BESIII Analysis Memo

Transcription

1 Memo version. BESIII Analysis Memo DocDBXXX BAMXXX June, Measurement of e e π π J/ψ in the vicinity of the X() mass Leonard Koch a, Wolfgang Kühn a, and Sören Lange a, and Yutie Liang a a JustusLiebigUniversitaet Giessen, II. Physikalisches Institut Internal Referee Committee Ref xx (Chair) b, Ref xx c, and Ref xx d d XXX e XXX f XXX DocDB : Hypernews : Abstract We report the measurement of the cross section of the process e e π π J/ψ close to the X() mass in search for the direct formation e e X(). Due to the axialvector properties of the X() state, at least two virtual photons are required. No signal is observed and an upper limit on the product of electronic width and branching fraction of X() π π J/ψ is determined: Γ ee B(X() π π J/ψ) < mev at the % confidence level. This is an improvement of a factor of about compared to existing limits.

2 June, : BES MEMO Contents Introduction Samples Monte Carlo Simulations of e e π π J/ψ. Signal Process e e X() π π J/ψ Continuum Process e e π π J/ψ Event Selection Background Study. Monte Carlo Simulations of Backgrounds Background Rejection Cuts Cross Section Measurent of e e π π J/ψ. Fit to the m(l l ) Distribution Systematic Uncertainties Result Upper Limit on Γ X ee. Lineshape Bayesian Formalism Incorporation of Systematic Uncertainties Uncertainty of the Cross Section Uncertainty of the Lineshape Parameterization Result Conclusion Appendix A Determination of the CenterofMass Energy of the A. BEMS Result A. Cross Check via the Analysis of the Dimuon Process A.. Samples

3 June, : BES MEMO A.. Invariant Dimuon Mass A.. Radiative Correction A.. Momentum Callibration A.. Result A.. Run Dependency of the CenterofMass Energy B Determination of the Integrated Luminosity of the B. Monte Carlo Set B. Event Selection B. Systematic Uncertainties B. Result B. Cross Check: Luminosity Determination via Dimuon Process Introduction The observation of the X() resonance by the Belle Collaboration in [] and its quick confirmation by other experiments [ ] mark the beginning of XYZ spectroscopy. Being inconsistent with the predictions of the q q meson model, the XYZ states are candidates for tetraquarks, meson molecules, hybrid mesons and more []. The X() state is probably the best known representative of this class. It has been observed in B decays, in radiative transitions of the Y() resonance, as well as in inclusive pp and p p collisions. Up to now, decays into five different final states are established [, ]. Its mass is very close to the D D threshold, which is also a decay channel. This supports the meson molecule model []. However, one important disrcriminant between the different models is the width, which is still unknown and only an upper limit of. MeV at the % confidence level exists []. The quantum numbers of the X() particle [] allow only a suppressed formation in e e collisions via two virtual photons. Nevertheless, modern e e colliders, such as BEPCII, may have sufficient luminosity to observe direct X() formation. A scan of the lineshape would provide important information about the width. The electronic width Γ ee, which itself may help to reveal the X() s nature, serves as a measure of the feasability of such a scan. The current upper limit is Γ ee B(X() π π J/ψ) < mev at % confidence level, determined by BESIII using the ISR technique []. A theoretical prediction using Vector Meson Dominance yields Γ ee mev []. Combined with the lower limit on B(X() π π J/ψ) of. % [], this results in a lower bound of Γ ee B(X() π π J/ψ). mev. The aim of this analysis is to measure the cross section σ(e e π π J/ψ) around the the X()

4 June, : BES MEMO mass and the subsequent determination of the product Γ ee B(X() π π J/ψ) or an improved limit on this quantity. Although the production of a hadronic nonvector final state has been observed in e e collisions [], it would be the first observation of the formation of a nonvector resonance in e e annihilations. The next sections describe the used data sets and the Monte Carlo Simulations of the e e π π J/ψ process. The basic event selection is outlined in Section. In Section, the further analysis strategy based on a background study is illustrated. The cross section σ(e e π π J/ψ) is deter mined in Section and the upper limit calculation is explained in Section. The analysis is concluded in Section. Samples This analysis aims for measuring (or setting an improved upper limit on) the product Γee X() B(X() π π J/ψ) of the X(). Therefore, two dedicated data sets were recorded in June in the vicin ity of the X() mass (. MeV). The intented centerofmass for one data set were directly the central X() mass (refered to as onresonance sample) and ca. MeV below that value for the other data set (refered to as offresonance sample). During data taking the Beam Energy Measurement System (BEMS) was running for a precise realtime measurment of s. During the first two runs, the beam energies were slightly too low. In order to correct for this, they were increased slightly. As a result, the offresonance sample contains two runs with slightly different s. Additionally, the s obtained from the BEMS is more precise than result of the usual analysis of dimuon events and is thus used for the offline analysis as well. The BEMS is also capable of measuring the energy spread of s. See Section A for a detailed description of the BEMS measurement. Furthermore, the two data sets from the XYZ scan, that have s closest to the X() mass, are used in this analysis. For these data sets, the centerofmass has been determined via the analysis of dimuon events []. All four data sets are reconstructed with the BOSS version... Hence, this version is used for the whole analysis. The luminosity of all data sets have been determined by the analysis of Bhabha events. For the data the values are taken from []. A description of the luminosity determination of the data can be found in Section B. An overview on the data sets is given in Table. The BEMS result is delayed by ca. min, thus it is not a strict realtime measurement.

