A New State of Baryonium

The recent discovery of a narrow resonance in the decay $J/\psi \to \gamma p \bar{p}$ is described as a zero baryon number, ``deuteron-like singlet ${}^1S_0 $'' state. The difference in binding energy of the deuteron (-2.225 MeV) and of the new state (-17.5 MeV) can be accounted for in a simple potential model with a $\lambda \cdot \lambda$ confining interaction.


I. INTRODUCTION
There has been a recent observation of a near-threshold narrow enhancement in the PP invariant mass spectrum from the radiative decays J/ψ → γPP by the BES Collaboration [1] who also report seeing nothing similar in the decay J/ψ → π 0 PP . The enhancement can be fit with either an S-or P-wave Breit Wigner resonance function. In the case of the S-wave fit, the peak mass is below 2m P at M = 1859 +3 −10 (stat) +5 −25 (sys)MeV /c 2 with a total width Γ < 30MeV /c 2 at the 90% percent confidence level. The structure has properties consistent with either a J P C = 0 −+ or J P C = 0 ++ quantum number assignment. The mass and width values are not consistent with any known meson resonance near this mass. Recently Belle has reported also observations of the decays showing enhancements in the PP invariant mass distributions near 2m P . In addition to this probable spin zero state, there is also the report [4] of a narrow, S-wave triplet PP resonance at a mass of 1870MeV /c 2 with a width of 10MeV /c 2 and J P C = 1 −− .
There have been some signs of an anomalous behavior in the proton-antiproton system at a mass of 2m P and since the 1960's there have been suggestions of states of nucleonantinucleon, sometimes called baryonium. The name has also been invoked for states containing two quarks and two antiquarks. An example is the MIT bag model by Jaffe [5] which postulates the existence of baryonium for states made up of two quarks and two antiquarks. For a historical review see [6]. In fact, the recent observations of an unexpectedly light narrow resonance in D + s π 0 with a mass of 2317 MeV by the BaBar collaboration [7], together with a possible second narrow resonance in D s π 0 γ with a mass 2460MeV /c 2 have led, among other explanations [8,9], to a multi-quark anti-quark model [10]. The mass difference between the D * s (2317) and the well established lightest charm-strange meson, D s , is ∆M = 350MeV /c 2 . This is less than the kaon mass, thus kinematically forbidding the decay D * s (2317) → D u,d + K. The possible resonance at 2460MeV /c 2 also has such a mass difference when taken with the lighter D * state; while this may be a artifact of a "feed-up" or "feed-down" mechanism [11] it is quite likely that both states may exist independently.

II. NUCLEON-NUCLEON AND NUCLEON ANTI-NUCLEON INTERACTIONS
For over fifty years there has been a general understanding of the nucleon-nucleon interaction as one in which there is, in potential model terms, a strong repulsive short distance core together with a longer range weaker attraction. Also, there have been many indications that in the nucleon anti-nucleon system, there should be a strong attractive NN bound state near threshold [12,13]. This understanding evolved to attribute the long force to be that of pion exchanges and the repulsive short-range interaction to that of ω exchange [14].
In a nuclear physics approach this idea of meson exchange has evolved into an accurate phenomenological way to describe experiments.
Later potential models, such as the Bonn potential [15], were based on quantum chromodynamics (QCD). However, the Bonn potential model ended up as a ten parameter model [16] and its connection with concepts such as one-gluon exchange are tenuous. Many of these models were based on the non-relativistic quark model or on the MIT bag model [17,18].
The modern view is that there is a color interaction of a λ · λ type between pairs of quarks.
The nucleon-nucleon or nucleon-antinucleon effective potential then arises from the residual color forces. However to establish the connection between the effective potential and the color forces in practice requires somewhat ad hoc assumptions involving either resonatinggroup methods [19], variational techniques [20] or quark Born perturbative methods [21].
In nuclear physics models the potential for NN is more attractive than that for NN; this is usually considered due to strong omega exchange which is repulsive for NN and attractive in NN . However, the idea of ω exchange should not be taken literally [21] since there is a mismatch of the ranges involved; 1/m ω ≈ 0.2f m whereas the nucleon radius is about 1f m.
In the early String/Regge approach to NN [22,23] there are narrow NN states based on selection rules. For example, there is a large baryon anti-baryon effect near threshold. In all of these approaches there is a trade-off between ranges -the annihilation radius is shortrange of about 1/2M N ≈ 0.1fm, while the long range potentials are dominated by meson exchanges.

