The e^{+}e^{-}\rightarrow K^{+}K^{-}\pi^{+}\pi^{-}, K^{+}K^{-}\pi^{0}\pi^{0} and K^{+}K^{-}K^{+}K^{-} Cross Sections Measured with Initial-State Radiation

The , and Cross Sections Measured with Initial-State Radiation

Abstract

We study the processes , and , where the photon is radiated from the initial state. About 34600, 4400 and 2300 fully reconstructed events, respectively, are selected from 232  of BABAR data. The invariant mass of the hadronic final state defines the effective 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. Studying the structure of these events, we find contributions from a number of intermediate states, and we extract their cross sections where possible. In particular, we isolate the contribution from and study its structure near threshold. In the charmonium region, we observe the in all three final states and several intermediate states, as well as the in some modes, and measure the corresponding branching fractions. We see no signal for the and obtain an upper limit of at 90% C.L.

pacs:
13.66.Bc, 14.40.Cs, 13.25.Gv, 13.25.Jx, 13.20.Jf

BABAR-PUB-07/021

SLAC-PUB-12435

Phys. Rev. D76, 012008 (2007)

1

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 energies, annihilation is dominated by quark-level processes producing two or more hadronic jets. However, low-multiplicity exclusive processes dominate at energies below about 2 , and the region near charm threshold, 3.0–4.5 , features a number of resonances PDG (). These allow us to probe a wealth of physics parameters, including cross sections, spectroscopy and form factors.

Of particular current interest are the recently observed states in the charmonium region, such as the  y4260 (), and a possible discrepancy between the measured value of the anomalous magnetic moment of the muon, , and that predicted by the Standard Model dehz (). Charmonium and other states with can be observed as resonances in the cross section, and intermediate states may be present in the hadronic system. Measurements of the decay modes and their branching fractions are important in understanding the nature of these states. For example, the glue-ball model shin () predicts a large branching fraction for into . The prediction for is based on hadronic-loop corrections measured from low-energy hadrons data, and these dominate the uncertainty on the prediction. Improving this prediction requires not only more precise measurements, but also measurements over the entire energy range and inclusion of all the important subprocesses in order to understand possible acceptance effects. ISR events at factories provide independent and contiguous measurements of hadronic cross sections from the production threshold to about 5 .

The cross section for the radiation of a photon of energy followed by the production of a particular hadronic final state is related to the corresponding direct cross section by

(1)

where is the initial c.m. energy, is the fractional energy of 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 combines with the increasing to produce a sizable cross section at very low . The angular distribution of the ISR photon peaks along the beam directions, but 10–15% ivanch () of the photons are within a typical detector acceptance.

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 simultaneously in one experiment, so that no structure is missed and the relative normalization uncertainties in data from different experiments or accelerator parameters are avoided. Furthermore, for large values of the hadronic system is collimated, reducing acceptance issues and allowing measurements at energies down to production threshold. The mass resolution is not as good as a typical beam energy spread used in direct measurements, but the resolution and absolute energy scale can be monitored by 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 reported Druzhinin1 (); isr3pi (); isr4pi (); isr6pi (), demonstrating the viability of such measurements.

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 . We recently reported phif0prd () an updated measurement of the final state with a larger data sample, along with the first measurement of the final state, in which we observed a structure near threshold in the intermediate state. 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 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 events and measure cross sections for a number of intermediate states. We study the charmonium region, measure several and branching fractions, 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. The total integrated luminosity used is 232 , which includes 211  collected at the peak, , and 21  collected below the resonance, at .

The BABAR detector is described elsewhere babar (). Here we use charged particles reconstructed in the tracking system, which comprises the five-layer silicon vertex tracker (SVT) and the 40-layer drift chamber (DCH) in a 1.5 T axial magnetic field. Separation of charged pions, kaons and protons uses a combination of Cherenkov angles measured in the detector of internally reflected Cherenkov light (DIRC) and specific ionization measured 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 the CsI(Tl) electromagnetic calorimeter (EMC). We use muon identification provided by the instrumented flux return (IFR) to select the final state.

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 Kühnkuehn2 (). 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 accuracy of the radiative corrections is about 1%.

