The energy dependence of the electron - positron hadronic cross section can
be measured not only by a straightforward energy scan, but also by means of the
radiative return method. To provide extensive comparisons between theory and
experiment a Monte Carlo event generator is an indispensable tool. We have
developed such a generator called PHOKHARA, which simulates e+e- -> mesons +
photon(s) processes. In this paper we present its latest tests and upgrades.Comment: 5 pages, 7 figures, presented at RADCOR 2002, Kloster Banz, Sept.
  8-13, 200

Bennett

Berends

Binner

Brown

Burkhardt

Czyż

Davier

Decker

Denig

Eidelman

Erler

Germán Rodrigo

Henryk Czyż

Hughes

Johann H. Kühn

Khoze

Kühn

Melnikov

Narison

Rodrigo

Solodov

Spagnolo

English

arXiv

The energy dependence of the electron - positron hadronic cross section can be measured not only by a straightforward energy scan, but also by means of the radiative return method. To provide extensive comparisons between theory and experiment a Monte Carlo event generator is an indispensable tool. We have developed such a generator called PHOKHARA, which simulates e+e- -> mesons + photon(s) processes. In this paper we present its latest tests and upgrades.The energy dependence of the electron - positron hadronic cross section can be measured not only by a straightforward energy scan, but also by means of the radiative return method. To provide extensive comparisons between theory and experiment a Monte Carlo event generator is an indispensable tool. We have developed such a generator called PHOKHARA, which simulates e+e- -> mesons + photon(s) processes. In this paper we present its latest tests and upgrades

Czyz, Henryk

Kuhn, Johann H.

