Phase Difference Between the Electromagnetic and Strong Amplitudes for and Decays into Pairs of Pseudoscalar Mesons
Using the data for produced in annihilations at MeV at the CESR-c collider and produced in the decay , the branching fractions for and decays to pairs of pseudoscalar mesons, , , and , have been measured using the CLEO-c detector. We obtain branching fractions , , , and , , , where the first errors are statistical and the second errors are systematic. The phase differences between the amplitudes for electromagnetic and strong decays of and to pseudoscalar pairs are determined by a Monte Carlo method to be and . The difference between the two is
Interest in final state interaction (FSI) phases originally arose from CP violation in decays and decays. However, recently interest has focused on the suggestion that large FSI phases are a general feature of strong decays. The electromagnetic and strong decays of the vector states of charmonium, and , offer good testing ground for this possibility. In a series of papers Suzuki suzuki (); suzuki1 (); suzuki2 () has studied FSI phase differences between the electromagnetic amplitude and the three-gluon strong amplitude . In the decays of to vector-pseudoscalar (VP) pairs Suzuki obtained suzuki (), and for the pseudoscalar-pseudoscalar (PP) pairs he obtained suzuki1 (). Rosner rosner () confirmed Suzuki’s results, obtaining , and . Further, Gerard and Weyers gerard () have argued that these differences are manifestations of what they call “universal incoherence”, i.e., phase difference between electromagnetic and every exclusive annihilation decay channel of and . In order to arrive at a deeper understanding of the origin of large phase differences it is therefore necessary to examine if what has been observed for decays persists in the corresponding decays of . As Suzuki has noted suzuki2 (), this is particularly important in the context of the curious suppression of the ratio (particularly notable for decays).
One of the best places to study the phase difference between electromagnetic and strong decays of and is in their decays to pseudoscalar pairs, , and . This is because the three decays sample the interactions in quite different ways. The decay is essentially purely electromagnetic, with strong decay being forbidden by isospin invariance, the decay is essentially purely strong and due only to SU(3) violation, and the decay can proceed through both the electromagnetic and strong interactions.
A particularly simple and transparent determination of the relative phase angle can be made by measuring it as the interior angle of the triangle in the complex plane with the amplitudes for the decays to , and as its three sides. This is illustrated in Fig. 1 for the and decays.
With the neglect of the small effect of SU(3)–breaking and interference between resonance and continuum amplitudes, the relative phase angle is given by
where the phase space correction factor ; , and . Thus it is required to measure the three branching fractions, , and , which are proportional to the squares of the respective amplitudes.
Previous measurements of were made by BES bes1 (); bes () and CLEO cleo (). Their results, recalculated to correspond to the internal phase angle defined by Eq. (1), are (BES), and (CLEO). The results for mentioned earlier were obtained by Suzuki and Rosner using the PDG 1998 pdg98 () evaluation of the branching fractions measured in 1985 by Mark III spear () in the decay of their sample of produced directly in annihilations at . In this paper we present much more accurate results for branching fractions and using the CLEO-c data of , eight times larger than that used in the previous CLEO measurement cleo (), and for branching fractions and using the data set of tagged by recoils in the decay . In all cases large improvement in precision over previous measurements is obtained.
The data used in this analysis were collected at the CESR storage ring using the CLEO-c detector cleodet (). The detector has a cylindrically symmetric configuration, and it provides 93 coverage of solid angle for charged and neutral particle identification. The detector components important for the present analysis are the vertex drift chamber, the main drift chamber (DR), the Ring Imaging Cherenkov (RICH) counter, and the CsI crystal calorimeter (CC).
Ii Event Selection
Event selection, particle identification, and branching fraction determination in the present paper follow closely those used by us in our published paper on the determination of cleo ().
