Contributions from the reaction e+e- --> pi+pi- e+e- to the pion form factor
measurement via radiative return method are discussed basing on the results of
a Monte Carlo generator (EKHARA). The generator contains contributions from the
initial and final state emission of a e+e- pair from e+e- --> pi+pi- production
diagrams and the pi+pi- pair production from space-like and time-like Bhabha
diagrams. A detailed study is performed for the Phi- factory energy. Tests of
the generation procedure are also presented

Czyz, Henryk

Nowak-Kubat, Elzbieta

Acta Physica Polonica B

English

arXiv

Contributions from the reaction e+e− !  + −e+e− to the pion form factor measurement via radiative return method are discussed basing on the results of a Monte Carlo generator (EKHARA). The generator contains contributions from the initial and final state emission of a e+e− pair from e+e− !  + − production diagrams and the  + − pair production from space-like and time-like Bhabha diagrams. A detailed study is performed for the  factory energy. Tests of the generation procedure are also presented

Czyż, Henryk

Nowak-Kubat, Elżbieta

Repozytorium Uniwersytetu Śląskiego RE-BUŚ

             Title: Radiative return via electron pair production : Monte Carlo simulation of the process e+ e- ---> pi+ pi- e+ e  Author: Henryk Czyż, Elżbieta Nowak-Kubat  Citation style: Czyż Henryk, Nowak-Kubat Elżbieta. (2005). Radiative return via electron pair production : Monte Carlo simulation of the process e+ e- ---> pi+ pi- e+ e. "Acta Physica Polonica B" (Vol. 36, no. 11 (2005), s. 3425-3433).   Vol. 36 (2005) ACTA PHYSICA POLONICA B No 11RADIATIVE RETURN VIA ELECTRON PAIRPRODUCTION: MONTE CARLO SIMULATIONOF THE PROCESS e+e− → pi+pi−e+e− ∗ ∗∗Henryk Czyż, Elżbieta Nowak-KubatInstitute of Physics, University of SilesiaUniwersytecka 4, 40-007 Katowice, Poland(Received October 17, 2005)Contributions from the reaction e+e− → pi+pi−e+e− to the pion formfactor measurement via radiative return method are discussed basing onthe results of a Monte Carlo generator (EKHARA). The generator con-tains contributions from the initial and final state emission of a e+e− pairfrom e+e− → pi+pi− production diagrams and the pi+pi− pair productionfrom space-like and time-like Bhabha diagrams. A detailed study is per-formed for the Φ-factory energy. Tests of the generation procedure are alsopresented.PACS numbers: 13.40.Ks, 13.66.Bc1. IntroductionThe radiative return method [1] is a powerful tool in the measurementof σ(e+e− → hadrons) and detailed studies of hadronic interactions [2–4].Very accurate knowledge of the hadronic cross section is essential for predic-tions of the hadronic contributions to aµ, the anomalous magnetic moment ofthe muon, and to the running of the electromagnetic coupling from its valueat low energy up to MZ (for recent reviews see [5–7]). Due to a complicatedexperimental setup, the use of Monte Carlo (MC) event generators [8, 9],which include various radiative corrections [10] is indispensable. A moreextensive analysis of that subject can be found also in [11]. The most im-portant hadronic mode, i.e. pi+pi−, was recently measured by KLOE [12]∗ Presented by E. Nowak-Kubat at the XXIX International Conference of TheoreticalPhysics, “Matter to the Deepest”, Ustroń, Poland, September 8–14, 2005.∗∗ Work supported in part by EC 5-th Framework Program under contractHPRN-CT-2002-00311 (EURIDICE network), TARI project RII3-CT-2004-506078and the Polish State Committee for Scientific Research (KBN) under contract1 P03B 003 28.(3425)3426 H. Czyż, E. Nowak-Kubatby means of radiative return method. In this measurement only pions(+ missing momenta) in the final state were observed. For that partic-ular measurement there is no difference between photon(s) and the pairproduction and one should estimate the contribution of the process e+e− →pi+pi−e+e− to the measured cross section. As suggested in [13] the contribu-tion from the e+e− production to the e+e− → pi+pi−γ is sizable and comesmostly from the t-channel Bhabha-like diagrams. In this paper a MonteCarlo study is performed to test if this claim remains true for a realistic(KLOE) experimental setup.2. Monte Carlo simulation and its testsIn Fig. 1 