Rodrigo, German

CERN Document Server

1CERN-TH/2002-316TTP02-35Radiative return at e+e− factories. Henryk Czy_za, yJohann H. Ku¨hnb, zGerman Rodrigoc, xaInstitute of Physics, University of Silesia, PL-40007 Katowice, Poland.bInstitut fu¨r Theoretische Teilchenphysik, Universita¨t Karlsruhe, D-76128 Karlsruhe, Germany.cTheory Division, CERN, CH-1211 Geneva 23, Switzerland.The energy dependence of the electron - positron hadronic cross section can be measured not only by a straight-forward energy scan, but also by means of the radiative return method. To provide extensive comparisons betweentheory and experiment a Monte Carlo event generator is an indispensable tool. We have developed such a gen-erator called PHOKHARA, which simulates e+e− → mesons + photon(s) processes. In this paper we present itslatest tests and upgrades.1. INTRODUCTION AND PHOKHARAUPGRADESIn view of the precision of the recent measure-ments of the muon anomalous magnetic momentaµ  (g − 2)µ=2 at BNL [1], hadronic contri-butions are crucial for the interpretation of thismeasurement, in particular for the isolation ofthe electroweak or of non-Standard Model physicscontributions [2]. Their understanding becomeseven more important, as it seems [3] that thee+e− annihilation data are not consistent withthe  decay data. A new aµ measurement, whichis under way, will challenge the theoretical pre-dictions even more.An important ingredient and the dominantsource of uncertainties in the theoretical predic-tion for the muon anomalous magnetic momentis the hadronic vacuum polarization [4]. It isin turn related via dispersion relations to thecross section for electron{positron annihilationWork supported in part by BMBF under grant number05HT9VKB0, EC 5-th Framework EURIDICE networkproject HPRN-CT2002-00311 and TARI project HPRI-CT-1999-00088; presented by H. Czyz˙ at RADCOR 2002,Kloster Banz, September 8 - 13, 2002.ySupported in part by EC 5-th Framework, contractHPRN-CT-2000-00149; e-mail:czyz@us.edu.plze-mail:jk@particle.uni-karlsruhe.dexSupported in part by E.U. TMR grant HPMF-CT-2000-00989; e-mail: german.rodrigo@cern.chinto hadrons had = (e+e− ! hadrons) andin some cases via isospin symmetry to the decaywidth  ! τ + hadrons. This quantity plays animportant role also in the evolution of the elec-tromagnetic coupling QED from the Thompsonlimit to high energies [4,5]. The interpretation ofimproved measurements at high energy colliderssuch as LEP, Tevatron, the LHC or TESLA there-fore depends signicantly on the precise knowl-edge of had.The feasibility of using tagged photon events athigh luminosity electron{positron storage rings,such as the -factory, DANE, CLEO-C or B-factories, to measure had over a wide range ofenergies has been proposed and studied in detailin [6{8] (see also [9,10]). The machine is operatingat a xed energy of the e+e− centre-of-mass sys-tem (cms) and the initial state radiation (ISR) isused to reduce the invariant mass of the hadronicsystem.The radiation of photons from the hadronicsystem (the nal state radiation, FSR) shouldbe considered as a background and can be sup-pressed by choosing suitable kinematical cuts,or controlled by the simulation, once a suitablemodel for this amplitude has been adopted. Onends that selecting events with the tagged pho-tons close to the beam axis and well separatedfrom the hadrons indeed reduces FSR drastically.As demonstrated in Fig. 1, the FSR contribution2to the total cross section can be easily