We simulate the final states both according to phase space and with models that include the and/or channels, and the final state both according to phase space and including the channel. The generated events go through a detailed detector simulation GEANT4 (), and we reconstruct them with the same software chain as the experimental data. Variations in detector and background conditions are taken into account.

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

Iii Event Selection and Kinematic Fit

In the initial selection of candidate events, we consider photon candidates in the EMC with energy above 0.03  and charged tracks reconstructed in the DCH or SVT or both that extrapolate within 0.25 cm of the beam axis in the transverse plane and within 3 cm of the nominal collision point along the axis. These criteria are looser than in our previous analysis isr4pi (), and have been chosen to maximize efficiency. We require a high-energy photon in the event with an energy in the initial c.m. frame of , and either exactly four charged tracks with zero net charge and total momentum roughly opposite to the photon direction, or exactly two oppositely charged tracks that combine with a set of other photons to roughly balance the highest-energy photon momentum. We fit a vertex to the set of charged tracks and use it as the point of origin to calculate the photon direction. Most events contain additional soft photons due to machine background or interactions in the detector material.

We subject each of these candidate events 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. We assume the photon with the highest in the c.m. frame is the ISR photon, and the kinematic fits use its direction along with the four-momenta and covariance matrices of the initial and the set of selected tracks and photons. Because of excellent resolution for the momenta in the DCH and good angular resolution for the photons in the EMC, the ISR photon energy is determined with better resolution through four-momentum conservation than through measurement in the EMC. Therefore we do not use its measured energy in the fits, eliminating the systematic uncertainty due to the EMC calibration for high energy photons. The fitted three-momenta for each charged track and photon are used in further kinematical calculations.

For the four-track candidates, the fits have three constraints (3C). We first fit to the hypothesis, obtaining a . If the four tracks include one identified and one , 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 it is lower than . If the event contains three or four identified , we fit to the hypothesis and retain the event as a candidate.

For the events with two charged tracks and five or more photon candidates, we require both tracks to 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 as candidates. We perform a six-constraint (6C) fit to each set of two non-overlapping candidates plus the ISR photon direction, the two tracks and the beam particles. Both candidates are constrained to the mass, and we retain the combination with the lowest .

Iv The final state

iv.1 Final Selection and Backgrounds

Figure 1: Distribution of from the three-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 cross-hatched (hatched) histogram represents the background from non-ISR events (plus that from ISR events), estimated as described in the text.

The experimental distribution for the candidates is shown in Fig. 1 as points, and the open histogram is the distribution for the simulated events. The simulated distribution is normalized to the data in the region where the backgrounds and radiative corrections are insignificant. The experimental distribution has contributions from background processes, but the simulated distribution is also broader than the expected 3C distribution. This is due to multiple soft-photon emission from the initial state and radiation from the final-state charged particles, which are not taken into account by the fit, but are present in both data and simulation. The shape of the distribution at high values was studied in detail  isr4pi (); isr6pi () using specific ISR processes for which a very clean sample can be obtained without any limit on the value.

The cross-hatched histogram in Fig. 1 represents the background from events, which is based on the JETSET simulation. It is dominated by events with a hard producing a fake ISR photon, and the similar kinematics cause it to peak 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 by an overall factor. The hatched histogram represents the sum of this background and that from ISR events with one or two misidentified , which also contributes at low values. We estimate the contribution as a function of from a simulation using the known cross section isr4pi ().

All remaining background sources are either negligible or give a distribution that is nearly uniform over the range shown in Fig. 1. We therefore define a signal region , and estimate the sum of the remaining backgrounds from the difference between the number of data and simulated entries in a control region, . This difference is normalized to the corresponding difference in the signal region, as described in detail in Refs. isr4pi (); isr6pi (). The signal region contains 34635 data and 14077 simulated events, and the control region contains 4634 data and 723 simulated events.

Figure 2: The invariant mass distribution for candidates in the data (points): the cross-hatched, hatched and open histograms represent, cumulatively, the non-ISR background, the contribution from ISR events, and the ISR background from the control region of Fig. 1.

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 cross-hatched histogram represents the background, which is negligible at low mass but becomes large at higher masses. The hatched region represents the ISR contribution, which we estimate to be 2.4% of the selected events on average. The open histogram represents the sum of all backgrounds, including those estimated from the control region. They total 6–8% at low mass but account for 20-25% of the observed data near 4 and become the largest contribution near 5 .

