbed into the auxil
iary amplitude while the technically most nontrivial cal
culation of the full amplitude is performed in the massless
approximation. The matching of the massive and massless
e?= 0 limit from the dimension
3It can be shown that the terms suppressed by a positive
power of m2
at very low c.m. energies,√s ∼ 10 MeV. Moreover, the terms
m2
(backward) region, unreachable for the experimental setup.
e/s do not play any phenomenological role already
e/t (or m2
e/u) become important in the extremely forward
Fig. 9.
“electron loop” NNLO corrections. The additional photons or
electronpositron pair in the final state are soft.
Some of the diagrams belonging to the class of the
results is then necessary only for the auxiliary amplitude
and is straightforward. Thus the twoloop massless result
for the scattering amplitude along with the twoloop mas
sive electron form factor [150] are sufficient to obtain the
twoloop photonic correction to the differential cross sec
tion in the small electron mass limit.
A method based on a similar principle was subsequently
developed in [137,138]; the authors of [138] confirmed the
result of [135,136] for the NNLO photonic corrections to
the Bhabha scattering differential cross section.
Electron loop corrections
The NNLO electron loop corrections arise from the inter
ference of twoloop Feynman diagrams with the treelevel
amplitude as well as from the interference of oneloop dia
grams, as long as one of the diagrams contributing to each
term involves a closed electron loop. This set of corrections
presents a single twoloop box topology and is therefore
technically less challenging to evaluate with respect to the
photonic correction set. The calculation of the electron
loop corrections was completed a few years ago [151,152,
153,154]; the final result retains the full dependence of
the differential cross section on the electron mass me. The
MIs involved in the calculation were identified by means of
the Laporta algorithm and evaluated with the differential
equation method. As expected, after UV renormalisation
the differential cross section contained only residual IR
poles which were removed by adding the contribution of
the soft photon emission diagrams. The resulting NNLO
differential cross section could be conveniently written in
terms of 1 and 2dimensional Harmonic Polylogarithms
(HPLs) of maximum weight three. Expanding the cross
section in the limit s,t ≫ m2
corrections to the Born cross section can be written as in
Eq. (17):
e, the ratio of the NNLO
dσ(2,EL)
dσ(Born)=
?α
π
?2
3
?
i=0
δ(EL,i)(Le)i+ O
?m2
e
s,m2
e
t
?
. (18)
Note that the series now contains a cubic collinear log
arithm. This logarithm appears, with an opposite sign,
Page 13
12
in the corrections due to the production of an electron
positron pair (the softpair production was considered in
[102]). When the two contributions are considered together
in the full NNLO, the cubic collinear logarithms cancel.
Therefore, the physical cross section includes at most a
double logarithm, as in Eq. (17).
The explicit expression of all the coefficients δ(EL,i),
obtained by expanding the results of [151,152,153], was
confirmed by two different groups [138,154]. In [138] the
small electron mass expansion was performed within the
softcollinear effective theory (SCET) framework, while
the analysis in [154] employed the asymptotic expansion
of the MIs.
Heavyflavor and hadronic corrections
Finally, we consider the corrections originating from two
loop Feynman diagrams involving a heavy flavour fermion
loop.4Since this set of corrections involves one more mass
scale with respect to the corrections analysed in the previ
ous sections, a direct diagrammatic calculation is in prin
ciple a more challenging task. Recently, in [138] the au
thors applied their technique based on SCET to Bhabha
scattering and obtained the heavy flavour NNLO correc
tions in the limit in which s,t,u ≫ m2
m2
fis the mass of the heavy fermion running in the loop.
Their result was very soon confirmed in [154] by means of
a method based on the asymptotic expansion of Mellin
Barnes representations of the MIs involved in the calcula
tion. However, the results obtained in the approximation
s,t,u ≫ m2
which√s < mf (as in the case of a tau loop at√s ∼ 1
GeV), and they apply only to a relatively narrow angular
region perpendicular to the beam direction when√s is
not very much larger than mf(as in the case of topquark
loops at the ILC). It was therefore necessary to calculate
the heavy flavour corrections to Bhabha scattering assum
ing only that the electron mass is much smaller than the
other scales in the process, but retaining the full depen
dence on the heavy mass, s,t,u,m2
The calculation was carried out in two different ways:
in [155,156] it was done analytically, while in [157,158] it
was done numerically with dispersion relations.
The technical problem of the diagrammatic calculation
of Feynman integrals with four scales can be simplified
by considering carefully, once more, the structure of the
collinear singularities of the heavyflavourcorrections. The
ratio of the NNLO heavy flavour corrections to the Born
cross section is given by
f≫ m2
e, where
f≫ m2
ecannot be applied to the case in
f≫ m2
e.
dσ(2,HF)
dσ(Born)=
?α
π
?2
1
?
i=0
δ(HF,i)(Le)i+ O
?m2
e
s,m2
e
t
?
, (19)
where now the coefficients δ(i)are functions of the scat
tering angle θ and, in general, of the mass of the heavy
4Here by “heavy flavour” we mean a muon or a τlepton,
as well as a heavy quark, like the top, the b or the cquark,
depending on the c.m. energy range that we are considering.
Fig. 10. Some of the diagrams belonging to the class of the
“heavy fermion” NNLO corrections. The additional photons in
the final state are soft.
fermions involved in the virtual corrections. It is possi
ble to prove that, in a physical gauge, all the collinear
singularities factorise and can be absorbed in the exter
nal field renormalisation [159]. This observation has two
consequences in the case at hand. The first one is that
box diagrams are free of collinear divergences in a phys
ical gauge; since the sum of all boxes forms a gauge in
dependent block, it can be concluded that the sum of
all box diagrams is free of collinear divergences in any
gauge. The second consequence is that the single collinear
logarithm in Eq. (19) arises from vertex corrections only.
Moreover, if one chooses onshell UV renormalisation con
ditions, the irreducible twoloop vertex graphs are free of
collinear singularities. Therefore, among all the twoloop
diagrams contributing to the NNLO heavy flavour cor
rections to Bhabha scattering, only the reducible vertex
corrections are logarithmically divergent in the me → 0
limit.5The latter are easily evaluated even if they depend
on two different masses. By exploiting these two facts,
one can obtain the NNLO heavyflavour corrections to
the Bhabha scattering differential cross section assuming
only that s,t,u,m2
me= 0 from the beginning in all the twoloop diagrams
with the exception of the reducible ones. This procedure
allows one to effectively eliminate one mass scale from
the twoloop boxes, so that these graphs can be evalu
ated with the techniques already employed in the dia
grammatic calculation of the electron loop corrections.6
In the case in which the heavy flavour fermion is a quark,
it is straightforward to modify the calculation of the two
loop selfenergy diagrams to obtain the mixed QEDQCD
corrections to Bhabha scattering [156].
An alternative approach to the calculation of the heavy
flavour corrections to Bhabha scattering is based on dis
persion relations. This method also applies to hadronic
corrections. The hadronic and heavy fermion corrections
to the Bhabhascattering cross section can be obtained by
f≫ m2
e. In particular, one can set
5Additional collinear logarithms arise also from the inter
ference of oneloop diagrams in which at least one vertex is
present.
6The necessary MIs can be found in [156,160,161,162].
Page 14
13
appropriately inserting the renormalised irreducible pho
ton vacuumpolarisation function Π in the photon propa
gator:
gµν
q2+ iδ
→
gµα
q2+ iδ
?q2gαβ− qαqβ?Π(q2)
gβν
q2+ iδ.
(20)
The vacuum polarisation Π can be represented by a once
subtracted dispersion integral [12],
Π(q2) = −q2
π
?∞
4M2dzImΠ(z)
z
1
q2− z + iδ.
(21)
The contributions to Π may then be determined from a
(properly normalised) production cross section by the op
tical theorem [163],
ImΠhad(z) = −α
3R(z). (22)
In this way, the hadronic vacuum polarisation may be ob
tained from the experimental data for R:
R(z) =
σ0
had(z)
(4πα2)/(3z), (23)
where σ0
lowenergy region the inclusive experimental data may be
used [35,164]. Around a narrow hadronic resonance with
mass Mresand width Γe+e−
res
had(z) ≡ σ({e+e−→ γ⋆→ hadrons};z). In the
one may use the relation
Rres(z) =9π
α2MresΓe+e−
res
δ(z − M2
res), (24)
and in the remaining regions the perturbative QCD pre
diction [165]. Contributions to Π arising from leptons and
heavy quarks with mass mf, charge Qfand colour Cfcan
be computed directly in perturbation theory. In the lowest
order it reads
Rf(z;mf) = Q2
fCf
?
1 + 2m2
f
z
??
1 − 4m2
f
z
. (25)
As a result of the above formulas, the massless photon
propagator gets replaced by a massive propagator, whose
effective mass z is subsequently integrated over:
?∞
gµν
q2+ iδ→
α
3π
4M2
dz Rtot(z)
z(q2− z + iδ)
?
gµν−
qµqν
q2+ iδ
?
,
(26)
where Rtot(z) contains hadronic and leptonic contribu
tions.
For selfenergy corrections to Bhabha scattering at one
loop order, the dispersion relation approach was first em
ployed in [166]. Twoloop applications of this technique,
prior to Bhabha scattering, are the evaluation of the had
ronic vertex correction [167] and of twoloop hadronic cor
rections to the lifetime of the muon [168]. The approach
was also applied to the evaluation of the twoloop form
factors in QED in [169,170,171].
The fermionic and hadronic corrections to Bhabha scat
tering at oneloop accuracy come only from the selfenergy
diagram; see for details Section 6. At twoloop level there
are reducible and irreducible selfenergy contributions, ver
tices and boxes. The reducible corrections are easily treat
ed. For the evaluation of the irreducible twoloop dia
grams, it is advantageous that they are oneloop diagrams
with selfenergy insertions because the application of the
dispersion technique as described here is possible.
The kernel function for the irreducible twoloop vertex
was derived in [167] and verified e.g. in [158]. The three
kernel functions for the twoloop box functions were first
obtained in [172,157,158] and verified in [173]. A complete
collection of all the relevant formulae may be found in
[158], and the corresponding Fortran code bhbhnnlohf is
publicly available at the web page [174]
wwwzeuthen.desy.de/theory/research/bhabha/ .
In [158], the dependence of the various heavy fermion
NNLO corrections on ln(s/m2
studied. The irreducible vertex behaves (before a combi
nation with real pair emission terms) like ln3(s/m2
while the sum of the various infrared divergent diagrams
as a whole behaves like ln(s/m2
cordance with Eq. (19), but the limit plays no effective
role at the energies studied here.
As a result of the efforts of recent years we now have at
least two completely independent calculations for all the
nonphotonic virtual twoloop contributions. The net re
sult, as a ratio of the NNLO corrections to the Born cross
section in per mill, is shown in Fig. 12 for KLOE and in
Fig. 13 for BaBar/Belle.7While the nonphotonic correc
tions stay at one per mill or less for KLOE, they reach a
few per mill at the BaBar/Belle energy range. The NNLO
photonic corrections are the dominant contributions and
amount to some per mill, both at φ and B factories. How
ever, as already emphasised, the bulk of both photonic and
nonphotonic corrections is incorporated into the genera
tors used by the experimental collaborations. Hence, the
consistent comparison between the results of NNLO cal
culations and the MC predictions at the same perturba
tive level enables one to assess the theoretical accuracy of
the luminosity tools, as will be discussed quantitatively in
Section 2.8.
f) for s,t,u ≫ m2
fwas
f) [167],
f)ln(s/m2
e). This is in ac
2.3.2 Fixedorder calculation of the hard photon emission at
one loop
The oneloop matrix element for the process e+e−→
e+e−γ is one of the contributions to the complete set of
NNLO corrections to Bhabha scattering. Its evaluation
requires the nontrivial computation of oneloop tensor in
tegrals associated with pentagon diagrams.
According to the standard PassarinoVeltman (PV)
approach [176], oneloop tensor integrals can be expressed
in terms of MIs with trivial numerators that are indepen
dent of the loop variable, each multiplied by a Lorentz
7The pure selfenergy corrections deserve a special discus
sion and are thus omitted in the plots.
Page 15
14
Fig. 11. Some of the diagrams belonging to the class of the
“hadronic” corrections. The additional photons in the final
state are soft.
20 40
60
80100 120 140
160
θ
0
2
4
6
103 * dσ2/dσ0
photonic
muon
electron
total nonphotonic
hadronic
s = 1.04 GeV
2
Fig. 12. Twoloop photonic and nonphotonic corrections to
Bhabha scattering at√s = 1.02 GeV, normalised to the QED
treelevel cross section, as a function of the electron polar angle;
no cuts; the parameterisations of Rhad from [175] and [35,164,
165] are very close to each other.
structure depending only on combinations of the external
momenta and the metric tensor. The achievement of the
complete PVreduction amounts to solving a nontrivial
system of equations. Due to its size, it is reasonable to re
place the analytic techniques by numerical tools. It is dif
ficult to implement the PVreduction numerically, since it
gives rise to Gram determinants. The latter naturally arise
in the procedure of inverting a system and they can vanish
at special phase space points. This fact requires a proper
modification of the reduction algorithm [177,178,179,180,
181,182,183]. A viable solution for the complete algebraic
reduction of tensorpentagon (and tensorhexagon) inte
grals was formulated in [184,185,186], by exploiting the
algebra of signed minors [187]. In this approach the can
cellation of powers of inverse Gram determinants was per
formed recently in [188,189].
Alternatively, the computation of the oneloop five
point amplitude e+e−→ e+e−γ can be performed by
20 40
60
80 100120140
160
θ
0
5
103 * dσ2/d σ0
photonic
muon
electron
total nonphotonic
hadronic
s1/2 = 10.56 GeV
Fig. 13. Twoloop photonic and nonphotonic corrections to
Bhabha scattering at√s = 10.56 GeV, normalised to the QED
treelevel cross section, as a function of the electron polar angle;
no cuts; the parameterisations of Rhad is from [175].
using generalisedunitarity cutting rules (see [190] for a
detailed compilation of references). In the following we
propose two ways to achieve the result, via an analyti
cal and via a seminumerical method. The application of
generalised cutting rules as an onshell method of calcula
tion is based on two fundamental properties of scattering
amplitudes: i) analyticity, according to which any ampli
tude is determined by its singularity structure [191,192,
193,163,194]; and ii) unitarity, according to which the
residues at the singularities are determined by products
of simpler amplitudes. Turning these properties into a
tool for computing scattering amplitudes is possible be
cause of the underlying representation of the amplitude
in terms of Feynman integrals and their PVreduction,
which grants the existence of a representation of any one
loop amplitudes as linear combination of MIs, each mul
tiplied by a rational coefficient. In the case of e+e−→
e+e−γ, pentagonintegralsmay be expressed, through PV
reduction, by a linear combination of 17 MIs (including 3
boxes, 8 triangles, 5 bubbles and 1 tadpole). Since the re
quired MIs are analytically known [195,196,197,185,179,
198,199], the determination of their coefficients is needed
for reconstructing the amplitude as a whole. Matching the
generalised cuts of the amplitude with the cuts of the
MIs provides an efficient way to extract their (rational)
coefficients from the amplitude itself. In general the ful
filment of multiplecut conditions requires loop momenta
with complex components. The effect of the cut conditions
is to freeze some or all of its components, depending on
the number of the cuts. With the quadruplecut [200] the
loop momentum is completely frozen, yielding the alge
braic determination of the coefficients of npoint functions
with n ≥ 4. In cases where fewer than four denominators
are cut, like triplecut [201,202,203], doublecut [204,205,
206,207,208,202] and singlecut [209], the loop momen
Page 16
15
tum is not frozen: the free components are left over as
phasespace integration variables.
For each multiplecut, the evaluation of the phase
space integral would generate, in general, logarithms and
a nonlogarithmic term. The coefficient of a given npoint
MI finally appears in the nonlogarithmic term of the cor
responding nparticle cut, where all the internal lines are
onshell (while the logarithms correspond to the cuts of
higherpoint MIs which share that same cut). Therefore
all the coefficients of MIs can be determined in a top
down algorithm, starting from the quadruplecuts for the
extraction of the fourpoint coefficients, and following with
the triple, double and singlecuts for the coefficients of
three, two and onepoint, respectively. The coefficient of
an npoint MI (n ≥ 2) can also be obtained by specialising
the generating formulas given in [210] for general oneloop
amplitudes to the case at hands.
Instead of the analytic evaluation of the multiplecut
phasespace integrals, it is worth considering the feasibil
ity of computing the process e+e−→ e+e−γ with a semi
numerical technique by now known as OPPreduction[211,
212], based on the decomposition of the numerator of any
oneloop integrand in terms of its denominators [213,214,
215,216]. Within this approach the coefficients of the MIs
can be found simply by solving a system of numerical
equations, avoiding any explicit integration. The OPP
reduction algorithm exploits the polynomial structures of
the integrand when evaluated at values of the loopmo
mentum fulfilling multiple cutconditions: i) for each n
point MI one considers the nparticle cut obtained by set
ting all the propagating lines onshell; ii) such a cut is
associated with a polynomial in terms of the free com
ponents of the loopmomentum, which corresponds to the
numerator of the integrand evaluated at the solution of the
onshell conditions; iii) the constantterm of that polyno
mial is the coefficient of the MI.
Hence the difficult task of evaluating oneloop Feynman
integrals is reduced to the much simpler problem of poly
nomial fitting, recently optimised by using a projection
technique based on the Discrete Fourier Transform [217].
In general the result of a dimensionalregulated ampli
tude in the 4dimensional limit, with D (= 4−2ǫ) the regu
lating parameter, is expected to contain (poly)logarithms,
often referred to as the cutconstructible term, and a pure
rational term. In a later paper [218], which completed
the OPPmethod, the rising of the rational term was at
tributed to two potential sources (of UVdivergent inte
grals): one, defined as R1, due to the Ddimensional com
pletion of the 4dimensional contribution of the numera
tor; a second one, called R2, due to the (−2ǫ)dimensional
algebra of Diracmatrices. Therefore in the OPPapproach
the calculation of the oneloop amplitude e+e−→ e+e−γ
can proceed through two computational stages:
1. the coefficients of the MIs that are responsible both
for the cutconstructible and for the R1rational terms
can be determined by applying the OPPreduction dis
cussed above [211,212,217];
2. the R2rational term can be computed by using addi
tional treelevellike diagrammatic rules, very much re
Table 2. The NNLO lepton and pion pair corrections to the
Bhabha scattering Born cross section σB: virtual corrections σv
, soft and hard real photon emissions σs,σh, and pair emission
contributions σpairs. The total pair correction cross sections
are obtained from the sum σs+v+h+σpairs. All cross sections,
according to the cuts given in the text, are given in nanobarns.
Electron pair corrections
σh
529.469 9.502
6.744 0.246
Muon pair corrections
σB
σh
529.4691.494
6.7440.091
Tau pair corrections
σB
σh
529.4690.020
6.7440.016
Pion pair corrections
σB
σh
529.4691.174
6.7440.062
σB
σv+s
11.567
0.271
σv+s+h
2.065
0.025
σpairs
0.271
0.017
KLOE
BaBar
σv+s
1.736
0.095
σv+s+h
0.241
0.004
σpairs
–
0.0005
KLOE
BaBar
σv+s
0.023
0.017
σv+s+h
0.003
0.0007
σpairs
–
< 10−7
KLOE
BaBar
σv+s
1.360
0.065
σv+s+h
0.186
0.003
σpairs
–
0.00003
KLOE
BaBar
sembling the computation of the counter terms needed
for the renormalisation of UVdivergences [218].
The numerical influence of the radiative loop diagrams,
including the pentagon diagrams, is expected not to be
particularly large. However, the calculation of such correc
tions would greatly help to assess the physical precision of
existing luminosity programs.8
2.3.3 Pair corrections
As was mentioned in the paragraph on virtual heavy fla
vour and hadronic corrections of Section 2.3.1, these vir
tual corrections have to be combined with real correc
tions in order to get physically sensible results. The virtual
NNLO electron, muon, tau and pion corrections have to
be combined with the emission of real electron, muon, tau
and pion pairs, respectively. The real pair production cross
sections are finite, but cut dependent. We consider here
the pion pair production as it is the dominant part of the
hadronic corrections and can serve as an estimate of the
role of the whole set of hadronic corrections. The descrip
tion of all relevant hadronic contributions is a much more
involved task and will not be covered in this review. As
was first explicitly shown for Bhabha scattering in [102]
for electron pairs, and also discussed in [158], there ap
pear exact cancellations of terms of the order ln3(s/m2
or ln3(s/m2
f), so that the leading terms are at most of
order ln2(s/m2
f).
e)
e),ln2(s/m2
8As already remarked, the exact calculation of oneloop cor
rections to hard photon emission in Bhabha scattering became
available [101] during the completion of the report, exactly ac
cording to the methods described in the present Section.
Page 17
16
In Table 2 we show NNLO lepton and pion pair con
tributions with typical kinematical cuts for the KLOE
and BaBar experiments. Besides contributions from un
resolved pair emissions σpairs, we also add unresolved real
hard photon emission contributions σh. The corrections
σpairs from fermions have been calculated with the For
tran package HELACPHEGAS [219,220,221,222], the real
pion corrections with EKHARA [223,224], the NNLO hard
photonic corrections σhwith a program [225] based on the
generator BHAGEN1PH [226]. The latter depend, tech
nically, on the soft photon cutoff Emin
up with σv+s, the sum of the two σv+s+his independent
of that; in fact here we use ω/Ebeam= 10−4. In order to
cover also pion pair corrections σv+sis determined with an
updated version of the Fortran package bhbhnnlohf [158,
174]. The cuts applied in Table 2 for the KLOE experi
ment are
–√s = 1.02 GeV,
– Emin= 0.4 GeV,
– 55◦< θ±< 125◦,
– ξmax= 9◦,
and for the BaBar experiment
–√s = 10.56 GeV,
– cos(θ±) < 0.7 and
cos(θ+) < 0.65 or cos(θ−) < 0.65,
– p+/Ebeam> 0.75 and p−/Ebeam> 0.5 or
p−/Ebeam> 0.75 and p+/Ebeam> 0.5,
– ξ3d
Here Eminis the energy threshold for the finalstate elec
tron/positron, θ± are the electron/positron polar angles
and ξmaxis the maximum allowed polar angle acollinear
ity:
γ
= ω. After adding
max= 30◦.