reduced tothe 1% level. The model dependence of the FSRin the case of the +− nal state can be con-trolled by the same experiment through studiesof the forward{backward asymmetry of the an-gular charged pion distribution. The asymmetrycomes from FSR{ISR interference and integratesto zero for ’charge blind’ congurations. As a re-sult it does not contributes to rates in Fig. 1.It can be used, however, to calibrate the FSRamplitude and more detailed tests of its modeldependence.020406080100120dσ(e+e−→pi+pi−γ)/dQ 2  (nb/GeV 2 )1.02 GeVθγ < 15o or θγ > 165o  40o < θpi < 140oISR+FSR  ≈  ISR onlyISR+FSRISR only10-410-20 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Q 2 (GeV 2)FSR/ISRθγ < 15o or θγ > 165o40o < θpi < 140oFigure 1. The role of the cuts in the suppressionof the FSR contributions to the cross section. Re-sults from the PHOKHARA generator. No cuts(upper curves) and suitable cuts applied (lowercurves).When running at higher energies, the FSR issuppressed with respect to the ISR by the dif-ferent behaviour of various propagators and formfactors relevant to the problem. In practice itmeans that no special angular cuts are needed tosuppress the FSR contribution when running athigh energies.The suppression of the FSR overcomes theproblem of its model dependence, which must betaken into account in a completely inclusive mea-surement [11].Preliminary experimental results using thismethod have been presented recently by theKLOE collaboration at DANE [12{15]. Largeevent rates were also observed by the BaBar col-laboration [16].In the rst version of the newly developedMonte Carlo program PHOKHARA [17] we haveconsidered the full next-to-leading order (NLO)QED corrections to the ISR in the annihilationprocess e+e− ! γ + hadrons, for the case wherethe photon is observed under a non-vanishing an-gle relative to the beam direction. The virtualand soft photon corrections were presented in [18]and the contribution of the emission of a secondhard photon in [17]. The nal hadronic state waslimited to the +−, with the hadronic currentmodeled as in [19], and the nal state emissionwas not included. The program allowed also forthe generation of +−γ(γ) nal states, againlimited to the emission of photon(s) from the ini-tial leptons.Radiative corrections proportional to (m2e )relevant to congurations with photons emittedat very low (’ me=ps) angles relative to thebeam direction were calculated in [20] and areincluded in the new version of PHOKHARA [21].The leading order corrections proportional to m2eare typically of the order of a few per cent [22],while the non-leading ones are of order of 0.1%,as seen from Fig.2. They will be important whenthe precision of the measurement will be below1%. Their eect depends on Q2 and thus aectsthe Q2 distribution from which the hadronic crosssection is extracted.Another new feature of the PHOKHARAevent generator is the inclusion of the four-pionhadronic nal states (2+2− and 20+− ).The description of the hadronic current in thatcase is based on the paper [23], with changes de-scribed in [8]. The comparison with the MonteCarlo, which simulates the same process at lead-ing order [8] and includes additional collinearradiation through structure function (SF) tech-niques, shows typical dierence of order of 1% asseen in Fig. 3 (a similar behaviour can be ob-served for 20+− nal state). The non-leading3ps = 10 GeVps = 4 GeVps = 1 GeVe+e  ! + ()pQ2 GeVd(NLOmass)dpQ2=ddpQ22:521:510:500 0:0002 0:0004 