We subtract the sum of backgrounds in each mass bin to obtain a 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 less than 3% in the 1.6–3  mass region, but increases to 3–5% in the region above 3 .

iv.2 Selection Efficiency

The selection procedures applied to the data are 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 phase space simulation. The broad, smooth mass distribution is chosen to facilitate the estimation of the efficiency as a function of mass, and this model reproduces the observed distributions of kaon and pion momenta and polar angles. We divide the number of reconstructed simulated events in each mass interval by the number generated in that interval to obtain the efficiency shown as the points in Fig. 3(b). The 3 order polynomial fit 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, which is about 30% wider than the EMC acceptance. This efficiency is for this fiducial region, but 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 simulated events in the phase space model, 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 order polynomial fit).

The simulations including the and/or channels have very different mass and angular distributions in the rest frame. However, the angular acceptance is quite uniform for ISR events, and the efficiencies are consistent with those from the phase space simulation within 3%. To study possible mis-modeling of the acceptance, we repeat the analysis with the tighter requirements that all charged tracks be within the DIRC acceptance, radians, and the ISR photon be well away from the edges of the EMC, radians. The fraction of selected data events satisfying the tighter requirements differs from the simulated ratio by 3.7%. We conservatively take the sum in quadrature of this variation and the 3% model variation (5% total) as a systematic uncertainty due to acceptance and model dependence.

(GeV) (nb) (GeV) (nb) (GeV) (nb) (GeV) (nb) (GeV) (nb)
1.4125 0.00 0.02 2.1375 2.83 0.13 2.8625 0.50 0.05 3.5875 0.12 0.03 4.3125 0.04 0.02
1.4375 0.01 0.02 2.1625 2.71 0.12 2.8875 0.51 0.05 3.6125 0.13 0.03 4.3375 0.04 0.02
1.4625 0.00 0.02 2.1875 2.46 0.12 2.9125 0.54 0.05 3.6375 0.12 0.03 4.3625 0.03 0.02
1.4875 0.04 0.02 2.2125 1.84 0.10 2.9375 0.46 0.05 3.6625 0.11 0.03 4.3875 0.06 0.02
1.5125 0.03 0.02 2.2375 1.66 0.10 2.9625 0.45 0.05 3.6875 0.25 0.03 4.4125 0.01 0.02
1.5375 0.11 0.03 2.2625 1.59 0.09 2.9875 0.46 0.05 3.7125 0.07 0.03 4.4375 0.03 0.02
1.5625 0.15 0.04 2.2875 1.66 0.09 3.0125 0.36 0.04 3.7375 0.08 0.02 4.4625 0.06 0.02
1.5875 0.32 0.05 2.3125 1.50 0.09 3.0375 0.39 0.04 3.7625 0.11 0.03 4.4875 0.03 0.02
1.6125 0.48 0.06 2.3375 1.65 0.09 3.0625 0.31 0.04 3.7875 0.11 0.03 4.5125 0.04 0.02
1.6375 0.85 0.08 2.3625 1.56 0.09 3.0875 2.95 0.10 3.8125 0.10 0.03 4.5375 0.01 0.02
1.6625 1.42 0.10 2.3875 1.49 0.09 3.1125 1.51 0.08 3.8375 0.08 0.02 4.5625 0.02 0.02
1.6875 1.86 0.11 2.4125 1.46 0.09 3.1375 0.37 0.04 3.8625 0.12 0.03 4.5875 0.05 0.02
1.7125 2.36 0.13 2.4375 1.48 0.09 3.1625 0.35 0.04 3.8875 0.09 0.02 4.6125 0.02 0.02
1.7375 2.67 0.13 2.4625 1.17 0.08 3.1875 0.28 0.04 3.9125 0.09 0.02 4.6375 0.01 0.02
1.7625 3.51 0.15 2.4875 1.16 0.08 3.2125 0.35 0.04 3.9375 0.08 0.02 4.6625 0.04 0.02
1.7875 3.98 0.16 2.5125 1.21 0.08 3.2375 0.31 0.04 3.9625 0.10 0.02 4.6875 0.02 0.02
1.8125 4.10 0.16 2.5375 0.94 0.07 3.2625 0.30 0.04 3.9875 0.04 0.02 4.7125 0.03 0.02
1.8375 4.68 0.17 2.5625 0.95 0.07 3.2875 0.24 0.04 4.0125 0.06 0.02 4.7375 0.01 0.02
1.8625 4.49 0.17 2.5875 0.84 0.07 3.3125 0.22 0.04 4.0375 0.07 0.02 4.7625 0.02 0.02
1.8875 4.26 0.17 2.6125 0.85 0.07 3.3375 0.25 0.04 4.0625 0.05 0.02 4.7875 0.01 0.02
1.9125 4.30 0.16 2.6375 0.90 0.07 3.3625 0.16 0.03 4.0875 0.06 0.02 4.8125 0.00 0.02
1.9375 4.20 0.16 2.6625 0.82 0.06 3.3875 0.17 0.03 4.1125 0.06 0.02 4.8375 0.02 0.02
1.9625 4.13 0.16 2.6875 0.70 0.06 3.4125 0.18 0.03 4.1375 0.05 0.02 4.8625 0.00 0.02
1.9875 3.74 0.15 2.7125 0.86 0.06 3.4375 0.12 0.03 4.1625 0.06 0.02 4.8875 0.04 0.02
2.0125 3.45 0.15 2.7375 0.81 0.06 3.4625 0.17 0.03 4.1875 0.05 0.02 4.9125 0.05 0.02
2.0375 3.38 0.14 2.7625 0.76 0.06 3.4875 0.17 0.03 4.2125 0.05 0.02 4.9375 0.02 0.02
2.0625 3.17 0.14 2.7875 0.73 0.06 3.5125 0.21 0.03 4.2375 0.08 0.02 4.9625 0.00 0.02
2.0875 3.23 0.14 2.8125 0.64 0.05 3.5375 0.14 0.03 4.2625 0.04 0.02 4.9875 0.04 0.02
2.1125 3.15 0.14 2.8375 0.56 0.05 3.5625 0.16 0.03 4.2875 0.08 0.02
Table 1: Measurements of the cross section (errors are statistical only).
Figure 4: The cross section as a function of the effective 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.