ξ = θ++ θ−− 180◦,(27)
and ξ3d
linearity:
maxis the maximum allowed three dimensional acol
ξ3d=
????arccos
?
p+· p−
(p−p+
?
×180◦
π
− 180◦
????. (28)
For e+e−→ e+e−µ+µ−, cuts are applied only to the e+e−
pair. In the case of e+e−→ e+e−e+e−, all possible e±e∓
combinations are checked and if at least one pair fulfils
the cuts the event is accepted.
At KLOE the electron pair corrections contribute about
3×10−3and at BaBar about 1×10−3, while all the other
contributions of pair production are even smaller. Like in
smallangle Bhabha scattering at LEP/SLC the pair cor
rections [227] are largely dominated by the electron pair
contribution.
2.4 Multiple photon effects and matching with NLO
corrections
2.4.1 Universal methods for leading logarithmic corrections
From inspection of Eqs. (10) and (13) for the SV NLO
QED corrections to the cross section of the Bhabha scat
tering and e+e−→ γγ process, it can be seen that large
logarithms L = ln(s/m2
sion, are present. Similar large logarithmic terms arise af
ter integration of the hard photon contributions from the
kinematical domains of photon emission at small angles
with respect to charged particles. For the energy range
of meson factories the logarithm is large numerically, i.e.
L ∼ 15 at the φ factories and L ∼ 20 at the B factories,
and the corresponding terms give the bulk of the total ra
diative correction. These contributions represent also the
dominant part of the NNLO effects discussed in Section
2.3. Therefore, to achieve the required theoretical accu
racy, the logarithmically enhanced contributions due to
emission of soft and collinear photons must be taken into
account at all orders in perturbation theory. The meth
ods for the calculation of higherorder (HO) QED correc
tions on the basis of the generators employed nowadays at
flavour factories were already widely and successfully used
in the 90s at LEP/SLC for electroweak tests of the SM.
They were adopted for the calculation of both the small
angle Bhabha scattering cross section (necessary for the
highprecision luminosity measurement) and Zboson ob
servables. Hence, the theory accounting for the control of
HO QED corrections at meson factories can be considered
particularly robust, having passed the very stringent tests
of the LEP/SLC era.
The most popular and standard methods to keep mul
tiple photon effects under control are the QED Structure
Function (SF) approach [228,229,230,231] and Yennie
FrautschiSuura (YFS) exponentiation [232]. The former
is used in all the versions of the generator BabaYaga [233,
234,235] and MCGPJ [236] (albeit according to differ
ent realisations), while the latter is the theoretical recipe
adopted in BHWIDE [237]. Actually, analytical QED SFs
D(x,Q2), valid in the strictly collinear approximation,
are implemented in MCGPJ, whereas BabaYaga is based
on a MC Parton Shower (PS) algorithm to reconstruct
D(x,Q2) numerically.
e), due to collinear photon emis
The Structure Function approach
Let us consider the annihilation process e−e+→ X,
where X is some given final state and σ0(s) its LO cross
section. Initialstate (IS) QED radiative corrections can
be described according to the following picture. Before
arriving at the annihilation point, the incoming electron
(positron) of fourmomentum p−(+)radiates real and vir
tual photons. These photons, due to the dynamical fea
tures of QED, are mainly radiated along the direction of
motion of the radiating particles, and their effect is mainly
to reduce the original fourmomentum of the incoming
electron (positron) to x1(2)p−(+). After this preemission,
the hard scattering process e−(x1p−)e+(x2p+) → X takes
place, at a reduced squared c.m. energy ˆ s = x1x2s. The
resulting cross section, corrected for IS QED radiation,
can be represented in the form [228,229,230]
σ(s) =
?1
0
dx1dx2D(x1,s)D(x2,s)σ0(x1x2s)Θ(cuts),
(29)
Page 18
17
where D(x,s) is the electron SF, representing the prob
ability that an incoming electron (positron) radiates a
collinear photon, retaining a fraction x of its original mo
mentum at the energy scale Q2= s, and Θ(cuts) stands
for a rejection algorithm taking care of experimental cuts.
When considering photonic radiation only the nonsinglet
part of the SF is of interest. If the running of the QED
coupling constant is neglected, the nonsinglet part of the
SF is the solution of the following Renormalisation Group
(RG) equation, analogous to the DokshitzerGribovLipa
tovAltarelliParisi (DGLAP) equation of QCD [238,239,
240]:
s∂
∂sD(x,s) =
α
2π
?1
x
dz
zP+(z)D
?x
z,s
?
, (30)
where P+(z) is the regularised AltarelliParisi (AP) split
ting function for the process electron → electron+photon,
given by
P+(z) = P(z) − δ(1 − z)
P(z) =1 + z2
1 − z.
Equation (30) can be also transformed into an integral
equation, subject to the boundary condition D(x,m2
δ(1 − x):
?s
?1
0
dxP(x),
(31)
e) =
D(x,s) = δ(1−x)+α
2π
m2
e
dQ2
Q2
?1
x
dz
zP+(z)D
?x
z,Q2?
.
(32)
Equation (32) can be solved exactly by means of nu
merical methods, such as the inverse Mellin transform
method. However, this derivation of D(x,s) turns out be
problematic in view of phenomenologicalapplications. There
fore, approximate (but very accurate) analytical repre
sentations of the solution of the evolution equation are
of major interest for practical purposes. This type of so
lution was the one typically adopted in the context of
LEP/SLC phenomenology. A first analytical solution can
be obtained in the soft photon approximation, i.e. in the
limit x ≃ 1. This solution, also known as GribovLipatov
(GL) approximation, exponentiates the large logarithmic
contributions of infrared and collinear origin at all per
turbative orders, but it does not take into account hard
photon (collinear) effects. This drawback can be overcome
by solving the evolution equation iteratively. At the nth
step of the iteration, one obtains the O(αn) contribution
to the SF for any value of x. By combining the GL solu
tion with the iterative one, in which the softphoton part
has been eliminated in order to avoid double counting, one
can build a hybrid solution of the evolution equation. It
exploits all the positive features of the two kinds of so
lutions and is not affected by the limitations intrinsic to
each of them. Two classes of hybrid solutions, namely the
additive and factorised ones, are known in the literature,
and both were adopted for applications to LEP/SLC pre
cision physics. A typical additive solution, where the GL
approximation DGL(x,s) is supplemented by finiteorder
terms present in the iterative solution, is given by [241]
DA(x,s) =
3
?
i=0
d(i)
A(x,s),
d(0)
A(x,s) =exp?1
d(1)
2β?3
4− γE
??
Γ?1 +1
4β(1 + x),
2β?
1
2β(1 − x)
1
2β−1,
A(x,s) = −1
d(2)
A(x,s) =
1
32β2[(1 + x)(−4ln(1 − x) + 3lnx)
−4lnx
1
384β3{(1 + x)[18ζ(2) − 6Li2(x)
−12ln2(1 − x)?+
+1
2(1 + 7x2)ln2x − 12(1 + x2)lnxln(1 − x)
−6(x + 5)(1 − x)ln(1 − x)
−1
1 − x− 5 − x
?
,
d(3)
A(x,s) =
1
1 − x
?
−3
2(1 + 8x + 3x2)lnx
4(39 − 24x − 15x2)
??
,(33)
where Γ is the Euler gammafunction, γE ≈ 0.5772 the
EulerMascheroni constant, ζ the Riemann ζfunction and
β is the large collinear factor
?
Explicit examples of factorised solutions, which are
obtained by multiplying the GL solution by finiteorder
terms in such a way that, order by order, the iterative
contributions are exactly recovered, can be found in [242].
For the calculation of HO corrections with a per mill ac
curacy analytical SFs in additive and factorised form con
taining up to O(α3) finiteorder terms are sufficient and
in excellent agreement. They also agree with an accuracy
much better than 0.1 with the exact numerical solution of
the QED evolution equation. Explicit solutions up to the
fifth order in α were calculated in [243,244].
The RG method described above was applied in [245]
for the treatment of LL QED radiative corrections to var
ious processes of interest for physics at meson factories.
Such a formulation was later implemented in the genera
tor MCGPJ. For example, according to [245], the Bhabha
scattering cross section, accounting for LL terms in all
orders, O(αnLn), n = 1,2,..., of perturbation theory, is
given by
?1
?1
?
β =2α
π
ln
?
s
m2
e
?
− 1
?
.(34)
dσBhabha
LLA
=
?
a,b,c,d=e±,γ
¯ z1
dz1
?1
¯ z2
dz2Dstr
ae−(z1)Dstr
be+(z2)
×dσab→cd
0
(z1,z2)
¯ y1
dy1
Y1
Dfrg
e−c(y1
Y1)
?1
¯ y2
dy2
Y2
Dfrg
e+d(y2
Y2)
+Oα2L,αm2
e
s
?
.(35)
Page 19
18
Here dσab→cd
the process ab → cd, with energy fractions of the incoming
particles being scaled by factors z1and z2with respect to
the initial electron and positron, respectively. In the nota
tion of [245], the electron SF Dstr
the electron fragmentation function Dfrg
the role played by IS radiation (described by Dstr
respect to the one due to finalstate radiation (described
by Dfrg
ab(z)). However, because of their probabilistic mean
ing, the electron structure and fragmentation functions
coincide. In Eq. (35) the quantities Y1,2 are the energy
fractions of particles c and d with respect to the beam
energy. Explicit expressions for Y1,2 = Y1,2(z1,z2,cosθ)
and other details on the kinematics can be found in [245].
The lower limits of the integrals, ¯ z1,2and ¯ y1,2, should be
defined according to the experimental conditions of par
ticle detection and kinematical constraints. For the case
of the e+e−→ γγ process one has to change the mas
ter formula (35) by picking up the twophoton final state.
Formally this can be done by just choosing the proper
fragmentation functions, Dfrg
The photonic part of the nonsinglet electron structure
(fragmentation) function in O(αnLn) considered in [245]
reads
0
(z1,z2) is the differential LO cross section of
ab(z) is distinguished from
ab(z) to point out
ab(z)) with
γcand Dfrg
γd.
DNS,γ
ee
(z) = δ(1 − z) +
α
2π(L − 1)Pγe(z) + O(α2L2),
Deγ(z) =
2πLPeγ(z) + O(α2L2),
?1 + z2
?
?
z
n
?
i=1
?α
2π(L − 1)
?i1
i!
?
P(0)
ee(z)
?⊗i
,
Dγe(z) =
α
P(0)
ee(z) =
1 − z
δ(1 − z)(2ln∆ +3
?
?
+
= lim
∆→0
2) + Θ(1 − z − ∆)1 + z2
1 − z
?
,
P(0)
ee(z)
?⊗i
=
1
dt
tP(i−1)
ee
(t)P(0)
ee
?z
t
?
,(36)
Pγe(z) = z2+ (1 − z)2,Peγ(z) =1 + (1 − z)2
z
.
Starting from the second order in α there appear also non
singlet and singlet e+e−pair contributions to the struc
ture function:
?α
DS,e+e−
ee
2!
R(z) = Peγ⊗ Pγe(z) =1 − z
+2(1 + z)lnz.
DNS,e+e−
ee
(z) =1
32πL
?2
?2
P(1)
ee(z) + O(α3L3),
(z) =1
?α
2πL
R(z) + O(α3L3),
3z
(4 + 7z + 4z2)
(37)
Note that radiation of a real pair, i.e. appearance of addi
tional electrons and positrons in the final state, require the
application of nontrivial conditions of experimental par
ticle registration. Unambiguously, that can be done only
within a MC event generator based on fourparticle matrix
elements, as already discussed in Section 2.3.
In the same way as in QCD, the LL cross sections de
pend on the choice of the factorisation scale Q2in the
argument of the large logarithm L = ln(Q2/m2
not fixed a priori by the theory. However, the scale should
be taken of the order of the characteristic energy trans
fer in the process under consideration. Typical choices
are Q2= s, Q2= −t and Q2= st/u. The first one is
good for annihilation channels like e+e−→ µ+µ−, the
second one is optimal for smallangle Bhabha scattering
where the tchannel exchange dominates, see [246]. The
last choice allows to exponentiate the leading contribu
tion due to initialfinal state interference [247] and is par
ticularly suited for largeangle Bhabha scattering in QED.
The option Q2= st/u is adopted in all the versions of the
generator BabaYaga. Reduction of the scale dependence
can be achieved by taking into account nexttoleading
corrections in O(αnLn−1), nexttonexttoleading ones in
O(αnLn−2) etc.
e), which is
The Parton Shower algorithm
The PS algorithm is a method for providing a MC it
erative solution of the evolution equation and, at the same
time, for generating the fourmomenta of the electron and
photon at a given step of the iteration. It was developed
within the context of QCD and later applied in QED too.
In order to implement the algorithm, it is first nec
essary to assume the existence of an upper limit for the
energy fraction x in such a way that the AP splitting func
tion is regularised by writing
P+(z) = θ(x+− z)P(z) − δ(1 − z)
?x+
0
dxP(x).(38)
Of course, in the limit x+→ 1, Eq. (38) recovers the usual
definition of the AP splitting function given in Eq. (31).
By inserting the modified AP vertex into Eq. (30), one
obtains
?x+
−α
s∂
∂sD(x,s) =
α
2π
x
dz
zP(z)D
?x+
?x
dzP(z).
z,s
?
2πD(x,s)
x
(39)
Separating the variables and introducing the Sudakov form
factor
?
which is the probability that the electron evolves from
virtuality −s2to −s1without emitting photons of energy
fraction larger than 1 − x+≡ ǫ (ǫ ≪ 1), Eq. (39) can be
recast into the integral form
Π(s1,s2) = exp−α
2π
?s1
s2
ds′
s′
?x+
0
dzP(z)
?
, (40)
D(x,s) = Π(s,m2
+α
e)D(x,m2
ds′
s′Π(s,s′)
e)
2π
?s
m2
e
?x+
x
dz
zP(z)D
?x
z,s′?
.
(41)
Page 20
19
The formal iterative solution of Eq. (41) can be repre
sented by the infinite series
??si−1
×α
2π
x/(z1···zi−1)
D(x,s) =
∞
?
n=0
n
?
i=1
m2
e
dsi
si
?
Π(si−1,si)
?x+
dzi
zi
P(zi)Π(sn,m2
e)D
?
x
z1···zn,m2
e
?
.
(42)
The particular form of Eq. (42) allows to exploit a MC
method for building the solution iteratively. The steps of
the algorithm are as follows:
1 – set Q2= m2
condition D(x,m2
2 – generate a random number ξ in the interval [0,1];
3 – if ξ < Π(s,Q2) stop the evolution; otherwise
4 – compute Q′2as solution of the equation ξ = Π(Q′2,Q2);
5 – generate a random number z according to the proba
bility density P(z) in the interval [0,x+];
6 – substitute x → xz and Q2→ Q′2; go to 2.
The x distribution of the electron SF as obtained by
means of the PS algorithm and a numerical solution (based
on the inverse Mellin transform method) of the QED evo
lution equation is shown in Fig. 14. Perfect agreement is
seen. Once D(x,s) has been reconstructed by the algo
rithm, the master formula of Eq. (29) can be used for
the calculation of LL corrections to the cross section of
interest. This cross section must be independent of the
softhard photon separator ǫ in the limit of small values
for ǫ. This can be clearly seen in Fig. 15, where the QED
corrected Bhabha cross section as a function of the fic
titious parameter ε is shown for DAΦNE energies with
the cuts of Eq. (15), but for an angular acceptance θ±
of 55◦÷ 125◦. The cross section reaches a plateau for ǫ
smaller than 10−4.
The main advantage of the PS algorithm with respect
to the analytical solutions of the electron evolution is the
possibility of going beyond the strictly collinear approxi
mation and generating transverse momentum p⊥of elec
trons and photons at each branching. In fact, the kine
matics of the branching process e(p) → e′(p′) + γ(q) can
be written as
e, and fix x = 1 according to the boundary
e) = δ(1 − x);
p = (E,0,pz),
p′= (zE,p⊥,p′
q = ((1 − z)E,−p⊥,qz).
z),
(43)
Once the variables p2, p′2and z are generated by the PS
algorithm, the onshell condition q2= 0, together with the
longitudinal momentum conservation, allows to obtain an
expression for the p⊥variable:
p2
⊥= (1 − z)(zp2− p′2),(44)
valid at first order in p2/E2≪ 1, p2
However, due to the approximationsinherent to Eq. (44),
this PS approach can lead to an incorrect behaviour of the
⊥/E2≪ 1.
Fig. 14. Comparison for the x distribution of the electron SF
as obtained by means of a numerical solution of the QED evo
lution equation (solid line) and the PS algorithm (histogram).
From [233].
467.48
467.49
467.5
467.51
467.52
467.53
467.54
467.55
467.56
1e081e071e061e051e040.001 0.01
σ(nb)
ε
Fig. 15. QED corrected Bhabha cross section at DAΦNE as
a function of the infrared regulator ε of the PS approach, ac
cording to the setup of Eq. (15). The error bars correspond to
1σ MC errors. From [235].
reconstruction of the exclusive photon kinematics. First
of all, since within the PS algorithm the generation of p′2
and z are independent, it can happen that in some branch
ings the p2
⊥as given by Eq. (44) is negative. In order to
avoid this problem, the introduction of any kinematical
cut on the p2or z generation (or the regeneration of the
whole event) would prevent the correct reconstruction of
the SF x distribution, which is important for a precise
cross section calculation. Furthermore, in the PS scheme,
each fermion produces its photon cascade independently
of the other ones, missing the effects due to the interfer
ence of radiation coming from different charged particles.
As far as inclusive cross sections (i.e. cross sections with
no cuts imposed on the generated photons) are concerned,
Page 21
20
these effects are largely integrated out. However, as shown
in [248], they become important when more exclusive vari
ables distributions are considered.
The first problem can be overcome by choosing the
generated p⊥of the photons different from Eq. (44). For
example, one can choose to extract the photon cosϑγac
cording to the universal leading poles 1/p · k present in
the matrix element for photon emission. Namely, one can
generate cosϑγas
cosϑγ∝
1
1 − β cosϑγ
,(45)
where β is the speed of the emitting particle. In this way,
photon energy and angle are generated independently, dif
ferent from Eq. (44). The nice feature of this prescription
is that p2
distribution reproduces exactly the SF, because no further
kinematical cuts have to be imposed to avoid unphysical
events. At this stage, the PS is used only to generate the
energies and multiplicity of the photons. The problem of
including the radiation interference is still unsolved, be
cause the variables of photons emitted by a fermion are
still uncorrelated with those of the other charged particles.
The issue of including photon interference can be success
fully worked out looking at the YFS formula [232]:
⊥= E2
γsin2ϑγ is always well defined, and the x
dσn≈ dσ0e2n
n!
n
?
l=1
d3kl
(2π)32k0
l
N
?
i,j=1
ηiηj
−pi· pj
(pi· kl)(pj· kl)
.
(46)
It gives the differential cross section dσnfor the emission
of n photons, whose momenta are k1,··· ,kn, from a kernel
process described by dσ0and involving N fermions, whose
momenta are p1,··· ,pN. In Eq. (46) ηiis a charge factor,
which is +1 for incoming e−or outgoing e+and −1 for
incoming e+or outgoing e−. Note that Eq. (46) is valid
in the soft limit (ki→ 0). The important point is that it
also accounts for coherence effects. From the YFS formula
it is straightforward to read out the angular spectrum of
the lthphoton:
cosϑl∝ −
N
?
i,j=1
ηiηj
1 − βiβjcosϑij
(1 − βicosϑil)(1 − βjcosϑjl)
.
(47)
It is worth noticing that in the LL prescription the
same quantity can be written as
cosϑl∝
N
?
i=1
1
1 − βicosϑil,(48)
whose terms are of course contained in Eq. (47).
In order to consider also coherence effects in the an
gular distribution of the photons, one can generate cosϑγ
according to Eq. (47), rather than to Eq. (48). This recipe
[248] is adopted in BabaYaga v3.5 and BabaYaga@NLO.
YennieFrautschiSuura exponentiation
The YFS exponentiation procedure, implemented in
the code BHWIDE, is a technique for summing up all the
infrared (IR) singularities present in any process accompa
nied by photonic radiation [232]. It is inherently exclusive,
i.e. all the summations of the IR singular contributions are
done before any phasespace integration over the virtual or
real photon fourmomenta are performed. The method was
mainly developed by S. Jadach, B.F.L. Ward and collab
orators to realise precision MC tools. In the following, the
general ideas underlying the procedure are summarised.
Let us consider the scattering process e+(p1)e−(p2) →
f1(q1)···fn(qn), where f1(q1)···fn(qn) represents a given
arbitrary final state, and let M0 be its treelevel matrix
element. By using standard Feynmandiagram techniques,
it is possible to show that the same process, when accom
panied by l additional real photons radiated by the IS
particles, and under the assumption that the l additional
photons are soft, i.e. their energy is much smaller that any
energy scale involved in the process, can be described by
the factorised matrix element built up by the LO one, M0,
times the product of l eikonal currents, namely
M ≃ M0
l?
i=1
?
e
?εi(ki) · p2
ki· p2
−εi(ki) · p1
ki· p1
??
,(49)
where e is the electron charge, ki are the momenta of
the photons and εi(ki) their polarisation vectors. Tak
ing the square of the matrix element in Eq. (49) and
multiplying by the proper flux factor and the Lorentz
invariant phase space volume, the cross section for the
process e+(p1)e−(p2) → f1(q1)···fn(qn) + lrealphotons
can be written as
?
?
By summing over the number of finalstate photons, one
obtains the cross section for the original process accom
panied by an arbitrary number of real photons, namely
dσ(l)
r
= dσ01
l!
l?
e2
i=1
kidkidcosϑidϕi
1
2(2π)3
×
εi
?εi(ki) · p2
ki· p2
−εi(ki) · p1
ki· p1
?2?
