Quest for precision in hadronic cross sections at low energy: Monte Carlo tools vs. experimental data
S.Actis, A.Arbuzov, G.Balossini, P.Beltrame, C.Bignamini, R.Bonciani, C.M.Carloni Calame, V.Cherepanov, M.Czakon, H.Czyz, A.Denig, S.Eidelman, G.V.Fedotovich, A.Ferroglia, J.Gluza, A.Grzelinska, M.Gunia, A.Hafner, F.Ignatov, S.Jadach, F.Jegerlehner, A.Kalinowski, W.Kluge, A.Korchin, J.H.Kuhn, E.A.Kuraev, P.Lukin, P.Mastrolia, G.Montagna, S.E.Muller, F.Nguyen, O.Nicrosini, D.Nomura, G.Pakhlova, G.Pancheri, M.Passera, A.Penin, F.Piccinini, W.Placzek, T.Przedzinski, E.Remiddi, T.Riemann, G.Rodrigo, P.Roig, O.Shekhovtsova, C.P.Shen, A.L.Sibidanov, T.Teubner, L.Trentadue, G.Venanzoni, J.J.van der Bij, P.Wang, B.F.L.Ward, Z.Was, M.Worek, C.Z.Yuan
aa r X i v : . [ h e p - ph ] D ec 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. Actis , A. Arbuzov , , G. Balossini , , P. Beltrame , C. Bignamini , , R. Bonciani , C. M. Carloni Calame ,V. Cherepanov , , M. Czakon , H. Czy˙z , , , , A. Denig , S. Eidelman , , , G. V. Fedotovich , , ,A. Ferroglia , J. Gluza , A. Grzeli´nska , M. Gunia , A. Hafner , F. Ignatov , S. Jadach , F. Jegerlehner , , ,A. Kalinowski , W. Kluge , A. Korchin , J. H. K¨uhn , E. A. Kuraev , P. Lukin , P. Mastrolia ,G. Montagna , , , , S. E. M¨uller , , F. Nguyen , , O. Nicrosini , D. Nomura , , G. Pakhlova ,G. Pancheri , M. Passera , A. Penin , F. Piccinini , W. P laczek , T. Przedzinski , E. Remiddi , , T. Riemann ,G. Rodrigo , P. Roig , O. Shekhovtsova , C. P. Shen , A. L. Sibidanov , T. Teubner , , L. Trentadue , ,G. Venanzoni , , , J. J. van der Bij , P. Wang , B. F. L. Ward , Z. Was , , M. Worek , , and C. Z. Yuan Institut f¨ur Theoretische Physik E, RWTH Aachen Universit¨at, D-52056 Aachen, Germany Institue of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China Institut f¨ur Physik Humboldt-Universit¨at zu Berlin, D-12489 Berlin, Germany Dipartimento di Fisica dell’Universit`a di Bologna, I-40126 Bologna, Italy INFN, Sezione di Bologna, I-40126 Bologna, Italy The Faculty of Physics, Astronomy and Applied Computer Science, Jagiellonian University, Reymonta 4, 30-059 Cracow,Poland Marian Smoluchowski Institute of Physics, Jagiellonian University, Reymonta 4, 30-059 Cracow, Poland Institute of Nuclear Physics Polish Academy of Sciences, PL-31342 Cracow, Poland Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980 Dubna, Russia Department of Physics, University of Alberta, Edmonton, AB T6G 2J1, Canada Laboratori Nazionali di Frascati dell’INFN, I-00044 Frascati, Italy Physikalisches Institut, Albert-Ludwigs-Universit¨at Freiburg, D-79104 Freiburg, Germany CERN, Physics Department, CH-1211 Gen`eve, Switzerland CERN, Theory Department, CH-1211 Gen`eve, Switzerland Laboratoire de Physique Subatomique et de Cosmologie, Universit´e Joseph Fourier/CNRS-IN2P3/INPG,F-38026 Grenoble, France University of Hawaii, Honolulu, Hawaii 96822, USA Institut f¨ur Experimentelle Kernphysik, Universit¨at Karlsruhe, D-76021 Karlsruhe, Germany Institut f¨ur Theoretische Teilchenphysik, Universit¨at Karlsruhe, D-76128 Karlsruhe, Germany. Institute of Physics, University of Silesia, PL-40007 Katowice, Poland National Science Center “Kharkov Institute of Physics and Technology”, 61108 Kharkov, Ukraine Department of Mathematical Sciences, University of Liverpool, Liverpool L69 3BX, U.K. Institut f¨ur Kernphysik, Johannes Gutenberg - Universit¨at Mainz, D-55128 Mainz, Germany Institut f¨ur Physik (THEP), Johannes Gutenberg-Universit¨at, D-55099 Mainz, Germany Institute for Theoretical and Experimental Physics, Moscow, Russia Budker Institute of Nuclear Physics, 630090 Novosibirsk, Russia Novosibirsk State University, 630090 Novosibirsk, Russia Laboratoire de Physique Th´eorique (UMR 8627),Universit´e de Paris-Sud XI, Bˆatiment 210, 91405 Orsay Cedex, France INFN, Sezione di Padova, I-35131 Padova, Italy LLR-Ecole Polytechnique, 91128 Palaiseau, France Dipartimento di Fisica, Universit`a di Parma, I-43100 Parma, Italy INFN, Gruppo Collegato di Parma, I-43100 Parma, Italy Dipartimento di Fisica Nucleare e Teorica, Universit`a di Pavia, I-27100 Pavia, Italy INFN, Sezione di Pavia, I-27100 Pavia, Italy Dipartimanto di Fisica dell’Universit`a “Roma Tre” and INFN Sezione di Roma Tre, I-00146 Roma, Italy School of Physics and Astronomy, University of Southampton, Southampton SO17 1BJ, U.K. Theory Center, KEK, Tsukuba, Ibaraki 305-0801, Japan Instituto de Fisica Corpuscular (IFIC), Centro mixto UVEG/CSIC, Edificio Institutos de Investigacion, Apartado de Correos22085, E-46071 Valencia, Espanya Paul Scherrer Institut, W¨urenlingen and Villigen, CH-5232 Villigen PSI, Switzerland Department of Physics, Baylor University, Waco, Texas 76798-7316, USA Fachbereich C, Bergische Universit¨at Wuppertal, D-42097 Wuppertal, Germany Deutsches Elektronen-Synchrotron, DESY, D-15738 Zeuthen, Germany Section 2 conveners Section 3 conveners Section 4 conveners Section 5 conveners Section 6 conveners Working group conveners Corresponding authors: [email protected], [email protected], [email protected]: date / Revised version: date
Abstract.
We present the achievements of the last years of the experimental and theoretical groups workingon 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 theyears to come. We emphasise the status and the precision of the Monte Carlo generators used to analysethe hadronic cross section measurements obtained as well with energy scans as with radiative return, todetermine luminosities and τ decays. The radiative corrections fully or approximately implemented in thevarious codes and the contribution of the vacuum polarisation are discussed. PACS. e − e + interactions – 13.35.Dx Decays of taus – 12.10.Dm Unifiedtheories and models of strong and electroweak interactions – 13.40.Ks Electromagnetic corrections tostrong- and weak-interaction processes – 29.20.-c Accelerators Contents R measurement from energy scan . . . . . . . . . . . 354 Radiative return . . . . . . . . . . . . . . . . . . . . 445 Tau decays . . . . . . . . . . . . . . . . . . . . . . . 726 Vacuum polarisation . . . . . . . . . . . . . . . . . . 827 Summary . . . . . . . . . . . . . . . . . . . . . . . . 88 The systematic comparison of Standard Model ( SM ) pre-dictions with precise experimental data served in the lastdecades as an invaluable tool to test the theory at thequantum level. It has also provided stringent constraintson “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 fewobservables 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) aswell as strong theoretical arguments hinting at the pres-ence of physics beyond the SM . New colliders, like theLHC or a future e + e − International Linear Collider (ILC),will hopefully answer many questions, offering at the sametime 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 basicinput 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 q is actually described by an effective electromagneticcoupling α ( q ) rather than by the low-energy constant α itself. The shift of the fine-structure constant from theThomson limit to high energy involves low energy non-perturbative hadronic effects which spoil this precision.In particular, the effective fine-structure constant at thescale M Z , α ( M Z ) = α/ [1 − ∆α ( M Z )], plays a crucial rolein basic EW radiative corrections of the SM . An importantexample is the EW mixing parameter sin θ , related to α , the Fermi coupling constant G F and M Z via the Sirlinrelation [3,4,5]sin θ S cos θ S = πα √ G F M Z (1 − ∆r S ) , (1)where the subscript S identifies the renormalisation scheme. ∆r S incorporates the universal correction ∆α ( M Z ), largecontributions that depend quadratically on the top quarkmass m t [6] (which led to its indirect determination beforethis quark was discovered), plus all remaining quantum ef-fects. In the SM , ∆r S depends on various physical param-eters, including M H , the mass of the Higgs boson. As thisis the only relevant unknown parameter in the SM , impor-tant indirect bounds on this missing ingredient can be setby comparing the calculated quantity in Eq. (1) with theexperimental value of sin θ S (e.g. the effective EW mixingangle sin θ lepteff measured at LEP and SLC from the on-resonance asymmetries) once ∆α ( M Z ) and other experi-mental inputs like m t are provided. It is important to notethat an error of δ∆α ( M Z ) = 35 × − [7] in the effectiveelectromagnetic coupling constant dominates the uncer-tainty of the theoretical prediction of sin θ lepteff , inducingan error δ (sin θ lepteff ) ∼ × − (which is comparablewith the experimental value δ (sin θ lepteff ) EXP = 16 × − determined by LEP-I and SLD [8,9]) and affecting the up-per bound for M H [8,9,10]. Moreover, as measurements ofthe effective EW mixing angle at a future linear collidermay improve its precision by one order of magnitude, amuch smaller value of δ∆α ( M Z ) will be required (see be-low). It is therefore crucial to assess all viable options tofurther reduce this uncertainty.The shift ∆α ( M Z ) can be split in two parts: ∆α ( M Z ) = ∆α lep ( M Z )+ ∆α (5)had ( M Z ). The leptonic contribution is cal-culable in perturbation theory and known up to three-loop accuracy: ∆α lep ( M Z ) = 3149 . × − [11]. Thehadronic contribution ∆α (5)had ( M Z ) of the five light quarks( u , d , s , c , and b ) can be computed from hadronic e + e − annihilation data via the dispersion relation [12] ∆α (5)had ( M Z ) = − (cid:18) αM Z π (cid:19) Re Z ∞ m π d s R ( s ) s ( s − M Z − iǫ ) , (2)where R ( s ) = σ ( s ) / (4 πα / s ) and σ ( s ) is the to-tal cross section for e + e − annihilation into any hadronicstates, with vacuum polarisation and initial state QED corrections subtracted off. The current accuracy of thisdispersion integral is of the order of 1%, dominated bythe error of the hadronic cross section measurements inthe 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)had ∼ × − , needed for precision physics at a futurelinear collider, requires the measurement of the hadroniccross section with a precision of O (1%) from threshold upto the Υ peak.Like the effective fine-structure constant at the scale M Z , the SM determination of the anomalous magnetic mo-ment of the muon a µ is presently limited by the evaluation δ∆α (5)had × δ (sin θ lepteff ) × Request on R
22 7.9 Present7 2.5 δR/R ∼
1% up to
J/ψ δR/R ∼
1% up to Υ Table 1.
Values of the uncertainties δ∆α (5)had (first column)and the errors induced by these uncertainties on the theoreticalSM prediction for sin θ lepteff (second column). The third columnindicates the corresponding requirements for the R measure-ment. From Ref. [16]. of the hadronic vacuum polarisation effects, which cannotbe computed perturbatively at low energies. However, us-ing analyticity and unitarity, it was shown long ago thatthis term can be computed from hadronic e + e − annihila-tion data via the dispersion integral [24]: a HLO µ = 14 π Z ∞ m π d s K ( s ) σ ( s )= α π Z ∞ m π d s K ( s ) R ( s ) /s . (3)The kernel function K ( s ) decreases monotonically withincreasing s . This integral is similar to the one enteringthe evaluation of the hadronic contribution ∆α (5)had ( M Z )in Eq. (2). Here, however, the weight function in the inte-grand gives a stronger weight to low-energy data. A recentcompilation of e + e − data gives [25]: a HLO µ = (695 . ± . × − . (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 a µ ,which uses the hadronic light-by-light result from [29] gives[25,30]: a SM µ = 116591834(49) × − . The difference be-tween the experimental average [31], a exp µ = 116592080(63) × − and the SM prediction is then ∆a µ = a exp µ − a SM µ =+246(80) × − , i.e. 3.1 standard deviations (adding allerrors in quadrature). Slightly higher discrepancies areobtained in Refs. [23,27,28]. As in the case of α ( M Z ),the uncertainty of the theoretical evaluation of a SM µ is stilldominated by the hadronic contribution at low energies,and a reduction of the uncertainty is necessary in order tomatch the increased precision of the proposed muon g-2experiments at FNAL [32] and J-PARC [33].The precise determination of the hadronic cross sec-tions (accuracy . 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 reachedon 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 measurement of the luminosity, MCGPJ for the simula-tion of the exclusive QED channels, and PHOKHARA forthe 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 higherorder contributions. From this point of view, the greatprogress in the calculation of two-loop corrections to theBhabha scattering cross section was essential to establishthe high theoretical accuracy of the existing generatorsfor the luminosity measurement. However, usually onlyanalytical or semi-analytical estimates of missing termsexist which don’t take into account realistic experimentalcuts. In addition, MC event generators include differentparametrisations for the VP which affect the prediction(and the precision) of the cross sections and also the RCare usually implemented differently.These arguments evidently imply the importance ofcomparisons 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 forthe validation of the generators, since they allow to checkthat the details entering the complex structure of the gen-erators are under control and free of possible bugs. Thiswas the main motivation for the “Working Group on Ra-diative Corrections and Monte Carlo Generators for LowEnergies” (Radio MontecarLow) , which was formed a fewyears ago bringing together experts (theorists and experi-mentalists) working in the field of low energy e + e − physicsand partly also the τ community.In addition to tuned comparisons, technical details ofthe MC generators, recent progress (like new calculations)and remaining open issues were also discussed in regularmeetings.This report is a summary of all these efforts: it pro-vides a self-contained and up-to-date description of theprogress which occurred in the last years towards preci-sion hadronic physics at low energies, together with newresults like comparisons and estimates of high order effects(e.g. of the pion pair correction to the Bhabha process) inthe presence of realistic experimental cuts.The report is divided into five sections: Sections 2, 3and 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 usedto estimate a HLO 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 -baseddeterminations of a HLO µ , even if isospin violation correc-tions are taken into account, shows that further commontheoretical and experimental efforts are necessary to un-derstand this phenomenon. In Section 5 the experimentalstatus 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 developmentsare 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 thedescription and comparison of available parametrisations.Finally, Section 7 contains a brief summary of the report. The present Section addresses the most important exper-imental and theoretical issues involved in the precisiondetermination of the luminosity at meson factories. Theluminosity is the key ingredient underlying all the mea-surements and studies of the physics processes discussedin the other Sections. Particular emphasis is put on thetheoretical accuracy inherent to the event generators usedin the experimental analyses, in comparison with the mostadvanced perturbative calculations and experimental pre-cision requirements. The effort done during the activityof the working group to perform tuned comparisons be-tween the predictions of the most accurate programs isdescribed 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 andup-to-date description of the progress occurred during thelast few years towards high-precision luminosity monitor-ing at flavour factories, as well as of the still open issuesnecessary 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 ofmajor interest, i.e. Bhabha scattering and photon pairproduction, are presented in Section 2.2, together withthe formulae for the next-to-leading-order (NLO) pho-tonic corrections to the above processes. The remarkableprogress on the calculation of next-to-next-leading-order(NNLO) QED corrections to the Bhabha cross section, asoccurred in the last few years, is reviewed in Section 2.3.In particular, this Section presents new exact results onlepton and hadron pair corrections, taking into accountrealistic event selection criteria. Section 2.4 is devotedto the description of the theoretical methods used in theMonte Carlo (MC) generators for the simulation of multi-ple photon radiation. The matching of such contributionswith NLO corrections is also described in Section 2.4. Themain 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 intothe MC generators are shown in Section 2.6 for both theBhabha 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. Thetheoretical accuracy presently reached by the luminositytools 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 the Bhabha cross section is given, as the main conclusionof 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 2.9as well.
The luminosity of a collider is the normalisation constantbetween the event rate and the cross section of a givenprocess. For an accurate measurement of the cross sectionof an electron-positron ( e + e − ) annihilation process, theprecise knowledge of the collider luminosity is mandatory.The luminosity depends on three factors: beam-beamcrossing frequency, beam currents and the beam overlaparea in the crossing region. However, the last quantity isdifficult to determine accurately from the collider optics.Thus, experiments prefer to determine the luminosity bythe counting rate of well selected events whose cross sec-tion is known with good accuracy, using the formula [38] Z L d t = Nǫ σ , (5)where N is the number of events of the chosen referenceprocess, ǫ the experimental selection efficiency and σ thetheoretical cross section of the reference process. There-fore, the total luminosity error will be given by the sum inquadrature of the fractional experimental and theoreticaluncertainties.Since the advent of low luminosity e + e − colliders, agreat effort was devoted to obtain good precision in thecross section of electromagnetic processes, extending thepioneering work of the earlier days [12]. At the e + e − col-liders operating in the c.m. energy range 1 GeV < √ s < /s ,while elastic e + e − scattering has a steep dependence onthe polar angle, ∼ /θ , thus providing a high rate forsmall values of θ .Also at high-energy, accelerators running in the ’90saround the Z pole to perform precision tests of the Stan-dard Model (SM), such as LEP at CERN and SLC atStanford, the experiments used small-angle Bhabha scat-tering events as a luminosity monitoring process. Indeed,for the very forward angular acceptances considered bythe 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 (orbetter) was achieved [42,43,44,45,46,47,48], thanks to thework of different theoretical groups and the excellent per-formance of precision luminometers. 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-angleBhabha scattering produces a large counting rate and canbe exploited for a precise measurement of the luminosity.Therefore, also in this latter case of e ± scattered atlarge angles, e.g. larger than 55 ◦ for the KLOE experi-ment [38] running at DAΦNE in Frascati, and larger than40 ◦ for the CLEO-c experiment [49] running at CESR inCornell, the main advantages of Bhabha scattering arepreserved:1. large statistics. For example at DAΦNE, a statisticalerror δ L / L ∼ .
3% is reached in about two hours ofdata taking, even at the lowest luminosities;2. high accuracy for the calculated cross section;3. clean event topology of the signal and small amount ofbackground.In Eq. (5) the cross section is usually evaluated byinserting 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 (geometricalacceptance, reconstruction efficiency and resolution of themeasured 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 errorof the luminosity measurement is due to a different polarangle resolution between data and MC which is observedat the edges of the accepted interval for Bhabha scatter-ing events. Fig. 1 shows a comparison between large angleBhabha KLOE data and MC, at left for the polar angleand at right for the acollinearity ζ = | θ e + + θ e − − ◦ | .One observes a very good agreement between data andMC, but also differences (of about 0.3 %) at the sharpinterval edges. The analysis cut, ζ < ◦ , applied to theacollinearity distribution is very far from the bulk of thedistribution and does not introduce noteworthy system-atic errors. Also in the CLEO-c luminosity measurementwith Bhabha scattering events, the detector modelling isthe 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 ofthe electron shower, as well as the steep e ± polar angledistribution.The luminosity measured with Bhabha scattering eventsis 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 . e + e − → µ + µ − events are also used, and the luminositydetermined from γγ ( µ + µ − ) is found to be 2 .
1% (0 . θ (degrees) / N d N / d θ ( d e g r ee s ) - ζ (degrees) / N d N / d ζ ( . d e g r ee s ) - -3 -2 -1 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 = | θ e + + θ e − − ◦ | (right), where the flight direction of the e ± is given by the position ofclusters 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 atVEPP-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 anglesand an experimental accuracy of a few per cent. However,BES-III aims at reaching an error of a few per mill in theirluminosity measurement in the near future [51].The need of precision, namely better than 1%, and pos-sibly redundant measurements of the collider luminosity isof utmost importance to perform accurate measurementsof the e + e − → hadrons cross sections, which are the keyingredient for evaluating the hadronic contribution to therunning of the electromagnetic coupling constant α andthe muon anomaly g − As remarked in Section 2.1, the processes of interest for theluminosity measurement at meson factories are Bhabhascattering and electron-positron annihilation into two pho-tons and muon pairs. Here we present the LO formulaefor the cross section of the processes e + e − → e + e − and e + e − → γγ , as well as the QED corrections to their crosssections in the NLO approximation of perturbation the-ory. The reaction e + e − → µ + µ − is discussed in Section3. For the Bhabha scattering process e − ( p − ) + e + ( p + ) → e − ( p ′− ) + e + ( p ′ + ) (6)at Born level with simple one-photon exchange (see Fig. 3)the differential cross section readsd σ Bhabha0 d Ω − = α s (cid:18) c − c (cid:19) + O (cid:18) m e s (cid:19) , (7)where s = ( p − + p + ) , c = cos θ − . (8)The angle θ − is defined between the initial and final elec-tron three-momenta, d Ω − = d φ − d cos θ − , and φ − is theazimuthal angle of the outgoing electron. The small masscorrection terms suppressed by the ratio m e /s are neg-ligible for the energy range and the angular acceptanceswhich 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 otherelectroweak effects are suppressed at least by a factor s/M Z . In particular, for large-angle Bhabha scatteringwith a c.m. energy √ s = 1 GeV the Z boson contribu-tion amounts to about − × − . For √ s = 3 GeV itamounts to − × − and − × − for √ s = 10 GeV.So only at B factories the electroweak effects should betaken into account at tree level, when aiming at a per millprecision level.The LO differential cross section of the two-photonannihilation channel (see Fig. 4) e + ( p + ) + e − ( p − ) → γ ( q ) + γ ( q ) Fig. 2.
Distributions of CLEO-c √ s = 3 .
774 GeV data (cir-cles) and MC simulations (histograms) for the polar angle ofthe positive lepton (upper two plots) in e + e − and µ + µ − events,and for the mean value of | cos θ γ | of the two photons in γγ events (lower panel). MC histograms are normalised to thenumber 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 Comptonscattering cross section computed by Brown and Feyn-man [52]. It readsd σ γγ d Ω = α s (cid:18) c − c (cid:19) + O (cid:18) m e s (cid:19) , (9)where d Ω denotes the differential solid angle of the firstphoton. 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 ( θ , ≫ m e /E ,where E is the beam energy). The complete set of NLO radiative corrections, emergingat O ( α ) of perturbation theory, to Bhabha scattering andtwo-photon annihilation can be split into gauge-invariantsubsets: QED corrections, due to emission of real photonsoff the charged leptons and exchange of virtual photonsbetween them, and purely weak contributions arising fromthe 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. AtNNLO, the leading virtual weak corrections from the topquark were derived first in [57] and are available in thefitting 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. Theweak 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 arenegligible at low energies and are not implemented yet innumerical packages for Bhabha scattering at meson facto-ries. In pure QED, the situation is considerably differentdue to the remarkable progress made on NNLO correctionsin recent years, as emphasised and discussed in detail inSection 2.3.As usual, the photonic corrections can be split intotwo parts according to their kinematics. The first partpreserves the Born-like kinematics and contains the ef-fects due to one-loop amplitudes (virtual corrections) andsingle soft-photon emission. Examples of Feynman dia-grams giving rise to such corrections are represented inFig. 5. The energy of a soft photon is assumed not to ex-ceed an energy ∆E , where E is the beam energy and theauxiliary parameter ∆ ≪ ∆E and corresponds to the radiative pro-cess e + e − → e + e − γ . Following [94,95], the soft plus virtual (SV) correctioncan be cast into the formd σ Bhabha B + S + V d Ω − = d σ Bhabha0 d Ω − (cid:26) απ ( L − (cid:20) ∆ + 32 (cid:21) − απ ln(ctg θ ∆ + απ K Bhabha SV (cid:27) , (10)where the factor K Bhabha SV is given by K Bhabha SV = − − (sin θ (cos θ c ) (cid:20) π c − c − c ) + 2(2 c − c + 9 c +3 c + 21) ln (sin θ − c + c − c ) ln (cos θ − c + 4 c + 5 c + 6) ln (tg θ c − c + 7 c −
5) ln(cos θ c + 9 c + 5 c + 31) ln(sin θ (cid:21) , (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 correctioncontains a leading logarithmic (LL) part enhanced by thecollinear logarithm L = ln( s/m e ). Among the virtual cor-rections there is also a numerically important effect dueto 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 resummedvacuum polarisation operators in the photon propagatorsof the Born-level Bhabha amplitudes.The differential cross section of the single hard brems-strahlung process e + ( p + ) + e − ( p − ) → e + ( p ′ + ) + e − ( p ′− ) + γ ( k )for scattering angles up to corrections of order m e /E readsd σ Bhabhahard = α π s R e ¯ eγ d Γ e ¯ eγ , (12)d Γ e ¯ eγ = d p ′ + d p ′− d kε ′ + ε ′− k δ (4) ( p + + p − − p ′ + − p ′− − k ) ,R e ¯ eγ = W T − m e ( χ ′ + ) (cid:18) st + ts + 1 (cid:19) − m e ( χ ′− ) (cid:18) st + t s + 1 (cid:19) − m e χ (cid:18) s t + ts + 1 (cid:19) − m e χ − (cid:18) s t + t s + 1 (cid:19) , where W = sχ + χ − + s χ ′ + χ ′− − t χ ′ + χ + − tχ ′− χ − + uχ ′ + χ − + u χ ′− χ + ,T = ss ( s + s ) + tt ( t + t ) + uu ( u + u ) ss tt , Fig. 5.
Examples of Feynman diagrams for real and virtualNLO QED initial-state corrections to the s -channel contribu-tion of the Bhabha process. and the invariants are defined as s = 2 p ′− p ′ + , t = − p − p ′− , t = − p + p ′ + ,u = − p − p ′ + , u = − p + p ′− , χ ± = kp ± , χ ′± = kp ′± . NLO QED radiative corrections to the two-photon an-nihilation channel were obtained in [96,97,98,99], whileweak corrections were computed in [100].In the one-loop approximation the part of the differ-ential cross section with the Born-like kinematics readsd σ γγB + S + V d Ω = d σ γγ d Ω ( απ (cid:20) ( L − (cid:18) ∆ + 32 (cid:19) + K γγSV (cid:21)) ,K γγSV = π − c c ) (cid:20)(cid:18) c − c (cid:19) ln 1 − c (cid:18) − c c + 12 1 + c − c (cid:19) ln − c c → − c ) (cid:21) ,c = cos θ , θ = d q p − . (13)In addition, the three-photon production process e + ( p + ) + e − ( p − ) → γ ( q ) + γ ( q ) + γ ( q )must be included. Its cross section is given byd σ e + e − → γ = α π s R γ d Γ γ , (14) R γ = s χ + ( χ ′ ) χ χ χ ′ χ ′ − m e (cid:20) χ + χ χ χ ( χ ′ ) + ( χ ′ ) + ( χ ′ ) χ ′ χ ′ χ (cid:21) + (cyclic permutations) , d Γ γ = d q d q d q q q q δ (4) ( p + + p − − q − q − q ) , where χ i = q i p − , χ ′ i = q i p + , i = 1 , , . The process has to be treated as a radiative correctionto the two-photon production. The energy of the thirdphoton should exceed the soft-photon energy threshold ∆E . In practice, the tree photon contribution, as well asthe radiative Bhabha process e + e − → e + e − γ , should besimulated with the help of a MC event generator in orderto take into account the proper experimental criteria of agiven event selection. Fig. 6.
Feynman diagrams for the NLO QED box correctionsto the s -channel contribution of the Bhabha process. In addition to the corrections discussed above, alsothe effect of vacuum polarisation, due to the insertion offermion loops inside the photon propagators, must be in-cluded in the precise calculation of the Bhabha scatteringcross section. Its theoretical treatment, which faces thenon-trivial problem of the non-perturbative contributiondue to hadrons, is addressed in detail in Section 6. How-ever, numerical results for such a correction are presentedin Section 2.6 and Section 2.8. σ ( nb ) LO e + e − NLO e + e − LO γγ NLO γγ -16-14-12-10-8-6-4 0 2 4 6 8 10 σ ( N L O ) − σ ( L O ) σ ( L O ) ( % ) √ 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 ofthe c.m. energy at meson factories (upper panel). In the lowerpanel, 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 ≃ m π and √ s ≃
10 GeV (upper panel). The resultswere obtained imposing the following cuts for the Bhabhaprocess: θ min ± = 45 ◦ , θ max ± = 135 ◦ ,E min ± = 0 . √ s , ξ max = 10 ◦ , (15)where θ min,max ± are the angular acceptance cuts, E min ± arethe minimum energy thresholds for the detection of thefinal-state electron/positron and ξ max is the maximum e + e − acollinearity. For the photon pair production pro-cesses we used correspondingly: θ min γ = 45 ◦ , θ max γ = 135 ◦ ,E min γ = 0 . √ s , ξ max = 10 ◦ , (16)where, as in Eq. (15), θ min,max γ are the angular acceptancecuts, E min γ is the minimum energy threshold for the de-tection of at least two photons and ξ max is the maximumacollinearity between the most energetic and next-to-mostenergetic photon.The cross sections display the typical 1 /s QED be-haviour. The relative effect of NLO corrections is shownin the lower panel. It can be seen that the NLO correctionsare largely negative and increase with increasing c.m. en-ergy, because of the growing importance of the collinearlogarithm L = ln( s/m e ). The corrections to e + e − → γγ are about one half of those to Bhabha scattering, becauseof the absence of final-state radiation effects in photonpair production. Beyond the NLO corrections discussed in the previous Sec-tion, in recent years a significant effort was devoted to thecalculation of the perturbative corrections to the Bhabhaprocess at NNLO in QED.The calculation of the full NNLO corrections to theBhabha 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 matrixelements for e + e − → e + e − γγ , with two soft or two hardphotons, or one soft and one hard photon. Also the pro-cess e + e − → e + e − e + e − , with one of the two e + e − pairsremaining undetected, contributes to the Bhabha signa-ture at NNLO. Depending on the kinematics, other finalstates like, e.g., e + e − µ + µ − or those with hadrons are alsopossible.The advent of new calculational techniques and a deeperunderstanding of the IR structure of unbroken gauge the-ories, such as QED or QCD, made the calculation of thecomplete set of two-loop QED corrections possible. Thehistory of this calculation will be presented in Section 2.3.1.Some remarks on the one-loop matrix elements withthree particles in the final state are in order now. The di-agrams involving the emission of a soft photon are knownand they were included in the calculations of the two-loopmatrix elements, in order to remove the IR soft diver-gences. However, although the contributions due to a hardcollinear 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-spaceconfiguration is still missing. In Section 2.3.2, we shall comment on the possible strategies which can be adoptedin order to calculate these corrections. As a general comment, it must be noticed that thefixed-order corrections calculated up to NNLO are takeninto account at the LL, and, partially, next-to-leading-log (NLL) level in the most precise MC generators, whichinclude, as will be discussed in Section 2.4 and Section2.5, the logarithmically enhanced contributions of soft andcollinear photons at all orders in perturbation theory.Concerning the tree level graphs with four particlesin the final state, the production of a soft e + e − pair wasconsidered in the literature by the authors of [102] by fol-lowing the evaluation of pair production [103,104] withinthe calculation of the O ( α L ) single-logarithmic accuratesmall-angle Bhabha cross section [43], and it is includedin the two-loop calculation (see Section 2.3.1). New re-sults on lepton and hadron pair corrections, which are atpresent approximately included in the available Bhabhacodes, are presented in Section 2.3.3. e + e − → e + e − process The calculation of the virtual two-loop QED corrections tothe Bhabha scattering differential cross section was carriedout in the last 10 years. This calculation was made possibleby an improvement of the techniques employed in the eval-uation of multi-loop Feynman diagrams. An essential toolused to manage the calculation is the Laporta algorithm[105,106,107,108], which enables one to reduce a genericcombination of dimensionally-regularised scalar integralsto a combination of a small set of independent integralscalled the “Master Integrals” (MIs) of the problem underconsideration. The calculation of the MIs is then pursuedby means of a variety of methods. Particularly importantare 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 provedto be very useful in the evaluation of virtual correctionsto Bhabha scattering because they are especially effectivein problems with a small number of different kinematicparameters. They both allow one to obtain an analytic ex-pression for the integrals, which must be written in termsof a suitable functional basis. A basis which was exten-sively employed in the calculation of multi-loop Feynmandiagrams of the type discussed here is represented by theHarmonic Polylogarithms [126,127,128,129,130,131,132,133,134] and their generalisations. Another fundamentalachievement which enabled one to complete the calcula-tion of the QED two-loop corrections was an improvedunderstanding of the IR structure of QED. In particular,the relation between the collinear logarithms in which theelectron mass m e plays the role of a natural cut-off andthe corresponding poles in the dimensionally regularisedmassless theory was extensively investigated in [135,136,137,138]. As emphasised in Section 2.8 and Section 2.9, the completecalculation 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 canbe found in [139]. However, this result was obtained inthe fully massless approximation ( m e = 0) by employ-ing dimensional regularisation (DR) to regulate both softand collinear divergences. Today, the complete set of two-loop corrections to Bhabha scattering in pure QED havebeen evaluated using m e as a collinear regulator, as re-quired in order to include these fixed-order calculations inavailable Monte Carlo event generators. The Feynman di-agrams involved in the calculation can be divided in threegauge-independent sets: i) diagrams without fermion loops(“photonic” diagrams), ii) diagrams involving a closedelectron loop, and iii) diagrams involving a closed loopof hadrons or a fermion heavier than the electron. Someof the diagrams belonging to the aforementioned sets areshown in Figs. 8–11. These three sets are discussed in moredetail below. Photonic corrections
A large part of the NNLO photonic corrections can beevaluated in a closed analytic form, retaining the full de-pendence on m e [140], by using the Laporta algorithmfor the reduction of the Feynman diagrams to a combina-tion of MIs, and then the differential equations method fortheir analytic evaluation. With this technique it is possi-ble to calculate, for instance, the NNLO corrections to theform factors [141,142,143,144]. However, a calculation ofthe two-loop photonic boxes retaining the full dependenceon m e seems to be beyond the reach of this method. Thisis due to the fact that the number of MIs belonging tothe same topology is, in some cases, large. Therefore, onemust solve analytically large systems of first-order ordi-nary linear differential equations; this is not possible ingeneral. Alternatively, in order to calculate the differentMIs involved, one could use the Mellin-Barnes techniques,as shown in [122,123,144,145,146,147], or a combinationof both methods. The calculation is very complicated anda full result is not available yet. However, the full depen-dence on m e is 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 safelyneglected everywhere, with the exception of the terms inwhich it acts as a collinear regulator. The ratio of the pho-tonic NNLO corrections to the Born cross section is givenbyd σ (2 , PH ) d σ ( Born ) = (cid:16) απ (cid:17) X i =0 δ ( PH ,i ) ( L e ) i + O (cid:18) m e s , m e t (cid:19) , (17)where L e = ln ( s/m e ) and the coefficients δ ( PH ,i ) containinfrared logarithms and are functions of the scattering an-gle θ . The approximation given by Eq. (17) is sufficient For the planar double box diagrams, all the MIs are known[145] for small m e , while the MIs for the non-planar doublebox diagrams are not completed.1 Fig. 8.
Some of the diagrams belonging to the class of the“photonic” NNLO corrections to the Bhabha scattering differ-ential cross section. The additional photons in the final stateare soft. for a phenomenological description of the process. Thecoefficients of the double and single collinear logarithmin Eq. (17), δ ( PH , and δ ( PH , , were obtained in [148,149]. However, the precision required for luminosity mea-surements at e + e − colliders demands the calculation ofthe non-logarithmic coefficient, δ ( PH , . The latter was ob-tained in [135,136] by reconstructing the differential crosssection in the s ≫ m e = 0 limit from the dimension-ally regularised massless approximation [139]. The mainidea of the method developed in [135,136] is outlined be-low: As far as the leading term in the small electron massexpansion is considered, the difference between the mas-sive and the dimensionally regularised massless Bhabhascattering can be viewed as a difference between two reg-ularisation schemes for the infrared divergences. With theknown massless two-loop result at hand, the calculationof the massive one is reduced to constructing the infraredmatching term which relates the two above mentioned reg-ularisation schemes. To perform the matching an auxiliaryamplitude is constructed, which has the same structure ofthe infrared singularities but is sufficiently simple to beevaluated at least at the leading order in the small massexpansion. The particular form of the auxiliary amplitudeis dictated by the general theory of infrared singularitiesin QED and involves the exponent of the one-loop correc-tion as well as the two-loop corrections to the logarithmof the electron form factor. The difference between thefull and the auxiliary amplitudes is infrared finite. It canbe evaluated by using dimensional regularisation for eachamplitude and then taking the limit of four space-timedimensions. The infrared divergences, which induce theasymptotic dependence of the virtual corrections on theelectron 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 masslessapproximation. The matching of the massive and massless It can be shown that the terms suppressed by a positivepower of m e /s do not play any phenomenological role alreadyat very low c.m. energies, √ s ∼
10 MeV. Moreover, the terms m e /t (or m e /u ) become important in the extremely forward(backward) region, unreachable for the experimental setup. Fig. 9.
Some of the diagrams belonging to the class of the“electron loop” NNLO corrections. The additional photons orelectron-positron pair in the final state are soft. results is then necessary only for the auxiliary amplitudeand is straightforward. Thus the two-loop massless resultfor the scattering amplitude along with the two-loop mas-sive electron form factor [150] are sufficient to obtain thetwo-loop photonic correction to the differential cross sec-tion in the small electron mass limit.A method based on a similar principle was subsequentlydeveloped in [137,138]; the authors of [138] confirmed theresult of [135,136] for the NNLO photonic corrections tothe 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-levelamplitude as well as from the interference of one-loop dia-grams, as long as one of the diagrams contributing to eachterm involves a closed electron loop. This set of correctionspresents a single two-loop box topology and is thereforetechnically less challenging to evaluate with respect to thephotonic correction set. The calculation of the electronloop corrections was completed a few years ago [151,152,153,154]; the final result retains the full dependence ofthe differential cross section on the electron mass m e . TheMIs involved in the calculation were identified by means ofthe Laporta algorithm and evaluated with the differentialequation method. As expected, after UV renormalisationthe differential cross section contained only residual IRpoles which were removed by adding the contribution ofthe soft photon emission diagrams. The resulting NNLOdifferential cross section could be conveniently written interms of 1- and 2-dimensional Harmonic Polylogarithms(HPLs) of maximum weight three. Expanding the crosssection in the limit s, | t | ≫ m e , the ratio of the NNLOcorrections to the Born cross section can be written as inEq. (17):d σ (2 , EL ) d σ ( Born ) = (cid:16) απ (cid:17) X i =0 δ ( EL,i ) ( L e ) i + O (cid:18) m e s , m e t (cid:19) . (18)Note that the series now contains a cubic collinear log-arithm. This logarithm appears, with an opposite sign, 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 togetherin the full NNLO, the cubic collinear logarithms cancel.Therefore, the physical cross section includes at most adouble logarithm, as in Eq. (17).The explicit expression of all the coefficients δ ( EL,i ) ,obtained by expanding the results of [151,152,153], wasconfirmed by two different groups [138,154]. In [138] thesmall electron mass expansion was performed within thesoft-collinear effective theory (SCET) framework, whilethe analysis in [154] employed the asymptotic expansionof the MIs. Heavy-flavor and hadronic corrections
Finally, we consider the corrections originating from two-loop Feynman diagrams involving a heavy flavour fermionloop. Since this set of corrections involves one more massscale 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 Bhabhascattering and obtained the heavy flavour NNLO correc-tions in the limit in which s, | t | , | u | ≫ m f ≫ m e , where m f is the mass of the heavy fermion running in the loop.Their result was very soon confirmed in [154] by means ofa 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 | ≫ m f ≫ m e cannot be applied to the case inwhich √ s < m f (as in the case of a tau loop at √ s ∼ √ s isnot very much larger than m f (as in the case of top-quarkloops at the ILC). It was therefore necessary to calculatethe heavy flavour corrections to Bhabha scattering assum-ing only that the electron mass is much smaller than theother scales in the process, but retaining the full depen-dence on the heavy mass, s, | t | , | u | , m f ≫ m e .The calculation was carried out in two different ways:in [155,156] it was done analytically, while in [157,158] itwas done numerically with dispersion relations.The technical problem of the diagrammatic calculationof Feynman integrals with four scales can be simplifiedby considering carefully, once more, the structure of thecollinear singularities of the heavy-flavour corrections. Theratio of the NNLO heavy flavour corrections to the Borncross section is given byd σ (2 , HF ) d σ ( Born ) = (cid:16) απ (cid:17) X i =0 δ ( HF,i ) ( L e ) i + O (cid:18) m e s , m e t (cid:19) , (19)where now the coefficients δ ( i ) are functions of the scat-tering angle θ and, in general, of the mass of the heavy Here 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 inthe final state are soft. fermions involved in the virtual corrections. It is possi-ble to prove that, in a physical gauge, all the collinearsingularities factorise and can be absorbed in the exter-nal field renormalisation [159]. This observation has twoconsequences in the case at hand. The first one is thatbox 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 ofall box diagrams is free of collinear divergences in anygauge. The second consequence is that the single collinearlogarithm 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 ofcollinear singularities. Therefore, among all the two-loopdiagrams contributing to the NNLO heavy flavour cor-rections to Bhabha scattering, only the reducible vertexcorrections are logarithmically divergent in the m e → The latter are easily evaluated even if they dependon two different masses. By exploiting these two facts,one can obtain the NNLO heavy-flavour corrections tothe Bhabha scattering differential cross section assumingonly that s, | t | , | u | , m f ≫ m e . In particular, one can set m e = 0 from the beginning in all the two-loop diagramswith the exception of the reducible ones. This procedureallows one to effectively eliminate one mass scale fromthe 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. 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-QCDcorrections to Bhabha scattering [156].An alternative approach to the calculation of the heavyflavour corrections to Bhabha scattering is based on dis-persion relations. This method also applies to hadroniccorrections. The hadronic and heavy fermion correctionsto the Bhabha-scattering cross section can be obtained by Additional collinear logarithms arise also from the inter-ference of one-loop diagrams in which at least one vertex ispresent. The necessary MIs can be found in [156,160,161,162].3 appropriately inserting the renormalised irreducible pho-ton vacuum-polarisation function Π in the photon propa-gator: g µν q + i δ → g µα q + i δ (cid:0) q g αβ − q α q β (cid:1) Π ( q ) g βν q + i δ . (20)The vacuum polarisation Π can be represented by a once-subtracted dispersion integral [12], Π ( q ) = − q π Z ∞ M d z Im Π ( z ) z q − z + i δ . (21)The contributions to Π may then be determined from a(properly normalised) production cross section by the op-tical theorem [163],Im Π had ( z ) = − α R ( z ) . (22)In this way, the hadronic vacuum polarisation may be ob-tained from the experimental data for R : R ( z ) = σ ( z )(4 πα ) / (3 z ) , (23)where σ ( z ) ≡ σ ( { e + e − → γ ⋆ → hadrons } ; z ). In thelow-energy region the inclusive experimental data may beused [35,164]. Around a narrow hadronic resonance withmass M res and width Γ e + e − res one may use the relation R res ( z ) = 9 πα M res Γ e + e − res δ ( z − M ) , (24)and in the remaining regions the perturbative QCD pre-diction [165]. Contributions to Π arising from leptons andheavy quarks with mass m f , charge Q f and colour C f canbe computed directly in perturbation theory. In the lowestorder it reads R f ( z ; m f ) = Q f C f m f z ! s − m f z . (25)As a result of the above formulas, the massless photonpropagator gets replaced by a massive propagator, whoseeffective mass z is subsequently integrated over: g µν q + iδ → α π Z ∞ M d z R tot ( z ) z ( q − z + iδ ) (cid:18) g µν − q µ q ν q + iδ (cid:19) , (26)where R tot ( 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 approachwas also applied to the evaluation of the two-loop formfactors in QED in [169,170,171]. The fermionic and hadronic corrections to Bhabha scat-tering at one-loop accuracy come only from the self-energydiagram; see for details Section 6. At two-loop level thereare 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 diagramswith self-energy insertions because the application of thedispersion technique as described here is possible.The kernel function for the irreducible two-loop vertexwas derived in [167] and verified e.g. in [158]. The threekernel functions for the two-loop box functions were firstobtained in [172,157,158] and verified in [173]. A completecollection of all the relevant formulae may be found in[158], and the corresponding Fortran code bhbhnnlohf ispublicly available at the web page [174] .In [158], the dependence of the various heavy fermionNNLO corrections on ln( s/m f ) for s, | t | , | u | ≫ m f wasstudied. The irreducible vertex behaves (before a combi-nation with real pair emission terms) like ln ( s/m f ) [167],while the sum of the various infrared divergent diagramsas a whole behaves like ln( s/m f ) ln( s/m e ). This is in ac-cordance with Eq. (19), but the limit plays no effectiverole at the energies studied here.As a result of the efforts of recent years we now have atleast two completely independent calculations for all thenon-photonic virtual two-loop contributions. The net re-sult, as a ratio of the NNLO corrections to the Born crosssection in per mill, is shown in Fig. 12 for KLOE and inFig. 13 for BaBar/Belle. While the non-photonic correc-tions stay at one per mill or less for KLOE, they reach afew per mill at the BaBar/Belle energy range. The NNLOphotonic corrections are the dominant contributions andamount to some per mill, both at φ and B factories. How-ever, as already emphasised, the bulk of both photonic andnon-photonic corrections is incorporated into the genera-tors used by the experimental collaborations. Hence, theconsistent 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 ofthe luminosity tools, as will be discussed quantitatively inSection 2.8. The one-loop matrix element for the process e + e − → e + e − γ is one of the contributions to the complete set ofNNLO corrections to Bhabha scattering. Its evaluationrequires 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 expressedin terms of MIs with trivial numerators that are indepen-dent of the loop variable, each multiplied by a Lorentz The pure self-energy corrections deserve a special discus-sion and are thus omitted in the plots.4
Fig. 11.
