A Monte Carlo generator (EKHARA) has been constructed to simulate the reaction e+e− !  + −e+e− based on initial and final state emission of a e+e− pair from e+e− !  + − production diagram. A detailed study of
the process, as a potential background for  (e+e− !  + −) measurement via radiative return method, is presented for  - and B-factory energies

Czyż, Henryk

Nowak, Elżbieta

Acta Physica Polonica B

English

Repozytorium Uniwersytetu Śląskiego RE-BUŚ

             Title: e+ e- ---> pi+ pi- e+ e- : A Potential background for sigma (e+ e- ---> pi+ pi-) measurement via radiative return method  Author: Henryk Czyż, Elżbieta Nowak  Citation style: Czyż Henryk, Nowak Elżbieta. (2003). e+ e- ---> pi+ pi- e+ e- : A Potential background for sigma (e+ e- ---> pi+ pi-) measurement via radiative return method. "Acta Physica Polonica B" (Vol. 34, no. 11 (2003), s. 5231-5238).   Vol. 34 (2003) ACTA PHYSICA POLONICA B No 11e+e−→ pi+pi−e+e−: A POTENTIAL BACKGROUNDFOR σ(e+e− → pi+pi−) MEASUREMENTVIA RADIATIVE RETURN METHOD∗ ∗∗Henryk Czyż and Elżbieta NowakInstitute of Physics, University of SilesiaUniwersytecka 4, 40-007 Katowice, Poland(Received October 24, 2003)A Monte Carlo generator (EKHARA) has been constructed to simulatethe reaction e+e− → pi+pi−e+e− based on initial and final state emission ofa e+e− pair from e+e− → pi+pi− production diagram. A detailed study ofthe process, as a potential background for σ(e+e− → pi+pi−) measurementvia radiative return method, is presented for Φ- and B-factory energies.PACS numbers: 13.40.Ks, 13.66.Bc1. IntroductionThe radiative return method [1] (see [2] for a short introduction) is apowerful tool in the measurement of σ(e+e− → hadrons), crucial for pre-dictions of the hadronic contributions to aµ, the anomalous magnetic mo-ment of the muon, and to the running of the electromagnetic coupling fromits value at low energy up to MZ (for recent reviews look [3–5]). Dueto a complicated experimental setup, the use of Monte Carlo (MC) eventgenerators [1, 6–9], which includes various radiative corrections [10, 11] isindispensable. Some more extensive analysis of that subject can be foundalso in [12–14]. The most important hadronic mode, i.e. pi+pi−, is currentlymeasured by KLOE [15–18], and BaBar [19] by means of radiative returnmethod. This measurement can suffer from a background from the processe+e− → pi+pi−e+e−, as suggested in [20], for at least two reasons: (i) Atpresent KLOE measures only pions (+ missing momenta) in the final state∗ Presented at the XXVII International Conference of Theoretical Physics, “Matter tothe Deepest”, Ustroń, Poland, September 15–21, 2003.∗∗ Work supported in part by EC 5-th Framework Program under contract HPRN-CT-2002-00311 (EURIDICE network) and the Polish State Committee for ScientificResearch (KBN) under contract 2 P03B 017 24.(5231)5232 H. Czyż, E. Nowakand for that particular measurement there is no difference between pho-ton(s) and pair production. (ii) The e+e− pair can escape detection, beinglost outside a detector, e.g. in the beam pipe, or having energy below adetection threshold. Again a Monte Carlo study is unavoidable, if one likesto know the actual value of the pair production contribution in a given ex-perimental setup, as the analytical, completely inclusive, calculations mightlead to misleading results.2. Monte Carlo program EKHARA and its testsIn Fig. 1 different types of diagrams contributing to process e+e− →pi+pi−e+e− are shown schematically. In the present version of the MonteCarlo program we include only two gauge invariant sets of diagrams fromFig. 