. (50)
dσ(∞)
r
=
∞
?
l=0
dσ(l)
r
= dσ0exp
?
kdkdcosϑdϕ
1
2(2π)3
×
?
ε
e2
?ε(k) · p2
k · p2
−ε(k) · p1
k · p1
?2?
. (51)
Equation (51), being limited to real radiation only, is IR
divergent once the phase space integrations are performed
down to zero photon energy. This problem, as is well known,
finds its solution in the matching between real and vir
tual photonic radiation. Equation (51) already shows the
key feature of exclusive exponentiation, i.e. summing up
all the perturbative contributions before performing any
phase space integration.
In order to get meaningful radiative corrections it is
necessary to consider, besides IS real photon corrections,
Page 22
21
also IS virtual photon corrections, i.e. the corrections due
to additional internal photon lines connecting the IS elec
tron and positron. For a vertextype amplitude, the result
can be written as
?
×Γ
MV1= −i
e2
(2π)4
d4k
1
k2+ iε¯ v(p1)γµ−(/ p1+ / k) + m
2p1· k + k2+ iε
(/ p2+ / k) + m
2p2· k + k2+ iεγµu(p2), (52)
where Γ stands for the Dirac structure of the LO process,
in such a way that M0 = ¯ v(p1)Γu(p2). The softphoton
part of the amplitude can be extracted by taking kµ≃ 0 in
all the numerators. In this approximation, the amplitude
of Eq. (52) becomes
MV1= M0× V,
V =
(2π)3
2iα
?
d4k
4p1· p2
(2p1· k + k2+ iε)(2p2· k + k2+ iε)
×
1
k2+ iε.
(53)
It can be seen that, as in the real case, the IR virtual
correction factorises off the LO matrix element so that it
is universal, i.e. independent of the details of the process
under consideration, and divergent in the IR portion of
the phase space.
The correction given by n soft virtual photons can be
seen to factorise with an additional factor 1/n!, namely
MVn= M0×1
n!Vn,(54)
so that by summing over all the additional soft virtual
photons one obtains
MV = M0× exp[V ].(55)
As already noticed both the real and virtual factors
are IR divergent. In order to obtain meaningful expres
sions one has to adopt some regularisation procedure. One
possibility is to give the photon a (small) mass λ and to
modify Eqs. (50) and (53) accordingly. Once all the ex
pressions are properly regularised, one can write down a
YFS master formula that takes into account real and vir
tual photonic corrections to the LO process. In virtue of
the factorisation properties discussed above, the master
formula can be obtained from Eq. (51) with the substitu
tion dσ0→ dσ0exp(V )2, i.e.
dσ = dσ0exp(V )2exp
?
As a last step it is possible to analytically perform the
IR cancellation between virtual and very soft real pho
tons. Actually, since very soft real photons do not affect
the kinematics of the process, the real photon exponent
?
−ε(k) · p1
k · p1
kdkdcosϑdϕ
1
2(2π)3
×
ε
e2
?ε(k) · p2
k · p2
?2?
. (56)
can be split into a contribution coming from photons with
energy less than a cutoff kmin plus a contribution from
photons with energy above it. The first contribution can
be integrated over all its phase space and can then be
combined with the virtual exponent. After this step it is
possible to remove the regularising photon mass by taking
the limit λ → 0, so that Eq. (56) becomes
?
?
where Y is given by
?
?
The explicit form of Y can be derived by performing all
the details of the calculation, and reads
dσ = dσ0exp(Y )expkdkdΘ(k − kmin)cosϑdϕ
?2?
1
2(2π)3
×
ε
e2
?ε(k) · p2
k · p2
−ε(k) · p1
k · p1
,(57)
Y = 2V +
kdkdΘ(kmin− k)cosϑdϕ
?ε(k) · p2
1
2(2π)3
×
ε
e2
k · p2
−ε(k) · p1
k · p1
?2
. (58)
Y = β lnkmin
E
+ δY FS,
?π2
δY FS=1
4β +α
π3
−1
2
?
.(59)
2.4.2 Matching NLO and higherorder corrections
As will be shown numerically in Section 2.6, NLO cor
rections must be combined with multiple photon emission
effects to achieve a theoretical accuracy at the per mill
level. This combination, technically known as matching,
is a fundamental ingredient of the most precise genera
tors used for luminosity monitoring, i.e. BabaYaga@NLO,
BHWIDE and MCGPJ. Although the matching is im
plemented according to different theoretical details, some
general aspects are common to all the recipes and must
be emphasised:
1. It is possible to match NLO and HO corrections consis
tently, avoiding double counting of LL contributions at
order α and preserving the advantages of resummation
of soft and collinear effects beyond O(α).
2. The convolution of NLO corrections with HO terms
allows to include the dominant part of NNLO correc
tions, given by infraredenhanced α2L subleading con
tributions. This was argued and demonstrated analyt
ically and numerically in [44] through comparison with
the available O(α2) corrections to schannel processes
and tchannel Bhabha scattering. Such an aspect of
the matching procedure is crucial to settle the theo
retical accuracy of the generators by means of explicit
comparisons with the exact NNLO perturbative cor
rections discussed in Section 2.3, and will be addressed
in Section 2.8.
Page 23
22
3. BabaYaga@NLO and BHWIDE implement a fully fac
torised matching recipe, while MCGPJ includes some
terms in additive form, as will be visible in the formu
lae reported below.
In the following we summarise the basic features of the
matching procedure as implemented in the codes MCGPJ,
BabaYaga@NLO and BHWIDE.
The matching approach realised in the MC event gen
erator MCGPJ was developed in [236]. In particular, Bha
bha scattering with complete O(α) and HO LL photonic
corrections can written as
dσe+e−→e+e−(γ)
dΩ−
=
1
?
¯ z1
dz1
1
?
¯ z2
dz2DNS,γ
ee
(z1)DNS,γ
ee
(z2)
×dˆ σBhabha
?
yth
?
∆
×2dσBhabha
dΩ−
?
?3 + c2
+α3
2π2s
k0>∆ε
θi>θ0
0
(z1,z2)
dΩ−
Y2
?
yth
???
?
1 +α
πKSV
?
Θ(cuts)
×
Y1
dy1
Y1
dy2
Y2
DNS,γ
ee
(y1
Y1)DNS,γ
ee
(y2
Y2)
+α
π
1
dx
x
1 − x +x2
??
(1 − x,1)
dΩ−
?28α
WT
4
2
?
lnθ2
0(1 − x)2
4
+x2
2
?
0
+1 − x +x2
2
?
lnθ2
0
4+x2
2
?
×
dˆ σBhabha
0
+dˆ σBhabha
0
(1,1 − x)
dΩ−
2)ln∆ε
ε
??
Θ(cuts)
−α2
4s1 − c
?
π
ln(ctgθ
Θ(cuts)dΓe¯ eγ
dΩ−
. (60)
Here the step functions Θ(cuts) stand for the particular
cuts applied. The auxiliary parameter θ0 defines cones
around the directions of the motion of the charged parti
cles in which the emission of hard photons is approximated
by the factorised form by convolution of collinear radiation
factors [249] with the Born cross section. The dependence
on the parameters ∆ and θ0cancels out in the sum with
the last term of Eq. (60), where the photon energy and
emission angles with respect to all charged particles are
limited from below (k0> ∆ε,θi > θ0). Taking into ac
count vacuum polarisation, the Born level Bhabha cross
section with reduced energies of the incoming electron and
positron can be cast in the form
dˆ σBhabha
0
(z1,z2)
dΩ−
1
1 − Π(ˆ s)2
−Re
(1 − Π(ˆt))(1 − Π(ˆ s))∗
ˆ s = z1z2s,
ˆt = −
=4α2
sa2
z2
?
1
1 − Π(ˆt)2
2(1 + c)2
2a2
z2
az1(1 − c)
1z2(1 − c)
z1+ z2− (z1− z2)c,
a2+ z2
2z2
2(1 + c)2
1(1 − c)2
+
1(1 − c)2+ z2
1
2(1 + c)2
?
dΩ−,
sz2
(61)
where Π(Q2) is the photon selfenergy correction. Note
that in the cross section above the cosine of the scattering
angle, c, is given for the original c.m. reference frame of
the colliding beams.
For the twophoton production channel, a similar rep
resentation is used in MCGPJ:
dσe+e−→γγ(γ)=
1
?
¯ z1
dz1DNS,γ
ee
(z1)
1
?
¯ z2
dz2DNS,γ
ee
(z2)
×dˆ σγγ
??
0(z1,z2)
?
1 +α
πKγγ
SV
?
+
α
π
1
?
∆
dx
x
×1 − x +x2
2
?
lnθ2
0
4+x2
4α3
π2s2
2
??
?
zi≥∆
dˆ σ0(1 − x,1)
+dˆ σ0(1,1 − x)
?
+
1
3
π−θ0≥θi≥θ0
dΓ3γ
×
?
z2
3(1 + c2
2(1 − c2
ci= cosθi,
3)
z2
1z2
1)(1 − c2
2)+ two cyclic permutations
?
,
zi=q0
i
ε,
θi= ?
z2
(1 − c2
p−qi,(62)
where the cross section with reduced energies has the form
dˆ σγγ
0(z1,z2)
dΩ1
=2α2
s
1(1 − c1)2+ z2
1)(z1+ z2+ (z2− z1)c1)2,
2(1 + c1)2
and the factor 1/3 in the last term of Eq. (62) takes into
account the identity of the finalstate photons. The sum
of the last two terms does not depend on ∆ and θ0.
Concerning BabaYaga@NLO,the matching starts from
the observation that Eq. (29) for the QED corrected all
order cross section can be rewritten in terms of the PS
ingredients as
dσ∞
LL= Π(Q2,ε)
∞
?
n=0
1
n!Mn,LL2dΦn.(63)
By construction, the expansion of Eq. (63) at O(α) does
not coincide with the exact O(α) result. In fact
?
≡ [1 + Cα,LL]M02dΦ0+ M1,LL2dΦ1,
dσα
LL=1 −α
2πI+ lnQ2
m2
?
M02dΦ0+ M1,LL2dΦ1
(64)
Page 24
23
where I+ ≡
section can always be cast in the form
?1−ǫ
0
P(z)dz, whereas the exact NLO cross
dσα= [1 + Cα]M02dΦ0+ M12dΦ1.(65)
The coefficients Cαcontain the complete O(α) virtual and
softbremsstrahlung corrections in units of the squared
Born amplitude, and M12is the exact squared matrix
element with the emission of one hard photon. We remark
that Cα,LLhas the same logarithmic structure as Cαand
that M1,LL2has the same singular behaviour as M12.
In order to match the LL and NLO calculations, the
following correction factors, which are by construction in
frared safe and free of collinear logarithms, are introduced:
FSV = 1+(Cα− Cα,LL),FH = 1+M12− M1,LL2
M1,LL2
.
(66)
With them the exact O(α) cross section can be expressed,
up to terms of O(α2), in terms of its LL approximation as
dσα= FSV(1 + Cα,LL)M02dΦ0 + FHM1,LL2dΦ1.
(67)
Driven by Eq. (67), Eq. (63) can be improved by writing
the resummed matched cross section as
dσ∞
matched= FSV Π(Q2,ε)
×
∞
?
n=0
1
n!
?n
i=0
?
FH,i
?
Mn,LL2dΦn.(68)
The correction factors FH,ifollow from the definition (66)
for each photon emission. The O(α) expansion of Eq. (68)
now coincides with the exact NLO cross section of Eq. (65),
and all HO LL contributions are the same as in Eq. (63).
This formulation is implemented in BabaYaga@NLO for
both Bhabha scattering and photon pair production, us
ing, of course, the appropriate SV and hard bremsstrah
lung formulae. This matching formulation has also been
applied to the study of DrellYanlike processes, by com
bining the complete O(α) electroweak corrections with
QED shower evolution in the generator HORACE [250,
251,252,253].
As far as BHWIDE is concerned, this MC event gen
erator realises the process
e+(p1)+e−(q1) −→ e+(p2)+e−(q2) +γ1(k1)+...+γn(kn)
(69)
via the YFS exponentiated cross section formula
dσ = e2αReB+2α˜ B
∞
?
n=0
1
n!
?
n
?
j=1
d3kj
k0
j
?
d4y
(2π)4
×eiy(p1+q1−p2−q2−P
jkj)+D¯βn(k1,...,kn)d3p2d3q2
p0
2q0
2
,
(70)
where the real infrared function˜B and the virtual infrared
function B are given in [237]. Here we note the usual con
nections
2α˜B =
?k≤Kmaxd3k
?
k0
˜S(k),
D =d3k
˜S(k)
k0
?e−iy·k− θ(Kmax− k)?
(71)
for the standard YFS infrared real emission factor
?
˜S(k) =
α
4π2
QfQf′
?
p1
p1· k−
q1
q1· k
?2
+ ...
?
, (72)
and where Qf is the electric charge of f in units of the
positron charge. In Eq. (72) the “...” represent the re
maining terms in˜S(k), obtained from the given one by
respective of Qf, p1, Qf′, q1 with corresponding values
for the other pairs of the external charged legs according
to the YFS prescription of Ref. [232,254] (wherein due at
tention is taken to obtain the correct relative sign of each
of the terms in˜S(k) according to this latter prescription).
The explicit representation is given by
2αReB(p1,q1,p2,q2) + 2α˜B(p1,q1,p2,q2;km) =
R1(p1,q1;km) + R1(p2,q2;km) + R2(p1,p2;km) +
R2(q1,q2;km) − R2(p1,q2;km) − R2(q1,p2;km), (73)
with
R1(p,q;km) = R2(p,q;km) +
?α
π
?π2
2
(74)
and
R2(p,q;km) =α
π
2ln2p0
??
ln2pq
m2
q0−1
?∆ + ω
?∆ − ω
?
e
4ln2(∆ + δ)2
?
?
− 1
?
lnk2
m
p0q0+1
2ln2pq
−1
?∆ + ω
?∆ − ω
m2
e
−1
4p0q0
4ln2(∆ − δ)2
?
?
4p0q0
−ReLi2
∆ + δ
− ReLi2
∆ − δ
−ReLi2
+π2
3
?2pq + (p0− q0)2, ω = p0+ q0, δ = p0− q0,
km≪ Ebeam).
The YFS hard photon residuals¯βiin Eq. (70), i = 0,1,
to O(α) are given exactly in Ref. [237] for BHWIDE.
Therefore this event generator calculates the YFS expo
nentiated exact O(α) cross section for e+e−→ e+e−+
n(γ) with multiple initial, initialfinal and final state radi
ation, using a corresponding MC realisation of Eq. (70) in
the wide angle regime. The library for O(α) electroweak
corrections, relevant for higher energies, is taken from [95,
255].
The result (70) is an exact rearrangement of the loop
expansion for the respective cross section and is indepen
dent of the dummy parameter Kmax. To derive this, one
∆ + δ
− ReLi2
∆ − δ
− 1,(75)
where ∆ =
and kmis a soft photon cutoff in the c.m. system (Esoft
γ
<
Page 25
24
may proceed as follows. Let the amplitude for the emission
of n real photons in the Bhabha process be
M(n)=
?
ℓ
M(n)
ℓ
,(76)
where M(n)
diagrams with ℓ virtual loops. The key result in the YFS
theory of Ref. [232,254] on virtual corrections is that we
may rewrite Eq. (76) as the exact representation
ℓ
is the contribution to M(n)from Feynman
M(n)= eαB
∞
?
j=0
m(n)
j,(77)
where we have defined
αB =
?
d4k
(k2− λ2+ iǫ)S(k),(78)
with the virtual infrared emission factor given by
S(k) =−iα
8π2
?
i′<j
Zi′θi′Zjθj
?
(2¯ pi′θi′ − k)µ
k2− 2k¯ pi′θi′ + iǫ
+
(2¯ pjθj+ k)µ
k2+ 2k¯ pjθj+ iǫ
?2
.(79)
Here, λ is an infrared regulatormass, and following Refs. [232,
254] we identify the sign of the jth external line charge
here as Zj= Qj and θj= +(−) for outgoing (incoming)
4momentum ¯ pj, so that here ¯ p1 = p1, ¯ p2 = q1, ¯ p3 =
p2, ¯ p4= q2, Z1= +1, θ1= −, Z2= −1, θ2= −, Z3=
+1, θ3= +, Z4= −1, θ4= +. The amplitudes {m(n)
are free of all virtual infrared divergences.
Using the result (77) for M(n), we get the attendant
differential cross section by the standard methods as
j}
dˆ σn=e2αReB
n!
?
n
?
l=1
d3kl
l+ λ2)1/2
(k2
×¯ ρ(n)(p1,q1,p2,q2,k1,··· ,kn)d3p2d3q2
?
p0
?
2q0
2
×δ(4)
p1+ q1− p2− q2−
n
?
i=1
ki
, (80)
where we have defined
¯ ρ(n)(p1,q1,p2,q2,k1,··· ,kn) =
?
spin
??????
∞
?
j=0
m(n)
j
??????
2
,(81)
in the incoming e+e−c.m. system. Here we have absorbed
the remaining kinematical factors for the initial state flux
and spin averaging into the normalisation of the ampli
tudes M(n)for pedagogical reasons, so that the ¯ ρ(n)are
averaged over initial spins and summed over final spins.
We then use the key result of Ref. [232,254] on real cor
rections to write the exact result
¯ ρ(n)(p1,q1,p2,q2,k1,··· ,kn) =
n
?
i=1
˜S(ki)¯β0+ ··· +
n
?
+¯βn(k1,...,kn),
i=1
˜S(ki)¯βn−1(k1,...,ki−1,ki+1,...,kn)
(82)
where the hard photon residuals¯βj are determined re
cursively [232,254] and are free of all virtual and all real
infrared singularities to all orders in α. Introducing the
result (82) into Eq. (80) and summing over the number of
real photons n leads directly to master formula (70). We
see that it allows for exact exclusive treatment of hard
photonic effects on an eventbyevent basis.
2.5 Monte Carlo generators
To measure the luminosity, event generators, rather than
analytical calculations, are mandatory to provide theoreti
cal results of real experimental interest. The software tools
used in early measurements of the luminosity at flavour
factories (and sometimes still used in recent experimen
tal publications) include generators such as BHAGENF
[256], BabaYaga v3.5 [234] and BKQED [257,258]. These
MC programs, however, are based either on a fixed NLO
calculation (such as BHAGENF and BKQED) or include
corrections to all orders in perturbation theory, but in the
LL approximation only (like BabaYaga v3.5). Therefore
the precision of these codes can be estimated to lie in the
range 0.5÷1%, depending on the adopted experimental
cuts.
The increasing precision reached on the experimental
side during the last years led to the development of new
dedicated theoretical tools, such as BabaYaga@NLO and
MCGPJ, and the adoption of already welltested codes,
like BHWIDE, the latter extensively used at the high
energy LEP/SLC colliders for the simulation of the large
angle Bhabha process. As already emphasised in Section
2.4.2, all these three codes include NLO corrections in
combination with multiple photon contributions and have,
therefore, a precision tag of ∼ 0.1%. As described in the
following, the experiments typically use more than one
generator, to keep the luminosity theoretical error under
control through the comparison of independent predic
tions.
A list of the MC tools used in the luminosity mea
surement at meson factories is given in Table 3, which
summarises the main ingredients of their formulation for
radiative corrections and the estimate of their theoretical
accuracy.
The basic theoretical and phenomenological features of
the different generators are summarised in the following.
1. BabaYaga v3.5 – It is a MC generator developed by
the Pavia group at the start of the DAΦNE opera
tion using a QED PS approach for the treatment of
Page 26
25
Table 3. MC generators used for luminosity monitoring at
meson factories.
Generator
BabaYaga v3.5
BabaYaga@NLO
BHAGENF
BHWIDE
BKQED
MCGPJ
Theory
Parton Shower
O(α) + PS
O(α)
O(α)YFS
O(α)
O(α) + SF
Accuracy
∼ 0.5 ÷ 1%
∼ 0.1%
∼ 1%
∼ 0.5%(LEP1)
∼ 1%
< 0.2%
LL QED corrections to luminosity processes and later
improved to account for the interference of radiation
emitted by different charged legs in the generation of
the momenta of the finalstate particles. The main
drawback of BabaYaga v3.5 is the absence of O(α)
nonlogarithmic contributions, resulting in a theoret
ical precision of ∼ 0.5% for largeangle Bhabha scat
tering and of about 1% for γγ and µ+µ−final states.
It is used by the CLEOc collaboration for the study
of all the three luminosity processes.
2. BabaYaga@NLO – It is the presently released ver
sion of BabaYaga, based on the matching of exact
O(α) corrections with QED PS, as described in Sec
tion 2.4.2. The accuracy of the current version is esti
mated to be at the 0.1% level for largeangle Bhabha
scattering, twophoton and µ+µ− 9production. It is
presently used by the KLOE and BaBar collabora
tions, and under consideration by the BESIII exper
iment. Like BabaYaga v3.5, BabaYaga@NLO is avail
able at the web page of the Paviaphenomenology group
www.pv.infn.it/~hepcomplex/babayaga.html .
3. BHAGENF/BKQED – BKQED is the event generator
developed by Berends and Kleiss and based on the clas
sical exact NLO calculations of [257,258] for all QED
processes. It was intensively used at LEP to perform
tests of QED through the analysis of the e+e−→ γγ
process and is adopted by the BaBar collaboration for
the simulation of the same reaction. BHAGENF is a
code realised by Drago and Venanzoni at the beginning
of the DAΦNE operation to simulate Bhabha events,
adapting the calculations of [257] to include the con
tribution of the φ resonance. Both generators lack the
effect of HO corrections and, as such, have a precision
accuracy of about 1%. The BHAGENF code is avail
able at the web address
www.lnf.infn.it/~graziano/bhagenf/bhabha.html.