We correct for mis-modeling of the shape of the distribution by ()% and the track finding efficiency following the procedures described in detail in Ref. isr4pi (). We use a comparison of data and simulated distributions in the much larger samples of ISR events. We consider data and simulated events that contain a high-energy photon plus exactly three charged tracks and satisfy a set of kinematical criteria, including a good from a kinematic fit under the hypothesis that there is exactly one missing 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.

We correct the simulated kaon identification efficiency using events. Events with a hard ISR photon and two charged tracks, one of which is identified as a kaon, with a invariant mass near the mass provide a very clean sample, and we compare the fractions of data and simulated events with the other track also identified as a kaon, as a function of momentum. The data-simulation efficiency ratio averages in the 1–5  momentum range with variations at the 0.01 level. We conservatively apply a correction of % per kaon, or % to the signal yield.

Source Correction Uncertainty
Rad. Corrections
Backgrounds ,
,
Model Dependence
Distn.
Tracking Efficiency
Kaon ID Efficiency
ISR Luminosity
Total ,
,
Table 2: Summary of corrections and systematic uncertainties on the cross section. The total correction is the linear sum of the components and the total uncertainty is the sum in quadrature.

iv.3 Cross Section for

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

(2)

where , is the measured invariant mass of the system, is the number of selected events after background subtraction in the interval , and is the corrected detection efficiency. We calculate the differential luminosity, , in each interval from ISR events with the photon in the same fiducial range used for the simulation; the procedure is described in Refs. isr4pi (); isr6pi (). From data-simulation comparison we conservatively estimate a systematic uncertainty on of 3%. This has been corrected for vacuum polarization and final-state soft-photon emission; the former should be excluded when using these data in calculations of .