Some of the diagrams belonging to the class of the“hadronic” corrections. The additional photons in the finalstate are soft.
20 40 60 80 100 120 140 160 θ * d σ / d σ photonicmuonelectrontotal non-photonichadronic s = 1.04 GeV Fig. 12.
Two-loop photonic and non-photonic corrections toBhabha scattering at √ s = 1 .
02 GeV, normalised to the QEDtree-level cross section, as a function of the electron polar angle;no cuts; the parameterisations of R had from [175] and [35,164,165] are very close to each other. structure depending only on combinations of the externalmomenta and the metric tensor. The achievement of thecomplete PV-reduction amounts to solving a nontrivialsystem 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 itgives rise to Gram determinants. The latter naturally arisein the procedure of inverting a system and they can vanishat special phase space points. This fact requires a propermodification of the reduction algorithm [177,178,179,180,181,182,183]. A viable solution for the complete algebraicreduction of tensor-pentagon (and tensor-hexagon) inte-grals was formulated in [184,185,186], by exploiting thealgebra 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 80 100 120 140 160 θ * d σ / d σ photonicmuonelectrontotal non-photonichadronic s = 10.56 GeV Fig. 13.
Two-loop photonic and non-photonic corrections toBhabha scattering at √ s = 10 .
56 GeV, normalised to the QEDtree-level cross section, as a function of the electron polar angle;no cuts; the parameterisations of R had is from [175]. using generalised-unitarity cutting rules (see [190] for adetailed compilation of references). In the following wepropose two ways to achieve the result, via an analyti-cal and via a semi-numerical method. The application ofgeneralised cutting rules as an on-shell method of calcula-tion is based on two fundamental properties of scatteringamplitudes: 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 theresidues at the singularities are determined by productsof simpler amplitudes. Turning these properties into atool for computing scattering amplitudes is possible be-cause of the underlying representation of the amplitudein 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-integrals may be expressed, through PV-reduction, by a linear combination of 17 MIs (including 3boxes, 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 neededfor reconstructing the amplitude as a whole. Matching thegeneralised cuts of the amplitude with the cuts of theMIs provides an efficient way to extract their (rational)coefficients from the amplitude itself. In general the ful-filment of multiple-cut conditions requires loop momentawith complex components. The effect of the cut conditionsis to freeze some or all of its components, depending onthe number of the cuts. With the quadruple-cut [200] theloop momentum is completely frozen, yielding the alge-braic determination of the coefficients of n -point functionswith n ≥
4. In cases where fewer than four denominatorsare cut, like triple-cut [201,202,203], double-cut [204,205,206,207,208,202] and single-cut [209], the loop momen- tum is not frozen: the free components are left over asphase-space integration variables.For each multiple-cut, the evaluation of the phase-space integral would generate, in general, logarithms anda non-logarithmic term. The coefficient of a given n -pointMI finally appears in the non-logarithmic term of the cor-responding n -particle cut, where all the internal lines areon-shell (while the logarithms correspond to the cuts ofhigher-point MIs which share that same cut). Thereforeall the coefficients of MIs can be determined in a top-down algorithm, starting from the quadruple-cuts for theextraction of the four-point coefficients, and following withthe triple-, double- and single-cuts for the coefficients ofthree-, two- and one-point, respectively. The coefficient ofan n -point MI ( n ≥
2) can also be obtained by specialisingthe generating formulas given in [210] for general one-loopamplitudes to the case at hands.Instead of the analytic evaluation of the multiple-cutphase-space integrals, it is worth considering the feasibil-ity of computing the process e + e − → e + e − γ with a semi-numerical technique by now known as OPP-reduction [211,212], based on the decomposition of the numerator of anyone-loop integrand in terms of its denominators [213,214,215,216]. Within this approach the coefficients of the MIscan be found simply by solving a system of numericalequations, avoiding any explicit integration. The OPP-reduction algorithm exploits the polynomial structures ofthe integrand when evaluated at values of the loop-mo-mentum fulfilling multiple cut-conditions: i) for each n -point MI one considers the n -particle cut obtained by set-ting all the propagating lines on-shell; ii) such a cut isassociated with a polynomial in terms of the free com-ponents of the loop-momentum, which corresponds to thenumerator of the integrand evaluated at the solution of theon-shell conditions; iii) the constant-term of that polyno-mial is the coefficient of the MI.Hence the difficult task of evaluating one-loop Feynmanintegrals is reduced to the much simpler problem of poly-nomial fitting, recently optimised by using a projectiontechnique based on the Discrete Fourier Transform [217].In general the result of a dimensional-regulated ampli-tude in the 4-dimensional limit, with D (= 4 − ǫ ) the regu-lating parameter, is expected to contain (poly)logarithms,often referred to as the cut-constructible term, and a purerational term. In a later paper [218], which completedthe OPP-method, the rising of the rational term was at-tributed to two potential sources (of UV-divergent inte-grals): one, defined as R , due to the D -dimensional com-pletion of the 4-dimensional contribution of the numera-tor; a second one, called R , due to the ( − ǫ )-dimensionalalgebra of Dirac-matrices. Therefore in the OPP-approachthe calculation of the one-loop amplitude e + e − → e + e − γ can proceed through two computational stages:1. the coefficients of the MIs that are responsible bothfor the cut-constructible and for the R -rational termscan be determined by applying the OPP-reduction dis-cussed above [211,212,217];2. the R -rational term can be computed by using addi-tional tree-level-like diagrammatic rules, very much re- Table 2.
The NNLO lepton and pion pair corrections to theBhabha scattering Born cross section σ B : virtual corrections σ v , soft and hard real photon emissions σ s , σ h , and pair emissioncontributions σ pairs . The total pair correction cross sectionsare obtained from the sum σ s + v + h + σ pairs . All cross sections,according to the cuts given in the text, are given in nanobarns. Electron pair corrections σ B σ h σ v + s σ v + s + h σ pairs KLOE 529.469 9.502 -11.567 -2.065 0.271BaBar 6.744 0.246 -0.271 -0.025 0.017
Muon pair corrections σ B σ h σ v + s σ v + s + h σ pairs KLOE 529.469 1.494 -1.736 -0.241 –BaBar 6.744 0.091 -0.095 -0.004 0.0005
Tau pair corrections σ B σ h σ v + s σ v + s + h σ pairs KLOE 529.469 0.020 -0.023 -0.003 –BaBar 6.744 0.016 -0.017 -0.0007 < − Pion pair corrections σ B σ h σ v + s σ v + s + h σ pairs KLOE 529.469 1.174 -1.360 -0.186 –BaBar 6.744 0.062 -0.065 -0.003 0.00003 sembling the computation of the counter terms neededfor the renormalisation of UV-divergences [218].The numerical influence of the radiative loop diagrams,including the pentagon diagrams, is expected not to beparticularly large. However, the calculation of such correc-tions would greatly help to assess the physical precision ofexisting luminosity programs. As was mentioned in the paragraph on virtual heavy fla-vour and hadronic corrections of Section 2.3.1, these vir-tual corrections have to be combined with real correc-tions in order to get physically sensible results. The virtualNNLO electron, muon, tau and pion corrections have tobe combined with the emission of real electron, muon, tauand pion pairs, respectively. The real pair production crosssections are finite, but cut dependent. We consider herethe pion pair production as it is the dominant part of thehadronic corrections and can serve as an estimate of therole of the whole set of hadronic corrections. The descrip-tion of all relevant hadronic contributions is a much moreinvolved task and will not be covered in this review. Aswas first explicitly shown for Bhabha scattering in [102]for electron pairs, and also discussed in [158], there ap-pear exact cancellations of terms of the order ln ( s/m e )or ln ( s/m f ), so that the leading terms are at most oforder ln ( s/m e ) , ln ( s/m f ). As already remarked, the exact calculation of one-loop cor-rections to hard photon emission in Bhabha scattering becameavailable [101] during the completion of the report, exactly ac-cording to the methods described in the present Section.6
In Table 2 we show NNLO lepton and pion pair con-tributions with typical kinematical cuts for the KLOEand BaBar experiments. Besides contributions from un-resolved pair emissions σ pairs , we also add unresolved realhard photon emission contributions σ h . The corrections σ pairs from fermions have been calculated with the For-tran package HELAC-PHEGAS [219,220,221,222], the realpion corrections with EKHARA [223,224], the NNLO hardphotonic corrections σ h with a program [225] based on thegenerator BHAGEN-1PH [226]. The latter depend, tech-nically, on the soft photon cut-off E min γ = ω . After addingup with σ v + s , the sum of the two σ v + s + h is independentof that; in fact here we use ω/E beam = 10 − . In order tocover also pion pair corrections σ v + s is determined with anupdated version of the Fortran package bhbhnnlohf [158,174]. The cuts applied in Table 2 for the KLOE experi-ment are – √ s = 1 .
02 GeV , – E min = 0 . – ◦ < θ ± < ◦ , – ξ max = 9 ◦ ,and for the BaBar experiment – √ s = 10 .
56 GeV , – | cos( θ ± ) | < . | cos( θ + ) | < .
65 or | cos( θ − ) | < .
65 , – | p + | /E beam > .
75 and | p − | /E beam > . | p − | /E beam > .
75 and | p + | /E beam > . – ξ d max = 30 ◦ .Here E min is the energy threshold for the final-state elec-tron/positron, θ ± are the electron/positron polar anglesand ξ max is the maximum allowed polar angle acollinear-ity: ξ = | θ + + θ − − ◦ | , (27)and ξ d max is the maximum allowed three dimensional acol-linearity: ξ d = (cid:12)(cid:12)(cid:12)(cid:12) arccos (cid:18) p + · p − ( | p − || p + | (cid:19) × ◦ π − ◦ (cid:12)(cid:12)(cid:12)(cid:12) . (28)For e + e − → e + e − µ + µ − , cuts are applied only to the e + e − pair. In the case of e + e − → e + e − e + e − , all possible e ± e ∓ combinations are checked and if at least one pair fulfilsthe cuts the event is accepted.At KLOE the electron pair corrections contribute about3 × − and at BaBar about 1 × − , while all the othercontributions of pair production are even smaller. Like insmall-angle Bhabha scattering at LEP/SLC the pair cor-rections [227] are largely dominated by the electron paircontribution. From inspection of Eqs. (10) and (13) for the SV NLOQED corrections to the cross section of the Bhabha scat-tering and e + e − → γγ process, it can be seen that large logarithms L = ln( s/m e ), due to collinear photon emis-sion, are present. Similar large logarithmic terms arise af-ter integration of the hard photon contributions from thekinematical domains of photon emission at small angleswith respect to charged particles. For the energy rangeof meson factories the logarithm is large numerically, i.e. L ∼
15 at the φ factories and L ∼
20 at the B factories,and the corresponding terms give the bulk of the total ra-diative correction. These contributions represent also thedominant part of the NNLO effects discussed in Section2.3. Therefore, to achieve the required theoretical accu-racy, the logarithmically enhanced contributions due toemission of soft and collinear photons must be taken intoaccount at all orders in perturbation theory. The meth-ods for the calculation of higher-order (HO) QED correc-tions on the basis of the generators employed nowadays atflavour factories were already widely and successfully usedin the 90s at LEP/SLC for electroweak tests of the SM.They were adopted for the calculation of both the small-angle Bhabha scattering cross section (necessary for thehigh-precision luminosity measurement) and Z -boson ob-servables. Hence, the theory accounting for the control ofHO QED corrections at meson factories can be consideredparticularly robust, having passed the very stringent testsof the LEP/SLC era.The most popular and standard methods to keep mul-tiple photon effects under control are the QED StructureFunction (SF) approach [228,229,230,231] and Yennie-Frautschi-Suura (YFS) exponentiation [232]. The formeris used in all the versions of the generator BabaYaga [233,234,235] and MCGPJ [236] (albeit according to differ-ent realisations), while the latter is the theoretical recipeadopted in BHWIDE [237]. Actually, analytical QED SFs D ( x, Q ), valid in the strictly collinear approximation,are implemented in MCGPJ, whereas BabaYaga is basedon a MC Parton Shower (PS) algorithm to reconstruct D ( x, Q ) numerically. The Structure Function approach
Let us consider the annihilation process e − e + → X ,where X is some given final state and σ ( s ) its LO crosssection. Initial-state (IS) QED radiative corrections canbe described according to the following picture. Beforearriving at the annihilation point, the incoming electron(positron) of four-momentum p − (+) radiates real and vir-tual photons. These photons, due to the dynamical fea-tures of QED, are mainly radiated along the direction ofmotion of the radiating particles, and their effect is mainlyto reduce the original four-momentum of the incomingelectron (positron) to x p − (+) . After this pre-emission,the hard scattering process e − ( x p − ) e + ( x p + ) → X takesplace, at a reduced squared c.m. energy ˆ s = x x s . Theresulting cross section, corrected for IS QED radiation,can be represented in the form [228,229,230] σ ( s ) = Z d x d x D ( x , s ) D ( x , s ) σ ( x x s ) Θ (cuts) , (29) where D ( x, s ) is the electron SF, representing the prob-ability that an incoming electron (positron) radiates acollinear photon, retaining a fraction x of its original mo-mentum at the energy scale Q = s , and Θ (cuts) standsfor a rejection algorithm taking care of experimental cuts.When considering photonic radiation only the non-singletpart of the SF is of interest. If the running of the QEDcoupling constant is neglected, the non-singlet part of theSF is the solution of the following Renormalisation Group(RG) equation, analogous to the Dokshitzer-Gribov-Lipa-tov-Altarelli-Parisi (DGLAP) equation of QCD [238,239,240]: s ∂∂s D ( x, s ) = α π Z x d zz P + ( z ) D (cid:16) xz , s (cid:17) , (30)where P + ( z ) is the regularised Altarelli-Parisi (AP) split-ting function for the process electron → electron+photon,given by P + ( z ) = P ( z ) − δ (1 − z ) Z d xP ( x ) ,P ( z ) = 1 + z − z . (31)Equation (30) can be also transformed into an integralequation, subject to the boundary condition D ( x, m e ) = δ (1 − x ): D ( x, s ) = δ (1 − x ) + α π Z sm e d Q Q Z x d zz P + ( z ) D (cid:16) xz , Q (cid:17) . (32)Equation (32) can be solved exactly by means of nu-merical methods, such as the inverse Mellin transformmethod. However, this derivation of D ( x, s ) turns out beproblematic in view of phenomenological applications. There-fore, approximate (but very accurate) analytical repre-sentations of the solution of the evolution equation areof major interest for practical purposes. This type of so-lution was the one typically adopted in the context ofLEP/SLC phenomenology. A first analytical solution canbe obtained in the soft photon approximation, i.e. in thelimit x ≃
1. This solution, also known as Gribov-Lipatov(GL) approximation, exponentiates the large logarithmiccontributions of infrared and collinear origin at all per-turbative orders, but it does not take into account hard-photon (collinear) effects. This drawback can be overcomeby solving the evolution equation iteratively. At the n -thstep of the iteration, one obtains the O ( α n ) contributionto the SF for any value of x . By combining the GL solu-tion with the iterative one, in which the soft-photon parthas been eliminated in order to avoid double counting, onecan build a hybrid solution of the evolution equation. Itexploits all the positive features of the two kinds of so-lutions and is not affected by the limitations intrinsic toeach of them. Two classes of hybrid solutions, namely theadditive and factorised ones, are known in the literature,and both were adopted for applications to LEP/SLC pre-cision physics. A typical additive solution, where the GL approximation D GL ( x, s ) is supplemented by finite-orderterms present in the iterative solution, is given by [241] D A ( x, s ) = X i =0 d ( i ) A ( x, s ) ,d (0) A ( x, s ) = exp (cid:2) β (cid:0) − γ E (cid:1)(cid:3) Γ (cid:0) β (cid:1) β (1 − x ) β − ,d (1) A ( x, s ) = − β (1 + x ) ,d (2) A ( x, s ) = 132 β [(1 + x ) ( − − x ) + 3 ln x ) − x − x − − x (cid:21) ,d (3) A ( x, s ) = 1384 β { (1 + x ) [18 ζ (2) − ( x ) −
12 ln (1 − x ) (cid:3) + 11 − x (cid:20) −
32 (1 + 8 x + 3 x ) ln x + 12 (1 + 7 x ) ln x − x ) ln x ln(1 − x ) − x + 5)(1 − x ) ln(1 − x ) −
14 (39 − x − x ) (cid:21)(cid:27) , (33)where Γ is the Euler gamma-function, γ E ≈ . ζ the Riemann ζ -function and β is the large collinear factor β = 2 απ (cid:20) ln (cid:18) sm e (cid:19) − (cid:21) . (34)Explicit examples of factorised solutions, which areobtained by multiplying the GL solution by finite-orderterms in such a way that, order by order, the iterativecontributions are exactly recovered, can be found in [242].For the calculation of HO corrections with a per mill ac-curacy analytical SFs in additive and factorised form con-taining up to O ( α ) finite-order terms are sufficient andin excellent agreement. They also agree with an accuracymuch better than 0.1 with the exact numerical solution ofthe QED evolution equation. Explicit solutions up to thefifth order in α were calculated in [243,244].The RG method described above was applied in [245]for the treatment of LL QED radiative corrections to var-ious processes of interest for physics at meson factories.Such a formulation was later implemented in the genera-tor MCGPJ. For example, according to [245], the Bhabhascattering cross section, accounting for LL terms in allorders, O ( α n L n ) , n = 1 , , . . . , of perturbation theory, isgiven byd σ BhabhaLLA = X a,b,c,d = e ± ,γ Z z d z Z z d z D str ae − ( z ) D str be + ( z ) × d σ ab → cd ( z , z ) Z y d y Y D frg e − c ( y Y ) Z y d y Y D frg e + d ( y Y )+ O (cid:18) α L, α m e s (cid:19) . (35) Here d σ ab → cd ( z , z ) is the differential LO cross section ofthe process ab → cd , with energy fractions of the incomingparticles being scaled by factors z and z with respect tothe initial electron and positron, respectively. In the nota-tion of [245], the electron SF D str ab ( z ) is distinguished fromthe electron fragmentation function D frg ab ( z ) to point outthe role played by IS radiation (described by D str ab ( z )) withrespect to the one due to final-state radiation (describedby D frg ab ( z )). However, because of their probabilistic mean-ing, the electron structure and fragmentation functionscoincide. In Eq. (35) the quantities Y , are the energyfractions of particles c and d with respect to the beamenergy. Explicit expressions for Y , = Y , ( z , z , cos θ )and other details on the kinematics can be found in [245].The lower limits of the integrals, ¯ z , and ¯ y , , should bedefined according to the experimental conditions of par-ticle detection and kinematical constraints. For the caseof the e + e − → γγ process one has to change the mas-ter formula (35) by picking up the two-photon final state.Formally this can be done by just choosing the properfragmentation functions, D frg γc and D frg γd .The photonic part of the non-singlet electron structure(fragmentation) function in O ( α n L n ) considered in [245]reads D NS,γee ( z ) = δ (1 − z ) + n X i =1 (cid:16) α π ( L − (cid:17) i i ! h P (0) ee ( z ) i ⊗ i ,D γe ( z ) = α π ( L − P γe ( z ) + O ( α L ) ,D eγ ( z ) = α π LP eγ ( z ) + O ( α L ) ,P (0) ee ( z ) = (cid:20) z − z (cid:21) + = lim ∆ → (cid:26) δ (1 − z )(2 ln ∆ + 32 ) + Θ (1 − z − ∆ ) 1 + z − z (cid:27) , h P (0) ee ( z ) i ⊗ i = Z z d tt P ( i − ee ( t ) P (0) ee (cid:16) zt (cid:17) , (36) P γe ( z ) = z + (1 − z ) , P eγ ( z ) = 1 + (1 − z ) z . Starting from the second order in α there appear also non-singlet and singlet e + e − pair contributions to the struc-ture function: D NS,e + e − ee ( z ) = 13 (cid:16) α π L (cid:17) P (1) ee ( z ) + O ( α L ) ,D S,e + e − ee ( z ) = 12! (cid:16) α π L (cid:17) R ( z ) + O ( α L ) ,R ( z ) = P eγ ⊗ P γe ( z ) = 1 − z z (4 + 7 z + 4 z )+2(1 + z ) ln z. (37)Note that radiation of a real pair, i.e. appearance of addi-tional electrons and positrons in the final state, require theapplication of nontrivial conditions of experimental par-ticle registration. Unambiguously, that can be done only within a MC event generator based on four-particle matrixelements, as already discussed in Section 2.3.In the same way as in QCD, the LL cross sections de-pend on the choice of the factorisation scale Q in theargument of the large logarithm L = ln( Q /m e ), which isnot fixed a priori by the theory. However, the scale shouldbe taken of the order of the characteristic energy trans-fer in the process under consideration. Typical choicesare Q = s , Q = − t and Q = st/u . The first one isgood for annihilation channels like e + e − → µ + µ − , thesecond one is optimal for small-angle Bhabha scatteringwhere the t -channel exchange dominates, see [246]. Thelast choice allows to exponentiate the leading contribu-tion due to initial-final state interference [247] and is par-ticularly suited for large-angle Bhabha scattering in QED.The option Q = st/u is adopted in all the versions of thegenerator BabaYaga. Reduction of the scale dependencecan be achieved by taking into account next-to-leadingcorrections in O ( α n L n − ), next-to-next-to-leading ones in O ( α n L n − ) etc.The Parton Shower algorithm The PS algorithm is a method for providing a MC it-erative solution of the evolution equation and, at the sametime, for generating the four-momenta of the electron andphoton at a given step of the iteration. It was developedwithin the context of QCD and later applied in QED too.In order to implement the algorithm, it is first nec-essary to assume the existence of an upper limit for theenergy fraction x in such a way that the AP splitting func-tion is regularised by writing P + ( z ) = θ ( x + − z ) P ( z ) − δ (1 − z ) Z x + d xP ( x ) . (38)Of course, in the limit x + →
1, Eq. (38) recovers the usualdefinition of the AP splitting function given in Eq. (31).By inserting the modified AP vertex into Eq. (30), oneobtains s ∂∂s D ( x, s ) = α π Z x + x d zz P ( z ) D (cid:16) xz , s (cid:17) − α π D ( x, s ) Z x + x d zP ( z ) . (39)Separating the variables and introducing the Sudakov formfactor Π ( s , s ) = exp (cid:20) − α π Z s s d s ′ s ′ Z x + d zP ( z ) (cid:21) , (40)which is the probability that the electron evolves fromvirtuality − s to − s without emitting photons of energyfraction larger than 1 − x + ≡ ǫ ( ǫ ≪ D ( x, s ) = Π ( s, m e ) D ( x, m e )+ α π Z sm e d s ′ s ′ Π ( s, s ′ ) Z x + x d zz P ( z ) D (cid:16) xz , s ′ (cid:17) . (41) The formal iterative solution of Eq. (41) can be repre-sented by the infinite series D ( x, s ) = ∞ X n =0 n Y i =1 (Z s i − m e d s i s i Π ( s i − , s i ) × α π Z x + x/ ( z ··· z i − ) d z i z i P ( z i ) ) Π ( s n , m e ) D (cid:18) xz · · · z n , m e (cid:19) . (42)The particular form of Eq. (42) allows to exploit a MCmethod for building the solution iteratively. The steps ofthe algorithm are as follows:1 – set Q = m e , and fix x = 1 according to the boundarycondition D ( x, m e ) = δ (1 − x );2 – generate a random number ξ in the interval [0 , ξ < Π ( s, Q ) stop the evolution; otherwise4 – compute Q ′ as solution of the equation ξ = Π ( Q ′ , Q );5 – generate a random number z according to the proba-bility density P ( z ) in the interval [0 , x + ];6 – substitute x → xz and Q → Q ′ ; go to 2.The x distribution of the electron SF as obtained bymeans of the PS algorithm and a numerical solution (basedon the inverse Mellin transform method) of the QED evo-lution equation is shown in Fig. 14. Perfect agreement isseen. Once D ( x, s ) has been reconstructed by the algo-rithm, the master formula of Eq. (29) can be used forthe calculation of LL corrections to the cross section ofinterest. This cross section must be independent of thesoft-hard photon separator ǫ in the limit of small valuesfor ǫ . This can be clearly seen in Fig. 15, where the QEDcorrected Bhabha cross section as a function of the fic-titious parameter ε is shown for DAΦNE energies withthe cuts of Eq. (15), but for an angular acceptance θ ± of 55 ◦ ÷ ◦ . The cross section reaches a plateau for ǫ smaller than 10 − .The main advantage of the PS algorithm with respectto the analytical solutions of the electron evolution is thepossibility of going beyond the strictly collinear approxi-mation and generating transverse momentum p ⊥ of elec-trons and photons at each branching. In fact, the kine-matics of the branching process e ( p ) → e ′ ( p ′ ) + γ ( q ) canbe written as p = ( E, , p z ) ,p ′ = ( zE, p ⊥ , p ′ z ) ,q = ((1 − z ) E, − p ⊥ , q z ) . (43)Once the variables p , p ′ and z are generated by the PSalgorithm, the on-shell condition q = 0, together with thelongitudinal momentum conservation, allows to obtain anexpression for the p ⊥ variable: p ⊥ = (1 − z )( zp − p ′ ) , (44)valid at first order in p /E ≪ p ⊥ /E ≪ Fig. 14.
Comparison for the x distribution of the electron SFas obtained by means of a numerical solution of the QED evo-lution equation (solid line) and the PS algorithm (histogram).From [233]. σ ( nb ) ε Fig. 15.
QED corrected Bhabha cross section at DAΦNE asa function of the infrared regulator ε of the PS approach, ac-cording to the setup of Eq. (15). The error bars correspond to1 σ MC errors. From [235]. reconstruction of the exclusive photon kinematics. Firstof all, since within the PS algorithm the generation of p ′ and z are independent, it can happen that in some branch-ings the p ⊥ as given by Eq. (44) is negative. In order toavoid this problem, the introduction of any kinematicalcut on the p or z generation (or the regeneration of thewhole event) would prevent the correct reconstruction ofthe SF x distribution, which is important for a precisecross section calculation. Furthermore, in the PS scheme,each fermion produces its photon cascade independentlyof the other ones, missing the effects due to the interfer-ence of radiation coming from different charged particles.As far as inclusive cross sections (i.e. cross sections withno cuts imposed on the generated photons) are concerned, these effects are largely integrated out. However, as shownin [248], they become important when more exclusive vari-ables distributions are considered.The first problem can be overcome by choosing thegenerated p ⊥ of the photons different from Eq. (44). Forexample, one can choose to extract the photon cos ϑ γ ac-cording to the universal leading poles 1 /p · k present inthe matrix element for photon emission. Namely, one cangenerate cos ϑ γ ascos ϑ γ ∝ − β cos ϑ γ , (45)where β is the speed of the emitting particle. In this way,photon energy and angle are generated independently, dif-ferent from Eq. (44). The nice feature of this prescriptionis that p ⊥ = E γ sin ϑ γ is always well defined, and the x distribution reproduces exactly the SF, because no furtherkinematical cuts have to be imposed to avoid unphysicalevents. At this stage, the PS is used only to generate theenergies and multiplicity of the photons. The problem ofincluding the radiation interference is still unsolved, be-cause the variables of photons emitted by a fermion arestill uncorrelated with those of the other charged particles.The issue of including photon interference can be success-fully worked out looking at the YFS formula [232]:d σ n ≈ d σ e n n ! n Y l =1 d k l (2 π ) k l N X i,j =1 η i η j − p i · p j ( p i · k l )( p j · k l ) . (46)It gives the differential cross section d σ n for the emissionof n photons, whose momenta are k , · · · , k n , from a kernelprocess described by d σ and involving N fermions, whosemomenta are p , · · · , p N . In Eq. (46) η i is a charge factor,which is +1 for incoming e − or outgoing e + and − e + or outgoing e − . Note that Eq. (46) is validin the soft limit ( k i → l th photon:cos ϑ l ∝ − N X i,j =1 η i η j − β i β j cos ϑ ij (1 − β i cos ϑ il )(1 − β j cos ϑ jl ) . (47)It is worth noticing that in the LL prescription thesame quantity can be written ascos ϑ l ∝ N X i =1 − β i cos ϑ il , (48)whose terms are of course contained in Eq. (47).In order to consider also coherence effects in the an-gular distribution of the photons, one can generate cos ϑ γ according to Eq. (47), rather than to Eq. (48). This recipe[248] is adopted in BabaYaga v3.5 and BabaYaga@NLO. Yennie-Frautschi-Suura exponentiation
The YFS exponentiation procedure, implemented inthe code BHWIDE, is a technique for summing up all the infrared (IR) singularities present in any process accompa-nied by photonic radiation [232]. It is inherently exclusive,i.e. all the summations of the IR singular contributions aredone before any phase-space integration over the virtual orreal photon four-momenta are performed. The method wasmainly developed by S. Jadach, B.F.L. Ward and collab-orators to realise precision MC tools. In the following, thegeneral ideas underlying the procedure are summarised.Let us consider the scattering process e + ( p ) e − ( p ) → f ( q ) · · · f n ( q n ), where f ( q ) · · · f n ( q n ) represents a givenarbitrary final state, and let M be its tree-level matrixelement. By using standard Feynman-diagram techniques,it is possible to show that the same process, when accom-panied by l additional real photons radiated by the ISparticles, and under the assumption that the l additionalphotons are soft, i.e. their energy is much smaller that anyenergy scale involved in the process, can be described bythe factorised matrix element built up by the LO one, M ,times the product of l eikonal currents, namely M ≃ M l Y i =1 (cid:20) e (cid:18) ε i ( k i ) · p k i · p − ε i ( k i ) · p k i · p (cid:19)(cid:21) , (49)where e is the electron charge, k i are the momenta ofthe photons and ε i ( k i ) their polarisation vectors. Tak-ing the square of the matrix element in Eq. (49) andmultiplying by the proper flux factor and the Lorentz-invariant phase space volume, the cross section for theprocess e + ( p ) e − ( p ) → f ( q ) · · · f n ( q n ) + l real photonscan be written asd σ ( l ) r = d σ l ! l Y i =1 (cid:20) k i d k i d cos ϑ i d ϕ i π ) × X ε i e (cid:18) ε i ( k i ) · p k i · p − ε i ( k i ) · p k i · p (cid:19) . (50)By summing over the number of final-state photons, oneobtains the cross section for the original process accom-panied by an arbitrary number of real photons, namelyd σ ( ∞ ) r = ∞ X l =0 d σ ( l ) r = d σ exp (cid:20) k d k d cos ϑ d ϕ π ) × X ε e (cid:18) ε ( k ) · p k · p − ε ( k ) · p k · p (cid:19) . (51)Equation (51), being limited to real radiation only, is IRdivergent once the phase space integrations are performeddown to zero photon energy. This problem, as is well known,finds its solution in the matching between real and vir-tual photonic radiation. Equation (51) already shows thekey feature of exclusive exponentiation, i.e. summing upall the perturbative contributions before performing anyphase space integration.In order to get meaningful radiative corrections it isnecessary to consider, besides IS real photon corrections, also IS virtual photon corrections, i.e. the corrections dueto additional internal photon lines connecting the IS elec-tron and positron. For a vertex-type amplitude, the resultcan be written as M V = − i e (2 π ) Z d k k + iε ¯ v ( p ) γ µ − (/ p + / k ) + m p · k + k + iε × Γ (/ p + / k ) + m p · k + k + iε γ µ u ( p ) , (52)where Γ stands for the Dirac structure of the LO process,in such a way that M = ¯ v ( p ) Γ u ( p ). The soft-photonpart of the amplitude can be extracted by taking k µ ≃ M V = M × V,V = 2 iα (2 π ) Z d k p · p (2 p · k + k + iε )(2 p · k + k + iε ) × k + iε . (53)It can be seen that, as in the real case, the IR virtualcorrection factorises off the LO matrix element so that itis universal, i.e. independent of the details of the processunder consideration, and divergent in the IR portion ofthe phase space.The correction given by n soft virtual photons can beseen to factorise with an additional factor 1 /n !, namely M V n = M × n ! V n , (54)so that by summing over all the additional soft virtualphotons one obtains M V = M × exp[ V ] . (55)As already noticed both the real and virtual factorsare IR divergent. In order to obtain meaningful expres-sions one has to adopt some regularisation procedure. Onepossibility is to give the photon a (small) mass λ and tomodify Eqs. (50) and (53) accordingly. Once all the ex-pressions are properly regularised, one can write down aYFS master formula that takes into account real and vir-tual photonic corrections to the LO process. In virtue ofthe factorisation properties discussed above, the masterformula can be obtained from Eq. (51) with the substitu-tion d σ → d σ | exp( V ) | , i.e.d σ = d σ | exp( V ) | exp (cid:20) k d k d cos ϑ d ϕ π ) × X ε e (cid:18) ε ( k ) · p k · p − ε ( k ) · p k · p (cid:19) . (56)As a last step it is possible to analytically perform theIR cancellation between virtual and very soft real pho-tons. Actually, since very soft real photons do not affectthe kinematics of the process, the real photon exponent can be split into a contribution coming from photons withenergy less than a cutoff k min plus a contribution fromphotons with energy above it. The first contribution canbe integrated over all its phase space and can then becombined with the virtual exponent. After this step it ispossible to remove the regularising photon mass by takingthe limit λ →
0, so that Eq. (56) becomesd σ = d σ exp( Y ) exp (cid:20) k d k d Θ ( k − k min ) cos ϑ d ϕ π ) × X ε e (cid:18) ε ( k ) · p k · p − ε ( k ) · p k · p (cid:19) , (57)where Y is given by Y = 2 V + Z k d k d Θ ( k min − k ) cos ϑ d ϕ π ) × X ε e (cid:18) ε ( k ) · p k · p − ε ( k ) · p k · p (cid:19) . (58)The explicit form of Y can be derived by performing allthe details of the calculation, and reads Y = β ln k min E + δ Y F S ,δ Y F S = 14 β + απ (cid:18) π − (cid:19) . (59) As will be shown numerically in Section 2.6, NLO cor-rections must be combined with multiple photon emissioneffects to achieve a theoretical accuracy at the per milllevel. This combination, technically known as matching ,is a fundamental ingredient of the most precise genera-tors used for luminosity monitoring, i.e. BabaYaga@NLO,BHWIDE and MCGPJ. Although the matching is im-plemented according to different theoretical details, somegeneral aspects are common to all the recipes and mustbe emphasised:1. It is possible to match NLO and HO corrections consis-tently, avoiding double counting of LL contributions atorder α and preserving the advantages of resummationof soft and collinear effects beyond O ( α ).2. The convolution of NLO corrections with HO termsallows to include the dominant part of NNLO correc-tions, given by infrared-enhanced α L sub-leading con-tributions. This was argued and demonstrated analyt-ically and numerically in [44] through comparison withthe available O ( α ) corrections to s -channel processesand t -channel Bhabha scattering. Such an aspect ofthe matching procedure is crucial to settle the theo-retical accuracy of the generators by means of explicitcomparisons with the exact NNLO perturbative cor-rections discussed in Section 2.3, and will be addressedin Section 2.8.