4. BHWIDE – It is a MC code realised in KrakowKnox
wille at the time of the LEP/SLC operation and de
scribed in [237]. In this generator exact O(α) correc
tions are matched with the resummation of the in
frared virtual and real photon contributions through
the YFS exclusive exponentiation approach. Accord
ing to the authors the precision is estimated to be
9At present, finite mass effects in the virtual corrections to
e+e−→ µ+µ−, which should be included for precision simula
tions at the φ factories, are not included in BabaYaga@NLO.
about 0.5% for c.m. energies around the Z resonance.
This accuracy estimate was derived through detailed
comparisons of the BHWIDE predictions with those of
other LEP tools in the presence of the full set of NLO
corrections, including purely weak corrections. How
ever, since the latter are phenomenologically unim
portant at e+e−accelerators of moderately high en
ergies and since the QED theoretical ingredients of
BHWIDE are very similar to the formulation of both
BabaYaga@NLO and MCGPJ, one can argue that the
accuracy of BHWIDE for physics at flavour factories
is at the level of 0.1%. It is adopted by the KLOE,
BaBar and BES collaborations. The code is available
at
placzek.home.cern.ch/placzek/bhwide/.
5. MCGPJ – It is the generator developed by the Dubna
Novosibirsk collaboration and used at the VEPP2M
collider. This program includes exact O(α) corrections
supplemented with HO LL contributions related to
the emission of collinear photon jets and taken into
account through analytical QED collinear SF, as de
scribed in Section 2.4.2. The theoretical precision is
estimated to be better than 0.2%. The generator is
available at the web address
cmd.inp.nsk.su/~sibid/ .
It is worth noticing that the theoretical uncertainty
of the most accurate generators based on the matching
of exact NLO with LL resummation starts at the level of
O(α2) NNL contributions, as far as photonic corrections
are concerned. Other sources of error affecting their phys
ical precision are discussed in detail in Section 2.8.
2.6 Numerical results
Before showing the results which enable us to settle the
technical and theoretical accuracy of the generators, it is
worth discussing the impact of various sources of radiative
corrections implemented in the programs used in the ex
perimental analysis. This allows one to understand which
corrections are strictly necessary to achieve a precision at
the per mill level for both the calculation of integrated
cross sections and the simulation of more exclusive distri
butions.
2.6.1 Integrated cross sections
The first set of phenomenological results about radiative
corrections refer to the Bhabha cross section, as obtained
by means of the code BabaYaga@NLO, according to dif
ferent perturbative and precision levels. In Table 4 we
show the values for the Born cross section σ0, the O(α)
PS and exact cross section, σPS
as well as the LL PS cross section σPSand the matched
cross section σmatched. Furthermore, the cross section in
the presence of the vacuum polarisation correction, σVP
is also shown. The results correspond to the c.m. ener
gies√s = 1,4,10 GeV and were obtained with the se
lection criteria of Eq. (15), but for an angular acceptance
α
and σNLO
α
, respectively,
0 ,
Page 27
26
Table 4. Bhabha cross section (in nb) at meson factories
according to different precision levels and using the cuts of
Eq. (15), but with an angular acceptance of 55◦≤ θ± ≤ 125◦.
The numbers in parentheses are 1σ MC errors.
√s(GeV)
σ0
σVP
0
σNLO
σPS
α
σmatched
σPS
1.02
529.4631(2)
542.657(6)
451.523(6)
454.503(6)
455.858(5)
458.437(4)
4
44.9619(1)
46.9659(1)
37.1654 (6)
37.4186 (6)
37.6731 (4)
37.8862 (4)
10
5.5026(2)
5.85526(3)
4.4256(2)
4.4565(1)
4.5046(3)
4.5301(2)
of 55◦≤ θ± ≤ 125◦resembling realistic data taking at
meson factories. One should keep in mind that the cuts
of Eq. (15) tend to single out quasielastic Bhabha events
and that the energy of final state electron/positron cor
responds to a socalled “bare” event selection (i.e. with
out photon recombination), which corresponds to what
is done in practice at flavour factories. In particular the
rather stringent energy and acollinearity cuts enhance the
impact of soft and collinear radiation with respect to a
more inclusive setup.
From these cross section values, it is possible to cal
culate the relative effect of various corrections, namely
the contribution of vacuum polarisation and exact O(α)
QED corrections, of nonlogarithmic (NLL) terms enter
ing the O(α) cross section, of HO corrections in the O(α)
matched PS scheme, and finally of NNL effects beyond
order α largely dominated by O(α2L) contributions. The
above corrections are shown in Table 5 in per cent and
can be derived from the cross section results of Table 4
with the following definitions:
δVP≡σVP
≡σNLO− σPS
δα2L≡σmatched− σNLO− σPS+ σPS
0
− σ0
σ0
,δα≡σNLO− σ0
δHO≡σmatched− σNLO
σ0
,
δNLL
α
α
σNLO
,
σNLO
,
α
σNLO
.
From Table 5 it can be seen that O(α) corrections
decrease the Bhabha cross section by about 15÷17% at
the φ and τcharm factories, and by about 20% at the
B factories. Within the full set of O(α) corrections, non
logarithmic terms are of the order of 0.5%, as expected
almost independent of the c.m. energy, and with a mild
dependence on the angular acceptance cuts due to box and
interference contributions. The effect of HO corrections
due to multiple photon emission is about 1% at the φ and
τcharm factories and reaches about 2% at the B factories.
The contribution of (approximate) O(α2L) corrections is
at the 0.1% level, while vacuum polarisation increases the
cross section by about 2% around 1 GeV, and by about
5% and 6% at 4 GeV and 10 GeV, respectively. Concern
ing the latter correction the nonperturbative hadronic
contribution to the running of α was parameterised in
terms of the HADR5N routine [259,260,18] included in
BabaYaga@NLO both in the LO and NLO diagrams. We
have checked that the results obtained for the vacuum
polarisation correction in terms of the parametrisation
[164] agree at the 10−4level with those obtained with
HADR5N, as shown in detail in Section 2.8. Those rou
tines return a data driven error, thus affecting the the
oretical precision of the calculation of the Bhabha cross
cross section as will be discussed in Section 2.9.
Analogous results for the size of radiative corrections
to the process e+e−→ γγ are given in Table 6 [261].
They were obtained using BabaYaga@NLO, according to
the experimental cuts of Eq. (16) for the c.m. energies
√s = 1,3,10 GeV.
Table 5. Relative size of different sources of corrections (in
per cent) to the largeangle Bhabha cross section for typical
selection cuts at φ, τcharm and B factories.
√s(GeV)
δα
δNLL
α
δHO
δα2L
δVP
1.02
−14.73
−0.66
0.97
0.09
2.43
4.
−17.32
−0.68
1.35
0.09
4.46
10.
−19.57
−0.70
1.79
0.11
6.03
Table 6. Photon pair production cross sections (in nb) at dif
ferent accuracy levels and relative corrections (in per cent) for
the setup of Eq. (16) and the c.m. energies√s = 1,3,10 GeV.
√s (GeV)
σ0
σNLO
σPS
α
σmatched
σPS
δα
δNLL
α
δHO
1
137.53
129.45
128.55
129.77
128.92
−5.87
0.70
0.24
3
15.281
14.211
14.111
14.263
14.169
−7.00
0.71
0.37
10
1.3753
1.2620
1.2529
1.2685
1.2597
−8.24
0.73
0.51
The numerical errors coming from the MC integration
are not shown in Table 5 because they are beyond the
quoted digits. From Table 5 it can be seen that the exact
O(α) corrections lower the Born cross section by about
5.9% (at the φ resonance), 7.0% (at√s = 3 GeV) and
8.2% (at the Υ resonance). The effect due to O(αnLn)
(with n ≥ 2) terms is quantified by the contribution δHO,
which is a positive correction of about 0.2% (at the φ
resonance), 0.4% (τcharm factories) and 0.5% (at the Υ
resonance), and therefore important in the light of the per
mill accuracy aimed at. On the other hand, also nextto
leading O(α) corrections, quantified by the contribution
δNLL
α
, are necessary at the precision level of 0.1%, since
their contribution is of about 0.7% almost independent
Page 28
27
1
10
100
1000
10000
100000
1e+06
0.80.85 0.90.951
dσ
dMe+e−(pb/GeV)
Me+e− (GeV)
NEW
OLD
O(α)
0
1
2
3
4
5
6
7
0.80.850.9
Me+e−
0.951
δ (%)
OLD−NEW
NEW
× 100
Fig. 16. Invariant mass distribution of the Bhabha process at
KLOE, according to BabaYaga v3.5 (OLD), BabaYaga@NLO
(NEW) and an exact NLO calculation. The inset shows the
relative effect of NLO corrections, given by the difference of
BabaYaga v3.5 and BabaYaga@NLO predictions. From [235].
of the c.m. energy. To further corroborate the precision
reached in the cross section calculation of e+e−→ γγ, we
also evaluated the effect due to the most important sub
leading O(α2) photonic corrections given by order α2L
contributions. It turns out that the effect due to O(α2L)
corrections does not exceed the 0.05% level. Obviously, the
contribution of vacuum polarisation is absent in γγ pro
duction. This is an advantage for particularly precise pre
dictions, as the uncertainty associated with the hadronic
part of vacuum polarisation does not affect the cross sec
tion calculation.
2.6.2 Distributions
Besides the integrated cross section, various differential
cross sections are used by the experimentalists to monitor
the collider luminosity. In Figs. 16 and 17 we show two
distributions which are particularly sensitive to the de
tails of photon radiation, i.e. the e+e−invariant mass and
acollinearity distribution, in order to quantify the size of
NLO and HO corrections. The distributions are obtained
according to the exact O(α) calculation and with the two
BabaYaga versions, BabaYaga v3.5 and BabaYaga@NLO.
From Figs. 16 and 17 it can be clearly seen that multiple
photon corrections introduce significant deviations with
respect to an O(α) simulation, especially in the hard tails
of the distributions where they amount to several per cent.
To make the contribution of exact O(α) nonlogarithmic
terms clearly visible, the inset shows the relative differ
ences between the predictions of BabaYaga v3.5 (denoted
as OLD) and BabaYaga@NLO (denoted as NEW). Actu
ally, as discussed in Section 2.4.2, these differences mainly
come from nonlogarithmic NLO contributions and to a
smaller extent from O(α2L) terms. Their effect is flat and
at the level of 0.5% for the acollinearity distribution, while
they reach the several per cent level in the hard tail of the
invariant mass distribution.
1
10
100
1000
10000
0246810
dσ
dξ(pb/deg)
ξ (deg)
NEW
OLD
O(α)
0.4
0.5
0.6
0.7
0.8
1 0 1 2 3 4 5 6 7 8 9 10
ξ (deg)
δ (%)
OLD−NEW
NEW
× 100
Fig. 17.
cess at KLOE, according to BabaYaga v3.5 (OLD) and
BabaYaga@NLO (NEW). The inset shows the relative effect
of NLO corrections, given by the difference of BabaYaga v3.5
and BabaYaga@NLO predictions. From [235].
Acollinearity distribution of the Bhabha pro
50
40
30
20
10
0
10
20
02468 10
δ (%)
ξ (deg.)
σ∞−σα
σ∞
σ∞−σα2
σ∞
× 100
× 100
Fig. 18. Relative effect of HO corrections α2L2and αnLn
(n ≥ 3) to the acollinearity distribution of the Bhabha process
at KLOE. From [235].
It is also worth noticing that LL radiative corrections
beyond α2can be quite important for accurate simula
tions, at least when considering differential distributions.
This means that even with a complete NNLO calculation
at hand it would be desirable to match such corrections
with the resummation of all the remaining LL effects. In
Fig. 18, the relative effect of HO corrections beyond α2
dominated by the α3contributions (dashed line) is shown
in comparison with that of the α2corrections (solid line)
on the acollinearity distribution for the Bhabha process
at DAΦNE. As can be seen, the α3effect can be as large
as 10% in the phase space region of soft photon emission,
corresponding to small acollinearity angles with almost
backtoback final state fermions.
Concerning the process e+e−→ γγ we show in Fig. 19
the energy distribution of the most energetic photon, while
the acollinearity distribution of the two most energetic
photons is represented in Fig. 20. The distributions refer
to exact O(α) corrections matched with the PS algorithm
(solid line), to the exact NLO calculation (dashed line)
Page 29
28
0.01
0.1
1
10
100
1000
10000
100000
0.25 0.30.35 0.40.45 0.50.55 0.6
dσ
dE(nb/GeV)
E (GeV)
20
10
0
10
20
0.4 0.42 0.44 0.46 0.48 0.5
E (GeV)
δ (%)
exp
O (α)
BabaYaga 3.5
δexp
δNLL
∞
Fig. 19. Energy distribution of the most energetic photon
in the process e+e−→ γγ, according to the PS matched
with O(α) corrections denoted as exp (solid line), the exact
O(α) calculation (dashed line) and the pure allorder PS as
in BabaYaga v3.5 (dasheddotted line). lnset: relative effect
(in per cent) of multiple photon corrections (solid line) and
of nonlogarithmic contributions of the matched PS algorithm
(dashed line). From [261].
and to allorder pure PS predictions of BabaYaga v3.5
(dasheddotted line). In the inset of each plot, the rel
ative effect due to multiple photon contributions (δHO)
and nonlogarithmic terms entering the improved PS al
gorithm (δNLL
α
) is also shown, according to the definitions
given in Eq. (83).
For the energy distribution of the most energetic pho
ton particularly pronounced effects due to exponentiation
are present. In the statistically dominant region, HO cor
rections reduce the O(α) distribution by about 20%, while
they give rise to a significant hard tail close to the energy
threshold of 0.3√s as a consequence of the higher pho
ton multiplicity of the resummed calculation with respect
to the fixedorder NLO prediction. Needless to say, the
relative effect of multiple photon corrections below about
0.46 GeV not shown in the inset is finite but huge. This
representation with the inset was chosen to make also the
contribution of O(α) nonlogarithmic terms visible, which
otherwise would be hardly seen in comparison with the
multiple photon corrections. Concerning the acollinearity
distribution, the contribution of higherorder corrections
is positive and of about 10% for quasibacktobackphoton
events, whereas it is negative and decreasing from ∼ −30%
to ∼ −10% for increasing acollinearity values. As far as
the contributions of nonlogarithmic effects dominated by
nexttoleading O(α) corrections are concerned, they con
tribute at the level of several per mill for the acollinearity
distribution, while they lie in the range of several per cent
for the energy distribution.
As a whole, the results of the present Section empha
sise that, for a 0.1% theoretical precision in the calculation
of both the cross sections and distributions, both exact
O(α) and HO photonic corrections are necessary, as well
as the running of α.
1
10
100
1000
02468 10
dσ
dξ(nb/deg)
ξ (deg)
30
20
10
0
10
20
02468 10
δ (%)
ξ (deg)
exp
O (α)
BabaYaga 3.5
δexp
δNLL
∞
Fig. 20. Acollinearity distribution for the process e+e−→ γγ,
according to the PS matched with O(α) corrections denoted
as exp (solid line), the exact O(α) calculation (dashed line)
and the pure allorder PS as in BabaYaga v3.5 (dasheddotted
line). lnset: relative effect (in per cent) of multiple photon cor
rections (solid line) and of nonlogarithmic contributions of the
matched PS algorithm (dashed line). From [261].
2.7 Tuned comparisons
The typical procedure followed in the literature to estab
lish the technical precision of the theoretical tools is to
perform tuned comparisons between the predictions of in
dependent programs using the same set of input parame
ters and experimental cuts. This strategy was initiated in
the 90s during the CERN workshops for precision physics
at LEP and is still in use when considering processes of
interest for physics at hadron colliders demanding partic
ularly accurate theoretical calculations. The tuning proce
dure is a key step in the validation of generators, because
it allows to check that the different details entering the
complex structure of the generators, e.g. the implementa
tion of radiative corrections, event selection routines, MC
integration and event generation, are under control, and
to fix possible mistakes.
The tuned comparisons discussed in the following were
performed switching off the vacuum polarisation correc
tion to the Bhabha scattering cross section. Actually, the
generators implement the nonperturbative hadronic con
tribution to the running of α according to different pa
rameterisations, which differently affect the cross section
prediction (see Section 6 for discussion). Hence, this sim
plification is introduced to avoid possible bias in the in
terpretation of the results and allows to disentangle the
effect of pure QED corrections. Also, in order to provide
useful results for the experiments, the comparisons take
into account realistic event selection cuts.
The present Section is a merge of results available in
the literature [235] with those of new studies. The results
refer to the Bhabha process at the energies of φ, τcharm
Page 30
29
Table 7. Cross section predictions [nb] of BabaYaga@NLO
and BHWIDE for the Bhabha cross section corresponding to
two different angular acceptances, for the KLOE experiment
at DAΦNE, and their relative differences (in per cent).
angular acceptance
20◦÷ 160◦
55◦÷ 125◦
BabaYaga@NLO
6086.6(1)
455.85(1)
BHWIDE
6086.3(2)
455.73(1)
δ(%)
0.005
0.030
and B factories. No tuned comparisons for the two photon
production process have been carried out.
2.7.1 φ and τcharm factories
First we show comparisons between BabaYaga@NLO and
BHWIDE accordingto the KLOE selection cuts of Eq. (15),
considering also the angular range 20◦≤ ϑ± ≤ 160◦for
cross section results. The predictions of the two codes are
reported in Table 7 for the two acceptance cuts together
with their relative deviations. As can be seen the agree
ment is excellent, the relative deviations being well below
the 0.1%. Comparisonsbetween BabaYaga@NLOand BH
WIDE at the level of differential distributions are given in
Figs. 21 and 22 where the inset shows the relative devi
ations between the predictions of the two codes. As can
be seen there is very good agreement between the two
generators, and the predicted distributions appear at a
first sight almost indistinguishable. Looking in more de
tail, there is a relative difference of a few per mill for the
acollinearity distribution (Fig. 22) and of a few per cent
for the invariant mass (Fig. 21), but only in the very hard
tails, where the fluctuations observed are due to limited
MC statistics. These configurations however give a negli
gible contribution to the integrated cross section, a factor
103÷ 104smaller than that around the very dominant
peak regions. In fact these differences on differential dis
tributions translate into agreement on the cross section
values well below the one per mill, as shown in Table 7.
Similar tuned comparisons were performed between
the results of BabaYaga@NLO, BHWIDE and MCGPJ
in the presence of cuts modelling the event selection cri
teria of the CMD2 experiment at the VEPP2M collider,
for a c.m. energy of√s = 900 MeV. The cuts used in this
case are
θ−+ θ+− π ≤ ∆θ,
φ−+ φ+ − π ≤ 0.15,
p−sin(θ−) ≥ 90 MeV,
(p−+ p+)/2 ≥ 90 MeV,
where θ−,θ+ are the electron/positron polar angles, re
spectively, φ±their azimuthal angles, and p±the moduli
of their threemomenta. ∆θ stands for an acollinearity cut.
Figure 23 shows the relative differences between the
results of BHWIDE and MCGPJ according to the criteria
of Eq. (83), as a function of the acollinearity cut ∆θ. The
1.1 ≤ (θ+− θ−+ π)/2 ≤ π − 1.1,
p+sin(θ+) ≥ 90 MeV,
(83)
Table 8. Cross section predictions [nb] of BabaYaga@NLO
and MCGPJ for the Bhabha cross section at τcharm factories
(√s = 3.5 GeV) and their relative difference (in per cent).
BabaYaga@NLO
35.20(2)
MCGPJ
35.181(5)
δ(%)
0.06
1
10
100
1000
10000
100000
1e+06
0.80.85 0.90.951
dσ
dMe+e−(pb/GeV)
Me+e− (GeV)
BABAYAGA
BHWIDE
1
0
1
2
3
4
0.80.850.9
Me+e−
0.951
δ (%)
BHWIDE−BABAYAGA
BABAYAGA
× 100
Fig. 21. Invariant mass distribution of the Bhabha process
according to BHWIDE and BabaYaga@NLO, for the KLOE
experiment at DAΦNE, and relative differences of the program
predictions (inset). From [235].
relative deviations between the results of BabaYaga@NLO
and MCGPJ for the same cuts are given in Fig. 24. It can
be seen that the predictions of the three generators lie
within a 0.2% band with differences of ∼ 0.3% for ex
treme values of the acollinearity cut. This agreement can
be considered satisfactory since for the acollinearity cut of
real experimental interest (∆θ ≈ 0.2 rad) the generators
agree within one per mill.
A number of comparisons were also performed for a
c.m. energy of 3.5 GeV relevant to the experiments at τ
charm factories. An example is given in Table 8 where
the predictions of BabaYaga@NLO and MCGPJ are com
pared, using cuts similar to those of Eq. (83) and for an
acollinearity cut of ∆θ = 0.25 rad. The agreement between
the two codes is below one per mill. Comparisons between
the two codes were also done at the level of differential
cross sections, showing satisfactory agreement in the sta
tistically relevant phase space regions. Preliminary results
[262] for a c.m. energy on top of the J/Ψ resonance show
good agreement between BabaYaga@NLO and BHWIDE
predictions too.
2.7.2 B factories
Concerning the B factories, a considerable effort was done
to establish the level of agreement between the genera
tors BabaYaga@NLO and BHWIDE in comparison with
BabaYagav3.5 too. This study made use of the realistic lu
minosity cuts quoted in Section 2.3.3 for the BaBar exper
iment. The cross sections predicted by BabaYaga@NLO
and BHWIDE are shown in Table 9, together with the
Page 31
30
1
10
100
1000
10000
02468 10
dσ
dξ(pb/deg)
ξ (deg)
BABAYAGA
BHWIDE
0.2
0
0.2
0.4
0.6
0.8
1
1 0 1 2 3 4 5 6 7 8 9 10
ξ (deg)
δ (%)
BHWIDE−BABAYAGA
BABAYAGA
× 100
Fig. 22. Acollinearity distribution of the Bhabha process ac
cording to BHWIDE and BabaYaga@NLO, for the KLOE ex
periment at DAΦNE, and relative differences of the program
predictions (inset). From [235].