different types of diagrams contributing to the reaction e+e− →pi+pi−e+e− are shown schematically. In the present version of the MonteCarlo program we include all but the γ∗γ∗ pion pair production processdiagrams (Fig. 1(d)), which were estimated to be negligible for DAΦNEenergy [14]. We use scalar QED to model the FSR e+e− pair emission andthe ρ dominance model for the γ∗(ρ∗)pipi coupling (see for details [9, 15]and [16] from where the model of the pion form factor was implemented).The contribution from initial state radiation (ISR) diagrams Fig. 1(a), andfinal state radiation (FSR) diagrams Fig. 1(b) was discussed in details in [9].e+e + e+e a)e+e + e+e b)e+e + e+e c)e+e + e+e d)e+(p1)e (p2)e+(q1)e (q2)+(1) (2)k2Qe)Fig. 1. Diagrams contributing to process e+(p1)e−(p2)→pi+(pi1)pi−(pi2)e+(q1)e−(q2):initial state electron pair emission (a), final state electron pair emission (b), pionpair emission from t-channel Bhabha process (c), γ∗γ∗ pion pair production (d)and pion pair emission from s-channel Bhabha process (e).Radiative Return via Electron Pair Production 3427Because of a heavy pair emission, the contribution from diagrams presentedschematically in Fig. 1(e) is completely negligible for any event selectionused in the analysis presented below.We use multi-channel variance reduction method to improve efficiencyof the generator and the generation is split into four channels, where two ofthem absorb peaks present in t-channel diagrams and other two take care ofthe s-channel peaks. All details will be given in a separate publication [15].To check correctness of the program code, we have performed a num-ber of tests of the new part of program. Gauge invariance of the sum ofthe amplitudes was checked analytically for set of diagrams from Fig. 1(c),(e). To cross check the helicity amplitudes, used in the program to cal-culate square of the matrix element, we have used also the standard tracemethod for independent calculation. Both results, summed over polarisa-tions of initial and final leptons, were compared numerically scanning thephysical phase space. The biggest relative difference between the two re-sults, which was found, was at the level of 10−11 for diagrams in Fig. 1(c)and 10−23 for diagrams in Fig. 1(e). The computer code has been writtenin quadruple precision not to lose accuracy as sever numerical cancellationsoccur even when using helicity amplitudes. The phase space volume as afunction of the square of the invariant mass of the pion system calculatedby the Monte Carlo program was compared with a one dimensional Gaussintegration. The result is presented in Fig. 2(a). The relative differencestays well within errors, which are at most at the level of a few per mil.ps = 1:02 GeVa)Q2 (GeV2)PHSP(MC)=PHSP(gauss) 110:80:60:40:200:0040:0020 0:002 0:004 0:006 0:008ps = 1:02 GeVb)Q2 (GeV2)PHSPp2q2(MC)=PHSPp2q2(AV) 110:80:60:40:200:0040:0030:0020:0010 0:001 0:002 0:003 0:004 0:005Fig. 2. Relative difference between phase space calculated by MC and by gaussmethod. The errors come from MC integration (left). Relative difference betweenMC result and average for phase space × simple |M |2 (right).3428 H. Czyż, E. Nowak-KubatBecause of the complexity of the generation we performed an additional testwhere a very simple matrix element was introduced. The matrix elementwas chosen in a way that only a four dimensional numerical integration wasnecessary. The matrix element was integrated by the Monte Carlo programwith full phase space generation and compared with the results obtained with4-dimensional Gauss integration, where the integration region was dividedinto (2× 107–6× 107) rectangular solids (or more complicated shapes at theborder of integration region). In each rectangular solid the 8-point Gaussprocedure was used recursively. The error of the Gauss method was esti-mated as the maximal difference of the results with different subdivisionsand the central value was taken as a mean of the smallest and the biggestobtained values. The relative difference between MC result and the Gaussmethod is presented in Fig. 2(b). Again the relative difference stays wellwithin errors, which are at most at the level of a few per mil.A comparison with existing analytic results [17, 18] was also performed.In the Table I the ratio (denoted by Rt+int) of differential cross sectionsfor t-channel (space-like Bhabha with radiated pion pair) plus the interfer-ence between t-channel and s-channel (only the ISR pion pair radiated frome+e− → e+e− is included) to the differential cross section of the reactione+e− → pi+pi−γ (Born: ISR+FSR) is presented as a function of the invari-ant mass of the two-pion system (Q2). No cuts are imposed as the analyticformulae exist only for that case. A discrepancy at the level of about 10 %between the Monte Carlo and the analytic results can be observed in theregion were the pair contribution is sizable and it is much larger in the regionwere the pair contribution is at the level of 1 per mil or lower. For s-channelTABLE IRatios Rt+int and R in % (see text for definitions): BN — results based on formulaefrom [17], BNK— results based on formulae from [17] and [18], MC —Monte Carloresults.Q2 [GeV2] Rt+int(BN) (%) Rt+int(MC) (%) R(BNK) (%) R(MC) (%)0.09 8.87 8.06(15) 9.35 8.54(15)0.16 3.60 3.35(5) 4.08 3.84(5)0.25 1.36 1.46(2) 1.83 1.92(2)0.36 0.46 0.581(7) 0.90 1.028(7)0.49 0.095 0.219(2) 0.51 0.639(2)0.5776 0.0056 0.01110(8) 0.40 0.5074(8)0.64 −0.024 0.0658(4) 0.35 0.4400(5)0.81 −0.036 0.01151(5) 0.25 0.3001(2)1.0 −0.012 0.000091(1) 0.10 0.10916(5)Radiative Return via Electron Pair Production 3429diagrams we have found previously [9] an agreement with analytic resultsof [18] at the level of 1 per mil. We shall observe, that even if we take the rel-ative difference as the error of the MC code, the accuracy of the code is goodenough for an estimation of the contribution of the pair production to theprocess with the photon(s) radiation. To estimate the complete contributionfor the (unrealistic) situation with no cuts imposed, the ratio (denoted by R)of the t- and the s-channel pair production to the differential cross sectionof e+e− → pi+pi−γ (Born: ISR+FSR) is also presented in Table I both forthe MC based and the analytic results [17, 18].3. An event selection dependence of the pion pair productioncontribution to the radiative return measurementAll results from Monte Carlo simulation, presented in this section, arefor DAΦNE energy 1.02 GeV. As observed already in the paper [13] t-channel diagrams contribute significantly in the small Q2 region if no cutsare imposed. That is shown in Fig. 3, where Fig. 3(a) presents the ratio oft-channel to s-channel contributions to the differential cross section of thereaction e+e− → pi+pi−e+e− and Fig. 3(b) presents the ratio of the differen-tial cross sections of the reactions e+e− → pi+pi−e+e− and e+e− → pi+pi−γ.The t-channel diagrams contribution, if no cuts are imposed, is up to 12times bigger than the s-channel contribution in the small Q2 region and itis negligible for large Q2 values.ps = 1:02 GeVno cutsa)e+e ! + e+e Q2 (GeV2)(ddQ2)h t=(ddQ2)h s10:80:60:40:20121086420b)no cutsps = 1:02 GeVe+e ! + e+e Q2 (GeV2)(ddQ2)pair=(ddQ2)ph10:80:60:40:200:080:070:060:050:040:030:020:010Fig. 3. (a) Ratio of differential cross sections of t-channel and s-channel of thereaction e+e− → pi+pi−e+e−. (b) Ratio of differential cross sections of the reactionse+e− → pi+pi−e+e− (pair) and e+e− → pi+pi−γ (ph).3430 H. Czyż, E. Nowak-KubatThe contribution from the pair emission to the photon emission depends,however, a lot on the event selection, even if the t-channel contribution stilldominates at low Q2. It is seen from Fig. 4, where for KLOE event selection(Fig. 4(a)) one gets only up to 1.2% contribution from the pair emissionand up to 2.5% if one uses only the angular cuts on pions polar angles(Fig. 4(b)). The pair contribution at the ρ peak is about 0.3% for KLOEcuts and 0.5% when only the angular cuts are imposed. Even if the paircontribution remains small it has to be taken into account if the aimedexperimental precision is of the order of 1% or better.a)ps = 1:02 GeVMtrcut++  < 15Æ or > 165Æ50Æ<  < 130Æe+e ! + e+e Q2 (GeV2)(ddQ2)pair=(ddQ2)ph10:80:60:40:200:0140:0120:010:0080:0060:0040:0020b)ps = 1:02 GeV50Æ<  < 130Æe+e ! + e+e Q2 (GeV2)(ddQ2)pair=(ddQ2)ph10:80:60:40:200:0450:040:0350:030:0250:020:0150:010:0050Fig. 4. Ratio of differential cross sections of the process