For the cross section measurement we use the tighter angular criteria on the charged tracks and the ISR photon, discussed in Sec. IV.2, to exclude possible errors from incorrect simulation of the EMC and DCH edge effects. We show the cross section as a function of in Fig. 4, with statistical errors only, and provide a list of our results in Table 1. The result is consistent with the direct measurement by DM1 2k2pidm1 (), and with our previous measurement of this channel isr4pi () but has much better statistical precision. 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 4.7 nb near 1.85 , then generally decreases with increasing energy. In addition to narrow peaks at the and masses, 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 simulated line shapes give a resolution on the measured mass that varies between 4.2  in the 1.5–2.5  region and 5.5  in the 2.5–3.5  region. The resolution function is not purely Gaussian due to soft-photon radiation, but less than 10% of the signal is outside the 25  mass bin. Since the cross section has no sharp structure other than the and peaks discussed in Sec. VIII below, we apply no correction for resolution.

iv.4 Substructure in the Final State

Our previous study isr4pi () showed many intermediate resonances in the final state. With the larger data sample used here, they can be seen more clearly and, in some cases, studied in detail. Figure 5(a) shows a scatter plot of the invariant mass of the pair versus that of the pair, and Fig. 5(b) shows the sum of the two projections. Here we have suppressed the contributions from and by requiring   and  , where and values are taken from the Particle Data Group (PDG) tables PDG (). Bands and peaks corresponding to the and are visible. In Fig. 5(c) we show the sum of projections of the bands, defined by lines in Fig. 5(a), with events in the overlap region plotted only once. No signal is seen, confirming that the cross section is small. We observe associated production, but it is mostly from decays (see Sec. VIII).

Figure 5: (a) Invariant mass of the pair versus that of the pair; (b) sum of projections of (a); (c) sum of projections of the bands of (a), with events in the overlap region taken only once. The and are vetoed.

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(a), and the vs.  mass scatter plot in Fig. 6(b). The bulk of Fig. 6(b) shows a strong positive correlation, characteristic of final states with no higher resonances. The horizontal band in Fig. 6(b) corresponds to the peak region in Fig. 6(a), and is consistent with contributions from the and resonances. There is also an indication of a vertical band in Fig. 6(b), perhaps corresponding to a resonance at 1.5 .

Figure 6: (a) The invariant mass distribution; (b) the mass versus mass.

We now suppress by considering only events outside the lines in Fig. 5(a). In Fig. 7 we show the invariant mass (two entries per event) vs. that of the pair, along with its two projections. There is a strong signal, and the mass projection shows further indications of the and resonances, both of which decay into . There are suggestions of additional structure in the mass distribution, including possible shoulder and a possible enhancement near the , however the current statistics do not allow us to make definitive statements.

Figure 7: (a) Invariant mass of the combinations versus that of the pair; (b) the and (c) mass projections of (a).

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

iv.5 The Cross Section

Signals for the and are clearly visible in the mass distributions in Fig. 5(b) and, with a different bin size, in Fig. 8(a). We perform a fit to this distribution using P-wave Breit-Wigner (BW) functions for the and signals and a third-order polynomial function for the remainder of the distribution taking into account the threshold. The result is shown in Fig. 8(a). The fit yields a signal of events with   and  , and a signal of events with   and  . These values are consistent with current world averages PDG (), and the fit describes the data well, indicating that contributions from any other resonances decaying into are small.

