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Structure of the a1ρπ vertex
Nuclear Physics B (Proc. Suppl.) 198 (2010) 212–215 www.elsevier.com/locate/npbps
Structure of the a1 ρπ vertex P. Lichardab∗ and M. Voj´ıka a
Institute of Physics, Silesian University in Opava, Bezruˇcovo n´am. 13, 74601 Opava, Czech Republic
b
Institute of Experimental and Applied Physics, Czech Technical University in Prague, Horsk´ a 3a/22, 12800 Prague, Czech Republic Experimental data on the three-pion decay of the τ lepton are used to determine the mixing angle in a twocomponent a1 ρπ Lagrangian. The result is compatible with the recent analysis based on the electron-positron annihilation into four pions.
1. Introduction The axial-vector isovector resonance a1 (1260) plays an important role in many phenomena of the nuclear and particle physics. There is a strong evidence that the a1 resonance intensively couples to the ρπ system, which gives origin to three pions in the experimentally observed final states. The a1 resonance have been studied in many processes, but even its basic parameters are not very well known. The values of the a1 (1260) resonance mass determined from different processes or by different experimental groups often contradict one another. The same applies, even to a larger extent, to the a1 width. Its values coming from the analyzes of the hadronic processes are systematically lower than those coming from the three-pion decay of the tau lepton. The origin of those problems lies in the very nature of the resonances with their short lifetimes and large widths. The usual definition of the mass and usual procedures for its measurement are not applicable. The mass and width of a resonance enter the formulas for experimentally accessible quantities via the assumed form of the resonance propagator. Another ingredient that defines a particular model of a physical process with the three pions coming from the a1 in the final states is the
a1 ρπ vertex. The models used by different groups are based on different a1 ρπ vertices. They are sometimes simply constructed as allowed combinations of the metric tensor and participating four-momenta. A more rigorous way lies in deriving them from interaction Lagrangians among the axial, vector and pseudoscalar fields. But even here one cannot rely very much on guidance from the theory yet, because various theoretical concepts lead to different Lagrangians and/or to Lagrangians that contain free parameters. The present effort of the particle theory group at the Silesian University in Opava is concentrated on reanalyzing various experimental data using the cross section and decay width formulas based on the same a1 ρπ phenomenological Lagrangian. The latter consists of two parts and contains two free parameters: an overall coupling constant and a mixing angle. We hope that good fits to the experimental data on various processes can be achieved for the same values of those two parameters. The results of the analyzes of the electron-positron annihilation into four pions have already been published [1,2]. Here, we report on the preliminary results of the analysis of the three-pion decay of the tau lepton. 2. Phenomenological a1 ρπ vertex
∗ This
work was supported by the Czech Ministry of Education, Youth and Sports under Contract No. MSM6840770029 and No. LC07050.
0920-5632/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysbps.2009.12.042
In the literature, one can find several prescriptions for the a1 ρπ vertex used in the calculation
P. Lichard, M. Vojík / Nuclear Physics B (Proc. Suppl.) 198 (2010) 212–215
of the decay rate of the tau lepton into three pions and neutrino. The simplest one X αμ ∝ g αμ (index α (μ) couples with the a1 (ρ) line), comes from the interaction Lagrangian among the a1 , ρ and π fields without derivatives. It was used, e. g., in Ref. [3]. An opposite extreme of complexity is the vertex used in the model of Isgur, Morningstar and Reader [4], which is transversal both to the a1 and ρ four-momenta. Xiong, Shuryak, and Brown [5] assessed the role of the a1 meson in photon production from πρ collisions in a meson gas. They chose the a1 ρπ vertex in the form μ X αμ = Gρ (pρ pπ )g αμ − pα ρ pπ ,
213
In Refs. [1,2], several existing models of the electron-positron annihilation into four pions were supplemented with the contributions from the a1 π intermediate states and compared with various data. These analyzes showed that the sine of the mixing angle in Eqs. (3) and (4) should lie between 0.40 and 0.52. 3. Details of the model Assuming the a1 dominance, the decay τ − → π π π ντ is described by two Feynman diagrams. One of them is shown in Figure 1, the other is obtained by exchanging the identical pions. − + −
where pρ and pπ are the momenta of the outgoing ρ meson and pion, respectively. This vertex can be generated from the Lagrangian that is proportional to L1 = Aμ · (Vμν × ∂ ν φ) ,
(1)
where Vμν = ∂μ Vν − ∂ν Vμ . Symbols Aμ , Vμ , and φ denote the field operators of the a1 , ρ, and π mesons, respectively. Janssen, Holinde, and Speth [6], when evaluating the amplitude of the πρ scattering, chose the Lagrangian proportional to L2 = Vμν · (∂ μ Aν × φ) .