5 June, : BES MEMO Table : Overview of the data sets. The centerofmass energy s and its spread δ s together with the method, how these values are obtained, are listed. In addition, the integrated luminosity L dt, the year of data taking and the run numbers are shown. s / MeV δ s / MeV s Determination L dt / pb Year Run numbers. ±. not measured Dimuon. ±.. ±.. ±. BEMS. ±.. ±.. ±.. ±.. ±. BEMS. ±.. ±. not measured Dimuon. ±. Monte Carlo Simulations of e e π π J/ψ The event selection is optimized and the backgrounds are estimated by the use of Monte Carlo () simulations. As already mentioned in the previous section, all steps of the analysis are performed with the BOSS version... The goal of this analysis is the measurement of (or an improved upper limit on) the electronic width Γ X() ee of the X()in the reaction e e X() π π J/ψ and the subsequent J/ψ l l decay, where l = e, µ. This resonant signal process is accompanied by the non resonant continuum process e e π π J/ψ (c.f. Figure ). The corresponding Feynman diagrams are shown in Figure.. Signal Process e e X() π π J/ψ The X() decay into π π J/ψ is known to proceed via the intermediate ρ J/ψ state [,, ] and the J/ψ is reconstructed via its dilepton decay. Thus, the signal process is simulated within EvtGen [] according to e e ρ J/ψ (PHSP) π π e e µ µ (VLL) (VLL) (VSS) The acronyms in parantheses indicate the EvtGen decay model used for the subdecay in the corresponding line. The ISR is simulated with KK [] with a cross section lineshape assumed to be flat, i.e. constant. Final state radiation (FSR) is simulated with PHOTOS []. For each of the four different data samples, events with J/ψ e e and events with J/ψ µ µ are simulated.

6 June, : BES MEMO J/ψ) / pb π π expected cross section e σ(e total resonant X() contribution ( ) nonresonant continuum contribution ( ) s / GeV Figure : Expected cross section of e e π π J/ψ assuming a constant continuum of pb, an electronic witdth of Γee X() =. ev, a total width of Γ X() tot =. MeV and a spread of s of. MeV. The black markers indicate the centerofmass energies of the used data sets. The error bars indicate an estimation of the expected uncertainty. Swave ρ (Pwave) π π J/ψ π π J/ψ X() e e e e Figure : Feynman diagrams of the nonresonant continuum process e e π π J/ψ (left) and the resonant signal process e e X() π π J/ψ (right), of which it is known, that the π π pair forms a ρ meson [,, ].

7 June, : BES MEMO. Continuum Process e e π π J/ψ The nonresonant e e π π J/ψ reaction proceeds via a single virtual photon and the final state is therefore constrained to the quantum numbers. The relative small phasespace available for the π π pair suggests that it exists in an S wave. The different possibilities to model this within EvtGen are presented in the following. Again, ISR is simulated with KK and FSR is simulated with PHOTOS. In the EvtGen version within BOSS, there is the JPIPI model which models the decay of a state to π π J/ψ and is tuned to reproduce the distributions of the ψ π π J/ψ decay observed in real data. The mass of the ψ is MeV below the s values of the data sets used in this analysis making this model a good candidate for the continuum process. However, during simulation the warning message prob > maxprob was printed several times implying that the distributions are not sampled correctly. The full decay chain is modelled according to e e π π J/ψ (JPIPI) e e µ µ (VLL) (VLL) Very similar to the JPIPI model, there is the VVPIPI model, designed to model the decay of a state to π π and a state, where the π π system is dominated by an S wave. This is a natural choice for the continuum process. Unfortunately, the same warning message as with the JPIPI model was shown. The implication is the same, i.e. that the distributions are not sampled correctly. The full chain is e e π π J/ψ (VVPIPI) e e µ µ (VLL) (VLL) The S wave character of the π π system suggests that they form the intermediate state of a σ meson. This corresponds to e e σ J/ψ (PHSP) π π e e µ µ (VLL) (VLL) (PHSP)

8 June, : BES MEMO Another possibility to incorporate the σ meson is the VVS PWAVE model. It models the decay of a state to another state and a (pseudo) scalar meson. The amplitudes and phases for the S, P, and Dwave contributions can be specified, although only a pure Pwave configuration has been tested by the developers. Here, only the S wave contributes. The full chain is e e σ J/ψ (VVS PWAVE) π π e e µ µ (VLL) (VLL) (PHSP) The simplest (and probably the least realistic) way to simulate the reaction is the PHSP model: e e π π J/ψ (PHSP) e e µ µ (VLL) (VLL) For all the different models and data sets, events are simulated for each of the two J/ψ decay modes. The difference between the models is best visualized in the m(π π ) distributions (c.f. Figures and ). The limited statistics of the used data sets doesn t allow a reliable judgement about which of the models reflects reality most (c.f. Figure ). Therefore, another data set of BESIII taken at s =. MeV in is analyzed. This set has pb (compared to the total integrated luminosity of. pb of the used four sets) and is only MeV above the highest energy point of this analysis. The m(π π ) distribution of this data set is shown in Figure. The majority of events is clustered towards higher dipion masses. This favours the VVPIPI and JPIPI model over the models with the σ resonance and the PHSP model. In this analysis uses the VVPIPI model for the continuum, but the difference to the σ model with PHSP decay is used as a systematic uncertainty (c.f. section.). Event Selection The final state π π l l (l = e, µ) consists of four charged tracks with zero net charge. Events with the following criteria are selected: Each charged track is required to to fulfill the standard tracking cuts of BESIII. They need to originate from a confined volume around the interaction point and lie within the detectors acceptance: z poca < cm

9 June, : BES MEMO. GeV. GeV counts ρ (X() signal) σ PHSP σ VVSPwave Jpipi VVpipi PHSP counts ρ (X() signal) σ PHSP σ VVSPwave Jpipi VVpipi PHSP GeV mass(π π ) / (GeV/c ) GeV mass(π π ) / (GeV/c ) counts ρ (X() signal) σ PHSP σ VVSPwave Jpipi VVpipi PHSP counts ρ (X() signal) σ PHSP σ VVSPwave Jpipi VVpipi PHSP mass(π π ) / (GeV/c ) mass(π π ) / (GeV/c ) Figure : Simulated distribution of m(π π ) for the different data sets and models of the reaction e e π π J/ψ described in Sections. and. with J/ψ e e.. GeV. GeV counts ρ (X() signal) σ PHSP σ VVSPwave Jpipi VVpipi PHSP counts ρ (X() signal) σ PHSP σ VVSPwave Jpipi VVpipi PHSP mass(π π ) / (GeV/c ) mass(π π ) / (GeV/c ). GeV. GeV counts ρ (X() signal) σ PHSP σ VVSPwave Jpipi VVpipi PHSP counts ρ (X() signal) σ PHSP σ VVSPwave Jpipi VVpipi PHSP mass(π π ) / (GeV/c ) mass(π π ) / (GeV/c ) Figure : Simulated distribution of m(π π ) for the different data sets and models of the reaction e e π π J/ψ described in Sections. and. with J/ψ µ µ.