0:0006 0:0008 0:001 0:0012 0:0014 0:0016 0:0018Figure 2. The relative contributions of the non-leading mass corrections to the dierential crosssection atps = 1 GeV, 4 GeV and 10 GeV.contributions to the cross section of the four-pionnal states are of the expected size and of thesame order as for the two-pion nal state [17].The program now includes also the contribu-tions from the nal state emitted photons to-gether with the ISR{FSR interference calculatedat the lowest order for +− and +− nalstates, while for the four-pion nal states the FSRcontribution is not taken into account.2. TESTS OF PHOKHARAAn obvious and one of the most importanttasks in the construction of a Monte Carlo eventgenerator is to demonstrate that its technical ac-curacy is much better than the desired physi-cal accuracy. The tests that were performed forthe previous version of PHOKHARA [17] are stillvalid, within the limitations of this version, whichwas applicable for non-vanishing photon angles.In the rst step we demonstrate the indepen-dence of total and dierential cross sections ofthe separation parameter  (called w in [17]) be-tween soft and hard photon regions. The soft pho-ton contribution is calculated analytically, whilethe additional hard photon is treated via MonteCarlo simulation. The parameter that speciese+e ! 2+2 ()pQ2GeV1 d(NLO)dpQ2=d(LL)dpQ243:532:521:510:500:020:0150:010:0050 0:005 0:01 0:015 0:02 0:025 0:03Figure 3. The relative non-leading contributionsto the dierential cross section atps = 4 GeV.NLO - full next-to-leading result, LL - leadinglogarithmic approximation.the separation between the two regions of thephase space has to be kept small enough to val-idate the soft photon approximation and largeenough to avoid negative weights. We performedthe tests for a +− hadronic nal state in [17],while for one of the four-pion modes the resultsare collected in Figs. 4 and 5. From Fig. 4it is clear that the choice  = 10−3 is still toobig, whereas Fig. 5 demonstrates the stabilityof the results between  = 10−4 and  = 10−5.This also proves that the Monte Carlo integra-tion works well in the soft photon region. TheMonte Carlo integration of the part of the pro-gram that simulates one hard large-angle photonemission was tested in [17] against a Gauss nu-merical integration. As shown in Fig. 6 a techni-cal precision of the program at the level of 10−4was demonstrated. Analytical results exist for thedierential (in Q2) cross section integrated overthe whole angular range of the photon(s) for bothone and two emitted photons [24]. The compar-ison with PHOKHARA can be found in Fig. 7,4e+e ! 20+ ()pQ2GeV1 d(=10 3)dpQ2=d(=10 4)dpQ210:90:80:70:60:50:40:0030:0020:0010 0:001 0:002 0:003Figure 4. The relative dierence of the dieren-tial cross sections for two dierent values of theseparation parameter .demonstrating again an excellent technical pre-cision also for the two-photon nal state. Theresults presented in Fig. 7 refer to the sum of vir-tual and hard corrections to the e+e− ! +−γcross section, while more detailed tests can befound in [21].3. CONCLUSIONSThe PHOKHARA Monte Carlo event genera-tor was upgraded, allowing for simulation in thesmall photon angles region. Besides the +−hadronic nal state, its present version includesalso 2+2− and 20+− nal states. For the+− and +− nal states, the FSR photoniccontributions were implemented at the lowest or-der, including ISR{FSR interference. Further up-grades are in progress.REFERENCES1. H. N. Brown et al. [Muon g − 2 Collabora-tion], Phys. Rev. Lett. 86 (2001) 2227 [hep-ex/0102017]; G.W.Bennett et al. [Muon g−2e+e ! 