3. BabaYaga@NLO and BHWIDE implement a fully fac-torised matching recipe, while MCGPJ includes someterms in additive form, as will be visible in the formu-lae reported below.In the following we summarise the basic features of thematching procedure as implemented in the codes MCGPJ,BabaYaga@NLO and BHWIDE.The matching approach realised in the MC event gen-erator MCGPJ was developed in [236]. In particular, Bha-bha scattering with complete O ( α ) and HO LL photoniccorrections can written asd σ e + e − → e + e − ( γ ) d Ω − = Z ¯ z d z Z ¯ z d z D NS,γee ( z ) D NS,γee ( z ) × dˆ σ Bhabha0 ( z , z )d Ω − (cid:16) απ K SV (cid:17) Θ (cuts) × Y Z y th d y Y Y Z y th d y Y D NS,γee ( y Y ) D NS,γee ( y Y )+ απ Z ∆ d xx ((cid:20)(cid:18) − x + x (cid:19) ln θ (1 − x ) x (cid:21) × σ Bhabha0 d Ω − + (cid:20)(cid:18) − x + x (cid:19) ln θ x (cid:21) × " dˆ σ Bhabha0 (1 − x, Ω − + dˆ σ Bhabha0 (1 , − x )d Ω − Θ (cuts) − α s (cid:18) c − c (cid:19) απ ln(ctg θ ∆εε + α π s Z k >∆ε θ i >θ W T Θ (cuts) d Γ e ¯ eγ d Ω − . (60)Here the step functions Θ (cuts) stand for the particularcuts applied. The auxiliary parameter θ defines conesaround the directions of the motion of the charged parti-cles in which the emission of hard photons is approximatedby the factorised form by convolution of collinear radiationfactors [249] with the Born cross section. The dependenceon the parameters ∆ and θ cancels out in the sum withthe last term of Eq. (60), where the photon energy andemission angles with respect to all charged particles arelimited from below ( k > ∆ε, θ i > θ ). Taking into ac-count vacuum polarisation, the Born level Bhabha crosssection with reduced energies of the incoming electron and positron can be cast in the formdˆ σ Bhabha0 ( z , z )d Ω − = 4 α sa (cid:26) | − Π (ˆ t ) | a + z (1 + c ) z (1 − c ) + 1 | − Π (ˆ s ) | z (1 − c ) + z (1 + c ) a − Re 1(1 − Π (ˆ t ))(1 − Π (ˆ s )) ∗ z (1 + c ) az (1 − c ) (cid:27) d Ω − , ˆ s = z z s, ˆ t = − sz z (1 − c ) z + z − ( z − z ) c , (61)where Π ( Q ) is the photon self-energy correction. Notethat in the cross section above the cosine of the scatteringangle, c , is given for the original c.m. reference frame ofthe colliding beams.For the two-photon production channel, a similar rep-resentation is used in MCGPJ:d σ e + e − → γγ ( γ ) = Z ¯ z d z D NS,γee ( z ) Z ¯ z d z D NS,γee ( z ) × dˆ σ γγ ( z , z ) (cid:16) απ K γγSV (cid:17) + απ Z ∆ d xx × (cid:20)(cid:18) − x + x (cid:19) ln θ x (cid:21) (cid:20) dˆ σ (1 − x, σ (1 , − x ) (cid:21) + 13 4 α π s Z zi ≥ ∆ π − θ ≥ θ i ≥ θ d Γ γ × (cid:20) z (1 + c ) z z (1 − c )(1 − c ) + two cyclic permutations (cid:21) ,z i = q i ε , c i = cos θ i , θ i = d p − q i , (62)where the cross section with reduced energies has the formdˆ σ γγ ( z , z )d Ω = 2 α s z (1 − c ) + z (1 + c ) (1 − c )( z + z + ( z − z ) c ) , and the factor 1 / ∆ and θ .Concerning BabaYaga@NLO, the matching starts fromthe observation that Eq. (29) for the QED corrected all-order cross section can be rewritten in terms of the PSingredients asd σ ∞ LL = Π ( Q , ε ) ∞ X n =0 n ! | M n,LL | d Φ n . (63)By construction, the expansion of Eq. (63) at O ( α ) doesnot coincide with the exact O ( α ) result. In factd σ αLL = (cid:20) − α π I + ln Q m (cid:21) | M | d Φ + | M ,LL | d Φ ≡ [1 + C α,LL ] | M | d Φ + | M ,LL | d Φ , (64) where I + ≡ R − ǫ P ( z )d z , whereas the exact NLO crosssection can always be cast in the formd σ α = [1 + C α ] | M | d Φ + | M | d Φ . (65)The coefficients C α contain the complete O ( α ) virtual andsoft-bremsstrahlung corrections in units of the squaredBorn amplitude, and | M | is the exact squared matrixelement with the emission of one hard photon. We remarkthat C α,LL has the same logarithmic structure as C α andthat | M ,LL | has the same singular behaviour as | M | .In order to match the LL and NLO calculations, thefollowing correction factors, which are by construction in-frared safe and free of collinear logarithms, are introduced: F SV = 1+( C α − C α,LL ) , F H = 1+ | M | − | M ,LL | | M ,LL | . (66)With them the exact O ( α ) cross section can be expressed,up to terms of O ( α ), in terms of its LL approximation asd σ α = F SV (1 + C α,LL ) | M | d Φ + F H | M ,LL | d Φ . (67)Driven by Eq. (67), Eq. (63) can be improved by writingthe resummed matched cross section asd σ ∞ matched = F SV Π ( Q , ε ) × ∞ X n =0 n ! n Y i =0 F H,i ! | M n,LL | d Φ n . (68)The correction factors F H,i follow from the definition (66)for each photon emission. The O ( α ) expansion of Eq. (68)now coincides with the exact NLO cross section of Eq. (65),and all HO LL contributions are the same as in Eq. (63).This formulation is implemented in BabaYaga@NLO forboth Bhabha scattering and photon pair production, us-ing, of course, the appropriate SV and hard bremsstrah-lung formulae. This matching formulation has also beenapplied to the study of Drell-Yan-like processes, by com-bining the complete O ( α ) electroweak corrections withQED shower evolution in the generator HORACE [250,251,252,253].As far as BHWIDE is concerned, this MC event gen-erator realises the process e + ( p )+ e − ( q ) −→ e + ( p )+ e − ( q ) + γ ( k )+ . . . + γ n ( k n )(69)via the YFS exponentiated cross section formulad σ = e α Re B +2 α ˜ B ∞ X n =0 n ! Z n Y j =1 d k j k j Z d y (2 π ) × e iy ( p + q − p − q − P j k j )+ D ¯ β n ( k , . . . , k n ) d p d q p q , (70)where the real infrared function ˜ B and the virtual infraredfunction B are given in [237]. Here we note the usual con- nections2 α ˜ B = Z k ≤ K max d kk ˜ S ( k ) ,D = Z d k ˜ S ( k ) k (cid:0) e − iy · k − θ ( K max − k ) (cid:1) (71)for the standard YFS infrared real emission factor˜ S ( k ) = α π " Q f Q f ′ (cid:18) p p · k − q q · k (cid:19) + . . . , (72)and where Q f is the electric charge of f in units of thepositron charge. In Eq. (72) the “ . . . ” represent the re-maining terms in ˜ S ( k ), obtained from the given one byrespective of Q f , p , Q f ′ , q with corresponding valuesfor the other pairs of the external charged legs accordingto the YFS prescription of Ref. [232,254] (wherein due at-tention is taken to obtain the correct relative sign of eachof the terms in ˜ S ( k ) according to this latter prescription).The explicit representation is given by2 α Re B ( p , q , p , q ) + 2 α ˜ B ( p , q , p , q ; k m ) = R ( p , q ; k m ) + R ( p , q ; k m ) + R ( p , p ; k m ) + R ( q , q ; k m ) − R ( p , q ; k m ) − R ( q , p ; k m ) , (73)with R ( p, q ; k m ) = R ( p, q ; k m ) + (cid:16) απ (cid:17) π R ( p, q ; k m ) = απ (cid:26)(cid:18) ln 2 pqm e − (cid:19) ln k m p q + 12 ln 2 pqm e −
12 ln p q −
14 ln ( ∆ + δ ) p q −
14 ln ( ∆ − δ ) p q − Re Li (cid:18) ∆ + ω∆ + δ (cid:19) − Re Li (cid:18) ∆ + ω∆ − δ (cid:19) − Re Li (cid:18) ∆ − ω∆ + δ (cid:19) − Re Li (cid:18) ∆ − ω∆ − δ (cid:19) + π − (cid:27) , (75)where ∆ = p pq + ( p − q ) , ω = p + q , δ = p − q ,and k m is a soft photon cut-off in the c.m. system ( E soft γ MC generators used for luminosity monitoring atmeson factories.Generator Theory AccuracyBabaYaga v3.5 Parton Shower ∼ . ÷ O ( α ) + PS ∼ . O ( α ) ∼ O ( α ) YFS ∼ . O ( α ) ∼ O ( α ) + SF < . LL QED corrections to luminosity processes and laterimproved to account for the interference of radiationemitted by different charged legs in the generation ofthe momenta of the final-state particles. The maindrawback of BabaYaga v3.5 is the absence of O ( α )non-logarithmic contributions, resulting in a theoret-ical precision of ∼ . 5% for large-angle Bhabha scat-tering and of about 1% for γγ and µ + µ − final states.It is used by the CLEO-c collaboration for the studyof all the three luminosity processes.2. BabaYaga@NLO – It is the presently released ver-sion of BabaYaga, based on the matching of exact O ( α ) corrections with QED PS, as described in Sec-tion 2.4.2. The accuracy of the current version is esti-mated to be at the 0.1% level for large-angle Bhabhascattering, two-photon and µ + µ − production. It ispresently used by the KLOE and BaBar collabora-tions, and under consideration by the BES-III exper-iment. Like BabaYaga v3.5, BabaYaga@NLO is avail-able at the web page of the Pavia phenomenology group .3. BHAGENF/BKQED – BKQED is the event generatordeveloped by Berends and Kleiss and based on the clas-sical exact NLO calculations of [257,258] for all QEDprocesses. It was intensively used at LEP to performtests of QED through the analysis of the e + e − → γγ process and is adopted by the BaBar collaboration forthe simulation of the same reaction. BHAGENF is acode realised by Drago and Venanzoni at the beginningof the DAΦNE operation to simulate Bhabha events,adapting the calculations of [257] to include the con-tribution of the φ resonance. Both generators lack theeffect of HO corrections and, as such, have a precisionaccuracy of about 1%. The BHAGENF code is avail-able at the web address .4. BHWIDE – It is a MC code realised in Krakow-Knox-wille at the time of the LEP/SLC operation and de-scribed in [237]. In this generator exact O ( α ) correc-tions are matched with the resummation of the in-frared virtual and real photon contributions throughthe YFS exclusive exponentiation approach. Accord-ing to the authors the precision is estimated to be At present, finite mass effects in the virtual corrections to e + e − → µ + µ − , which should be included for precision simula-tions at the φ factories, are not included in BabaYaga@NLO. about 0.5% for c.m. energies around the Z resonance.This accuracy estimate was derived through detailedcomparisons of the BHWIDE predictions with those ofother LEP tools in the presence of the full set of NLOcorrections, including purely weak corrections. How-ever, since the latter are phenomenologically unim-portant at e + e − accelerators of moderately high en-ergies and since the QED theoretical ingredients ofBHWIDE are very similar to the formulation of bothBabaYaga@NLO and MCGPJ, one can argue that theaccuracy of BHWIDE for physics at flavour factoriesis at the level of 0.1%. It is adopted by the KLOE,BaBar and BES collaborations. The code is availableat placzek.home.cern.ch/placzek/bhwide/ .5. MCGPJ – It is the generator developed by the Dubna-Novosibirsk collaboration and used at the VEPP-2Mcollider. This program includes exact O ( α ) correctionssupplemented with HO LL contributions related tothe emission of collinear photon jets and taken intoaccount through analytical QED collinear SF, as de-scribed in Section 2.4.2. The theoretical precision isestimated to be better than 0.2%. The generator isavailable at the web address cmd.inp.nsk.su/~sibid/ .It is worth noticing that the theoretical uncertaintyof the most accurate generators based on the matchingof exact NLO with LL resummation starts at the level of O ( α ) NNL contributions, as far as photonic correctionsare concerned. Other sources of error affecting their phys-ical precision are discussed in detail in Section 2.8. Before showing the results which enable us to settle thetechnical and theoretical accuracy of the generators, it isworth discussing the impact of various sources of radiativecorrections implemented in the programs used in the ex-perimental analysis. This allows one to understand whichcorrections are strictly necessary to achieve a precision atthe per mill level for both the calculation of integratedcross sections and the simulation of more exclusive distri-butions. The first set of phenomenological results about radiativecorrections refer to the Bhabha cross section, as obtainedby means of the code BabaYaga@NLO, according to dif-ferent perturbative and precision levels. In Table 4 weshow the values for the Born cross section σ , the O ( α )PS and exact cross section, σ PS α and σ NLO α , respectively,as well as the LL PS cross section σ PS and the matchedcross section σ matched . Furthermore, the cross section inthe presence of the vacuum polarisation correction, σ VP0 ,is also shown. The results correspond to the c.m. ener-gies √ s = 1 , , 10 GeV and were obtained with the se-lection criteria of Eq. (15), but for an angular acceptance Table 4. Bhabha cross section (in nb) at meson factoriesaccording to different precision levels and using the cuts ofEq. (15), but with an angular acceptance of 55 ◦ ≤ θ ± ≤ ◦ .The numbers in parentheses are 1 σ MC errors. √ s (GeV) 1.02 4 10 σ . . . σ VP0 . . . σ NLO . 523 (6) 37 . . σ PS α . 503 (6) 37 . . σ matched . 858 (5) 37 . . σ PS . 437 (4) 37 . . of 55 ◦ ≤ θ ± ≤ ◦ resembling realistic data taking atmeson factories. One should keep in mind that the cutsof Eq. (15) tend to single out quasi-elastic Bhabha eventsand that the energy of final state electron/positron cor-responds to a so-called “bare” event selection (i.e. with-out photon recombination), which corresponds to whatis done in practice at flavour factories. In particular therather stringent energy and acollinearity cuts enhance theimpact of soft and collinear radiation with respect to amore inclusive setup.From these cross section values, it is possible to cal-culate the relative effect of various corrections, namelythe contribution of vacuum polarisation and exact O ( α )QED corrections, of non-logarithmic (NLL) terms enter-ing the O ( α ) cross section, of HO corrections in the O ( α )matched PS scheme, and finally of NNL effects beyondorder α largely dominated by O ( α L ) contributions. Theabove corrections are shown in Table 5 in per cent andcan be derived from the cross section results of Table 4with the following definitions: δ VP ≡ σ VP0 − σ σ , δ α ≡ σ NLO − σ σ ,δ NLL α ≡ σ NLO − σ PS α σ NLO , δ HO ≡ σ matched − σ NLO σ NLO ,δ α L ≡ σ matched − σ NLO − σ PS + σ PS α σ NLO . From Table 5 it can be seen that O ( α ) correctionsdecrease the Bhabha cross section by about 15 ÷ 17% atthe φ and τ -charm factories, and by about 20% at the B factories. Within the full set of O ( α ) corrections, non-logarithmic terms are of the order of 0.5%, as expectedalmost independent of the c.m. energy, and with a milddependence on the angular acceptance cuts due to box andinterference contributions. The effect of HO correctionsdue to multiple photon emission is about 1% at the φ and τ -charm factories and reaches about 2% at the B factories.The contribution of (approximate) O ( α L ) corrections isat the 0.1% level, while vacuum polarisation increases thecross section by about 2% around 1 GeV, and by about5% and 6% at 4 GeV and 10 GeV, respectively. Concern-ing the latter correction the non-perturbative hadroniccontribution to the running of α was parameterised in terms of the HADR5N routine [259,260,18] included inBabaYaga@NLO both in the LO and NLO diagrams. Wehave checked that the results obtained for the vacuumpolarisation correction in terms of the parametrisation[164] agree at the 10 − level with those obtained withHADR5N, as shown in detail in Section 2.8. Those rou-tines return a data driven error, thus affecting the the-oretical precision of the calculation of the Bhabha crosscross section as will be discussed in Section 2.9.Analogous results for the size of radiative correctionsto the process e + e − → γγ are given in Table 6 [261].They were obtained using BabaYaga@NLO, according tothe experimental cuts of Eq. (16) for the c.m. energies √ s = 1 , , 10 GeV. Table 5. Relative size of different sources of corrections (inper cent) to the large-angle Bhabha cross section for typicalselection cuts at φ , τ -charm and B factories. √ s (GeV) 1.02 4. 10. δ α − . − . − . δ NLL α − . − . − . δ HO . 97 1 . 35 1 . δ α L . 09 0 . 09 0 . δ VP . 43 4 . 46 6 . Table 6. Photon pair production cross sections (in nb) at dif-ferent accuracy levels and relative corrections (in per cent) forthe setup of Eq. (16) and the c.m. energies √ s = 1 , , 10 GeV. √ s (GeV) 1 3 10 σ . 53 15 . 281 1 . σ NLO . 45 14 . 211 1 . σ PS α . 55 14 . 111 1 . σ matched . 77 14 . 263 1 . σ PS . 92 14 . 169 1 . δ α − . − . − . δ NLL α . 70 0 . 71 0 . δ HO . 24 0 . 37 0 . The numerical errors coming from the MC integrationare not shown in Table 5 because they are beyond thequoted digits. From Table 5 it can be seen that the exact O ( α ) corrections lower the Born cross section by about5 . 9% (at the φ resonance), 7 . 0% (at √ s = 3 GeV) and8 . 2% (at the Υ resonance). The effect due to O ( α n L n )(with n ≥ 2) terms is quantified by the contribution δ HO ,which is a positive correction of about 0 . 2% (at the φ resonance), 0 . 4% ( τ -charm factories) and 0 . 5% (at the Υ resonance), and therefore important in the light of the permill accuracy aimed at. On the other hand, also next-to-leading O ( α ) corrections, quantified by the contribution δ NLL α , are necessary at the precision level of 0.1%, sincetheir contribution is of about 0 . 7% almost independent d σ d M e + e − ( pb / G e V ) M e + e − (GeV) NEWOLD O ( α )012345670.8 0.85 0.9 0.95 1 δ ( % ) M e + e − OLD − NEWNEW × Fig. 16. Invariant mass distribution of the Bhabha process atKLOE, according to BabaYaga v3.5 (OLD), BabaYaga@NLO(NEW) and an exact NLO calculation. The inset shows therelative effect of NLO corrections, given by the difference ofBabaYaga v3.5 and BabaYaga@NLO predictions. From [235]. of the c.m. energy. To further corroborate the precisionreached in the cross section calculation of e + e − → γγ , wealso evaluated the effect due to the most important sub-leading O ( α ) photonic corrections given by order α L contributions. It turns out that the effect due to O ( α L )corrections does not exceed the 0.05% level. Obviously, thecontribution of vacuum polarisation is absent in γγ pro-duction. This is an advantage for particularly precise pre-dictions, as the uncertainty associated with the hadronicpart of vacuum polarisation does not affect the cross sec-tion calculation. Besides the integrated cross section, various differentialcross sections are used by the experimentalists to monitorthe collider luminosity. In Figs. 16 and 17 we show twodistributions which are particularly sensitive to the de-tails of photon radiation, i.e. the e + e − invariant mass andacollinearity distribution, in order to quantify the size ofNLO and HO corrections. The distributions are obtainedaccording to the exact O ( α ) calculation and with the twoBabaYaga versions, BabaYaga v3.5 and [email protected] Figs. 16 and 17 it can be clearly seen that multiplephoton corrections introduce significant deviations withrespect to an O ( α ) simulation, especially in the hard tailsof the distributions where they amount to several per cent.To make the contribution of exact O ( α ) non-logarithmicterms clearly visible, the inset shows the relative differ-ences between the predictions of BabaYaga v3.5 (denotedas OLD) and BabaYaga@NLO (denoted as NEW). Actu-ally, as discussed in Section 2.4.2, these differences mainlycome from non-logarithmic NLO contributions and to asmaller extent from O ( α L ) terms. Their effect is flat andat the level of 0.5% for the acollinearity distribution, whilethey reach the several per cent level in the hard tail of theinvariant mass distribution. d σ d ξ ( pb / d e g ) ξ (deg) NEWOLD O ( α ) 0.40.50.60.70.8-1 0 1 2 3 4 5 6 7 8 9 10 δ ( % ) ξ (deg) OLD − NEWNEW × Fig. 17. Acollinearity distribution of the Bhabha pro-cess at KLOE, according to BabaYaga v3.5 (OLD) andBabaYaga@NLO (NEW). The inset shows the relative effectof NLO corrections, given by the difference of BabaYaga v3.5and BabaYaga@NLO predictions. From [235]. -50-40-30-20-1001020 0 2 4 6 8 10 δ ( % ) ξ (deg.) σ ∞ − σ α σ ∞ × σ ∞ − σ α σ ∞ × Fig. 18. Relative effect of HO corrections α L and α n L n ( n ≥ 3) to the acollinearity distribution of the Bhabha processat KLOE. From [235]. It is also worth noticing that LL radiative correctionsbeyond α can be quite important for accurate simula-tions, at least when considering differential distributions.This means that even with a complete NNLO calculationat hand it would be desirable to match such correctionswith the resummation of all the remaining LL effects. InFig. 18, the relative effect of HO corrections beyond α dominated by the α contributions (dashed line) is shownin comparison with that of the α corrections (solid line)on the acollinearity distribution for the Bhabha processat DAΦNE. As can be seen, the α effect can be as largeas 10% in the phase space region of soft photon emission,corresponding to small acollinearity angles with almostback-to-back final state fermions.Concerning the process e + e − → γγ we show in Fig. 19the energy distribution of the most energetic photon, whilethe acollinearity distribution of the two most energeticphotons is represented in Fig. 20. The distributions referto exact O ( α ) corrections matched with the PS algorithm(solid line), to the exact NLO calculation (dashed line) d σ d E ( nb / G e V ) E (GeV)-20-10010200.4 0.42 0.44 0.46 0.48 0.5 δ ( % ) E (GeV) exp O ( α ) BabaYaga δ exp δ NLL ∞ Fig. 19. Energy distribution of the most energetic photonin the process e + e − → γγ , according to the PS matchedwith O ( α ) corrections denoted as exp (solid line), the exact O ( α ) calculation (dashed line) and the pure all-order PS asin BabaYaga v3.5 (dashed-dotted line). lnset: relative effect(in per cent) of multiple photon corrections (solid line) andof non-logarithmic contributions of the matched PS algorithm(dashed line). From [261]. and to all-order pure PS predictions of BabaYaga v3.5(dashed-dotted line). In the inset of each plot, the rel-ative effect due to multiple photon contributions ( δ HO )and non-logarithmic terms entering the improved PS al-gorithm ( δ NLL α ) is also shown, according to the definitionsgiven in Eq. (83).For the energy distribution of the most energetic pho-ton particularly pronounced effects due to exponentiationare present. In the statistically dominant region, HO cor-rections reduce the O ( α ) distribution by about 20%, whilethey give rise to a significant hard tail close to the energythreshold of 0.3 √ s as a consequence of the higher pho-ton multiplicity of the resummed calculation with respectto the fixed-order NLO prediction. Needless to say, therelative effect of multiple photon corrections below about0.46 GeV not shown in the inset is finite but huge. Thisrepresentation with the inset was chosen to make also thecontribution of O ( α ) non-logarithmic terms visible, whichotherwise would be hardly seen in comparison with themultiple photon corrections. Concerning the acollinearitydistribution, the contribution of higher-order correctionsis positive and of about 10% for quasi-back-to-back photonevents, whereas it is negative and decreasing from ∼ − ∼ − 10% for increasing acollinearity values. As far asthe contributions of non-logarithmic effects dominated bynext-to-leading O ( α ) corrections are concerned, they con-tribute at the level of several per mill for the acollinearitydistribution, while they lie in the range of several per centfor the energy distribution.As a whole, the results of the present Section empha-sise that, for a 0.1% theoretical precision in the calculationof both the cross sections and distributions, both exact O ( α ) and HO photonic corrections are necessary, as wellas the running of α . d σ d ξ ( nb / d e g ) ξ (deg)-30-20-1001020 0 2 4 6 8 10 δ ( % ) ξ (deg) exp O ( α ) BabaYaga δ exp δ NLL ∞ Fig. 20. Acollinearity distribution for the process e + e − → γγ ,according to the PS matched with O ( α ) corrections denotedas exp (solid line), the exact O ( α ) calculation (dashed line)and the pure all-order PS as in BabaYaga v3.5 (dashed-dottedline). lnset: relative effect (in per cent) of multiple photon cor-rections (solid line) and of non-logarithmic contributions of thematched PS algorithm (dashed line). From [261]. The typical procedure followed in the literature to estab-lish the technical precision of the theoretical tools is toperform tuned comparisons between the predictions of in-dependent programs using the same set of input parame-ters and experimental cuts. This strategy was initiated inthe 90s during the CERN workshops for precision physicsat LEP and is still in use when considering processes ofinterest for physics at hadron colliders demanding partic-ularly accurate theoretical calculations. The tuning proce-dure is a key step in the validation of generators, becauseit allows to check that the different details entering thecomplex structure of the generators, e.g. the implementa-tion of radiative corrections, event selection routines, MCintegration and event generation, are under control, andto fix possible mistakes.The tuned comparisons discussed in the following wereperformed switching off the vacuum polarisation correc-tion to the Bhabha scattering cross section. Actually, thegenerators implement the non-perturbative hadronic con-tribution to the running of α according to different pa-rameterisations, which differently affect the cross sectionprediction (see Section 6 for discussion). Hence, this sim-plification is introduced to avoid possible bias in the in-terpretation of the results and allows to disentangle theeffect of pure QED corrections. Also, in order to provideuseful results for the experiments, the comparisons takeinto account realistic event selection cuts.The present Section is a merge of results available inthe literature [235] with those of new studies. The resultsrefer to the Bhabha process at the energies of φ , τ -charm Table 7. Cross section predictions [nb] of BabaYaga@NLOand BHWIDE for the Bhabha cross section corresponding totwo different angular acceptances, for the KLOE experimentat DAΦNE, and their relative differences (in per cent).angular acceptance BabaYaga@NLO BHWIDE δ (%)20 ◦ ÷ ◦ ◦ ÷ ◦ and B factories. No tuned comparisons for the two photonproduction process have been carried out. φ and τ -charm factories First we show comparisons between BabaYaga@NLO andBHWIDE according to the KLOE selection cuts of Eq. (15),considering also the angular range 20 ◦ ≤ ϑ ± ≤ ◦ forcross section results. The predictions of the two codes arereported in Table 7 for the two acceptance cuts togetherwith their relative deviations. As can be seen the agree-ment is excellent, the relative deviations being well belowthe 0.1%. Comparisons between BabaYaga@NLO and BH-WIDE at the level of differential distributions are given inFigs. 21 and 22 where the inset shows the relative devi-ations between the predictions of the two codes. As canbe seen there is very good agreement between the twogenerators, and the predicted distributions appear at afirst sight almost indistinguishable. Looking in more de-tail, there is a relative difference of a few per mill for theacollinearity distribution (Fig. 22) and of a few per centfor the invariant mass (Fig. 21), but only in the very hardtails, where the fluctuations observed are due to limitedMC statistics. These configurations however give a negli-gible contribution to the integrated cross section, a factor10 ÷ smaller than that around the very dominantpeak regions. In fact these differences on differential dis-tributions translate into agreement on the cross sectionvalues well below the one per mill, as shown in Table 7.Similar tuned comparisons were performed betweenthe results of BabaYaga@NLO, BHWIDE and MCGPJin the presence of cuts modelling the event selection cri-teria of the CMD-2 experiment at the VEPP-2M collider,for a c.m. energy of √ s = 900 MeV. The cuts used in thiscase are | θ − + θ + − π | ≤ ∆θ, . ≤ ( θ + − θ − + π ) / ≤ π − . , || φ − + φ + | − π | ≤ . ,p − sin( θ − ) ≥ 90 MeV , p + sin( θ + ) ≥ 90 MeV , ( p − + p + ) / ≥ 90 MeV , (83)where θ − , θ + are the electron/positron polar angles, re-spectively, φ ± their azimuthal angles, and p ± the moduliof their three-momenta. ∆θ stands for an acollinearity cut.Figure 23 shows the relative differences between theresults of BHWIDE and MCGPJ according to the criteriaof Eq. (83), as a function of the acollinearity cut ∆θ . The Table 8. Cross section predictions [nb] of BabaYaga@NLOand MCGPJ for the Bhabha cross section at τ -charm factories( √ s = 3 . δ (%)35.20(2) 35.181(5) 0.06 d σ d M e + e − ( pb / G e V ) M e + e − (GeV) BABAYAGABHWIDE -1012340.8 0.85 0.9 0.95 1 δ ( % ) M e + e − BHWIDE − BABAYAGABABAYAGA × Fig. 21. Invariant mass distribution of the Bhabha processaccording to BHWIDE and BabaYaga@NLO, for the KLOEexperiment at DAΦNE, and relative differences of the programpredictions (inset). From [235]. relative deviations between the results of BabaYaga@NLOand MCGPJ for the same cuts are given in Fig. 24. It canbe seen that the predictions of the three generators liewithin a 0 . 2% band with differences of ∼ . 3% for ex-treme values of the acollinearity cut. This agreement canbe considered satisfactory since for the acollinearity cut ofreal experimental interest ( ∆θ ≈ . τ -charm factories. An example is given in Table 8 wherethe predictions of BabaYaga@NLO and MCGPJ are com-pared, using cuts similar to those of Eq. (83) and for anacollinearity cut of ∆θ = 0 . 25 rad. The agreement betweenthe two codes is below one per mill. Comparisons betweenthe two codes were also done at the level of differentialcross sections, showing satisfactory agreement in the sta-tistically relevant phase space regions. Preliminary results[262] for a c.m. energy on top of the J/Ψ resonance showgood agreement between BabaYaga@NLO and BHWIDEpredictions too. B factories Concerning the B factories, a considerable effort was doneto establish the level of agreement between the genera-tors BabaYaga@NLO and BHWIDE in comparison withBabaYaga v3.5 too. This study made use of the realistic lu-minosity cuts quoted in Section 2.3.3 for the BaBar exper-iment. The cross sections predicted by BabaYaga@NLOand BHWIDE are shown in Table 9, together with the d σ d ξ ( pb / d e g ) ξ (deg) BABAYAGABHWIDE -0.200.20.40.60.81-1 0 1 2 3 4 5 6 7 8 9 10 δ ( % ) ξ (deg) BHWIDE − BABAYAGABABAYAGA × Fig. 22. Acollinearity distribution of the Bhabha process ac-cording to BHWIDE and BabaYaga@NLO, for the KLOE ex-periment at DAΦNE, and relative differences of the programpredictions (inset). From [235]. , rad θ∆ , % M C G P J σ ) / M C G P J σ - B H W I D E σ ( -0.4-0.3-0.2-0.100.10.20.30.4 Fig. 23. Relative differences between BHWIDE and MCGPJBhabha cross sections as a function of the acollinearity cut, forthe CMD-2 experiment at VEPP-2M. corresponding relative differences as a function of the con-sidered angular range. The latter are also shown in Fig. 25,where the 1 σ numerical error due to MC statistics is alsoquoted. As can be seen, the two codes agree nicely, thepredictions for the central value being in general in agree-ment at the 0.1% level or statistically compatible when-ever a two to three per mill difference is present.To further investigate how the two generators comparewith each other a number of differential cross sections werestudied. The results of this study are shown in Figs. 26 and27 for the distribution of the electron energy and the polarangle, respectively, and in Fig. 28 for the acollinearity. Forboth the energy and scattering angle distribution, the twoprograms agree within the statistical errors showing de-viations below 0.5%. For the acollinearity dependence ofthe cross section, BabaYaga@NLO and BHWIDE agree , rad θ∆ , % M C G P J σ ) / M C G P J σ - B a b a Y a g a @ N L O σ ( -0.4-0.3-0.2-0.100.10.20.30.4 Fig. 24. Relative differences between BabaYaga@NLO andMCGPJ Bhabha cross sections as a function of the acollinearitycut, for the CMD-2 experiment at VEPP-2M. Table 9. Cross section predictions [nb] of BabaYaga@NLOand BHWIDE for the Bhabha cross section as a function ofthe angular selection cuts for the BaBar experiment at PEP-IIand absolute value of their relative differences (in per cent).angular range (c.m.s.) BabaYaga@NLO BHWIDE | δ (%) | ◦ ÷ ◦ ◦ ÷ ◦ ◦ ÷ ◦ ◦ ÷ ◦ ◦ ÷ ◦ ◦ ÷ ◦ ◦ ÷ ◦ within ∼ φ factories.The main conclusions emerging from the tuned com-parisons discussed in the present Section can be sum-marised as follows: – The predictions for the Bhabha cross section of themost precise tools, i.e. BabaYaga@NLO, BHWIDE andMCGPJ, generally agree within 0.1%. If (slightly) lar-ger differences are present they show up for particu-larly tight cuts or are due to limited MC statistics.When statistically meaningful discrepancies are ob-served they can be ascribed to the different theoret-ical recipes for the treatment of radiative correctionsand their technical implementation. For example, asalready emphasised, BabaYaga@NLO and BHWIDEadopt a fully factorised prescription for the matchingof NLO and HO corrections, whereas MCGPJ imple-ment some pieces of the radiative corrections in addi- angular range (from x to 180-x degrees) 10 20 30 40 50 60 70 80 r e l. d i ff e r e n ce i n p e r ce n t -0.8-0.6-0.4-0.200.20.4 Fig. 25. Relative differences between BabaYaga@NLO andBHWIDE Bhabha cross sections as a function of the angularacceptance cut for the BaBar experiment at PEP-II. From [50]. [ GeV ] - e E [ nb / . G e V ] d E σ d -1 BHWIDE [ GeV ] - e E [ nb / . G e V ] d E σ d -1 Babayaga@NLO [ GeV ] - e E [ nb / . G e V ] d E σ d -1 Babayaga.3.5 [ GeV ] - e E d i ff e r e n ce i n p e r ce n t / . G e V -80-70-60-50-40-30-20-100 relative difference ± BHWIDEBHWIDE - Babayaga@NLO BHWIDEBHWIDE - Babayaga.3.5 [ GeV ] - e E d i ff e r e n ce i n p e r ce n t / . G e V zoom in Fig. 26. Electron energy distributions according to BHWIDE,BabaYaga@NLO and BabaYaga v3.5 for the BaBar experimentat PEP-II and relative differences of the predictions of theprograms. From [50]. tive form. This can give rise to discrepancies betweenthe programs’ predictions, especially in the presence oftight cuts enhancing the effect of soft radiation. Fur-thermore, different choices are adopted in the genera-tors for the scale entering the collinear logarithms inHO corrections beyond O ( α ), which are another pos-sible source of the observed differences. To go beyondthe present situation, a further nontrivial effort shouldbe done by comparing, for instance, the programs inthe presence of NLO corrections only (technical test)and by analysing their different treatment of the expo-nentiation of soft and collinear logarithms. This wouldcertainly shed light on the origin of the (small) dis-crepancies still registered at present. Fig. 27. Electron polar angle distributions according to BH-WIDE, BabaYaga@NLO and BabaYaga v3.5 for the BaBar ex-periment at PEP-II and relative differences of the predictionsof the programs. From [50]. ] o acol [ ] o [ nb / d ( ac o l ) σ d -3 -2 -1 BHWIDE ] o acol [ ] o [ nb / d ( ac o l ) σ d -3 -2 -1 Babayaga@NLO ] o acol [ ] o [ nb / d ( ac o l ) σ d -3 -2 -1 Babayaga.3.5 ] o acol [ o d i ff e r e n ce i n p e r ce n t / -400-300-200-1000100 relative difference ± BHWIDEBHWIDE - Babayaga@NLOBHWIDEBHWIDE - Babayaga.3.5 ] o acol [ 150 155 160 165 170 175 180 o d i ff e r e n ce i n p e r ce n t / -10-8-6-4-202 zoom in Fig. 28. Acollinearity distributions according to BHWIDE,BabaYaga@NLO and BabaYaga v3.5 for the BaBar experimentat PEP-II and relative differences of the predictions of theprograms. From [50].2 – Also the distributions predicted by the generators agreewell, with relative differences below the 1% level. Slight-ly larger discrepancies are only seen in sparsely popu-lated phase space regions corresponding to very hardphoton emission which do not influence the luminositymeasurement noticeably. As discussed in Section 2.1, the total luminosity errorcrucially depends on the theoretical accuracy of the MCprograms used by the experimentalists. As emphasisedin Section 2.5, some of these generators like BHAGENF,BabaYaga v3.5 and BKQED miss theoretical ingredientswhich are unavoidable for cross section calculations witha precision at the per mill level. Therefore, they are in-adequate for a highly accurate luminosity determination.BabaYaga@NLO, BHWIDE and MCGPJ include, how-ever, both NLO and multiple photon corrections, and theiraccuracy aims at a precision tag of 0.1%. But also thesegenerators are affected by uncertainties which must becarefully considered in the light of the very stringent crite-ria of per mill accuracy. The most important componentsof the theoretical error of BabaYaga@NLO, BHWIDE andMCGPJ are mainly due to approximate or partially in-cluded pieces of radiative corrections and come from thefollowing sources:1. The non-perturbative hadronic contributions to therunning of α . It can be reliably evaluated only usingthe data of the hadron cross section at low energies.Hence, the vacuum polarisation correction receives adata driven error which affects in turn the predictionof the Bhabha cross section, as emphasised in Section6.2. The complete set of O ( α ) QED corrections. In spiteof the impressive progress in this area, as reviewed inSection 2.3, an important piece of NNLO corrections,i.e. the exact NLO SV QED corrections to the sin-gle hard bremsstrahlung process e + e − → e + e − γ , isstill missing for the full s + t Bhabha process. How-ever, partial results obtained for t -channel small-angleBhabha scattering [263,47] and large-angle annihila-tion processes are available [264,265].3. The O ( α ) contribution due to real and virtual (leptonand hadron) pairs. The virtual contributions originatefrom the NNLO electron, heavy flavour and hadronicloop corrections discussed in Section 2.3, while the realcorrections are due to the conversion of an externalphoton into pairs. The latter, as discussed in Section2.3.3, gives rise to a final state with four particles, twoof which to be considered as undetected to contributeto the Bhabha signature.The uncertainty relative to the first point can be esti-mated by using the routines available in the literature for As already remarked and further discussed in the follow-ing, the complete calculation of the NLO corrections to hardphoton emission in Bhabha scattering was performed duringthe completion of this report [101]. the calculation of the non-perturbative hadronic contribu-tion ∆α (5)hadr ( q ) to the running α . Actually these routinesreturn, in addition to ∆α (5)hadr ( q ), an error δ hadr on itsvalue. Therefore an estimate of the induced error can besimply obtained by computing the Bhabha cross sectionwith ∆α (5)hadr ( q ) ± δ hadr and taking the difference as thetheoretical uncertainty due to the hadronic contributionto vacuum polarisation. In Table 10, the Bhabha cross sec-tions, as obtained in the presence of the vacuum polarisa-tion correction according to the parameterisations of [259,260,18] (denoted as J) and of [164] (denoted as HMNT),respectively, are shown for φ , τ -charm and B factories.The applied angular cuts refer to the typically adoptedacceptance 55 ◦ ≤ θ ± ≤ ◦ . Table 10. Bhabha scattering cross section in the presenceof the vacuum polarisation correction, according to [259,260,18] (J) and [164] (HMNT), at meson factories. The notationJ − /HMNT − , J/HMNT and J + /HMNT + indicates minimum,central and maximum value of the two parametrisations.Parametrisation φ τ -charm B J − + − + From Table 10 it can be seen that the two treatmentsof ∆α (5)hadr ( q ) induce effects on the Bhabha cross sectionin very good agreement, the relative differences betweenthe central values being 0.05% ( φ factories), 0.005% ( τ -charm factories) and 0.02% ( B factories). This can beunderstood in terms of the dominance of t -channel ex-change for large-angle Bhabha scattering at meson fac-tories. Indeed, the two routines provide results in excel-lent agreement for space-like momenta, as we explicitlychecked, whereas differences in the predictions show upfor time-like momenta which, however, contribute onlymarginally to the Bhabha cross section. Also the spreadbetween the minimum/maximum values and the centralone as returned by the two routines agrees rather well, alsoa consequence of the dominance of t -channel exchange.This spread amounts to a few units in 10 − and is pre-sented in detail in Table 11 in the next Section.Concerning the second point a general strategy to eval-uate the size of missing NNLO corrections consists in de-riving a cross section expansion up to O ( α ) from thetheoretical formulation implemented in the generator ofinterest. It can be cast in general into the following form σ α = σ α SV + σ α SV , H + σ α HH , (84)where in principle each of the O ( α ) contributions is af-fected by an uncertainty to be properly estimated. In Eq. (84)the first contribution is the cross section including O ( α ) SV corrections, whose uncertainty can be evaluated througha comparison with some of the available NNLO calcula-tions reviewed in Section 2.3. In particular, in [235] the σ α SV of the BabaYaga@NLO generator was compared withthe calculation of photonic corrections by Penin [135,136]and the calculations by Bonciani et al. [140,141,151,152,153] who computed two-loop fermionic corrections (in theone-family approximation N F = 1) with finite mass termsand the addition of soft bremsstrahlung and real pair con-tributions. The results of such comparisons are shownin Figs. 29 and 30 for realistic cuts at the φ factories. InFig. 