1(a) and 1(b). The former represents initial state radiation (ISR),and the latter final state radiation (FSR), of an e+e− pair from e+e− →pi+pi− production diagram. We use scalar QED (sQED) to model FSRe+e− pair emission and ρ dominance model for γ∗(ρ∗)pipi coupling (see [1]for details). The diagrams from Fig. 1(c), representing pion pair emissionfrom t-channel Bhabha process, together with s-channel Bhabha pion pairemission (not shown in Fig. 1), will be included in the new version [21]of the presented generator, completing the discussion of this paper. Thecontribution from γ∗γ∗ pion pair production process (Fig. 1(d)) is negligiblefor DAΦNE energy [17], and as its interference with other diagrams doesnot contribute to the cross section integrated over charge symmetric cuts,these contributions are not relevant, at least for Φ-factories.For parametrisation of the phase space we use the following variables:Q2 = (pi1 + pi2)2–invariant mass of pi+pi− system, k21 = (q1 + q2)2–invariantmass of e+e− system, polar and azimuthal angles of ~Q momentum, definedin the centre-of-mass (CM) frame of initial e+e− pair, polar and azimuthalangles of ~pi1 momentum, defined in Q- rest frame and polar and azimuthalangles of ~q1 momentum, defined in k1-rest frame. All four vectors are boostedinto the initial e+e− CM frame after being generated and all necessary cutscan be applied at this stage of the generation. Multi–channel variance reduc-tion method was used to improve efficiency of the generator and all detailswill be given in a separate publication [21].We have performed a number of ‘standard’ tests to ensure that the writ-ten FORTRAN code is correct. Gauge invariance of the sum of the ampli-tudes was checked analytically separately for set of diagrams from Fig. 1(a)and 1(b). We use helicity amplitudes in EKHARA to calculate square ofthe matrix element describing the e+e− → pi+pi−e+e− process, but as across check, we have used also the standard trace technique to calculatethe square of the matrix element, summed over polarisations of initial ande+e− → pi+pi−e+e−: a Potential Background for σ(e+e− → pi+pi−) . . . 5233e+e + e+e a)e+e + e+e b)e+e + e+e c)e+e + e+e d)Fig. 1. Diagrams contributing to the processe+(p1)e−(p2)→ pi+(pi1)pi−(pi2)e+(q1)e−(q2): initial state electron pair emission (a),final state electron pair emission (b), pion pair emission from t-channel Bhabhaprocess (c) and γ∗γ∗ pion pair production (d).final leptons. Both results were compared numerically scanning the physi-cal phase space, and the biggest relative difference between the two resultsfound was at the level of 10−9. It was necessary to use quadrupole precisionof the real numbers for the trace technique result, as one can observe severecancellations between various terms. The phase space volume calculated byMonte Carlo program was cross checked with the Gauss integration and therelative difference at the level of 10−5 was well within the errors of the MCresult, which were of the same order.Inclusive analytical formulae from [22] provide additional, nontrivial testsof the implementation of the contributions from Fig. 1(a). Formula (23)from [22] provides Q2 differential cross section (other variables are integratedover the whole allowed range) valid for large Q2. In Fig. 2(a), we comparethe values of the integrals, over 10 equally spaced intervals ofQ2, obtained bymeans of MC program and one-dimensional Gauss integration of the above5234 H. Czyż, E. Nowakmentioned analytical formula. The Gauss routine, which we use, guaranteesprecision of 12 significant digits. One observes a relatively good agreementfor values of Q2 ∼ s and worse for Q2 nearby pi+pi− production threshold, asexpected from the applicability of the analytical formula. EKHARA resultsagree much better (see Fig. 2(b)) with known analytically doubly differentialcross section dσ/(dQ2dk21) [22, 23], integrated over the whole allowed rangeof k21 and 10 equally spaced intervals of Q2. The exact analytical resultwas integrated