(2)
Inspired by Eqs. (1) and (2), we construct the following phenomenological Lagrangian: ga ρπ (3) La1 ρπ = √1 (L1 cos θ + L2 sin θ) , 2 which has already been used in the recent study of the electron-positron annihilation into four pions [1,2]. The coupling constant and mixing angle in (3) are free parameters, which have to be fixed by requiring the best description of the experimental data. The Lagrangian (3) implies the following a1 ρπ vertex iga1 ρπ μ αμ √ X αμ = cosθ pα ρ pπ − (pπ pρ ) g 2 μ αμ − sin θ pα , (4) ρ pa1 − (pa1 pρ ) g where pa1 is the four-momentum of the incoming a1 meson.
Figure 1. One of two diagrams considered.
We choose the propagator of the a1 (1260) in the form −iGμν a1 (p) =
−g μν + pμ pν /m2a1 , s − Ma21 (s) + ima1 Γa1 (s)
where s = p2 , ma1 is the nominal mass of the a1 , and Γa1 (s) is the energy-dependent total width. The running mass is normalized by Ma21 (m2a1 ) = m2a1 . A similar propagator was used in the seminal paper by Isgur, Morningstar, and Reader [4] and recently also by the CLEO collaboration [7]. In our work, Γa1 (s) is the sum of the partial decay widths of the a1 → ρπ → 3π, ¯ a1 → K ∗ K ¯ → K Kπ ¯ and a1 → K K¯∗ → K Kπ, a1 → f0 π → 3π channels. Ma21 (s) has been calculated from a twice-subtracted dispersion relation with Γa1 (s) as input.
214
P. Lichard, M. Vojík / Nuclear Physics B (Proc. Suppl.) 198 (2010) 212–215
The ρ(770) propagator was taken from [8]. It reads −iGμν ρ (pρ ) =
−g μν + pμρ pνρ /m2ρ , t − Mρ2 (t) + imρ Γρ (t)
where t = p2ρ and Γρ (t) is the total width of the √ ρ-resonance with off-shell mass t, normalized at t = m2ρ to the nominal width Γρ . Furthermore, Mρ2 (t) is the running mass squared satisfying condition Mρ2 (t = m2ρ ) = m2ρ . The total decay width Γρ (t) includes, besides the ρ → ππ contribution, ¯ and ηππ channels. These adalso the ωπ, K K ditions proved to be important for getting a reasonable behavior of Mρ2 (t). For details, see [8]. The last refinement concerns the a1 ρπ and ρππ vertices. The effective Lagrangian approach takes hadrons as elementary quanta of the corresponding fields, ignoring thus their internal structure. As a consequence, the interaction strength is overestimated at higher momentum transfers. To describe the interaction among participating mesons more realistically, we explore the chromoelectric flux-tube breaking model of Kokoski and Isgur [9]. Each interaction vertex is modified by the factor exp{−p∗ 2 /(12β 2 )}, where p∗ is the three-momentum magnitude of the lighter particles in the rest frame of the heaviest one (virtual masses are taken in the intermediate states). The authors of [9] advocated the value of β ≈ 0.4 GeV, but we consider β a free parameter. 4. Data and results We will compare the three-pion mass distribution calculated from our model with various data sets. There are five parameters we want to determine by minimizing the usual χ2 : (i) the a1 mass ma1 , (ii) the a1 width Γa1 , (iii) the Lagrangian mixing parameter sin θ, (iv ) the Kokoski-Isgur cutoff β and, finally, (v ) an overall multiplicative factor, which does not have physical meaning because none of the data sets is absolutely normalized. As an experimental input we first use the data by the ARGUS Collaboration at DESY [10], namely their background and acceptance corrected three pion mass distribution. The data set consists of twenty-eight points in the mass range
0.425–1.775 GeV. We got a very good fit characterized by χ2 /N DF = 0.663 and the confidence level of 89.1%. Unfortunately, we also found a set of parameters with only a little worse χ2 , but with parameters incompatible with those from the main minimum. To resolve the ambiguity mentioned above, we took into consideration another set of data on the three-charged-pion decay of the tau lepton, namely that of the OPAL collaboration [11] at the CERN Large Electron-Positron Collider (LEP). It consists of twenty points in the three-pionmass-squared plot corrected for background and efficiency. The number of free parameters of our model increased to six because each data set requires their own multiplication factor. The minimizing of χ2 =χ2 (ARGUS)+χ2 (OPAL) yielded the following results: χ2 /NDF=30.66/42, Confidence Level = 90%, ma1 = (1.37 ± 0.01) GeV, Γa1 = (0.61 ± 0.01) GeV, sin θ = 0.54 ± 0.02, β = (0.24 ± 0.02) GeV. The errors are defined in the usual way [12]. The mass and width of the a1 we obtained are a little higher than the values in current Review of Particle Physics [13]. The central value of the Lagrangian mixing parameter sin θ lies outside the interval found in [1,2] (0.40–0.52), but, keeping in mind the error, it is still compatible with the e+ e− → 4π results. The τ -lepton decay into three pions and neutrino was also investigated by the CLEO Collaboration at the Cornell Electron Storage Ring (CESR). Unfortunately, their results on the allcharged-pions channel still exist only in a preliminary form [14]. We therefore used their data on the π − π 0 π 0 channel [7], which is, in our approach, described by the same parameterization as the charged channel. Unfortunately, we were not able to get a reasonable fit, value of χ2 remained very high. The probable reason is the ab− sence of the a− 1 → f0 π intermediate state, which differentiates between π − π + π − and π − π 0 π 0 systems in the final state.
P. Lichard, M. Vojík / Nuclear Physics B (Proc. Suppl.) 198 (2010) 212–215
Figure 2 shows the scan of χ2 over all values of sin θ for all three data sets we considered. The a1 mass and width were kept at their optimal values. It is obvious that the quality of the fit is very sensitive to the value of the mixing parameter. Only a relatively narrow interval around sin θ ≈ 0.5 can provide a good fit, as testified by all three data sets. This observation agrees with the result of the electron-positron annihilation into four pions, depicted in Figure 2 by a short horizontal abscissa.
215
termine those free parameters from fit to data on several different processes. In this contribution we considered the three-pion (plus neutrino) decay of the tau lepton measured by three experimental groups. The best fit to all data requires the Lagrangian mixing parameter falling roughly in the same region as the fit to completely different process–the electron-positron annihilation into four pions. It indicates that the form of the effective Lagrangian we use is reasonable. In the future we will proceed in two directions. Firstly, we intend to improve our model in order to describe the decay τ − → π − π 0 π 0 ντ better. Secondly, we will combine our model of the threepion decay of the τ lepton with the model of the electron-positron annihilation into four pions [1,2] to get a common fit to those two processes and thus determine the a1 ρπ vertex more precisely. REFERENCES
Figure 2. Dependence of χ2 on the a1 ρπ Lagrangian mixing parameter for fixed values both of the a1 mass and width.
5. Conclusions and comments Our investigation shows that the form of the a1 ρπ vertex effects strongly the agreement of a model with data and also the values of the fitted parameters. In the situation when we do not know the “true” form of the a1 ρπ Lagrangian (and of the corresponding vertex) from the first principles it is necessary to work with an effective Lagrangian that contains free parameters and de-
1. P. Lichard and J. Jur´ an ˇ , Phys. Rev. D 76 (2007) 094030. 2. J. Jur´ an ˇ and P. Lichard, Phys. Rev. D 78 (2008) 017501. 3. T. N. Pham, C. Roiesnel and Tran N. Truong, Phys. Lett. B 78 (1978) 623. 4. N. Isgur, C. Morningstar and C. Reader, Phys. Rev. D 39 (1989) 1357. 5. L. Xiong, E. Shuryak and G. E. Brown, Phys. Rev. D 46 (1992) 3798. 6. G. Janssen, K. Holinde, and J. Speth, Phys. Rev. C 49 (94) 2763. 7. D. M. Asner et al. (CLEO Collaboration), Phys. Rev. D 61 (1999) 012002. 8. P. Lichard, Phys. Rev. D 60 (1999) 053007. 9. R. Kokoski and N. Isgur, Phys. Rev. D 35 (1987) 907. 10. H. Albrecht et al. (ARGUS Collaboration), Z. Phys. C 58 (1993) 61. 11. R. Akers et al., Z. Phys. C 67 (1995) 45 12. F. James and M. Roos, Comput. Phys. Commun. 10, 343 (1975). 13. C. Amsler et al. (Particle Data Group), Phys. Lett. B 667 (2008) 1. 14. E. I. Shibata (representing the CLEO Collaboration), Nucl. Phys. B (Proc. Suppl.) 123 (2003) 40.