, rad
θ∆
00.2 0.40.6 0.81
, %
MCGPJ
σ
)/
MCGPJ
σ

BHWIDE
0.1
σ
(
0.4
0.3
0.2
0
0.1
0.2
0.3
0.4
Fig. 23. Relative differences between BHWIDE and MCGPJ
Bhabha cross sections as a function of the acollinearity cut, for
the CMD2 experiment at VEPP2M.
corresponding relative differences as a function of the con
sidered angular range. The latter are also shown in Fig. 25,
where the 1σ numerical error due to MC statistics is also
quoted. As can be seen, the two codes agree nicely, the
predictions for the central value being in general in agree
ment at the 0.1% level or statistically compatible when
ever a two to three per mill difference is present.
To further investigate how the two generators compare
with each other a number of differential cross sections were
studied. The results of this study are shown in Figs. 26 and
27 for the distribution of the electron energy and the polar
angle, respectively, and in Fig. 28 for the acollinearity. For
both the energy and scattering angle distribution, the two
programs agree within the statistical errors showing de
viations below 0.5%. For the acollinearity dependence of
the cross section, BabaYaga@NLO and BHWIDE agree
, rad
θ∆
00.2 0.40.60.81
, %
MCGPJ
σ
)/
MCGPJ
σ

BabaYaga@NLO
0.2
σ
(
0.4
0.3
0.1
0
0.1
0.2
0.3
0.4
Fig. 24.
MCGPJ Bhabha cross sections as a function of the acollinearity
cut, for the CMD2 experiment at VEPP2M.
Relative differences between BabaYaga@NLO and
Table 9. Cross section predictions [nb] of BabaYaga@NLO
and BHWIDE for the Bhabha cross section as a function of
the angular selection cuts for the BaBar experiment at PEPII
and absolute value of their relative differences (in per cent).
angular range (c.m.s.)
15◦÷ 165◦
30◦÷ 150◦
40◦÷ 140◦
50◦÷ 130◦
60◦÷ 120◦
70◦÷ 110◦
80◦÷ 100◦
BabaYaga@NLO
119.5(1)
24.17(2)
11.67(3)
6.31(3)
1.928(2)
3.554(6)
0.824(2)
BHWIDE
119.53(8)
24.22(2)
11.660(8)
6.289(4)
1.931(3)
3.549(3)
0.822(1)
δ(%)
0.025
0.207
0.086
0.332
0.141
0.155
0.243
within ∼ 1%. Therefore, the level of the agreement be
tween the two codes around 10 GeV is the same as that
observed at the φ factories.
The main conclusions emerging from the tuned com
parisons discussed in the present Section can be sum
marised as follows:
– The predictions for the Bhabha cross section of the
most precise tools, i.e. BabaYaga@NLO,BHWIDE and
MCGPJ, generally agree within 0.1%. If (slightly) lar
ger differences are present they show up for particu
larly tight cuts or are due to limited MC statistics.
When statistically meaningful discrepancies are ob
served they can be ascribed to the different theoret
ical recipes for the treatment of radiative corrections
and their technical implementation. For example, as
already emphasised, BabaYaga@NLO and BHWIDE
adopt a fully factorised prescription for the matching
of NLO and HO corrections, whereas MCGPJ imple
ment some pieces of the radiative corrections in addi
Page 32
31
angular range (from x to 180x degrees)
10 2030 405060 7080
rel. difference in percent
0.8
0.6
0.4
0.2
0
0.2
0.4
Fig. 25. Relative differences between BabaYaga@NLO and
BHWIDE Bhabha cross sections as a function of the angular
acceptance cut for the BaBar experiment at PEPII. From [50].
[ GeV ]

e
E
12345
[ nb / 0.05 GeV ]
σ
d
dE
1
10
1
10
2
10
BHWIDE
[ GeV ]

e
E
12345
[ nb / 0.05 GeV ]
dE
σ
d
1
10
1
10
2
10
Babayaga@NLO
[ GeV ]

e
E
12345
[ nb / 0.05 GeV ]
σ
d
dE
1
10
1
10
2
10
Babayaga.3.5
[ GeV ]
e
E
12345
difference in percent / 0.05 GeV
80
70
60
50
40
30
20
10
0
relative difference
0.03
±
0.09
BHWIDE
BHWIDE  Babayaga@NLO
BHWIDE
BHWIDE  Babayaga.3.5
[ GeV ]
e
E
4.95 5.15.25.3
difference in percent / 0.05 GeV
0
0.5
1
1.5
2
zoom in
Fig. 26. Electron energy distributions according to BHWIDE,
BabaYaga@NLO and BabaYaga v3.5 for the BaBar experiment
at PEPII and relative differences of the predictions of the
programs. From [50].
tive form. This can give rise to discrepancies between
the programs’ predictions, especially in the presence of
tight cuts enhancing the effect of soft radiation. Fur
thermore, different choices are adopted in the genera
tors for the scale entering the collinear logarithms in
HO corrections beyond O(α), which are another pos
sible source of the observed differences. To go beyond
the present situation, a further nontrivial effort should
be done by comparing, for instance, the programs in
the presence of NLO corrections only (technical test)
and by analysing their different treatment of the expo
nentiation of soft and collinear logarithms. This would
certainly shed light on the origin of the (small) dis
crepancies still registered at present.
Fig. 27. Electron polar angle distributions according to BH
WIDE, BabaYaga@NLO and BabaYaga v3.5 for the BaBar ex
periment at PEPII and relative differences of the predictions
of the programs. From [50].
]
o
acol [
0 20 40 60 80 100 120 140 160 180
]
o
[ nb / 3
d(acol)
σ
d
3
10
2
10
1
10
1
10
2
10
BHWIDE
]
o
acol [
020 40 60 80 100 120 140 160 180
]
o
[ nb / 3
d(acol)
σ
d
3
10
2
10
1
10
1
10
2
10
Babayaga@NLO
]
o
acol [
0 20 40 60 80 100 120 140 160 180
]
o
[ nb / 3
d(acol)
σ
d
3
10
2
10
1
10
1
10
2
10
Babayaga.3.5
]
o
acol [
0 20 40 60 80 100 120 140 160 180
o
difference in percent / 3
400
300
200
100
0
100
relative difference
0.03
±
0.10
BHWIDE
BHWIDE  Babayaga@NLO
BHWIDE
BHWIDE  Babayaga.3.5
]
o
acol [
150155 160165170175180
o
difference in percent / 3
10
8
6
4
2
0
2
zoom in
Fig. 28. Acollinearity distributions according to BHWIDE,
BabaYaga@NLO and BabaYaga v3.5 for the BaBar experiment
at PEPII and relative differences of the predictions of the
programs. From [50].
Page 33
32
– Also the distributions predicted by the generatorsagree
well, with relative differences below the 1% level. Slight
ly larger discrepancies are only seen in sparsely popu
lated phase space regions corresponding to very hard
photon emission which do not influence the luminosity
measurement noticeably.
2.8 Theoretical accuracy
As discussed in Section 2.1, the total luminosity error
crucially depends on the theoretical accuracy of the MC
programs used by the experimentalists. As emphasised
in Section 2.5, some of these generators like BHAGENF,
BabaYaga v3.5 and BKQED miss theoretical ingredients
which are unavoidable for cross section calculations with
a precision at the per mill level. Therefore, they are in
adequate for a highly accurate luminosity determination.
BabaYaga@NLO, BHWIDE and MCGPJ include, how
ever, both NLO and multiple photon corrections, and their
accuracy aims at a precision tag of 0.1%. But also these
generators are affected by uncertainties which must be
carefully considered in the light of the very stringent crite
ria of per mill accuracy. The most important components
of the theoretical error of BabaYaga@NLO, BHWIDE and
MCGPJ are mainly due to approximate or partially in
cluded pieces of radiative corrections and come from the
following sources:
1. The nonperturbative hadronic contributions to the
running of α. It can be reliably evaluated only using
the data of the hadron cross section at low energies.
Hence, the vacuum polarisation correction receives a
data driven error which affects in turn the prediction
of the Bhabha cross section, as emphasised in Section
6.
2. The complete set of O(α2) QED corrections. In spite
of the impressive progress in this area, as reviewed in
Section 2.3, an important piece of NNLO corrections,
i.e. the exact NLO SV QED corrections to the sin
gle hard bremsstrahlung process e+e−→ e+e−γ, is
still missing for the full s + t Bhabha process.10How
ever, partial results obtained for tchannel smallangle
Bhabha scattering [263,47] and largeangle annihila
tion processes are available [264,265].
3. The O(α2) contribution due to real and virtual (lepton
and hadron) pairs. The virtual contributions originate
from the NNLO electron, heavy flavour and hadronic
loop corrections discussed in Section 2.3, while the real
corrections are due to the conversion of an external
photon into pairs. The latter, as discussed in Section
2.3.3, gives rise to a final state with four particles, two
of which to be considered as undetected to contribute
to the Bhabha signature.
The uncertainty relative to the first point can be esti
mated by using the routines available in the literature for
10As already remarked and further discussed in the follow
ing, the complete calculation of the NLO corrections to hard
photon emission in Bhabha scattering was performed during
the completion of this report [101].
the calculation of the nonperturbative hadronic contribu
tion ∆α(5)
return, in addition to ∆α(5)
value. Therefore an estimate of the induced error can be
simply obtained by computing the Bhabha cross section
with ∆α(5)
theoretical uncertainty due to the hadronic contribution
to vacuum polarisation. In Table 10, the Bhabha cross sec
tions, as obtained in the presence of the vacuum polarisa
tion correction according to the parameterisations of [259,
260,18] (denoted as J) and of [164] (denoted as HMNT),
respectively, are shown for φ, τcharm and B factories.
The applied angular cuts refer to the typically adopted
acceptance 55◦≤ θ±≤ 125◦.
hadr(q2) to the running α. Actually these routines
hadr(q2), an error δhadr on its
hadr(q2) ± δhadrand taking the difference as the
Table 10. Bhabha scattering cross section in the presence
of the vacuum polarisation correction, according to [259,260,
18] (J) and [164] (HMNT), at meson factories. The notation
J−/HMNT−, J/HMNT and J+/HMNT+ indicates minimum,
central and maximum value of the two parametrisations.
Parametrisation
J−
J
J+
HMNT−
HMNT
HMNT+
φ
542.662(4)
542.662(4)
542.662(4)
542.500(5)
542.391(5)
542.283(5)
τcharm
46.9600(1)
46.9658(1)
46.9715(1)
46.9580(1)
46.9638(1)
46.9697(1)
B
5.85364(2)
5.85529(2)
5.85693(2)
5.85496(1)
5.85621(1)
5.85746(2)
From Table 10 it can be seen that the two treatments
of ∆α(5)
in very good agreement, the relative differences between
the central values being 0.05% (φ factories), 0.005% (τ
charm factories) and 0.02% (B factories). This can be
understood in terms of the dominance of tchannel ex
change for largeangle Bhabha scattering at meson fac
tories. Indeed, the two routines provide results in excel
lent agreement for spacelike momenta, as we explicitly
checked, whereas differences in the predictions show up
for timelike momenta which, however, contribute only
marginally to the Bhabha cross section. Also the spread
between the minimum/maximum values and the central
one as returned by the two routines agrees rather well, also
a consequence of the dominance of tchannel exchange.
This spread amounts to a few units in 10−4and is pre
sented in detail in Table 11 in the next Section.
Concerning the second point a general strategy to eval
uate the size of missing NNLO corrections consists in de
riving a cross section expansion up to O(α2) from the
theoretical formulation implemented in the generator of
interest. It can be cast in general into the following form
hadr(q2) induce effects on the Bhabha cross section
σα2= σα2
SV+ σα2
SV,H+ σα2
HH,(84)
where in principle each of the O(α2) contributions is af
fected by an uncertainty to be properly estimated. In Eq. (84)
the first contribution is the cross section including O(α2)
Page 34
33
SV corrections,whose uncertainty can be evaluated through
a comparison with some of the available NNLO calcula
tions reviewed in Section 2.3. In particular, in [235] the
σα2
SVof the BabaYaga@NLO generator was compared with
the calculation of photonic corrections by Penin [135,136]
and the calculations by Bonciani et al. [140,141,151,152,
153] who computed twoloop fermionic corrections (in the
onefamily approximation NF= 1) with finite mass terms
and the addition of soft bremsstrahlung and real pair con
tributions.11The results of such comparisons are shown
in Figs. 29 and 30 for realistic cuts at the φ factories. In
Fig. 29 δσ is the difference between σα2
and the cross sections of the two O(α2) calculations, de
noted as photonic (Penin) and NF = 1 (Bonciani et al.),
as a function of the logarithm of the infrared regulator ǫ. It
can be seen that the differences are given by flat functions,
demonstrating that such differences are infraredsafe, as
expected, a consequence of the universality and factori
sation properties of the infrared divergences. In Fig. 30,
δσ is shown as a function of the logarithm of a fictitious
electron mass and for a fixed value of ǫ = 10−5. Since
the difference with the calculation by Penin is given by a
straight line, this indicates that the soft plus virtual two
loop photonic corrections missing in BabaYaga@NLO are
O(α2L) contributions, as already remarked. On the other
hand, the difference with the calculation by Bonciani et al.
is fitted by a quadratic function, showing that the electron
twoloop effects missing in BabaYaga@NLO are of the or
der of α2L2. However, it is important to emphasise that,
as shown in detail in [235], the sum of the relative differ
ences with the two O(α2) calculations does not exceed the
2 × 10−4level for experiments at φ and B factories.
The second term in Eq. (84) is the cross section con
taining the oneloop corrections to single hard photon
emission, and its uncertainty can be estimated by relying
on partial results existing in the literature. Actually the
exact perturbative expression of σα2
for full s+t Bhabha scattering, but using the results valid
for smallangle Bhabha scattering [263,47] and largeangle
annihilation processes [264,265] the relative uncertainty of
the theoretical tools in the calculation of σα2
servatively estimated to be at the level of 0.05%. Indeed
the papers [263,47,264,265] show that a YFS matching of
NLO and HO corrections gives SV oneloop results for the
tchannel process e+e−→ e+e−γ and schannel annihila
tion e+e−→ f¯fγ (f = fermion) differing from the exact
perturbative calculations by a few units in 10−4at most.
This conclusion also holds when photon energy cuts are
varied. It is worth noting that during the completion of
the present work a complete calculation of the NLO QED
corrections to hard bremsstrahlung emission in full s + t
Bhabha scattering appeared in the literature [101], along
SVof BabaYaga@NLO
SV,His not yet available
SV,Hcan be con
11To provide meaningful results, the contribution of the vac
uum polarisation was switched off in BabaYaga@NLO to com
pare with the calculation by Penin consistently. For the same
reason the real soft and some pieces of virtual electron pair cor
rections were neglected in the comparison with the calculation
by Bonciani et al.
1
0.5
0
0.5
1
1e161e141e121e101e081e06
δσ (nb)
ε
NF=1
photonic
fit
fit
Fig. 29. Absolute differences (in nb) between the σα2
diction of BabaYaga@NLO and the NNLO calculations of the
photonic corrections [135,136] (photonic) and of the electron
loop corrections [140,141,151,152,153] (NF = 1) as a function
of the infrared regulator ǫ for typical KLOE cuts. From [235].
SVpre
5
1e10
4
3
2
1
0
1
2
3
4
5
1e091e081e071e061e051e040.0010.01
δσ (nb)
me(GeV)
NF=1
photonic
fit
fit
Fig. 30. Absolute differences (in nb) between the σα2
diction of BabaYaga@NLO and the NNLO calculations of the
photonic corrections [135,136] (photonic) and of the electron
loop corrections [140,141,151,152,153] (NF = 1) as a function
of a fictitious electron mass for typical KLOE cuts. From [235].
SVpre
the lines described in Section 2.3.2. Explicit comparisons
between the results of such an exact calculation with the
predictions of the most accurate MC tools according to
the typical luminosity cuts used at meson factories would
be worthwhile to make the present error estimate related
to the calculation of σα2
SV,Hmore robust.
The third contribution in Eq. (84) is the double hard
bremsstrahlung cross section whose uncertainty can be
directly evaluated by explicit comparison with the exact
e+e−→ e+e−γγ cross section. It was shown in [235] that
the differences between σα2
HHas in BabaYaga@NLO and
the matrix element calculation, which exactly describes
the contribution of two hard photons, are really negligi
ble, being at the 10−5level.
The relative effect due to lepton (e,µ,τ) and hadron
(π) pairs has been numerically analysed in Section 2.3.3,
in the presence of realistic selection cuts. This evalua
tion makes use of the complete NNLO virtual corrections
Page 35
34
combined with an exact matrix element calculation of
the fourparticle production processes. It supersedes previ
ous approximate estimates which underestimated the im
pact of those corrections. According to this new evalua
tion, the pair contribution, dominated by the electron pair
correction, amounts to about 0.3% for KLOE and 0.1%
for BaBar. These contributions are partially included in
the BabaYaga@NLO code, as well as in other generators,
through the insertion of the vacuum polarisation correc
tion in the NLO diagrams, and detailed comparisons be
tween the exact calculation and the BabaYaga@NLO pre
dictions are in progress [266].
2.9 Conclusions and open issues
During the last few years a remarkable progress occurred
in reducing the error of the luminosity measurements at
flavour factories.
Dedicated event generators like BabaYaga@NLO and
MCGPJ were developed in 2006 to provide predictions
for the cross section of the largeangle Bhabha process, as
well as for other QED reactions of interest, with a theoret
ical accuracy at the level of 0.1%. In parallel codes well
known since the time of the LEP/SLC operation such as
BHWIDE were extensively used by the experimentalists
in data analyses. All these MC programs include, albeit
according to different formulations, exact O(α) QED cor
rections matched with LL contributions describing multi
ple photon emission. Such ingredients, together with the
vacuum polarisation correction, are strictly necessary to
achieve a physical precision down to the per mill level.
Indeed, when considering typical selection, cuts the NLO
photonic corrections amount to about 15÷20%, vacuum
polarisation contributes at the several per cent level and
HO effects lie between 1÷2%.
The generators mentioned are, however, affected by
an uncertainty due to HO effects neglected in their for
mulation, such as light pair corrections or exact pertur
bative contributions present in NNLO calculations. From
this point of view the great progress in the calculation of
twoloop corrections to the Bhabha scattering cross sec
tion was essential to establish the theoretical accuracy of
the existing generators and will be crucial if an improve
ment of the precision below the one per mill level will be
required.
A particular effort was done to compare the predictions
of the generators consistently, in order to assess the techni
cal precision obtained by the implementation of radiative
corrections and related computational details. These com
parisons were performed in the presence of realistic event
selection criteria and at different c.m. energies. For the
KLOE and CMD2 experiments around the φresonance,
where the statistics of Bhabha events is the highest and
the experimental luminosity error at a few per mill level,
the cross section results of BabaYaga@NLO, BHWIDE
and MCGPJ agree within ∼ 0.1%. If (slightly) larger dis
crepancies are observed, they show up only for particularly
tight cuts or exclusive distributions in specific phase space
regions which do not influence the luminosity determina
tion. Very similar results were obtained for τcharm and
B factories. The main conclusion of the work on tuned
comparisons is that the technical precision of MC pro
grams is well under control, the discrepancies being due
to different details in the treatment of the same sources
of radiative corrections and their technical implementa
tion. For example, BabaYaga@NLO and BHWIDE adopt
a fully factorised prescription for the matching of NLO and
HO corrections, whereas MCGPJ implement some radia
tive corrections pieces in additive form. This can give rise
to some discrepancies between their predictions, especially
in the presence of tight cuts enhancing the effect of soft ra
diation. Furthermore, different choices are adopted in the
generators for the energy scale in the treatment of HO cor
rections beyond O(α), which are another possible source
of the observed differences. To go beyond the present sit
uation, a further, nontrivial effort should be done by com
paring, for instance, the programs in the presence of NLO
corrections only (technical test) and for the specific effect
due to the exponentiation of soft and collinear logarithms.
This would certainly shed light on the origin of the (mi
nor) discrepancies still registered at present.
On the theoretical side, a new exact evaluation of lep
ton and hadron pair corrections to the Bhabha scattering
cross section was carried out, taking into account realistic
cuts. This calculation provides results in substantial agree
ment with estimates based on singlet SF but supersedes
previous evaluations in the softphoton approximation.
The results of the new exact calculation were preliminarily
compared with the predictions of BabaYaga@NLO, which
includes the bulk of such corrections (due to reducible
contributions) through the insertion of the vacuum polar
isation correction in the NLO diagrams, but neglects the
effect of real pair radiation and twoloop form factors. It
turns out that the error induced by the approximate treat
ment of pair corrections amounts to a few units in 10−4,
both at KLOE and BaBar. Further work is in progress to
arrive at a more solid and quantitative error estimate for
these corrections when considering other selection criteria
and c.m. energies too [266]. Also, the contribution induced
by the uncertainty related to the nonperturbative contri
bution to the running of α was revisited, making use of
and comparing the two independent parameterisations de
rived in [259,260,18] and [164].
A summary of the different sources of theoretical er
ror and their relative impact on the Bhabha cross section
is given in Table 11. In Table 11, δerr
duced by the hadronic component of the vacuum polar
isation, δerr
δerr
δerr
bremsstrahlung process and δerr
oneloop corrections to single hard bremsstrahlung. As can
be seen, pair corrections and exact NLO corrections to
e+e−→ e+e−γ are the dominant sources of error.
The total theoretical uncertainty as obtained by sum
ming the different contributions linearly is 0.12÷0.14%
at the φ factories, 0.18% at the τcharm factories and
VP is the error in
pairs the error due to missing pair corrections,
SV the uncertainty coming from SV NNLO corrections,
HH the uncertainty in the calculation of the double hard
SV,H the error estimate for
Page 36
35
Table 11. Summary of different sources of theoretical uncer
tainty for the most precise generators used for luminosity mea
surements and the corresponding total theoretical errors for the
calculation of the largeangle Bhabha cross section at meson
factories.