10 June, : BES MEMO J/ψ e e J/ψ µ µ counts counts mass(π π ) / (GeV/c ) mass(π π ) / (GeV/c ) (a) J/ψ e e mode. (b) J/ψ µ µ mode. Figure : Invariant dipion mass distribution for the data at s =. MeV. The shown events correspond to the J/ψ peak region as defind in section. and Figure. The accumulation of events in the low m(π π ) region in the e e mode is background from e e γe e as discussed in Section. r poca < cm cos θ <. Here, z poca is the zcomponent of the point of closest approach of the track with respect to the interaction point, which in turn is determined for easch run independently. The radial component of the point of closest approach is r poca = x poca y poca. For the third requirement, θ is the polar angle of the momentum vector of the track at the point of closest approach. Charged tracks fulfilling these requirements are labeled good tracks. Each candidate event needs to have a total number of good tracks with net zero charge. The lepton tracks coming from the J/ψ decay have a large momentum (in the lab frame), while the pion tracks have a relative low momentum (c.f. Figure a). Tracks with lower momentum than. GeV/c are identified as pion candidates and lepton candidates are required to have larger momentum than. GeV/c. Each candidate event needs to have two pion candidates with opposite charge and two lepton candidates with opposite charge. The two J/ψ decay modes can be distinguished by the energy deposition in the E associated to the lepton tracks. Electrons deposit a large fraction of their energy, while the muons pass the E almost undisturbed leaving only a small energy deposition (c.f. Figure b). Each event must have either two electron candidates or two muon candidates.

11 June, : BES MEMO counts All Events with Tracks ( ) π ± l ± counts µ ± Lepton Candidates e ± track momentum/ (GeV/c) (a) Pion / lepton discrimination by a cut on the lab momentum edep in E / GeV (b) Electron / muon discrimination by a cut on the energy deposit in the E. Figure : simulation of the signal process at s =. MeV. Background Study. Monte Carlo Simulations of Backgrounds In order to estimate the background contamination from nonπ π J/ψ events, the following backgrounds were simulated. The most dominant background for the events, in which the J/ψ candidate is reconstructed in the e e mode, is the radiative Bhabha process e e γe e with subsequent conversion of the photon in the detector material. Here, the resulting e e pair might be reconstructed as π ± candidates. This background is simulated with the Babayaga. event generator []. A similar background appears in the µ µ reconstruction mode of the J/ψ. Here, ther photon from radiative dimuon events (e e γµ µ ) converts and the created e e pair is again misidentified as a π π pair. Compared to the radiative Bhabha process, this reaction has a much lower cross section. This process is simulated with the Phokhara event generator []. Another QED process with four charged tracks in the final state ist the reaction e e e e e e. Here, one of the e e pairs might be misidentified as a pion pair and the total event is accepted as a π π e e event. This process is simulated with the BesTwogam event generator ref. Similar to the previous process the reaction e e e e µ µ could contribute to the overall background. Here, either the e e or the µ µ pair might be misidentified as a pion pair and the total event is accepted as a π π e e or a π π µ µ event. This process is also simulated with the BesTwogam event generator.

12 June, : BES MEMO Similar to the previous two processes the reaction e e e e q q could contribute to the overall background. The q q stands for an uū or a d d quark pair. Among others, they can hadronise into a π π pair. The final state is identical to the signal in the J/ψ e e mode. This process is simulated with the BesTwogam event generator as well. The most dominant background for the µ µ reconstruction mode is the reaction e e π π π π. Pions have a similar mass as muons and deposit a similar amount of energy in the E. They are easily misidentified as muons. This reaction is simulated with the ConExc event generator ref, which also simulates ISR. Another hadronic background with four charged tracks is the reaction e e KS K± π with the subsequent KS decay into two charged pions. Again, these events might be identified as π π µ µ events. This reaction is also simulated with ConExc. Another hadronic background with four charged tracks and a non negligible cross section is the reaction e e K K π π. Again, these events might be identified as π π µ µ events. This reaction is also simulated with ConExc. The ψ state is only MeV below the s of the used data sets. This implies a nonnegligible cross section for its production in ISR and its subsequent decay to the signal final state e e γ IS R ψ γ IS R π π J/ψ γ IS R π π l l. Without the detection of the ISR photon, this process has the same signature as the signal process. This background is simulated in EvtGen using the model VECTORISR for the emission of the ISR photon, the model JPIPI for the ψ decay and the VLL model for the J/ψ l l decay. An overview of the background samples is given in Table. In the following, the background samples are scaled to the integrated luminosity of data and added up to one cocktail sample per energy point.. Background Rejection Cuts Each event passing the basic selection criteria (c.f. Section ) is subjected to a kinematic fit. The kinematic constraints are the conservation of total four momentum, i.e. the four momenta of all tracks should add up to ( s,,, ) in the centerofmass frame. Additionally, the tracks are constrained to originate from a common vertex. For the data sets, the measured beam energy spread is included in the fit. For the data, there is no information about the beam spread, so it is not included. Due to inconsitencies between the estimated error of the track parameters in data and in, the resulting χ