Figure 8: (a) The mass distribution (two entries per event) for all selected events: the solid line represents a fit including two resonances and a polynomial function (see text), shown separately as the dashed line; (b) the cross section obtained from the signal by a similar fit in each 25  mass bin.
(GeV) (nb) (GeV) (nb) (GeV) (nb) (GeV) (nb)
1.5875 0.16 0.11 2.0875 2.36 0.16 2.5875 0.54 0.07 3.0875 1.73 0.10
1.6125 0.31 0.08 2.1125 1.92 0.16 2.6125 0.63 0.06 3.1125 0.92 0.07
1.6375 0.81 0.13 2.1375 1.99 0.14 2.6375 0.57 0.06 3.1375 0.21 0.04
1.6625 0.79 0.12 2.1625 1.19 0.15 2.6625 0.46 0.06 3.1625 0.24 0.04
1.6875 1.33 0.15 2.1875 1.24 0.14 2.6875 0.46 0.06 3.1875 0.08 0.03
1.7125 1.63 0.15 2.2125 1.25 0.11 2.7125 0.64 0.06 3.2125 0.15 0.03
1.7375 1.87 0.14 2.2375 0.90 0.10 2.7375 0.56 0.06 3.2375 0.14 0.04
1.7625 2.12 0.17 2.2625 0.79 0.11 2.7625 0.46 0.06 3.2625 0.16 0.03
1.7875 2.51 0.20 2.2875 1.15 0.10 2.7875 0.36 0.06 3.2875 0.13 0.03
1.8125 2.96 0.21 2.3125 0.99 0.09 2.8125 0.31 0.05 3.3125 0.12 0.03
1.8375 4.35 0.20 2.3375 0.91 0.11 2.8375 0.35 0.05 3.3375 0.14 0.03
1.8625 4.11 0.20 2.3625 1.11 0.09 2.8625 0.27 0.04 3.3625 0.12 0.06
1.8875 3.26 0.23 2.3875 0.83 0.09 2.8875 0.27 0.05 3.3875 0.09 0.03
1.9125 3.90 0.20 2.4125 0.87 0.09 2.9125 0.34 0.05 3.4125 0.10 0.03
1.9375 3.53 0.20 2.4375 1.00 0.09 2.9375 0.29 0.04 3.4375 0.11 0.03
1.9625 3.42 0.21 2.4625 0.86 0.08 2.9625 0.25 0.04 3.4625 0.10 0.05
1.9875 2.81 0.18 2.4875 0.88 0.09 2.9875 0.38 0.05 3.4875 0.08 0.03
2.0125 2.47 0.17 2.5125 0.69 0.07 3.0125 0.21 0.04
2.0375 2.26 0.16 2.5375 0.62 0.07 3.0375 0.24 0.04
2.0625 2.00 0.16 2.5625 0.55 0.07 3.0625 0.22 0.04
Table 3: Measurements of the cross section (errors are statistical only).

We perform a similar fit to the data in bins of the invariant mass, with the resonance masses and widths fixed to the values obtained by the overall fit. Since there is at most one per event, we convert the resulting yield in each bin into an “inclusive” cross section, following the procedure described in Sec. IV.3. This cross section is shown in Fig. 8(b) and listed in Table 3 for the effective c.m. energies from threshold up to 3.5 . 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 , followed by a very rapid decrease with increasing energy. There are suggestions of narrow structure in the peak region, but the only statistically significant structure is the peak, which is discussed below.

The contribution is a large fraction of the total cross section at all energies above its threshold, and dominates in the 1.8–2.0  region. We are unable to extract a meaningful measurement of the cross section from this data sample because it is more than ten times smaller. The intermediate state makes up the majority of the remainder of the cross section and it can be estimated as a difference of the values in Table 1 and Table 3 for the and final states.

iv.6 The Intermediate State

Intermediate states containing relatively narrow resonances can be studied more easily. Figure 9(a) shows a scatter plot of the invariant mass of the pair versus that of the pair. Horizontal and vertical bands corresponding to the and , respectively, are visible, and there is a concentration of entries on the band corresponding to the correlated production of and . The signal is also visible in the mass projection, Fig. 9(c). The large contribution from the , coming from the decay, is nearly uniform in the mass, and the cross-hatched histogram shows the non- background estimated from the control region in . The cross-hatched histogram also shows a peak, but this is a small fraction of the events. Subtracting this background and fitting the remaining data gives 170656 events produced via the intermediate state.

To study the channel, we select candidate events with a invariant mass within 10  of the mass, indicated by the inner vertical lines in Figs. 9(a,c), and estimate the non- contribution from the mass sidebands between the inner and outer vertical lines. In Fig. 9(b) we show the invariant mass distributions for candidate events, sideband events and control region events as the open, hatched and cross-hatched histograms, respectively, and in Fig. 9(d) we show the numbers of entries from the candidate events minus those from the sideband and control region. There is a clear peak over a broad mass distribution, with no indication of associated production.

Figure 9: (a) The vs. the invariant masses for all selected events; (b) the invariant mass projections for events in the peak (open histogram), sidebands (hatched) and background control region (cross-hatched); (c) the mass projections for all events (open) and control region (cross-hatched); (d) the difference between the open and the sum of the other histograms in (b).