Source of error (%)
δerr
δerr
δerr
δerr
δerr
δerr
δerr
φ
0.00
0.02
0.02
0.00
0.05
0.05
0.12÷0.14
τcharm
0.01
0.01
0.02
0.00
0.05
0.1
0.18
B
0.03
0.02
0.02
0.00
0.05
0.02
0.11÷0.12
VP [259,260,18]
VP [164]
SV
HH
SV,H
pairs
total
0.11 ÷ 0.12% at the B factories. As can be seen from Ta
ble 11, the slightly larger uncertainty at the τcharm fac
tories is mainly due to the pair contribution error, which
is presently based on a very preliminary evaluation and
for which a deeper analysis is ongoing [266]. The total
uncertainty is slightly affected by the particular choice of
the routine for the calculation of ∆α(5)
two parameterisations considered here give rise to simi
lar errors, with the exception of the φ factories for which
the two recipes return uncertainties differing by 2×10−4.
However the “parametric” error induced by the hadronic
contribution to the vacuum polarisation may become a rel
evant source of uncertainty when considering predictions
for a c.m. energy on top of and closely around very narrow
resonances. For such a specific situation of interest, for in
stance for the BES experiment, the appropriate treatment
of the running α in the calculation of the Bhabha cross
section should be scrutinised deeper because of the differ
ences observed between the predictions for ∆α(5)
tained by means of the different parametrisation routines
available (see Section 6 for a more detailed discussion).
Although the theoretical uncertainty quoted in Ta
ble 11 could be put on firmer ground thanks to further
studies in progress, it appears to be quite robust and suf
ficient for present and planned precision luminosity mea
surements at meson factories, where the experimental er
ror currently is about a factor of two or three larger.
Adopting the strategy followed during the LEP/SLC op
eration one could arrive at a more aggressive error es
timate by summing the relative contributions in quadra
ture. However, for the time being, this does not seem to be
necessary in the light of the current experimental errors.
In conclusion, the precision presently reached by large
angle Bhabha programs used in the luminosity measure
ment at meson factories is comparable with that achieved
about ten years ago for luminosity monitoring through
smallangle Bhabha scattering at LEP/SLC.
Some issues are still left open. In the context of tuned
comparisons, no effort was done to compare the available
codes for the process of photon pair production. Since it
contributes relevantly to the luminosity determination and
as precise predictions for its cross section can be obtained
by means of the codes BabaYaga@NLO and MCGPJ, this
hadr(q2), since the
hadr(q2) ob
work should be definitely carried out. This would lead
to a better understanding of the luminosity on the ex
perimental side. In the framework of new theoretical ad
vances, an evaluation of NNLO contributions to the pro
cess e+e−→ γγ would be worthwhile to better assess the
precision of the generators which, for the time being, do
not include such corrections exactly. More importantly,
the exact oneloop corrections to the radiative process
e+e−→ e+e−γ should be calculated going beyond the
partial results scattered in the literature (and referring to
selection criteria valid for highenergy e+e−colliders) or
limited to the softphoton approximation.12Furthermore,
to get a better control of the theoretical uncertainty in
the sector of NNLO corrections to Bhabha scattering, the
radiative Bhabha process at oneloop should be evaluated
taking into account the typical experimental cuts used at
meson factories. Incidentally this calculation would be also
of interest for other studies at e+e−colliders of moderately
high energy, such as the search for new physics phenomena
(e.g. dark matter candidates), for which radiative Bhabha
scattering is a very important background.
3 R measurement from energy scan
In this section we will consider some theoretical and exper
imental aspects of the direct R measurement and related
quantities in experiments with energy scan. As discussed
in the Introduction, the cross section of e+e−annihilation
into hadrons is involved in evaluations of various problems
of particle physics and, in particular, in the definition of
the hadronic contribution to vacuum polarisation, which
is crucial for the precision tests of the Standard Model
and searches for new physics.
The ratio of the radiationcorrected hadronic cross sec
tions to the cross section for muon pair production, cal
culated in the lowest order, is usually denoted as (see
Eq. (23))
R ≡ R(s) =
σ0
had(s)
4πα2/(3s).(85)
In the numerator of Eq. (85) one has to use the so called
undressed hadronic cross section which does not include
vacuum polarisation corrections.
The value of R has been measured in many experi
ments in different energy regions from the pion pair pro
duction threshold up to the Z mass. Practically all electron
positron colliders contributed to the global data set on the
hadronic annihilation cross section [267]. The value of R
12As already remarked in Section 2.8, during the completion
of the present work a complete calculation of the NLO QED
corrections to hard bremsstrahlung emission in full s+t Bhabha
scattering was performed in [101]. However, explicit compar
isons between the predictions of this new calculation and the
corresponding results of the most precise luminosity tools are
still missing and would be needed to better assess the theoret
ical error induced by such contributions in the calculation of
the luminosity cross section.
Page 37
36
extracted from the experimental data is then widely used
for various QCD tests as well as for the calculation of
dispersion integrals. At high energies and away from res
onances, the experimentally determined values of R are
in good agreement with predictions of perturbative QCD,
confirming, in particular, the hypothesis of three colour
degrees of freedom for quarks. On the other hand, for the
low energy range the direct R measurement [267,268] at
experiments with energy scan is necessary.13Matching be
tween the two regions is performed at energies of a few
GeV, where both approaches for the determination of R
are in fair agreement.
For the best possible compilation of R(s), data from
different channels and different experiments have to be
combined. For√s ≤ 1.4 GeV, the total hadronic cross
section is a sum of about 25 exclusive final states. At the
present level of precision, a careful treatment of the ra
diative corrections is required. As mentioned above, it
is mandatory to remove VP effects from the observed
cross sections, but the final state radiation off hadrons
should be kept. The major contribution to the uncertainty
comes from the systematic error of the R(s) measurement
at low energies (s < 2 GeV2), which, in turn, is domi
nated by the systematic error of the measured cross sec
tion e+e−→ π+π−.
3.1 Leadingorder annihilation cross sections
Here we present the lowestorder expressions for the pro
cesses of electronpositronannihilation into pairs of muons,
pions and kaons.
For the muon production channel
e−(p−) + e+(p+) → µ−(p′
within the Standard Model at Born level we have
dσµµ
0
dΩ−
s = (p−+ p+)2= 4ε2,
?
suppressed by the factor m2
resents contributions due to Zboson intermediate states
including Z − γ interference, see, e.g., Refs. [270,271]:
W=s2(2 − β2
(s − M2
?
?
ca= −
−) + µ+(p′
+) (86)
=α2βµ
4s
?2 − β2
µ(1 − c2)?(1 + Kµµ
c = cosθ−,
W),(87)
θ−= ?
p−p′
−,
where βµ=
1 − m2
µ/ε2is the muon velocity. Small terms
e/s are omitted. Here Kµµ
Wrep
Kµµ
µ(1 − c2))−1
Z)2+ M2
3 − 2M2
s
?
1
2sin2θW,
ZΓ2
Z
?
?
−1 − β2
??
(2 − β2
µ(1 − c2))
×c2
v
?
Z
?
+ c2
?
cv= ca(1 − 4sin2θW),
a
µ
2
(c2
a+ c2
v)
+cβµ
41 −M2
Z
s
c2
a+ 8c2
ac2
v
,
(88)
13Lattice QCD computations (see, e.g., Ref. [269]) of the
hadronic vacuum polarisation are in progress, but they are not
yet able to provide the required precision.
where θW is the weak mixing angle.
The contribution of Z boson exchange is suppressed,
in the energy range under consideration, by a factor s/M2
which reaches per mill level only at B factories.
In the Born approximation the differential cross sec
tion of the process
Z
e+(p+) + e−(p−) → π+(q+) + π−(q−)
has the form
dσππ
0
dΩ
βπ=1 − m2
The pion form factor Fπ(s) takes into account nonpertur
bative virtual vertex corrections due to strong interac
tions [272,256]. We would like to emphasise that in the
approach under discussion the final state QED corrections
are not included into Fπ(s). The form factor is extracted
from the experimental data on the same process as dis
cussed below.
The annihilation process with three pions in the final
state was considered in Refs. [273,274] including radia
tive corrections relevant to the energy region close to the
threshold. A standalone Monte Carlo event generator for
this channel is available [273]. The channel was also in
cluded in the MCGPJ generator [236] on the same footing
as other processes under consideration in this report.
In the case of KLKSmeson pair production the differ
ential cross section in the Born approximation reads
(89)
(s) =α2β3
?
π
8s
π/ε2,
sin2θ Fπ(s)2,
θ = ?
(90)
p−q−.
dσ0(s)KLKS
dΩL
=α2β3
K
4s
sin2θ FLS(s)2. (91)
Here, as well as in the case of pion production, we as
sume that the form factor FLS also includes the vacuum
polarisation operator of the virtual photon. The quantity
βK=?1 − 4m2
living kaon and the initial electron.
In the case of K+K−meson production near thresh
old, the SakharovSommerfeld factor for the Coulomb fi
nal state interaction should additionally be taken into ac
count:
dσ0(s)K+K−
dΩ−
Z =2πα
vs
K/s is the K meson c.m.s. velocity, and θ
is the angle between the directions of motion of the long
=α2β3
?
K
4s
sin2θFK(s)2
?
Z
1 − exp(−Z),
1 +s − 4m2
s
, v = 2
s − 4m2
K
K
?−1
,(92)
where v is the relative velocity of the kaons [275] which
has the proper nonrelativistic and ultrarelativistic lim
its. When s = m2
φ, we have v ≈ 0.5 and the final state
interaction correction gives about 5% enhancement in the
cross section.
3.2 QED radiative corrections
Oneloop radiative corrections(RC) for the processes(86,89)
can be separated into two natural parts according to the
parity with respect to the substitution c → −c.
Page 38
37
The ceven part of the oneloop soft and virtual contri
bution to the muon pair creation channel can be combined
from the well known Dirac and Pauli form factors and the
soft photon contributions. It reads
dσB+S+V
µµ−even
dΩ
=dσµµ
dΩ
?
−5
?
0
1
1 − Π(s)2
ln∆ε
ε
?1 + β2
1
2βµ
?
1 +2α
π
??
??
?
L − 2
+1 + β2
2βµ
µ
lβ
+3
4(L − 1) + Kµµ
even
, (93)
Kµµ
even=π2
64+ ρ
µ
2βµ
−1
?
2+
1
4βµ
+ln1 + βµ
2
+1 + β2
µ
βµ
(94)
−1 − β2
2βµ
+1 + β2
2βµ
+ρln1 + βµ
µ
lβ
2 − β2
?π2
µ(1 − c2)
µ
6
+ 2Li2
?1 − βµ
1 + βµ
?
2β2
µ
+ 2ln1 + βµ
2
ln1 + βµ
2β2
µ
?
,
lβ= ln1 + βµ
1 − βµ
,ρ = ln
s
m2
µ
L = ln
s
m2
e
,
where Li2(z) = −?z
centre–of–mass (c.m.) system. Π(s) is the vacuum polar
isation operator. Here we again see that the terms with
the large logarithm L dominate numerically.
The codd part of the one–loop correction comes from
the interference of Born and box Feynman diagrams and
from the interference part of the soft photon emission con
tribution. It causes the charge asymmetry of the process:
0dtln(1 − t)/t is the dilogarithm and
∆ε ≪ ε is the maximum energy of soft photons in the
η =dσ(c) − dσ(−c)
dσ(c) + dσ(−c)?= 0.(95)
The codd part of the differential cross section has the
following form [245]:
dσS+V
odd
dΩ
=dσµµ
dΩ
0
2α
π
?
2ln∆ε
ε
ln1 − βc
1 + βc+ Kµµ
odd
?
. (96)
The expression for the codd form factor can be found
in Ref. [245]. Note that in most cases the experiments
have a symmetric angular acceptance, so that the odd part
of the cross section does not contribute to the measured
quantities.
Consider now the process of hard photon emission
e+(p+) + e−(p−) → µ+(q+) + µ−(q−) + γ(k). (97)
It was studied in detail in Refs. [245,276]. The photon
energy is assumed to be larger than ∆ε. The differential
cross section has the form
dσµµγ=
α3
2π2s2RdΓ,(98)
dΓ =d3q−d3q+d3k
q0
s
16(4πα)3
−q0
+k0
δ(4)(p++ p−− q−− q+− k),
R =
?
spins
M2= Re+ Rµ+ Reµ.
The quantities Riare found directly from the matrix ele
ments and read
Re=
s
χ−χ+B −m2
−m2
2χ2
+
?
t1
χ+χ′
+
s1
χ′
+
B =u2+ u2
e
2χ2
−
(t2
1+ u2
1+ 2m2
s2
1
+m2
s2
1
t
χ−χ′
µs1)
e
(t2+ u2+ 2m2
µs1)
s2
1
µ
∆s1s1,
Reµ= B
u
χ−χ′
?
B +m2
+
+
u1
χ+χ′
−
−
−
−
+m2
ss1∆ss1,
µ
Rµ=
−χ′
µ
s2∆ss,
1+ t2+ t2
4ss1
1
,
∆s1s1= −(t + u)2+ (t1+ u1)2
∆ss= −u2+ t2
2(χ′
1
χ′
+
∆ss1=s + s1
2
?
2χ−χ+
1+ 2sm2
−)2
?ss1− s2+ tu + t1u1− 2sm2
?
+2(u − t1)
χ′
−
,
µ
−u2
1+ t2+ 2sm2
2(χ′
µ
+)2
+
−χ′
µ
?,
u
χ−χ′
+
+
u1
χ+χ′
+2(u1− t)
χ′
−
−
t
χ−χ′
−
−
t1
χ+χ′
++
,
where
s1= (q++ q−)2, t = −2p−q−, t1= −2p+q+,
u = −2p−q+, u1= −2p+q−, χ±= p±k, χ′
±= q±k.
The bulk of the hard photon radiation comes from ISR
in collinear regions. If we consider photon emission inside
two narrow cones along the beam axis with restrictions
?
p±k = θ ≤ θ0≪ 1,θ0≫me
ε,
(99)
Page 39
38
we see that the corresponding contribution takes the fac
torised form
?dσµµ
dΩ−
?
coll
?
?
∆
= Ce+ De,(100)
Ce=
α
2π
ln
s
m2
e
− 1
?
1
?
∆
dx1 + (1 − x)2
x
A0,
De=
α
2π
?d˜ σ0(1 − x,1)
1
dx
?
x +1 + (1 − x)2
x
lnθ2
0
4
?
A0,
A0=
dΩ−
+d˜ σ0(1,1 − x)
dΩ−
?
,
where the shifted Born differential cross section describes
the process e+(p+(1−x2))+e−(p−(1−x1)) → µ+(q+)+
µ−(q−),
d˜ σµµ
0(z1,z2)
dΩ−
×y1[z2
=α2
4s
1(Y1− y1c)2+ z2
z3
2[z1+ z2− (z1− z2)cY1/y1]
1,2−4m2
s
Y1=4m2
s
?
+
z1+ z2− c(z1− z2).
2(Y1+ y1c)2+ 8z1z2m2
µ/s]
1z3
,
y2
1,2= Y2
µ
,Y1,2=q0
−,+
ε
,z1,2= 1 − x1,2,
µ
(z2− z1)c
?
2z1z2
+
4z2
1z2
2− 4(m2
2z1z2
µ/s)((z1+ z2)2− (z1− z2)2c2)
?−1
(101)
One can recognise that the large logarithms related to the
collinear photon emission appear in Cein agreement with
the structure function approach discussed in the Luminos
ity Section. In analogy to the definition of the QCD struc
ture functions, one can move the factorised logarithmic
corrections Ceinto the QED electron structure function.
Adding the higherorder radiative corrections in the lead
ing logarithmic approximation to the complete oneloop
result, the resulting cross section can be written as
dσe+e−→µ+µ−(γ)
dΩ−
×d˜ σµµ
dΩ−
?
=
1
?
zmin
1 +α
1
?
zmin
dz1dz2D(z1,s)D(z2,s)
1 − Π(sz1z2)2
0(z1,z2)
?
πKµµ
Reme=0
1 − Π(s1)2
odd+α
πKµµ
dΓ
dΩ−
even
?
+
α3
2π2s2
?
k0>∆ε
c
kp±>θ0
?
k0>∆ε
+
De
1 − Π(s1)2
?
+
?
α3
2π2s2
?
Re
Reµ
(1 − Π(s1))(1 − Π(s))∗
+
Rµ
1 − Π(s)2
Cµ
1 − Π(s)2
Cµ=2α
π
Ceµ=4α
π
?dΓ
?
ln∆ε
dΩ−
+ Re
Ceµ
(1 − Π(s1))(1 − Π(s))∗
+
,(102)
dσµµ
dΩ−
dσµµ
dΩ−
0
ε
?1 + β2
ln1 − βc
1 + βc,
2β
ln1 + β
1 − β− 1
?
,
0
ln∆ε
ε
where De, Ceµ and Cµ are compensating terms, which
provide the cancellation of the auxiliary parameters ∆ and
θ0inside the curly brackets. In the first term, containing D
functions, we collect all the leading logarithmic terms. A
part of nonleading terms proportional to the Born cross
section is written as the Kfactor. The rest of the non
leading contributions are written as two additional terms.
The compensating term De(see Eq. (100)) comes from the
integration in the collinear region of hard photon emission.
The quantities Cµand Ceµcome from the even and odd
parts of the differential cross section (arising from soft
and virtual corrections), respectively. Here we consider the
phase space of two (dΩ−) and three (dΓ) final particles as
those that already include all required experimental cuts.
Using specific experimental conditions one can determine
the lower limits of the integration over z1and z2instead
of the kinematical limit zmin= 2mµ/(2ε − mµ).
Matching of the complete O(α) RC with higherorder
leading logarithmic corrections can be performed in differ
ent schemes. The above approach is implemented in the
MCGPJ event generator [236]. The solution of the QED
evolution equations in the form of parton showers (see the
Luminosity Section), matched again with the first order
corrections, is implemented in the BabaYaga@NLO gen
erator [234]. Another possibility is realised in the KKMC
code [277,278] with the YennieFrautschiSuura exponen
tiated representation of the photonic higherorder correc
tions. A good agreement was obtained in [236] for various
differential distributions for the µ+µ−channel between
MCGPJ, BabaYaga@NLO and KKMC, see Fig. 31 for an
example.
Since the radiative corrections to the initial e+e−state
are the same for annihilation into hadrons and muons as
well as that into pions, they cancel out in part in the ra
Page 40
39
E, MeV
×
2
400600 8001000 1200 1400
Cross section difference, %
1
0.8
0.6
0.4
0.2
0
0.2
0.4
0.6
0.8
1
Fig. 31. Comparison of the e+e−→ µ+µ−total cross sections
computed by the MCGPJ and KKMC generators versus the
c.m. energy.
tio (106). However, the experimental conditions and sys
tematic are different for the muon and hadron channels.
Therefore, a separate treatment of the processes has to be
performed and the corrections to the initial states have to
be included in the analysis.
For the π+π−channel the complete set of O(α) correc
tions matched with the leading logarithmic electron struc
ture functions can be found in Ref. [279]. There the RC
calculation was performed within scalar QED.
Taking into account only final state corrections calcu
lated within scalar QED, it is convenient to introduce the
bare e+e−→ π+π−(γ) cross section as
πFπ(s)21−Π(s)2?
where the factor 1 − Π(s)2with the polarisation opera
tor Π(s) gives the effect of leptonic and hadronic vacuum
polarisation. The final state radiation (FSR) correction is
denoted by Λ(s). For an inclusive measurement without
cuts it reads [280,281,282,283]
σ0
ππ(γ)=πα2
3sβ3
1 +α
πΛ(s)
?
, (103)
Λ(s) =1 + β2
?
π
βπ
?
4Li2(1 − βπ
1 + βπ) + 2Li2(−1 − βπ
?
4(1 + β2
1 + βπ)
?
ln1 + βπ
1 − βπ
−3ln(
2
1 + βπ) + 2lnβπ
1
β3
π
1 + β2
π
β2
π
ln1 + βπ
1 − βπ
π)2− 2
− 3ln(
4
1 − β2
π
)
−4lnβπ+
+3
2
?5
?
.(104)
For the neutral kaon channel the corrected cross sec
tion has the form
dσe+e−→KLKS(s)
dΩL
=
∆
?
0
dxdσe+e−→KLKS
0
(s(1 − x))
dΩL
F(x,s).
The radiation factor F takes into account radiative cor
rections to the initial state within the leading logarith
mic approximation with exponentiation of the numeri
cally important contribution of soft photon radiation, see
Ref. [228]:
?
−37
42
+1
x(1 + 3(1 − x)2)ln
?α
+b
3m2
e
3
+(2 − x)ln(1 − x) +x
2
F(x,s) = bxb−1
??
1 +3
4b +α
?
π
?π2
?
3
4(2 − x)ln1
−1
2
?
−b2
24
?1
3L − 2π2
− b
?
1 −x
+1
8b2
1
x
1 − x− 6 + x
?b?
+1
2L2
??
?
+
π
?
?2?1
lnsx2
6x
?
x −2me
?3?
ε
(2 − 2x + x2)
?2
Θ(x −2me
?
lnsx2
m2
e
−5
3
?2
−5
3
1 − (1 − x)3
1 − x
ε
).
Radiative corrections to the K+K−channel in the
pointlike particle approximation are the same as for the
case of charged pion pair (with the substitution mπ →
mK). Usually, for the kaon channel we deal with the en
ergy range close to φ mass. There one may choose the
maximal energy of a radiated photon as
ω ≤ ∆E = mφ− 2mK≪ mK,∆ ≡∆E
mK
≈
1
25. (105)
For these photons one can use the soft photon approxima
tion.
3.3 Experimental treatment of hadronic cross sections
and R
For older low energy data sets obtained at various e+e−
colliders, the correct treatment of radiative corrections is
difficult and sometimes ambiguous. So, to avoid uncon
trolled possible systematic errors, it may be reasonable
not to include all previous results except the recent data
from CMD2 and SND. Both experiments at the VEPP
2M collider in Novosibirsk have delivered independent new
measurements. The covered energy range is crucial for (gµ
2)/2 of muon and for running α. As for the twopion chan
nel π+π−, which gives more than 70% of the total hadronic
contribution, both experiments have very good agreement
over the whole energy range. The relative deviation “SND
 CMD2” is (0.3 ± 1.6)% only, well within the quoted
errors.