13 June, : BES MEMO Table : Overview of the generated background samples. For each process and centerofmass energy, the cross section σ as calculated by the event generator is shown. From the cross section, the integrated luminosity for each s, and the number of simulated events N sim, the ratio of N sim to the number of expected events is calculated. The expected number of entries within the J/ψ peak region columns, each standing for one J/ψ mode. For the process e e (γ IS R )π π π π, ConExc gives a cross section which is ca. a factor of too low compared to the measurement by BABAR []. For this analysis, the cross section from ConExc is magnified by a factor of. The process e e (γ IS R )KS K± π is generated with the exclusive decay KS π π, so the corresponding branching fraction is included in the cross section. The process e e γ IS R ψ is generated with the exclusive decays of ψ > π π J/ψ and J/ψ l l. Its cross section is calculated by hand. as defined in Figure after all cuts listed in the N peak,ll exp peak, ee peak, µµ Process s / MeV σ / nb Nsim N sim /N exp Nexp Nexp e e γe e.. M.. ±.. ±. e e γe e... M.. ±.. ±. e e γe e... M.. ±.. ±. e e γe e.. M.. ±.. ±. e e γµ µ.. k. ±.. ±. e e γµ µ.. k. ±.. ±. e e γµ µ.. k. ±.. ±. e e γµ µ.. k. ±.. ±. e e e e e e.. M.. ±.. ±. e e e e e e.. M.. ±.. ±. e e e e e e.. M.. ±.. ±. e e e e e e.. M.. ±.. ±. e e e e µ µ.. M.. ±.. ±. e e e e µ µ.. M.. ±.. ±. e e e e µ µ.. M.. ±.. ±. e e e e µ µ.. M.. ±.. ±. e e e e q q.. M.. ±.. ±. e e e e q q.. k.. ±.. ±. e e e e q q.. M.. ±.. ±. e e e e q q.. M.. ±.. ±. e e (γ IS R )π π π π.. M.. ±.. ±. e e (γ IS R )π π π π.. M.. ±.. ±. e e (γ IS R )π π π π.. M.. ±.. ±. e e (γ IS R )π π π π.. M.. ±.. ±. e e (γ IS R )KS K± π.. M. ±,. ±. e e (γ IS R )KS K± π.. M. ±,. ±. e e (γ IS R )KS K± π.. M. ±,. ±. e e (γ IS R )KS K± π.. M. ±,. ±. e e (γ IS R )K K π π.. k.. ±.. ±. e e (γ IS R )K K π π.. k.. ±.. ±. e e (γ IS R )K K π π.. k.. ±.. ±. e e (γ IS R )K K π π.. k.. ±.. ±. e e γ IS R ψ.. M. ±.. ±. e e γ IS R ψ.. M. ±.. ±. e e γ IS R ψ.. M. ±.. ±. e e γ IS R ψ.. M. ±.. ±.

14 June, : BES MEMO All Points All Points events /. events /. π π cosθ data Signal Background events /. events /. π π cosθ data Signal Background cosθ π π cosθ π π (a) J/ψ e e mode. (b) J/ψ µ µ mode. Figure : Comparisson between data, signal, and background distributions of cos θ π π. All four data sets are combined. The cut cos θ π π <. is indicated by the arrow. The other cuts on χ and cos π ± e are applied as well as a cut on the fitting range of the invariant dilepton mass (. GeV/c < m(l l ) <. GeV/c, c.f. Figure). The distributions are scaled to match the integrated luminosity of the data. distributions differ, too. This effect is compensated by a correction of the track parameter errors of the events, as described in []. The background contamination is reduced by the following cuts: The background from gamma conversion can be reduced a lot by requiring cos θ π π <., when θ is the opening angle between the two pion candidates. The as pion misidentified electrons ππ from gamma conversion carry the boost of the initial photon and are therefore almost parallel and peak at cos θ π π = (c.f. Figure ). This cut is applied to both J/ψ reconstruction modes. It reduces almost all of the radiative dimuon background, while the large abundancy of radiative Bhabha events requires an additional cut. The electrons from gamma conversion can also be identified as a π ± l pair. The cut cos θ π ± e <. is applied only to the electron mode (c.f. Figure ). All events must have a χ of the kinematic fit lower than. The corresponding distribution is shown in Figure. Figure show the distributions of the invariant dipion mass. The peak at the lower edge of the spectrum in the e e reconstruction mode is caused by the gamma conversion background. The position and shape of this peak is reproduced in. However, the absolute number of events is underestimated by a factor of ca.. In principle at cut on m(π π ) would remove this background. Unfortunately, the different models of the continuum process e e π π J/ψ differ significantly in the lower region

15 ± June, : BES MEMO All Points All Points events /. events /. data Signal events /. data Signal Background Background cosθ π ±e (a) J/ψ e e mode. cosθ π ± e ± (b) J/ψ µ µ mode. cosθ π ± µ ± Figure : Comparisson between data, signal, and background distributions of cos θ π ± l. All four data sets are combined. The cut cos θ π ± e <. is indicated by the arrow. The other cuts on χ and cos π π are applied as well as a cut on the fitting range of the invariant dilepton mass (. GeV/c < m(l l ) <. GeV/c, c.f. Figure). The distributions are scaled to match the integrated luminosity of the data. All Points All Points events / data events / data Signal Signal Background Background χ (a) J/ψ e e mode. χ (b) J/ψ µ µ mode. Figure : Comparisson between data, signal, and background distributions of χ. All four data sets are combined. The cut χ < is indicated by the arrow. The other cuts on cos π π and cos θ π ± e are applied as well as a cut on the fitting range of the invariant dilepton mass (. GeV/c < m(l l ) <. GeV/c, c.f. Figure). The distributions are scaled to match the integrated luminosity of the data.