20+ ()pQ2GeV1 d(=10 4)dpQ2=d(=10 5)dpQ210:90:80:70:60:50:40:0030:0020:0010 0:001 0:002 0:003Figure 5. The relative dierence of the dieren-tial cross sections for two dierent values of theseparation parameter .Collaboration], Phys. Rev. Lett. 89:101804,2002, Erratum,ibid. 89:129903, 2002 [hep-ex/0208001] and M. Deile, these proceedings.2. V. W. Hughes and T. Kinoshita, Rev. Mod.Phys. 71 (1999) S133.3. M. Davier, S. Eidelman, A. Hocker and Z.Zhang [hep-ph/0208177].4. S. Eidelman and F. Jegerlehner, Z. Phys. C67 (1995) 585 [hep-ph/9502298]. D. H. Brownand W. A. Worstell, Phys. Rev. D 54(1996) 3237 [hep-ph/9607319]. M. Davier andA. Ho¨cker, Phys. Lett. B 435 (1998) 427 [hep-ph/9805470]. S. Narison, Phys. Lett. B 513(2001) 53 [hep-ph/0103199]. F. Jegerlehner,[hep-ph/0104304]. J. F. De Troconiz andF. J. Yndurain, [hep-ph/0106025].5. H. Burkhardt and B. Pietrzyk, Phys. Lett. B513 (2001) 46. A. D. Martin, J. Outhwaiteand M. G. Ryskin, [hep-ph/0012231]; Phys.Lett. B 492 (2000) 69 [hep-ph/0008078].J. Erler, Phys. Rev. D 59 (1999) 054008 [hep-ph/9803453]. J. H. Ku¨hn and M. Steinhauser,[hep-ph/0109084]; Phys. Lett. B 437 (1998)5-0.4-0.3-0.2-0.100.10.20.30.4x 10-30 0.25 0.5 0.75 1Q2 (GeV2)(σMC -σGauss)/σGauss-0.4-0.3-0.2-0.100.10.20.30.4x 10-3-1 -0.5 0 0.5 1cos θγFigure 6. The relative dierence between dieren-tial cross sections obtained by the PHOKHARAMonte Carlo generator (MC) and a Gauss numer-ical integration (Gauss).425 [hep-ph/9802241].6. S. Binner, J. H. Ku¨hn and K. Melnikov, Phys.Lett. B 459 (1999) 279 [hep-ph/9902399].7. K. Melnikov, F. Nguyen, B. Valeriani andG. Venanzoni, Phys. Lett. B 477 (2000) 114[hep-ph/0001064].8. H. Czy_z and J. H. Ku¨hn, Eur. Phys. J. C 18(2001) 497 [hep-ph/0008262].9. S. Spagnolo, Eur. Phys. J. C 6 (1999) 637.10. V. A. Khoze, M. I. Konchatnij,N. P. Merenkov, G. Pancheri, L. Trentadueand O. N. Shekhovzova, Eur. Phys. J. C 18(2001) 481 [hep-ph/0003313].11. A. Ho¨fer, J. Gluza and F. Jegerlehner, Eur.Phys. J. C 24 (2002) 51 [hep-ph/0107154].12. A. Aloisio et al. [KLOE Collaboration], [hep-ex/0107023].13. A. Denig et al. [KLOE Collaboration], eConfC010430 (2001) T07 [hep-ex/0106100].14. M. Adinol et al. [KLOE Collaboration],[hep-ex/0006036].Q2 (GeV2)dσNLO(e+e−→pi+pi−γ)/dQ2  (nb/GeV2 )0204060801001200 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Q2 (GeV2)-0.003-0.002-0.00100.0010.0020.0030.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Figure 7. A comparison between PHOKHARAand analytical [24] results.15. A. Denig et al. [KLOE Collaboration], theseproceedings, hep-ex/0211024.16. E. P. Solodov [BABAR collaboration], eConfC010430 (2001) T03 [hep-ex/0107027].17. G. Rodrigo , H. Czy_z, J.H. Ku¨hn and M.Szopa Eur. Phys. J. C 24 (2002) 71 [hep-ph/0112184].18. G. Rodrigo, A. Gehrmann-De Ridder,M. Guilleaume and J. H. Ku¨hn, Eur. Phys.J. C 22 (2001) 81 [hep-ph/0106132].19. J. H. Ku¨hn and A. Santamaria, Z. Phys. C48 (1990) 445.20. G. Rodrigo and J.H. Ku¨hn, Eur. Phys. J. C25 (2002) 215 [hep-ph/0204283].21. H. Czy_z, A. Grzelinska, J. H. Ku¨hn andG. Rodrigo, in preparation.22. G. Rodrigo, Acta Phys. Polon. B 32 (2001)3833 [hep-ph/0111151].23. R. Decker, M. Finkemeier, P. Heiliger and H.H. Jonson, Z. Phys. C 70 (1996) 247.24. F.A. Berends, W.L. van Neerven and G.J.H.Burgers, Nucl.Phys. B 297 (1988) 429, Erra-tum, ibid. B 304 (1988) 921; Phys. Lett. B177 (1986) 191.