Page 41
40
The CMD2 and SND detectors were located in the
opposite straight sections of VEPP2M and were taking
data in parallel until the year 2000 when the collider was
shut down to prepare for the construction of the new col
lider VEPP2000. Some important features of the CMD
2 detector allowed one to select a sample of the “clean”
collinear backtoback events. The drift chamber (DC) was
used to separate e+e−, µ+µ−, π+π−and K+K−events
from other particles. The Zchamber allowed one to sig
nificantly improve the determination of the polar angle of
charged particle tracks in the DC that, in turn, provided
the detector acceptance with 0.2% precision. The barrel
electromagnetic calorimeter based on CsI crystals helped
to separate the Bhabha from other collinear events.
The SND detector consisted of three spherical lay
ers of the electromagnetic calorimeter with 1620 crystals
(NaI) and a total weight of 3.6 tons. The solid angle
of the calorimeter is about 90% of 4π steradians, which
makes the detector practically hermetic for photons com
ing from the interaction point. The angular and energy
resolution for photons was found to be 1.5◦and σ(E)/E =
4.2%/E(GeV)1/4, respectively. More detail about CMD2
and SND can be found elsewhere [284,285].
3.3.1 Data taking and analysis of the π+π−channel
The detailed data on the pion form factor are crucial for
a number of problems in hadronic physics and they are
used to extract ρ(770) meson parameters and its radial
excitations. Besides, the detailed data allow to extrapolate
the pion form factor to the point s = 0 and determine the
value of the pion electromagnetic radius.
From the experimental point of view the form factor
can be defined as [268]
Fπ2=
Nππ
Nee+ Nµµ
− ∆3π,
σee(1 + δee)εee+ σµµ(1 + δµµ)εµµ
σππ(1 + δππ)(1 + ∆N)(1 + ∆D)εππ
(106)
where the ratio Nππ/(Nee+ Nµµ) is derived from the ob
served numbers of events, σ are the corresponding Born
cross sections, δ are the radiative corrections (see below),
ǫ are the detection efficiencies, ∆D and ∆N are the cor
rections for the pion losses caused by decays in flight
and nuclear interactions respectively, and ∆3πis the cor
rection for misidentification of ω → π+π−π0events as
e+e−→ π+π−. In the case of the latter process, σππcor
responds to pointlike pions.
The data were collected in the whole energy range of
VEPP2M and the integrated luminosity of about 60 pb−1
was recorded by both detectors. The beam energy was con
trolled and measured with a relative accuracy not worse
than ∼ 10−4by using the method of resonance depolari
sation. A sample of the e+e−, µ+µ−and π+π−events was
selected for analysis. As for CMD2, the procedure of the
e/µ/π separation for energies 2E ≤ 600 MeV was based on
the momentum measurement in the DC. For these ener
gies the average difference between the momenta of e/µ/π
is large enough with respect to the momentum resolution
(Fig. 32). On the contrary, for energies 2E ≥ 600 MeV,
the energy deposition of the particles in the calorime
ter is quite different and allows one to separate electrons
from muons and pions (Fig. 33). At the same time, muons
and pions cannot be separated by their energy deposi
tions in the calorimeter. So, the ratio N(µ+µ−)/N(e+e−)
was fixed according to QED calculations taking into ac
count the detector acceptance and the radiative correc
tions. Since the selection criteria were the same for all
collinear events, many effects of the detector imperfec
tions were partly cancelled out. It allowed one to measure
the cross section of the process e+e−→ π+π−with better
precision than that of the luminosity.
, MeV/c
 P
100120140160180200220
, MeV/c
+
P
100
120
140
160
180
200
220
e
+
e
π
+
π
µ
+
µ
cosmic
Fig. 32. Twodimensional plot of the e/µ/π events. Cosmic
events are distributed predominantly along a corridor which
extends from the right upper to the left bottom corner. Points
in this plot correspond to the momenta of particles for the
beam energy of 195 MeV.
Separation of e+e−, µ+µ−and π+π−events was based
on the minimisation of the unbinned likelihood function.
This method is described in detail elsewhere [286]. To sim
plify the error calculation of the pion form factor, the
likelihood function had the global fit parameters (Nee+
Nµµ) and Nππ/(Nee+ Nµµ), through Fπ(s)2given by
Eq. (106). The pion form factor measured by CMD2 has
a systematic error of about 0.60.8% for√s ≤ 1 GeV. For
energies above 1 GeV it varies from 1.2% to 4.2%.
Since at low energies all three final states could be
separated independently, the cross section of the process
e+e−→ µ+µ−was also measured, providing an additional
consistency test. The experimental value σexp
µµ/σQED
µµ
=
Page 42
41
E, MeV
E+, MeV
400
0
200
0200400
cosmic
µ
e
π
Fig. 33. Energy deposition of collinear events for the beam
energy of 460 MeV.
(0.980 ± 0.013 ± 0.007) is in good agreement with the
expected value of 1 within 1.4 statistical deviations.
Another method to discriminate electrons and pions
from other particles was used in SND. The event sepa
ration was based on the difference in longitudinal energy
deposition profiles (energy deposition in three calorime
ter layers) for these particles. To use the correlations be
tween energy depositions in calorimeter layers in the most
complete way, the separation parameter was based on the
neural network approach [287,288]. The network had an
input layer consisting of 7 neurons, two hidden layers with
20 neurons each, and the output layer with one neuron. As
input data, the network used the energy depositions of the
particles in calorimeter layers and the polar angle of one of
the particles. The output signal Re/πis a discrimination
parameter between different particles. The network was
tuned by using simulated events and was checked with
experimental 3π and e+e−events. The misidentification
ratio between electrons and pions was found to be 0.5 
1%. SND measured the e+e−→ π+π−cross section in the
energy range 0.36  0.87 GeV with a systematic error of
1.3%.
The GounarisSakurai (GS) parametrisation was used
to fit the pion form factor. Results of the fit are shown in
Fig. 34. The χ2was found to be χ2
that corresponds to the probability P(χ2
The average deviation between SND [287,288] and CMD
2 [289] data is: ∆(SND – CMD2) ∼ (1.3 ± 3.6)% for the
energy range√s ≤ 0.55 GeV and ∆(SND – CMD2) ∼
(−0.53±0.34)% for the energy range√s ≥ 0.55 GeV. The
obtained ρ meson parameters are:
CMD2 – Mρ= 775.97± 0.46 ± 0.70 MeV,
Γρ= 145.98± 0.75 ± 0.50 MeV,
min/n.d.f. = 122.9/111
min/n.d.f.) = 0.21.
Γee= 7.048 ± 0.057 ± 0.050 keV,
Br(ω → π+π−) = (1.46 ± 0.12 ± 0.02)%;
SND – Mρ= 774.6 ± 0.4 ± 0.5 MeV,
Γρ= 146.1± 0.8 ± 1.5 MeV,
Γee= 7.12 ± 0.02 ± 0.11 keV,
Br(ω → π+π−) = (1.72 ± 0.10 ± 0.07)%.
The systematic errors were carefully studied and are listed
in Table 12.
Energy, MeV
400600800 10001200
2
F
π
1
10
1
10
770 780790800810
25
30
35
40
45
50
CMD2, 1995 data (published)
CMD2, 1996 data
CMD2, 1997 data
CMD2, 1998 data
Fig. 34. Pion form factor data from CMD2 and GS fit. The
energy range around the ω meson is scaled up and presented
in the inset.
The comparison of the ρ meson parameters determined
by CMD2 and SND with the values from the PDG is pre
sented in Fig. 35. Good agreement is observed for all pa
rameters, except for the branching fraction of ω decaying
to π+π−, where a difference ∼ 1.6 standard deviations is
observed.
CMD2’01 PDG’05SND’05CMD2’05
775
776
CMD2’01PDG’05SND’05CMD2’05
142
144
146
148
CMD2’01PDG’05SND’05CMD2’05
6.8
7
7.2
7.4
7.6
CMD2’01PDG’05SND’05CMD2’05
1
1.2
1.4
1.6
1.8
2
Fig. 35. Comparison of ρ meson parameters from CMD2 and
SND with corresponding PDG values. The panels (topleft to
bottomright) refer to the mass (MeV), width (MeV), leptonic
width (keV) and the branching fraction of the decay ω → π+π−
(%).
Page 43
42
Sources of errorsCMD2
√s < 1 GeV
0.2  0.4%
0.2%
0.2  0.5%
0.2%
0.3  0.4%
0.1  0.3%
0.2%
0.6  0.8%
SNDCMD2
1.4 >√s > 1 GeV
0.2 − 1.5%
0.2 − 0.5%
0.5 − 2%
0.2%
0.5 − 2%
0.7 − 1.1%
0.6 − 2.2%
1.2 − 4.2%
Event separation method
Fiducial volume
Detection efficiency
Corrections for pion losses
Radiative corrections
Beam energy determination
Other corrections
The total systematic error
0.5%
0.8%
0.6%
0.2%
0.2%
0.3%
0.5%
1.3 %
Table 12. The main sources of the systematic errors for different energy regions.
3.3.2 Cross section of the process e+e−→ π+π−π0
This channel was studied by SND in the energy range√s
from 0.6 to 1.4 GeV [290,291], while CMD2 has reported
results of the measurements in vicinity of the ω [289] and φ
meson peaks [292]. For both the ω and φ resonances CMD
2 and SND obtain consistent results for the product of the
resonance branching fractions into e+e−and π+π−π0, for
which they have the world’s best accuracy (SND for the
ω and CMD2 for the φ resonance).
CMD2 has also performed a detailed Dalitz plot anal
ysis of the dynamics of φ decaying to π+π−π0. Two models
of 3π production were used: a ρπ mechanism and a contact
amplitude. The result obtained for the ratio of the con
tact and ρπ amplitudes is in good agreement with that of
KLOE [293].
The systematic accuracy of the measurements is about
1.3% around the ω meson energy region, 2.5% in the φ
region, and about 5.6% for higher energies. The results of
different experiments are collected in Fig. 36. The curve
is the fit which takes into account the ρ, ω, φ, ω′and ω′′
mesons.
, GeVs
0.7 0.80.911.11.21.31.4
, nb
σ
1
10
2
10
3
10
0
π
π
π
+
π
π
+
π
π
+
π
SND
CMD2
CMD2
BABAR
0
π
π
π
0
+
0
π
Fig. 36. Cross section of the process e+e−→ π+π−π0.
3.3.3 Cross section of the process e+e−→ 4π
This cross section becomes important for energies above
the φ meson region. CMD2 showed that the a1(1260)π
mechanism is dominant for the process e+e−→ π+π−π+π−,
whereas for the channel e+e−→ π+π−π0π0in addition
the intermediate state ωπ is required to describe the en
ergy dependence of the cross section [294]. The SND anal
ysis confirmed these conclusions [295]. The knowledge of
the dynamics of 4π production allowed to determine the
detector acceptance and efficiencies with better precision
compared to the previous measurements.
The cross section of the process e+e−→ π+π−π+π−
was measured with a total systematic error of 15% for
CMD2 and 7% for SND. For the channel e+e−→ π+π−π0π0
the systematic uncertainty was 15 and 8%, respectively.
The CMD2 reanalysis of the process e+e−→ π+π−π+π−,
with a better procedure for the efficiency determination,
reduced the systematic error to (57)% [296], and these
new results are now in remarkable agreement with other
experiments.
3.3.4 Other modes
CMD2 and SND have also measured the cross sections of
the processes e+e−→ KSKLand e+e−→ K+K−from
threshold and up to 1.38 GeV with much better accu
racy than before [297,298,299]. These cross sections were
studied thoroughly in the vicinity of the φ meson, and
their systematic errors were determined with a precision
of about 1.7% (SND) and 4% (CMD2), respectively. The
analyses were based on two decay modes of the KS: KS→
π0π0and π+π−. As for the process e+e−→ K+K−, the
systematic uncertainty was studied in detail and found to
be 2.2% (CMD2) and 7% (SND).
At energies√s above 1.04 GeV the cross sections of
the processes e+e−→ KSKL,K+K−were measured with
a statistical accuracy of about 4% and systematic errors of
about 46% and 3%, respectively, and are in good agree
ment with other experiments.
To summarise, the experiments performed in 1995–
2000 with the CMD2 and SND detectors at VEPP2M al
lowed one to measure the exclusive cross sections of e+e−
annihilation into hadrons in the energy range√s = 0.36
Page 44
43
 1.38 GeV with larger statistics and smaller systematic
errors compared to the previous experiments. Figure 37
summarises the cross section measurements from CMD2
ans SND. The results of these experiments determine the
, GeVs
0.4 0.60.81 1.2
, nb
σ
1
10
1
10
2
10
3
10
118
2
π
π
π
CMD2 F
65
π
96
π
21
21
66
+
π
π
π

+
π
π
π
CMD2
CMD2
0
+
19
00π
K
K
K
84
γη
γπ
πη
π
45
π
π
π
π

K
K
95
γη
44
γπ
0
ππ
+
CMD2
+
CMD2 K
CMD2 K
CMD2 K
CMD2
CMD2
CMD2
+
L
S
51
γ
3
6
→
π
0
+
19
e
+
e
0
CMD2
SND F
2
48
π
125
35
π
62
66
+
π
+
π
π
π
SND
0π
+
SND
SND
00π
+
+
SND K
SND K
SND
SND
SND
L
S
0
45
γ
0
Fig. 37. Hadronic cross sections measured by CMD2 and
SND in the whole energy range of VEPP2M. The curve rep
resents a smooth spline of the sum of all data.
current accuracy of the calculation of the muon anomaly,
and they are one of the main sources of information about
physics of vector mesons at low energies.
3.3.5 R measurement at CLEO
Two important measurements of the R ratio have been
recently reported by the CLEO Collaboration [300,301].
In the energy range just above the open charm thresh
old, they collected statistics at thirteen c.m. energy points
from 3.97 to 4.26 GeV [301]. Hadronic cross sections in this
region exhibit a rich structure, reflecting the production of
c¯ c resonances. Two independent measurements have been
performed. In one of them they determined a sum of the
exclusive cross sections for final states consisting of two
charm mesons (D¯D, D∗¯D, D∗¯D∗, D+
D∗+
is accompanied by a pion. In the second one they measured
the inclusive cross section with a systematic uncertainty
between 5.2 and 6.1%. The results of both measurements
are in excellent agreement, which leads to the important
conclusion that in this energy range the sum of the two
and threebody cross sections saturates the total cross sec
tion for charm production. In Fig. 38 the inclusive cross
section measured by CLEO is compared with the previous
measurements by Crystal Ball [302] and BES [303]. Good
agreement is observed between the data.
CLEO has also performed a new measurement of R
at higher energy. They collected statistics at seven c.m.
energy points from 6.964 to 10.538 GeV [300] and reached
a very small systematic uncertainty of 2% only. Results
of their scan are presented in Fig. 39 and are in good
agreement with those of Crystal Ball [302], MD1 [304] and
the previous measurement of CLEO [305]. However, they
are obviously inconsistent with those of the old MARK I
measurement [306].
sD−
s, D∗+
sD−
s, and
sD∗−
s) and of processes in which the charmmeson pair
Fig. 38. Comparison of the R values from CLEO (the inclusive
determination) with those from Crystal Ball and BES.
Fig. 39. Top plot: comparison of the R values from CLEO
with those from MARK I, Crystal Ball and MD1; bottom plot:
comparison of the new CLEO results with the QCD prediction
at Λ =0.31 GeV.
Page 45
44
3.3.6 R measurement at BES
Above 2 GeV the number of final states becomes too large
for completely exclusive measurements, so that the values
of R are measured inclusively.
In 1998, as a feasibility test of R measurements, BES
took data at six c.m. energy points between 2.6 and 5.0
GeV [307]. The integrated luminosity collected at each
energy point changed from 85 to 292 nb−1. The statistical
error was around 3% per point and the systematic error
ranged from 7 to 10%.
Later, in 1999, BES performed a systematic fine scan
over the c.m. energy range from 2 to 4.8 GeV [303]. Data
were taken at 85 energy points, with an integrated lumi
nosity varying from 9.2 to 135 nb−1per point. In this
experiment, besides the continuum region below the char
monium threshold, the high charmonium states from 3.77
to 4.50 GeV were studied [308] in detail. The statistical
error was between 2 to 3%, while the systematic error
ranged from 5 to 8%, due to improvement on hadronic
event selection and Monte Carlo simulation of hadroni
sation processes. The uncertainty due to the luminosity
determination varied from 2 to 5.8%.
More recently, in 2003 and 2004, before BESII was
shut down for the upgrade to BESIII, a highstatistics
data sample was taken at 2.6, 3.07 and 3.65 GeV, with
an integrated luminosity of 1222, 2291 and 6485 nb−1,
respectively [309]. The systematic error, which exceeded
the statistical error, was reduced to 3.5% due to further
refinement on hadronic event selection and Monte Carlo
simulation.
For BESIII, the main goal of the R measurement
is to perform a fine scan over the whole energy region
which BEPCII can cover. For a continuum region (below
3.73 GeV), the step size should not exceed 100 MeV, and
for the resonance region (above 3.73 GeV), the step size
should be 10 to 20 MeV. Since the luminosity of BEPCII
is two orders of magnitude higher than at BEPC, the scan
of the resonance region will provide precise information on
the 1−−charmonium states up to 4.6 GeV.
3.4 Estimate of the theoretical accuracy
Let us discuss the accuracy of the description of the pro
cesses under consideration. This accuracy can be subdi
vided into two major parts: theoretical and technical one.
The first one is related to the precision in the actual
computer codes. It usually does not take into account
all known contributions in the best approximation. The
technical precision can be verified by special tests within
a given code (e.g., by looking at the numerical cancella
tion of the dependence on auxiliary parameters) and tuned
comparisons of different codes.
The pure theoretical precision consists of unknown high
erorder corrections, of uncertainties in the treatment of
photon radiation off hadrons, and of errors in the phe
nomenological definition of such quantities as the hadronic
vacuum polarisation and the pion form factor.
Many of the codes used at meson factories do not in
clude contributions from weak interactions even at Born
level. As discussed above, these contributions are sup
pressed at least by a factor of s/M2
the precision up to the energies of B factories.
Matching the complete oneloop QED corrections with
the higherorder corrections in the leading logarithmic ap
proximation, certain parts of the secondorder nextto
leading corrections are taken into account [235]. For the
case of Bhabha scattering, where, e.g., soft and virtual
photonic corrections in O(α2L) are known analytically,
one can see that their contribution in the relevant kine
matic region does not exceed 0.1%.14
The uncertainty coming from the the hadronic vacuum
polarisation has been estimated [13] to be of order 0.04%.
For measurements performed with the c.m. energy at a
narrow resonance (like at the φmeson factories), a sys
tematic error in the determination of the resonance con
tribution to vacuum polarisation is to be added.
The next point concerns nonleading terms of order
(α/π)2L. There are several sources of them. One is the
emission of two extra hard photons, one in the collinear
region and one at large angles. Others are related to vir
tual and softphoton radiative corrections to single hard
photon emission and Born processes. Most of these con
tributions were not considered up to now. Nevertheless
we can estimate the coefficient in front of the quantity
(α/π)2L ≈ 1·10−4to be of order one. This was indirectly
confirmed by our complete calculations of these terms for
the case of small–angle Bhabha scattering.
Considering all sources of uncertainties mentioned above
as independent, we conclude that the systematic error of
our formulae is about 0.2% or better, both for muons and
pions. For the former it is a rather safe estimate. Com
parisons between different codes which treat higherorder
QED corrections in different ways typically show agree
ment at the 0.1% level. Such comparisons test the techni
cal and partially the theoretical uncertainties. As for the
π+π−and two kaon channels, the uncertainty is enhanced
due to the presence of form factors and due to the appli
cation of the pointlike approximation for the final state
hadrons.
Zand do not spoil
4 Radiative return
4.1 History and evolution of radiative return in
precision physics
The idea to use Initial State Radiation to measure hadronic
cross sections from the threshold of a reaction up to the
centreofmass (c.m.) energy of colliders with fixed en
ergies√s, to reveal reaction mechanisms and to search
for new mesonic states consists in exploiting the process
e+e−→ hadrons + nγ, thus reducing the c.m. energy of
the colliding electrons and positrons and consequently the
14The proper choice of the factorisation scale [246] is impor
tant here.
Page 46
45
mass squared M2
tem in the final state by emission of one or more photons.
The method is particularly well suited for modern meson
factories like DAΦNE (detector KLOE), running at the
φresonance, BEPCII (detector BESIII), commissioned
in 2008 and running at the J/ψ and ψ(2S)resonances,
PEPII (detector BaBar) and KEKB (detector Belle) at
the Υ(4S)resonance. Their high luminosities compensate
for the α/π suppression of the photon emission. DAΦNE,
BEPCII, PEPII and KEKB cover the regions in Mhad
up to 1.02, 3.8 (maximally 4.6) and 10.6 GeV, respec
tively (for the latter actually restricted to 4–5 GeV if
hard photons are detected). A big advantage of ISR is the
low pointtopoint systematic errors of the hadronic en
ergy spectra. This is because the luminosity, the energy of
the electrons and positrons and many other contributions
to the detection efficiencies are determined once for the
whole spectrum. As a consequence, the overall normalisa
tion error is the same for all energies of the hadronic sys
tem. The term Radiative Return alternately used for ISR
refers to the appearance of pronounced resonances (e.g.
ρ, ω, φ, J/ψ, Z) with energies below the collider energy.
Reviews and updated results can be found in the Proceed
ings of the International Workshops in Pisa (2003) [310],
Nara (2004) [311], Novosibirsk (2006) [312], Pisa (2006)
[313], Frascati (2008) [314], and Novosibirsk (2008) [315].
Calculations of ISR date back to the sixties to seven
ties of the 20thcentury. For example, photon emission for
muon pair production in electronpositron collisions has
been calculated in Ref. [316], for the 2πfinal state in Refs.