The events for the three decay modes, are required to have two charged particles for decays, and four charged particles for decays, and zero net charge. The event selection criteria are identical to those in our earlier paper cleo (). To recapitulate, all charged particles are required to meet the standard criteria for track quality. The , , and from decays (with vertex displaced by mm) are accepted in the regions and 0.93, respectively. The invariant mass of the from decay is required to be within 10 MeV of MeV.
Particle identification is done by combining information from the drift chamber (DR) and the likelihood information from the RICH detector for the particle species , , and . The variable for the information is , and for the RICH information it is the likelihood function, -2log. To distinguish between two particle species a joint function, is constructed. Charged kaons in decays are distinguished from protons, pions, and leptons by requiring . Looser criteria are used for pions in decay and daughters from in decay, . In addition, electrons are rejected in all decays by requiring . All these requirements are identical to those in Ref. cleo ().
In Ref. zweber () a detailed study was made to distinguish pions from the much more prolific yield of muons. It was determined that the energy deposited in the central calorimeter by pions due to their hadronic interactions provides a very efficient means of distinguishing them from muons which deposit much smaller energy due only to . For decay, it was determined that requiring every pion to deposit GeV reduced muon contamination to level. This requirement was used in Ref. cleo (), and we impose it also in the present analysis for the channel . For , the pion yield is much larger, and the corresponding requirement is determined to be GeV. For decay an explicit veto was made, as in Ref. cleo ().
Iii Determination of , , and
For and , it was required that the total momentum be less than 75 MeV. As expected, this requirement removes most of the , peaks and the background which is present without it in the event distributions plotted as a function of . The resulting distributions are shown in Figs. 2(a,b) in the extended region of . For with only observed no such total momentum cut can be imposed, and the different criteria developed in Ref. cleo () were implemented to take account of the unobserved .
For , the direction of is inferred from that of the observed . We require that there be no shower associated with neutrals closest to this direction with energy GeV. Further, we require that in a cone of 0.35 radians around the direction there be no more than one shower with MeV. We require that outside this cone there be no single shower with MeV and the sum of all showers MeV. These selections remove events with neutral particles other than accompanying the detected . As shown in Fig. 2 (c), the remaining background is featureless, and very small in the signal region, .
In Figs. 2 (d,e,f) we show the distributions of Figs. 2 (a,b,c) in detail in the smaller region of in which we fit the data with MC–generated peak shapes and linear backgrounds. The fit results are presented in Table 1. The observed peak counts, , are , , and for , , and , respectively.
The MC signal events were generated assuming angular distributions, where is the angle between a meson and the positron beam. In Figs. 2 (g,h,i) we show that the angular distributions of the data events are in excellent agreement with the MC distributions in all three cases for , and decays.
The observed peaks for and decays contain contributions from continuum, or form factor production in addition to the resonance contributions. Continuum contribution in decays can, however, only arise from higher order processes, and is expected to be very small. The continuum contributions have to be estimated, and subtracted from the observed peak yields of and before the corresponding branching fractions can be determined. This is particularly challenging for the decays.
The main limitation in the measurement of the interference phase difference angle in earlier publications came from the determination of the branching fraction for decay, . As mentioned earlier, the three–gluon strong decay is forbidden by isospin conservation, and the branching fraction is consequently small. The problem of the intrinsically small branching fraction is compounded by the fact that no good–statistics measurements of the continuum contribution were available. As a result all three previous measurements had very large errors:
In the DASP dasp () and BES bes1 () measurements no attempt was made to subtract the continuum contribution. In our published paper cleo () with 3 million (corresponding to integrated luminosity ), only 11 counts were observed. From these the scaled continuum contribution of 7 counts, based on data with integrated luminosity of 20.7 taken off– at MeV, was subtracted to get . This led to the poor determination of cleo ().
Our present analysis has two big advantages over the old analysis. We now have available a much larger data set of 24.5 million corresponding to an integrated luminosity of 48 . Also, we are able to make a much better estimate of the continuum contributions in the yields of and based on our large-statistics form factor measurements with luminosity of 805 at MeV and 586 pb at MeV.