29 δσ is the difference between σ α SV of BabaYaga@NLOand the cross sections of the two O ( α ) calculations, de-noted as photonic (Penin) and N F = 1 (Bonciani et al. ),as a function of the logarithm of the infrared regulator ǫ . Itcan be seen that the differences are given by flat functions,demonstrating that such differences are infrared-safe, asexpected, a consequence of the universality and factori-sation properties of the infrared divergences. In Fig. 30, δσ is shown as a function of the logarithm of a fictitiouselectron mass and for a fixed value of ǫ = 10 − . Sincethe difference with the calculation by Penin is given by astraight line, this indicates that the soft plus virtual two-loop photonic corrections missing in BabaYaga@NLO are O ( α L ) contributions, as already remarked. On the otherhand, the difference with the calculation by Bonciani et al. is fitted by a quadratic function, showing that the electrontwo-loop effects missing in BabaYaga@NLO are of the or-der of α L . However, it is important to emphasise that,as shown in detail in [235], the sum of the relative differ-ences with the two O ( α ) calculations does not exceed the2 × − level for experiments at φ and B factories.The second term in Eq. (84) is the cross section con-taining the one-loop corrections to single hard photonemission, and its uncertainty can be estimated by relyingon partial results existing in the literature. Actually theexact perturbative expression of σ α SV , H is not yet availablefor full s + t Bhabha scattering, but using the results validfor small-angle Bhabha scattering [263,47] and large-angleannihilation processes [264,265] the relative uncertainty ofthe theoretical tools in the calculation of σ α SV , H can be con-servatively estimated to be at the level of 0.05%. Indeedthe papers [263,47,264,265] show that a YFS matching ofNLO and HO corrections gives SV one-loop results for the t -channel process e + e − → e + e − γ and s -channel annihila-tion e + e − → f ¯ f γ ( f = fermion) differing from the exactperturbative calculations by a few units in 10 − at most.This conclusion also holds when photon energy cuts arevaried. It is worth noting that during the completion ofthe present work a complete calculation of the NLO QEDcorrections to hard bremsstrahlung emission in full s + t Bhabha scattering appeared in the literature [101], along To provide meaningful results, the contribution of the vac-uum polarisation was switched off in BabaYaga@NLO to com-pare with the calculation by Penin consistently. For the samereason the real soft and some pieces of virtual electron pair cor-rections were neglected in the comparison with the calculationby Bonciani et al. -1-0.500.51 1e-16 1e-14 1e-12 1e-10 1e-08 1e-06 δ σ ( nb ) ε NF=1photonicfitfit Fig. 29. Absolute differences (in nb) between the σ α SV pre-diction of BabaYaga@NLO and the NNLO calculations of thephotonic corrections [135,136] (photonic) and of the electronloop corrections [140,141,151,152,153] ( N F = 1) as a functionof the infrared regulator ǫ for typical KLOE cuts. From [235]. -5-4-3-2-10123451e-10 1e-09 1e-08 1e-07 1e-06 1e-05 1e-04 0.001 0.01 δ σ ( nb ) m e (GeV) NF=1photonicfitfit Fig. 30. Absolute differences (in nb) between the σ α SV pre-diction of BabaYaga@NLO and the NNLO calculations of thephotonic corrections [135,136] (photonic) and of the electronloop corrections [140,141,151,152,153] ( N F = 1) as a functionof a fictitious electron mass for typical KLOE cuts. From [235]. the lines described in Section 2.3.2. Explicit comparisonsbetween the results of such an exact calculation with thepredictions of the most accurate MC tools according tothe typical luminosity cuts used at meson factories wouldbe worthwhile to make the present error estimate relatedto the calculation of σ α SV , H more robust.The third contribution in Eq. (84) is the double hardbremsstrahlung cross section whose uncertainty can bedirectly evaluated by explicit comparison with the exact e + e − → e + e − γγ cross section. It was shown in [235] thatthe differences between σ α HH as in BabaYaga@NLO andthe matrix element calculation, which exactly describesthe contribution of two hard photons, are really negligi-ble, being at the 10 − level.The relative effect due to lepton ( e, µ, τ ) and hadron( π ) pairs has been numerically analysed in Section 2.3.3,in the presence of realistic selection cuts. This evalua-tion makes use of the complete NNLO virtual corrections combined with an exact matrix element calculation ofthe four-particle production processes. It supersedes previ-ous approximate estimates which underestimated the im-pact of those corrections. According to this new evalua-tion, the pair contribution, dominated by the electron paircorrection, amounts to about 0.3% for KLOE and 0.1%for BaBar. These contributions are partially included inthe BabaYaga@NLO code, as well as in other generators,through the insertion of the vacuum polarisation correc-tion in the NLO diagrams, and detailed comparisons be-tween the exact calculation and the BabaYaga@NLO pre-dictions are in progress [266]. During the last few years a remarkable progress occurredin reducing the error of the luminosity measurements atflavour factories.Dedicated event generators like BabaYaga@NLO andMCGPJ were developed in 2006 to provide predictionsfor the cross section of the large-angle Bhabha process, aswell as for other QED reactions of interest, with a theoret-ical accuracy at the level of 0.1%. In parallel codes well-known since the time of the LEP/SLC operation such asBHWIDE were extensively used by the experimentalistsin data analyses. All these MC programs include, albeitaccording to different formulations, exact O ( α ) QED cor-rections matched with LL contributions describing multi-ple photon emission. Such ingredients, together with thevacuum polarisation correction, are strictly necessary toachieve a physical precision down to the per mill level.Indeed, when considering typical selection, cuts the NLOphotonic corrections amount to about 15 ÷ ÷ φ -resonance,where the statistics of Bhabha events is the highest andthe experimental luminosity error at a few per mill level,the cross section results of BabaYaga@NLO, BHWIDEand MCGPJ agree within ∼ . τ -charm and B factories. The main conclusion of the work on tunedcomparisons is that the technical precision of MC pro-grams is well under control, the discrepancies being dueto different details in the treatment of the same sourcesof radiative corrections and their technical implementa-tion. For example, BabaYaga@NLO and BHWIDE adopta fully factorised prescription for the matching of NLO andHO corrections, whereas MCGPJ implement some radia-tive corrections pieces in additive form. This can give riseto some discrepancies between their predictions, especiallyin the presence of tight cuts enhancing the effect of soft ra-diation. Furthermore, different choices are adopted in thegenerators for the energy scale in the treatment of HO cor-rections beyond O ( α ), which are another possible sourceof the observed differences. To go beyond the present sit-uation, a further, nontrivial effort should be done by com-paring, for instance, the programs in the presence of NLOcorrections only (technical test) and for the specific effectdue to the exponentiation of soft and collinear logarithms.This would certainly shed light on the origin of the (mi-nor) discrepancies still registered at present.On the theoretical side, a new exact evaluation of lep-ton and hadron pair corrections to the Bhabha scatteringcross section was carried out, taking into account realisticcuts. This calculation provides results in substantial agree-ment with estimates based on singlet SF but supersedesprevious evaluations in the soft-photon approximation.The results of the new exact calculation were preliminarilycompared with the predictions of BabaYaga@NLO, whichincludes the bulk of such corrections (due to reduciblecontributions) through the insertion of the vacuum polar-isation correction in the NLO diagrams, but neglects theeffect of real pair radiation and two-loop form factors. Itturns out that the error induced by the approximate treat-ment of pair corrections amounts to a few units in 10 − ,both at KLOE and BaBar. Further work is in progress toarrive at a more solid and quantitative error estimate forthese corrections when considering other selection criteriaand c.m. energies too [266]. Also, the contribution inducedby the uncertainty related to the non-perturbative contri-bution to the running of α was revisited, making use ofand comparing the two independent parameterisations de-rived in [259,260,18] and [164].A summary of the different sources of theoretical er-ror and their relative impact on the Bhabha cross sectionis given in Table 11. In Table 11, | δ errVP | is the error in-duced by the hadronic component of the vacuum polar-isation, | δ errpairs | the error due to missing pair corrections, | δ errSV | the uncertainty coming from SV NNLO corrections, | δ errHH | the uncertainty in the calculation of the double hardbremsstrahlung process and | δ errSV , H | the error estimate forone-loop corrections to single hard bremsstrahlung. As canbe seen, pair corrections and exact NLO corrections to e + e − → e + e − γ are the dominant sources of error.The total theoretical uncertainty as obtained by sum-ming the different contributions linearly is 0.12 ÷ φ factories, 0.18% at the τ -charm factories and Table 11. Summary of different sources of theoretical uncer-tainty for the most precise generators used for luminosity mea-surements and the corresponding total theoretical errors for thecalculation of the large-angle Bhabha cross section at mesonfactories.Source of error (%) φ τ -charm B | δ errVP | [259,260,18] 0.00 0.01 0.03 | δ errVP | [164] 0.02 0.01 0.02 | δ errSV | | δ errHH | | δ errSV , H | | δ errpairs | | δ errtotal | ÷ ÷ . ÷ . 12% at the B factories. As can be seen from Ta-ble 11, the slightly larger uncertainty at the τ -charm fac-tories is mainly due to the pair contribution error, whichis presently based on a very preliminary evaluation andfor which a deeper analysis is ongoing [266]. The totaluncertainty is slightly affected by the particular choice ofthe routine for the calculation of ∆α (5)hadr ( q ), since thetwo parameterisations considered here give rise to simi-lar errors, with the exception of the φ factories for whichthe two recipes return uncertainties differing by 2 × − .However the “parametric” error induced by the hadroniccontribution to the vacuum polarisation may become a rel-evant source of uncertainty when considering predictionsfor a c.m. energy on top of and closely around very narrowresonances. For such a specific situation of interest, for in-stance for the BES experiment, the appropriate treatmentof the running α in the calculation of the Bhabha crosssection should be scrutinised deeper because of the differ-ences observed between the predictions for ∆α (5)hadr ( q ) ob-tained by means of the different parametrisation routinesavailable (see Section 6 for a more detailed discussion).Although the theoretical uncertainty quoted in Ta-ble 11 could be put on firmer ground thanks to furtherstudies in progress, it appears to be quite robust and suf-ficient for present and planned precision luminosity mea-surements at meson factories, where the experimental er-ror currently is about a factor of two or three larger.Adopting the strategy followed during the LEP/SLC op-eration one could arrive at a more aggressive error es-timate by summing the relative contributions in quadra-ture. However, for the time being, this does not seem to benecessary in the light of the current experimental errors.In conclusion, the precision presently reached by large-angle Bhabha programs used in the luminosity measure-ment at meson factories is comparable with that achievedabout ten years ago for luminosity monitoring throughsmall-angle Bhabha scattering at LEP/SLC.Some issues are still left open. In the context of tunedcomparisons, no effort was done to compare the availablecodes for the process of photon pair production. Since itcontributes relevantly to the luminosity determination andas precise predictions for its cross section can be obtainedby means of the codes BabaYaga@NLO and MCGPJ, this work should be definitely carried out. This would leadto a better understanding of the luminosity on the ex-perimental side. In the framework of new theoretical ad-vances, an evaluation of NNLO contributions to the pro-cess e + e − → γγ would be worthwhile to better assess theprecision of the generators which, for the time being, donot include such corrections exactly. More importantly,the exact one-loop corrections to the radiative process e + e − → e + e − γ should be calculated going beyond thepartial results scattered in the literature (and referring toselection criteria valid for high-energy e + e − colliders) orlimited to the soft-photon approximation. Furthermore,to get a better control of the theoretical uncertainty inthe sector of NNLO corrections to Bhabha scattering, theradiative Bhabha process at one-loop should be evaluatedtaking into account the typical experimental cuts used atmeson factories. Incidentally this calculation would be alsoof interest for other studies at e + e − colliders of moderatelyhigh energy, such as the search for new physics phenomena(e.g. dark matter candidates), for which radiative Bhabhascattering is a very important background. R measurement from energy scan In this section we will consider some theoretical and exper-imental aspects of the direct R measurement and relatedquantities in experiments with energy scan. As discussedin the Introduction, the cross section of e + e − annihilationinto hadrons is involved in evaluations of various problemsof particle physics and, in particular, in the definition ofthe hadronic contribution to vacuum polarisation, whichis crucial for the precision tests of the Standard Modeland searches for new physics.The ratio of the radiation-corrected hadronic cross sec-tions to the cross section for muon pair production, cal-culated in the lowest order, is usually denoted as (seeEq. (23)) R ≡ R ( s ) = σ ( s )4 πα / (3 s ) . (85)In the numerator of Eq. (85) one has to use the so called undressed hadronic cross section which does not includevacuum polarisation corrections.The value of R has been measured in many experi-ments in different energy regions from the pion pair pro-duction threshold up to the Z mass. Practically all electron-positron colliders contributed to the global data set on thehadronic annihilation cross section [267]. The value of R As already remarked in Section 2.8, during the completionof the present work a complete calculation of the NLO QEDcorrections to hard bremsstrahlung emission in full s + t Bhabhascattering was performed in [101]. However, explicit compar-isons between the predictions of this new calculation and thecorresponding results of the most precise luminosity tools arestill missing and would be needed to better assess the theoret-ical error induced by such contributions in the calculation ofthe luminosity cross section.6 extracted from the experimental data is then widely usedfor various QCD tests as well as for the calculation ofdispersion integrals. At high energies and away from res-onances, the experimentally determined values of R arein good agreement with predictions of perturbative QCD,confirming, in particular, the hypothesis of three colourdegrees of freedom for quarks. On the other hand, for thelow energy range the direct R measurement [267,268] atexperiments with energy scan is necessary. Matching be-tween the two regions is performed at energies of a fewGeV, where both approaches for the determination of R are in fair agreement.For the best possible compilation of R ( s ), data fromdifferent channels and different experiments have to becombined. For √ s ≤ R ( s ) measurementat low energies ( s < ), which, in turn, is domi-nated by the systematic error of the measured cross sec-tion e + e − → π + π − . Here we present the lowest-order expressions for the pro-cesses of electron-positron annihilation into pairs of muons,pions and kaons.For the muon production channel e − ( p − ) + e + ( p + ) → µ − ( p ′− ) + µ + ( p ′ + ) (86)within the Standard Model at Born level we haved σ µµ d Ω − = α β µ s (cid:0) − β µ (1 − c ) (cid:1) (1 + K µµW ) , (87) s = ( p − + p + ) = 4 ε , c = cos θ − , θ − = d p − p ′− , where β µ = q − m µ /ε is the muon velocity. Small termssuppressed by the factor m e /s are omitted. Here K µµW rep-resents contributions due to Z -boson intermediate statesincluding Z − γ interference, see, e.g., Refs. [270,271]: K µµW = s (2 − β µ (1 − c )) − ( s − M Z ) + M Z Γ Z (cid:26) (2 − β µ (1 − c )) × (cid:18) c v (cid:18) − M Z s (cid:19) + c a (cid:19) − − β µ c a + c v )+ cβ µ (cid:20) (cid:18) − M Z s (cid:19) c a + 8 c a c v (cid:21)(cid:27) ,c a = − 12 sin 2 θ W , c v = c a (1 − θ W ) , (88) Lattice QCD computations (see, e.g., Ref. [269]) of thehadronic vacuum polarisation are in progress, but they are notyet able to provide the required precision. where θ W is the weak mixing angle.The contribution of Z boson exchange is suppressed,in the energy range under consideration, by a factor s/M Z which reaches per mill level only at B factories.In the Born approximation the differential cross sec-tion of the process e + ( p + ) + e − ( p − ) → π + ( q + ) + π − ( q − ) (89)has the form d σ ππ d Ω ( s ) = α β π s sin θ | F π ( s ) | , (90) β π = p − m π /ε , θ = d p − q − . The pion form factor F π ( s ) takes into account non-pertur-bative virtual vertex corrections due to strong interac-tions [272,256]. We would like to emphasise that in theapproach under discussion the final state QED correctionsare not included into F π ( s ). The form factor is extractedfrom the experimental data on the same process as dis-cussed below.The annihilation process with three pions in the finalstate was considered in Refs. [273,274] including radia-tive corrections relevant to the energy region close to thethreshold. A stand-alone Monte Carlo event generator forthis channel is available [273]. The channel was also in-cluded in the MCGPJ generator [236] on the same footingas other processes under consideration in this report.In the case of K L K S meson pair production the differ-ential cross section in the Born approximation readsd σ ( s ) K L K S d Ω L = α β K s sin θ | F LS ( s ) | . (91)Here, as well as in the case of pion production, we as-sume that the form factor F LS also includes the vacuumpolarisation operator of the virtual photon. The quantity β K = p − m K /s is the K meson c.m.s. velocity, and θ is the angle between the directions of motion of the longliving kaon and the initial electron.In the case of K + K − meson production near thresh-old, the Sakharov-Sommerfeld factor for the Coulomb fi-nal state interaction should additionally be taken into ac-count:d σ ( s ) K + K − d Ω − = α β K s sin θ | F K ( s ) | Z − exp( − Z ) ,Z = 2 παv , v = 2 r s − m K s (cid:18) s − m K s (cid:19) − , (92)where v is the relative velocity of the kaons [275] whichhas the proper non-relativistic and ultra-relativistic lim-its. When s = m φ , we have v ≈ . One-loop radiative corrections (RC) for the processes (86,89)can be separated into two natural parts according to theparity with respect to the substitution c → − c . The c -even part of the one-loop soft and virtual contri-bution to the muon pair creation channel can be combinedfrom the well known Dirac and Pauli form factors and thesoft photon contributions. It readsd σ B + S + Vµµ − even d Ω = d σ µµ d Ω | − Π ( s ) | ( απ "(cid:20) L − 2+ 1 + β µ β µ l β (cid:21) ln ∆εε + 34 ( L − 1) + K µµ even , (93) K µµ even = π − 54 + ρ (cid:18) β µ β µ − 12 + 14 β µ (cid:19) + ln 1 + β µ β µ + 1 + β µ β µ ! (94) − − β µ β µ l β − β µ (1 − c )+ 1 + β µ β µ (cid:20) π (cid:18) − β µ β µ (cid:19) + ρ ln 1 + β µ β µ + 2 ln 1 + β µ β µ β µ (cid:21) ,l β = ln 1 + β µ − β µ , ρ = ln sm µ L = ln sm e , where Li ( z ) = − R z dt ln(1 − t ) /t is the dilogarithm and ∆ε ≪ ε is the maximum energy of soft photons in thecentre–of–mass (c.m.) system. Π ( s ) is the vacuum polar-isation operator. Here we again see that the terms withthe large logarithm L dominate numerically.The c -odd part of the one–loop correction comes fromthe interference of Born and box Feynman diagrams andfrom the interference part of the soft photon emission con-tribution. It causes the charge asymmetry of the process: η = d σ ( c ) − d σ ( − c )d σ ( c ) + d σ ( − c ) = 0 . (95)The c -odd part of the differential cross section has thefollowing form [245]:d σ S + V odd d Ω = d σ µµ d Ω απ " ∆εε ln 1 − βc βc + K µµ odd (cid:21) . (96)The expression for the c -odd form factor can be foundin Ref. [245]. Note that in most cases the experimentshave a symmetric angular acceptance, so that the odd partof the cross section does not contribute to the measuredquantities.Consider now the process of hard photon emission e + ( p + ) + e − ( p − ) → µ + ( q + ) + µ − ( q − ) + γ ( k ) . (97)It was studied in detail in Refs. [245,276]. The photonenergy is assumed to be larger than ∆ε . The differential cross section has the formd σ µµγ = α π s R d Γ, (98)d Γ = d q − d q + d kq − q k δ (4) ( p + + p − − q − − q + − k ) ,R = s πα ) X spins | M | = R e + R µ + R eµ . The quantities R i are found directly from the matrix ele-ments and read R e = sχ − χ + B − m e χ − ( t + u + 2 m µ s ) s − m e χ ( t + u + 2 m µ s ) s + m µ s ∆ s s ,R eµ = B (cid:18) uχ − χ ′ + + u χ + χ ′− − tχ − χ ′− − t χ + χ ′ + (cid:19) + m µ ss ∆ ss ,R µ = s χ ′− χ ′ + B + m µ s ∆ ss ,B = u + u + t + t ss ,∆ s s = − ( t + u ) + ( t + u ) χ − χ + ,∆ ss = − u + t + 2 sm µ χ ′− ) − u + t + 2 sm µ χ ′ + ) + 1 χ ′− χ ′ + (cid:0) ss − s + tu + t u − sm µ (cid:1) ,∆ ss = s + s (cid:18) uχ − χ ′ + + u χ + χ ′− − tχ − χ ′− − t χ + χ ′ + (cid:19) + 2( u − t ) χ ′− + 2( u − t ) χ ′ + , where s = ( q + + q − ) , t = − p − q − , t = − p + q + ,u = − p − q + , u = − p + q − , χ ± = p ± k, χ ′± = q ± k. The bulk of the hard photon radiation comes from ISRin collinear regions. If we consider photon emission insidetwo narrow cones along the beam axis with restrictions d p ± k = θ ≤ θ ≪ , θ ≫ m e ε , (99) we see that the corresponding contribution takes the fac-torised form (cid:18) d σ µµ d Ω − (cid:19) coll = C e + D e , (100) C e = α π (cid:18) ln sm e − (cid:19) Z ∆ d x − x ) x A ,D e = α π Z ∆ d x (cid:26) x + 1 + (1 − x ) x ln θ (cid:27) A ,A = (cid:20) d˜ σ (1 − x, Ω − + d˜ σ (1 , − x )d Ω − (cid:21) , where the shifted Born differential cross section describesthe process e + ( p + (1 − x )) + e − ( p − (1 − x )) → µ + ( q + ) + µ − ( q − ),d˜ σ µµ ( z , z )d Ω − = α s × y [ z ( Y − y c ) + z ( Y + y c ) + 8 z z m µ /s ] z z [ z + z − ( z − z ) cY /y ] ,y , = Y , − m µ s , Y , = q − , + ε , z , = 1 − x , ,Y = 4 m µ s ( z − z ) c (cid:20) z z + q z z − m µ /s )(( z + z ) − ( z − z ) c ) (cid:21) − + 2 z z z + z − c ( z − z ) . (101)One can recognise that the large logarithms related to thecollinear photon emission appear in C e in agreement withthe structure function approach discussed in the Luminos-ity Section. In analogy to the definition of the QCD struc-ture functions, one can move the factorised logarithmiccorrections C e into the QED electron structure function.Adding the higher-order radiative corrections in the lead-ing logarithmic approximation to the complete one-loop result, the resulting cross section can be written asd σ e + e − → µ + µ − ( γ ) d Ω − = Z z min Z z min d z d z D ( z , s ) D ( z , s ) | − Π ( sz z ) | × d˜ σ µµ ( z , z )d Ω − (cid:18) απ K µµ odd + απ K µµ even (cid:19) + (cid:26) α π s Z k >∆ε c kp ± >θ R e | m e =0 | − Π ( s ) | d Γ d Ω − + D e | − Π ( s ) | (cid:27) + (cid:26) α π s Z k >∆ε (cid:18) Re R eµ (1 − Π ( s ))(1 − Π ( s )) ∗ + R µ | − Π ( s ) | (cid:19) d Γ d Ω − + Re C eµ (1 − Π ( s ))(1 − Π ( s )) ∗ + C µ | − Π ( s ) | (cid:27) , (102) C µ = 2 απ d σ µµ d Ω − ln ∆εε (cid:18) β β ln 1 + β − β − (cid:19) ,C eµ = 4 απ d σ µµ d Ω − ln ∆εε ln 1 − βc βc , where D e , C eµ and C µ are compensating terms, whichprovide the cancellation of the auxiliary parameters ∆ and θ inside the curly brackets. In the first term, containing D functions, we collect all the leading logarithmic terms. Apart of non-leading terms proportional to the Born crosssection is written as the K -factor. The rest of the non-leading contributions are written as two additional terms.The compensating term D e (see Eq. (100)) comes from theintegration in the collinear region of hard photon emission.The quantities C µ and C eµ come from the even and oddparts of the differential cross section (arising from softand virtual corrections), respectively. Here we consider thephase space of two (d Ω − ) and three (d Γ ) final particles asthose that already include all required experimental cuts.Using specific experimental conditions one can determinethe lower limits of the integration over z and z insteadof the kinematical limit z min = 2 m µ / (2 ε − m µ ).Matching of the complete O ( α ) RC with higher-orderleading logarithmic corrections can be performed in differ-ent schemes. The above approach is implemented in theMCGPJ event generator [236]. The solution of the QEDevolution equations in the form of parton showers (see theLuminosity Section), matched again with the first ordercorrections, is implemented in the BabaYaga@NLO gen-erator [234]. Another possibility is realised in the KKMCcode [277,278] with the Yennie-Frautschi-Suura exponen-tiated representation of the photonic higher-order correc-tions. A good agreement was obtained in [236] for variousdifferential distributions for the µ + µ − channel betweenMCGPJ, BabaYaga@NLO and KKMC, see Fig. 31 for anexample.Since the radiative corrections to the initial e + e − stateare the same for annihilation into hadrons and muons aswell as that into pions, they cancel out in part in the ra- E, MeV × 400 600 800 1000 1200 1400 C r o ss s ec ti on d i ff e r e n ce , % -1-0.8-0.6-0.4-0.2-00.20.40.60.81 Fig. 31. Comparison of the e + e − → µ + µ − total cross sectionscomputed by the MCGPJ and KKMC generators versus thec.m. energy. tio (106). However, the experimental conditions and sys-tematic are different for the muon and hadron channels.Therefore, a separate treatment of the processes has to beperformed and the corrections to the initial states have tobe included in the analysis.For the π + π − channel the complete set of O ( α ) correc-tions matched with the leading logarithmic electron struc-ture functions can be found in Ref. [279]. There the RCcalculation was performed within scalar QED.Taking into account only final state corrections calcu-lated within scalar QED, it is convenient to introduce the bare e + e − → π + π − ( γ ) cross section as σ ππ ( γ ) = πα s β π | F π ( s ) | | − Π ( s ) | (cid:16) απ Λ ( s ) (cid:17) , (103)where the factor | − Π ( s ) | with the polarisation opera-tor Π ( s ) gives the effect of leptonic and hadronic vacuumpolarisation. The final state radiation (FSR) correction isdenoted by Λ ( s ). For an inclusive measurement withoutcuts it reads [280,281,282,283] Λ ( s ) = 1 + β π β π (cid:26) ( 1 − β π β π ) + 2Li ( − − β π β π ) − (cid:20) β π ) + 2 ln β π (cid:21) ln 1 + β π − β π (cid:27) − − β π ) − β π + 1 β π (cid:20) 54 (1 + β π ) − (cid:21) ln 1 + β π − β π + 32 1 + β π β π . (104) For the neutral kaon channel the corrected cross sec-tion has the formd σ e + e − → K L K S ( s )d Ω L = ∆ Z d x d σ e + e − → K L K S ( s (1 − x ))d Ω L F ( x, s ) . The radiation factor F takes into account radiative cor-rections to the initial state within the leading logarith-mic approximation with exponentiation of the numeri-cally important contribution of soft photon radiation, seeRef. [228]: F ( x, s ) = bx b − (cid:20) b + απ (cid:18) π − (cid:19) − b (cid:18) L − π − (cid:19)(cid:21) − b (cid:18) − x (cid:19) + 18 b (cid:20) − x ) ln 1 x + 1 x (1 + 3(1 − x ) ) ln 11 − x − x (cid:21) + (cid:18) απ (cid:19) (cid:26) x (cid:18) x − m e ε (cid:19) b (cid:20) (2 − x + x ) (cid:18) ln sx m e − (cid:19) + b (cid:18) ln sx m e − (cid:19) (cid:21) + 12 L (cid:20) 23 1 − (1 − x ) − x +(2 − x ) ln(1 − x ) + x (cid:21)(cid:27) Θ ( x − m e ε ) . Radiative corrections to the K + K − channel in thepoint-like particle approximation are the same as for thecase of charged pion pair (with the substitution m π → m K ). Usually, for the kaon channel we deal with the en-ergy range close to φ mass. There one may choose themaximal energy of a radiated photon as ω ≤ ∆E = m φ − m K ≪ m K , ∆ ≡ ∆Em K ≈ . (105)For these photons one can use the soft photon approxima-tion. R For older low energy data sets obtained at various e + e − colliders, the correct treatment of radiative corrections isdifficult and sometimes ambiguous. So, to avoid uncon-trolled possible systematic errors, it may be reasonablenot to include all previous results except the recent datafrom CMD-2 and SND. Both experiments at the VEPP-2M collider in Novosibirsk have delivered independent newmeasurements. The covered energy range is crucial for ( g µ -2)/2 of muon and for running α . As for the two-pion chan-nel π + π − , which gives more than 70% of the total hadroniccontribution, both experiments have very good agreementover the whole energy range. The relative deviation “SND- CMD-2” is (-0.3 ± The CMD-2 and SND detectors were located in theopposite straight sections of VEPP-2M and were takingdata in parallel until the year 2000 when the collider wasshut down to prepare for the construction of the new col-lider VEPP-2000. Some important features of the CMD-2 detector allowed one to select a sample of the “clean”collinear back-to-back events. The drift chamber (DC) wasused to separate e + e − , µ + µ − , π + π − and K + K − eventsfrom other particles. The Z-chamber allowed one to sig-nificantly improve the determination of the polar angle ofcharged particle tracks in the DC that, in turn, providedthe detector acceptance with 0.2% precision. The barrelelectromagnetic calorimeter based on CsI crystals helpedto separate the Bhabha from other collinear events.The SND detector consisted of three spherical lay-ers of the electromagnetic calorimeter with 1620 crystals(NaI) and a total weight of 3.6 tons. The solid angleof the calorimeter is about 90% of 4 π steradians, whichmakes the detector practically hermetic for photons com-ing from the interaction point. The angular and energyresolution for photons was found to be 1 . ◦ and σ ( E ) /E =4 . /E (GeV) / , respectively. More detail about CMD-2and SND can be found elsewhere [284,285]. π + π − channel The detailed data on the pion form factor are crucial fora number of problems in hadronic physics and they areused to extract ρ (770) meson parameters and its radialexcitations. Besides, the detailed data allow to extrapolatethe pion form factor to the point s = 0 and determine thevalue of the pion electromagnetic radius.From the experimental point of view the form factorcan be defined as [268] | F π | = N ππ N ee + N µµ σ ee (1 + δ ee ) ε ee + σ µµ (1 + δ µµ ) ε µµ σ ππ (1 + δ ππ )(1 + ∆ N )(1 + ∆ D ) ε ππ − ∆ π , (106)where the ratio N ππ / ( N ee + N µµ ) is derived from the ob-served numbers of events, σ are the corresponding Borncross sections, δ are the radiative corrections (see below), ǫ are the detection efficiencies, ∆ D and ∆ N are the cor-rections for the pion losses caused by decays in flightand nuclear interactions respectively, and ∆ π is the cor-rection for misidentification of ω → π + π − π events as e + e − → π + π − . In the case of the latter process, σ ππ cor-responds to point-like pions.The data were collected in the whole energy range ofVEPP-2M and the integrated luminosity of about 60 pb − was recorded by both detectors. The beam energy was con-trolled and measured with a relative accuracy not worsethan ∼ − by using the method of resonance depolari-sation. A sample of the e + e − , µ + µ − and π + π − events wasselected for analysis. As for CMD-2, the procedure of the e/µ/π separation for energies 2E ≤ 600 MeV was based on the momentum measurement in the DC. For these ener-gies the average difference between the momenta of e/µ/π is large enough with respect to the momentum resolution(Fig. 32). On the contrary, for energies 2E ≥ 600 MeV,the energy deposition of the particles in the calorime-ter is quite different and allows one to separate electronsfrom muons and pions (Fig. 33). At the same time, muonsand pions cannot be separated by their energy deposi-tions in the calorimeter. So, the ratio N ( µ + µ − ) /N ( e + e − )was fixed according to QED calculations taking into ac-count the detector acceptance and the radiative correc-tions. Since the selection criteria were the same for allcollinear events, many effects of the detector imperfec-tions were partly cancelled out. It allowed one to measurethe cross section of the process e + e − → π + π − with betterprecision than that of the luminosity. , MeV/c - P 100 120 140 160 180 200 220 , M e V / c + P - e + e - π + π - µ + µ cosmic Fig. 32. Two-dimensional plot of the e/µ/π events. Cosmicevents are distributed predominantly along a corridor whichextends from the right upper to the left bottom corner. Pointsin this plot correspond to the momenta of particles for thebeam energy of 195 MeV. Separation of e + e − , µ + µ − and π + π − events was basedon the minimisation of the unbinned likelihood function.This method is described in detail elsewhere [286]. To sim-plify the error calculation of the pion form factor, thelikelihood function had the global fit parameters ( N ee + N µµ ) and N ππ / ( N ee + N µµ ), through | F π ( s ) | given byEq. (106). The pion form factor measured by CMD-2 hasa systematic error of about 0.6-0.8% for √ s ≤ e + e − → µ + µ − was also measured, providing an additionalconsistency test. The experimental value σ exp µµ /σ QED µµ = E-, MeV E + , M e V cosmic µ e π Fig. 33. Energy deposition of collinear events for the beamenergy of 460 MeV. (0.980 ± ± R e/π is a discriminationparameter between different particles. The network wastuned by using simulated events and was checked withexperimental 3 π and e + e − events. The misidentificationratio between electrons and pions was found to be 0.5 -1%. SND measured the e + e − → π + π − cross section in theenergy range 0.36 - 0.87 GeV with a systematic error of1.3%.The Gounaris-Sakurai (GS) parametrisation was usedto fit the pion form factor. Results of the fit are shown inFig. 34. The χ was found to be χ / n . d . f . = 122 . / χ / n . d . f . ) = 0.21.The average deviation between SND [287,288] and CMD-2 [289] data is: ∆ (SND – CMD-2) ∼ (1 . ± . √ s ≤ ∆ (SND – CMD-2) ∼ ( − . ± . √ s ≥ ρ meson parameters are:CMD-2 – M ρ = 775 . ± . ± . 70 MeV, Γ ρ = 145 . ± . ± . 50 MeV, Γ ee = 7 . ± . ± . 050 keV,Br( ω → π + π − ) = (1 . ± . ± . M ρ = 774 . ± . ± . Γ ρ = 146 . ± . ± . Γ ee = 7 . ± . ± . 11 keV,Br( ω → π + π − ) = (1 . ± . ± . Energy, MeV 400 600 800 1000 1200 | π | F -1 770 780 790 800 810253035404550 CMD-2, 1995 data (published)CMD-2, 1996 dataCMD-2, 1997 dataCMD-2, 1998 data Fig. 34. Pion form factor data from CMD-2 and GS fit. Theenergy range around the ω meson is scaled up and presentedin the inset. The comparison of the ρ meson parameters determinedby CMD-2 and SND with the values from the PDG is pre-sented in Fig. 35. Good agreement is observed for all pa-rameters, except for the branching fraction of ω decayingto π + π − , where a difference ∼ CMD2’01 PDG’05 SND’05 CMD2’05 CMD2’01 PDG’05 SND’05 CMD2’05 CMD2’01 PDG’05 SND’05 CMD2’05 CMD2’01 PDG’05 SND’05 CMD2’05 Fig. 35. Comparison of ρ meson parameters from CMD-2 andSND with corresponding PDG values. The panels (top-left tobottom-right) refer to the mass (MeV), width (MeV), leptonicwidth (keV) and the branching fraction of the decay ω → π + π − (%).2 Sources of errors CMD-2 SND CMD-2 √ s < . > √ s > . − . . − . . − . − . − . . − . . − . Table 12. The main sources of the systematic errors for different energy regions. e + e − → π + π − π This channel was studied by SND in the energy range √ s from 0.6 to 1.4 GeV [290,291], while CMD-2 has reportedresults of the measurements in vicinity of the ω [289] and φ meson peaks [292]. For both the ω and φ resonances CMD-2 and SND obtain consistent results for the product of theresonance branching fractions into e + e − and π + π − π , forwhich they have the world’s best accuracy (SND for the ω and CMD-2 for the φ resonance).CMD-2 has also performed a detailed Dalitz plot anal-ysis of the dynamics of φ decaying to π + π − π . Two modelsof 3 π production were used: a ρπ mechanism and a contactamplitude. The result obtained for the ratio of the con-tact and ρπ amplitudes is in good agreement with that ofKLOE [293].The systematic accuracy of the measurements is about1.3% around the ω meson energy region, 2.5% in the φ region, and about 5.6% for higher energies. The results ofdifferent experiments are collected in Fig. 36. The curveis the fit which takes into account the ρ, ω, φ, ω ′ and ω ′′ mesons. , GeVs0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 , nb σ π - π + π SND π - π + π CMD2 π - π + π CMD2 π - π + π BABAR Fig. 36. Cross section of the process e + e − → π + π − π . e + e − → π This cross section becomes important for energies abovethe φ meson region. CMD-2 showed that the a (1260) π mechanism is dominant for the process e + e − → π + π − π + π − ,whereas for the channel e + e − → π + π − π π in additionthe intermediate state ωπ is required to describe the en-ergy dependence of the cross section [294]. The SND anal-ysis confirmed these conclusions [295]. The knowledge ofthe dynamics of 4 π production allowed to determine thedetector acceptance and efficiencies with better precisioncompared to the previous measurements.The cross section of the process e + e − → π + π − π + π − was measured with a total systematic error of 15% forCMD-2 and 7% for SND. For the channel e + e − → π + π − π π the systematic uncertainty was 15 and 8%, respectively.The CMD-2 reanalysis of the process e + e − → π + π − π + π − ,with a better procedure for the efficiency determination,reduced the systematic error to (5-7)% [296], and thesenew results are now in remarkable agreement with otherexperiments. CMD-2 and SND have also measured the cross sections ofthe processes e + e − → K S K L and e + e − → K + K − fromthreshold and up to 1.38 GeV with much better accu-racy than before [297,298,299]. These cross sections werestudied thoroughly in the vicinity of the φ meson, andtheir systematic errors were determined with a precisionof about 1.7% (SND) and 4% (CMD-2), respectively. Theanalyses were based on two decay modes of the K S : K S → π π and π + π − . As for the process e + e − → K + K − , thesystematic uncertainty was studied in detail and found tobe 2.2% (CMD-2) and 7% (SND).At energies √ s above 1.04 GeV the cross sections ofthe processes e + e − → K S K L , K + K − were measured witha statistical accuracy of about 4% and systematic errors ofabout 4-6% and 3%, respectively, and are in good agree-ment with other experiments.To summarise, the experiments performed in 1995–2000 with the CMD-2 and SND detectors at VEPP-2M al-lowed one to measure the exclusive cross sections of e + e − annihilation into hadrons in the energy range √ s = 0.36 - 1.38 GeV with larger statistics and smaller systematicerrors compared to the previous experiments. Figure 37summarises the cross section measurements from CMD-2ans SND. The results of these experiments determine the , GeVs0.4 0.6 0.8 1 1.2 , nb σ -1 | π CMD2 |F 65 - π + π - π + π CMD2 96 π - π + π CMD2 19 π π - π + π CMD2 21 - K + CMD2 K 21 - K + CMD2 K 66 L K S CMD2 K 84 γη CMD2 51 γ →γ π CMD2 6 - π + πη CMD2 19 - e + e π CMD2 45 | π SND |F 48 - π + π - π + π SND 125 π - π + π SND 35 π π - π + π SND 62 - K + SND K 66 L K S SND K 95 γη SND 44 γ π SND 45 γ π π SND Fig. 37. Hadronic cross sections measured by CMD-2 andSND in the whole energy range of VEPP-2M. The curve rep-resents a smooth spline of the sum of all data. current accuracy of the calculation of the muon anomaly,and they are one of the main sources of information aboutphysics of vector mesons at low energies. R measurement at CLEO Two important measurements of the R ratio have beenrecently reported by the CLEO Collaboration [300,301].In the energy range just above the open charm thresh-old, they collected statistics at thirteen c.m. energy pointsfrom 3.97 to 4.26 GeV [301]. Hadronic cross sections in thisregion exhibit a rich structure, reflecting the production of c ¯ c resonances. Two independent measurements have beenperformed. In one of them they determined a sum of theexclusive cross sections for final states consisting of twocharm mesons ( D ¯ D , D ∗ ¯ D , D ∗ ¯ D ∗ , D + s D − s , D ∗ + s D − s , and D ∗ + s D ∗− s ) and of processes in which the charm-meson pairis accompanied by a pion. In the second one they measuredthe inclusive cross section with a systematic uncertaintybetween 5.2 and 6.1%. The results of both measurementsare in excellent agreement, which leads to the importantconclusion that in this energy range the sum of the two-and three-body cross sections saturates the total cross sec-tion for charm production. In Fig. 38 the inclusive crosssection measured by CLEO is compared with the previousmeasurements by Crystal Ball [302] and BES [303]. Goodagreement is observed between the data.CLEO has also performed a new measurement of R at higher energy. They collected statistics at seven c.m.energy points from 6.964 to 10.538 GeV [300] and reacheda very small systematic uncertainty of 2% only. Resultsof their scan are presented in Fig. 39 and are in goodagreement with those of Crystal Ball [302], MD-1 [304] andthe previous measurement of CLEO [305]. However, theyare obviously inconsistent with those of the old MARK Imeasurement [306]. Fig. 38. Comparison of the R values from CLEO (the inclusivedetermination) with those from Crystal Ball and BES. Fig. 39. Top plot: comparison of the R values from CLEOwith those from MARK I, Crystal Ball and MD-1; bottom plot:comparison of the new CLEO results with the QCD predictionat Λ =0.31 GeV.4 R measurement at BES Above 2 GeV the number of final states becomes too largefor completely exclusive measurements, so that the valuesof R are measured inclusively.In 1998, as a feasibility test of R measurements, BEStook data at six c.m. energy points between 2.6 and 5.0GeV [307]. The integrated luminosity collected at eachenergy point changed from 85 to 292 nb − . The statisticalerror was around 3% per point and the systematic errorranged from 7 to 10%.Later, in 1999, BES performed a systematic fine scanover the c.m. energy range from 2 to 4.8 GeV [303]. Datawere taken at 85 energy points, with an integrated lumi-nosity varying from 9.2 to 135 nb − per point. In thisexperiment, besides the continuum region below the char-monium threshold, the high charmonium states from 3.77to 4.50 GeV were studied [308] in detail. The statisticalerror was between 2 to 3%, while the systematic errorranged from 5 to 8%, due to improvement on hadronicevent selection and Monte Carlo simulation of hadroni-sation processes. The uncertainty due to the luminositydetermination varied from 2 to 5.8%.More recently, in 2003 and 2004, before BES-II wasshut down for the upgrade to BES-III, a high-statisticsdata sample was taken at 2.6, 3.07 and 3.65 GeV, withan integrated luminosity of 1222, 2291 and 6485 nb − ,respectively [309]. The systematic error, which exceededthe statistical error, was reduced to 3.5% due to furtherrefinement on hadronic event selection and Monte Carlosimulation.For BES-III, the main goal of the R measurementis to perform a fine scan over the whole energy regionwhich BEPC-II can cover. For a continuum region (below3.73 GeV), the step size should not exceed 100 MeV, andfor the resonance region (above 3.73 GeV), the step sizeshould be 10 to 20 MeV. Since the luminosity of BEPC-IIis two orders of magnitude higher than at BEPC, the scanof the resonance region will provide precise information onthe 1 −− charmonium states up to 4.6 GeV. Let us discuss the accuracy of the description of the pro-cesses under consideration. This accuracy can be subdi-vided into two major parts: theoretical and technical one.The first one is related to the precision in the actualcomputer codes. It usually does not take into accountall known contributions in the best approximation. Thetechnical precision can be verified by special tests withina given code (e.g., by looking at the numerical cancella-tion of the dependence on auxiliary parameters) and tunedcomparisons of different codes.The pure theoretical precision consists of unknown high-er-order corrections, of uncertainties in the treatment ofphoton radiation off hadrons, and of errors in the phe-nomenological definition of such quantities as the hadronicvacuum polarisation and the pion form factor. Many of the codes used at meson factories do not in-clude contributions from weak interactions even at Bornlevel. As discussed above, these contributions are sup-pressed at least by a factor of s/M Z and do not spoilthe precision up to the energies of B factories.Matching the complete one-loop QED corrections withthe higher-order corrections in the leading logarithmic ap-proximation, certain parts of the second-order next-to-leading corrections are taken into account [235]. For thecase of Bhabha scattering, where, e.g., soft and virtualphotonic corrections in O ( α L ) are known analytically,one can see that their contribution in the relevant kine-matic region does not exceed 0.1%. The uncertainty coming from the the hadronic vacuumpolarisation has been estimated [13] to be of order 0.04%.For measurements performed with the c.m. energy at anarrow resonance (like at the φ -meson factories), a sys-tematic error in the determination of the resonance con-tribution to vacuum polarisation is to be added.The next point concerns non-leading terms of order( α/π ) L . There are several sources of them. One is theemission of two extra hard photons, one in the collinearregion and one at large angles. Others are related to vir-tual and soft-photon radiative corrections to single hardphoton emission and Born processes. Most of these con-tributions were not considered up to now. Neverthelesswe can estimate the coefficient in front of the quantity( α/π ) L ≈ · − to be of order one. This was indirectlyconfirmed by our complete calculations of these terms forthe case of small–angle Bhabha scattering.Considering all sources of uncertainties mentioned aboveas independent, we conclude that the systematic error ofour formulae is about 0.2% or better, both for muons andpions. For the former it is a rather safe estimate. Com-parisons between different codes which treat higher-orderQED corrections in different ways typically show agree-ment at the 0.1% level. Such comparisons test the techni-cal and partially the theoretical uncertainties. As for the π + π − and two kaon channels, the uncertainty is enhanceddue to the presence of form factors and due to the appli-cation of the point-like approximation for the final statehadrons. The idea to use Initial State Radiation to measure hadroniccross sections from the threshold of a reaction up to thecentre-of-mass (c.m.) energy of colliders with fixed en-ergies √ s , to reveal reaction mechanisms and to searchfor new mesonic states consists in exploiting the process e + e − → hadrons + nγ , thus reducing the c.m. energy ofthe colliding electrons and positrons and consequently the The proper choice of the factorisation scale [246] is impor-tant here.5 mass squared M = s − √ s E γ of the hadronic sys-tem in the final state by emission of one or more photons.The method is particularly well suited for modern mesonfactories like DAΦNE (detector KLOE), running at the φ -resonance, BEPC-II (detector BES-III), commissionedin 2008 and running at the J/ψ and ψ (2 S )-resonances,PEP-II (detector BaBar) and KEKB (detector Belle) atthe Υ (4 S )-resonance. Their high luminosities compensatefor the α/π suppression of the photon emission. DAΦNE,BEPC-II, PEP-II and KEKB cover the regions in M had up to 1.02, 3.8 (maximally 4.6) and 10.6 GeV, respec-tively (for the latter actually restricted to 4–5 GeV ifhard photons are detected). A big advantage of ISR is thelow point-to-point systematic errors of the hadronic en-ergy spectra. This is because the luminosity, the energy ofthe electrons and positrons and many other contributionsto the detection efficiencies are determined once for thewhole spectrum. As a consequence, the overall normalisa-tion error is the same for all energies of the hadronic sys-tem. The term Radiative Return alternately used for ISRrefers to the appearance of pronounced resonances (e.g. ρ, ω, φ, J/ψ, Z ) with energies below the collider energy.Reviews and updated results can be found in the Proceed-ings of the International Workshops in Pisa (2003) [310],Nara (2004) [311], Novosibirsk (2006) [312], Pisa (2006)[313], Frascati (2008) [314], and Novosibirsk (2008) [315].Calculations of ISR date back to the sixties to seven-ties of the 20 th century. For example, photon emission formuon pair production in electron-positron collisions hasbeen calculated in Ref. [316], for the 2 π -final state in Refs.