Cross Sections for the Reactions e^{+}e^{-}\rightarrow K^{+}K^{-}\pi^{+}\pi^{-}, K^{+}K^{-}\pi^{0}\pi^{0}, and K^{+}K^{-}K^{+}K^{-} Measured using Initial-State Radiation Events

Cross Sections for the Reactions , , and Measured using Initial-State Radiation Events


We study the processes , , and , where the photon is radiated from the initial state. About 84000, 8000, and 4200 fully-reconstructed events, respectively, are selected from 454  of BABAR data. The invariant mass of the hadronic final state defines the center-of-mass energy, so that the data can be compared with direct measurements of the reaction. No direct measurements exist for the or reactions, and we present an update of our previous result based on a data sample that is twice as large. Studying the structure of these events, we find contributions from a number of intermediate states, and extract their cross sections. In particular, we perform a more detailed study of the reaction, and confirm the presence of the resonance in the and modes. In the charmonium region, we observe the in all three final states and in several intermediate states, as well as the in some modes, and measure the corresponding products of branching fraction and electron width.

13.66.Bc, 14.40.-n, 13.25.Jx




Phys. Rev. D86, 012008 (2012)

The BABAR Collaboration

I Introduction

Electron-positron annihilation at fixed center-of-mass (c.m.) energies has long been a mainstay of research in elementary particle physics. The idea of utilizing initial-state radiation (ISR) to explore reactions below the nominal c.m. energies was outlined in Ref. baier (), and discussed in the context of high-luminosity and factories in Refs. arbus (); kuehn (); ivanch (). At high c.m. energies, annihilation is dominated by quark-level processes producing two or more hadronic jets. Low-multiplicity processes dominate below or around 2 , and the region near the charm threshold, 3.0–4.5 , features a number of resonances PDG (). Thus, studies with ISR events allow us to probe a wealth of physics topics, including cross sections, spectroscopy and form factors. Charmonium and other states with can be observed, and intermediate states may contribute to the final state hadronic system. Measurements of their decay modes and branching fractions are important to an understanding of the nature of such states.

Of particular current interest (see Ref. y2175theory ()) is the state observed to decay to in our previous study isr2k2pi () and confirmed by the BES y2175bes () and Belle belle_phif0 () Collaborations. With twice the integrated luminosity (compared to Ref. isr2k2pi ()) in the present analysis, we perform a more detailed study of this structure.

The study of hadrons reactions in data is also critical to hadronic-loop corrections to the muon magnetic anomaly, . The theoretical predictions of this anomaly rely on these measurements dehz (). Improving this prediction requires not only more precise measurements, but also measurements from threshold to the highest c.m. energy possible. In addition, all the important sub-processes should be studied in order to properly incorporate possible acceptance effects. Events produced via ISR at factories provide independent and contiguous measurements of hadronic cross sections from the production threshold to a c.m. energy of . With more data we also are able to reduce systematic uncertainties in the cross section measurements.

The cross section for the radiation of a photon of energy in the c.m. frame, followed by the production of a particular hadronic final state , is related to the corresponding direct cross section by


where is the nominal c.m. energy, is the fraction of the beam energy carried by the ISR photon, and is the effective c.m. energy at which the final state is produced. The probability density function for ISR photon emission has been calculated with better than 1% precision (see, e.g.  Ref. ivanch ()). It falls rapidly as increases from zero, but has a long tail, which in combination with the increasing produces a sizable event rate at very low . The angular distribution of the ISR photon peaks along the beam directions. For a typical detector, around 10-15% of the ISR photons fall within the experimental acceptance ivanch () .