numerically, using recursively one-dimensional 8-point Gaussprocedure and dividing the region of integration into pieces small enoughto guarantee the overall accuracy of 10 significant digits. From Fig. 2(b)it is clear that a technical precision of EKHARA of the order of 0.1% wasachieved.a)ps = 1.02 GeVQ2 (GeV2)MC=theory 11:210:80:60:40:200:0140:0120:010:0080:0060:0040:0020b)ps = 1.02 GeVQ2 (GeV2)MC=theory 11:210:80:60:40:200:0010:00050 0:0005 0:001 0:0015 0:002 0:0025 0:003 0:0035 0:004Fig. 2. EKHARA results compared with analytical results of [22] (see text fordetails). The errors come from MC integration.3. Monte Carlo data analysisFor ISR of e+e− pair ( Fig. 1(a)), a factorisation similar to photon emis-sion holds [22] and adding ISR pair production to ISR photon productionresults just in a change of the radiator function, thus radiative return methodstill can be used [2]. On the other hand, FSR of e+e− pair ( Fig. 1(b)) ismodel dependent, the same way as it is the emission of a real photon, andthe question of its relative, to ISR, contribution to the cross section is asimportant as for the photon emission. One can observe, that e+e− pairemission resembles a lot photon emission, with big contributions of FSRfor inclusive configurations (Fig. 3(a)) of a Φ-factory, which can be easilyreduced, by suitable cuts, to a negligible level (Fig. 3(b)). Moreover, thee+e− → pi+pi−e+e−: a Potential Background for σ(e+e− → pi+pi−) . . . 5235cuts which reduce photon FSR reduces as well the e+e− pair FSR. In addi-tion, analogously to photon FSR, e+e− pair FSR is completely negligible atB-factories.a)ps = 1.02 GeVno cutsQ2 (GeV2)(ddQ2)ISR+FSR=(ddQ2)ISR 11:210:80:60:40:20:50:450:40:350:30:250:20:150:10:050b)50Æ< < 130Æps = 1.02 GeV++ < 15Æor ++ > 165ÆQ2 (GeV2)(ddQ2)ISR+FSR=(ddQ2)ISR 11:210:80:60:40:20:070:060:050:040:030:020:010 0:01 0:02 0:03Fig. 3. Comparisons of ISR and FSR contributions to e+e− → pi+pi−e+e− crosssection at DAΦNE energy.The most relevant information, how big is the contribution of the processe+e− → pi+pi−e+e− in comparison to the main process used in the radiativereturn method, mainly e+e− → pi+pi−+photon(s), is presented in Fig. 4 forDAΦNE energy, both without any cuts (Fig. 4a), and with cuts resemblingKLOE event selection [16, 18] (Fig. 4(b)). The results of e+e− → pi+pi− +photon(s) cross section were obtained using PHOKHARA 3.0 MC generator[9] and in the following, whenever we refer to e+e− → pi+pi−+photon(s) crosssection we mean cross section obtained using PHOKHARA 3.0. As one cansee from Fig. 4, the contribution of the e+e− pair production is below 0.5%,independently on the cuts. It is Q2 dependant, being big for low Q2 values.Even if it is small, this 0.5% contribution can become important, whenaiming at the precision below, or of the order of 1%, for the e+e− → pi+pi−cross section measurement. At B-factories, the relative contribution of thepair production might be as big as 0.9% (Fig. 5(a)) and it is again Q2 andcut dependent (Fig. 5(b)).As stated already, ISR of electron pairs can be treated in a similar wayas ISR of photons resulting in the change of radiator function in the ra-diative return method. However, one can alternatively try to treat it as abackground to the process with photon(s) emission. In this case, the mostnatural way of reducing that background is to veto the electron (positron) inthe final state. In Fig. 6 we show an example of such a procedure performedfor Φ-factory energy. We assume here that an electron or positron can be5236 H. Czyż, E. Nowaka)ps = 1.02 GeVno cutsQ2 (GeV2)(ddQ2)pair=(ddQ2)ph1:210:80:60:40:20:0050:00450:0040:00350:0030:00250:002b)with Mtr50Æ< < 130Æps = 1.02 GeV++ < 15Æor ++ > 165ÆQ2 (GeV2)(ddQ2)pair=(ddQ2)ph1:210:80:60:40:20:00550:0050:00450:0040:00350:0030:00250:0020:00150:0010:0005Fig. 