A coherent sum of two Breit-Wigner functions is sufficient to describe the invariant mass distribution of the pair recoiling against a in Fig. 9(d). We fit with the function:

(3)

where is the invariant mass, and are the parameters of the resonance, is their relative phase and are normalization parameters, corresponding to the number of events under each BW. One BW corresponds to the , but a wide range of values of the other parameters can describe the data. Fixing the relative phase to and the parameters of the first BW to   and   (which could be interpreted as the  PDG ()), we obtain the fit shown in Fig. 9(d). It describes the data well and gives an signal of events, with and , consistent with the PDG values PDG (). There is a suggestion of an peak in the data, but it is much smaller than the peak and we do not consider it further.

We obtain the number of events in bins of invariant mass by fitting the invariant mass projection in that bin after subtracting non- background. Each projection is a subset of Fig. 9(c), where the curve represent a fit to the full sample. In each mass bin, all parameters are fixed to the values obtained from the overall fit except the numbers of events in the peak and the non- component.

The efficiency may depend on the details of the production mechanism. Using the two-pion mass distribution in Fig. 9(d) as input, we simulate the system as an S-wave comprising two scalar resonances, with parameters set to the values given above. To describe the mass distribution we use a simple model with one resonance, the , of mass 1.68  and width 0.2 , decaying to . The simulated reconstructed spectrum is shown in Fig. 10(a). There is a sharp increase at about 2  due to the threshold. All other structure is determined by phase space and a falloff with increasing mass.

Figure 10: (a) The invariant mass distributions from the simulation described in the text, reconstructed in the signal (open) and control (hatched) regions; (b) net reconstruction and selection efficiency as a function of mass: the solid line represents a cubic fit, and the dashed line the corresponding fit for the space phase model shown in Fig. 3.

Dividing the number of reconstructed events in each bin by the number of generated ones, we obtain the efficiency as a function of mass shown in Fig. 10(b). The solid line represents a fit to a third order polynomial, and the dashed line the corresponding fit to the phase space model from Fig. 3. The model dependence is weak, giving confidence in the efficiency calculation. We calculate the cross section as described in Sec. IV.3 but using the efficiency from the fit to Fig. 10(b) and dividing by the branching fraction of 0.491 PDG (). We show our results as a function of energy in Fig. 11 and list them in Table 4. The cross section has a peak value of about 0.6 nb at about 1.7 , then decreases with increasing energy until threshold, around 2.0 . From this point it rises, falls sharply at about 2.2 , and then decreases slowly. Except in the charmonium region, the results at energies above 2.9  are not meaningful due to small signals and potentially large backgrounds, and are omitted from Table 4. Figure 11 displays the cross-section up to 4.5  to show the signals from the and decays. They are discussed in Sec. VIII. There are no previous measurements of this cross section, and our results are consistent with the upper limits given in Ref. 2k2pidm1 ().

Figure 11: The cross section as a function of the effective c.m. energy.
(GeV) (nb) (GeV) (nb) (GeV) (nb) (GeV) (nb)
1.4875 0.01 0.02 1.8375 0.39 0.10 2.1875 0.32 0.06 2.5375 0.09 0.03
1.5125 0.03 0.03 1.8625 0.44 0.10 2.2125 0.22 0.05 2.5625 0.03 0.02
1.5375 0.09 0.04 1.8875 0.23 0.08 2.2375 0.15 0.04 2.5875 0.06 0.02
1.5625 0.13 0.04 1.9125 0.34 0.09 2.2625 0.10 0.03 2.6125 0.07 0.02
1.5875 0.21 0.06 1.9375 0.37 0.08 2.2875 0.11 0.04 2.6375 0.08 0.03
1.6125 0.23 0.06 1.9625 0.31 0.08 2.3125 0.08 0.03 2.6625 0.06 0.02
1.6375 0.54 0.08 1.9875 0.36 0.07 2.3375 0.13 0.03 2.6875 0.04 0.02
1.6625 0.61 0.09 2.0125 0.38 0.07 2.3625 0.10 0.04 2.7125 0.08 0.03
1.6875 0.64 0.10 2.0375 0.29 0.07 2.3875 0.13 0.04 2.7375 0.06 0.02
1.7125 0.38 0.09 2.0625 0.42 0.07 2.4125 0.12 0.04 2.7625 0.07 0.02
1.7375 0.64 0.10 2.0875 0.30 0.06 2.4375 0.15 0.04 2.7875 0.02 0.02