Page 1
arXiv:0912.0749v1 [hep-ph] 3 Dec 2009
EPJ manuscript No.
(will be inserted by the editor)
BIHEP-TH-2009-005, BU-HEPP-09-08,
CERN-PH-TH/2009-201, DESY 09-092,
FNT/T 2009/03, Freiburg-PHENO-09/07,
HEPTOOLS 09-018, IEKP-KA/2009-33,
LNF-09/14(P), LPSC 09/157,
LPT-ORSAY-09-95, LTH 851, MZ-TH/09-38,
PITHA-09/14, PSI-PR-09-14,
SFB/CPP-09-53, WUB/09-07
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, D-52056 Aachen, Germany
2Institue of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China
3Institut f¨ ur Physik Humboldt-Universit¨ at zu Berlin, D-12489 Berlin, Germany
4Dipartimento di Fisica dell’Universit` a di Bologna, I-40126 Bologna, Italy
5INFN, Sezione di Bologna, I-40126 Bologna, Italy
6The Faculty of Physics, Astronomy and Applied Computer Science, Jagiellonian University, Reymonta 4, 30-059 Cracow,
Poland
7Marian Smoluchowski Institute of Physics, Jagiellonian University, Reymonta 4, 30-059 Cracow, Poland
8Institute of Nuclear Physics Polish Academy of Sciences, PL-31342 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, I-00044 Frascati, Italy
12Physikalisches Institut, Albert-Ludwigs-Universit¨ at Freiburg, D-79104 Freiburg, Germany
13CERN, Physics Department, CH-1211 Gen` eve, Switzerland
14CERN, Theory Department, CH-1211 Gen` eve, Switzerland
15Laboratoire de Physique Subatomique et de Cosmologie, Universit´ e Joseph Fourier/CNRS-IN2P3/INPG,
F-38026 Grenoble, France
16University of Hawaii, Honolulu, Hawaii 96822, USA
17Institut f¨ ur Experimentelle Kernphysik, Universit¨ at Karlsruhe, D-76021 Karlsruhe, Germany
18Institut f¨ ur Theoretische Teilchenphysik, Universit¨ at Karlsruhe, D-76128 Karlsruhe, Germany.
19Institute of Physics, University of Silesia, PL-40007 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, D-55128 Mainz, Germany
23Institut f¨ ur Physik (THEP), Johannes Gutenberg-Universit¨ at, D-55099 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 Paris-Sud XI, Bˆ atiment 210, 91405 Orsay Cedex, France
28INFN, Sezione di Padova, I-35131 Padova, Italy
Page 2
2
29LLR-Ecole Polytechnique, 91128 Palaiseau, France
30Dipartimento di Fisica, Universit` a di Parma, I-43100 Parma, Italy
31INFN, Gruppo Collegato di Parma, I-43100 Parma, Italy
32Dipartimento di Fisica Nucleare e Teorica, Universit` a di Pavia, I-27100 Pavia, Italy
33INFN, Sezione di Pavia, I-27100 Pavia, Italy
34Dipartimanto di Fisica dell’Universit` a “Roma Tre” and INFN Sezione di Roma Tre, I-00146 Roma, Italy
35School of Physics and Astronomy, University of Southampton, Southampton SO17 1BJ, U.K.
36Theory Center, KEK, Tsukuba, Ibaraki 305-0801, Japan
37Instituto de Fisica Corpuscular (IFIC), Centro mixto UVEG/CSIC, Edificio Institutos de Investigacion, Apartado de Correos
22085, E-46071 Valencia, Espanya
38Paul Scherrer Institut, W¨ urenlingen and Villigen, CH-5232 Villigen PSI, Switzerland
39Department of Physics, Baylor University, Waco, Texas 76798-7316, USA
40Fachbereich C, Bergische Universit¨ at Wuppertal, D-42097 Wuppertal, Germany
41Deutsches Elektronen-Synchrotron, DESY, D-15738 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 weak-interaction 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, matter-antimatter 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 fine-structure 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 low-energy constant α
itself. The shift of the fine-structure constant from the
Thomson limit to high energy involves low energy non-
perturbative hadronic effects which spoil this precision.