4. Ratio of differential cross sections of the process e+e− → pi+pi−e+e− (pair)and e+e− → pi+pi− + photon(s) (ph).a)ps = 10.52 GeVno cutsQ2 (GeV2)(ddQ2)pair=(ddQ2)ph1:210:80:60:40:200:00920:0090:00880:00860:00840:00820:0080:0078b)50Æ< < 130Æps = 10.52 GeVQ2 (GeV2)(ddQ2)pair=(ddQ2)ph1:210:80:60:40:20:00750:0070:00650:0060:00550:0050:0045Fig. 5. Ratio of differential cross sections of the process e+e− → pi+pi−e+e− (pair)and e+e− → pi+pi− + photon(s) (ph).seen, and the event rejected, if its angle with respect to the beam axis isbigger then 20◦. Fig. 6(a) shows that up to 50% of the events pass the re-jection procedure, when no other cuts are applied. However, in the case ofKLOE event selection, which requires that the e+e− pair is emitted alongthe beam axis, one rejects only a small fraction of these events (Fig. 6(b)).The situation is completely different at B-factories, where one can almostcompletely reduce the background coming from e+e− pairs ( Fig. 7), byrejecting the events with at least one charged lepton, electron or positron,in the detector.e+e− → pi+pi−e+e−: a Potential Background for σ(e+e− → pi+pi−) . . . 5237ps = 1.02 GeVa)Q2 (GeV2)(ddQ2)c2=(ddQ2)tot1:210:80:60:40:20:50:450:40:350:30:250:20:150:10:050ps = 1.02 GeVb)Q2 (GeV2)(ddQ2)c2=(ddQ2)tot1:210:80:60:40:21:0510:950:90:850:80:750:70:65Fig. 6. Non-reducible pair production background at DAΦNE energy : (a) nocuts in ( dσdQ2)tot; (b) for both (dσdQ2)c2 and (dσdQ2)tot the following cuts are imposed:50◦ < θpi± < 130◦, θpi++pi− < 15◦ or θpi++pi− > 165◦ and Mtr cut. For (a) and(b) c2 denotes additional cuts: (θe+ < 20◦ or θe+ > 160◦) and (θe− < 20◦ orθe− > 160◦).a)ps = 10.52 GeVQ2 (GeV2)(ddQ2)c2=(ddQ2)tot1:210:80:60:40:200:0180:0160:0140:0120:010:0080:0060:0040:0020 0:002b)ps = 10.52 GeVQ2 (GeV2)(ddQ2)c2=(ddQ2)tot1:210:80:60:40:200:020:0180:0160:0140:0120:010:0080:0060:0040:0020Fig. 7. Non-reducible pair production background at B-factory energy : (a) no cutsin ( dσdQ2)tot; (b) for both (dσdQ2)c2 and (dσdQ2)tot cuts on pion angles are imposed:50◦ < θpi± < 130◦. For (a) and (b) c2 denotes additional cuts: (θe+ < 20◦ orθe+ > 160◦) and (θe− < 20◦ or θe− > 160◦).4. ConclusionsWe have constructed the Monte Carlo generator EKHARA, which simu-lates the reaction e+e− → pi+pi−e+e− based on initial and final state emis-sion of a e+e− pair from e+e− → pi+pi− production diagram. A detailed5238 H. Czyż, E. Nowakstudy of this process, as a potential background for σ(e+e− → pi+pi−) mea-surement via radiative return method, shows that it can become important,when the experimental precision will reach 1%, or better.We thank J.H. Kühn for discussion and we are grateful for the supportand the kind hospitality of the Institut für Theoretische Teilchenphysik ofthe Universität Karlsruhe.REFERENCES[1] S. Binner, J.H. Kühn, K. Melnikov, Phys. Lett. B459, 279 (1999).[2] H. Czyż, A. Grzelińska, Acta Phys. Pol B 34, 5219 (2003).[3] M. Davier, S. Eidelman, A. Höcker, Z. Zhang, hep-ph/0308213, to be sub-mitted to Phys. Lett. B.[4] A. Nyffeler, Acta Phys. Pol. B 34, 5197 (2003).[5] F. Jegerlehner, hep-ph/0310234.[6] H. Czyż, J.H. Kühn, Eur. Phys. J. C18, 497 (2001).[7] G. Rodrigo, H. Czyż, J.H. Kühn, M. Szopa, Eur. Phys. J. C24, 71 (2002).[8] H. Czyż, A. Grzelińska, J.H. Kühn, G. Rodrigo, Eur. Phys. J. C27, 563 (2003).[9] H. Czyż, A. Grzelińska, J.H. Kühn, G. Rodrigo, hep-ph/0308312.[10] G. Rodrigo, A. 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e+ e- ---> pi+ pi- e+ e- : A Potential background for sigma (e+ e- ---> pi+ pi-) measurement via radiative return method

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e+ e- ---> pi+ pi- e+ e- : A Potential background for sigma (e+ e- ---> pi+ pi-) measurement via radiative return method

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