Radiative return at $e^+ e^-$ factories

The energy dependence of the electron - positron hadronic cross section can be measured not only by a straightforward energy scan, but also by means of the radiative return method. To provide extensive comparisons between theory and experiment a Monte Carlo event generator is an indispensable tool. We have developed such a generator called PHOKHARA, which simulates e+e- -> mesons + photon(s) processes. In this paper we present its latest tests and upgrades

Kühn, Johann H.

Rodrigo García, Germán Vicente

Repositori d'Objectes Digitals per a l'Ensenyament la Recerca i la Cultura

arXiv:hep-ph/0211186v1  12 Nov 20021CERN-TH/2002-316TTP02-35Radiative return at e+e− factories. ∗Henryk Czyz˙a, †Johann H. Ku¨hnb, ‡Germa´n Rodrigoc, §aInstitute of Physics, University of Silesia, PL-40007 Katowice, Poland.bInstitut fu¨r Theoretische Teilchenphysik, Universita¨t Karlsruhe, D-76128 Karlsruhe, Germany.cTheory Division, CERN, CH-1211 Geneva 23, Switzerland.The energy dependence of the electron - positron hadronic cross section can be measured not only by a straight-forward energy scan, but also by means of the radiative return method. To provide extensive comparisons betweentheory and experiment a Monte Carlo event generator is an indispensable tool. We have developed such a gen-erator called PHOKHARA, which simulates e+e− → mesons + photon(s) processes. In this paper we present itslatest tests and upgrades.1. INTRODUCTION AND PHOKHARAUPGRADESIn view of the precision of the recent measure-ments of the muon anomalous magnetic momentaµ ≡ (g − 2)µ/2 at BNL [1], hadronic contri-butions are crucial for the interpretation of thismeasurement, in particular for the isolation ofthe electroweak or of non-Standard Model physicscontributions [2]. Their understanding becomeseven more important, as it seems [3] that thee+e− annihilation data are not consistent withthe τ decay data. A new aµ measurement, whichis under way, will challenge the theoretical pre-dictions even more.An important ingredient and the dominantsource of uncertainties in the theoretical predic-tion for the muon anomalous magnetic momentis the hadronic vacuum polarization [4]. It isin turn related via dispersion relations to thecross section for electron–positron annihilation∗Work supported in part by BMBF under grant number05HT9VKB0, EC 5-th Framework EURIDICE networkproject HPRN-CT2002-00311 and TARI project HPRI-CT-1999-00088; presented by H. Czyz˙ at RADCOR 2002,Kloster Banz, September 8 - 13, 2002.†Supported in part by EC 5-th Framework, contractHPRN-CT-2000-00149; e-mail:czyz@us.edu.pl‡e-mail:jk@particle.uni-karlsruhe.de§Supported in part by E.U. TMR grant HPMF-CT-2000-00989; e-mail: german.rodrigo@cern.chinto hadrons σhad = σ(e+e− → hadrons) andin some cases via isospin symmetry to the decaywidth τ → ντ + hadrons. This quantity plays animportant role also in the evolution of the elec-tromagnetic coupling αQED from the Thompsonlimit to high energies [4,5]. The interpretation ofimproved measurements at high energy colliderssuch as LEP, Tevatron, the LHC or TESLA there-fore depends significantly on the precise knowl-edge of σhad.The feasibility of using tagged photon events athigh luminosity electron–positron storage rings,such as the φ-factory, DAΦNE, CLEO-C or B-factories, to measure σhad over a wide range ofenergies has been proposed and studied in detailin [6–8] (see also [9,10]). The machine is operatingat a fixed energy of the e+e− centre-of-mass sys-tem (cms) and the initial state radiation (ISR) isused to reduce the invariant mass of the hadronicsystem.The radiation of photons from the hadronicsystem (the final state radiation, FSR) shouldbe considered as a background and can be sup-pressed by choosing suitable kinematical cuts,or controlled by the simulation, once a suitablemodel for this amplitude has been adopted. Onefinds that selecting events with the tagged pho-tons close to the beam axis and well separatedfrom the hadrons indeed reduces FSR drastically.As demonstrated in Fig. 1, the FSR contribution2to the total cross section can be easily reduced tothe 1% level. The model dependence of the FSRin the case of the π+π− final state can be con-trolled by the same experiment through studiesof the forward–backward asymmetry of the an-gular charged pion distribution. The asymmetrycomes from FSR–ISR interference and integratesto zero for ’charge blind’ configurations. As a re-sult it does not contributes to rates in Fig. 1.It can be used, however, to calibrate the FSRamplitude and more detailed tests of its modeldependence.020406080100120ds(e+e-→p+p-g)/dQ 2  (nb/GeV 2 )1.02 GeVqg < 15o or qg > 165o  40o < qp < 140oISR+FSR  ≈  ISR onlyISR+FSRISR only10-410-20 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Q 2 (GeV 2)FSR/ISRqg < 15o or qg > 165o40o < qp < 140oFigure 1. The role of the cuts in the suppressionof the FSR contributions to the cross section. Re-sults from the PHOKHARA generator. No cuts(upper curves) and suitable cuts applied (lowercurves).When running at higher energies, the FSR issuppressed with respect to the ISR by the dif-ferent behaviour of various propagators and formfactors relevant to the problem. In practice itmeans that no special angular cuts are needed tosuppress the FSR contribution when running athigh energies.The suppression of the FSR overcomes theproblem of its model dependence, which must betaken into account in a completely inclusive mea-surement [11].Preliminary experimental results using thismethod have been presented recently by theKLOE collaboration at DAΦNE [12–15]. Largeevent rates were also observed by the BaBar col-laboration [16].In the first version of the newly developedMonte Carlo program PHOKHARA [17] we