

Low-Mass Proton-Antiproton Enhancement: Belle and Bes Results, Premises of Lear and Expectations From Clas

B.Kerbikov, A.Stavinsky, V.Fedotov
State Research Center
Institute of Theoretical and Experimental Physics,
Moscow, Russia

We present a simple explanation for the recently observed near-threshold proton-antiproton enhancement. It is described by a set of low-energy parameters deduced from the analysis of experiments at LEAR. We predict a related effect in photoproduction reaction under study by CLAS collaboration.

1 Introduction

Low-mass baryon-antibaryon enhancement has recently been observed in the decays [1], [2] and [3]. On the theoretical side this discovery has been discussed by several authors [4, 5, 6]. In [4] the near-threshold effect has been considered either as a gluonic state or as a result of the quark fragmentation process. In [5] it has been attributed to the Breit colorspin interaction while in [6] it has been regarded as caused by peripheral one-pion-exchange potential.

The history of bound-states and resonances in system (”baryonium”) started before the discovery of the antiproton and is full of controversial experimental findings and theoretical claims. The interested reader is addressed to review papers [7]-[11]. At the end of eighties an ample supply of diverse information of low-energy interaction has been obtained mainly due to the commissioning of the Low Energy Antiproton Ring (LEAR) at CERN. In the situation when theoretical models were at variance with each other and unable to explain the whole set of the observed phenomena the semi-phenomenological effective range analysis [12]-[14] turned out to be a coherent approach to low-energy interaction. In this approach the lack of the dynamical picture is traded for the possibility to reconcile within a unique scheme different pieces of information on interaction.

As we shall see the effective range analysis [12]-[14] based on the old LEAR data enables to explain Belle [1, 2] and BES [3] results. In particular the observation by BES of a strong effect in the decay and the absence of a similar structure in perfectly fits into the solution for the low-energy parameters obtained 15 years ago [12]-[14]. We shall return to this point below. An immediate consequence of the proposed scheme is the prediction of a related effect in the reaction . This process can be investigated by CLAS collaboration at Jefferson Lab. Unlike - and decays the photoproduction reaction has not been discussed from the theoretical side. Therefore we shall choose it as a starting point in order to introduce the effective range formalism which also applies to Belle and BES data.

The double differential cross section for the reaction is given by the well-known Chew-Low expression


where is the nucleon mass, is the energy of ,


with and being the 4-momenta of and forming the low-mass pair, being the 4-momentum of the remaining proton, is the standard kinematical function. The angle is defined in the CM system of the particles (2,3). The variable is the square of the invariant mass of the (2,3) pair. It follows from (1) that the angular distribution of particles 2 and 3 ( the low-mass pair) is isotropic in their CM system provided is independent of which is the case for and states. Equation (1) yields the following invariant mass distribution


where is the angle between and the direction of the incident ( we remind that the index 1 is attributed to the proton which remains outside the correlated pair).

In the spirit of the Migdal-Watson FSI theory we single out from the matrix element a factor responsible for the low energy interaction. Within Migdal-Watson approach one has


where is the CM momenta, , is the Jost function corresponding to the interaction at low energy, is called the enhancement factor. As it was already mentioned the dynamics of interaction is much more complicated than that of [7] -[11]. Annihilation dominates at short and possibly intermediate distances, - and other odd -parity exchanges lead to a strong attraction to be added to the one-pion exchange in the outer region [15]. Therefore the approximation of by a Born term from one-pion exchange [6] misses the essential features of dynamics. On the other hand effective range solution for the low-energy amplitude describes the whole set of data with a fair accuracy [12]-[14].

Ignoring for the moment complications due to annihilation, Coulomb interaction and spin-isospin structure we may write the following expression for Jost function in the scattering length approximation


where is the scattering length and is a well defined smooth function with a property . A similar expression for system is not as simple for three reasons:

(i) system is a combination of the two isospin states with ; isospin invariance is violated by the mass difference of proton and neutron;

(ii) a powerful annihilation results in the complexity of the scattering lengths;

(iii) the Coulomb attraction acts in the system.

One should also keep in mind that there are two spin states with . In the absence of polarization experiments only spin-averages quantities were extracted from the experimental data [12]-[14] (see also [16]).

The enhancement factor in which all three points (i) -(iii) inherent for the system are accounted for has the following form [12]-[14]




is the Sakharov Coulomb attraction factor [17] with fm being the Bohr radius of the atom,


where are -wave scattering lengths with isospin ,


with The quantity is the momentum in the channel,


(recall that is the proton mass).

The point MeV/c corresponds to the threshold; below this point momentum becomes imaginary, Parameter is the Schwinger correction to the scattering length [18], fm [12]-[14].

The enhancement factor (6) does not include the effective range term and the contribution from nonzero orbital momenta. According to [12]-[14] the effective range term is of minor importance up to MeV/c i.e. to MeV. We note in passing that for a multichannel system the effective range may be negative or even complex [19]. The -wave comes into play a little earlier especially as far as the total cross section is concerned. Inclusion of -wave brings about two problems. First, the -wave scattering length (it has a dimension of fm) is very sensitive to the fitting procedure (see [12]-[14]) and the present set of the experimental data do not warrant a stable solution for the amplitude. The second problem is the following. With -wave included equations become rather cumbersome and the number of parameters increases substantially. This may cause unnecessary doubts in the reliability of the proposed approach. We plan to consider effective range terms and amplitudes in the next publication. This would allow to analyze the correlation function in a wider energy range.

The most remarkable result of all fits [12]-[14] for low energy parameters is a clear dominance of the -wave with over that with . The corresponding scattering lengths read


where (11) and (12) are respectively the results of [13] and [14]. The sign convention in [12]-[14] was . We see that the absolute value of is much larger than that of . This is completely in line with the observation by BES of a strong low-mass effect in the decay process and the absence of a similar structure in [3].

The negative sign of may be interpreted as an indication that the potential in this channel is either repulsive or on the contrary attractive and strong enough to produce a bound state somewhere below the threshold. However such a simple interpretation is not obvious in strong annihilation regime. This problem is beyond the scope of the present work.

Coming back to Eq.(3) for the effective mass distribution we note that due to Coulomb attraction it has a tiny jump at threshold since at and it has cusp at due to the opening of the channel.

As it is clear from Eqs. (3-4) the near threshold region is dominated by the interplay of the phase-space factor and the enhancement factor . The first one is trivial and plays a role of a universal background.

The two solutions given by (11) and (12) are somewhat different. This is due to several reasons described in [12]-[14]. The main source of ambiguity is the lack of the experimental data at very low energies. Therefore, Belle, BES and future CLAS data which we have interpreted in the framework of the effective range solution may in their turn help to update the solution itself.

In Fig.1 we plot the enhancement factor as a function of


The solid curve corresponds to the solution (11) while the dashed one to the solution (12).

The authors are indebted to L.N.Bogdanova, K.Michailov and Yu.A.Simonov for useful discussions and clarifying remarks.


Fig.1 The near-threshold behaviour of the enhabcement function , .See the text for the description of the curves.