Experimentally, the measured invariant mass of the hadronic final state defines . An important feature of ISR data is that a wide range of energies is scanned continuously in a single experiment, so that no structure is missed, and the relative normalization uncertainties in data from different experiments are avoided. Furthermore, for large values of the hadronic system is collimated, reducing acceptance issues and allowing measurements down to production threshold. The mass resolution is not as good as the typical beam energy spread used in direct measurements, but resolution and absolute energy scale can be monitored by means of the measured values of the width and mass of well-known resonances, such as the produced in the reaction . Backgrounds from hadrons events at the nominal and from other ISR processes can be suppressed by a combination of particle identification and kinematic fitting techniques. Studies of and several multi-hadron ISR processes using BABAR data have been performed  isr2k2pi (); Druzhinin1 (); isr3pi (); isr4pi (); isr6pi (); isr5pi (); isrkkpi (); isr2pi (), demonstrating the viability of such measurements. These analyses have led to improvements in background reduction procedures for more rare ISR processes.

The final state has been measured directly by the DM1 Collaboration 2k2pidm1 () for , and we have previously published ISR measurements of the and final states isr4pi () for . Later we reported an updated measurement of the final state with a larger data sample, together with the first measurement of the final state, in which we observed a structure near threshold in the intermediate state isr2k2pi ().

In this paper we present a more detailed study of these two final states along with an updated measurement of the final state. In all cases we require the detection of the ISR photon and perform a set of kinematic fits. We are able to suppress backgrounds sufficiently to study these final states from their respective production thresholds up to =5 . In addition to measuring the overall cross sections, we study the internal structure of the final states and measure cross sections for a number of intermediate states that contribute to them. We also study the charmonium region, measure several and products of branching fraction and electron width, and set limits on other states.

Ii The BABAr detector and dataset

The data used in this analysis were collected with the BABAR detector at the PEP-II asymmetric-energy  storage rings at the SLAC National Accelerator Laboratory. The total integrated luminosity used is 454.2 , which includes 413.1  collected at the peak, , and 41.1  collected at about .

The BABAR detector is described elsewhere babar (). In the present work, we use charged-particle tracks reconstructed in the tracking system, which is comprised of a five double-sided-layer silicon vertex tracker (SVT) and a 40-layer drift chamber (DCH) in a 1.5 T axial magnetic field. Separation of charged pions, kaons, and protons is achieved using a combination of Cherenkov angles measured in the detector of internally-reflected Cherenkov light (DIRC) and specific-ionization measurements in the SVT and DCH. For the present study we use a kaon identification algorithm that provides 90–95% efficiency, depending on momentum, and pion and proton rejection factors in the 20–100 range. Photon and electron energies are measured in a CsI(Tl) electromagnetic calorimeter (EMC). We use muon identification provided by an instrumented flux return (IFR) to select the final state used for photon efficiency studies.

To study the detector acceptance and efficiency, we use a simulation package developed for radiative processes. The simulation of hadronic final states, including , and , is based on the approach suggested by Czyż and Kühn kuehn2 (). Multiple soft-photon emission from the initial-state charged particles is implemented with a structure-function technique kuraev (); strfun (), and photon radiation from the final-state particles is simulated by the PHOTOS package PHOTOS (). The precision of the radiative corrections is about 1% kuraev (); strfun ().

We simulate the two () final states uniformly in phase space, and also according to models that include the and/or channels. The final state is simulated according to phase space, and also including the channel. The generated events are subjected to a detailed detector simulation GEANT4 (), and we reconstruct them with the same software chain used for the experimental data. Variations in detector and background conditions over the course of the experiment are taken into account.

We also generate a large number of potential background processes, including the ISR reactions , , and , which can contribute due to particle misidentification. We also simulate , , and , which have larger cross sections and can contribute background via missing or spurious tracks or photons. In addition, we study non-ISR backgrounds resulting from generated using JETSET jetset () and from generated using KORALB koralb (). The cross sections for these processes are known to about 10% accuracy or better, which is sufficiently precise for the purposes of the measurements in this paper. The contribution from decays is found to be negligible.