16 June, : BES MEMO All Points All Points ) events / ( MeV / c data Continuum Background ) events / ( MeV / c data Continuum Background (a) J/ψ e e mode. mass(π π ) / (GeV / c ) (b) J/ψ µ µ mode. mass(π π ) / (GeV / c ) Figure : Comparisson between data, continuum, and background distributions of m(π π ). All four data sets are combined. All cuts as well as a cut on the fitting range of the invariant dilepton mass (. GeV/c < m(l l ) <. GeV/c, c.f. Figure) are applied. The distributions are scaled to match the integrated luminosity of the data. of the distribution (c.f. Figure ). Due to this model ambiguity, a cut would introduce a large systematic uncertainty. The invariant mass distributions of the dilepton pairs are shown in Figure. The J/ψ peak is clearly visible and the peak region. GeV/c < m(l l ) <. GeV/c is indicated by arrows. Figures show the same distributions as Figures, but only with the events in the J/ψ peak region instead of the whole fit range. Cross Section Measurent of e e π π J/ψ The cross section of the process e e π π J/ψ is determined using σ(e e π π J/ψ) = N obs L dt ɛ ( δ) B(J/ψ l l ) () with: N obs is the number of observed events, extracted from a fit to the l l mass spectrum, c.f. next section. The number of events in the J/ψ peak is equal to N obs. L dt is the integrated luminosity (c.f. Table ). The efficiency ɛ is determined from the analysis of the signal and continuum samples. In the beginning it is assumed, that there is no X() signal and the efficiency is solely determined from the continuum. After the observation of an enhancement at the X() mass, the efficiency

17 ± June, : BES MEMO All Points All Points ) events / ( MeV / c data Signal ) events / ( MeV / c data Signal Background Background (a) J/ψ e e mode. mass(e e ) / (GeV / c ) (b) J/ψ µ µ mode. mass(µ µ ) / (GeV / c ) Figure : Comparisson between data, signal, and background distributions of m(l l ). All four data sets are combined. All cuts are applied. The distributions are scaled to match the integrated luminosity of the data. The arrows indicate the peak region. All Points All Points events /. events /. π π cosθ data Signal Background events /. events /. π π cosθ data Signal Background cosθ π π cosθ π π (a) J/ψ e e mode. (b) J/ψ µ µ mode. Figure : The same as Figure, but only containing events in the J/ψ peak region, c.f. Figure. All Points All Points events /. events /. data Signal events /. data Signal Background Background cosθ π ±e (a) J/ψ e e mode. cosθ π ± e ± (b) J/ψ µ µ mode. cosθ π ± µ ± Figure : The same as Figure, but only containing events in the J/ψ peak region, c.f. Figure.

18 June, : BES MEMO All Points All Points events / data events / data Signal Background Signal Background χ (a) J/ψ e e mode. χ (b) J/ψ µ µ mode. Figure : The same as Figure, but only containing events in the J/ψ peak region, c.f. Figure. All Points All Points ) events / ( MeV / c data Continuum Background ) events / ( MeV / c data Continuum Background (a) J/ψ e e mode. mass(π π ) / (GeV / c ) (b) J/ψ µ µ mode. mass(π π ) / (GeV / c ) Figure : The same as Figure, but only containing events in the J/ψ peak region, c.f. Figure.

19 June, : BES MEMO for the. MeV data set would be the weighted average of the efficiencies of the continuum and the signal model according to the cross section fraction attributed to continuum and the X/) signal. However, there is no such enhancement at the X() mass and the efficiency is just the one obtained from the continuum. ( δ) is the radiative correction factor to account for ISR. It is calculated from the KK event generator assuming a constant lineshape of the cross section. After the measurement of the true lineshape, the new one would be used as input to KK and a new value of ( δ) would be obtained to calculate a new value for the cross section. This procedure would be iterated until the values of ( δ) and σ converge. However, the measured lineshape cannot be distinguished from a constant, so the iterative procedure doesn t need to be done. Finally, the exclusive reconstruction of the J/ψ in its dilepton decays needs to be incorporated by the branching fraction for this decay B(J/ψ l l ). The values are taken from the PDG [].. Fit to the m(l l ) Distribution As mentioned above, the number of signal events is extracted from a maximum likelihood fit to the invariant dilepton mass spectrum. This is done independently for both J/ψ decay modes in the range of. GeV/c < m(l l ) <. GeV/c. The lineshape is used as the signal pdf, while the background is modeled as a linear function. The complete model p and the likelihood function L are p(m, a, f ) = f s(m) ( f ) b(m, a) () N L(a, f ) = p(m i, a, f ) () i= where m = m(l l ) is the invariant dilepton mass, f is the signal fraction and a is a parameter of the linear function. N is the total number of events included in the fit and the invariant dilepton mass of event i is m i. Instead of maximizing the likelihood funtion directly, the negative loglikelihood function NLL = log N is minimized, which is equivalent but numerically much more stable. The fit is performed in the ROOT framework [] using the RooFit [] library and the Minuit minimizer []. The number of signal events is N obs = ˆ f N when ˆ f is the value of f at the global minimum of NLL. Its error is estimated from the second derivative of NLL min with respect to the fit parameters. Alternatively, an extended maximum likelihood fit could have been performed. Here, the expected number of events ν is included and ν s = ν f and ν b = ν( f ) directly substituted. The extended likelihood function contains the additional Poissonian term:

20 ) ) ) ) June, : BES MEMO. GeV. GeV Events / (. GeV / c Events / (. GeV / c mass(e e ) (GeV / c ) mass(e e ) (GeV / c ). GeV. GeV Events / (. GeV / c Events / (. GeV / c mass(e e ) (GeV / c ) mass(e e ) (GeV / c ) Figure : Fit of the m(e e ) distribution for each data set. The markers with error bars is the data distribution. The red line represents the full fit pdf, while the dashed gray line is the background contribution to the pdf. ν νn L(a, ν s, ν b ) = e N! e ν s ν b N p(m i, a, ν s, ν b ) () i= N [ν s s(m) ν b b(m, a)] () i= As long as the model has no explicit dependency on ν, the minimization of the corresponding NLL will yield the same result as the standard maximum likelihood fit. Figures and show the m(l l ) distribution with the fit pdf overlayed. The result of the fit is summarized in Table.. Systematic Uncertainties In the following, the systematic uncertainties affecting the cross section measurement are discussed. The uncertainty of the integrated luminosity for each data set is listed in Table (also in Table ). For the measurement, four charged tracks are analyzed. It was shown in [? ], that the uncertainty of the tracking efficiency is % per track yielding a % uncertainty.