In particular, the effective fine-structure 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
Page 4
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 LEP-I 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 fine-structure 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 low-energy 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 light-by-light 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 g-2
experiments at FNAL [32] and J-PARC [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 non-perturbative 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, PEP-II 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
Page 5
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, well-tested 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 two-loop 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 semi-analytical 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 self-contained and up-to-date 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 R-scan 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 τ-based
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 self-contained and
up-to-date description of the progress occurred during the
last few years towards high-precision 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 next-to-leading-order (NLO) pho-
tonic corrections to the above processes. The remarkable
progress on the calculation of next-to-next-leading-order
(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 two-photon 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 centre-of-mass
(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
Page 6
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 electron-positron (e+e−) annihilation process, the
precise knowledge of the collider luminosity is mandatory.
The luminosity depends on three factors: beam-beam
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, VEPP-II at Novosibirsk
and Adone at Frascati, the luminosity measurement was
based on Bhabha scattering [39,40] with final-state 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 high-energy, 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 small-angle 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 intermediate-energy high-lumino-
sity meson factories, the small polar angle region is diffi-
cult to access due to the presence of the low-beta inser-
tions close to the beam crossing region, while wide-angle
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 CLEO-c 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 CLEO-c 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 CLEO-c,
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
CLEO-c 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 large-angle 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 PEP-II collider, Stanford, yielding a lu-
minosity determination with an error of about 1% [50].
Large-angle Bhabha scattering is the normalisation pro-
cess adopted by the CMD-2 and SND collaborations at
VEPP-2M, 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 final-state particles detected at wide polar angles
and an experimental accuracy of a few per cent. However,
BES-III 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 electron-positronannihilation 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 one-photon 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 three-momenta, 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 t-channel photon exchange,
followed by s-t interference and s-channel annihilation.
Furthermore, Z-boson exchange contributions and other
electroweak effects are suppressed at least by a factor
s/M2
Z. In particular, for large-angle 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 two-photon
annihilation channel (see Fig. 4)
e/s are neg-
e+(p+) + e−(p−) → γ(q1) + γ(q2)
Page 8
7
Fig. 2. Distributions of CLEO-c√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 s-channel annihilation and t-channel 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
two-photon annihilation can be split into gauge-invariant
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
ρ-parameter [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 Born-like kinematics and contains the ef-
fects due to one-loop amplitudes (virtual corrections) and
single soft-photon 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 soft-photon 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 initial-final-state 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 Born-level 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 initial-state corrections to the s-channel contribu-
tion of the Bhabha process.
and the invariants are defined as
s1= 2p′
u = −2p−p′
NLO QED radiative corrections to the two-photon an-
nihilation channel were obtained in [96,97,98,99], while
weak corrections were computed in [100].
In the one-loop approximation the part of the differ-
ential cross section with the Born-like kinematics reads
?
π
??
SV=π2
32(1 + c2
1)
?
c1= cosθ1,θ1= ?
In addition, the three-photon 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 two-photon production. The energy of the third
photon should exceed the soft-photon 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 s-channel 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
non-trivial problem of the non-perturbative 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
final-state 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 next-to-most
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 final-state 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 two-loop matrix elements for the e+e−→
e+e−process; ii) the one-loop 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 tree-level 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 two-loop QED corrections possible. The
history of this calculation will be presented in Section 2.3.1.