Iii Event Selection and Kinematic Fit

In the selection of candidate events, we consider photon candidates in the EMC with energy above 0.03 , and charged-particle tracks reconstructed in either or both of the DCH and SVT, that extrapolate within 0.25 cm of the collision axis in the transverse plane and within 3 cm of the nominal collision point along this axis. We require a photon with c.m. energy in each event, and either four charged-particle tracks with zero net charge and total momentum roughly (within 0.3 radians) opposite to the photon direction, or two oppositely-charged tracks that combine with other photons to roughly balance the high-energy photon momentum. We assume that the photon with the largest value of is the ISR photon. We fit the set of charged-particle tracks to a common vertex and use this as the point of origin in calculating the photon direction(s). If additional well-reconstructed tracks exist, the nearest four (two) to the interaction region are chosen for the four-track (two-track) analysis. Most events contain additional soft photons due to machine background or interactions in the detector material.

We subject each candidate event to a set of constrained kinematic fits and use the fit results, along with charged-particle identification, both to select the final states of interest and to measure backgrounds from other processes. The kinematic fits use the ISR photon direction and energy along with the four-momenta and covariance matrices of the initial and the set of selected tracks and photons. The ISR photon energy and position are additionally aligned and calibrated using the ISR process, since the two well-identified muons predict precisely the position and energy of the photon. This process is also used to identify and measure data - Monte Carlo (MC) simulation differences in the photon detection efficiency and resolution. The fitted three-momentum for each charged-particle track and the photon are used in further kinematical calculations.

For the four-track event candidates the fits have four constraints (4C). We first fit to the hypothesis, obtaining the chi-squared value . If the four tracks include one identified and one identified , we fit to the hypothesis and retain the event as a candidate. For events with one identified kaon, we perform fits with each of the two oppositely charged tracks given the kaon hypothesis, and the combination with the lower is retained if its value is less than . If the event contains three or four identified , we fit to the hypothesis and retain the event as a candidate with chi-squared value .

For the events with two charged-particle tracks and five or more photon candidates, we require that both tracks be identified as kaons to suppress background from ISR and events. We then pair all non-ISR photon candidates and consider combinations with invariant mass within 30  of the mass PDG () as candidates. We perform a six-constraint (6C) fit to each set of two non-overlapping candidates, the ISR photon, the two charged-particle tracks, and the beam particles. Both candidates are constrained to the mass, and we retain the combination with the lowest chi-squared value, .

Iv The final state

iv.1 Final Selection and Backgrounds

Figure 1: Distribution of from the four-constraint fit for candidates in the data (points). The open histogram is the distribution for simulated signal events, normalized as described in the text. The shaded, cross-hatched, and hatched regions represent, respectively, the background from non-ISR events, from the ISR process, and backgrounds with dominant contribution from mis-identified ISR events. Signal and control regions are indicated.

The distribution in data for the candidates is shown in Fig. 1 (points); the open histogram is the distribution for the simulated events. The distributions are broader than those for a typical 4C distribution due to higher order ISR, and the experimental distribution has contributions from background processes. The simulated distribution is normalized to the data in the region where the contributions of the backgrounds and radiative corrections do not exceed 10%.

The shaded histogram in Fig. 1 represents the background from non-ISR events obtained from the JETSET simulation. It is dominated by events with a hard that results in a fake ISR photon. These events otherwise have kinematics similar to the signal, resulting in the peaking structure at low values of . We evaluate this background in a number of ranges by combining the ISR photon candidate with another photon candidate in both data and simulated events, and comparing the signals in the resulting invariant mass distributions. The simulation gives an -dependence consistent with the data, so we normalize it using an overall factor. The cross-hatched region in Fig. 1 represents events with decays close to the interaction region, and one pion mis-identified as a kaon. The process has similar kinematics to the signal process, and a contribution of about 1% is estimated using the cross section measured in our previous study isrkkpi (). The hatched region represents the contribution from ISR events with one or two misidentified pions; this process contributes mainly at low values. We estimate the contribution as a function of from a simulation using the cross section value and shape from our previous study isr4pi ().