Structure of the a1 ρπ vertex P. Lichardab∗ and M. Voj´ıka a
Institute of Physics, Silesian University in Opava, Bezruˇcovo n´am. 13, 74601 Opava, Czech Republic
b
Institute of Experimental and Applied Physics, Czech Technical University in Prague, Horsk´ a 3a/22, 12800 Prague, Czech Republic Experimental data on the three-pion decay of the τ lepton are used to determine the mixing angle in a twocomponent a1 ρπ Lagrangian. The result is compatible with the recent analysis based on the electron-positron annihilation into four pions.
1. Introduction The axial-vector isovector resonance a1 (1260) plays an important role in many phenomena of the nuclear and particle physics. There is a strong evidence that the a1 resonance intensively couples to the ρπ system, which gives origin to three pions in the experimentally observed final states. The a1 resonance have been studied in many processes, but even its basic parameters are not very well known. The values of the a1 (1260) resonance mass determined from different processes or by different experimental groups often contradict one another. The same applies, even to a larger extent, to the a1 width. Its values coming from the analyzes of the hadronic processes are systematically lower than those coming from the three-pion decay of the tau lepton. The origin of those problems lies in the very nature of the resonances with their short lifetimes and large widths. The usual definition of the mass and usual procedures for its measurement are not applicable. The mass and width of a resonance enter the formulas for experimentally accessible quantities via the assumed form of the resonance propagator. Another ingredient that defines a particular model of a physical process with the three pions coming from the a1 in the final states is the
a1 ρπ vertex. The models used by different groups are based on different a1 ρπ vertices. They are sometimes simply constructed as allowed combinations of the metric tensor and participating four-momenta. A more rigorous way lies in deriving them from interaction Lagrangians among the axial, vector and pseudoscalar fields. But even here one cannot rely very much on guidance from the theory yet, because various theoretical concepts lead to different Lagrangians and/or to Lagrangians that contain free parameters. The present effort of the particle theory group at the Silesian University in Opava is concentrated on reanalyzing various experimental data using the cross section and decay width formulas based on the same a1 ρπ phenomenological Lagrangian. The latter consists of two parts and contains two free parameters: an overall coupling constant and a mixing angle. We hope that good fits to the experimental data on various processes can be achieved for the same values of those two parameters. The results of the analyzes of the electron-positron annihilation into four pions have already been published [1,2]. Here, we report on the preliminary results of the analysis of the three-pion decay of the tau lepton. 2. Phenomenological a1 ρπ vertex
∗ This
work was supported by the Czech Ministry of Education, Youth and Sports under Contract No. MSM6840770029 and No. LC07050.
0920-5632/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysbps.2009.12.042
In the literature, one can find several prescriptions for the a1 ρπ vertex used in the calculation
P. Lichard, M. Vojík / Nuclear Physics B (Proc. Suppl.) 198 (2010) 212–215
of the decay rate of the tau lepton into three pions and neutrino. The simplest one X αμ ∝ g αμ (index α (μ) couples with the a1 (ρ) line), comes from the interaction Lagrangian among the a1 , ρ and π fields without derivatives. It was used, e. g., in Ref. [3]. An opposite extreme of complexity is the vertex used in the model of Isgur, Morningstar and Reader [4], which is transversal both to the a1 and ρ four-momenta. Xiong, Shuryak, and Brown [5] assessed the role of the a1 meson in photon production from πρ collisions in a meson gas. They chose the a1 ρπ vertex in the form μ X αμ = Gρ (pρ pπ )g αμ − pα ρ pπ ,
213
In Refs. [1,2], several existing models of the electron-positron annihilation into four pions were supplemented with the contributions from the a1 π intermediate states and compared with various data. These analyzes showed that the sine of the mixing angle in Eqs. (3) and (4) should lie between 0.40 and 0.52. 3. Details of the model Assuming the a1 dominance, the decay τ − → π π π ντ is described by two Feynman diagrams. One of them is shown in Figure 1, the other is obtained by exchanging the identical pions. − + −
where pρ and pπ are the momenta of the outgoing ρ meson and pion, respectively. This vertex can be generated from the Lagrangian that is proportional to L1 = Aμ · (Vμν × ∂ ν φ) ,
(1)
where Vμν = ∂μ Vν − ∂ν Vμ . Symbols Aμ , Vμ , and φ denote the field operators of the a1 , ρ, and π mesons, respectively. Janssen, Holinde, and Speth [6], when evaluating the amplitude of the πρ scattering, chose the Lagrangian proportional to L2 = Vμν · (∂ μ Aν × φ) .