III. A SIMPLE TOY MODEL
Within the modern QCD approach, it is the λ · λ color interaction that plays an important role in trying to understand the few nucleon problem [20]. Nevertheless, the actual calculational details rely on other time-honored techniques as mentioned above. Here we wish to propose a model that has as its basis the 6-quark state making up the deuteron. It is known that in the triplet neutron-proton system there is only one bound state (the deuteron -3 S 1 ) with a binding energy of −2.225 MeV. There is also a large singlet scattering state, the virtual 1 S 0 , often called a virtual 1 S 0 state, with an energy just above zero of 0.0382 MeV [24,25]. A simple phenomenological model of the deuteron consists of using a square well potential [24,25,26] with a depth sufficient to bind the isoscalar 3 S 1 state but not quite deep enough to bind the 1 S 0 state. Then the equation for a bound state is and β = √ 2ME, where V is the depth of the potential, E, the binding energy and a the size Our approach uses the fact the potential between two quarks due to the λ · λ color interaction gives an attraction factor of −2/3. In the case of qq, the potential becomes even more attractive by a factor of two [20,27]. Whether this factor of two translates into a similar doubling of the phenomenological potential is not obvious. We will solve for the attractive force to fit the binding energy of the new PP state (17.5MeV ). It turns out from Eq. (1) the solution is V = 64MeV , surprisingly just a factor of 1.76, very close to two, deeper! Such a stronger attractive force, such as that expected from the color factor in the potential would seem to be consistent with the new 0 −+ being a real 1 S 0 bound state. In the case of the deuteron, we might assign the role of hyperfine interactions to raise the effective potential from that which binds the 3 S 1 "deuteron" to a value which just fails to bind the virtual 1 S 0 state. For NN baryonium we expect that the annihilation is a short range phenomenon, which can modify the affect of the short range hyperfine interactions making its role in the PP system unclear. Hence there is no simple way to predict the potential for the 3 S 1 PP state. Also, while there is a clear distinction between the spin-one P N deuteron ( 3 S 1 ) being isoscalar and the spin-zero 1 S 0 being isovector, no similar distinction can be made for the nucleon anti-nucleon state since both I = 0 and I = 1 states can exist with either spin-zero or spin-one [28] However, this does not mean they should all be seen in the J/ψ → γPP as we show in table 1.

IV. SUMMARY AND CONCLUSIONS
We have presented a simple model where we account for the new 0 −+ state of PP being a bound state of baryonium comparable to the 1 S 0 virtual bound state of the deuteron.
This would imply that the "deuteron" equivalent 3 S 1 state may also exist although we do not have any guidance on how to derive the size of the equivalent potential. The 1 −− state at 1870MeV /c 2 seen in the e + e − → PP would appear to be a suitable candidate. It also would appear likely that similar types of baryonia should exist; for example, a calculation similar to that described above but with the Λ mass substituted for that of the proton predict another ΛΛ 0 −+ state with a binding energy of 31MeV at a mass of 2200MeV /c 2 .
Therefore, the idea that these resonances could be analogous to the "virtual bound state" in the N-P system implies that further resonances should be expected. Rosner has also looked at baryon anti-baryon enhancements in B decays [29]. He also notes that there is a whole new interesting set of B decays possible involving exotic mesons and baryons.

V. ACKNOWLEDGMENTS
We thank Steve Olsen for bringing this new effect to our attention and for discussions.
This work is supported by the Natural Science and Engineering Council of Canada (NSERC) under grant number A3828.