All remaining background sources are either negligible or give a distribution that is nearly uniform over the range shown in Fig. 1. We define the signal region by requiring , and estimate the sum of the remaining backgrounds from the difference between the number of data and simulated entries in the control region, , as shown in Fig. 1. The background contribution to any distribution other than is estimated as the difference between the distributions in the relevant quantity for data and MC events from the control region of Fig. 1, normalized to the difference between the number of data and MC events in the signal region. The non-ISR background is subtracted separately. The signal region contains 85598 data and 63784 simulated events; the control region contains 9684 data and 4315 simulated events.

Figure 2: The invariant mass distribution for candidates in the data (points): the shaded, cross-hatched and hatched regions show, respectively, the non-ISR background from JETSET simulation, the background with a small contribution from the control region of Fig. 1, and the dominant contribution resulting from ISR mis-identified events.

Figure 2 shows the invariant mass distribution from threshold up to 5.0  for events in the signal region. Narrow peaks are apparent at the and masses. The shaded histogram represents the background, which is negligible at low mass but dominates at higher masses. The cross-hatched region represents the background from the channel (which exhibits a peak isrkkpi ()) and from the control region. The hatched region represents the contribution from mis-identified ISR , and is dominant for masses below 3.0 . The total background is 6–8% at low mass, but accounts for 20-25% of the observed distribution near 4 , and increases further for higher masses.

We subtract the sum of backgrounds in each mass interval to obtain the number of signal events. Considering uncertainties in the cross sections for the background processes, the normalization of events in the control region, and the simulation statistics, we estimate a systematic uncertainty on the signal yield that is 2% or less in the 1.6–3.3  mass region, but increases linearly to 10% in the 3.3-5.0  region, and is about 20% for the masses below 1.6 .

iv.2 Selection Efficiency

The selection procedure applied to the data is also applied to the simulated signal samples. The resulting invariant-mass distributions in the signal and control regions are shown in Fig. 3(a) for the uniform phase space simulation. This model reproduces the observed distributions of kaon and pion momenta and polar angles. A broad, smooth mass distribution is chosen to facilitate the estimation of the efficiency as a function of mass. We divide the number of reconstructed simulated events in each mass interval by the number generated in that interval to obtain the efficiency shown by the points in Fig. 3(b). The result of fitting a third-order polynomial to the points is used for further calculations. We simulate events with the ISR photon confined to the angular range 20–160 with respect to the electron beam in the c.m. frame; this angular range is wider than the actual EMC acceptance. The calculated efficiency is for this fiducial region, and includes the acceptance for the final-state hadrons, the inefficiencies of the detector subsystems, and event loss due to additional soft-photon emission.

Figure 3: (a) The invariant mass distributions for MC events that are simulated uniformly in phase space, reconstructed in the signal (open) and control (hatched) regions of Fig. 1; (b) net reconstruction and selection efficiency as a function of mass obtained from this simulation (the curve represents a third-order polynomial fit). The dashed curve is obtained for the final state.

The simulations including the and/or channels give very different mass and angular distributions in the rest frame. However, the angular acceptance is quite uniform for ISR events (see Ref. isr4pi ()), and the efficiencies are within of those from the uniform phase space simulation, as shown by the dashed curve in Fig. 3(b) for the final state.

To study possible mis-modeling of the acceptance, we repeat the analysis with tighter requirements. All charged tracks are required to lie within the DIRC acceptance, radians, and the ISR photon must not appear near the edges of the EMC, radians. The fraction of selected data events satisfying the tighter requirements differs from the simulated ratio by 1.5%. We take the sum in quadrature of this variation and the 1% model variation (2% total) as the systematic uncertainty due to acceptance and model dependence.