[317,318]; the resonances ρ, ω and φ have been imple-mented in Ref. [318], the excitations ψ (3100) and ψ ′ (3700)in Ref. [319], and the possibility to determine the pionform factor was discussed in Ref. [320]. The application ofISR to the new high luminosity meson factories, originallyaimed at the determination of the hadronic contribution tothe vacuum polarisation, more specifically the pion formfactor, has materialised in the late nineties. Early calcula-tions of ISR for the colliders DAΦNE, PEP-II and KEKBcan be found in [321,322,323,324]. In Ref. [279] calcula-tions of radiative corrections for pion and kaon productionbelow energies of 2 GeV have been reported. An impres-sive example of ISR is the Radiative Return to the regionof the Z -resonance at LEP-2 with collider energies around200 GeV [325,326,327,328] (see Fig. 40).ISR became a powerful tool for the analysis of exper-iments at low and intermediate energies with the devel-opment of EVA-PHOKHARA, a Monte Carlo generatorwhich is user friendly, flexible and easy to implement intothe software of the existing detectors [329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345].EVA and its successor PHOKHARA allow to simulatethe process e + e − → hadrons + γ for a variety of exclusivefinal states. As a starting point EVA was constructed [329]to simulate leading order ISR and FSR for the π + π − chan-nel, and additional soft and collinear ISR was included onthe basis of structure functions taken from [346]. Subse- Fig. 40. The reconstructed distribution of e + e − → q ¯ q eventsas a function of the invariant mass of the quark-antiquark sys-tem. The data has been taken for a collider energy range of 182- 209 GeV. The prominent peak around 90 GeV represents theZ-resonance, populated after emission of photons in the initialstate [326]. quently EVA was extended to include the four-pion state[330], albeit without FSR. Neglecting FSR and radiativecorrections, i.e. including one-photon emission from theinitial state only, the cross section for the radiative re-turn can be cast into the product of a radiator function H ( M , s ) and the cross section σ ( M ) for the reaction e + e − → hadrons : s d σ ( e + e − → hadrons γ ) / d M = σ ( M ) H ( M , s ).However, for a precise evaluation of σ ( M ), the lead-ing logarithmic approximation inherent in EVA is insuffi-cient. Therefore, in the next step, the exact one-loop cor-rection to the ISR process was evaluated analytically, firstfor large angle photon emission [331], then for arbitraryangles, including collinear configurations [332]. This wasand is one of the key ingredients of the generator calledPHOKHARA [333,334], which also includes soft and hardreal radiation, evaluated using exact matrix elements for-mulated within the framework of helicity amplitudes [333].FSR in NLO approximation was addressed in [335] andincorporated in [336,337]. The importance of the chargeasymmetry, a consequence of interference between ISRand FSR amplitudes, for a test of the (model dependent)description of FSR has been emphasised already in Ref.[329] and was further studied in [337].Subsequently the generator was extended to allow forthe generation of many more channels with mesons, like K + K − , K ¯ K , π + π − π , for an improved description ofthe 4 π modes [338,339] and for improvements in the de-scription of FSR for the µ + µ − channel [336,337]. Also thenucleon channels p ¯ p and n ¯ n were implemented [340], andit was demonstrated that the separation of electric andmagnetic proton form factors is feasible for a wide en-ergy range. In fact, for the case of Λ ¯ Λ and including thepolarisation-sensitive weak decay of Λ into the simulation, it was shown that even the relative phase between the twoindependent form factors could be disentangled [341].Starting already with [347], various improvements weremade to include the direct decay φ → π + π − γ as a specificaspect of FSR into the generator, a contribution of specificimportance for data taken on top of the φ resonance.This was further pursued in the event generators FEVAand FASTERD based on EVA-PHOKHARA. FEVA in-cludes the effects of the direct decay φ → π − π + γ and thedecay via the ρ -resonance φ → ρ ± π ∓ → π − π + γ [348,349,350]. The code FASTERD takes into account Final StateRadiation in the framework of both Resonance Pertur-bation Theory and sQED, Initial State Radiation , theirinterference and also the direct decays e + e − → φ → ( f o ; f o + σ ) γ → π + π − γ , e + e − → φ → ρ ± π ∓ → π + π − γ and e + e − → ρ → ωπ o → π o π o γ [351], with the possibilityto include additional models.EVA-PHOKHARA was applied for the first time to anexperiment to determine the cross section e + e − → π + π − from the reaction threshold up to the maximum energy ofthe collider with the detector KLOE at DAΦNE [352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376] (Section 4.4.1). Themotivation was the determination of the 2 π final state con-tribution to the hadronic vacuum polarisation.The determination of the hadronic contribution to thevacuum polarisation, which arises from the coupling ofvirtual photons to quark-antiquark pairs, γ ⋆ → q ¯ q → γ ⋆ ,is possible by measuring the cross section of electron-positron annihilation into hadrons, e + e − → γ ∗ → q ¯ q → hadrons , and applying the optical theorem. It is of greatimportance for the interpretation of the precision measure-ment of the anomalous magnetic moment of the muon a µ in Brookhaven (E821) [377,378,31,379] and for the de-termination of the value of the running QED coupling atthe Z o resonance, α ( m Z ), which contributes to precisiontests of the Standard Model of particle physics, for detailssee e.g. Jegerlehner [380], also Davie and Marciano [381],or Teubner et al. [382,26,383]. The hadronic contributionto a µ below about 2 GeV is dominated by the 2 π finalstate, which contributes about 70% due to the dominanceof the ρ − resonance. Other major contributions come fromthe three- and four-pion final states. These hadronic finalstates constitute at present the largest error to the Stan-dard Model values of a µ and α ( m Z ) and can be determinedonly experimentally. This is because calculations withinperturbative QCD are unrealistic, calculations on the lat-tice are not yet available with the necessary accuracy, andcalculations in the framework of chiral perturbation the-ory are restricted to values close to the reaction thresholds.At energies above about 2 to 2.5 GeV, perturbative QCDcalculations start to become possible and reliable, see e.g.Refs. [384,385], and also [386].The Novosibirsk groups CMD-2 [312,268,297,387,289,388,389,390,391,392] and SND [291,287,393,288,299,298]measured hadronic cross sections below 1.4 GeV by chang-ing the collider energy ( energy scan , see the preceding Sec-tion 3). The Initial State Radiation method used by KLOErepresents an alternative, independent and complemen- tary way to determine hadronic cross sections with differ-ent systematic errors. KLOE has determined the cross sec-tion for the reaction e + e − → π + π − in the energy regionbetween 0.63 and 0.958 GeV by measuring the reaction e + e − → π + π − γ and applying a radiator function basedon PHOKHARA. For the hadronic contribution to theanomalous magnetic moment of the muon due to the 2 π final state it obtained a ππµ = (356 . ± . stat+syst ) · − [374]. This value is in good agreement with those fromSND [298] and CMD-2 [392], a ππµ = (361 . ± . stat+syst ) · − and a ππµ = (361 . ± . stat + syst ) · − , respectively,leading to an evaluation of a µ [380,381,382,26,383,37]which differs by about three standard deviations from theBNL experiment [31]. A different evaluation using τ de-cays into two pions results in a reduced discrepancy [381,37]. The difference between e + e − and τ based analyses isat present not understood. But one has to be aware thatthe evaluation with τ data needs more theoretical input.Soon after the application of EVA-PHOKHARA toKLOE [352], the BaBar collaboration also started the mea-surement of hadronic cross sections exploiting ISR [394]and using PHOKHARA (Section 4.4.2). In recent years aplethora of final states has been studied, starting with thereaction e + e − → J/ψ γ → µ + µ − γ [395]. While detect-ing a hard photon, the upper energy for the hadron crosssections is limited to roughly 4.5 GeV. Final states with3, 4, 5, 6 charged and neutral pions, 2 pions and 2 kaons,4 kaons, 4 pions and 2 kaons, with a φ and an f o (980), J/ψ and 2 pions or 2 kaons, pions and η , kaons and η , butalso baryonic final states with protons and antiprotons, Λ o and ¯ Λ o , Λ o and ¯ Σ o , Σ o and ¯ Σ o , D ¯ D , D ¯ D ∗ , and D ∗ ¯ D ∗ mesons, etc. have been investigated [396,397,398,399,400,401,402,403,404,405,406,407,408]. In preparation are fi-nal states with 2 pions [409] and 2 kaons. Particularlyimportant final states are those with 4 pions (including ωπ o ). They contribute significantly to the muon anoma-lous magnetic moment and were poorly known before theISR measurements. In many of these channels additionalinsights into isospin symmetry breaking are expected fromthe comparison between e + e − annihilation and τ decays.More recently also Belle joined the ISR programmewith emphasis on final states containing mesons with hid-den and open charm: J/ψ and ψ (2 S ), D ( ∗ ) and ¯ D ( ∗ ) , Λ c + Λ c − [410,411,412,413,414,415,416,417] (Section 4.4.3).A major surprise in recent years was the opening ofa totally new field of hadron spectroscopy by applyingISR. Several new, relatively narrow highly excited stateswith J P C = 1 −− , the quantum numbers of the photon,have been discovered (preliminarily denoted as X, Y, Z )at the B factories PEP-II and KEKB with the detec-tors BaBar and Belle, respectively. The first of them wasfound by BaBar in the reaction e + e − → Y (4260) γ → J/ψ π + π − γ [418], a state around 4260 MeV with a widthof 90 MeV, later confirmed by Belle via ISR [419,410] andby CLEO in an direct energy scan [420] and a radiativereturn [421]. Another state was detected at 2175 MeV byBaBar in the reaction e + e − → Y (2175) γ → φf o (980) γ [400]. Belle found new states at 4050, 4360, 4660 MeV inthe reactions e + e − → Y γ → J/ψ π + π − γ and e + e − → Y γ → ψ (2 S ) π + π − γ [410,411]. The structure of basicallyall of these new states (if they will survive) is unknown sofar. Four-quark states, e.g. a [ cs ][¯ c ¯ s ] state for Y (4260), a[ ss ][¯ s ¯ s ] state for Y (2175), hybrid and molecular structuresare discussed, see also [422].Detailed analyses allow, in addition, also the identi-fication of intermediate states, and consequently a studyof reaction mechanisms. For instance, in the case of thefinal state with 2 charged and 2 neutral pions ( e + e − → π + π − π o π o γ ), the dominating intermediate states are ωπ o and a (1260) π , while ρ + ρ − and ρ o f o (980) contribute sig-nificantly less.Many more highly excited states with quantum num-bers different from those of the photon have been found indecay chains of the primarily produced heavy mesons atthe B factories PEP-II and KEKB. These analyses with-out ISR have clearly been triggered and encouraged bythe unexpected discovery of highly excited states with J P C = 1 −− found with ISR.Also baryonic final states with protons and antipro-tons, Λ o and ¯ Λ o , Λ o and ¯ Σ o , Σ o and ¯ Σ o have been investi-gated using ISR. The effective proton form factor (see Sec-tion 4.4.2) shows a strong increase down to the p ¯ p thresh-old and nontrivial structures at invariant p ¯ p masses of 2.25and 3.0 GeV, so far unexplained [398,423,424,425,426].Furthermore, it should be possible to disentangle electricand magnetic form factors and thus shed light on discrep-ancies between different measurements of these quantitiesin the space-like region [427].Prospects for the Radiative Return at the Novosibirskcollider VEPP-2000 and BEPC-II are discussed in Sec-tions 4.4.4 and 4.4.5. We consider the e + e − annihilation process e + ( p ) + e − ( p ) → hadrons + γ ( k ) , (107)where the real photon is emitted either from the initial(Fig. 41a) or the final state (Fig. 41b). The former processis denoted initial state radiation (ISR), while the latter iscalled final state radiation (FSR).The differential rate for the ISR process can be castinto the product of a leptonic L µν and a hadronic H µν tensor and the corresponding factorised phase spaced σ ISR = 12 s L µν ISR H µν × d Φ ( p , p ; Q, k )d Φ n ( Q ; q , · , q n ) d Q π , (108)where d Φ n ( Q ; q , · , q n ) denotes the hadronic n -body phase-space with all the statistical factors coming from the hadro-nic final state included, Q = P q i and s = ( p + p ) . e + e − ¯ hhγ (a) e + e − ¯ hhγ (b) Fig. 41. Leading order contributions to the reaction e + e − → h ¯ h + γ from ISR (a) and FSR (b). Final state particles arepions or muons, or any other multi-hadron state. The blobrepresents the hadronic form factor. For an arbitrary hadronic final state, the matrix ele-ment for the diagrams in Fig. 41a is given by A (0)ISR = M (0)ISR · J (0) == − e Q ¯ v ( p ) (cid:18) ε/ ∗ ( k )[ k/ − p/ + m e ] γ µ k · p + γ µ [ p/ − k/ + m e ] ε/ ∗ ( k )2 k · p (cid:19) u ( p ) J (0) µ , (109)where J µ is the hadronic current. The superscript (0) in-dicates that the scattering amplitude is evaluated at tree-level. Summing over the polarisations of the final real pho-ton, averaging over the polarisations of the initial e + e − state, and using current conservation, Q · J (0) = 0, theleptonic tensor L (0) ,µν ISR = M (0) , µ ISR ( M (0) , ν ISR ) † can be written in the form L (0) , µν ISR = (4 πα ) Q (cid:20) (cid:18) m q (1 − q ) y y − q + y + y y y (cid:19) g µν + (cid:18) m y − q y y (cid:19) p µ p ν s + (cid:18) m y − q y y (cid:19) p µ p ν s − (cid:18) m y y (cid:19) p µ p ν + p ν p µ s (cid:21) , (110)with y i = 2 k · p i s , m = m e s , q = Q s . (111)The leptonic tensor is symmetric under the exchange ofthe electron and the positron momenta. Expressing thebilinear products y i by the photon emission angle in thec.m. frame, y , = 1 − q ∓ β cos θ ) , β = p − m , and rewriting the two-body phase space asd Φ ( p , p ; Q, k ) = 1 − q π d Ω , (112) it is evident that expression (110) contains several singu-larities: soft singularities for q → θ → ± 1. The former are avoided by requir-ing a minimal photon energy. The latter are regulated bythe electron mass. For s ≫ m e the expression (110) cannevertheless be safely taken in the limit m e → m = m e /s areneglected prematurely.Physics of the hadronic system, whose description ismodel dependent, enters through the hadronic tensor H µν = J (0) µ ( J (0) ν ) † , (113)where the hadronic current has to be parametrised throughform factors. For two charged pions in the final state, thecurrent J (0) , µπ + π − = ieF π ( Q ) ( q − q ) µ , (114)where q and q are the momenta of the π + and π − , re-spectively, is determined by only one function, the pionform factor F π . The current for the µ + µ − final state isobviously defined by QED: J (0) , µµ + µ − = ie ¯ u ( q ) γ µ v ( q ) . (115)Integrating the hadronic tensor over the hadronic phasespace, one gets Z H µν d Φ n ( Q ; q , · , q n ) = e π ( Q µ Q ν − g µν Q ) R ( Q ) , (116)where R ( Q ) = σ ( e + e − → hadrons) /σ ( e + e − → µ + µ − ),with σ ( e + e − → µ + µ − ) = 4 π α Q (117)the tree-level muonic cross section in the limit Q ≫ m µ .After the additional integration over the photon angles,the differential distribution Q d σ ISR d Q = 4 α s R ( Q ) (cid:26) s + Q s ( s − Q ) ( L − (cid:27) , (118)with L = log( s/m e ) is obtained. If instead the photonpolar angle is restricted to be in the range θ min < θ <π − θ min , this differential distribution is given by Q d σ ISR d Q = 4 α s R ( Q ) (cid:26) s + Q s ( s − Q ) log 1 + cos θ min − cos θ min − s − Q s cos θ min (cid:27) . (119)In the latter case, the electron mass can be taken equalto zero before integration, since the collinear region is ex-cluded by the angular cut. The contribution of the two-pion exclusive channel can be calculated from Eq. (118)and Eq. (119) with R π + π − ( Q ) = 14 (cid:18) − m π Q (cid:19) / | F π ( Q ) | , (120) d σ ( e + e − → π + π − γ ) / d Q ( nb / G e V ) θ γ < 15 o or θ γ > 165 o o < θ π < 140 o ISR+FSR ≈ ISR onlyISR+FSRISR only -4 -2 Q (GeV ) F S R /I S R θ γ < 15 o or θ γ > 165 o o < θ π < 140 o Fig. 42. Suppression of the FSR contributions to the crosssection by a suitable choice of angular cuts; results from thePHOKHARA generator; no cuts (upper curves) and suitablecuts applied (lower curves). and the corresponding muonic contribution with R µ + µ − ( Q ) = s − m µ Q m µ Q ! . (121)A potential complication for the measurement of thehadronic cross section from the radiative return may arisefrom the interplay between photons from ISR and FSR[329]. Their relative strength is strongly dependent on thephoton angle relative to the beam and to the direction ofthe final state particles, the c.m. energy of the reactionand the invariant mass of the hadronic system. While ISRis independent of the hadronic final state, FSR is not.Moreover, it cannot be predicted from first principles andthus has to be modelled.The amplitude for FSR (Fig. 41b) factorises as well as A (0)FSR = M (0) · J (0)FSR , (122)where M (0) µ = es ¯ v ( p ) γ µ u ( p ) . (123)Assuming that pions are point-like, the FSR current fortwo pions in scalar QED (sQED) reads J (0) , µ FSR = − i e F π ( s ) × (cid:20) − g µσ + ( q + k − q ) µ (2 q + k ) σ k · q − ( q − k − q ) µ (2 q + k ) σ k · q (cid:21) ǫ ∗ σ ( k ) . (124) cos θ π+ d σ ( e + e − → π + π − γ ) / d c o s θ π+ ( nb ) θ γ < 15 o or θ γ > 165 o o < θ π < 140 o ISR+FSR ≈ ISR onlyISR+FSR ISR only cos θ µ+ d σ ( e + e − → µ + µ − γ ) / d c o s θ µ + ( nb ) θ γ < 15 o or θ γ > 165 o o < θ µ < 140 o ISR+FSR ≈ ISR onlyISR+FSRISR only Fig. 43. Angular distributions of π + and µ + at √ s = 1 . 02 GeV with and without FSR for different angular cuts. cos θ π+ d σ ( e + e − → π + π − γ ) / d c o s θ π+ ( nb ) o < θ γ , θ π < 150 o cos θ µ+ d σ ( e + e − → µ + µ − γ ) / d c o s θ µ + ( nb ) ISR+FSRISR only30 o < θ γ , θ µ < 150 o √ Q < 6 GeV √ Q < 3 GeV √ Q < 1 GeV Fig. 44. Angular distributions of π + (ISR ≃ FSR+ISR) and µ + at √ s =10.6 GeV for various Q cuts. Due to momentum conservation, p + p = q + q + k ,and current conservation, this expression can be simplifiedfurther to J (0) , µ FSR = 2 i e F π ( s ) (cid:20) g µσ + q µ q σ k · q + q µ q σ k · q (cid:21) ǫ ∗ σ ( k ) . (125)This is the basic model adopted in EVA [329] and in PHO-KHARA [331,332,333,334,335,336,337,338,341,428] to sim-ulate FSR off charged pions. The corresponding FSR cur-rent for muons is given by QED.The fully differential cross section describing photonemission at leading order can be split into three piecesd σ (0) = d σ (0)ISR + d σ (0)FSR + d σ (0)INT , (126)which originate from the squared ISR and FSR amplitudesand the interference term, respectively. The ISR–FSR in-terference is odd under charge conjugation,d σ (0)INT ( q , q ) = − d σ (0)INT ( q , q ) , (127) and its contribution vanishes after angular integration. Itgives rise, however, to a relatively large charge asymmetryand, correspondingly, to a forward–backward asymmetry A ( θ ) = N h ( θ ) − N h ( π − θ ) N h ( θ ) + N h ( π − θ ) . (128)The asymmetry can be used for the calibration of the FSRamplitude, and fits to the angular distribution A ( θ ) cantest details of its model dependence [329].The second option to disentangle ISR from FSR ex-ploits the markedly different angular distribution of thephoton from the two processes. This observation is com-pletely general and does not rely on any model like sQEDfor FSR. FSR is dominated by photons collinear to thefinal state particles, while ISR is dominated by photonscollinear to the beam direction. This suggests that weshould consider only events with photons well separatedfrom the charged final state particles and preferentiallyclose to the beam [329,333,334].This is illustrated in Fig. 42, which has been generatedrunning PHOKHARA at leading order (LO). After intro-ducing suitable angular cuts, the contamination of events h ¯ hγe + e − √ s = 1 . GeV γe + e − √ s = 10 . GeV Fig. 45. Typical kinematic configuration of the radiative re-turn at low and high energies. with FSR is easily reduced to less than a few per mill.The price to pay, however, is a suppression of the thresh-old region too. To have access to that region, photons atlarge angles need to be tagged and a better control of FSRis required. In Fig. 43 the angular distribution of π + and µ + at DAΦNE energies, √ s = 1 . 02 GeV, are shown fordifferent angular cuts. The angles are defined with respectto the incoming positron. If no angular cut is applied, theangular distribution in both cases is highly asymmetric asa consequence of the ISR–FSR interference contribution.If cuts suitable to suppress FSR, and therefore the ISR–FSR interference, are applied, the distributions becomesymmetric.Two complementary analyses are therefore possible (fordetails see Section 4.4.1): The small photon angle analysis,where the photon is untagged and FSR can be suppressedbelow some reasonable limit. This analysis is suitable forintermediate values of the invariant mass of the hadronicsystem. And the large photon angle analysis, giving accessto the threshold region, where FSR is more pronouncedand the charge asymmetry is a useful tool to probe itsmodel dependence.These considerations apply, however, only to low beamenergies, around 1 GeV. At high energies, e.g. at B facto-ries, very hard tagged photons are needed to access the re-gion with low hadronic invariant masses, and the hadronicsystem is mainly produced back-to-back to the hard pho-ton. The suppression of FSR is naturally accomplishedand no special angular cuts are needed. This kinemati-cal situation is illustrated in Fig. 45. The suppression ofFSR contributions to π + π − γ events is also a consequenceof the rapid decrease of the form factor above 1 GeV.The relative size of FSR is of the order of a few per mill(see Fig. 44). For µ + µ − in the final state, the amountof FSR depends on the invariant mass of the muons. For p Q < Q (see Fig. 44). The original and default version of EVA [329], simulatingthe process e + e − → π + π − γ at LO, allowed for additional initial state radiation of soft and collinear photons by thestructure function (SF) method [429,346].In the leading logarithmic approximation (LL), themultiple emission of collinear photons off an electron isdescribed by the convolution integral σ ( e − X → Y + nγ ) = Z d x f e ( x, Q ) σ ( e − X → Y ) , (129)where f e ( x, Q ) is the probability distribution of the elec-tron with longitudinal momentum fraction x , and Q isthe transverse momentum of the collinear photons. Thefunction f e ( x, Q ) fulfils the evolution equationdd log Q f e ( x, Q ) = Z x d zzαπ (cid:18) z (1 − z ) + + 32 δ (1 − z ) (cid:19) f e ( xz , Q ) (130)with initial conditions f e ( x, Q ) (cid:12)(cid:12) Q = m e = δ (1 − x ) , (131)and the + prescription defined as Z d x f ( x )(1 − x ) + = Z d x f ( x ) − f (1)(1 − x ) . (132)The analytic solution to Eq. (130) provided in Refs. [429,346] allows to resum soft photons to all orders in pertur-bation theory, accounting for large logarithms of collinearorigin, L = log( s/m e ), up to two loops. The resummedcross section, σ SF = Z d x Z d x D ( x ) D ( x ) σ e + e − → had . + γ ( x x s ) , (133)is thus obtained by convoluting the Born cross section ofthe hard photon emission process e + e − → hadrons + γ with the SF distribution [429,346] D ( x ) = [1 + δ N ] / β e − x ) βe − × (cid:26) 12 (1 + x ) + 12 (1 − x ) L − β e (cid:18) − 12 (1 + 3 x ) log x − (1 − x ) (cid:19) (cid:27) , (134)with β e = 2 απ ( L − 1) (135)and δ N = απ (cid:18) L + π − (cid:19) + β e π (cid:16) απ (cid:17) (cid:18) − π (cid:19) L . (136)In the SF approach, the additional emission of collinearphotons reduces the effective c.m. energy of the collision to √ x x s . Momentum conservation is not accomplishedbecause the extra radiation is integrated out. In order toreduce the kinematic distortion of the events, a minimalinvariant mass of the observed particles, hadrons plus thetagged photon, was required in [329], introducing in turna cut dependence. Therefore the SF predictions are notaccurate enough for a high precision measurement of thehadronic cross section from radiative return, and a next-to-leading order (NLO) calculation is in order. The NLOprediction contains the large logarithms L = log( s/m e ) atorder α and additional sub-leading terms, which are nottaken into account within the SF method. Furthermore,it allows for a better control of the kinematical configura-tions because momentum conservation is fulfilled. A com-parison between SF and NLO predictions can be foundin [333]. At NLO, the e + e − annihilation process in Eq. (107) re-ceives contributions from one-loop corrections and fromthe emission of a second real photon (see Fig. 46). Afterrenormalisation, the one-loop matrix elements still con-tain infrared divergences. These are cancelled by addingthe two-photon contributions to the one-loop corrections.There are several well established methods to perform thiscancellation. The slicing method, where amplitudes areevaluated in dimensional regularisation and the two pho-ton contribution is integrated analytically in phase spacefor one of the photon energies up to an energy cutoff E γ < w √ s far below √ s , was used in [331,332] to cal-culate the NLO corrections to ISR. Here the sum of thevirtual and soft contributions is finite, but it depends onthe soft photon cutoff. The contribution from the emis-sion of the second photon with energy E γ > w √ s , whichis evaluated numerically, completes the calculation andcancels this dependence.The size and sign of the NLO corrections do depend onthe particular choice of the experimental cuts. Hence, onlyusing a Monte Carlo event generator one can realisticallycompare theoretical predictions with experiment. This isthe main motivation behind PHOKHARA [331,332,333,334,335,336,337,338,341,428].The full set of scattering amplitudes at tree-level andone-loop can be constructed from the sub-amplitudes de-picted in Fig. 46. The one-loop amplitude with emissionof a single photon is given by A (1)1 γ = A (1)ISR + A (1)FSR + M (1) · J (0)FSR + M (0)ISR · J (1) + A γ ∗ ISR + A γ ∗ FSR , (137)where A (1)ISR = M (1)ISR · J (0) , A (1)FSR = M (0) · J (1)FSR , (138) e + e − γ ∗ M (0) ¯ hhγ ∗ J (0) M (1) J (1) M (0)ISR J (0)FSR M (0)2ISR J (0)2FSR M (1)ISR J (1)FSR A γ ∗ ISR A γ ∗ FSR Fig. 46. Typical sub-amplitudes describing virtual and realcorrections to the reaction e + e − → h ¯ h + γ ( γ ), where h = π − , µ − . The superscripts (0) and (1) denote tree-level and one-loop quantities, respectively. ISR and FSR indicate that realphotons are emitted from the initial or final state. The lasttwo diagrams, with exchange of two virtual photons, are non-factorisable. Permutations are omitted. while the amplitude with emission of two real photonsreads A (0)2 γ = A (0)2ISR + A (0)2FSR + (cid:16) M (0)ISR ( k ) · J (0)FSR ( k ) + ( k ↔ k ) (cid:17) , (139)where A (0)2ISR = M (0)2ISR · J (0) , A (0)2FSR = M (0) · J (0)2FSR . (140) PHOKHARA includes the full LO amplitudes and themost relevant C-even NLO contributions:d σ = d σ (0) + d σ (1)ISR + d σ (1)IFS , (141)where d σ (0) is the LO differential cross section (Eq. (126)),d σ (1)ISR = 12 s (cid:20) (cid:26) A (1)ISR (cid:16) A (0)ISR (cid:17) † (cid:27) d Φ ( p , p ; q , q , k )+ (cid:12)(cid:12)(cid:12) A (0)2ISR (cid:12)(cid:12)(cid:12) d Φ ( p , p ; q , q , k , k ) (cid:21) (142)is the second order radiative correction to ISR, andd σ (1)IFS = 12 s (cid:20) (cid:26) M (0)ISR · J (1) (cid:16) A (0)ISR (cid:17) † + M (1) · J (0)FSR (cid:16) A (0)FSR (cid:17) † (cid:27) d Φ ( p , p ; q , q , k )+ (cid:18)(cid:12)(cid:12)(cid:12) M (0)ISR ( k ) · J (0)FSR ( k ) (cid:12)(cid:12)(cid:12) + ( k ↔ k ) (cid:19) × d Φ ( p , p ; q , q , k , k ) (cid:21) (143)is the contribution of events with simultaneous emission ofone photon from the initial state and another one from thefinal state, together with ISR amplitudes with final stateone-loop vertex corrections, and FSR amplitudes with ini-tial state one-loop vertex corrections. We denote these cor-rections as IFS.Vacuum polarisation corrections are included in thehadronic currents multiplicatively: J ( i ) → C VP ( Q ) J ( i ) ,J ( i )FSR ( k j ) → C VP (( Q + k j ) ) J ( i )FSR ( k j ) ,J (0)2FSR → C VP ( s ) J (0)2FSR . (144)The virtual photon propagator is by definition included inthe leptonic sub-amplitudes M ( i ) , M ( i )ISR and M (0)2ISR : M ( i ) ∼ s ,M ( i )ISR ( k j ) ∼ p + p + k j ) ,M (0)2ISR ∼ Q . (145)Neither diagrams where two photons are emitted fromthe final state, nor final-state vertex corrections with as-sociated real radiation from the final state are included.These constitute radiative corrections to FSR and will givenon-negligible contributions only for those cases where atleast one photon is collinear with one of the final state par-ticles. Box diagrams with associated real radiation fromthe initial- or the final-state leptons, as well as pentagondiagrams, are also neglected. As long as one considerscharge symmetric observables only, their contribution isdivergent neither in the soft nor the collinear limit and is thus of order α/π without any enhancement factor.One should stress that PHOKHARA includes only C-evengauge invariant sets of diagrams at NLO. The missing con-tributions are either small or do not contribute for chargesymmetric cuts. Nevertheless their implementation is un-derway.The calculation of the NLO corrections to ISR, d σ (1)ISR ,is independent of the final state. These corrections areincluded by default for all the final state channels imple-mented in PHOKHARA, and can be easily added for anyother new channel, with the sole substitution of the tree-level final state current. The radiative corrections of theIFS process depend on the final state. The latest versionof PHOKHARA (version 6.0 [341]) includes these correc-tions for two charged pions, kaons and muons. Virtual and soft corrections to ISR The virtual and soft QED corrections to ISR in e + e − annihilation were originally implemented in PHOKHARAthrough the leptonic tensor. For future applications, how-ever, it will be more convenient to implement those cor-rections directly at the amplitude level (in preparation).In terms of sub-amplitudes, the leptonic tensor is given by L µν ISR = L (0) ,µν ISR + M (1) , µ ISR (cid:16) M (0) , ν ISR (cid:17) † + M (0) , µ ISR (cid:16) M (1) , ν ISR (cid:17) † + 12(2 π ) d − Z w √ s E d − d E d Ω M (0) , µ (cid:16) M (0) , ν (cid:17) † , (146)where E and Ω are the energy and the solid angle of thesoft photon, respectively, and d = 4 − ǫ is the numberof dimensions in dimensional regularisation. The leptonictensor has the general form L µν ISR = (4 πα ) Q (cid:20) a g µν + a p µ p ν s + a p µ p ν s + a p µ p ν + p µ p ν s + iπ a − p µ p ν − p µ p ν s (cid:21) , (147)where the scalar coefficients a ij and a − allow the follow-ing expansion: a ij = a (0) ij + απ a (1) ij , a − = απ a (1) − . (148)The imaginary antisymmetric piece, which is proportionalto a − , appears for the first time at second order and isparticularly relevant for those cases where the hadroniccurrent receives contributions from different amplitudeswith nontrivial relative phases. This is possible, e.g., forfinal states with three or more mesons, or for p ¯ p produc-tion. The LO coefficients a (0) ij can be read directly from Eq.(110) a (0)00 = 2 m q (1 − q ) y y − q + y + y y y ,a (0)11 = 8 m y − q y y , a (0)22 = a (0)11 ( y ↔ y ) ,a (0)12 = − m y y . (149)The NLO coefficients a (1) ij and a (1) − are obtained bycombining the one-loop and the soft contributions. It isconvenient to split the coefficients a (1) ij into a part thatcontributes at large photon angles and a part proportionalto m e and m e which is relevant only in the collinear re-gions. These coefficients are denoted by a (1 , ij and a (1 ,m ) ij ,respectively: a (1) ij = a (0) ij (cid:20) − log(4 w )[1 + log( m )] − 32 log( m q ) − π (cid:21) + a (1 , ij + a (1 ,m ) ij . (150)The factor proportional to the LO coefficients a (0) ij con-tains the usual soft and collinear logarithms. The quantity w denotes the dimensionless value of the soft photon en-ergy cutoff, E γ < w √ s . It is enough to present four out ofthe five coefficients because exchanging the positron withthe electron momenta leads to the symmetry relation a (1)22 = a (1)11 ( y ↔ y ) . (151)The large-angle contributions have been calculated inRef. [331]. The coefficient proportional to g µν reads a (1 , = 1 y y (cid:20) − q (1 − q )2 − y y − (cid:20) q + 2 y y − q (cid:21) log( q )+ (cid:26) y (cid:20) − y − q )1 − y (cid:21) log( y q ) − (cid:20) − y ) + y q y (cid:21) L ( y ) + ( y ↔ y ) (cid:27)(cid:21) , (152)where the function L is defined as L ( y i ) = Li ( − y i q ) − Li (1 − q )+ log( q + y i ) log( y i q ) , (153)with Li the Spence (or dilogarithmic) function definedbelow Eq. (94). The coefficient in front of the tensor struc- ture p µ p ν is given by a (1 , = 1 y y (cid:20) (1 + q ) (cid:18) − y − − q (cid:19) − − y ) y − q − q − q (cid:20) (1 − y ) (cid:18) y + q y + 2 y − q (cid:19) + 2 q − q (cid:21) log( q ) − q (cid:20) y (cid:21) log( y q ) − q (cid:20) (2 − y )(1 − y ) y (1 − y ) (cid:21) log( y q ) − q (cid:20) y (cid:21) L ( y ) − q (cid:20) q y + q y (cid:21) L ( y ) (cid:21) . (154)For the symmetric tensor structure ( p µ p ν + p µ p ν ) one gets a (1 , = 1 y y (cid:20) − q + ( y − y ) − q − q (cid:20) q y y + 1 + q − y y (1 − q ) (cid:21) log( q ) + (cid:26) q − y − q − y (cid:20) − y + q y − q − y ) (cid:21) log( y q ) − q (cid:20) q y + q y (cid:21) L ( y ) + ( y ↔ y ) (cid:27)(cid:21) . (155)Finally, the antisymmetric coefficient a − accompanying( p µ p ν − p µ p ν ) reads a (1 , − = q y y (cid:20) − y ) y + 1 − q − y + q (1 − y ) (cid:21) − ( y ↔ y ) . (156)The mass-suppressed coefficients a (1 ,m ) ij are given by [332] a (1 ,m )00 = m q y (cid:20) log( q ) log( y m q ) + 4Li (1 − q )+ Li (1 − y m ) − π (cid:21) − m (1 − q ) y (cid:20) − log( y m )+ m y (cid:18) Li (1 − y m ) − π (cid:19)(cid:21) + q n ( y , − q q )+ ( y ↔ y ) , (157) whereas a (1 ,m )11 = q − q (cid:26) m y (cid:20) − log( y m )+ m y (cid:18) Li (1 − y m ) − π (cid:19)(cid:21) − n ( y , m q y ( m (1 − q ) − y ) (cid:20) q log( y m ) + log( q )1 − q + (cid:18) m m (1 − q ) − y (cid:19) N ( y ) (cid:21)(cid:27) ++ 11 − q (cid:26) m (1 − q ) y (cid:20) log( q ) log( y m q )+ 4Li (1 − q ) + 2 (cid:18) Li (1 − y m ) − π (cid:19)(cid:21) + 4 m q y (cid:20) − log( y m ) + (cid:18) m y (cid:19)(cid:18) Li (1 − y m ) − π (cid:19)(cid:21) − − q q n ( y , − q + 6 q − q )+ 2 m y ( m (1 − q ) − y ) (cid:20) q log( y m ) + log( q )1 − q + (cid:18) m m (1 − q ) − y (cid:19) N ( y ) (cid:21)(cid:27) , (158)and a (1 ,m )12 = q − q (cid:26) m y (cid:20) − log( y m )+ (cid:18) 12 + m y (cid:19)(cid:18) Li (1 − y m ) − π (cid:19)(cid:21) − − q q n ( y , − q ) + 2 m y ( m (1 − q ) − y ) × (cid:20) q log( y m ) + log( q )1 − q + (cid:18) m m (1 − q ) − y (cid:19) N ( y ) (cid:21)(cid:27) + ( y ↔ y ) . (159)The asymmetric coefficient does not get mass corrections, a (1 ,m ) − = 0 . (160)The functions n ( y i , z ) and N ( y i ) are defined through n ( y i , z ) = m y i ( m − y i ) (cid:20) z log( y i m ) (cid:21) + m ( m − y i ) log( y i m ) , (161)and N ( y i ) = log( q ) log( y i m ) + Li (1 − q )+ Li (1 − y i m ) − π . (162) The apparent singularity of the function n ( y i , z ) insidethe phase space limits is compensated by the zero in thenumerator. In the region y i close to m it behaves as n ( y i , z ) (cid:12)(cid:12) y i → m = 1 y i (cid:20) z log( y i m ) (cid:21) − m X n =0 (cid:18) n + 2 + zn + 1 (cid:19) (cid:16) − y i m (cid:17) n . (163)Similarly, the function N ( y i ) guarantees that the coeffi-cients a (1) ij are finite in the limit y i → m (1 − q ): m N ( y i ) m (1 − q ) − y i (cid:12)(cid:12)(cid:12)(cid:12) y i → m (1 − q ) = − log(1 − q ) q − log( q )1 − q . (164) Virtual and soft corrections to IFS The virtual plus soft photon corrections of the initial-state and final-state vertex (see Eq. (143)) to FSR andISR, respectively, can be written as [430,431]d σ V+SIFS = απ h δ V+S ( w ) d σ (0)FSR ( s )+ η V+S ( s ′ , w ) d σ (0)ISR ( s ′ ) i , (165)where d σ (0)FSR and d σ (0)ISR are the leading order FSR and ISRdifferential cross sections, respectively, w = E cut γ / √ s with E cut γ the maximal energy of the soft photon in the e + e − c.m. rest frame, and s ′ corresponds to the squared massof the h ¯ hγ system. The function δ V+S ( w ) is independentof the final state. In the limit m e ≪ s , δ V+S ( w ) = 2 (cid:20) ( L − 1) log (2 w ) + 34 L − π (cid:21) , (166)where L = log( s/m e ). For two pions in the final state, thefunction η V+S ( s ′ , w ) is given by η V+S ( s ′ , w ) = − (cid:20) β π β π log( t π ) + 1 (cid:21) × (cid:20) log(2 w ) + 1 + s ′ s ′ − s log (cid:16) ss ′ (cid:17)(cid:21) + log (cid:18) m π s ′ (cid:19) − β π β π (cid:20) (1 − t π ) + log( t π ) log(1 + t π ) − π (cid:21) − β π β π log( t π ) − , (167)where β π = r − m π s ′ , t π = 1 − β π β π . (168)The function η V+S ( s ′ , w ) is equivalent to the famil-iar correction factor derived in [280,281] for the reaction e + e − → π + π − γ in the framework of sQED (see also [283])in the limit s → s ′ :log(2 w ) + 1 + s ′ s ′ − s log (cid:16) ss ′ (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) s → s ′ = log(2 w ′ ) (169)with w ′ = E cut γ / √ s ′ . The factor on the right hand side ofEq. (169) for s = s ′ arises from defining the soft photoncutoff in the e + e − laboratory frame.Correspondingly, the function η V+S ( s ′ , w ) for two muonsin the final state reads η V+S ( s ′ , w ) = − " β µ β µ log( t µ ) + 1 × (cid:20) log(2 w ) + 1 + s ′ s ′ − s log (cid:16) ss ′ (cid:17)(cid:21) + log m µ s ′ ! − β µ β µ (cid:20) (1 − t µ ) − t µ ) log (cid:18) β µ (cid:19) − π (cid:21) − β µ (cid:20) − β µ + β µ (cid:21) log( t µ ) − , (170)where β µ = r − m µ s ′ , t µ = 1 − β µ β µ . (171) Real corrections Matrix elements for the emission of two real photons, e + ( p ) + e − ( p ) → hadrons ( Q ) + γ ( k ) + γ ( k ) , (172)are calculated in PHOKHARA following the helicity am-plitude method with the conventions introduced in [432,433]. The Weyl representation for fermions is used wherethe Dirac matrices γ µ = (cid:18) σ µ + σ µ − (cid:19) , µ = 0 , , , , (173)are given in terms of the unit 2 × I and thePauli matrices σ i , i = 1 , , 3, with σ µ ± = ( I, ± σ i ). Thecontraction of any four-vector a µ with the γ µ matriceshas the form a/ = a µ γ µ = (cid:18) a + a − (cid:19) , (174)where the 2 × a ± are given by a ± = a µ σ ± µ = (cid:18) a ∓ a ∓ ( a − ia ) ∓ ( a + ia ) a ± a (cid:19) . (175)The helicity spinors u and v for a particle and anantiparticle of four-momentum p = ( E, p ) and helicity λ = ± / u ( p, λ = ± / 2) = (cid:18) p E ∓ | p | χ ( p , ± ) p E ± | p | χ ( p , ± ) (cid:19) ≡ (cid:18) u I u II (cid:19) ,v ( p, λ = ± / 2) = (cid:18) ∓ p E ± | p | χ ( p , ∓ ) ± p E ∓ | p | χ ( p , ∓ ) (cid:19) ≡ (cid:18) v I v II (cid:19) . (176) The helicity eigenstates χ ( p , λ ) can be expressed in termsof the polar and azimuthal angles of the momentum vector p as χ ( p , +) = (cid:18) cos ( θ/ e iφ sin ( θ/ (cid:19) ,χ ( p , − ) = (cid:18) − e − iφ sin ( θ/ θ/ (cid:19) . (177)Finally, complex polarisation vectors in the helicity basisare defined for the real photons: ε µ ( k i , λ i = ± ) = 1 √ (cid:0) , ∓ cos θ i cos φ i + i sin φ i , ∓ cos θ i sin φ i − i cos φ i , ± sin θ i (cid:1) , (178)with i = 1 , Phase space One of the key ingredients of any Monte Carlo simula-tion is an efficient generation of the phase space. The gen-eration of the multi-particle phase space in PHOKHARAis based on the Lorentz-invariant representationd Φ m + n ( p , p ; k , · , k m , q , · , q n ) =d Φ m ( p , p ; Q, k , · , k m )d Φ n ( Q ; q , · , q n ) d Q π , (179)where p and p are the four-momenta of the initial par-ticles, k . . . k m are the four momenta of the emitted pho-tons and q . . . q n , with Q = P q i , label the four-momentaof the final state hadrons.When two particles of the same mass are produced inthe final state, q i = M , their phase space is given byd Φ ( Q ; q , q ) = q − M Q π d Ω , (180)where d Ω is the solid angle of one of the final state parti-cles at, for instance, the Q rest frame.Single photon emission is described by the correspond-ing leptonic part of the phase space,d Φ ( p , p ; Q, k ) = 1 − q π d Ω , (181)with q = Q /s and dΩ the solid angle of the emittedphoton at the e + e − rest frame. The polar angle θ is de-fined with respect to the positron momentum p . In orderto make the Monte Carlo generation more efficient, thefollowing substitution is performed:cos θ = 1 β tanh( β t ) , t = 12 β log 1 + β cos θ − β cos θ , (182)with β = p − m e /s , which accounts for the collinearemission peaks d cos θ − β cos θ = d t . (183) With this the azimuthal angle and the new variable t aregenerated flat.Considering the emission of two real photons in thec.m. of the initial particles, the four-momenta of the posi-tron, the electron and the two emitted photons are givenby p = √ s , , , β ) , p = √ s , , , − β ) ,k = w √ s (1 , sin θ cos φ , sin θ sin φ , cos θ ) ,k = w √ s (1 , sin θ cos φ , sin θ sin φ , cos θ ) , (184)respectively. The polar angles θ and θ are again definedwith respect to the positron momentum p . Both photonsare generated with energies larger than the soft photoncutoff: w i > w with i = 1 , 2. At least one of these exceedsthe minimal detection energy: w > E min γ / √ s or w >E min γ / √ s . In terms of the solid angles d Ω and d Ω of thetwo photons and the normalised energy of one of them,e.g. w , the leptonic part of the phase space readsd Φ ( p , p ; Q, k , k ) = 12! s π ) × w w − q − w d w d Ω d Ω , (185)where the limits of the phase space are determined fromthe constraint q = 1 − w + w ) + 2 w w (1 − cos χ ) , (186)with χ being the angle between the two photonscos χ = sin θ sin θ cos( φ − φ ) + cos θ cos θ . (187)Again, the matrix element squared contains severalpeaks, soft and collinear, which should be softened bychoosing suitable substitutions in order to achieve an ef-ficient Monte Carlo generator. The leading behaviour ofthe matrix element squared is given by 1 / ( y y y y ),where y ij = 2 k i · p j s = w i (1 ∓ β cos θ i ) . (188)In combination with the leptonic part of the phase space,we haved Φ ( p , p ; Q, k , k ) y y y y ∼ d w w (1 − q − w ) × d Ω − β cos θ d Ω − β cos θ . (189)The collinear peaks are then flattened with the help of Eq.(182), with one change of variables for each photon polarangle. The remaining soft peak, w → w , is reabsorbedwith the following substitution w = 1 − q e − u , u = log w − q − w , (190) or d w w (1 − q − w ) = d u − q , (191)where the new variable u is generated flat. Multi-channe-ling is used to absorb simultaneously the soft and collinearpeaks, and the peaks of the form factors. NLO cross section and theoretical uncertainty The LO and NLO predictions for the differential crosssection of the process e + e − → π + π − γ ( γ ) at DAΦNE en-ergies, √ s = 1 . 02 GeV, are presented in Fig. 47 as a func-tion of the invariant mass of the hadronic system M ππ .We choose the same kinematical cuts as in the small an-gle analysis of KLOE [374]; pions are restricted to be inthe central region, 50 o < θ π < o , with | p T | > 160 MeVor | p z | > 90 MeV, the hard photon is not tagged and thesum of the momenta of the two pions, which flows in theopposite direction to the photon’s momenta, is close to thebeam ( θ ππ < o or θ ππ > o ). The track mass, whichis calculated from the equation (cid:18) √ s − q | p π + | + M − q | p π − | + M (cid:19) − ( p π + + p π − ) = 0 , (192)lies within the limits 130 MeV < M trk < 220 MeV and M trk < (250 − p − ( M ππ / . ) MeV, with M ππ inGeV, in order to reject µ + µ − and π + π − π events. Thecut on the track mass, however, does not have any effectfor single photon emission, as obviously M trk = m π forsuch events.The lower plot in Fig. 47 shows the relative size, withrespect to the LO prediction, of FSR at LO, ISR cor-rections at NLO, and IFS contributions. The NLO ISRradiative corrections are almost flat and of the order of − M ππ .To estimate the systematic uncertainty of the NLOprediction, we observe that leading logarithmic two-loop O ( α ) corrections and the associate real emission are notincluded. For samples with untagged photons the process e + e − → e + e − π + π − might also become a sizable back-ground. This process, however, can be simulated with theMonte Carlo event generator EKHARA [224,223]. Its con-tribution depends on the pion pair invariant mass, rangesfrom 0 . − . 8% for the KLOE event selection, and hasbeen taken into account in the KLOE analysis [374].From na¨ıve exponentiation one expects that LL correc-tions at next-to-next-to-leading order (NNLO) are of theorder of ( ( α/π ) log( s/m e )) ≈ . . 2% for inclusiveobservables. For less inclusive distributions, a larger erroris expected. The conservative estimate of the accuracy ofPHOKHARA from ISR is 0 . . 