Some remarks on the one-loop 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 two-loop
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 phase-space
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
fixed-order corrections calculated up to NNLO are taken
into account at the LL, and, partially, next-to-leading-
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) single-logarithmic accurate
small-angle Bhabha cross section [43], and it is included
in the two-loop 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 two-loop 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 multi-loop 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 dimensionally-regularised 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 Mellin-Barnes 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 multi-loop 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 two-loop 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 cut-off 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 fixed-order calculations in
available Monte Carlo event generators. The Feynman di-
agrams involved in the calculation can be divided in three
gauge-independent 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 two-loop 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 first-order ordi-
nary linear differential equations; this is not possible in
general. Alternatively, in order to calculate the different
MIs involved, one could use the Mellin-Barnes 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 non-planar 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 non-logarithmic 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 two-loop 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 one-loop correc-
tion as well as the two-loop 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 space-time
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
electron-positron 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 two-loop massless result
for the scattering amplitude along with the two-loop mas-
sive electron form factor [150] are sufficient to obtain the
two-loop 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 two-loop Feynman diagrams with the tree-level
amplitude as well as from the interference of one-loop 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 two-loop 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 2-dimensional 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 soft-pair 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
soft-collinear effective theory (SCET) framework, while
the analysis in [154] employed the asymptotic expansion
of the MIs.
Heavy-flavor 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 top-quark
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 heavy-flavourcorrections. 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 τ-lepton,
as well as a heavy quark, like the top, the b- or the c-quark,
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 on-shell UV renormalisation con-
ditions, the irreducible two-loop vertex graphs are free of
collinear singularities. Therefore, among all the two-loop
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 heavy-flavour corrections to
the Bhabha scattering differential cross section assuming
only that s,|t|,|u|,m2
me= 0 from the beginning in all the two-loop diagrams
with the exception of the reducible ones. This procedure
allows one to effectively eliminate one mass scale from
the two-loop 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 self-energy diagrams to obtain the mixed QED-QCD
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 Bhabha-scattering cross section can be obtained by
f≫ m2
e. In particular, one can set
5Additional collinear logarithms arise also from the inter-
ference of one-loop 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 vacuum-polarisation 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
low-energy 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 self-energy corrections to Bhabha scattering at one-
loop order, the dispersion relation approach was first em-
ployed in [166]. Two-loop applications of this technique,
prior to Bhabha scattering, are the evaluation of the had-
ronic vertex correction [167] and of two-loop hadronic cor-
rections to the lifetime of the muon [168]. The approach
was also applied to the evaluation of the two-loop form
factors in QED in [169,170,171].
The fermionic and hadronic corrections to Bhabha scat-
tering at one-loop accuracy come only from the self-energy
diagram; see for details Section 6. At two-loop level there
are reducible and irreducible self-energy contributions, ver-
tices and boxes. The reducible corrections are easily treat-
ed. For the evaluation of the irreducible two-loop dia-
grams, it is advantageous that they are one-loop diagrams
with self-energy insertions because the application of the
dispersion technique as described here is possible.
The kernel function for the irreducible two-loop vertex
was derived in [167] and verified e.g. in [158]. The three
kernel functions for the two-loop 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]
www-zeuthen.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
non-photonic virtual two-loop 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 non-photonic 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
non-photonic 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 Fixed-order calculation of the hard photon emission at
one loop
The one-loop 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 one-loop tensor in-
tegrals associated with pentagon diagrams.
According to the standard Passarino-Veltman (PV)
approach [176], one-loop 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 self-energy 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 non-photonic
hadronic
s = 1.04 GeV
2
Fig. 12. Two-loop photonic and non-photonic corrections to
Bhabha scattering at√s = 1.02 GeV, normalised to the QED
tree-level 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 PV-reduction 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 PV-reduction 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 tensor-pentagon (and tensor-hexagon) 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 one-loop 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 non-photonic
hadronic
s1/2 = 10.56 GeV
Fig. 13. Two-loop photonic and non-photonic corrections to
Bhabha scattering at√s = 10.56 GeV, normalised to the QED
tree-level cross section, as a function of the electron polar angle;
no cuts; the parameterisations of Rhad is from [175].
using generalised-unitarity 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 semi-numerical method. The application of
generalised cutting rules as an on-shell 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 PV-reduction,
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, pentagon-integralsmay 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 multiple-cut 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 quadruple-cut [200] the
loop momentum is completely frozen, yielding the alge-
braic determination of the coefficients of n-point functions
with n ≥ 4. In cases where fewer than four denominators
are cut, like triple-cut [201,202,203], double-cut [204,205,
206,207,208,202] and single-cut [209], the loop momen-
Download full-text