(GeV) (nb) (GeV) (nb) (GeV) (nb) (GeV) (nb)
1.4125 0.000 0.004 2.3125 1.531 0.056 3.2125 0.357 0.025 4.1125 0.082 0.011
1.4375 0.009 0.008 2.3375 1.586 0.056 3.2375 0.328 0.023 4.1375 0.078 0.011
1.4625 0.018 0.008 2.3625 1.496 0.055 3.2625 0.339 0.023 4.1625 0.065 0.010
1.4875 0.014 0.010 2.3875 1.574 0.055 3.2875 0.304 0.022 4.1875 0.079 0.010
1.5125 0.075 0.017 2.4125 1.427 0.053 3.3125 0.292 0.022 4.2125 0.082 0.011
1.5375 0.078 0.018 2.4375 1.407 0.052 3.3375 0.295 0.021 4.2375 0.065 0.010
1.5625 0.135 0.022 2.4625 1.353 0.051 3.3625 0.257 0.020 4.2625 0.071 0.009
1.5875 0.297 0.030 2.4875 1.221 0.048 3.3875 0.242 0.020 4.2875 0.075 0.010
1.6125 0.550 0.040 2.5125 1.203 0.047 3.4125 0.245 0.020 4.3125 0.076 0.010
1.6375 0.975 0.053 2.5375 1.020 0.044 3.4375 0.199 0.018 4.3375 0.061 0.009
1.6625 1.363 0.061 2.5625 0.991 0.043 3.4625 0.254 0.019 4.3625 0.060 0.009
1.6875 1.808 0.069 2.5875 0.986 0.043 3.4875 0.212 0.019 4.3875 0.068 0.009
1.7125 2.291 0.078 2.6125 0.837 0.040 3.5125 0.265 0.020 4.4125 0.041 0.008
1.7375 2.500 0.083 2.6375 0.925 0.041 3.5375 0.176 0.018 4.4375 0.062 0.009
1.7625 3.376 0.094 2.6625 0.886 0.040 3.5625 0.186 0.017 4.4625 0.065 0.009
1.7875 3.879 0.099 2.6875 0.839 0.038 3.5875 0.190 0.018 4.4875 0.053 0.008
1.8125 4.160 0.101 2.7125 0.902 0.039 3.6125 0.170 0.016 4.5125 0.047 0.008
1.8375 4.401 0.103 2.7375 0.768 0.037 3.6375 0.173 0.016 4.5375 0.055 0.008
1.8625 4.630 0.105 2.7625 0.831 0.038 3.6625 0.195 0.017 4.5625 0.041 0.007
1.8875 4.219 0.101 2.7875 0.752 0.036 3.6875 0.272 0.019 4.5875 0.028 0.008
1.9125 4.016 0.098 2.8125 0.689 0.034 3.7125 0.161 0.016 4.6125 0.050 0.007
1.9375 4.199 0.099 2.8375 0.644 0.033 3.7375 0.147 0.015 4.6375 0.033 0.007
1.9625 3.942 0.095 2.8625 0.555 0.031 3.7625 0.156 0.015 4.6625 0.052 0.008
1.9875 3.611 0.091 2.8875 0.559 0.031 3.7875 0.133 0.015 4.6875 0.043 0.006
2.0125 3.403 0.088 2.9125 0.543 0.030 3.8125 0.143 0.015 4.7125 0.039 0.006
2.0375 3.112 0.085 2.9375 0.550 0.030 3.8375 0.112 0.013 4.7375 0.027 0.006
2.0625 3.249 0.085 2.9625 0.508 0.030 3.8625 0.121 0.015 4.7625 0.032 0.006
2.0875 3.165 0.083 2.9875 0.549 0.030 3.8875 0.135 0.014 4.7875 0.035 0.006
2.1125 3.036 0.080 3.0125 0.468 0.028 3.9125 0.126 0.013 4.8125 0.019 0.006
2.1375 2.743 0.077 3.0375 0.461 0.027 3.9375 0.114 0.013 4.8375 0.022 0.006
2.1625 2.499 0.073 3.0625 0.476 0.028 3.9625 0.130 0.013 4.8625 0.028 0.006
2.1875 2.351 0.070 3.0875 3.057 0.065 3.9875 0.099 0.012 4.8875 0.028 0.005
2.2125 1.785 0.062 3.1125 1.561 0.048 4.0125 0.117 0.013 4.9125 0.030 0.005
2.2375 1.833 0.061 3.1375 0.449 0.028 4.0375 0.075 0.011 4.9375 0.028 0.005
2.2625 1.641 0.059 3.1625 0.455 0.027 4.0625 0.090 0.011 4.9625 0.030 0.005
2.2875 1.762 0.059 3.1875 0.385 0.025 4.0875 0.099 0.012 4.9875 0.037 0.005
Table 1: Summary of the cross section measurements for . Errors are statistical only.
Figure 4: The cross section as a function of c.m. energy measured with ISR data at BABAR (dots). The direct measurements from DM1 2k2pidm1 () are shown as the open circles. Only statistical errors are shown.