The widths of and are MeV and MeV, respectively pdg10 (). They are about two orders of magnitude or more larger than MeV, and the estimates of the resonance branching fractions for the decays of and to , , and range from to . These lead to the conclusion that the resonance contributions in and decays of and are less than 0.01% of the total, i.e., the total observed counts are entirely due to continuum or form factor contribution. They can therefore be confidently extrapolated to estimate continuum contribution in the measured counts at . The extrapolation is done as
where MeV and MeV and 4160 MeV. Here are the luminosities, and are efficiencies determined by Monte Carlo (MC) simulations, and the factor is the conventional extrapolation based on the constancy of for vector meson form factors explain (). The observed and counts at MeV and 4170 MeV have statistical and systematic uncertainties zweber (). Using the counts at 3772 and 4170 MeV in Eq. (2) leads to estimated continuum contributions at MeV which are 184.108.40.206 and 220.127.116.11 counts respectively for kaons, and 18.104.22.168 and 22.214.171.124 counts respectively for pions (the first errors are statistical and the second errors are systematic). We use their averages, taking account of the fact that systematic errors are correlated, as 106.95.5 counts for kaons, and 40.44.6 counts for pions as our best estimates of continuum contributions at 3686 MeV.
These contributions are illustrated as dotted histograms in Figs. 2 (d,e).
In Table 1, for decays we list the number of counts observed in the , and peaks, the number of continuum counts estimated as described above, the net signal counts , the event selection efficiencies , and the branching fractions calculated as
where the number of is . We note that these branching fractions have been obtained without taking account of possible interference between continuum and resonance contributions.
We have considered various sources of systematic uncertainties in our branching fraction results. As in Ref. cleo (), for all three decay channels, , and the common uncertainties are in number of , per track in track finding, and per track in charged particle identification. Uncertainties in trigger efficiency are in and , and in . The systematic uncertainty in determination of the factor in Eq. (2) comes from the uncertainties in the total integrated luminosity values of the data taken at and at MeV and MeV, which correspond to 1 for each. Thus, we assign 1.4 systematic uncertainty to the value , determined in Eq. (2).
Variation of the total momentum 75 MeV requirement by MeV resulted in no statistically significant change in , and a change in , which we assign as systematic uncertainty. For changing the requirement GeV by resulted in change in . The effect of the implementation of -related constraints in decay was determined as the ratio of fitted peak counts in the spectra in Fig. 2(f)/Fig. 2(c). It was determined to be 0.8350.042. The 5 uncertainty in this determination was assigned as a systematic error in .
The total systematic uncertainties are 8.0, 5.2, and 6.5 for , and decays, respectively.
The resulting branching fractions are
These are listed in Table 1. In Table 2 the errors are listed with the statistical and systematic errors combined in quadrature, together with results from previous investigations by DASP dasp (), BES bes (); bes2 (), and CLEO cleo (). The uncertainties in our branching fractions results are factors two to five smaller than those in the published results. In Table 2, we also list the phase angle difference , calculated using Eq. (1). All the published values of are also listed, as recalculated using Eq. (1).
In Sec. V we present the determination of the phase angle differences using a MC method which allows us to take account of the distributions in the values of the branching fractions.
Iv Determination of and
The 24.5 million in our data set lead to 8.6 million events tagged by recoil in the decay . This sample is automatically free of any contamination of , , , or events produced in direct measurement at MeV. The subsequent decays to , and also do not contain any continuum contributions. As such, these data are cleaner and simpler to analyze than the data from collisions at MeV.
As stated earlier, to select events for decays our event selection criteria are modified to require four charged particles instead of two. Recoil mass is then constructed for every pair of two oppositely charged particles. This is dominated by the production of , as shown in Fig. 3 for the channel , which is similar to those obtained for the channels . We define the clean sample as consisting of events with MeV. Selection of events for , and is done exactly in the same manner as described for . As mentioned earlier, the most efficient cut to reject from decays is to require that each pion satisfy MeV. The MC–estimated muon contamination with this requirement is .
