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

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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

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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)

51.8

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

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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, 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

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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 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

50 607080 90 100110120130

ζ (degrees)

1/N dN/dζ (0.2 degrees)-1

10

-3

10

-2

10

-1

0123456789 10

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.

2040

60

80 100120140

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

2040

60

80100 120140

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-