Our data sample contains about 3000 events in the peak. Comparing this number with and without selection on we find less than a 1% difference between data and MC simulation due to mis-modeling of the shape of the distribution. This value is taken as an estimate of the systematic uncertainty associated with the selection criterion. To measure tracking efficiency, we consider data and simulated events that contain a high-energy photon and exactly three charged-particle tracks, which satisfy a set of kinematical criteria, including a good from a kinematic fit to the hypothesis, assuming one missing pion track in the event. We find that the simulated track-finding efficiency is overestimated by per track, so we apply a correction of to the signal yield.

The kaon identification efficiency is studied in BABAR  using many different test processes (e.g.  ), and we conservatively estimate a systematic uncertainty of % per kaon due to data-MC differences in our kaon momentum range.

The data-MC simulation correction due to ISR photon detection efficiency was studied with a sample of events and was found to be .

Source Correction Uncertainty
Rad. Corrections 1%
Backgrounds 2% ,
2-10 % ,
Model Acceptance 2%
Distribution 1%
Tracking Efficiency +3% 1%
Kaon ID Efficiency 2%
Photon Efficiency +1.0% 0.5%
ISR Luminosity 1%
Total +4.0% 4% ,
4-11% ,
Table 2: Summary of corrections and systematic uncertainties for the cross section measurements. The total correction is the linear sum of the contributions, and the total uncertainty is obtained by summing the individual uncertainties in quadrature.

iv.3 Cross Section for

We calculate the cross section as a function of the effective c.m. energy from


where with the measured invariant mass of the system, the number of selected events after background subtraction in the interval , the corrected detection efficiency, and a radiative correction.

We calculate the differential luminosity in each interval , with the photon in the same fiducial range as that used for the simulation, using the simple leading order (LO) formula described in Ref. isr3pi (). From the mass spectra, obtained from the MC simulation with and without extra-soft-photon (ISR and FSR) radiation, we extract , which gives a correction less than 1%. Our data, calculated according to Eq. 2, include vacuum polarization (VP) and exclude any radiative effects, as is conventional for the reporting of cross sections. Note that VP should be excluded and FSR included for calculations of . From data-simulation comparisons for the events we estimate a systematic uncertainty on of 1% isr2pi ().

We show the cross section as a function of in Fig. 4 with statistical errors only in comparison with the direct measurements from DM1 2k2pidm1 (), and list our results in Table 1. The results are consistent with our previous measurements for this reaction isr4pi (); isr2k2pi (), but have increased statistical precision. Our data lie systematically below the DM1 data for above 1.9 . The systematic uncertainties, summarized in Table 2, affect the normalization, but have little effect on the energy dependence.

The cross section rises from threshold to a peak value of about 4.6 nb near 1.86 , then generally decreases with increasing energy. In addition to narrow peaks at the and mass values, there are several possible wider structures in the 1.8–2.8  region. Such structures might be due to thresholds for intermediate resonant states, such as near 2 . Gaussian fits to the distributions of the mass difference between generated and reconstructed MC data yield mass resolution values that vary from 4.2  in the 1.5–2.5  region to 5.5  in the 2.5–3.5  region. The resolution functions are not purely Gaussian due to soft-photon radiation, but less than 10% of the signal is outside the 0.025  mass interval used in Fig. 4. Since the cross section has no sharp structure other than the and peaks discussed in Sec. IX below, we apply no correction for mass resolution.