[317,318]; the resonances ρ, ω and φ have been imple
mented in Ref. [318], the excitations ψ(3100) and ψ′(3700)
in Ref. [319], and the possibility to determine the pion
form factor was discussed in Ref. [320]. The application of
ISR to the new high luminosity meson factories, originally
aimed at the determination of the hadronic contribution to
the vacuum polarisation, more specifically the pion form
factor, has materialised in the late nineties. Early calcula
tions of ISR for the colliders DAΦNE, PEPII and KEKB
can be found in [321,322,323,324]. In Ref. [279] calcula
tions of radiative corrections for pion and kaon production
below energies of 2 GeV have been reported. An impres
sive example of ISR is the Radiative Return to the region
of the Zresonance at LEP2 with collider energies around
200 GeV [325,326,327,328] (see Fig. 40).
had= s − 2√s Eγ of the hadronic sys
ISR became a powerful tool for the analysis of exper
iments at low and intermediate energies with the devel
opment of EVAPHOKHARA, a Monte Carlo generator
which is user friendly, flexible and easy to implement into
the software of the existing detectors [329,330,331,332,
333,334,335,336,337,338,339,340,341,342,343,344,345].
EVA and its successor PHOKHARA allow to simulate
the process e+e−→ hadrons+γ for a variety of exclusive
final states. As a starting point EVA was constructed [329]
to simulate leading order ISR and FSR for the π+π−chan
nel, and additional soft and collinear ISR was included on
the basis of structure functions taken from [346]. Subse
Fig. 40. The reconstructed distribution of e+e−→ q¯ q events
as a function of the invariant mass of the quarkantiquark sys
tem. The data has been taken for a collider energy range of 182
 209 GeV. The prominent peak around 90 GeV represents the
Zresonance, populated after emission of photons in the initial
state [326].
quently EVA was extended to include the fourpion state
[330], albeit without FSR. Neglecting FSR and radiative
corrections, i.e. including onephoton emission from the
initial state only, the cross section for the radiative re
turn can be cast into the product of a radiator function
H(M2
e+e−→ hadrons:
s dσ(e+e−→ hadrons γ)/dM2
However, for a precise evaluation of σ(M2
ing logarithmic approximation inherent in EVA is insuffi
cient. Therefore, in the next step, the exact oneloop cor
rection to the ISR process was evaluated analytically, first
for large angle photon emission [331], then for arbitrary
angles, including collinear configurations [332]. This was
and is one of the key ingredients of the generator called
PHOKHARA [333,334], which also includes soft and hard
real radiation, evaluated using exact matrix elements for
mulated within the framework of helicity amplitudes [333].
FSR in NLO approximation was addressed in [335] and
incorporated in [336,337]. The importance of the charge
asymmetry, a consequence of interference between ISR
and FSR amplitudes, for a test of the (model dependent)
description of FSR has been emphasised already in Ref.
[329] and was further studied in [337].
Subsequently the generator was extended to allow for
the generation of many more channels with mesons, like
K+K−, K0¯K0, π+π−π0, for an improved description of
the 4π modes [338,339] and for improvements in the de
scription of FSR for the µ+µ−channel [336,337]. Also the
nucleon channels p¯ p and n¯ n were implemented [340], and
it was demonstrated that the separation of electric and
magnetic proton form factors is feasible for a wide en
ergy range. In fact, for the case of Λ¯Λ and including the
polarisationsensitive weak decay of Λ into the simulation,
had,s) and the cross section σ(M2
had) for the reaction
had= σ(M2
had) H(M2
had), the lead
had,s).
Page 47
46
it was shown that even the relative phase between the two
independent form factors could be disentangled [341].
Starting already with [347], various improvements were
made to include the direct decay φ → π+π−γ as a specific
aspect of FSR into the generator, a contribution of specific
importance for data taken on top of the φ resonance.
This was further pursued in the event generatorsFEVA
and FASTERD based on EVAPHOKHARA. FEVA in
cludes the effects of the direct decay φ → π−π+γ and the
decay via the ρresonance φ → ρ±π∓→ π−π+γ [348,349,
350]. The code FASTERD takes into account Final State
Radiation in the framework of both Resonance Pertur
bation Theory and sQED, Initial State Radiation, their
interference and also the direct decays e+e−→ φ →
(fo;fo+ σ)γ → π+π−γ, e+e−→ φ → ρ±π∓→ π+π−γ
and e+e−→ ρ → ωπo→ πoπoγ [351], with the possibility
to include additional models.
EVAPHOKHARA was applied for the first time to an
experiment to determine the cross section e+e−→ π+π−
from the reaction threshold up to the maximum energy of
the collider with the detector KLOE at DAΦNE [352,353,
354,355,356,357,358,359,360,361,362,363,364,365,366,367,
368,369,370,371,372,373,374,375,376] (Section 4.4.1). The
motivation was the determination of the 2π final state con
tribution to the hadronic vacuum polarisation.
The determination of the hadronic contribution to the
vacuum polarisation, which arises from the coupling of
virtual photons to quarkantiquark pairs, γ⋆→ q¯ q → γ⋆,
is possible by measuring the cross section of electron
positron annihilation into hadrons, e+e−→ γ∗→ q¯ q →
hadrons, and applying the optical theorem. It is of great
importance for the interpretation of the precision measure
ment of the anomalous magnetic moment of the muon aµ
in Brookhaven (E821) [377,378,31,379] and for the de
termination of the value of the running QED coupling at
the Zoresonance, α(m2
Z), which contributes to precision
tests of the Standard Model of particle physics, for details
see e.g. Jegerlehner [380], also Davie and Marciano [381],
or Teubner et al. [382,26,383]. The hadronic contribution
to aµ below about 2 GeV is dominated by the 2π final
state, which contributes about 70% due to the dominance
of the ρ−resonance. Other major contributions come from
the three and fourpion final states. These hadronic final
states constitute at present the largest error to the Stan
dard Model values of aµand α(m2
only experimentally. This is because calculations within
perturbative QCD are unrealistic, calculations on the lat
tice are not yet available with the necessary accuracy, and
calculations in the framework of chiral perturbation the
ory are restricted to values close to the reaction thresholds.
At energies above about 2 to 2.5 GeV, perturbative QCD
calculations start to become possible and reliable, see e.g.
Refs. [384,385], and also [386].
The Novosibirsk groups CMD2 [312,268,297,387,289,
388,389,390,391,392] and SND [291,287,393,288,299,298]
measured hadronic cross sections below 1.4 GeV by chang
ing the collider energy (energy scan, see the preceding Sec
tion 3). The Initial State Radiation method used by KLOE
represents an alternative, independent and complemen
Z) and can be determined
tary way to determine hadronic cross sections with differ
ent systematic errors. KLOE has determined the cross sec
tion for the reaction e+e−→ π+π−in the energy region
between 0.63 and 0.958 GeV by measuring the reaction
e+e−→ π+π−γ and applying a radiator function based
on PHOKHARA. For the hadronic contribution to the
anomalous magnetic moment of the muon due to the 2π
final state it obtained aππ
[374]. This value is in good agreement with those from
SND [298] and CMD2 [392], aππ
10−10and aππ
leading to an evaluation of aµ [380,381,382,26,383,37]
which differs by about three standard deviations from the
BNL experiment [31]. A different evaluation using τ de
cays into two pions results in a reduced discrepancy [381,
37]. The difference between e+e−and τ based analyses is
at present not understood. But one has to be aware that
the evaluation with τ data needs more theoretical input.
Soon after the application of EVAPHOKHARA to
KLOE [352], the BaBar collaborationalso started the mea
surement of hadronic cross sections exploiting ISR [394]
and using PHOKHARA (Section 4.4.2). In recent years a
plethora of final states has been studied, starting with the
reaction e+e−→ J/ψ γ → µ+µ−γ [395]. While detect
ing a hard photon, the upper energy for the hadron cross
sections is limited to roughly 4.5 GeV. Final states with
3, 4, 5, 6 charged and neutral pions, 2 pions and 2 kaons,
4 kaons, 4 pions and 2 kaons, with a φ and an fo(980),
J/ψ and 2 pions or 2 kaons, pions and η, kaons and η, but
also baryonic final states with protons and antiprotons,
Λoand¯Λo, Λoand¯Σo, Σoand¯
mesons, etc. have been investigated [396,397,398,399,400,
401,402,403,404,405,406,407,408]. In preparation are fi
nal states with 2 pions [409] and 2 kaons. Particularly
important final states are those with 4 pions (including
ωπo). They contribute significantly to the muon anoma
lous magnetic moment and were poorly known before the
ISR measurements. In many of these channels additional
insights into isospin symmetry breaking are expected from
the comparison between e+e−annihilation and τ decays.
More recently also Belle joined the ISR programme
with emphasis on final states containing mesons with hid
den and open charm: J/ψ and ψ(2S), D(∗)and¯D(∗),
Λc+Λc−[410,411,412,413,414,415,416,417] (Section 4.4.3).
A major surprise in recent years was the opening of
a totally new field of hadron spectroscopy by applying
ISR. Several new, relatively narrow highly excited states
with JPC= 1−−, the quantum numbers of the photon,
have been discovered (preliminarily denoted as X, Y, Z)
at the B factories PEPII and KEKB with the detec
tors BaBar and Belle, respectively. The first of them was
found by BaBar in the reaction e+e−→ Y (4260) γ →
J/ψ π+π−γ [418], a state around 4260 MeV with a width
of 90 MeV, later confirmed by Belle via ISR [419,410] and
by CLEO in an direct energy scan [420] and a radiative
return [421]. Another state was detected at 2175 MeV by
BaBar in the reaction e+e−→ Y (2175) γ → φfo(980)γ
[400]. Belle found new states at 4050, 4360, 4660 MeV in
the reactions e+e−→ Y γ → J/ψ π+π−γ and e+e−→
µ = (356.7 ± 3.1stat+syst) · 10−10
µ = (361.0±5.1stat+syst)·
µ = (361.5±3.4stat+syst)·10−10, respectively,
Σo, D¯D, D¯D∗, and D∗¯D∗
Page 48
47
Y γ → ψ(2S)π+π−γ [410,411]. The structure of basically
all of these new states (if they will survive) is unknown so
far. Fourquark states, e.g. a [cs][¯ c¯ s] state for Y (4260), a
[ss][¯ s¯ s] state for Y (2175), hybrid and molecular structures
are discussed, see also [422].
Detailed analyses allow, in addition, also the identi
fication of intermediate states, and consequently a study
of reaction mechanisms. For instance, in the case of the
final state with 2 charged and 2 neutral pions (e+e−→
π+π−πoπoγ), the dominating intermediate states are ωπo
and a1(1260)π, while ρ+ρ−and ρofo(980) contribute sig
nificantly less.
Many more highly excited states with quantum num
bers different from those of the photon have been found in
decay chains of the primarily produced heavy mesons at
the B factories PEPII and KEKB. These analyses with
out ISR have clearly been triggered and encouraged by
the unexpected discovery of highly excited states with
JPC= 1−−found with ISR.
Also baryonic final states with protons and antipro
tons, Λoand¯Λo, Λoand¯Σo, Σoand¯
gated using ISR. The effective proton form factor (see Sec
tion 4.4.2) shows a strong increase down to the p¯ p thresh
old and nontrivial structures at invariant p¯ p masses of 2.25
and 3.0 GeV, so far unexplained [398,423,424,425,426].
Furthermore, it should be possible to disentangle electric
and magnetic form factors and thus shed light on discrep
ancies between different measurements of these quantities
in the spacelike region [427].
Prospects for the Radiative Return at the Novosibirsk
collider VEPP2000 and BEPCII are discussed in Sec
tions 4.4.4 and 4.4.5.
Σohave been investi
4.2 Radiative return: a theoretical overview
4.2.1 Radiative return at leading order
We consider the e+e−annihilation process
e+(p1) + e−(p2) → hadrons + γ(k1) ,(107)
where the real photon is emitted either from the initial
(Fig. 41a) or the final state (Fig. 41b). The former process
is denoted initial state radiation (ISR), while the latter is
called final state radiation (FSR).
The differential rate for the ISR process can be cast
into the product of a leptonic Lµνand a hadronic Hµν
tensor and the corresponding factorised phase space
dσISR=
1
2sLµν
×dΦ2(p1,p2;Q,k1)dΦn(Q;q1,·,qn)dQ2
ISRHµν
2π
, (108)
where dΦn(Q;q1,·,qn) denotes the hadronic nbody phase
space with all the statistical factors coming from the hadro
nic final state included, Q =?qiand s = (p1+ p2)2.
e+
e−
¯h
h
γ
(a)
e+
e−
¯h
h
γ
(b)
Fig. 41. Leading order contributions to the reaction e+e−→
h¯h + γ from ISR (a) and FSR (b). Final state particles are
pions or muons, or any other multihadron state. The blob
represents the hadronic form factor.
For an arbitrary hadronic final state, the matrix ele
ment for the diagrams in Fig. 41a is given by
A(0)
ISR= M(0)
= −e2
ISR· J(0)=
Q2¯ v(p1)
+γµ[p /2− k /1+ me]ε /∗(k1)
2k1· p2
where Jµis the hadronic current. The superscript (0) in
dicates that the scattering amplitude is evaluated at tree
level. Summing over the polarisations of the final real pho
ton, averaging over the polarisations of the initial e+e−
state, and using current conservation, Q · J(0)= 0, the
leptonic tensor
?ε /∗(k1)[k /1− p /1+ me]γµ
2k1· p1
?
u(p2) J(0)
µ
, (109)
L(0),µν
ISR
= M(0),µ
ISR(M(0),ν
ISR)†
can be written in the form
L(0),µν
ISR
=(4πα)2
?8m2
?8m2
Q4
??2m2q2(1 − q2)2
−4q2
y1y2
?pµ
y2
1pν
s
1pµ
1y2
2
−2q2+ y2
?8m2
1+ y2
y1y2
−4q2
y1y2
2
?
2
gµν
+
y2
2
?pµ
2+ pν
s
1
+
?
y2
1
?pµ
2pν
s
−
y1y2
1pν
2
,(110)
with
yi=2k1· pi
s
,m2=m2
e
s
,q2=Q2
s
.(111)
The leptonic tensor is symmetric under the exchange of
the electron and the positron momenta. Expressing the
bilinear products yiby the photon emission angle in the
c.m. frame,
y1,2=1 − q2
2
(1 ∓ β cosθ) ,β =
?
1 − 4m2,
and rewriting the twobody phase space as
dΦ2(p1,p2;Q,k1) =1 − q2
32π2dΩ ,(112)
Page 49
48
it is evident that expression (110) contains several singu
larities: soft singularities for q2→ 1 and collinear singu
larities for cosθ → ±1. The former are avoided by requir
ing a minimal photon energy. The latter are regulated by
the electron mass. For s ≫ m2
nevertheless be safely taken in the limit me → 0 if the
emitted real photon lies far from the collinear region. In
general, however, one encounters spurious singularities in
the phase space integrations if powers of m2= m2
neglected prematurely.
Physics of the hadronic system, whose description is
model dependent, enters through the hadronic tensor
ethe expression (110) can
e/s are
Hµν= J(0)
µ(J(0)
ν)†,(113)
where the hadronic current has to be parametrised through
form factors. For two charged pions in the final state, the
current
J(0),µ
π+π−= ieF2π(Q2) (q1− q2)µ,
where q1and q2are the momenta of the π+and π−, re
spectively, is determined by only one function, the pion
form factor F2π. The current for the µ+µ−final state is
obviously defined by QED:
(114)
J(0),µ
µ+µ−= ie ¯ u(q2)γµv(q1) .(115)
Integrating the hadronic tensor over the hadronic phase
space, one gets
?
where R(Q2) = σ(e+e−→ hadrons)/σ0(e+e−→ µ+µ−),
with
σ0(e+e−→ µ+µ−) =4π α2
the treelevel muonic cross section in the limit Q2≫ 4m2
After the additional integration over the photon angles,
the differential distribution
?
with L = log(s/m2
e) is obtained. If instead the photon
polar angle is restricted to be in the range θmin < θ <
π − θmin, this differential distribution is given by
Q2dσISR
3sR(Q2)
−s − Q2
s
HµνdΦn(Q;q1,·,qn) =e2
6π(QµQν− gµνQ2)R(Q2) ,
(116)
3Q2
(117)
µ.
Q2dσISR
dQ2=4α3
3sR(Q2)
s2+ Q4
s(s − Q2)(L − 1)
?
,(118)
dQ2=4α3
?
s2+ Q4
s(s − Q2)log1 + cosθmin
cosθmin
.
1 − cosθmin
?
(119)
In the latter case, the electron mass can be taken equal
to zero before integration, since the collinear region is ex
cluded by the angular cut. The contribution of the two
pion exclusive channel can be calculated from Eq. (118)
and Eq. (119) with
Rπ+π−(Q2) =1
4
?
1 −4m2
π
Q2
?3/2
F2π(Q2)2,(120)
0
20
40
60
80
100
120
dσ(e+e−→π+π−γ)/dQ 2 (nb/GeV 2)
1.02 GeV
θγ < 15o or θγ > 165o 40o < θπ < 140o
ISR+FSR ≈ ISR only
ISR+FSR
ISR only
10
4
10
2
00.10.2 0.30.40.50.60.70.80.91
Q 2 (GeV 2)
FSR/ISR
θγ < 15o or θγ > 165o
40o < θπ < 140o
Fig. 42. Suppression of the FSR contributions to the cross
section by a suitable choice of angular cuts; results from the
PHOKHARA generator; no cuts (upper curves) and suitable
cuts applied (lower curves).
and the corresponding muonic contribution with
?
Rµ+µ−(Q2) =
1 −4m2
µ
Q2
?
1 +2m2
µ
Q2
?
.(121)
A potential complication for the measurement of the
hadronic cross section from the radiative return may arise
from the interplay between photons from ISR and FSR
[329]. Their relative strength is strongly dependent on the
photon angle relative to the beam and to the direction of
the final state particles, the c.m. energy of the reaction
and the invariant mass of the hadronic system. While ISR
is independent of the hadronic final state, FSR is not.
Moreover, it cannot be predicted from first principles and
thus has to be modelled.
The amplitude for FSR (Fig. 41b) factorises as well as
A(0)
FSR= M(0)· J(0)
FSR,(122)
where
M(0)
µ
=e
s¯ v(p1)γµu(p2) .(123)
Assuming that pions are pointlike, the FSR current for
two pions in scalar QED (sQED) reads
J(0),µ
FSR= −ie2F2π(s)
×
−(q1− k1− q2)µ(2q2+ k1)σ
?
−2gµσ+ (q1+ k1− q2)µ(2q1+ k1)σ
2k1· q1
?
2k1· q2
ǫ∗
σ(k1) .(124)
Page 50
49
0
5
10
15
20
25
10.8 0.60.40.200.20.40.60.81
cos θπ+
dσ(e+e−→π+π−γ)/dcos θπ+ (nb)
1.02 GeV
θγ < 15 o or θγ > 165 o 40 o < θπ < 140 o
ISR+FSR ≈ ISR only
ISR+FSR
ISR only
0
5
10
15
20
25
30
35
10.80.60.40.200.20.40.60.81
cos θµ+
dσ(e+e−→µ+µ−γ)/dcos θµ+ (nb)
1.02 GeV
θγ < 15 o or θγ > 165 o 40 o < θµ < 140 o
ISR+FSR ≈ ISR only
ISR+FSR
ISR only
Fig. 43. Angular distributions of π+and µ+at√s = 1.02 GeV with and without FSR for different angular cuts.
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
10.80.60.40.200.20.4 0.60.81
cos θπ+
dσ(e+e−→π+π−γ)/dcos θπ+ (nb)
10.6 GeV
30 o < θγ , θπ < 150 o
0
0.0025
0.005
0.0075
0.01
0.0125
0.015
0.0175
0.02
0.0225
10.80.60.40.200.20.40.60.81
cos θµ+
dσ(e+e−→µ+µ−γ)/dcos θµ+ (nb)
10.6 GeV
ISR+FSR
ISR only
30 o < θγ , θµ < 150 o
√Q 2 < 6 GeV
√Q 2 < 3 GeV
√Q 2 < 1 GeV
Fig. 44. Angular distributions of π+(ISR ≃ FSR+ISR) and µ+at√s=10.6 GeV for various Q2cuts.
Due to momentum conservation, p1+ p2= q1+ q2+ k1,
and current conservation, this expression can be simplified
further to
?
J(0),µ
FSR= 2ie2F2π(s)gµσ+qµ
2qσ
k1· q1
1
+qµ
k1· q2
1qσ
2
?
ǫ∗
σ(k1) .
(125)
This is the basic model adopted in EVA [329] and in PHO
KHARA [331,332,333,334,335,336,337,338,341,428] to sim
ulate FSR off charged pions. The corresponding FSR cur
rent for muons is given by QED.
The fully differential cross section describing photon
emission at leading order can be split into three pieces
dσ(0)= dσ(0)
ISR+ dσ(0)
FSR+ dσ(0)
INT,(126)
which originate from the squared ISR and FSR amplitudes
and the interference term, respectively. The ISR–FSR in
terference is odd under charge conjugation,
dσ(0)
INT(q1,q2) = −dσ(0)
INT(q2,q1) , (127)
and its contribution vanishes after angular integration. It
gives rise, however, to a relatively large charge asymmetry
and, correspondingly, to a forward–backward asymmetry
A(θ) =Nh(θ) − Nh(π − θ)
Nh(θ) + Nh(π − θ).(128)
The asymmetry can be used for the calibration of the FSR
amplitude, and fits to the angular distribution A(θ) can
test details of its model dependence [329].
The second option to disentangle ISR from FSR ex
ploits the markedly different angular distribution of the
photon from the two processes. This observation is com
pletely general and does not rely on any model like sQED
for FSR. FSR is dominated by photons collinear to the
final state particles, while ISR is dominated by photons
collinear to the beam direction. This suggests that we
should consider only events with photons well separated
from the charged final state particles and preferentially
close to the beam [329,333,334].
This is illustrated in Fig. 42, which has been generated
running PHOKHARA at leading order (LO). After intro
ducing suitable angular cuts, the contamination of events
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