e+e− → pi+pi−e+e− (pair)and e+e− → pi+pi−γ (ph) with two sets of cuts imposed (a) KLOE cuts (b) onlypion angular cuts are imposed.Another important observable used to control the FSR contribution tothe cross section of the reaction e+e− → pi+pi− + photons is the chargeasymmetry. Again the measured value contains the pair contributions ifonly pions are observed. However, one can see in Figs. 5(a) and 6(a) thatthis contribution is completely negligible both in the case where no eventselection is used (Fig. 5(a)) and in the case when KLOE event selection isapplied (Fig. 6(a)). That conclusion holds despite the fact that the asym-metry for pair production itself is sizable in both cases (Figs. 5(b) and 6(b)).As stated already in [9], ISR and t-channel of electron pairs can betreated likewise ISR of photons, what results in the change of the radiatorfunction in the radiative return method. Another possibility is to treat thepair production as a background to the process e+e− → pi+pi−γ(γ). In thatcase one would like to reduce that background. The way to do that is toveto the outgoing electrons and/or positrons. However, as shown in Fig. 7,Radiative Return via Electron Pair Production 3431a)no cutsps = 1:02 GeVe+e ! + (e+e + )Q2 (GeV2)N(+>90Æ) N(+<90Æ)N(+>90Æ)+N(+<90Æ)10:80:60:40:200:10:080:060:040:020 0:02 0:04 0:06b)no cutsps = 1:02 GeVe+e ! + e+e Q2 (GeV2)N(+>90Æ) N(+<90Æ)N(+>90Æ)+N(+<90Æ)10:80:60:40:200:250:20:150:10:050 0:05 0:1 0:15Fig. 5. Charge asymmetryN(θpi+>90◦)−N(θpi+<90◦)N(θpi+>90◦)+N(θpi+<90◦): (a) for e+e− → pi+pi−(γ +e+e−) (the solid line shows the charge asymmetry for e+e− → pi+pi−γ only) and(b) for e+e− → pi+pi−e+e− (no cuts imposed).a)ps = 1:02 GeVMtrcut++  < 15Æ or > 165Æ50Æ<  < 130Æe+e ! + (e+e + )Q2 (GeV2)N(+>90Æ) N(+<90Æ)N(+>90Æ)+N(+<90Æ)10:80:60:40:200:010:0080:0060:0040:0020 0:002 0:004 0:006b)ps = 1:02 GeVMtrcut++  < 15Æ or > 165Æ50Æ<  < 130Æe+e ! + e+e Q2 (GeV2)N(+>90Æ) N(+<90Æ)N(+>90Æ)+N(+<90Æ)10:80:60:40:200:050 0:05 0:1 0:15 0:2 0:25 0:3Fig. 6. Charge asymmetryN(θpi+>90◦)−N(θpi+<90◦)N(θpi+>90◦)+N(θpi+<90◦): for (a) e+e− → pi+pi−(γ +e+e−) (the solid line shows the charge asymmetry for e+e− → pi+pi−γ only) and(b) for e+e− → pi+pi−e+e− (for KLOE event selection).where the percentage of the electrons and positrons escaping the detection[(θe+ < 20◦ or θe+ > 160◦) and (θe− < 20◦ or θe− > 160◦)] is plotted, it isclear that it is practically impossible for KLOE event selection (Fig. 7(a)),as about 90% of electron–positron pairs escape detection. For looser cutsone can reduce up to 50% of the pair background (Fig. 7(b)).3432 H. Czyż, E. Nowak-Kubata)and (e  < 20Æ or > 160Æ)c2 is (e+ < 20Æ or > 160Æ)ps = 1:02 GeVMtrcut++  < 15Æ or > 165Æ50Æ<  < 130Æe+e ! + e+e Q2 (GeV2)(ddQ2)pair2/(ddQ2)pair10:80:60:40:20:950:90:850:80:750:70:650:60:55b)and (e  < 20Æ or > 160Æ)c2 is (e+ < 20Æ or > 160Æ)ps = 1:02 GeV50Æ<  < 130Æe+e ! + e+e Q2 (GeV2)(ddQ2)pair2/(ddQ2)pair10:80:60:40:20:50:450:40:350:30:250:20:150:10:050Fig. 7. Non-reducible pair production background.4. ConclusionsA detailed analysis of the contribution of the reaction e+e−→pi+pi−e+e−to the e+e− → pi+pi− cross section extraction via the radiative returnmethod from the σ(e+e− → pi+pi− + missing momentum) measurement ispresented basing on the Monte Carlo simulations with the EKHARA eventgenerator. It is shown that t-channel diagrams contributions dominate atsmall Q2 values independently on the event selection. The biggest possiblecontribution to the process e+e− → pi+pi−γ reaches about 8 % if no eventselection is applied and it is reduced to 1.2 % for KLOE event selection.The reaction e+e− → pi+pi−e+e− does not contribute significantly to chargeasymmetry measured in the reaction e+e− → pi+pi− +missing momentum.The authors are grateful to W.L. van Neerven for providing the correctformula for the t-channel pair production [17].REFERENCES[1] Min-Shih Chen, P.M. Zerwas, Phys. Rev. D11, 58 (1975).[2] H. Czyż, J.H. Kühn, E. Nowak, G. Rodrigo, Eur. 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Radiative return via electron pair production: Monte Carlo simulation of the process e+ e- --> pi+ pi- e+ e-

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Repozytorium Uniwersytetu Śląskiego RE-BUŚ