iv.4 Substructures in the Final State

Our previous study isr4pi (); isr2k2pi () showed evidence for many intermediate resonances in the final state. With the larger data sample used here, these can be seen more clearly and, in some cases, studied in detail. Figure 5(a) shows a plot of the invariant mass of the pair versus that of the pair. Signal for the is clearly visible. Figure 5(b) shows the mass distribution (two entries per event) for all selected events. As we show in our previous study isr2k2pi (), the signal at about 1400  has parameters consistent with . Therefore, we perform a fit to this distribution using P- and D-wave Breit-Wigner (BW) functions for the and signals, respectively, and a third-order polynomial function for the remainder of the distribution, taking into account the threshold. The fit result is shown by the curves in Fig. 5(b). The fit yields a signal of events with   and  , and a signal of events with   and  . These values are consistent with current world averages for and  PDG () , and the fit describes the data well, indicating that contributions from other resonances decaying into , like and/or , are small.

Figure 5: (a) Invariant mass of the pair versus that of the pair; (b) The mass distribution (two entries per event) for all selected events: the solid line represents a fit including two resonances and a polynomial background function, which is shown as the hatched region.

We combine candidates within the lines in Fig. 5(a) with the remaining pion and kaon to obtain the invariant mass distribution shown in Fig. 6(b), and the versus  mass plot in Fig. 6(a). The bulk of Fig. 6(a) shows a strong positive correlation, characteristic of final states with no higher resonances. The horizontal bands in Fig. 6(a) correspond to the peak regions of the projection plot of Fig. 6(b) and are consistent with the contribution from the and resonances. There is also an indication of a vertical band in Fig. 6(a), perhaps corresponding to a structure at 1.5 . The projection plot of Fig. 6(c) for events with   shows the enhancement not consistent with phase space behavior.

Figure 6: (a) Invariant mass of the system versus that of the system; (b) the projection plot of (a); (c) the projection plot of (a) for  .

We next suppress the contribution by considering only events outside the lines in Fig. 5(a). In Fig. 7(a) the invariant mass (two entries per event) shows evidence of the and resonances, both of which decay into , although the latter decay is very weak PDG (). In Fig. 7(b) we plot the invariant mass for events with  . There is a strong signal, and there are indications of additional structures in the and regions.

Figure 7: (a) The invariant mass of the combinations with events excluded; (b) the invariant mass for events from (a) with the region suppressed by requiring   as shown by vertical line in (a).

The separation of all these, and any other, intermediate states involving relatively broad resonances requires a partial wave analysis. This is beyond the scope of this paper. Instead we present the cross sections for the sum of all states that include , or signals, and study intermediate states that include a narrow or resonance.

iv.5 The , and Cross Sections

Signals for the and are clearly visible in the mass distributions in Fig. 5(a,b). To extract the number of events with correlated production of and , we perform the same fit as that shown in Fig. 5(b), but to the invariant mass distribution in each 0.04  interval of invariant mass. From each fit we obtain the number of and events and plot these values as a function of mass in Fig. 8(a) and Fig. 8(b), respectively. The fit to the data of Fig. 8(a) indicates that only events are associated with correlated production (about 1% of the total number of events), and that events correspond to pairs, compared to , the total number of events with a in the final state. The distribution of the events from the peak shows a strong signal at the mass in Fig. 8(b), which contains events, in agreement with the number of pairs obtained above.