3% in the invariant massregions which are not close to the nominal energies of the L d Σ H e + e - ® Π + Π - Γ H Γ LL (cid:144) d M ΠΠ H nb G e V - L (cid:143)!!!! s = Θ ΠΠ < o or Θ ΠΠ > o o < Θ Π < o p T > È p z È > 90 MeVLONLO0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 - - - M ΠΠ H GeV L d Σ H L FSR ’ d Σ H L d Σ H L IFS ’ d Σ H L d Σ H L ISR ’ d Σ H L Fig. 47. Differential cross section for the process e + e − → π + π − γ at LO and NLO for √ s = 1 . 02 GeV. The cuts are thesame as in the small angle analysis of KLOE, including the cuton the track mass. The lower plot shows the relative size ofFSR at LO, ISR at NLO and IFS contributions with respectto the full LO prediction. experiments. Improving the accuracy of PHOKHARA be-low 0 . ∗ VMD model The model for FSR from pions described in details in Sec-tions 4.2.1 and 4.2.3 will be called for short the sQED ∗ VMDmodel. The question arises how well it can reflect the data.As shown in [317], the first two terms in the expansion ofthe FSR amplitude as a function of k / p Q (i.e. the di-vergence and the constant) are fully given by the pion formfactor. Thus one could expect that going beyond this ap-proximation is necessary only for a hard photon emission.Moreover, the pion form factor is extremely big in the ρ resonance region, and thus the validity of this approxima-tion is further extended. In the kinematical regions whereresonance contributions are not contained in the pion form factor, and also near the π + π − threshold, where the emit-ted photon is hard and the pion form factor is relativelysmall, it is necessary to go beyond the sQED ∗ VMD modeland one needs a more general description of the amplitude M ( γ ∗ ( Q ) → γ ( k ) + π + ( q ) + π − ( q )).In the general case the amplitude of the reaction γ ∗ ( Q ) → γ ( k ) + π + ( q ) + π − ( q ) depends on three 4-momenta, which can be chosen as Q , k and l ≡ q − q . Thesecond-rank Lorentz tensor M µν ( Q, k, l ) that describesthe FSR amplitude can be decomposed through ten inde-pendent tensors [434,435]. Taking into account the chargeconjugation symmetry of the S-matrix element( h γ ( k ) , π + ( q ) π − ( q ) | S | γ ∗ ( Q ) i = h γ ( k ) , π − ( q ) π + ( q ) | S | γ ∗ ( Q ) i ),the photon crossing symmetry ( Q ↔ − k and µ ↔ ν ) andthe gauge invariance conditions Q µ M µν ( Q, k, l ) = 0 and M µνF ( Q, k, l ) k ν = 0, the number of independent tensorsdecreases to five. For a final real photon, i.e. k = 0 and k ν ǫ ν = 0 ( ǫ ν being the polarisation vector of the finalphoton) and the initial virtual photon produced in e + e − annihilation ( Q ≥ m π ), the FSR tensor can be rewrit-ten in terms of three gauge invariant tensors [434,435] M µν ( Q, k, l ) = τ µν f + τ µν f + τ µν f , (193)where the gauge invariant tensors τ µνi read τ µν = k µ Q ν − g µν k · Q, (194) τ µν = k · l ( l µ Q ν − g µν k · l ) + l ν ( k µ k · l − l µ k · Q ) ,τ µν = Q ( g µν k · l − k µ l ν ) + Q µ ( l ν k · Q − Q ν k · l ) . It thus follows that the evaluation of the FSR tensoramounts to the calculation of the scalar functions f i ( Q , Q · k, k · l ) ( i = 1 , , ∗ VMD model.Extensive theoretical studies of the role of the FSRemission beyond the sQED ∗ VMD model were performed[337,347,349,351,350]. They are important mainly for theKLOE measurements at DAΦNE, as at B factories FSRis naturally suppressed and the accuracy needed in itsmodelling is by far less demanding than that for KLOEpurposes.For DAΦNE, running on or near the φ resonance, thefollowing mechanisms of the π + π − final state photon emis-sion have to be considered:– bremsstrahlung process e + + e − → π + + π − + γ , (195)which is modelled by sQED ∗ VMD; – direct φ decay e + + e − → φ → ( f ; f + σ ) γ → π + + π − + γ , (196)and– double resonance process e + + e − → ( φ ; ω ′ ) → ρπ → π + + π − + γ . (197)The resonance chiral theory (R χ T) [436,437] was used in[349,350] to estimate the contributions beyond sQED ∗ VMD.They were implemented at leading order into the eventgenerator FASTERD [351]. Having in mind that at presentthese models still await accurate experimental tests, othermodels [438,439] were also implemented in the event gen-erator FASTERD. To include both next-to-leading-orderradiative corrections and the mechanisms discussed forFSR, a part of the FASTERD code, based on the mod-els [438,439], was implemented by O. Shekhovtsova inPHOKHARA v6.0 (PHOKHARA v6.1 [440]) and the stud-ies presented below are based on this code. The model usedthere, even if far from an ideal, is the best tested modelavailable in literature.We briefly describe main features of the models usedto describe processes contributing to FSR photon emis-sion listed above. For a more detailed description and thecalculation of the function f i we refer the reader to [337,347,351] (see also references therein).The sQED ∗ VMD part gives contributions to f and f . The direct φ decay is assumed to proceed through theintermediate scalar meson state: φ → ( f + σ ) γ → ππγ .Various models are proposed to describe the φ -scalar- γ vertex: either it is the direct decay φ → ( scalar ) γ , orthe vertex is generated dynamically through a loop of thecharged kaons. As shown in [347], in the framework of anymodel, the direct φ decay affects only the form factor f of Eq. (193).The double resonance contribution consists of the off-shell φ meson decay into ( ρ ± π ∓ ) and subsequent decay ρ → πγ . In the energy region around 1 GeV the tail of theexcited ω meson can also play a role, and γ ∗ → ω ′ → ρπ has to be considered. The double resonance mechanismaffects all three form factors f i of Eq. (193).Assuming isospin symmetry, this part can be deducedfrom the measurement of the neutral pion pair production.Various models [438,439] were confronted with data byKLOE [441] for the neutral mode. The model that wasreproducing the data in the best way was adopted to beused for the charged pion pair production relying on theisospin symmetry [440].In [337] it was shown that an important tool for testingthe various models of FSR is the charge asymmetry. Atleading order it originates from the fact that the pion paircouples to an even (odd) number of photons if the finalstate photon is emitted from the final (initial) state. Theinterference diagrams do not give any contribution to theintegrated cross section for C–even event selections, butproduce an asymmetry in the angular distribution. Thedefinitions and experimental studies based on the chargeasymmetry are presented in Section 4.3.2. Few strategies can be adopted to profit in the bestway from the KLOE data taken on and off peak. The’easiest’ part is to look for the event selections where theFSR contributions are negligible. This was performed byKLOE [374] (see Section 4.4.1), giving important informa-tion on the pion form factor relevant for the prediction ofthe hadronic contributions to the muon anomalous mag-netic moment a µ . Typical contributions of the FSR (1 –4%) to the differential cross section (Figs. 47 and 48) allowfor excellent control of the accuracy of these corrections.One disadvantage of using this event selection is that itdoes not allow to perform measurements near the pionproduction threshold. -0.05-0.04-0.03-0.02-0.0100.010.020.030.040.050.3 0.35 0.4 0.45 0.5 PSfrag replacements Q (GeV ) d σ ( I F S N L O ) d Q / d σ ( I S R N L O ) d Q − o < θ π ± < o θ ( ~p π + + ~p π − ) < o or < o M tr cutno M tr cut Fig. 48. Relative contribution of the FSR to the differentialcross section of the reaction e + e − → π + π − γ ( γ ) for √ s = m φ and low invariant masses of pion pairs. KLOE small angle eventselection [374] was used, and for this event selection the relativecontribution of the FSR is almost identical also for the off peakcross section. The effect of a trackmass cut (see Section 4.4.1)is shown. ISRNLO refers to initial state corrections at next-to-leading order (NLO). The IFSNLO cross section contains thefinal state emissions at NLO. The next step, partly discussed in Section 4.3.2, is toconfront the models based on isospin symmetry and theneutral channel data with charged pion data taken off-peak, where the contributions from models beyond thesQED ∗ VMD approximation are relatively small (Fig. 49).For the off-peak data [442] the region below Q = 0 . can be covered experimentally. However, the small statis-tics in this region makes it difficult to perform high-pre-cision tests of the models. For this analysis an accurateknowledge of the pion form factor at the nominal energy ofthe experiment is important, as it defines the sQED ∗ VMDpredictions and as the FSR corrections (Fig. 50) are size-able.The last step, which allows for the most accurate FSRmodel testing and profits from the knowledge of the pionform factor from previous analysis, is the on-peak large an-gle measurement. The large FSR corrections coming fromsources beyond the sQED ∗ VMD approximation (Figs. 49and 50) make these data [443] the most valuable source of -0.100.10.20.30.40.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 PSfrag replacements Q (GeV ) d σ ( I F S N L O ( T O T )) d Q / d σ ( I F S N L O ( S Q E D )) d Q − o < θ π ± < o o < θ γ < o s = m φ GeV s = 1 GeV Fig. 49. The contributions of FSR beyond the sQED ∗ VMDapproximation (see Eqs. (196) and (197)) for KLOE large angleevent selection [442,443] for √ s = m φ and for √ s = 1 GeV. information on these models. In this case, the accumulateddata set is much larger than the off-peak data set and oneis able to cover also the region below Q = 0 . . -0.4-0.200.20.40.60.811.21.40.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 PSfrag replacements Q (GeV ) d σ ( I F S N L O ( T O T )) d Q / d σ ( I S R N L O ) d Q − o < θ π ± < o o < θ γ < o s = m φ GeV s = 1 GeV Fig. 50. Relative contribution of FSR to the differential crosssection of the reaction e + e − → π + π − γ ( γ ) for √ s = m φ andfor √ s = 1 GeV. KLOE large angle event selection [442,443]was used. e + e − → π + π − γ with FSR withthe CMD-2 detector at VEPP-2M The process e + e − → π + π − γ with final state radiation canbe used to answer the question whether one can treat pions as point-like particles and apply scalar QED to calculatethe radiative corrections to the cross section. In particular,one can compare the photon spectra obtained using scalarQED with those found in data.The radiative corrections due to photon emission inthe final state (FSR) contribute about 1% to the crosssection. The hadronic contribution of the process e + e − → π + π − to the value a had µ amounts to ∼ 50 ppm, while theanomalous magnetic moment of the muon was measuredin the E821 experiment at BNL with an accuracy of 0.5ppm [31]. Therefore the theoretical precision of the crosssection calculation for this process should be several timessmaller than 1%. In this case we can neglect the error ofthis contribution to the value a had µ compared to 0.5 ppm.These facts are the main motivation to study this process. Event selection For the analysis, data were taken in a c.m. energy rangefrom 720 to 780 MeV, with one photon detected in the CsIcalorimeter. Events from the processes e + e − → e + e − γ and e + e − → µ + µ − γ have a very similar topology in thedetector, compared to e + e − → π + π − γ events. In addi-tion, the cross section of the process e + e − → π + π − γ withFSR is more than ten times smaller than the one for thesimilar process with ISR. On the other hand, the cross sec-tion of the process e + e − → π + π − γ has a strong energydependence due to the presence of the ρ -resonance. Thisfact allows to significantly enrich the fraction of the events e + e − → π + π − γ with FSR for energies below the ρ -peak.Indeed, ISR shifts the c.m. energy to smaller values and, asa result, the cross section falls down dramatically, whereasthe process with FSR is almost energy-independent. Sev-eral curves describing the ratio σ FSR+ISR π + π − γ /σ ISR π + π − γ plot-ted against the c.m. energy, are presented in Fig. 51 (a)for different energy thresholds for photons detected in thecalorimeter. It is clearly visible that the optimal energyrange to be used in this study goes from 720 MeV up to780 MeV.It is also seen that this ratio increases with the thresh-old energy for photons to be detected. This means thatthe fraction of the π + π − γ events with FSR (with respectto events without FSR) grows with increasing photon en-ergy. It allows to enrich the number of π + π − γ events withFSR. Let us recollect that the shape of the distributionof π + π − γ events, at photon energies of the same order asthe pion mass or larger, is of special interest. First of all,namely in that part of the photon spectrum we can meeta discrepancy with the sQED prediction.A typical π + π − γ event in the CMD-2 detector has twotracks in the drift chamber with two associated clustersin the CsI calorimeter and a third cluster representingthe radiated photon. To suppress multi-photon events andsignificantly cut off collinear π + π − events the followingrequirements were applied: the angle between the directionof photon momentum and missing momentum must belarger than 1 rad and the angle between one of the twotracks and the photon direction must be smaller than 0.2rad. 700 720 740 760 780 800 I S R σ / I S R + FS R σ (a)(b) Fig. 51. (a) Ratio σ ISR+FSR /σ ISR vs the c.m. energy. Theset of curves indicates how this ratio depends on the thresholdenergy for the detected photons. The threshold energy in MeVis stated over the curves. (b) Distributions of the parameter W for events of the processes e + e − → π + π − γ , e + e − → µ + µ − γ and e + e − → e + e − γ , for a c.m. energy of 780 MeV. To suppress e + e − γ events, a parameter W = p/E wasused, in which the particle momentum p (measured in thedrift chamber) is divided by the energy E (measured inthe CsI calorimeter). Simulation results are presented inFig. 51 (b). The condition W < . ∼ M > furtherrejects the number of electrons and muons by a factor of1.5. About 1% of the pion events are lost with these cuts. (a)(b) Fig. 52. (a) Distributions of the parameter M for events ofthe processes e + e − → π + π − γ , e + e − → µ + µ − γ and e + e − → e + e − γ for a c.m. energy of 780 MeV. (b) Distribution of the π + π − γ events against the photon energy in relative units. Alsostated is the fraction of π + π − γ events with FSR for each regionas indicated by the vertical lines. Preliminary results of the analysis The histogram of the number π + π − γ events againstthe photon energy in relative units is presented in Fig. 52(b). The histogram represents the simulation, while thepoints with error bars show the experimental data. Ver-tical dotted lines divide the plot area into three zones.The inscription inside each zone indicates the fraction of π + π − γ events with FSR with respect to others. The num-ber of the simulated events was normalised to the experi-mental one. The average deviation between the two distri-butions was found to be ( − . ± . Forthcoming experiments at VEPP-2000 will significantlyimprove the statistical error. e + e − → π + π − γ with FSR withKLOE detector As has been explained in Section 4.2, the forward-backwardasymmetry A F B ( Q ) = N ( θ π + > ◦ ) − N ( θ π + < ◦ ) N ( θ π + > ◦ ) + N ( θ π + < ◦ ) (cid:0) Q (cid:1) (198)can be used to test the validity of the description of thevarious mechanisms of the π + π − final state photon emis-sion, by confronting the output of the Monte Carlo gener-ator with data. In the following studies, the Monte Carlogenerator PHOKHARA v6.1 [440] was used. The parame-ters for the pion form factor were taken from [444], basedon the parametrisation of K¨uhn and Santamaria [445]. Theparameters for the description of the direct φ decay andthe double resonance contribution were taken from theKLOE analysis of the neutral mode [441].To suppress higher order effects, for which the interfer-ence and thus the asymmetry is not implemented in theMonte Carlo generator, a rather tight cut on the trackmass variable (see Section 4.4.1 and Fig. 60) of | M trk − M π ± | < 10 MeV has been applied in the data, in additionto the large angle selection cuts described in Section 4.4.1.This should reduce events with more than one hard pho-ton emitted and enhance the contribution of the final stateradiation processes under study over the dominant ISRprocess.The datasets used in the analysis were taken in twodifferent periods: • The data taken in 2002 were collected with DAΦNEoperating at the φ -peak, at √ s = M φ (240 pb − ). • The data taken in 2006 were collected with DAΦNEoperating 20 MeV below the φ -peak, at √ s = 1000MeV (230 pb − ).Since the 2006 data were taken more than 4 Γ φ be-low the resonant peak ( Γ φ = 4 . 26 MeV), one expects thecontributions from the direct φ decay and the double reso-nance contribution to be suppressed compared to the datataken on-peak in 2002 (see Fig. 49). In fact one observes avery different shape of the forward-backward asymmetryfor the two different datasets, as can be seen in Figs. 53and 54. Especially in the region below 0.4 GeV and in thevicinity of the f (980) at 0.96 GeV , one observes differenttrends in the asymmetries for the two datasets.One can also see that, qualitatively, the theoretical de-scription used to model the different FSR contributionsagrees well with the data, although, especially at low M ππ ,the data statistics becomes poor and the data points forthe asymmetry have large errors. In particular, the off-peak data in Fig. 54 show very good agreement above0.35 GeV . In this case, the asymmetry is dominated fullyby the bremsstrahlung-process, as the other processes donot contribute outside the φ -resonance. The assumption of point-like pions (sQED) used to describe the bremsstrahlungin the Monte Carlo generator seems to be valid above 0.35GeV , while below it is difficult to make a statement dueto the large statistical errors of the data points.However, to obtain a solid quantitative statement onthe validity of the models, as needed, e.g., in the radiativereturn analyses at the KLOE experiment, one needs tounderstand how a discrepancy between theory and datain the forward-backward asymmetry affects the cross sec-tion, as it is the cross section one wants to measure. Thisrequires further work, which at the moment is still inprogress.It should also be mentioned that the KLOE experi-ment has taken almost ten times more data in the years2004–2005 than what is shown in Fig. 53, with DAΦNEoperating at the φ -peak energy. This is unfortunately notthe case for the off-peak data, which is restricted to thedataset shown in Fig. 54. In the future, the larger datasetfrom 2004–2005 may be used, together with the resultsfrom the neutral channel and the assumption of isospinsymmetry, to determine the parameters of the direct φ decay and the double resonance contribution with highprecision. The KLOE experiment, in operation at the DAΦNE e + e − collider in Frascati between 1999 and 2006, utilises radia-tive return to obtain precise measurements of hadroniccross sections in the energy range below 1 GeV. As theDAΦNE machine was designed to operate as a meson fac-tory with collision energy equal to the mass of the φ -meson( m φ = 1.01946 GeV), with limited possibility to changethe energy of the colliding beams while maintaining sta-ble running conditions, the use of events with initial stateradiation of hard photons from the e + or the e − is theonly way to access energies below DAΦNE’s nominal col-lision energy. These low-energy cross sections are impor-tant for the theoretical evaluation of the muon magneticmoment anomaly a µ = ( g µ − / e + e − → π + π − gives the largest contribution tothe hadronic part a had µ of the anomaly. Therefore, so farKLOE efforts have concentrated on the derivation of thepion pair-production cross section σ ππ from measurementsof the differential cross section d σ ππγ ( γ ) d M ππ , in which M ππ isthe invariant mass squared of the di-pion system in thefinal state.The KLOE detector (shown in Fig. 55), which con-sists of a high resolution drift chamber ( σ p /p ≤ . σ t ∼ 54 ps / p E [GeV] ⊕ 100 ps) and good energy ( σ E /E ∼ . / p E [GeV]) resolution, is optimally suited for thiskind of analyses. Data 2002Monte Carlo M ππ [ GeV ] F . - B . asy mm e t r y -0.2-0.100.10.20.30.4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 (a) M ππ [ GeV ] -0.15-0.1-0.0500.050.10.150.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 (b) Fig. 53. (a) Preliminary Forward–Backward asymmetry fordata taken at √ s = M φ in 2002, and the corresponding MonteCarlo prediction using the PHOKHARA v6.1 generator. (b)Absolute difference between the asymmetries from data andMonte Carlo prediction. Used with permission of the KLOEcollaboration. The KLOE ππγ analyses The KLOE analyses for σ ππ use two different sets ofacceptance cuts: • In the small angle analysis, photons are emitted withina cone of θ γ < ◦ around the beamline (narrow conesin Fig. 55), and the two charged pion tracks have 50 ◦ <θ π < ◦ . The photon is not explicitly detected; itsdirection is reconstructed from the track momenta byclosing the kinematics: p γ ≃ p miss = − ( p π + + p π − ).In this analysis, the separation of pion- and photon se-lection regions greatly reduces the contamination fromthe resonant process e + e − → φ → π + π − π in whichthe π mimics the missing momentum of the photon(s)and from the final state radiation process e + e − → π + π − γ FSR . Since ISR-photons are mostly collinear withthe beam line, a high statistics for the ISR signal eventsremains. On the other hand, a high energy photonemitted at angles close to the incoming beams forcesthe pions also to have a small angle with respect to Data 2006Monte Carlo M ππ [ GeV ] F . - B . asy mm e t r y -0.4-0.3-0.2-0.100.10.20.30.4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 (a) M ππ [ GeV ] -0.15-0.1-0.0500.050.10.150.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 (b) Fig. 54. (a) Preliminary Forward–Backward asymmetry fordata taken at √ s ≃ the beamline (and thus outside the selection cuts),resulting in a kinematical suppression of events with M ππ < . 35 GeV . • The large angle analysis requires both photons and pi-ons to be emitted at 50 ◦ < θ π,γ < ◦ (wide cones inFig. 55), allowing for a detection of the photons in thebarrel of the calorimeter. This analysis allows to reachthe 2 π threshold region, at the price of higher back-ground contributions from the π + π − π final state andevents with final state radiation. In addition, eventsfrom the decays φ → f γ → π + π − γ and φ → π ± ρ ∓ → π ± π ∓ γ , which need to be described by model-dependentparameterisations, contribute to the spectrum of theselected events (running at the φ peak).Two analyses based on the small angle acceptance cutshave been carried out. The first one, using 140 pb − ofdata taken in the year 2001, was published in 2005 [373].The second one, based on 240 pb − of data taken in 2002,was published in 2008 [446]. Fig. 55. KLOE detector with the selection regions for smallangle photons (narrow cones) and for pion tracks and largeangle photons (wide cones). Used with permission of the KLOEcollaboration. The differential cross section is obtained from the spec-trum of selected events N sel subtracting the residual back-ground (mostly µµγ ( γ ), πππ and radiative Bhabha events)and dividing by the selection efficiencies and the inte-grated luminosity:d σ ππγ ( γ ) d M ππ = N sel − N bkg ∆M ππ · ε sel · R L d t . (199) ∆M ππ is the bin width used in the analysis (typically 0.01GeV ), and R L d t is the integrated luminosity obtainedfrom Bhabha events detected at large angles (55 ◦ < θ e < ◦ ) and the reference cross section from the BabaYagagenerator [233,235] (discussed in Section 2). The totalcross section is then obtained from the formula σ ππ ( M ππ ) = s · d σ ππγ ( γ ) d M ππ H ( s, M ππ ) . (200)In Eq. (200), s is the squared energy at which the DAΦNEcollider is operated during data taking, and H ( s, M ππ )is the radiator function describing the emission of pho-tons from the e + or the e − in the initial state. Note thatEq. (200) does not contain the effects from final state ra-diation from pions. These effects complicate the analy-sis, since the KLOE detector can not distinguish whetherphotons in an event were emitted in the initial or the fi-nal state. The PHOKHARA Monte Carlo generator [335],which includes final state radiation at next-to-leading or-der and in the pointlike-pion approximation, is used toproperly take into account final state radiation in the anal-yses. This is important because the bare cross section usedto evaluate a had µ via an appropriate dispersion integral should be inclusive with respect to final state radiation,and also needs to be undressed from vacuum polarisationeffects present in the virtual photon produced in the e + e − annihilation. For the latter, we use a function provided byF. Jegerlehner [447] (see Section 6), and correct the crosssection via σ bare ππ ( M ππ ) = σ dressed ππ ( M ππ ) (cid:18) α (0) α ( M ππ ) (cid:19) . (201)Here α (0) is the fine structure constant in the limit q = 0,and α ( M ππ ) represents the value of the effective couplingat the scale of the invariant mass of the di-pion system.Since the hadronic contributions to α ( M ππ ) are calculatedvia a dispersion integral which includes the hadronic crosssection itself in the integrand (see Section 6), the correctprocedure has to be iterative and should include the samedata that must be corrected. However, since the correctionis at the few percent level, the inclusion of the new KLOEdata will not change α ( M ππ ) at a level which would sig-nificantly affect the analyses. We therefore have used thevalues for α ( M ππ ) derived from the existing hadronic crosssection database. As an example, Fig. 56 shows the KLOEresult for d σ ππγ ( γ ) / d M ππ obtained from data taken in theyear 2002 [446]. Inserting this differential cross section intoEq. (200) and the result into Eq. (201), one derives σ bare ππ .Using the bare cross section to get the ππ -contribution to a had µ between 0.35 and 0.95 GeV then gives the value (inunits of 10 − ) a ππµ (0 . − . ) = (387 . ± . stat ± . exp ± . th ) . Table 13 shows the contributions to the systematic errorson a ππµ (0 . − . 95 GeV ). M ¼¼ GeV ( ) d () / d ¾ ee ¼¼ ° + ! -- + M ¼¼ ( nb / G e V ) Fig. 56. Differential radiative cross section dσ ππγ ( γ ) /dM ππ ,inclusive in θ π and with 0 o < θ γ < o or 165 o < θ γ < o measured by the KLOE experiment [446]. Used with permis-sion of the KLOE collaboration.4 Reconstruction Filter negligibleBackground subtraction 0.3 %Trackmass 0.2 %Particle ID negligibleTracking 0.3 %Trigger 0.1 %Unfolding negligibleAcceptance ( θ ππ ) 0.2 %Acceptance ( θ π ) negligibleSoftware Trigger (L3) 0.1 %Luminosity (0 . th ⊕ . exp )% 0.3 % √ s dep. of H H Table 13. List of systematic errors on the ππ -contribution to a had µ between 0.35 and 0.95 GeV when using the σ ππ cross sec-tion measured by the KLOE experiment in the correspondingdispersion integral [446]. M ππ [ GeV ] H ( s , M ππ ) Fig. 57. The dimensionless radiator function H ( s, M ππ ), in-clusive in θ π,γ . The value used for s in the Monte Carlo pro-duction was s = M φ = (1 . . Radiative corrections and Monte Carlo tools The radiator function is a crucial ingredient in thiskind of radiative return analyses, and is obtained usingthe relation H ( s, M ππ ) = s · M ππ πα β π · d σ ISR ππγ ( γ ) d M ππ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | F π | =1 , (202)in which d σ ISR ππγ ( γ ) d M ππ (cid:12)(cid:12)(cid:12)(cid:12) | F π | =1 is evaluated using the PHOK-HARA Monte Carlo generator in next-to-leading orderISR-only configuration, with the squared pion form factor | F π | set to 1. β π = q − m π M ππ is the pion velocity. WhileEq. (202) provides a convenient mechanism to extract thedimensionless quantity H ( s, M ππ ) also for specific angu-lar regions of pions and photons by applying the relevantcuts to d σ ISR ππγ ( γ ) d M ππ (cid:12)(cid:12)(cid:12)(cid:12) | F π | =1 , in the published KLOE analyses. H ( s, M ππ ) is evaluated fully inclusive for pion and pho-ton angles in the range 0 ◦ < θ π,γ < ◦ . Figure 57 showsthe radiator function in the range of 0 . < M ππ < . 95 GeV . As can be seen from Table 13, the 0.5% uncer-tainty of the radiator function quoted by the authors ofPHOKHARA translates into an uncertainty of 0.5% inthe ππ -contribution to a had µ between 0.35 and 0.95 GeV ,giving the largest individual contribution and dominatingthe theoretical systematic error.The presence of events with final state radiation in thedata sample affects the analyses in several ways: • Passing from M ππ to ( M ππ ) . The presence of finalstate radiation shifts the observed value of M ππ (evalu-ated from the momenta of the two charged pion tracksin the events) away from the value of the invariantmass squared of the virtual photon produced in thecollision of the electron and the positron, ( M ππ ) . Thetransition from M ππ to ( M ππ ) is performed usinga modified version of the PHOKHARA Monte Carlogenerator, which allows to (approximately) determinewhether a generated photon comes from the initial orthe final state [448]. Figure 58 shows the probabilitymatrix relating M ππ to ( M ππ ) by giving the prob-ability for an event in a bin of M ππ to end up in abin of ( M ππ ) . It can be seen that the shift is onlyin one direction, ( M ππ ) ≥ M ππ , so events with onephoton from initial state radiation and one photonfrom final state radiation move to a higher value of( M ππ ) . The entries lining up above ( M ππ ) ≃ . represent events with two pions and only onephoton, emitted in the final state. Events of this typehave ( M ππ ) = s , there is no hard photon from ini-tial state radiation present. Since in the KLOE analy-ses, the maximum value of ( M ππ ) for which the crosssections are measured is 0 . 95 GeV and sufficientlysmaller than s ≃ M φ of the DAΦNE collider, these leading-order final state radiation events need to betaken out in the analysis. By moving these events to( M ππ ) = s , the passage from M ππ to ( M ππ ) auto-matically performs this task. Figure 59 shows the frac-tion of events from leading-order final state radiationcontributing to the total number of events, evaluatedwith the PHOKHARA event generator. Since in the small angle analysis the angular regions for pions andphotons are separated, final state radiation, for whichthe photons are emitted preferably along the directionof the pions, is suppressed to less than 0.5%. Using large angle acceptance cuts, the effect is much bigger,especially above and below the ̺ -resonance, where itcan reach 20-30%. The correction of the shift in M ππ depends on the implementation of final state radiationin the Monte Carlo generator in terms of model depen-dence and missing contributions. It also relies on thecorrect assignment of photons coming from the initialor the final state; however, in case of symmetrical cutsin θ γ , interference effects between the two states vanishand the separation of initial and final state amplitudesis feasible. • The acceptance in θ γ . Since the direction of the pho-tons emitted in the final state is peaked along the di-rection of the pions, and the photons are emitted in the ( M ππ ) [ G e V ] M ππ [ G e V ] -2 -1 Fig. 58. Probability matrix relating the measured quantity M ππ to ( M ππ ) . To produce this plot, a private version of thePHOKHARA Monte Carlo generator was used [448]. The pho-ton angle is restricted to θ γ < ◦ ( θ γ > ◦ ). initial state along the e + / e − direction, the choice of theacceptance cuts affects the amount of final state radi-ation in the analyses. Using the small angle analysiscuts, a large part of final state radiation is suppressedby the separation of the pion and photon acceptanceregions, and consequently needs to be reintroduced us-ing corrections obtained from Monte Carlo simulationsto arrive at a result which is inclusive with respect tofinal state radiation (as needed in the dispersion inte-gral for a ππµ ). Even if in the large angle analysis thefraction of events with final state radiation survivingthe selection is larger, again the missing part has to beadded using Monte Carlo simulations. The acceptancecorrection for the cut in θ γ is evaluated for initial andfinal state radiation using the PHOKHARA generator,and the small differences found in the comparison ofdata and Monte Carlo distributions contribute to thesystematic uncertainty of the measurement (see Ta-ble 13 and [449]). • The distributions of kinematical variables. Cuts on thekinematical trackmass variable M trk (see Eq. (192)),introduced in the analyses to remove background fromthe process e + e − → φ → π + π − π , take out also afraction of the events with final state radiation, ne-cessitating a correction to obtain an inclusive result.Figure 60 shows the effect final state radiation hason the distribution of the trackmass variable. The ra-diative tail of multi-photon events to the right of thepeak at the π ± mass increases because the additionalradiation moves events from the peak to higher val-ues in M trk . The width of the peak at M π ± is dueto the detector resolution; the plot was produced us-ing the PHOKHARA event generator interfaced withthe KLOE detector simulation [450]. Between 150 and200 MeV, an M ππ -dependent cut is used in the eventselection to reject the π + π − π events which have avalue of M trk > M π ± . In this region, the cut also actson the signal events. Missing terms concerning finalstate radiation in the Monte Carlo simulation or the M ππ [ GeV ] (lo FSR) / (ISR + FSR)00.050.10.150.20.250.30.350.40.450.5x 10 -2 (a) M ππ [ GeV ] (lo FSR) / (ISR + FSR)00.10.20.30.40.50.60.7 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 (b) Fig. 59. (a) Fraction of events with leading order final stateradiation in the small angle selection: 50 ◦ < θ π < ◦ and θ γ < ◦ ( θ γ > ◦ ). (b) Fraction of events with leadingorder final state radiation in the large angle selection: 50 ◦ <θ π < ◦ and 50 ◦ < θ γ < ◦ . The PHOKHARA generatorwas used to produce the plots. ISRISR + FSR M Trk [ MeV ] a . u . -2 -1 Fig. 60. Modification of the distribution of the trackmass vari-able due to the presence of final state radiation (dark grey tri-angles) compared to the one with initial state radiation only(light grey triangles). The arrows indicate the region in whichthe M ππ -dependent cut is applied in the analysis. The plotwas created with the PHOKHARA generator interfaced to theKLOE detector simulation [450]. non-validity of the pointlike-pion approximation usedin PHOKHARA may affect the shape of the radiativetail in the trackmass variable. To overcome this, inthe KLOE analyses, small corrections are applied tothe momenta and the angles of the charged particlesin the event in the simulation, and good agreement inthe shape of M trk is obtained between Monte Carlosimulation and data [449]. • The division by the radiator function H ( s, M ππ ). Inthis case, one assumes perfect factorisation betweenthe ISR and the FSR process. This has been tested byperforming the analysis in an inclusive and exclusiveapproach with respect to final state radiation. The as-sumption was found to be valid within 0 . 2% [373,451].It has been argued that contributions from events withtwo hard photons in the final state, which are not includedin the PHOKHARA generator, may have an effect on theanalyses [380].The effect of the direct decay φ → π + π − γ on the ra-diative return analysis has been addressed already in [347].Running at √ s ≃ . 02 GeV, the amplitude for the pro-cesses φ → ( f (980) + f (600)) γ → π + π − γ interferes withthe amplitude for the final state radiation process. Dueto the yet unclear nature of the scalar states f (980) and f (600), the effect on the π + π − γ ( γ ) cross section dependson the model used to describe the scalar mesons. The pos-sibility to simulate φ decays together with the processesfor initial and final state radiation has been implementedin the PHOKHARA event generator in [337], using twocharacteristic models for the φ decays: the “no structure”model of [452] and the K + K − loop model of [453]. Arefined version of the K + K − loop model [439] and thedouble vector resonance φ → π ± ̺ ∓ ( → π ∓ γ ) have beenincluded as described in [350]. Using parameter values forthe different φ decays found in the analysis of the neutralchannel φ → ( f (980) + f (600)) γ → π π γ [439,441], onecan estimate the effect on the different analyses. While inthe small angle analysis there is no significant effect dueto the choice of the acceptance cuts, in the large angle se-lection the effect is of the order of several percent and canreach up to 20% in the vicinity of the f (980), see Fig. 61(a). While this allows to study the different models for thedirect decays of φ -mesons (see also Section 4.3.2), it pre-vents a precise measurement of σ ππ until the model andthe parameters are understood with better accuracy. Anobvious way out is to use data taken at a value of √ s out-side the narrow peak of the φ resonance ( Γ φ = 4 . ± . ∼ 250 pb − of data at √ s = 1 GeV, 20 MeV below M φ . Ascan be seen in Fig. 61 (b), this reduces the effect due tocontributions from f γ and ̺π decays of the φ -meson tobe within ± Normalisation with muon events An alternative method to extract the pion form factoris to normalise the differential cross section d σ ππγ ( γ ) / d M ππ directly to the process e + e − → µ + µ − γ ( γ ), d σ µµγ ( γ ) / d M µµ ,in each bin of ∆M ππ = ∆M µµ . Radiative corrections likethe effect of vacuum polarisation, the radiator functionand also the integrated luminosity R L d t cancel out in theratio of pions over muons, and only the effects from finalstate radiation (which is different for pions and muons)need to be taken into account consistently. An approachcurrently under way at KLOE uses the following equation M ππ [ GeV ] (a) M ππ [ GeV ] (b) Fig. 61. (a): d σ (ISR+FSR+ f + ̺π ) ππγ / d σ (ISR+FSR) ππγ for √ s = 1 . σ (ISR+FSR+ f + ̺πππγ ) / d σ (ISR+FSR) ππγ for √ s = 1 GeV.Both plots were produced with the PHOKHARA v6.1 eventgenerator using large angle acceptance regions for pions andphotons, with model parameters for the f and ̺π contribu-tions from [439,441]. to obtain | F π | : | F π ( s ′ ) | · (1+ η ( s ′ )) = 4(1 + 2 m µ /s ′ ) β µ β π · ( d σ ππγ ( γ ) d M ππ ) ISR+FSR ( d σ µµγ ( γ ) d M µµ ) ISR (203)In this formula, the measured differential cross sectiond σ ππγ ( γ ) / d M ππ should be inclusive with respect to pio-nic final state radiation, while the measured cross sec-tion d σ µµγ ( γ ) / d M µµ should be exclusive for muonic finalstate radiation. s ′ = M ππ = M µµ is the squared invari-ant mass of the di-pion or the di-muon system after therespective corrections for final state radiation. Using thisapproach, one gets on the left-hand side the pion formfactor times the factor (1 + η ( s ′ )), which describes theeffect of the pionic final state radiation. This bare formfactor is the quantity needed in the dispersion integral forthe ππ -contribution to a had µ . While the measurement ofd σ ππγ ( γ ) / d M ππ and its corrections for pionic final stateradiation are very similar to the one using the normali-sation with Bhabha events already performed at KLOE,the corrections needed to subtract the muonic final stateradiation from the d σ µµγ ( γ ) / d M µµ cross section are pureQED and can be obtained from the PHOKHARA gener-ator, which includes final state radiation for muon pairproduction at next-to-leading order [336]. Due to the factthat the KLOE detector does not provide particle IDs, pi-ons and muons have to be separated and identified usingkinematical variables (e.g. the aforementioned trackmassvariable) [367]. The analysis is in progress and a system- atic precision similar to the one obtained in the absolutemeasurement is expected. The BaBar radiative return program aims at the study ofall significant hadronic processes in electron-positron an-nihilation, e + e − → hadrons , for energies from thresholdup to about 4.5 GeV. Moreover, hadron spectroscopy ofthe initial J P C = 1 −− states, which are produced in e + e − collisions, and of their decay products is performed. Inthis chapter BaBar results for processes with 3, 4, 5 and6 hadrons in the final state, as well as measurements ofbaryon form factors in the time-like region are reported.A precision analysis of the pion form factor, i.e. of thecross section e + e − → π + π − , which is essential for an im-proved determination of the hadronic contribution to theanomalous magnetic moment of the muon, appeared mostrecently [454]. The results presented in this chapter arebased on a total integrated luminosity of 230 fb − , ex-cept for the 3 π and 4 hadron channels of Ref. [397], whichwere analysed using a data sample of 90 fb − . The totalBaBar data sample collected between the years 1999 to2008 amounts to 530 fb − . A typical feature common toall radiative return analyses at BaBar is a wide coverageof the entire mass range of interest in one single experi-ment, with reduced point-by-point uncertainties comparedto previous experiments. e + e − → pions The π + π − π mass spectrum has been measured from 1 . J/ψ mass region with a systematic er-ror of ∼ 5% below 2 . ∼ 20% at highermasses [396]. The spectrum is dominated by the ω , φ and J/ψ resonances. The BaBar measurement was able to sig-nificantly improve the world knowledge on the excited ω states. The spectrum has been fitted up to 1 . ω ′ and ω ′′ states have been found: M ( ω ′ ) = (1350 ± ± Γ ( ω ′ ) = (450 ± ± 70) MeV, M ( ω ′′ ) = (1660 ± ± 2) MeV, Γ ( ω ′′ ) = (230 ± ± 20) MeV. Note that below1 . e + e − → hadrons The π + π − π + π − , K + K − π + π − and K + K − K + K − exclu-sive final states have been measured from threshold up to4 . K + K − K + K − measurement is thefirst measurement of this process at all. Figure 62 showsthe mass distribution of the π + π − π + π − channel. We iden-tify an impressive improvement with respect to previousexperiments. Background is relatively low for all channelsunder study (e.g. a few percent at 1 . π + π − π + π − )and is dominated by ISR-events of higher multiplicitiesand by continuum non-ISR events at higher masses. The π + π − π + π − final state is dominated by the two-body in- NDOLYACMDSND M3NDM1GG2DM2CMD2 BaBar E C.M. (MeV) σ ( e + e - → π + π - ) ( nb ) Fig. 62. BaBar measurement of the energy dependence of the e + e − → π + π − π + π − cross section obtained by radiative returnin comparison with the world data set. NDOLYASND M3NGG2DM2CMD2BABAR beam , MeV Fig. 63. Preliminary BaBar data for the e + e − → π + π − π π cross section in comparison with previous experiments. termediate state a (1260) π ; the K + K − π + π − final stateshows no significant two-body states, but a rich three-body structure, including K ∗ (890) Kπ , φππ , ρKK and K ∗ (1430) Kπ .Figure 63 shows BaBar preliminary results for the process e + e − → π + π − π π . The current systematic error of themeasurement varies from 8% around the peak of the crosssection to 14% at 4.5 GeV. BaBar results are in agreementwith SND [456] in the energy range below 1.4 GeV andshow a significant improvement for higher energies ( > e + e − → π + π − π π final state isdominated by the ωπ , a (1260) π and ρ + ρ − intermediatechannels, where the latter channel has been observed forthe first time.A specific analysis was devoted to the intermediate struc-tures in the e + e − → K + K − π + π − and e + e − → K + K − π π c.m. (GeV) σ ( ( π + π - )) ( nb ) c.m. (GeV) σ ( ( π + π - π )) ( nb ) Fig. 64. The energy dependence of the cross sections for e + e − → π + π − ) (upper plot) and e + e − → π + π − )2 π (lower plot), obtained by BaBar (filled circles) by radiativereturn, in comparison with previous data. channels [401]. Of special interest is the intermediate state φf (980), where the decays f (980) → π + π − and f (980) → π π have been looked at. A peak is observed in the φf (980)channel at a mass M = 2175 ± 18 MeV and a width Γ = 58 ± K + K − f spec-trum. e + e − → π + π − ) π , π + π − ) η The e + e − → π + π − ) π cross section has been measuredby BaBar from threshold up to 4.5 GeV [403]. A large cou-pling of the J/ψ and ψ (2 S ) to this channel is observed.The systematic error of the measurement is about 7%around the peak of the mass spectrum. In the π + π − π mass distribution the ω and η peaks are observed; therest of the events have a 3 πρ structure.BaBar performed also the first measurement of the e + e − → π + π − ) η cross section. A peak value of about 1.2 nb atabout 2.2 GeV is observed, followed by a monotonic de-crease towards higher energies. Three intermediate statesare seen: ηρ (1450), η ′ ρ (770) and f (1285) ρ (770). FENICEDM2DM1BESCLEOPS170E835E760BABAR M pp (GeV/c ) P r o t on f o r m f ac t o r -2 -1 Fig. 65. The e + e − → p ¯ p cross section measured by BaBar(filled circles) in comparison with data from other e + e − collid-ers (blue points) and from ¯ pp experiments (red points). e + e − → hadrons The 6 hadron final state has been measured in the exclu-sive channels 3( π + π − ), 2( π + π − )2 π and K + K − π + π − )[399]. The cross section in the last case has never beenmeasured before; the precision in the first two cases is ∼ . π + π − ) and 2( π + π − )2 π are shown in Fig. 64. A clear dip is visible at about 1 . π + π − ) (2( π + π − )2 π )mode, BaBar obtains values of 1880 ± 30 MeV (1860 ± ± 30 MeV (160 ± o ± o ( − o ± o )for the phase shift between the resonance and continuum. e + e − → K + K − π , K + K − η , K S K ± π ∓ A recent BaBar ISR-analysis is dedicated to three hadronsin the final state, including a pair of kaons ( K + K − π , KK S π ); a peak near 1.7 GeV, which is mainly due tothe φ ′ (1680) state, is observed. A Dalitz plot analysisshows that the KK ∗ (892) and KK ∗ (1430) intermediatestates are dominating the K ¯ Kπ channel. A fit to the e + e − → K ¯ Kπ cross section assuming the expected contri-butions from the φ, φ ′ , φ ′′ , ρ , ρ ′ , ρ ′′ states was performed.The parameters of the φ ′ and other excited vector mesonstates are compatible with PDG values. Time-like proton form factor e + e − → p ¯ p , hyperon formfactors e + e − → Λ ¯ Λ , Λ ¯ Σ , Σ ¯ Σ BaBar has also performed a measurement of the e + e − → p ¯ p cross section [398]. This time-like process is parametri- sed by the electric and magnetic form factors, G E and G M : σ e + e − → p ¯ p = 4 πα βC s × ( | G M | + 2 m p s | G E | ) , where β = q − m p /s and the factor C = y/ (1 − e − y )(with y = παm p / ( β √ s )) accounts for the Coulomb in-teraction of the final state particles. The proton helicityangle θ p in the p ¯ p rest frame can be used to separate the | G E | and | G M | terms. Their respective variations areapproximately ∼ sin θ p and ∼ (1 + cos θ p ). By fittingthe cos θ p distribution to a sum of the two terms, the ra-tio | G E | / | G M | can be extracted. This is done separatelyin six bins of M p ¯ p . The results disagree significantly withprevious measurements from LEAR [458] above threshold.BaBar observes a ratio | G E | / | G M | > M p ¯ p the BaBar measurement finds | G E | / | G M | ≈ 1. LEAR data, on the contrary, show a be-haviour | G E | / | G M | < e + e − and ¯ pp experiments), the effective form factor G is introduced: G = q | G E | + 2 m p /s | G M | .The BaBar measurement of G is in good agreement withexisting results, as can be seen in Fig. 65. The structureof the form factor is rather complicated; the following ob-servations can be made: (i) BaBar confirms an increase of G towards threshold as seen before by other experiments;(ii) two sharp drops of the spectrum at M p ¯ p = 2 . 