21 ) ) ) ) June, : BES MEMO. GeV. GeV Events / (. GeV / c Events / (. GeV / c mass(µ µ ) (GeV / c ) mass(µ µ ) (GeV / c ). GeV. GeV Events / (. GeV / c Events / (. GeV / c mass(µ µ ) (GeV / c ) mass(µ µ ) (GeV / c ) Figure : Fit of the m(µ µ ) distribution for each data set. The markers with error bars is the data distribution. The red line represents the full fit pdf, while the dashed gray line is the background contribution to the pdf. Table : Result of the fit to the dilepton mass distribution. Shown are the results of the two independent J/ψ modes and a combined value. s / MeV. ±.. ±.. ±.. ±. L dt / pb. ±.. ±.. ±.. ±. ( δ). ±.. ±.. ±.. ±. B(J/ψ e e ) / %. ±.. ±.. ±.. ±. N e e obs ± ± ± ± ɛ e e / %. ±.. ±.. ±.. ±. σ e e / pb ±. ±.. ±. ± B(J/ψ µ µ ) / %. ±.. ±.. ±.. ±. N µ µ obs ± ± ± ± ɛ µ µ / %. ±.. ±.. ±.. ±. σ µ µ / pb. ±.. ±.. ±.. ±. σ l l / pb. ±.. ±.. ±.. ±.

22 June, : BES MEMO The radiative correction factor ( δ) is calculated by KK. The samples were generated in sub samples for each s and J/ψ mode. Thus, there are different values for ( δ). The mean value is used in the determination of the cross section (Equation ()). The standard deviation is used as the systematic uncertainty of ( δ). The values for B(J/ψ l l ) together with their uncertainties are taken from the PDG []. The values are given in Table. In [] it was shown, that the uncertainty associated to the kinematic fit can be estimated by half the efficiency difference of the analysis with and without the helix parameter correction (c.f. section.). In the fit to the m(l l ) distribution, the background is modeled as a linear function. This choice has a certain ambiguity and its impact on the extracted cross section is acompanied by a systematic uncertainty. It is estimated by the difference in N sig when the fit is performed with a quadratic background parameterization. The ambiguity of the modeling of the continuum process e e π π J/ψ is already mentioned in Section. and indicated in Figures and. The systematic uncertainty associated to this ambiguity is estimated by the efficiency difference when using the σ PHSP model instead of the VVPIPI model. The efficiency ɛ is determined by the analysis of events. The statistical limited set attaches an error on the efficiency. The relative large size of each signal and continuum set of k implies a relative small errror which can be seen in Table. It is less than. % and can be neglected. The contributions of each source of systematic uncertainty are listed in Table.. Result The final result of the cross section measuremnt is listed in Table and shown in Figure. The statistical uncertainty clearly dominates the overall uncertainty. There is no enhancement at the X() visible. In fact, there is a small dip in the cross section. However, this cannot be explained by a destructive interference between the X() and the continuum amplitudes, since they carry different quantum numbers (c.f. Figure ). One might argue that the process e e γ γ ρ J/ψ could happen via double vector meson dominance. It is noteworthy, that the threshold for this reaction is very close to the X() mass. A simplified calculation according to [] yields a global maximum of the e e ρ J/ψ

23 June, : BES MEMO Table : Relative systematic uncertainties (in %) affecting the measured cross section σ(e e π π J/ψ). The total uncertainty is the quadratic sum of the individual errors.. MeV. MeV. MeV. MeV Source e e µ µ e e µ µ e e µ µ e e µ µ L dt.... Tracking.... ( δ).... B(J/ψ l l ) Kinematic fit m(l l ) fit Decay model Total Table : Cross section of e e π π J/ψ as determined in this analysis. The first error is the statistical and the second one is the systematic uncertainty. s / MeV σ(e e π π J/ψ) e e mode µ µ mode both modes combined.. ±. ±.. ±. ±.. ±. ±... ±. ±.. ±. ±.. ±. ±... ±. ±.. ±. ±.. ±. ±... ±. ±.. ±. ±.. ±. ±. cross section of less than. pb, which in addition is dominated by the and states. Furthermore, the cross section is strongly peaked towards cos θ ±, which is outside the detectors acceptance. In total, there is little room for a amplitude intefering with the X(). The measured cross section is in good agreement with a constant cross section. Since ther is no sign of direct X() production, an upper limit on Γ X() ee is determined in the next section. A simple fit of a constant to the measured values gives χ /NDF =./ or alternatively a pvalue of..

24 June, : BES MEMO J/ψ) / pb π π e σ(e CenterofMass Energy / MeV Figure : Cross section of e e π π J/ψ. The results of both J/ψ decay modes are combined. The nominal mass of the X() as listed in the PDG [] is indicated by the vertical line. The error bars represent the statistical errors and the small horizontal lines above and below the error bars represent the total uncertainties, i.e. the quadratic sum of the statistical and systematic errors.