Figure 8: The invariant mass distribution corresponding to the number of (a) and (b) events obtained from the fits to the invariant mass distribution for each interval of mass. The curves result from the fits described in the text.
Figure 9: The cross section, obtained from the signal of Fig. 5(b).
Figure 10: The cross section, obtained from the signal of Fig. 5(b).
Figure 11: (a) The mass distribution for all selected events with the and regions excluded: the solid curve represents a fit as described in the text, and the background contribution is shown separately as the hatched region; (b) the cross section obtained from the signal from the fit in each 0.025  c.m. energy interval.
(GeV) (nb) (GeV) (nb) (GeV) (nb) (GeV) (nb)
1.5875 0.00 0.00 2.1875 1.40 0.09 2.7875 0.38 0.03 3.3875 0.11 0.02
1.6125 0.19 0.04 2.2125 1.26 0.08 2.8125 0.33 0.03 3.4125 0.16 0.02
1.6375 0.48 0.07 2.2375 1.17 0.08 2.8375 0.39 0.03 3.4375 0.12 0.02
1.6625 1.01 0.08 2.2625 0.96 0.07 2.8625 0.24 0.03 3.4625 0.15 0.02
1.6875 1.29 0.10 2.2875 1.14 0.07 2.8875 0.32 0.03 3.4875 0.13 0.02
1.7125 1.58 0.11 2.3125 0.90 0.07 2.9125 0.24 0.03 3.5125 0.15 0.02
1.7375 1.82 0.11 2.3375 0.98 0.07 2.9375 0.30 0.03 3.5375 0.08 0.01
1.7625 2.24 0.13 2.3625 0.90 0.06 2.9625 0.33 0.03 3.5625 0.12 0.01
1.7875 2.75 0.15 2.3875 0.85 0.06 2.9875 0.31 0.03 3.5875 0.12 0.01
1.8125 3.61 0.16 2.4125 0.85 0.06 3.0125 0.26 0.03 3.6125 0.09 0.01
1.8375 4.22 0.17 2.4375 0.83 0.06 3.0375 0.26 0.03 3.6375 0.12 0.02
1.8625 4.01 0.17 2.4625 0.86 0.06 3.0625 0.25 0.02 3.6625 0.09 0.01
1.8875 3.52 0.15 2.4875 0.83 0.05 3.0875 1.84 0.06 3.6875 0.15 0.02
1.9125 3.78 0.15 2.5125 0.63 0.05 3.1125 0.96 0.05 3.7125 0.08 0.01
1.9375 3.82 0.16 2.5375 0.58 0.05 3.1375 0.24 0.02 3.7375 0.07 0.01
1.9625 3.40 0.15 2.5625 0.60 0.04 3.1625 0.22 0.02 3.7625 0.11 0.01
1.9875 2.98 0.14 2.5875 0.55 0.04 3.1875 0.19 0.02 3.7875 0.09 0.01
2.0125 2.69 0.13 2.6125 0.55 0.04 3.2125 0.18 0.02 3.8125 0.09 0.01
2.0375 2.17 0.11 2.6375 0.52 0.04 3.2375 0.19 0.02 3.8375 0.06 0.01
2.0625 2.27 0.12 2.6625 0.48 0.04 3.2625 0.19 0.02 3.8625 0.06 0.01
2.0875 1.91 0.11 2.6875 0.41 0.04 3.2875 0.18 0.02 3.8875 0.08 0.01
2.1125 2.02 0.11 2.7125 0.57 0.04 3.3125 0.17 0.02 3.9125 0.05 0.01
2.1375 1.84 0.10 2.7375 0.47 0.04 3.3375 0.19 0.02 3.9375 0.06 0.01
2.1625 1.49 0.10 2.7625 0.46 0.04 3.3625 0.16 0.02 3.9625 0.06 0.01
Table 3: Summary of the cross section measurements for . Errors are statistical only.

We perform a fit similar to that shown in Fig. 5(b) to the data in intervals of invariant mass, with the resonance masses and widths fixed to the values obtained from the overall fit. Since correlated production is small, we convert the resulting yield in each interval into a cross section value for or , 1 following the procedure described in Sec. IV.3. These cross section values take into account only the decay of the and the .

Note that the () cross section includes a small contribution from the () channel, because the final state has not been taken into account. These cross sections are shown in Fig. 9 and Fig. 10, and the channel is listed in Table 3 for energies from threshold up to 4.0 . At higher energies the signals are small and contain an unknown, but possibly large, contribution from events. There is a rapid rise from threshold to a peak value of about 4 nb at 1.84  for the cross section, followed by a very rapid decrease with increasing energy. There are suggestions of narrow structures in the peak region, but the only statistically significant structure is the peak, which is discussed below. There are some structures in the cross section, but the signal size is too small to make any definite statement.