Figures 4 (g,h,i), show that the angular distributions of the data events for all thr ee decays are in excellent agreement with the MC distributions, as in the case of decays in Figs. 2 (g,h,i).
The systematic errors for decays were determined in exactly the same manner as for decays. Their totals are , , and for , , and , respectively.
Figure 4 shows the results for decays to , , as Fig. 2 does for the corresponding decays. Figures 4 (a,b,c) show the data in the extended range of with arbitrarily normalized MC predictions. The distribution of the events as a function of in the rest frame of , shown in Fig. 4 (c), needs comment. The peak at is clearly separated from the smaller peak at which arises from the decays into , , , despite rejection. The clear separation of the peak from the peak is confirmed by the MC simulation whose result is superposed on the data in Fig. 4 (c). In Figs. 4 (d,e,f), the event distributions of Figs. (a,b,c) are shown in detail in the smaller region of . Also shown are the fits using MC–determined peak shapes and linear backgrounds. The fits give , , and counts for , , and , respectively. These compare with 84, 107 and 74 counts in the Mark III measurements with a factor three smaller sample of spear (). These counts, the MC determined efficiencies , and njpsi (), lead to the branching fractions listed in Table 1.
|DASP dasp ()||BES bes1 (); bes ()||CLEO cleo ()||This analysis||This analysis|
|Mark IIIspear ()||BES bes2 ()||This analysis||This analysis|
Thus the final branching fractions are
These are listed in Table 1, and in Table 2 with the statistical and systematic errors combined in quadrature. The above branching fractions for and are consistent with those reported by Mark III spear (), and have factors two and three smaller uncertainties, respectively. Our branching fraction for based on well resolved events, as shown in Fig. 4 (f), is a factor 2.6 larger than reported by Mark III with 74 identified counts, obtained by “stringent cuts” to remove and decays, and is larger than reported by BES II bes () with identified events. In the large statistics BES II measurements the peak due to , , , overlapped with the direct , peak, and strong cuts had to be made to resolve the two peaks. As shown in Figs. 4 (c,f), in our measurements the two peaks are completely resolved. Further, MC calculations confirm that with our selections the decay , also does not make any contribution under the , peak at .
As for , we calculate using Eq. (1), and obtain , as compared to obtained using the branching fractions measured by Mark III. These are listed in Table 2.
V Monte Carlo Based Evaluation of and the Difference
The individual measured branching fraction values have distributions which are conventionally stated in terms of errors, as listed in Table I. Using only the central values to evaluate according to Eq. (1) is not correct. The more correct procedure is to make Monte Carlo evaluation of Eq. (1) taking account of random associations of the branching fraction values in the distributions for the three decays. We have made such toy MC evaluations of both and . As shown in Fig. 5 (left), the large error () in branching fraction leads to a very asymmetric MC distribution for , whereas the much smaller error () for results in a much smaller asymmetry in the distribution for . If we adopt the usual definition of the error as that which includes 68% of the area on each side of the peak of a distribution, our results are
The difference, whose MC distribution is illustrated in Fig. 5 (right), is
We consider the above estimates of , , and their difference to be our final results.
We have made large statistics measurements of the branching fractions for the decays of and to pseudoscalar pairs , , and . Our branching fraction results have errors which are factors two to five smaller than the previously published results. Using these branching fractions we have made calculations of the phase angle differences between the electromagnetic and strong decays for both and , taking proper account of the distributions of the branching fraction values. Our results are nearly a factor two more precise than the previously published results.
Acknowledgements.This measurement was done using CLEO data, and as members of the former CLEO Collaboration we thank it for this privilege. We wish to thank Jon Rosner for his helpful comments and suggestions. This research was supported by the U.S. Department of Energy and the National Science Foundation.
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