25 June, : BES MEMO Upper Limit on Γ X ee. Lineshape Due to the different quantum numbers of the continuum process and the resonant X() formation, the total e e π π J/ψ cross section is modeled as its incoherent sum. The continuum is assumed to be flat, i.e. a constant. The X() is modeled as a reletivistic BreitWigner resonance. Only the π π J/ψ decay mode is taken into account, so the corresponding branching fraction needs to be included in the lineshape parameterization (using c = = ): σ e e π π J/ψ( s) = σ cont π Γ totγ ee B(X() π π J/ψ) ( s m ) m Γ tot () where σ cont, Γ tot, and Γ ee are the constant continuum, the total width and the electronic width of the X(), respectively. The X() mass is m and s is the Mandelstam variable. Of the X() parameters, only the mass is known (m(x()) = (.±.) MeV/c []). Since the branching ratio B(X() π π J/ψ) has not been determined yet, this analysis treats the product Γ ee B(X() π π J/ψ) as one parameter and an upper limit on this product is set instead of a limit on Γ ee alone. In total, there are three unknown parameters. In each data set, s is Gaussian distributed according to the energy spread in Table. The resulting measured cross section for data set i is σ i = σ e e π π J/ψ( s i ) N( s i, δ s i ) () = σ e e π π J/ψ(x) N( s i x, δ s i ) dx () = σ cont πγ ee B(X() π π Γ tot J/ψ) ( ) x m N( s i x, δ s i ) dx () m Γ tot when N(x µ, σ) is the normal distribution with mean µ and variance σ. For the data, the BEMS information can be used, while the two data points are far away from the X() mass. As a result, they are only sensitive to the constant term σ const in the lineshape and a convolution with a moderate energy spread won t alter the cross section value. Nevertheless, a s spread of. MeV is assumed. With the lineshape parameterization, the model aquires an explicit dependency on the total number of events, so here, the extended likelihood function as defined in () is used. The expected number of signal events is substituted by the cross section with the relation given in (). As a result, the likelihood function for each data set i and J/ψ mode j is now depending on the cross section:

26 June, : BES MEMO L j i (a j i, ν j s,i, ν j b,i ) L j i (σ i, a j i, ν j b,i ) () The overall likelihood function is L = i= j=ee,µµ L j i (σ = σ i, a j i, ν j b,i ) (). Bayesian Formalism In Bayesian formalism, the likelihood function L(x, θ) is interpreted as the conditional pdf f (x θ), i.e. the likelihood of observing data x given the parameter θ is true []. This is not restricted to single parameter models, but θ R N and of course the data x R N consists of many measurements. Using Bayes Theorem [], this pdf can be turned into a conditional pdf giving the likelihood of the parameters θ giving the data: f (θ x) = f (x θ)π(θ) f (x θ)π(θ) dθ () f (x θ)π(θ) () The prior pdf π(θ) gives the likelihood of the parameters before the measurement x was obtained. In () the denominator is independent of θ and can be viewed as a normalization constant. In most cases, the prior pdf is unknown and very often a flat, i.e. constant one, is assumed. Although this is strictly speaking not a pdf and not in all cases the optimal choice, but the mode of the resulting posterior pdf (θ x) coincides with the maximum likelihood fit. In this analysis, the priors are constant in the physical region, i.e. Γ tot > and Γ ee B(X() π π J/ψ) >, and are set to zero in the unphysical region. Most of the time, there are parameters in a model, that are not of interest, e.g. the coefficients of a polynomial describing the background. Consider the parameter set θ = ( θ, θ n ) where θ is the interesting parameter set and θ n are the others, socalled nuissance parameters. The posterior pdf concerning only the parameters of interest is obtained by the marginalization of the nuissance parameters[]: f ( θ x) = f (( θ, θ n ) x) dθ n () A constant cannot be normalized and is therefore no pdf. Nevertheless, it can result in a proper, i.e. normalizable, posterior pdf (θ x). There are different approaches to construct socalled noninformative or objective priors, which are favourable, but very difficult to obtain in multiparameter models [, ].

27 June, : BES MEMO In this analysis, the two parameters of interest are Γ ee B := Γ ee B(X() π π J/ψ) and Γ tot, so the marginalized likelihood is ˆL(Γ ee B, Γ tot ) = i= j=ee,µµ L j i (σ = σ i, a j i, ν j b,i ) da j i dν j b,i dσ cont () There are each eight different a j i and ν j b,i variables and together with σ cont the integration needs to be carried out over dimensions. Fortunately, most of the variables are perpenticular to each other and the integration over dimensions can be reduced to eight twodimensional integrals. With ˆL j i (σ) = L j i (σ, a j i, ν j b,i ) da j i dν j b,i () () can be written as ˆL(Γ ee B, Γ tot ) = i= j=ee,µµ ˆL j i (σ = σ i) dσ cont () All the integrations are performed numerically within ROOT []. A % credible interval in Bayesian inference can be defined by the following two properties []: For each point in the interval, the posterior pdf is larger than for every other point outside the interval. The integral of the (normalized) posterior over the interval must be %. This definition can be extended to the multiparameter case, where the interval becomes a multidimensional region. An upper limit is the upper interval boundary, when the lower interval boundary coincides with the border to the unphysical region.. Incorporation of Systematic Uncertainties There are two different kinds of systematic uncertainties which have to be treated differently. The first kind affects the cross section measurement and is described in Section.. The second kind affects the lineshape parameterization and includes the uncertainties associated with the X() mass, s, and the beam spread.

28 June, : BES MEMO. GeV. GeV Marginalized Likelihood / pb... ee mode µµ mode ee mode with sys. error µµ mode with sys. error Both modes with sys. error Marginalized Likelihood / pb..... ee mode µµ mode ee mode with sys. error µµ mode with sys. error Both modes with sys. error..... σ / pb σ / pb. GeV. GeV Marginalized Likelihood / pb..... ee mode µµ mode ee mode with sys. error µµ mode with sys. error Both modes with sys. error Marginalized Likelihood / pb.... ee mode µµ mode ee mode with sys. error µµ mode with sys. error Both modes with sys. error σ / pb σ / pb Figure : The marginalized likelihoods before and after the concolution with the systematic uncertainties ˆL j j i (σ) and Li (σ) as well as the normalized product of both modes for all four data sets. The systematic uncertainties have only little effect as can also be seen in Figure... Uncertainty of the Cross Section The systematic uncertainties of the cross section are determined in Section. and listed in Tables and. They are incorporated by the convolution of () with a Gaussian with the corresponding variance: L j i (σ) = ˆL j i (σ) N(σ, j sys,iσ) () The marginalized likelihoods ˆL j j i (σ) and Li (σ) as well as the product L i(σ) := L ee i (σ) L µµ i (σ) are plotted in Figure... Uncertainty of the Lineshape Parameterization The uncertainties affecting the lineshape are the following. The mass of the X() is taken from the PDG together with its uncertainty m = (. ±.) MeV/c [].