25 and3 . M p ¯ p > e + e − → Λ ¯ Λ cross section [404].So far only one data point from DM2 [459] was existingfor this channel, which is in good agreement with BaBardata. About 360 Λ ¯ Λ events could be selected using the Λ → pπ decay. In two invariant mass bins an attempt hasbeen made to extract the ratio of the electric to magneticform factor | G E | / | G M | . In the mass range below 2 . | G E | / | G M | =1 . +0 . − . ). Above 2 . | G E | / | G M | = 0 . +0 . − . ). Also the Λ polarisationand the phase between G E and G M was studied usingthe slope of the angle between the polarisation axis andthe proton momentum in the Λ rest frame. The followinglimit on Λ polarisation is obtained: − . < ζ < . 28; therelative phase between the two form factors is measuredas − . < sin( φ ) < . 98, which is not yet significant dueto limited statistics.Finally, the first measurements of the e + e − → Σ ¯ Σ and e + e − → Σ ¯ Λ ( Λ ¯ Σ ) cross sections were performed. For thedetection of the Σ baryon, the decay Σ → Λγ → pπγ was used. About 40 candidate events were selected for thereaction Σ ¯ Σ and about 20 events for Λ ¯ Σ . All baryon form factors measured by BaBar have a similar size andmass shape, namely a rise towards threshold. The reasonfor this peculiar behaviour is not understood. ISR studies at Belle Until now most of the Belle analyses using radiativereturn focused on studies of the charmonium and charmo-nium-like states. They can be subdivided into final stateswith open and hidden charm. Final states with open charm Belle performed a systematic study of various exclusivechannels of e + e − annihilation into charmed mesons andbaryons using ISR, often based on the so called partialreconstruction to increase the detection efficiency.In Ref. [413] they measured the cross sections of theprocesses e + e − → D ∗± D ∗∓ and e + e − → D + D ∗− + c.c. .The shape of the former is complicated and has several lo-cal maxima and minima. The first two maxima are closeto the ψ (4040) and ψ (4160) states. The latter shows sig-nificant excess of events near the ψ (4040).The cross sections of the processes e + e − → D + D − and e + e − → D ¯ D show a signal of the ψ (3770), as wellas hints of the ψ (4040), ψ (4160) and ψ (4415) [414]. Thereis also an enhancement near 3.9 GeV, which qualitativelyagrees with the prediction of the coupled channel model[460].The cross section of the process e + e − → D D − π + has a prominent peak at the energy corresponding to the ψ (4415) [415]. From a study of the resonant substructurein the decay ψ (4415) → D D − π + they conclude that it isdominated by the intermediate D ¯ D ∗ (2460) mechanism.In contrast to expectations of some hybrid models pre-dicting Y (4260) → D ( ∗ ) ¯ D ( ∗ ) π decays, no clear structureswere observed in the cross section of the process e + e − → D D ∗− π + [417]. There is only some evidence ( ∼ . σ ) forthe ψ (4415).Finally, they measure the cross section of the reac-tion e + e − → Λ + c Λ − c and observe a significant peak nearthreshold that they dub X (4630) [416]. Assuming thatthe peak is a resonance, they find that its mass and widthare compatible within errors with those of the Y (4660)state found by Belle in the ψ (2 S ) π + π − final state viaISR [411]. However, interpretations other than X (4630) ≡ Y (4660) cannot be excluded. For example, peaks at thebaryon-antibaryon threshold are observed in various pro-cesses. According to other assumptions, the X (4630) isa ψ (5 S ) [461] or ψ (6 S ) [462] charmonium state, or, forexample, a threshold effect which is due to the ψ (3 D ),slightly below the Λ + c Λ − c threshold [463]. Figure 66 showsall cross sections mentioned above, with the vertical linesshowing positions of both well established states like ψ (4040), ψ (4160) and ψ (4415), and new charmonium-like states Y (4008), Y (4260), Y (4360) and Y (4660) discussed below. σ ( nb ) a)b)c)d)e)f) √ s, GeV/c Fig. 66. Cross sections of various exclusive processes measuredby Belle: a) e + e − → D ∗± D ∗∓ , b) e + e − → D + D ∗− + c.c. ,c) e + e − → D ¯ D , d) e + e − → D D − π + + c.c. , e) e + e − → D D ∗− π + + c.c. , and f) e + e − → Λ + c Λ − c . The dashed lines showthe position of the ψ states, while the dotted lines correspondto the Y (4008) , Y (4260) , Y (4360), and Y (4660) states. Summing the measured cross sections and taking intoaccount not yet observed final states on base of isospinsymmetry they find that the sum of exclusive cross sec-tions almost saturates the total inclusive cross sectionmeasured by BES [303]. Final states with hidden charm Studying the J/ψπ + π − final state, Belle confirmed the Y (4260) discovered by BaBar and in addition observed anew structure dubbed Y (4008) [410], see Fig. 67. Theyalso observe the reaction e + e − → J/ψK + K − and findfirst evidence for the reaction e + e − → J/ψK S K S [412].Studying the ψ (2 S ) π + π − final state, Belle confirmedthe Y (4360) discovered by BaBar and in addition observeda new structure dubbed Y (4660) [411], see Fig. 68.It is worth noting that the resonance interpretation ofvarious enhancements discussed above is not unambiguousand can be strongly affected by close thresholds of differentfinal states and rescattering effects.Various ISR studies performed at the Belle detector inthe charmonium region are summarised in Table 14. ISR studies of light quark states M( π + π - J/ ψ ) (GeV/c ) E n t r i e s / M e V / c Solution ISolution II Fig. 67. The J/ψπ + π − invariant mass distribution. M( π + π - ψ (2S)) (GeV/c ) E n t r i e s / M e V / c Solution ISolution II Fig. 68. The ψ (2 S ) π + π − invariant mass distribution. Table 14. Summary of ISR studies in the c ¯ c region at Belle.Final state R L d t , fb − Ref. D ∗ + D ∗− D ± D ∗∓ D ¯ D , D + D − 673 [414] D D − π + 673 [415] D D ∗− π + 695 [417] Λ + c Λ − c 695 [416] J/ψπ + π − 548 [410] ψ (2 S ) π + π − 673 [411] J/ψK + K − 673 [412] In one case the ISR method was used to study thelight quark states [464]. In this analysis the cross sectionsof the reactions e + e − → φπ + π − and e + e − → φf (980)are measured from threshold to 3 GeV, using a data sam-ple of 673 fb − , see Fig. 69 (a, b). In the φπ + π − modethe authors observe and measure for the first time the pa-rameters of the φ (1680); they also observe and measurethe parameters of the φ (2170). Also selected in this anal-ysis is the φf (980) final state, which shows a clear signalof the φ (2170). For Monte Carlo simulation they use aversion of PHOKHARA in which the produced resonancedecays into φπ + π − or φf (980) with the subsequent de-cays φ → K + K − and f (980) → π + π − . The π + π − systemis in the S -wave, the π + π − system and the φ are also ina relative S -wave. The π + π − mass distribution is gener- E C.M. (GeV) σ ( φ π + π - ) ( nb ) E C.M. (GeV) σ ( φ f ( )) ( nb ) Fig. 69. Cross sections of the processes e + e − → φπ + π − (a)and e + e − → φf (980) (b). ated according to phase space. They assign 0.1% as thesystematic uncertainty of the ISR photon radiator.In all the ISR studies the Monte Carlo simulation isperformed as follows. First, the kinematics of the initialstate radiation is generated using the PHOKHARA v5.0package for simulation of the process e + e − → V γ ISR ( γ ISR )[338]. Then a q ¯ q generator is used to generate V decays. As discussed above, the major hadronic leading-order con-tribution to a had µ comes from the energy range below 1GeV, where in turn the π + π − channel gives the dominantcontribution. Direct scan at VEPP-2000 will deliver hugestatistics at the experiments CMD-3 and SND, but theaccuracy of the cross sections will be determined by sys-tematic errors. Therefore, any other possibility to measurethe pion form factor, for example with ISR, will be a valu-able tool to provide a cross check for better understandingthe scale of systematic effects.The design luminosity of ∼ cm − c − is expectedat √ s = 2 GeV. The luminosity recalculated to the ρ -peak will be close to the one obtained with CMD-2. Letus recollect that the ISR method provides a continuous“low energy scan”, while taking data at fixed high energy.The threshold region, 2 m π – 0.5 GeV, gives about 13% ofthe total contribution to the muon anomaly. As a rule, thecollider luminosity dramatically decreases at low energies.To overcome the lack of data in the threshold region, theISR method can serve as a very efficient and unique wayto measure the pion form factor inside this energy region.Today, the theoretical precision for the cross section ofthe process e + e − → π + π − γ is dominated by the uncer-tainty of the radiator function (0.5%), and there is hopeto reduce it to a few per mill in the future. In the case ofthe pion form factor extraction from the π + π − γ/µ + µ − γ ratio, the dependence on theory will be significantly re-duced, since the main uncertainty of the radiator functionand vacuum polarisation effects cancel out in the ratio.With the integrated luminosity of several inverse femto-barn at 2 GeV, one can reach a fractional accuracy on thetotal error smaller than 0.5%.In direct scan experiments the data are collected atfixed energy points. Thus, some “empty” gaps withoutdata naturally arise. The experiments with ISR will coverthe whole energy scale, filling any existing gaps. Trigger and reconstruction efficiencies, detector imperfections andmany other factors will be identical for all data in thewhole energy range. Therefore, some systematic errors willbe cancelled out in part. Comparison of cross sections forthe process e + e − → µ + µ − , measured both with ISR anddirect scan, can serve as a benchmark to study and controlsystematic effects. It should confirm the validity of thismethod and help to determine the energy scale. A fit of the ω and φ resonances will also provide a calibration of theenergy scale – an important feature to achieve a systematicaccuracy of a few per mill for the pion form factor. The designed peak luminosity of BEPC-II is 1 × cm − s − at √ s = 3 . 77 GeV, i.e. the ψ (3770) peak. It hasreached 30% of the design luminosity now and is start-ing to deliver luminosity to BES-III for physics. Althoughthe physics programs at BES-III are rather rich [51], mostof the time, the machine will run at √ s = 3 . 77 GeV and4.17 GeV for charm physics, since the cross sections of J/ψ and ψ (2 S ) production are large and the required statisticscan be accumulated in short time, say, one year at eachenergy point. The estimated running time of BEPC-II at √ s = 3 . 77 and 4.17 GeV is around eight years, which cor-responds to an integrated luminosity of about 20 fb − ateach energy point.Data samples at √ s = 3 . 77 and 4.17 GeV can beused for radiative return studies, for the c.m. energies ofthe hadron system between the π + π − threshold to above2.0 GeV. This will allow for measurements of the pion,kaon and proton form factors, as well as of cross sectionsfor some multi-hadron final states. The good coverage ofthe muon detector at BES-III also allows the identificationof the µ + µ − final state, thus supplying a normalisationfactor for the other two-body final states.Figure 70 shows the expected luminosity at low en-ergies in 10 MeV bins for 10 fb − data accumulated onthe ψ (3770) peak. In terms of luminosity at the ρ peak,one can see that 10 fb − of data at √ s = 3 . 77 GeV isequivalent to 70 fb − at 10.58 GeV, i.e. at the B factories.With Monte Carlo generated e + e − → γ ISR π + π − datausing PHOKHARA [333], after a fast simulation and re-construction with the BES-III software, one found the ef-ficiency for events at the ρ peak to be around 5% if onerequires the detection of the ISR photon. This is higherthan the efficiency at BaBar [465]. Figure 71 shows the sig-nal for 10,000 generated π + π − events. One estimates thenumber of events in each 10 MeV bin to be around 20,000at the ρ peak, for 10 fb − of data at √ s = 3 . 77 GeV.This is comparable with the recent BaBar results basedon 232 fb − of data at the Υ (4 S ) peak [465].The most important work related to the pion form fac-tor measurement is the estimate of the systematic error.Since the cross section of good events at the ψ (3770) peakis not large (around 30 nb for the total hadronic crosssection, with about 400 nb cross section for the QED pro-cesses) compared to the highest trigger rates at J/ψ and ψ (2 S ) peak energies, a loose trigger is mandatory to allow Lum per 10 fb -1 data Ecm (GeV) Lu m / M e V ( nb - ) Ecm (GeV) Lu m / M e V ( nb - ) Fig. 70. Expected luminosity at low energies due to ISR for10 fb − data accumulated on the ψ (3770) peak. M( π + π - ) (GeV) E n t r i e s / M e V Fig. 71. Detected γ ISR π + π − in 10000 produced events at the ψ (3770) peak. The sample is generated with PHOKHARA. the ISR events to be recorded. In principle, the triggerrate for these events could reach 100%, with an allowedtrigger purity of less than 20%.With enough D ¯ D events accumulated at the same en-ergy, the tracking and particle ID efficiencies can be mea-sured with high precision (as has been done at CLEO-c [466]). In addition, a huge data sample at the ψ (2 S ) andthe well measured large branching fraction of ψ (2 S ) tran-sition modes, such as π + π − J/ψ , J/ψ → µ + µ − , can beused to study the tracking efficiency, µ -ID efficiency and so on. All this will greatly help to understand the detectorperformance and to pin down the systematic errors in theform factor measurement.The kaon and proton form factors can be measured aswell since they are even simpler than the measurement ofthe pion form factor. This will allow us to better under-stand the structure close to threshold and possible existinghigh-mass structures.Except for the lowest lying vector states ( ρ , ω , φ ), theparameters of other vector states are poorly known, andfurther investigations are needed. BES-III ISR analysesmay reach energies slightly above 2 GeV, while beyondthat BEPC-II can run by adjusting the beam energy. Thisallows BES-III to study the full range of vector mesonsbetween the π + π − threshold and 4.6 GeV, which is thehighest energy BEPC-II can reach, thus covering the ρ , ω and φ , as well as the ψ sector. One will have the chance tostudy the excited ρ , ω and φ states between 1 and about2.5 GeV. The final states include π + π − π , K ¯ K , 4 pions, ππKK , etc. Final states with more than four particleswill be hard to study using the ISR method, since the D ¯ D decay will contribute as background. After discovery of the τ lepton, which is a fundamentallepton, heavy enough to decay not only into leptons, butalso into dozens of various hadronic final states, it becameclear that corresponding Monte Carlo (MC) event gener-ators are needed for various purposes: – To calculate detector acceptance, efficiencies and vari-ous distributions for signal event selection and compar-ison to data. In general the acceptance is small (a fewpercent) and depends on the model; in principle, it is acomplicated function of invariant masses, angles, andresolutions. Analysis of publications shows that effectsof MC signal modelling are almost always neglected. – To estimate the number of background (BG) events N BGev and their distributions; in addition to backgroundcoming from τ + τ − pairs (so called cross-feed), theremight be BG events from q ¯ q continuum, γγ collisionsetc. – To unfold observed distributions to get rid of detectoreffects, important when extracting resonance parame-ters.Various computer packages like, e.g., KORALB [467],KKMC [468], TAUOLA [469,470,471] and PHOTOS [472]were developed to generate events for τ lepton produc-tion in e + e − annihilation and their subsequent decay, tak-ing into account the possibility of photon emission. Thesecodes became very important tools for experiments atLEP, CLEO, Tevatron and HERA.Simulation of hadronic decays requires the knowledgeof hadronic form factors. Various hadronic final stateswere considered in the 90’s, resulting in a large numberof specific hadronic currents [473]. However, already experiments at LEP and CLEO show-ed that with increase of the collected data sets a more pre-cise description is necessary. Some attempts were made toimprove the parametrisation of various hadronic currents.One should note the serious efforts of the ALEPH andCLEO Collaborations, which created their own parametri-sations of TAUOLA hadronic currents already in the late90’s, or a parametrisation of the hadronic current in the4 π decays [474], based on the experimental information on e + e − → π + π − , π + π − π from Novosibirsk [294], whichis now implemented in the presently distributed TAUOLAcode [475]. In this section we will briefly discuss the most precise re-cent experimental data on τ lepton decays, showing, wher-ever possible, their comparison with the existing MC gen-erators and discussing the decay dynamics. τ − → π − π ν τ at Belle Recently results of a study of the τ − → π − π ν τ decayby the Belle Collaboration were published [476]. Fromless than 10% of the dataset available the authors se-lected a huge statistics of 5.4M events, about two ordersof magnitude larger than in any previous experiment, de-termined the branching fraction and after the unfoldingobtained the hadronic mass spectrum, in which for thefirst time three ρ -like resonances were observed together: ρ (770) , ρ (1450) and ρ (1700). Their parameters were alsodetermined.The comparison of the obtained missing mass distri-butions with simulations for different polar angle ranges(Fig. 72) shows that there exist small discrepancies be-tween MC and data.Figure 73 shows various background contributions tothe di-pion mass distribution (upper panel) and underly-ing dynamics (lower panel), clearly demonstrating a pat-tern of the three interfering resonances ρ (770) , ρ (1450)and ρ (1700). τ − → ¯ K π − ν τ , K − π ν τ Two high-precision studies of the τ decay into the Kπν τ final state were recently published. The BaBar Collabo-ration reported a measurement of the branching fractionof the τ − → K − π ν τ decay [477]. They do not studyin detail the Kπ invariant mass distribution, noting onlythat the K ∗ (892) − resonance is seen prominently abovethe simulated background, see Fig. 74. Near 1.4 GeV /c decays to higher K ∗ mesons are expected, such as the K ∗ (1410) − and K ∗ (1430) − , but their branching fractionsare not yet measured well. These decays are not included M miss (GeV/c ) N u m be r o f en t r i e s / . ( G e V / c ) (a) (20 ° ≤ θ miss < ° ) DATA τ + τ - two-photon(hadron)two-photon(lepton) µ + µ - bhabha qq – (q=u,d,s,c)) M miss (GeV/c ) N u m be r o f en t r i e s / . ( G e V / c ) (b) (55 ° ≤ θ miss < ° ) DATA τ + τ - two-photon(hadron)two-photon(lepton) µ + µ - bhabha qq – (q=u,d,s,c)) M miss (GeV/c ) N u m be r o f en t r i e s / . ( G e V / c ) (c) (125 ° ≤ θ miss < ° ) DATA τ + τ - two-photon(hadron)two-photon(lepton) µ + µ - bhabha qq – (q=u,d,s,c)) Fig. 72. Projections to the missing mass and missing direc-tion for τ − → π − π ν τ decays at Belle: (a)–(c) correspond todifferent ranges of the missing polar angles. The solid circlesrepresent the data and the histograms the MC simulation (sig-nal + background). The open histogram shows the contributionfrom τ + τ − pairs, the vertical (horizontal) striped area showsthat from two-photon leptonic (hadronic) processes; the wide(narrow) hatched area shows that from Bhabha ( µ + µ − ), andthe shaded area that from the q ¯ q continuum.4 (M π±π ) (GeV/c ) N u m b er o f e n t r i e s / . ( G e V / c ) DATAMC(signal) τ - → h - (n π ) ν τ τ - → K - π ν τ τ - → ω π - ν τ ( ω → π γ )continuum B.G. (M ππ ) (GeV/c ) Belle N u m b er o f e n t r i e s / . ( G e V / c ) DataG & S Fit ( ρ (770) + ρ (1450) + ρ (1700) ) Fig. 73. Invariant-mass-squared distribution for τ − → π − π ν τ decay at Belle. (a) Contributions of different back-ground sources. The solid circles with error bars represent thedata, and the histogram represents the MC simulation (sig-nal + background). (b) Fully corrected distribution. The solidcurve is the result of a fit to the Gounaris-Sakurai model withthe ρ (770), ρ (1450) and ρ (1700) resonances. in the BaBar simulation of τ decays, but seem to be presentin the data around 1.4 GeV /c . It is also worth notingthat this decay mode is heavily contaminated by cross-feed backgrounds from other τ decays. For example, below0.7 GeV /c the background is dominated by K − π π ν τ and K − K π ν τ events, for which the branching fractionsare only known with large relative uncertainties of ≈ ≈ τ − → π − π ν τ decay, which has a largebranching fraction and thus should be simulated properly.Another charge combination of the final state particles,i.e., K S π − ν τ , was studied in the Belle experiment [478]. In ) (GeV/c π - K M0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 ) - ( x10 E ve n t s / . G e V / c -3 -2 -1 Fig. 74. The Kπ invariant mass distribution for the decay τ − → K − π ν τ at BaBar. The dots are the data, while thehistograms are background MC events with selection and effi-ciency corrections: τ background (dashed line), q ¯ q (dash-dottedline), µ + µ − (dotted line). a) √ s, GeV/c N EVE N T S / ( . M e V / c ) -1 b) √ s, GeV/c N EVE N T S / ( . M e V / c ) SignalK S K L π K S ππ K S K3 π non- ττ Fig. 75. The Kπ invariant mass distribution for the decay τ − → K − π ν τ at Belle. Points are experimental data, his-tograms are spectra expected for different models. (a) showsthe fitted result in the model with the K ∗ (892) alone. (b) showsthe fitted result in the K ∗ (892) + K ∗ (800) + K ∗ (1410) model.Also shown are different types of background.5 ,K) GeV/c φ M(1.5 1.55 1.6 1.65 1.7 1.75 1.8 1.85 1.9 C o m b i n a t i on s / M e V / c (a) Fig. 76. The φK invariant mass distribution for the decay τ − → φK − ν τ at Belle. Points with error bars are the data.The open histogram is the phase-space distributed signal MC,and dotted and dot-dashed histograms indicate the signal MCmediated by a resonance with mass and width of 1650 MeVand 100 MeV, and 1570 MeV and 150 MeV, respectively. this case a detailed analysis of the Kπ invariant mass dis-tribution has been performed. The authors also concludethat the decay dynamics differs from pure K ∗ (892): thebest fit includes K ∗ (800)+ K ∗ (892)+ K ∗ (1410) /K ∗ (1430),see Fig. 75. τ decays into three pseudoscalars Recently a measurement of the branching fractions of var-ious particle combinations in the decay to three chargedhadrons (any combination of pions and kaons) was re-ported by the BaBar Collaboration [479]. A similar studywas also performed by the Belle group [480]. However,both groups have not yet analysed the mass spectra indetail. In the K − K + K − ν τ final state BaBar [479] andBelle [481] reported the observation of the decay mode φK − ν τ , while in the K − K + π − ν τ final state BaBar ob-served the φπ − ν τ decay mode [479]. Belle analysed thespectrum of the φK − mass and concluded that it mighthave a complicated dynamics, see Fig. 76.The most detailed previous study of the mass spec-tra was done by the CLEO group [482]. With the statis-tics of about 8,000 events they conclude that the 3 π massspectrum is dominated by the a (1260) meson, and con-firmed that the decay of the latter is not saturated bythe ρπ intermediate state, having in addition a significant f (600) π − component observed earlier in e + e − annihila-tion into four charged pions [294].Recently the Belle Collaboration performed a detailedstudy of various decays with the η meson in the finalstate [483]. They measured the branching fractions of thefollowing decay modes: τ − → K − ην τ , τ − → K − π ην τ , τ − → π − π ην τ , τ − → π − K S ην τ , and τ − → K ∗− ην τ .They also set upper limits on the branching fractions ofthe decays into K − K S ην τ , π − K S π ην τ , K − ηην τ , π − ηην τ ,and non-resonant K − π ην τ final states. Figure 77 shows that there is reasonable agreementfor ηπ − π ν τ (a, b) and a worse one for ηK − ν τ (c) and ηK ∗− ν τ (d). τ decays to four pions There are two possible isospin combinations of this hadronicfinal state, 2 π − π + π and π − π . Both have not yet beenstudied at B factories, so the best existing results arebased on ALEPH [484] and CLEO [485] results.The theoretical description of such decays is basedon the CVC relations and the available low energy e + e − data [486,330,474,339]. τ − → h − h + ν τ at BaBar A new study of the τ − → h − h + ν τ decay ( h = π, K )has been performed by the BaBar Collaboration [487]. Alarge dataset of over 34,000 events (two orders of mag-nitude larger than in the best previous measurement atCLEO [488]) allows one a first search for resonant struc-tures and decay dynamics.The invariant mass distribution of the five charged par-ticles in Fig. 78 shows a clear discrepancy between thedata and the MC simulation, which uses the phase spacedistribution for τ − → π − π + ν τ .The mass of the h + h − pair combinations in Fig. 79(upper panel), with a prominent shoulder at 0.77 GeV /c ,suggests a strong contribution from the ρ meson. Notethat there are three allowed isospin states for this decay,of which two may have a ρ meson. The mass of the 2 h + h − combinations in Fig. 79 (lower panel) also shows a struc-ture at 1.285 GeV /c coming from the τ − → f (1285) π − ν τ decay.The first attempt to take into account the dynamics ofthis decay was recently performed in Ref. [489]. τ decays to six pions The six-pion final state was studied by the CLEO Collab-oration [490]. Two charge combinations, 3 π − π + π and2 π − π + π , were observed and it was found that the de-cays are saturated by intermediate states with η and ω mesons. Despite the rather limited statistics (about 260events altogether), it became clear that the dynamics ofthese decays is rather rich. More than 50 different Lepton-Flavour Violating (LFV)decays have been studied by the CLEO, BaBar and BelleCollaborations. Publications rarely describe how the sim-ulation of such decays is performed. Moreover, theoreticalpapers suggesting LFV in new models usually do not pro-vide differential cross sections. In some experimental pa-pers the authors claim that the production of final state Fig. 77. Invariant mass distributions: (a) ππ and (b) πηπ for τ → ππ ην τ ; (c) ηK for τ → Kην τ and (d) πK S η for τ → πK S ην τ at Belle. The points with error bars are the data. The normal and filled histograms indicate the signal and τ + τ − background MC distributions, respectively. hadrons with a phase space distribution is assumed. How-ever, the real meaning of this statement is not very clearsince LFV assumes New Physics and, therefore, matrix el-ements are not necessarily separated into weak and hadronicparts.However, there exist a few theoretical papers consid-ering differential cross sections. For example, angular cor-relations for τ − → µ − γ, µ − µ + µ − and µ − e + e − decayswere studied in Ref. [491]. An attempt to classify differenttypes of operators entering New Physics Lagrangians for τ decays to three charged leptons was made in [492]. τ production and decays High-statistics and high-precision experiments, as well assearches for rare processes, result in a new challenge: MonteCarlo generators based on an adequate theoretical descrip-tion of energy and angular distributions. In the followingwe will describe the status of the Monte Carlo programsused by experiments. We will review the building blocksused in the simulation with the goal in mind to localisethe points requiring most urgent attention.At present, for the production of τ pairs, the MonteCarlo programs KORALB [467] and KKMC [468] are the standard codes to be used. For the generation of brems-strahlung in decays, the Monte Carlo PHOTOS [472] isused. Finally, τ decays themselves are simulated with theprogram TAUOLA [469,470,471]. The EvtGen code waswritten and maintained for simulation of B meson decays,see . It offers aunique opportunity to specify, at run time, a list of thefinal state particles , without having to change and/orcompile the underlying code. In a multi-particle final statedominated by phase space considerations, this generatorprovides an adequate description of the final state mo-menta, for which the underlying form factor calculationis more involved and not presently available in a closedform. That is why it is used by experiments measuring τ decays too.So far, our discussion has been based on the com-parison of experimental data and theory embodied intoMonte Carlo programs treated as a black box. One couldsee that a typical signature of any given τ decay channelis matching rather poorly the publicly available MonteCarlo predictions. This should be of no surprise as effortsto compare data with predictions were completed for the E.g. τ lepton decay products including neutrinos.7 last time in late 90’s by the ALEPH and CLEO collabo-rations. The resulting hadronic currents were afterwardsimplemented in [475]. Since that time no efforts to preparea complete parametrisation of τ decay simulation for thepublic use were undertaken seriously.There is another important message which can be drawnfrom these comparisons. Starting from a certain precisionlevel, the study of a given decay mode can not be sepa-rated from the discussion of others. In the distributionsaimed at representing the given decay mode, a contribu-tion from the other τ decay modes can be large, up toeven 30%.It may be less clear that experiments differ significantlyin the way how they measure individual decay modes. Forinstance, ALEPH produced τ samples free of the non- τ backgrounds, but, on the other hand, strongly boosted,making the reconstruction of some angles in the hadronicsystem more difficult. This is important and affects prop-erties of the decay models which will be used for a parametri-sation. In particular, when the statistics is small, possi-ble fluctuations may affect the picture and there are notenough data to complete an estimate of the systematic er-rors. In this case, details of the description of the hadroniccurrent, as the inclusion of intermediate resonances, arenot important. Let us consider, as an example, τ − → K S π − π ν τ . The matrix element in the ALEPH parametri-sation is saturated by ρ − → π − π and K ∗ → K S π ,and a similar parametrisation is used for K ∗− → K S π − .In practice, the contribution of the ρ is more significantin the ALEPH parametrisation in contrast to the CLEOone where the K ∗ dominates. One has to admit that atthe time when both collaborations were preparing theirparametrisations to be used in TAUOLA, the data sam-ples of both experiments were rather small and the differ-ences were not of much significance. This can, however,affect possible estimates of backgrounds for searches ofrare decays, e.g. of B mesons at LHCb. Let us now go point by point and discuss examples ofMonte Carlo programs and fitting strategies. We will fo-cus on subjects requiring most attention and future work.We will review the theoretical constraints which are use-ful in the construction of the models used for the datadescription. Because of the relatively low multiplicity of final state par-ticles, it is possible to separate the description of τ pro-duction and decay into segments describing the matrix el-ements and the phase space. In the phase space no approx-imations are used, contrary to the matrix elements where LHCb performed MC studies for B s → µ + µ − and the ra-diative decays B → K ∗ γ and B s → φγ , but τ decays havenot yet been taken into account. These results are not publicand exist only as internal documents LHCB-ROADMAP1-002and LHCB-ROADMAP4-001. ) Mass (GeV/c0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 E ve n t s / . G e V / c DataSignal MCTau MCqqbar B A B AR ) Mass (GeV/c0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 E ve n t s / . G e V / c Fig. 78. Invariant mass of five charged particles for τ − → h − h + ν τ at BaBar. ) Mass (GeV/c0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 E ve n t s / . G e V / c DataSignal MC Bkgd MC ) Mass (GeV/c0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 E ve n t s / . G e V / c B A B AR ) Mass (GeV/c1.1 1.15 1.2 1.25 1.3 1.35 1.4 E ve n t s / . G e V / c B A B AR Fig. 79. Invariant mass distributions for τ − → h − h + ν τ atBaBar. Points with error bars are the data: Upper panel – h + h − ; the unshaded and shaded histograms are the signal andbackground predicted by MC. Lower panel – 2 π + π − ; the solidline is a fit to the data using a second-order polynomial (dashedline) for the background and a Breit-Wigner convoluted witha Gaussian for the peak region.8 all approximations and assumptions reside. The descrip-tion of the phase space used in TAUOLA is given in detailin [471]. The description of the phase space for τ produc-tion is given in [468]. Thanks to conformal symmetry itis exact for an arbitrary number of photons. Using expo-nentiation, see, for example, Yennie-Frautchi-Suura [493],the phase space description can be exact and the matrixelement can be refined order by order. For radiative correc-tions in the decay PHOTOS can be used. Its phase space isdescribed, for example, in the journal version of [494] andis exact. Approximations are made in the matrix elementonly. Benchmark comparisons with other calculations,which are actually based on second-order matrix elementsand exponentiation, found excellent agreement [495,496]. The lifetime of the τ lepton is orders of magnitude largernot only than its formation time in high energy experi-ments, but also than the time scale of all phenomena re-lated to higher-order corrections such as bremsstrahlung.The separation of τ production and decay is excellentdue to the small width of the τ lepton. Its propagator canbe well approximated by a delta function for phase spaceand matrix elements. The cross section for the process f ¯ f → τ + τ − Y ; τ + → X + ¯ ν τ ; τ − → l − ν l ν τ readsd σ = X spin | M | d Ω = X spin | M | d Ω prod d Ω τ + d Ω τ − , where Y and X + stand for particles produced togetherwith the τ + τ − and in the τ + decay, respectively; d Ω ,d Ω prod , d Ω τ + , d Ω τ − denote the phase space in the originalprocess, in production and decay, respectively.This formalism looks simple, but because of the over20 τ decay channels there are more than 400 distinct pro-cesses.Let us write the spin amplitude separated into theparts for τ pair production and decay: M = X λ λ =1 M prod λ λ M τ + λ M τ − λ . After integrating out the τ propagators, the formula forthe cross section can be rewritten asd σ = (cid:16)X spin | M prod | (cid:17)(cid:16)X spin | M τ + | (cid:17)(cid:16)X spin | M τ − | (cid:17) The purpose of this type of tests may vary. If two programsdiffer in their physics assumptions, it may help to control thephysics precision. If the physics assumptions are identical, butthe technical constructions differ, then the comparison checksthe correctness of the implementation of the algorithm. Fi-nally, the comparison of results from the same program, butinstalled on different computers, may check the correctness ofthe code’s implementation in new software environments. Suchcomparisons, or just the data necessary for comparisons, willbe referred to as physical, technical and installation bench-marks, respectively. They are indispensable for the reliable useof Monte Carlo programs. × wt d Ω prod d Ω τ + d Ω τ − , where wt = (cid:16) X i,j =0 , R ij h i + h j − (cid:17) ,R = 1 , < wt > = 1 , ≤ wt ≤ .R ij can be calculated from M λ λ , h i + and h j − from M τ + and M τ − , respectively. Bell inequalities (related tothe Einstein-Rosen-Podolsky paradox [497]) tell us thatin general it is impossible to rewrite wt in the followingfactorised form, wt factorized : wt = wt factorized = (cid:16) X i,j =0 , R Ai h i + (cid:17)(cid:16) X i,j =0 , R Bj h j − (cid:17) , where R Ai and R Bj are four-component objects calculatedfrom variables of the process of τ pair production. In theMonte Carlo construction it is thus impossible to gen-erate a τ + τ − pair, where each of the two is in somequantum state, and later to perform the decays of the τ + and the τ − independently. This holds at all orders ofthe perturbative expansion. τ production and decay arecorrelated through spin effects, which can be representedby the well-behaved factor wt introduced previously. Theabove formulae do not lead to any loss of precision andhold in presence of radiative corrections as well. Differ-ent options for the formalism, based on these expressions,are used in Monte Carlo programs and are basically wellfounded. This should be confronted with processes whereinstead of τ leptons short-lived intermediate states areconsidered. Then, in general, ambiguities appear and cor-rections proportional to the ratio of the resonance widthto its mass (or other energy scales of the process resulting,for example, from cut-offs) must be included. Interferingbackground diagrams may cause additional problems. Fordetails we refer to [498,469,468]. τ lepton production KORALB was published [467,499] more than twenty yearsago . It included first-order QED corrections and completemass and spin effects. It turned out to be very useful, andstill remains in broad use. On the other hand, some ofits ingredients are outdated and do not match the presentday requirements, even for technical tests. For examplethe function PIRET(S), which describes the real part ofthe photon hadronic vacuum polarisation as measured bythe data collected until the early 80’s should be replacedby one of the new precise codes (see Section 6 for details).Unfortunately, this replacement does not solve all nor-malisation problems of KORALB. For example, it is wellknown that the one-loop corrections are not sufficient. Thetwo major improvements which were developed during theLEP era are the introduction of higher-order QED correc-tions into Monte Carlo simulation and a better way tocombine loop corrections with the rest of the field theorycalculations. For energies up to 10 GeV (typical of the B factories), the KKMC Monte Carlo [468] provides a reali-sation of the above improvements. This program includeshigher-order QED matrix elements with the help of exclu-sive exponentiation, and explicit matrix elements up tothe second order. Also in this case the function calculat-ing the vacuum polarisation must be replaced by a versionappropriate for low energy (see Section 6).Once this is completed, and if the two-loop photon vac-uum polarisation can be neglected, KORALB and KKMCcan form a base for tests and studies of systematic errorsfor cross section normalisations at low energies. Using astrategy similar to the one for Bhabha scattering [500],the results obtained in [501,278] allow to expect a preci-sion of 0.35–0.45% using KKMC at Belle/BaBar energies.Certainly, a precision tag similar to that for linear collid-ers can also be achieved for lower energies. Work beyond[501] and explained in that paper would then be necessary. The matrix element used in TAUOLA for semi-leptonicdecays, τ ( P, s ) → ν τ ( N ) X , M = G √ u ( N ) γ µ ( v + aγ ) u ( P ) J µ (204)requires the knowledge of the hadronic current J µ . Theexpression is easy to manipulate. One obtains: | M | = G v + a ω + H µ s µ ) ,ω = P µ ( Π µ − γ va Π µ ) , ,H µ = 1 M ( M δ νµ − P µ P ν )( Π ν − γ va Π ν ) ,Π µ = 2[( J ∗ · N ) J µ + ( J · N ) J ∗ µ − ( J ∗ · J ) N µ ] ,Π µ = 2 Im ǫ µνρσ J ∗ ν J ρ N σ ,γ va = − vav + a . (205)If the τ coupling is v + aγ and m ν τ = 0 is allowed, onehas to add to ω and H µ :ˆ ω = 2 v − a v + a m ν M ( J ∗ · J ) , ˆ H µ = − v − a v + a m ν Im ǫ µνρσ J ∗ ν J ρ P σ . (206)The expressions are useful for Monte Carlo applicationsand are also calculable from first principles. The resultingexpression can be used to the precision level of the orderof 0.2–0.3%.In contrast to other parts, the hadronic current J µ stillcan not be calculated reliably from first principles. Sometheoretical constraints need to be fulfilled, but in generalit has to be obtained from experimental data. We willreturn to this point later (see Section 5.9). The PHOTOS Monte Carlo is widely used for generationof radiative corrections in cascade decays, starting fromthe early papers [502,503]. With time the precision of itspredictions improved significantly, but the main principleremains the same. Its algorithm is aimed to modify thecontent of the event record filled in with complete cascadedecays at earlier steps of the generation. PHOTOS modifiesthe content of the event record; it adds additional photonsto the decay vertices and at the same time modifies thekinematic configuration of other decay products.One could naively expect that this strategy is boundto substantial approximations. However, the algorithm iscompatible with NLO calculations, leads to a completecoverage of the phase space for multi-photon final statesand provides correct distributions in soft photon limits.For more details of the program organisation and its phasespace generation we address the reader to [494].The changes introduced over the last few years intothe PHOTOS Monte Carlo program itself were rathersmall and the work concentrated on its theoretical foun-dations. This wide and complex subject goes beyond thescope of this Review and the interested reader can con-sult [504], where some of the topics are discussed. Pre-vious tests of two-body decays of the Z into a pair ofcharged leptons [496] and a pseudoscalar B into a pairof scalars [494] were recently supplemented [505] with thestudy of W ± → l ± νγ . The study of the process γ ∗ → π + π − is on-going [506]. In all of these cases a universalkernel of PHOTOS was replaced with the one matchingthe exact first-order matrix element. In this way terms forthe NLO/NLL level are implemented. The algorithm cov-ers the full multi-photon phase space and it is exact in theinfrared region of the phase space. One should not forgetthat PHOTOS generates weight-one events.The results of all tests of PHOTOS with an NLO kernelare at a sub-per mill level. No differences with benchmarkswere found, even for samples of 10 events. When sim-pler physics assumptions were used, differences betweentotal rates at sub-per mill level were observed or they werematching a precision of the programs used for tests.This is very encouraging and points to the possibleextension of the approach beyond (scalar) QED, and inparticular to QCD and/or models with phenomenologicalLagrangians for interactions of photons with hadrons. Forthis work to be completed, spin amplitudes have to befurther studied [507].The refinements discussed above affect the practicalside of simulations for τ physics only indirectly. Changesin the kernels necessary for NLO may remain as optionsfor tests only. They are available from the PHOTOS webpage [505], but are not recommended for wider use. Thecorrections are small, and distributions visualising theirsize are available. On the other hand, their use could beperilous, as it requires control of the decaying particle spinstate. It is known (see, e.g., [508]) that this is not easybecause of technical reasons.We will show later that radiative corrections do notprovide a limitation in the quest for improved precision of matching theoretical models to experimental data untilissues discussed in subsection 5.12 are solved. So far all discussed contributions to the predictions werefound to be controlled to the precision level of 0.5% withrespect to the decay rate under study. This is not the case for the hadronic current, which isthe main source of our difficulties. It can not be obtainedfrom perturbative QCD as the energy scales involved aretoo small. On the other hand, for the low energy limits thescale is too large. Despite these difficulties one can obtaina theoretically clear object if enough effort is devoted. Thismay lead to a better understanding of the boundaries ofthe perturbative domain of QCD as well.The unquestionable property which hadronic currentsmust fulfil is Lorentz invariance. For example, if the finalstate consists of three scalars with momenta p , p , p ,respectively, it must take the form J µ = N ˘ T µν ˆ c ( p − p ) ν F + c ( p − p ) ν F + c ( p − p ) ν F ˜ + c q µ F − ic ǫ µ. νρσ π f π p ν p ρ p σ F ¯ , (207) where T µν = g µν − q µ q ν /q is the transverse projectorand q = p + p + p . The functions F i depend on threevariables that can be chosen as q = ( p + p + p ) and twoof the following three, s = ( p + p ) , s = ( p + p ) , s =( p + p ) . This form is obtained from Lorentz invarianceonly.Among the first four hadronic structure functions ( F , F , F , F ), only three are independent. We leave thestructure function F in the basis because, neglecting thepseudoscalar resonance production mechanism, the con-tribution due to F is negligible ( ∼ m π /q ) [509] and (de-pending on the decay channel) one of F , F and F dropsout, exactly as it is in TAUOLA since long.In each case, the number of independent functions isfour (rather than five) and not larger than the dimensionof our space-time. That is why the projection operatorscan be defined, for two- and three-scalar final states. Workin that direction has already been done in Ref. [473] andthen implemented in tests of TAUOLA too. Thanks tosuch a method, hadronic currents can be obtained fromdata without any need of phenomenological assumptions.Since long such methods were useful for data analysis, butonly in part. Experimental samples were simply too small.At present, for high statistics and precision the methodmay be revisited. That is why it is of great interest to ver-ify whether detector deficiencies will invalidate the methodor if adjustments due to incomplete phase space coverageare necessary. We will return to that question later. In This 0 . 