(2)
Inspired by Eqs. (1) and (2), we construct the following phenomenological Lagrangian: ga ρπ (3) La1 ρπ = √1 (L1 cos θ + L2 sin θ) , 2 which has already been used in the recent study of the electron-positron annihilation into four pions [1,2]. The coupling constant and mixing angle in (3) are free parameters, which have to be fixed by requiring the best description of the experimental data. The Lagrangian (3) implies the following a1 ρπ vertex iga1 ρπ μ αμ √ X αμ = cosθ pα ρ pπ − (pπ pρ ) g 2 μ αμ − sin θ pα , (4) ρ pa1 − (pa1 pρ ) g where pa1 is the four-momentum of the incoming a1 meson.
Figure 1. One of two diagrams considered.
We choose the propagator of the a1 (1260) in the form −iGμν a1 (p) =
−g μν + pμ pν /m2a1 , s − Ma21 (s) + ima1 Γa1 (s)
where s = p2 , ma1 is the nominal mass of the a1 , and Γa1 (s) is the energy-dependent total width. The running mass is normalized by Ma21 (m2a1 ) = m2a1 . A similar propagator was used in the seminal paper by Isgur, Morningstar, and Reader [4] and recently also by the CLEO collaboration [7]. In our work, Γa1 (s) is the sum of the partial decay widths of the a1 → ρπ → 3π, ¯ a1 → K ∗ K ¯ → K Kπ ¯ and a1 → K K¯∗ → K Kπ, a1 → f0 π → 3π channels. Ma21 (s) has been calculated from a twice-subtracted dispersion relation with Γa1 (s) as input.
214
P. Lichard, M. Vojík / Nuclear Physics B (Proc. Suppl.) 198 (2010) 212–215
The ρ(770) propagator was taken from [8]. It reads −iGμν ρ (pρ ) =
−g μν + pμρ pνρ /m2ρ , t − Mρ2 (t) + imρ Γρ (t)
where t = p2ρ and Γρ (t) is the total width of the √ ρ-resonance with off-shell mass t, normalized at t = m2ρ to the nominal width Γρ . Furthermore, Mρ2 (t) is the running mass squared satisfying condition Mρ2 (t = m2ρ ) = m2ρ . The total decay width Γρ (t) includes, besides the ρ → ππ contribution, ¯ and ηππ channels. These adalso the ωπ, K K ditions proved to be important for getting a reasonable behavior of Mρ2 (t). For details, see [8]. The last refinement concerns the a1 ρπ and ρππ vertices. The effective Lagrangian approach takes hadrons as elementary quanta of the corresponding fields, ignoring thus their internal structure. As a consequence, the interaction strength is overestimated at higher momentum transfers. To describe the interaction among participating mesons more realistically, we explore the chromoelectric flux-tube breaking model of Kokoski and Isgur [9]. Each interaction vertex is modified by the factor exp{−p∗ 2 /(12β 2 )}, where p∗ is the three-momentum magnitude of the lighter particles in the rest frame of the heaviest one (virtual masses are taken in the intermediate states). The authors of [9] advocated the value of β ≈ 0.4 GeV, but we consider β a free parameter. 4. Data and results We will compare the three-pion mass distribution calculated from our model with various data sets. There are five parameters we want to determine by minimizing the usual χ2 : (i) the a1 mass ma1 , (ii) the a1 width Γa1 , (iii) the Lagrangian mixing parameter sin θ, (iv ) the Kokoski-Isgur cutoff β and, finally, (v ) an overall multiplicative factor, which does not have physical meaning because none of the data sets is absolutely normalized. As an experimental input we first use the data by the ARGUS Collaboration at DESY [10], namely their background and acceptance corrected three pion mass distribution. The data set consists of twenty-eight points in the mass range
0.425–1.775 GeV. We got a very good fit characterized by χ2 /N DF = 0.663 and the confidence level of 89.1%. Unfortunately, we also found a set of parameters with only a little worse χ2 , but with parameters incompatible with those from the main minimum. To resolve the ambiguity mentioned above, we took into consideration another set of data on the three-charged-pion decay of the tau lepton, namely that of the OPAL collaboration [11] at the CERN Large Electron-Positron Collider (LEP). It consists of twenty points in the three-pionmass-squared plot corrected for background and efficiency. The number of free parameters of our model increased to six because each data set requires their own multiplication factor. The minimizing of χ2 =χ2 (ARGUS)+χ2 (OPAL) yielded the following results: χ2 /NDF=30.66/42, Confidence Level = 90%, ma1 = (1.37 ± 0.01) GeV, Γa1 = (0.61 ± 0.01) GeV, sin θ = 0.54 ± 0.02, β = (0.24 ± 0.02) GeV. The errors are defined in the usual way [12]. The mass and width of the a1 we obtained are a little higher than the values in current Review of Particle Physics [13]. The central value of the Lagrangian mixing parameter sin θ lies outside the interval found in [1,2] (0.40–0.52), but, keeping in mind the error, it is still compatible with the e+ e− → 4π results. The τ -lepton decay into three pions and neutrino was also investigated by the CLEO Collaboration at the Cornell Electron Storage Ring (CESR). Unfortunately, their results on the allcharged-pions channel still exist only in a preliminary form [14]. We therefore used their data on the π − π 0 π 0 channel [7], which is, in our approach, described by the same parameterization as the charged channel. Unfortunately, we were not able to get a reasonable fit, value of χ2 remained very high. The probable reason is the ab− sence of the a− 1 → f0 π intermediate state, which differentiates between π − π + π − and π − π 0 π 0 systems in the final state.