29 June, : BES MEMO The measured centerofmass energies have uncertainties which have to be taken into account. They are given in Table. The consideration of the beam energy spread in the cross section lineshape makes the overall result sensitive to the corresponding uncertainty. The BEMS provided the uncertainies on the energy spread for the data (also given in Table ). For the data there is no such information, but Section. discussed the negligible impact of the energy spread for those data points. Nevertheless, uncertainties of. MeV are assumed which is approximately twice as large as the uncertainty for the onresonance and four times as large as the one for the offresonance data set. A common approach to include these systematic uncertainties is the extension of the likelihood function by Gaussian prior pdfs for the systematically uncertain parameters and the subsequent marginalization over them []. This turns the onedimensional integral of () into a tendimensional one making it computational very expensive. This issue is resolved by the application of a integration technique. The following procedure is repeated several times.. The values for the parameters with a systematic uncertainty are sampled from the corresponding normal distributions.. The marginalized likelihood ˆL(Γ ee B, Γ tot ) is calculated (only onedimensional integral, c.f. ()). Finally, the likelihoods of each repetition are summed up. This is equivalent to an averaging, because the likelihoods aren t normalized. Effectively, a ninedimensional integral is converted into a sum: L(Γ ee B, Γ tot ) = ˆL(Γ ee B, Γ tot ) θ=θi () i where θ stands for the parameters with an uncertainty and θ i are the sampled parameters in iteration i. In Figure the obtained upper limit of Γ ee B assuming Γ tot. MeV depending on the number of iterations in the integration is shown. After already iterations a stable value is obtained. Of course, all iterations are used for the result.. Result The likelihood function () is shown in Figure. Since the total width Γ tot is unknown, the upper limit on Γ ee B can be determined as a function of Γ tot. For each value of Γ tot, the likelihood is integrated over Γ ee B until the intgral becomes %. This is shown in Figure for a fixed Γ tot =. The upper limit depending on the total width is shown in Figure.

30 BES MEMO June, : UL(Γ ee Br(X() ππj/ψ % C. L. / mev Γtot =. MeV Iteration Figure : Development of the obtained upper limit on Γee B at % C.L. depending on the number of iterations in the integration. A total width of Γtot. MeV is assumed, the current upper limit. Γ ee Br(X() ππj/ψ )) / mev Marginalized Likelihood... Γ tot(x()) / MeV Figure : Unnormalized marginalized likelihood function L(Γee B, Γtot ). The gray line indicates the current % upper limit on Γtot.

31 June, : BES MEMO Γ tot =. MeV Marginalized Likelihood Γ ee Br(X() π π J/ψ)) / mev Figure : Determination of the upper limit on Γ ee B for an assumed total width of approximately. MeV, its current upper limit. The gray area indicates the % integral. The value of Γ ee B at the right edge of that area is the upper limit on γ ee B at the % confidence level. % C. L. / mev π UL(Γ ee Br(X() π... Γ tot / MeV Figure : The upper limit on Γ ee B at the % confidence level as a function of the total width. The gray line indicates the current upper limit on Γ tot.

32 June, : BES MEMO J/ψ)) / mev π Γ ee Br(X() π Marginalized Likelihood Γ tot (X()) / MeV Figure : Likelihood function of Γ ee B and Γ tot with the nonuniform prior for Γ tot reflecting its known upper limit. A unique value for the upper limit of Γ ee B could be obtained by a marginalization of the likelihood over Γ tot. However, s significant fraction of the likelihood stretches far beyond the current upper limit of Γ tot (c.f. Figure ). Consequently, this marginalization is not feasible. A possibility to incorporate the current knowledge about Γ tot, namely the upper limit of. MeV, is to use a nonuniform prior for it. In the determination of the limit, the likelihood function of Γ tot is a zeromean Gaussian []. A standard deviation of. MeV ensures the % limit at. MeV. This Gaussian shape is used as the prior pdf of Γ tot. The resulting likelihood function is shown in Figure. After the marginalization of the obtained likelihood function over Γ tot, it depends only on Γ ee B. It is plotted in Figure and the % limit is determined. After rounding, it is Γ ee B < mev at % confidence level. A twodimensional credible region for (Γ tot, Γ ee B) can also be constructed when the nonuniform prior is used. It is shown in Figure. Conclusion In this analysis, the cross section of e e π π J/ψ was measured at four different s close to the X() mass. The result of this measurement is listed in Table. Since there is no enhancement at the X() mass, an upper limit for Γ ee B is determined. The % upper limit depending on the total width is shown in Figure. For an assumed total width of. MeV, the upper limit on Γ ee B is mev

33 June, : BES MEMO marginalized likelihood / a. u. UL(Γ ee Br(X() π π J/ψ))) =. % C. L. Γ ee Br(X() π π J/ψ)) / mev Figure : Likelihood function of Γ ee B after the marginalization over Γ tot with the nonuniform prior. The % integral is indicated as the gray area and its right edge corresponds to the upper limit. J/ψ)) / mev π % Credible Region Γ ee Br(X() π Γ tot (X()) / MeV Figure : Twodimensional credible region for (Γ tot, Γ ee B) at the % confidence level.

34 June, : BES MEMO at the % confidence level. When the current knowledge about the total width is incorporated, a twodimensional % credible region of (Γ tot, Γ ee B) can be constructed. Furthermore, a Γ tot independent upper limit on Γ ee B of mev is obtained, an improvement of a factor compared to the previous limit []. It is in no conflict to the theoretical prediction of Γ ee B. mev [].