5% uncertainty is for QED radiative effects. Oneshould bear in mind other mechanisms involving the produc-tion of photons, like, for example, the decay channel ω → πγ ,which occurs with a probability of (8 . ± . the mean time let us return to other theoretical consider-ations which constrain the form of hadronic currents, butnot always to the precision of today’s data. Once the allowed Lorentz structures are determined and aproper minimal set of them is chosen, one should imposethe QCD symmetries valid at low energies. The chiral sym-metry of massless QCD allows to develop an effective fieldtheory description valid for momenta much smaller thanthe ρ mass, χP T [510,511].Although χP T cannot provide predictions valid over thefull τ decay phase space, it constrains the form and thenormalisation of the form factors in such limits.The model, proposed in [445] for τ decaying to pions andused also for extensions to other decay channels, employsweighted products of Breit-Wigner functions to take intoaccount resonance exchange. The form factors used therehave the right chiral limit at LO. However, as it wasdemonstrated in [509], they do not reproduce the NLOchiral limit.The step towards incorporating the right low-energy limitup to NLO and the contributions from meson resonanceswhich reflect the experimental data was done within Reso-nance Chiral Theory ( RχT [437,436]). The current state-of-the-art for the hadronic form factors ( F i ) appearing inthe τ decays is described in [512,513]. Apart from the cor-rect low energy properties, it includes the right falloff [514,515] at high energies.The energy-dependent imaginary parts in the propagatorsof the vector and the axial-vector mesons, 1 / ( m − q − imΓ ( q )), were calculated in [516] at one-loop, exploitingthe optical theorem that relates the appropriate hadronicmatrix elements of τ decays and the cuts with on-shellmesons in the (axial-) vector-(axial-) vector correlators.This formalism has been shown to successfully describethe invariant mass spectra of experimental data in τ de-cays for the following hadronic systems: ππ [517,518,519], πK [520,521], 3 π [509,512,513,522] and KKπ [512,522].Other channels will be worked out along the same lines.It has already been checked that the RχT results pro-vide also a good description of the three-meson processes Γ ( τ → πν τ ) [523] and σ ( e + e − → KKπ ) I =1 [402].Both the spin-one resonance widths and the form factorsof the decays τ − → ( ππ, πK, π, KKπ ) − ν τ computedwithin RχT are being implemented in TAUOLA only now.Starting from a certain precision level, the predictions,like the ones presented above, may turn out to be not suf-ficiently precise. Nonetheless, even in such a case they canprovide some essential constraints on the form of the func-tions F i . Further refinements will require large and com-bined efforts of experimental and theoretical physicists.We will elaborate on possible technical solutions later inthe review. Such attempts turned out to be difficult inthe past and a long time was needed for parametrisa-tions given in [475] to become public. Even now they are semi-official and are not based on the final ALEPH and/orCLEO data. If one neglects quark masses, QCD is invariant under atransformation replacing quark flavours. As a consequence,hadronic currents describing vector τ decays (2 π, π, ηππ, . . . )and low energy e + e − annihilation into corresponding iso-vector final states are related and can be obtained fromone another [524,525]. This property, often referred to asconservation of the vector current (CVC) in τ decays, re-sults in the possibility to predict invariant mass distribu-tions of the hadronic system, as well as the correspond-ing branching fractions in τ decays using e + e − data. Asystematic check of these predictions showed that at the(5–10)% level they work rather well [526].In principle, the corrections due to mass and chargedifferences between u and d quarks are not expected toprovide significant and impossible to control effects [527,528]. However, the high-precision data of the CLEO [529],ALEPH [530], OPAL [531], Belle [476], CMD-2 [289,388,390,392], SND [288] and KLOE [374] collaborations in the2 π channel challenged this statement, and as it was shownin [35,20,17,532,36,380,27] that the spectral functions for τ − → π − π ν τ significantly differ from those obtained us-ing e + e − → π + π − data. Some evidence for a similar dis-crepancy is also observed in the τ − → π − π + π ν τ decay[533,534,339]. This effect remains unexplained. The mag-nitude of the isospin-breaking corrections has been up-dated recently, making the discrepancy in the 2 π channelsmaller [37].These CVC based relations were originally used in theTAUOLA form factors parametrisation, but they were of-ten modified to improve fits to the data. Let us point hereto an example where experimental e + e − data were usedfor the model of the τ → πν τ decay channels [474]. Inthis case, only a measurement of the distribution in thetotal invariant mass of the hadronic system was available.This is not enough to fix the distribution over the mul-tidimensional phase space. For other dimensions one hadto rely on phenomenological models or other experiments.In the future, this may not be necessary, but will alwaysremain as a method of benchmarks construction. As we have argued before, refined techniques for fits, in-volving simultaneous fits to many τ decay channels, arenecessary to improve the phenomenological description of τ decays. Complex backgrounds (where each channel con-tributes to signatures of other decay modes as well), differ-ent sensitivities of experiments for measurements of someangular distributions within the same hadronic system,and sometimes even an incomplete reconstruction of finalstates, are the main cause of this necessity. Moreover, the-oretical models based on the Lagrangian approach simul-taneously describe more than one τ decay channel with the same set of parameters, and only simultaneous fits al-low to establish their experimental constraints in a consis-tent way. Significant efforts are thus necessary and closecollaboration between phenomenologists and experimen-tal physicists is indispensable. As a result, techniques ofautomated calculations of hadronic currents may becomenecessary [535]. For the final states of up to three scalars, the use of pro-jection operators [473] is popular since long [533]. It en-ables, at least in principle, to obtain form factors used inhadronic currents directly from the data, for one scalarfunction defined in Eq. (207) at a time. Only recently ex-perimental samples became sufficiently large. However, toexploit this method one may have to improve it first bysystematically including the effects of a limited detectoracceptance. Implementation of the projection operatorsinto packages like MC-TESTER [536] may be useful. Ef-forts in that direction are being pursued now [538].On the theoretical side one may need to choose predic-tions from many models, before a sufficiently good agree-ment with data will be achieved. Some automated meth-ods of calculations may then become useful [539]. This isespecially important for hadronic multiplicities larger thanthree, when projector operators have never been defined.Certain automation of the methods is thus advisable.To discriminate from the broad spectrum of choices, newmethods of data analysis may become useful [540]. Suchmethods may require simulating samples of events whereseveral options for the matrix element calculation are usedsimultaneously. The neutrino coming from τ decays escapes detectionand as a result the τ rest frame can not be reconstructed.Nevertheless, as was shown in Ref. [473], angular distribu-tions can be used for the construction of projection opera-tors, which allow the extraction of the hadronic structurefunctions from the data. This is possible as they dependon s , s and q only.A dedicated module for the MC-TESTER [536], im-plementing the moments of different angular functions de-fined in Eqs. (39)–(47) of Ref. [473], is under development.The moments are proportional to combinations of the type α | F i | + β | F j | + γ Re(F i F ∗ j ), where the coefficients α , β and γ are functions of hadron four-momentum components inthe hadronic rest frame. Preliminary results obtained withlarge statistics of five million τ → a ν τ → πν τ decays,and assuming vanishing F and F form factors, show thatit is possible to extract | F | , | F | and | F · F ∗ | as func-tions of s , s and Q . This extraction requires solving a This may help to embed the method in the modern soft-ware for fits, see, e.g., [537]. Attempts to code such methods into TAUOLA, combinedwith programs for τ pair production and experimental detec-tor environment, were recently performed [541], but they wereapplied so far as prototypes only, see Fig. 1 of Ref. [542].2 set of equations. Since the solution is sensitive to the pre-cision of the estimation of the moments entering the equa-tion, large data samples of the order of O (10 − ) arenecessary. The calculation of the moments also requiresthe knowledge of the initial √ s of the τ pair, which makesthe analysis sensitive to initial state radiation (ISR) ef-fects. The same studies show that the analysis is easierif one, instead of extracting the form factors | F i | , com-pares the moments obtained from the experimental datawith theoretical predictions. Such a comparison does notrequire repetition of the Monte Carlo simulation of τ de-cays with different form factors, and only the calculationof combinations of | F i | and Re(F i F ∗ j ) is necessary. This ismuch simpler than comparing the kinematic distributionsobtained from data with distributions coming from MonteCarlo simulations with various theoretical models. Furthercomplications, for example, due to the presence of an ini-tial state bremsstrahlung or an incomplete acceptance ofdecay phase space, were not yet taken into account. Definitely the improvements of τ decay simulation pack-ages and fit strategies are of interest for phenomenologyof low energy. As a consequence, their input for such do-mains like phenomenology of the muon g − α QED , α QCD and their use in constraints of new physics wouldimprove.In this section, let us argue if possible benefits for LHCphenomenology may arise from a better understandingof τ decay channels in measurements as well. In the pa-pers [543,544] it was shown that spin effects can indeedbe useful to measure properties of the Higgs boson suchas parity. Moreover, such methods were verified to workwell when detector effects as proposed for a future linearcollider were taken into account. Good control of the de-cay properties is helpful. For example, in Ref. [545] it wasshown that for the τ → a ν τ → πν τ decay the sensitiv-ity to the τ polarisation increases about four times whenall angular variables are used compared with the usuald Γ/ d q , see also [546].Even though τ decays provide some of the most promi-nent signatures for the LHC physics program, see, e.g.,Ref. [547], for some time it was expected that methodsexploiting detailed properties of τ cascade decays are notpractical for LHC studies. Thanks to efforts on reconstruc-tion of π and ρ invariant mass peaks, this opinion evolves.Such work was done for studies of the CMS ECAL detectorinter-calibration [548], and in a relatively narrow p T range(5–10 GeV) some potentially encouraging results were ob-tained. Some work in context of searches for new particlesstarted recently [549]. There, improved knowledge of dis-tinct τ decay modes may become important at a certainpoint.One can conclude that the situation is similar to thatat the start of LEP, and some control of all τ decay chan-nels is important. Nonetheless, only if detector studies of π and ρ reconstruction will provide positive results, thegate to improve the sensitivity of τ spin measurements with most of its decay modes, as at LEP [550,551,552],will be open. At this moment, however, it is difficult tojudge about the importance of such improvements in thedescription of τ decays for LHC perspectives. The experi-ence of the first years of LHC must be consolidated first.In any case such an activity is important for the physicsof future Linear Colliders. We have shown that the most urgent challenge in the questfor a better understanding of τ decays is the developmentof efficient techniques for fitting multidimensional distri-butions, which take into account realistic detector con-ditions. This includes cross contamination of different τ decay modes, their respective signatures and detector ac-ceptance effects, which have to be simultaneously takeninto account when fitting experimental data. Moreover,at the current experimental precision, theoretical conceptshave to be reexamined. In contrast to the past, the pre-cision of predictions based on chiral Lagrangians and/orisospin symmetry can not be expected to always matchthe precision of the data. The use of model-independentdata analyses should be encouraged whenever possible inrealistic conditions.Good understanding of τ decays is crucial for under-standing the low energy regime of strong interactions andthe matching between the non-perturbative and the per-turbative domains. Further work on better simulations of τ decays at the LHC is needed to improve its potentialto study processes of new physics, especially in the Higgssector. In addition, an accurate simulation of τ decays isimportant for the control of backgrounds for very rare de-cays. For the project to be successful, this should lead tothe encapsulation of our knowledge on τ decays in formof a Monte Carlo library to be used by low-energy as wellas high-energy applications. The vacuum polarisation (VP) of the photon is a quantumeffect which leads, through renormalisation, to the scaledependence (‘running’) of the electromagnetic coupling, α ( q ). It therefore plays an important role in many phys-ical processes and its knowledge is crucial for many pre-cision analyses. A prominent example is the precision fitsof the Standard Model as performed by the electroweakworking group, where the QED coupling α ( q = M Z ) isthe least well known of the set of fundamental parame-ters at the Z scale, { G µ , M Z , α ( M Z ) } . Here we are moreconcerned about the VP at lower scales as it enters allphoton-mediated hadronic cross sections. These are used,e.g., in the determination of the strong coupling α s , thecharm and bottom quark masses from R had as well as inthe evaluation of the hadronic contributions to the muon g − α ( q ) itself. It also appears in Bhabha scattering in higher orders of perturbation theory needed for a pre-cise determination of the luminosity. It is hence clear thatVP also has to be included in the corresponding MonteCarlo programs.In the following we shall first define the relevant nota-tions, then briefly discuss the calculation of the leptonicand hadronic VP contributions, before comparing avail-able VP parametrisations. q γ ∗ Fig. 80. Photon vacuum polarisation Π ( q ). Conventionally the vacuum polarisation function is de-noted by Π ( q ) where q is a space- or time-like momen-tum. The shaded blob in Fig. 80 stands for all possibleone-particle irreducible leptonic or hadronic contributions.The full photon propagator is then the sum of the barephoton propagator and arbitrarily many iterations of VPinsertions,full photon propagator ∼ − iq · (1 + Π + Π · Π + Π · Π · Π + . . . ) . (208)The Dyson summation of the real part of the one-particleirreducible blobs then defines the effective QED coupling α ( q ) = α − ∆α ( q ) = α − Re Π ( q ) , (209)where α ≡ α (0) is the usual fine structure constant, α ∼ / a e , as measured bythe Harvard group to an amazing 0 . 24 ppb [1], in agree-ment with less precise determinations from caesium andrubidium atom experiments. The most precise value for α ,which includes the updated calculations of O ( α ) contri-butions to a e [553], is given by 1 /α = 137 . 035 999 084 (51).By using Eq. (209) we have defined Π to include theelectric charge squared, e for leptons, but note that differ-ent conventions are used in the literature, and sometimes Π is also defined with a different overall sign.Equation (209) is the usual definition of the runningeffective QED coupling and has the advantage that oneobtains a real coupling. However, the imaginary part ofthe VP function Π is completely neglected, which is nor-mally a good approximation as the contributions fromthe imaginary part are formally suppressed. This can beseen, e.g., in the case of the ‘undressing’ of the exper-imentally measured hadronic cross section σ had ( s ). Themeasured cross section e + e − → γ ∗ → hadrons contains | full photon propagator | , i.e. the modulus squared of theinfinite sum (208). Writing Π = e ( P + iA ) one easily seesthat | e ( P + iA ) + e ( P + iA ) + . . . | =1 + e P + e (3 P − A ) + e P ( P − A ) + . . . and that the imaginary part A enters only at order O ( e )compared to O ( e ) for the leading contribution from thereal part P . To account for the imaginary part of Π onemay therefore apply the summed form of the ‘(un)dressing’factor with the relation σ had ( s ) = σ ( s ) | − Π | (210)instead of the traditionally used relation with the real ef-fective coupling, σ had ( s ) = σ ( s ) (cid:18) α ( s ) α (cid:19) . (211)We shall return to a comparison of the different approachesbelow for the case of the hadronic VP.It should be noted that the summation breaks downand hence can not be used if | Π ( s ) | ∼ 1. This is the case if √ s is very close to or even at narrow resonance energies.In this case one can not include the narrow resonance inthe definition of the effective coupling but has to rely onanother formulation, e.g. through a Breit-Wigner prop-agator (or a narrow width approximation with a delta-function). For a discussion of this issue see [554]. Alsonote that the VP summation covers only the class of one-particle irreducible diagrams of factorisable bubbles de-picted in Fig. 80. This includes photon radiation withinand between single bubbles, but clearly does not take intoaccount higher-order corrections from initial state radia-tion or initial-final state interference effects in e + e − → hadrons .As will be discussed in the following, leptonic andhadronic contributions to ∆α are normally calculated sep-arately and then added, ∆α ( q ) = ∆α lep ( q )+ ∆α had ( q ).While the leptonic contributions can be predicted withinperturbation theory, the precise determination of the ha-dronic contributions relies on a dispersion relation usingexperimental data as input. The leptonic contributions ∆α lep have been calculated tosufficiently high precision. The leading order (LO) andnext-to-leading order (NLO) contributions are known asanalytic expressions including the full mass dependence [555],where LO and NLO refer to the expansion in terms of α . The next-to-next-to-leading order (NNLO) contribu-tion is available as an expansion in terms of m ℓ /q [11],where m ℓ is the lepton mass. To evaluate ∆α lep ( q ) for | q | < ∼ m τ , this expansion is not appropriate, but this isexactly the region where the hadronic uncertainties aredominant. Also from the smallness of the NNLO contribu-tion, we conclude that we do not need to further improvethe leptonic contributions beyond this approximation.The evaluation of the LO contribution is rather simple,and we briefly summarise the results below. Hereafter, it isunderstood that we impose the renormalisation condition Π (0) = 0 on Π ( q ). For q < 0, the VP function reads Π ( q ) = − e π (cid:16) − η (212)+3( − η ) p η ln √ η + 1 √ η − (cid:17) , where η ≡ m ℓ / ( − q ). For 0 ≤ q ≤ m ℓ one obtains Π ( q ) = − e π (cid:16) − η (213)+3( − η ) p − − η arctan √− − η − − η (cid:17) , and for q ≥ m ℓ Π ( q ) = − e π (cid:16) − η + 3( − η ) p η (214) · ln 1 + √ η − √ η (cid:17) − i e π (1 − η ) p η . An easily accessible reference which gives the NLO con-tributions is, for instance, Ref. [556,557]. As mentionedabove, the NNLO contribution is given in Ref. [11]. For allforeseeable applications the available formulae can be eas-ily implemented and provide a sufficient accuracy. Whilethe uncertainty from α is of course completely negligible,the uncertainty stemming from the lepton masses is onlytiny. Therefore the leptonic VP poses no problem. In contrast to the leptonic case, the hadronic VP Π had ( q )can not be reliably calculated using perturbation theory.This is clear for time-like momentum transfer q > Π had ( q ) ∼ σ ( e + e − → hadrons ) goes through all the resonances in the low energyregion. However, it is possible to use a dispersion relationto obtain the real part of Π from the imaginary part. Thedispersion integral is given by ∆α (5)had ( q ) = − q π α P Z ∞ m π σ ( s ) d ss − q , (215)where σ ( s ) is the (undressed) hadronic cross sectionwhich is determined from experimental data. Only awayfrom hadronic resonances and (heavy) quark thresholdsone can apply perturbative QCD to calculate σ ( s ). Inthis region the parametric uncertainties due to the val-ues of the quark masses and α s , and due to the choice ofthe renormalisation scale, are small. Therefore the uncer-tainty of the hadronic VP is dominated by the statisticaland systematic uncertainties of the experimental data for σ ( s ) used as input in (215).Note that the dispersion integral (215) leads to a smoothfunction for space-like momenta q < 0, whereas in thetime-like region it has to be evaluated using the principalvalue description and shows strong variations at resonance energies, as demonstrated e.g. in Fig. 81. In Eq. (215) ∆α (5)had denotes the five-flavour hadronic contribution. Atenergies we are interested in, i.e. far below the t ¯ t threshold,the contribution from the top quark is small and usuallyadded separately. The analytic expressions for ∆α top ( q )obtained in perturbative QCD are the same as for theleptonic contributions given above, up to multiplicativefactors taking into account the top quark charge and thecorresponding SU(3) colour factors, which read Q t N c atLO and Q t N c − N c at NLO.Contributions from narrow resonances can easily betreated using the narrow width approximation or a Breit-Wigner form. For the latter one obtains ∆α Breit − Wigner ( s ) = 3 Γ ee αM s ( s − M − Γ )( s − M ) + M Γ , (216)with M , Γ and Γ ee the mass, total and electronic widthof the resonance. For a discussion of the undressing of Γ ee see [554].Although the determination of ∆α (5)had ( q ) via the dis-persion integral (215) may appear straightforward, in prac-tice the data combination for σ ( s ) is far from triv-ial. In the low energy region up to about 1 . − σ ( s ) can be summed. For higher en-ergies the data for the fast growing number of possiblemulti-hadronic final states are far from complete, and in-stead inclusive (hadronic) measurements are used. For thedetails of the data input, the treatment of the data w.r.t.radiative corrections, the estimate of missing thresholdcontributions and unknown subleading channels (often viaisospin correlations) and the combination procedures werefer to the publications of the different groups cited be-low.In the following we shall briefly describe and then com-pare the evaluations of the (hadronic) VP available asparametrisations or tabulations from different groups. For many years Helmut Burkhardt and Bolek Pietrzykhave been providing the Fortran function named REPIfor the leptonic and hadronic VP [175,558,260,559,15].While the leptonic VP is coded in analytical form withone-loop accuracy, the hadronic VP is given as a very com-pact parametrisation in the space-like region, but does notcover the time-like region. For their latest update see [7].The code can be obtained from Burkhardt’s web-pageswhich contain also a short introduction and a list of olderreferences, see http://hbu.web.cern.ch/hbu/aqed/aqed.html .Similarly, Fred Jegerlehner has been providing a pack-age of Fortran routines for the running of the effectiveQED coupling [259,13,21,20,18,19]. It provides leptonicand hadronic VP both in the space- and time-like region. Fig. 81. Different contributions to ∆α ( s ) in the time-like re-gion as given by the routine from Fred Jegerlehner. Figureprovided with the package alphaQED.uu from his homepage. For the leptonic VP the complete one- and two-loop re-sults and the known high energy approximation for thethree-loop corrections are included. The hadronic con-tributions are given in tabulated form in the subroutineHADR5N. The full set of routines can be downloaded fromJegerlehner’s web-page ∼ fjeger/ . Theversion available from there is the one we use in the com-parisons below and was last modified in November 2003.It will be referred to as J03 in the following. An update isin progress and other versions may be available from theauthor upon request. Note that for quite some time hisroutine has been the only available code for the time-likehadronic VP. Fig. 81 shows the leptonic and hadronic con-tributions together with their sum as given by Jegerleh-ner’s routine.The experiments CMD-2 and SND at Novosibirsk areusing their own VP compilation to undress hadronic crosssections, and the values used are given in tables in some oftheir publications. Recently CMD-2 has made their com-pilation publicly available, see Fedor Ignatov’s web-page http://cmd.inp.nsk.su/ ∼ ignatov/vpl/ . There linksare given to a corresponding talk at the ‘4th meeting ofthe Working Group on Radiative Corrections and MonteCarlo Generators for Low Energies’ (Beijing 2008), to thethesis of Ignatov (in Russian) and to a file containing thetabulation, which can be used together with a download-able package. The tabulation is given for the real andimaginary parts of the sum of leptonic and hadronic VP,for both space- and time-like momenta, and for the corre-sponding errors. Fig. 82, also displayed on their web-page,shows the results from CMD-2 for | Π | both for thespace- and time-like momenta in the range − (15 GeV) 95 GeV. As for the discrepancy at 1GeV < ∼ √ s < ∼ . . < ∼ √ s < ∼ . 95 GeV is further scrutinised inFig. 84, where in addition to the two parametrisationsHMNT (solid (red) line) and J03 (dotted (blue) line), theresult for ∆α (5)had ( s ) /α obtained by integrating over the R -data as compiled by the PDG [267] is shown as thedashed (green) line. While the results from HMNT andthe one based on the PDG R -data agree rather well, theirdisagreement with the J03 compilation in the region 0 . < ∼ √ s < ∼ . 95 GeV is uncomfortably large comparedto the error but may be due to a different data input ofthe J03 parametrisation.In the following we shall compare the parametrisationfrom HMNT with the one from the CMD-2 collaborationwhich has become available very recently. Note that forundressing their experimentally measured hadronic crosssections, CMD-2 includes the imaginary part of the VPfunction Π ( q ) in addition to the real part. Before com-ing to the comparison with CMD-2, let us discuss somegeneralities about Im Π ( q ). If we are to include the imag-inary part, then the VP correction factor α ( q ) should bereplaced as (cid:18) α − ∆α ( q ) (cid:19) = (cid:18) α − Re Π ( q ) (cid:19) → (217) (cid:12)(cid:12)(cid:12)(cid:12) α − Π ( q ) (cid:12)(cid:12)(cid:12)(cid:12) = α (1 − Re Π ( q )) + (Im Π ( q )) . Note that, as mentioned already in the introduction, thecontribution from the real part appears at O ( e ) in thedenominator, while that from the imaginary part startsonly at O ( e ). Because of this suppression we expect theeffects from the imaginary part to be small. Neverthelesswe would like to stress two points. First, field-theoretically,it is more accurate to include the imaginary part which The actual compilation of the data is available in electronicform from http://pdg.lbl.gov/2008/hadronic-xsections/hadronicrpp page1001.dat . √ -Q (GeV) ∆ α h a d ( ) ( Q ) / α solid (red): HMNTdashed: BP05dotted: J03 dashed: (BP05-HMNT)/HMNT ( × × -2-101234 10 -1 √ s (GeV) ∆ α h a d ( ) ( s ) / α solid (red): HMNTdotted (blue): J03 -2-10123 -1 Fig. 83. Comparison of the results from Hagiwara et al.(HMNT [554]) for ∆α (5)had ( q ) in units of α with parametrisa-tions from Burkhardt and Pietrzyk (BP05 [7]) and Jegerlehner(J03). Upper panel: ∆α (5)had ( Q ) /α for space-like momentumtransfer ( Q < × ∆α (5)had ( s ) /α from J03and HMNT (as labelled) for time-like momenta ( q = s ). Forreadability, only the error band of HMNT is displayed. exists above threshold. Including only Re Π ( q ) in the VPcorrection is an approximation which may be sufficient inmost cases. Second, it is expected that the contributionfrom the imaginary part is of the order of a few per mill ofthe total VP corrections. While this seems small, it can benon-negligible at the ρ meson region where the accuracyof the cross section measurements reaches the order of (oreven less than) 1%. Similarly, in the region of the narrow φ resonance, the contributions from the imaginary partbecome non-negligible and should be taken into account.In Fig. 85 the VP correction factor, based on the com-pilation from HMNT, with and without Im Π ( q ) is com-pared to | − Π ( s ) | as used by the CMD-2 collaboration intheir recent analysis of the hadronic cross section in the -1.5-1-0.5 0 0.5 1 1.5 2 2.5 3 3.5 0.7 0.75 0.8 0.85 0.9 0.95 1 ∆ α ( ) h a d ( s ) / α √ s (GeV) HMNTRPP dataJegerlehner Fig. 84. Comparison of the results from Hagiwara et al.(HMNT, solid (red) line) for ∆α (5)had ( s ) /α with the parametrisa-tion from Jegerlehner (J03, dotted (blue) line) in the time-likeregion in the range √ s = 0 . − π channel in the ρ central region [392]. In the upperpanel the VP correction factors are given, whereas in thelower panel the differences are shown. As expected, thedifferences between the three are visible, and are about afew per mill at most. The difference between the CMD-2results and the one from HMNT including Im Π ( q ) (solid(red) curve in the lower panel of Fig. 85 shows a markeddip followed by a peak in the ρ − ω interference regionwhere the π + π − cross section falls sharply. This is mostprobably a direct consequence of the different data inputused. However, in most applications such a difference willbe partially cancelled when integrated over an energy re-gion including the ρ peak.In Figs. 86 and 87 we compare ∆α ( s ) in the time-like region as given by the parametrisation from CMD-2with the one from HMNT, where for HMNT we have cal-culated the leptonic contributions (up to including theNNLO corrections) as described above. The two panelsin Fig. 86 (upper panel: 0 < √ s < < √ s < 10 GeV) show ∆α ( s ) with the 1 σ er-ror band from CMD-2 as a solid (blue) band, whereas forHMNT the mean value for ∆α ( s ) is given by the dotted(red) line, which can hardly be distinguished. To high-light the differences between the two parametrisations,Fig. 87 displays the normalised difference ( ∆α CMD − ( s ) − ∆α HMNT ( s )) /∆α HMNT ( s ) as a solid (black) line, and alsoshows the relative errors of CMD-2 and HMNT as dashed(blue) and red (dotted) lines, respectively. As visible inFig. 87, the error as given by the CMD-2 parametrisa-tion is somewhat smaller than the one from HMNT. Bothparametrisations agree fairly well, and for most energiesthe differences between the parametrisations are about aslarge or smaller than the error bands. Close to narrow We thank Gennadiy Fedotovich for providing us with atable including the VP correction factors not included in [392]. | − Π ( s ) | √ s (MeV) CMD2HMNT w/o Im Π (s)HMNT w/ Im Π (s)-0.004-0.002 0 0.002 0.004 0.006 0.008 600 650 700 750 800 850 900 950 D i ff e r e n ce i n | − Π ( s ) | √ s (MeV)(CMD2) – (HMNT w/ Im Π )(CMD2) – (HMNT w/o Im Π )(HMNT w/ Im Π ) – (HMNT w/o Im Π ) Fig. 85. Upper panel: Correction factor | − Π ( s ) | as used for‘undressing’ by the CMD-2 collaboration in [392] (dashed line)compared to the same quantity using the HMNT compilationfor the e + e − → hadrons data (solid line). Also shown is thecorrection factor (1 − Re Π ) = ( α/α ( s )) , based on α ( s ) inthe time-like region from HMNT (dotted line). Lower panel:Differences of the quantities as indicated on the plot. resonances the estimated uncertainties are large, but asdiscussed above, there the approximation of the effectivecoupling α ( s ) breaks down and resonance contributionsshould be treated differently. Vacuum polarisation of the photon plays an importantrole in many physical processes. It has to be taken intoaccount, e.g., in Monte Carlo generators for hadronic crosssections or Bhabha scattering. When low energy data areused in dispersion integrals to predict the hadronic contri-butions to muon g − ∆α ( q ), undressed data have to beused, so VP has to be subtracted from measured cross sec-tions. The different VP contributions have been discussed,and available VP compilations have been briefly describedand compared. Until recently only one parametrisation √ s (GeV) ∆ α ( s ) dotted (red): HMNTband (blue): CMD-2 -0.04-0.0200.020.040.060.08 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 √ s (GeV) ∆ α ( s ) dotted (red): HMNTband (blue): CMD-2 -0.02-0.0100.010.020.030.040.050.06 2 3 4 5 6 7 8 9 10 Fig. 86. ∆α ( s ) in the time-like region as given by theparametrisation from CMD-2 (solid (blue) band) comparedto the same quantity from HMNT (dotted (red) line). Upperpanel: 0 < √ s < < √ s < 10 GeV. has been available in the time-like region, now three rou-tines in the space- and time-like regions exist, from Jegerlehner,CMD-2 and HMNT, and a fourth from Burkhardt andPietrzyk in the space-like region. While the accuracy ofthe hadronic cross section data themselves is the limitingfactor in the precise determination of g − ∆α ( M Z ),the error of the VP (or ∆α ( q )) is not the limiting fac-tor in its current applications. With the ongoing efforts tomeasure σ had ( s ) with even better accuracy in the wholelow energy region, further improvements of the various VPparametrisations are foreseen. In this Report we have summarised the achievements ofthe last years of the experimental and theoretical groupsworking on hadronic cross section measurements and tauphysics. In addition we have sketched the prospects in thisfield for the years to come. We have emphasised the im-portance of continuous and close collaboration betweenthe experimental and theoretical groups which is crucialin the quest for precision in hadronic physics. The plat-form set to simplify this collaboration is a Working Groupon Radiative Corrections and Monte Carlo Generators for √ s (GeV) ( C M D - - H M N T ) / H M N T , σ e rr o r b a nd s (r e l . ) -0.15-0.1-0.0500.050.10.15 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 √ s (GeV) ( C M D - - H M N T ) / H M N T , σ e rr o r b a nd s (r e l . ) -0.04-0.0200.020.040.060.08 2 3 4 5 6 7 8 9 10 Fig. 87. Solid (black) lines: Normalised difference( ∆α CMD − ( s ) − ∆α HMNT ( s )) /∆α HMNT ( s ) in the time-like region. The dashed (blue) and dotted (red) lines indicatethe relative error for the CMD-2 and HMNT parametrisations.Upper panel: 0 < √ s < < √ s < Low Energies (Radio MontecarLow) , for the better under-standing of the needs and limitations of both experimentaland theoretical communities and to facilitate the informa-tion flow between them. This Review is a result of theWorking Group.The Report was divided into five Sections covering theluminosity measurements at low energies (up to the energyof B factories) (Section 2), R measurement by energy scan(Section 3), R measurement using radiative return (Sec-tion 4), tau physics (Section 5), and the calculation of thevacuum polarisation with emphasis on the hadronic con-tributions (Section 6). In all the Sections, with the excep-tion of Section 6, we gave an overview of the experimentalresults and the status of the Monte Carlo event generatorsused in the experimental analyses with emphasis on theiraccuracy and tests.Concerning the work done on the topic of precisionluminosity measurement (Section 2), a particular effortwas paid to arrive at an up-to-date estimate of the ac-curacy of the most precise MC tools used by the exper-imentalists. Several tuned comparisons between the pre-dictions of independent generators were presented, consid-ering the large-angle Bhabha process with realistic event selection criteria and at different c.m. energies. It turnedout that the three most precise luminosity tools, i.e. theprograms BabaYaga@NLO, BHWIDE and MCGPJ, agreewithin 0.1% for the integrated cross sections and withinless than 1% for the differential distributions. Thereforethe main conclusion of the work on tuned comparisons isthat the technical precision of MC programs is well undercontrol, the (minor) discrepancies still observed being dueto slightly different details in the treatment of radiativecorrections and their implementation. The theoretical ac-curacy of the generators with regard to radiative correc-tions not fully taken into account was assessed by per-forming detailed comparisons between the results of thegenerators and those of exact perturbative calculations.In particular, explicit cross-checks with the predictions ofavailable NNLO QED calculations and with new exactresults for lepton and hadron pair corrections led to theconclusion that the total theoretical uncertainty is at theone per mill level for the large-angle Bhabha process atdifferent c.m. energies. Albeit this error estimate could beput on firmer grounds thanks to further work in progress,it appears to be already quite robust and sufficient for aprecise determination of the luminosity.In Section 3 we presented the current status of thestudies of e + e − annihilation into hadrons and muons atthe energies up to a few GeV. Accurate measurements ofthe ratio R , i.e. the ratio of the cross sections of hadronand muon channels, are crucial for the evaluation of thehadronic contribution to vacuum polarisation and subse-quently for various precision tests of the Standard Model.Results of several experimental collaborations have beenreviewed for the most important processes with the fi-nal states µ + µ − , π + π − , π + π − π , π + π − π , π + π − , twokaons and heavier mesons. In particular, R scans at theexperiments CMD-2, SND, CLEO and BES experimentshave been discussed. Analytic expressions for the Bornlevel cross sections of the main processes have been pre-sented. First-order QED radiative corrections have beengiven explicitly for the case of muon, pion and kaon pairproduction. The two latter cases are computed using scalarQED to describe interactions of pseudoscalar mesons withphotons in the final state. Matching with higher-orderQED corrections evaluated in the leading logarithmic ap-proximation have been discussed. Good agreement be-tween different Monte Carlo codes for the muon channelhas been shown. The theoretical uncertainty in the de-scription of these processes has been evaluated. For thetwo main channels, e + e − → µ + µ − and e + e − → π + π − ,this uncertainty has been estimated to be of the order of0 . τ lep-tons. The available programs have been discussed in thecontext of the required accuracy to match current high-statistics experimental data. After a review of the existingprograms used in the data analysis we have emphasisedthe topics which will require particular attention in thefuture. We have elaborated on the efforts which are goingon at present and focused on the necessary improvements.The techniques for fitting τ decay currents require partic-ular attention. The observed spectra and angular distri-butions are a convolution of theoretical predictions withexperimental effects which should be taken into account inthe fitting procedures. Background contributions also playan important role if high precision is requested. We havealso commented on the impact of these efforts for forth-coming high energy experiments (like at LHC), where τ decays are used to constrain hard processes rather thanto measure properties of τ decays.In Section 6 the different vacuum polarisation (VP)contributions have been discussed, and available parametri-sations have been compared. VP forms a universal part ofradiative corrections and as such is an important ingre-dient in Monte Carlo programs. In addition, to evaluatethe hadronic contributions to the muon g − ∆α ( q )via dispersion relations, one has to use the ‘undressed’hadronic cross section, i.e. data with the VP effects re-moved. Therefore the precise knowledge of VP is required.While in the space-like region the VP is a smooth functionand the parametrisations are in excellent agreement, inthe time-like region the VP is a fast varying function and differences exist between different parametrisations, espe-cially around resonances. However, the accuracy which istypically of the order of or below a few per mill and theagreement of the more recent compilations indicate thatthe current precision of VP is sufficient for the envisagedapplications. In the future better hadronic cross sectiondata will lead to further improved accuracy. Acknowledgements This work was supported in part by: – European Union Marie-Curie Research Training NetworksMRTN-CT-2006-035482 “FLAVIAnet” and MRTN-CT-2006-035505 “HEPTOOLS”; – European Union Research Programmes at LNF, FP7, Trans-national Access to Research Infrastructure (TARI), HadronPhysics2-Integrating Activity, Contract No. 227431; – Generalitat Valencianaunder Grant No. PROMETEO/2008/069; – German Federal Ministry of Education and Research (BMBF)grants 05HT4VKA/3, 06-KA-202 and 06-MZ-9171I; – German Research Foundation (DFG): ’Emmy Noether Pro-gramme’, contracts DE839/1-4, ’Heisenberg Programme’and Sonderforschungsbereich/Transregio SFB/TRR 9; – Initiative and Networking Fund of the Helmholtz Associa-tion, contract HA-101 (”Physics at the Terascale”); – INTAS project Nr 05-1000008-8328 “Higher-order effectsin e + e − annihilation and muon anomalous magnetic mo-ment”; – Ministerio de Ciencia e Innovaci´on under Grant No. FPA2007-60323, and CPAN (Grant No. CSD2007-00042); – National Natural Science Foundation of China under Con-tracts Nos. 10775142, 10825524 and 10935008; – Polish Government grant N202 06434 (2008-2010); – PST.CLG.980342 – Research Fellowship of the Japan Society for the Promotionof Science for Young Scientists; – RFBR grants 03-02-16477, 04-02-16217, 04-02-1623, 04-02-16443, 04-02-16181-a, 04-02-16184-a, 05-02-16250-a, 06-02-16192-a, 07-02-00816-a, 08-02-13516, 08-02-91969 and 09-02-01143; – Theory-LHC-France initiative of CNRS/IN2P3; – US DOE contract DE-FG02-09ER41600.We thank J. 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