P. Lichard, M. Vojík / Nuclear Physics B (Proc. Suppl.) 198 (2010) 212–215
Figure 2 shows the scan of χ2 over all values of sin θ for all three data sets we considered. The a1 mass and width were kept at their optimal values. It is obvious that the quality of the fit is very sensitive to the value of the mixing parameter. Only a relatively narrow interval around sin θ ≈ 0.5 can provide a good fit, as testified by all three data sets. This observation agrees with the result of the electron-positron annihilation into four pions, depicted in Figure 2 by a short horizontal abscissa.
215
termine those free parameters from fit to data on several different processes. In this contribution we considered the three-pion (plus neutrino) decay of the tau lepton measured by three experimental groups. The best fit to all data requires the Lagrangian mixing parameter falling roughly in the same region as the fit to completely different process–the electron-positron annihilation into four pions. It indicates that the form of the effective Lagrangian we use is reasonable. In the future we will proceed in two directions. Firstly, we intend to improve our model in order to describe the decay τ − → π − π 0 π 0 ντ better. Secondly, we will combine our model of the threepion decay of the τ lepton with the model of the electron-positron annihilation into four pions [1,2] to get a common fit to those two processes and thus determine the a1 ρπ vertex more precisely. REFERENCES
Figure 2. Dependence of χ2 on the a1 ρπ Lagrangian mixing parameter for fixed values both of the a1 mass and width.
5. Conclusions and comments Our investigation shows that the form of the a1 ρπ vertex effects strongly the agreement of a model with data and also the values of the fitted parameters. In the situation when we do not know the “true” form of the a1 ρπ Lagrangian (and of the corresponding vertex) from the first principles it is necessary to work with an effective Lagrangian that contains free parameters and de-
1. P. Lichard and J. Jur´ an ˇ , Phys. Rev. D 76 (2007) 094030. 2. J. Jur´ an ˇ and P. Lichard, Phys. Rev. D 78 (2008) 017501. 3. T. N. Pham, C. Roiesnel and Tran N. Truong, Phys. Lett. B 78 (1978) 623. 4. N. Isgur, C. Morningstar and C. Reader, Phys. Rev. D 39 (1989) 1357. 5. L. Xiong, E. Shuryak and G. E. Brown, Phys. Rev. D 46 (1992) 3798. 6. G. Janssen, K. Holinde, and J. Speth, Phys. Rev. C 49 (94) 2763. 7. D. M. Asner et al. (CLEO Collaboration), Phys. Rev. D 61 (1999) 012002. 8. P. Lichard, Phys. Rev. D 60 (1999) 053007. 9. R. Kokoski and N. Isgur, Phys. Rev. D 35 (1987) 907. 10. H. Albrecht et al. (ARGUS Collaboration), Z. Phys. C 58 (1993) 61. 11. R. Akers et al., Z. Phys. C 67 (1995) 45 12. F. James and M. Roos, Comput. Phys. Commun. 10, 343 (1975). 13. C. Amsler et al. (Particle Data Group), Phys. Lett. B 667 (2008) 1. 14. E. I. Shibata (representing the CLEO Collaboration), Nucl. Phys. B (Proc. Suppl.) 123 (2003) 40.