haveconsidered the full next-to-leading order (NLO)QED corrections to the ISR in the annihilationprocess e+e− → γ + hadrons, for the case wherethe photon is observed under a non-vanishing an-gle relative to the beam direction. The virtualand soft photon corrections were presented in [18]and the contribution of the emission of a secondhard photon in [17]. The final hadronic state waslimited to the π+π−, with the hadronic currentmodeled as in [19], and the final state emissionwas not included. The program allowed also forthe generation of µ+µ−γ(γ) final states, againlimited to the emission of photon(s) from the ini-tial leptons.Radiative corrections proportional to (αm2e )relevant to configurations with photons emittedat very low (≃ me/√s) angles relative to thebeam direction were calculated in [20] and areincluded in the new version of PHOKHARA [21].The leading order corrections proportional to m2eare typically of the order of a few per cent [22],while the non-leading ones are of order of 0.1%,as seen from Fig.2. They will be important whenthe precision of the measurement will be below1%. Their effect depends on Q2 and thus affectsthe Q2 distribution from which the hadronic crosssection is extracted.Another new feature of the PHOKHARAevent generator is the inclusion of the four-pionhadronic final states (2π+2π− and 2π0π+π− ).The description of the hadronic current in thatcase is based on the paper [23], with changes de-scribed in [8]. The comparison with the MonteCarlo, which simulates the same process at lead-ing order [8] and includes additional collinearradiation through structure function (SF) tech-niques, shows typical difference of order of 1% asseen in Fig. 3 (a similar behaviour can be ob-served for 2π0π+π− final state). The non-leading3ps = 10 GeVps = 4 GeVps = 1 GeVe+e  ! + ()pQ2 GeVd(NLOmass)dpQ2=ddpQ22:521:510:500 0:0002 0:0004 0:0006 0:0008 0:001 0:0012 0:0014 0:0016 0:0018Figure 2. The relative contributions of the non-leading mass corrections to the differential crosssection at√s = 1 GeV, 4 GeV and 10 GeV.contributions to the cross section of the four-pionfinal states are of the expected size and of thesame order as for the two-pion final state [17].The program now includes also the contribu-tions from the final state emitted photons to-gether with the ISR–FSR interference calculatedat the lowest order for π+π− and µ+µ− finalstates, while for the four-pion final states the FSRcontribution is not taken into account.2. TESTS OF PHOKHARAAn obvious and one of the most importanttasks in the construction of a Monte Carlo eventgenerator is to demonstrate that its technical ac-curacy is much better than the desired physi-cal accuracy. The tests that were performed forthe previous version of PHOKHARA [17] are stillvalid, within the limitations of this version, whichwas applicable for non-vanishing photon angles.In the first step we demonstrate the indepen-dence of total and differential cross sections ofthe separation parameter ǫ (called w in [17]) be-tween soft and hard photon regions. The soft pho-ton contribution is calculated analytically, whilethe additional hard photon is treated via MonteCarlo simulation. The parameter that specifiese+e ! 2+2 ()pQ2GeV1 d(NLO)dpQ2=d(LL)dpQ243:532:521:510:500:020:0150:010:0050 0:005 0:01 0:015 0:02 0:025 0:03Figure 3. The relative non-leading contributionsto the differential cross section at√s = 4 GeV.NLO - full next-to-leading result, LL - leadinglogarithmic approximation.the separation between the two regions of thephase space has to be kept small enough to val-idate the soft photon approximation and largeenough to avoid negative weights. We performedthe tests for a π+π− hadronic final state in [17],while for one of the four-pion modes the resultsare collected in Figs. 4 and 5. From Fig. 4it is clear that the choice ǫ = 10−3 is still toobig, whereas Fig. 5 demonstrates the stabilityof the results between ǫ = 10−4 and ǫ = 10−5.This also proves that the Monte Carlo integra-tion works well in the soft photon region. TheMonte Carlo integration of the part of the pro-gram that simulates one hard large-angle photonemission was tested in [17] against a Gauss nu-merical integration. As shown in Fig. 6 a techni-cal precision of the program at the level of 10−4was demonstrated. Analytical results exist for thedifferential (in Q2) cross section integrated overthe whole angular range of the photon(s) for bothone and two emitted photons [24]. The compar-ison with PHOKHARA can be found in Fig. 7,4e+e ! 20+ ()pQ2GeV1 d(=10 3)dpQ2=d(=10 4)dpQ210:90:80:70:60:50:40:0030:0020:0010 0:001 0:002 0:003Figure 4. The relative difference of the differen-tial cross sections for two different values of theseparation parameter ǫ.demonstrating again an excellent technical pre-cision also for the two-photon final state. Theresults presented in Fig. 7 refer to the sum of vir-tual and hard corrections to the e+e− → π+π−γcross section, while more detailed tests can befound in [21].3. CONCLUSIONSThe PHOKHARA Monte Carlo event genera-tor was upgraded, allowing for simulation in thesmall photon angles region. Besides the π+π−hadronic final state, its present version includesalso 2π+2π− and 2π0π+π− final states. For theπ+π− and µ+µ− final states, the FSR photoniccontributions were implemented at the lowest or-der, including ISR–FSR interference. Further up-grades are in progress.REFERENCES1. H. N. Brown et al. [Muon g − 2 Collabora-tion], Phys. Rev. Lett. 86 (2001) 2227 [hep-ex/0102017]; G.W.Bennett et al. 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Radiative return at e+e- factories

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Radiative return at e+e- factories

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