About the creation of proton-anti-proton pair at electron-positron collider in the energy range of the mass $\Psi(3770)$

The process of electron--positron annihilation into proton--antiproton pair is considered within the vicinity of $\psi(3770)$ resonance. The interference between the pure electromagnetic intermediate state and the $\psi(3770)$ state is evaluated. It is shown that this interference is destructive and the relative phase between these two contributions is large ($\phi_0 \approx 250^o$).


Introduction
Large statistics of J/ψ, ψ(2S) and ψ(3770) samples have been obtained in recent years by BEPCII/BESIII facility [1]. It provides the possibility to study many decay channels of J/ψ, ψ(2S) and ψ(3770) resonances. In a profound work, BESIII has measured the phase angle φ between the continuum and resonant amplitudes [2] and found two possible solutions, which are φ = (266.9 ± 6.1 ± 0.9) o or φ = (255.8 ± 37.9 ± 4.8) o . This means that the strong decay amplitude and electromagnetic decay amplitude are almost orthogonal. The BES III data were taken as an energy scan in the vicinity of ψ(3770). The data show some structure: clearly seen dip in the energy strip of size of the resonance ψ(3770) width, which had been observed previously by CLEO collaboration for some mesonic decay channels [3].
In this note we try to explain this rather specific behavior of the total cross section of process e + e − → pp in the energy range close to resonance ψ(3770) creation.
In contrast to the channel e + e − → ψ(3770) → µ + µ − , in process of hadron creation (i.e. e + e − → ψ(3770) → π + π − ,pp,nn), a QCD gluonic state contribution to the hadron (in particular nucleon) formfactor ψ →pp is to be investigated. Besides the Breit-Wigner character of the amplitude, one must take into account the specific character of interaction of quarkonium to nucleon-antinucleon pair mediated through 3 gluon intermediate state and the final state interaction of the created nucleon pair.
The second effect is the final state interaction phase of amplitude which arise mostly from large distances (or soft exchanges of final stable hadrons). It has the same form for γ * →pp and for ψ →pp vertexes and we can safely assume its cancellation in the interference of pure QED and quarkonium states.
On the contrary, the phase which arises from 3 gluon state can essentially affect on the Breit-Wigner character of pure QED final state.
It is the motivation of this paper to investigate the detailed behavior of the total cross section in the energy range within the mass of a narrow resonance ψ(3770).

Born approximation
We consider two mechanisms of creation of a pp in electron-positron collisions (see Fig. 1) One proceeds through virtual photon intermediate state (see Fig. 1(a)), leading to the contribution to matrix element where lepton J e µ and proton J p µ currents have a form: and G(s) is the model-dependent proton formfactor.
In the recent paper [4] the remarkable relation F 1 ( √ s ∼ 2 GeV) = 1, F 2 ( √ s ∼ 2 GeV) = 0 for proton form-factors near the threshold was obtained which meant, that proton in some environment near the √ s ∼ 2 − 3 GeV can be considered as a point-like particle. Assuming this facts and keeping in mind the closeness of the considered energy range to the pp threshold we put further G(s) = 1. The corresponding contribution to the differential cross section where m is the proton mass, √ s = 2E is the total energy in center of mass reference frame (cmf), E is the electron beam energy and the scattering angle θ is the cmf angle between the 3-momenta of the initial electron q − and the created proton p + . The total cross section then 3. The quarkonium ψ(3770) contribution: three gluon vertex The second mechanism (see Fig. 1(b)) describes the conversion of electronpositron pair to ψ(3770) with the subsequent conversion to the protonantiproton pair through three gluon intermediate state (see Fig. 2(a)).
For this aim we put the whole matrix element as where the contribution with ψ(3770) intermediate state is Here we assumed that vertex ψ → e + e − has the same structure as γ → e + e − , i.e.: and the constant g e is defined via ψ → e + e − decay (g 2 e = 12πΓ ψ→e + e − /M ψ ) thus giving g e = 1.6 · 10 −3 [5].
The current J ν (3g) which describes the transition of ψ(3770) with momentum q = 2p into proton-antiproton pair via three gluon intermediate state has the form (see Fig. 2(a)): where α s is the strong interaction coupling which is associated with each gluon line andÔ ν iŝ andÔ αβγ ν where p and M are the 4-momentum and the mass of the charmed quark (anti-quark) inside ψ(3770) state and one must take into account the contributions from all gluon lines permutations. Color factor describes the interaction of gluons with quarks of the proton. The quantity R is connected with wave function of ψ(3770) and is derived in Appendix A.
Thus the contribution to the total cross section arising from the interference of relevant amplitudes has the form where two-particle phase volume dΓ 2 is and θ is again the angle between the directions of initial electron q − and the produced proton p + . To perform the summation over spin states we use the method of invariant integration [6]: where Thus we get for the contribution to the total cross section where where H(β) and F (β) are correspondingly real and imaginary part of vertex ψ → 3g → pp, i.e. function Z(β). Our approach consists in calculation of the s-channel discontinuity of Z(β) with the subsequent restoration of real part H(β) with the use of dispersion relation. For this aim we use the Cutkosky rule for gluon propagators This allows us integrate over phase volume of three gluon intermediate state as where and c is the cosine of the angle between directions k 1 and k 2 . It is convenient to write the phase volume element in form where dΩ i is the phase volumes of the on mass shell gluons and c 1,2 ≡ cos(p + , k 1,2 ). So we obtain s-channel discontinuity of Z in the form: where C 1 = x 1 (1 − βc 1 ) and C 2 = x 2 (1 + βc 2 ) and the integration over phase volume dγ is performed in the kinematical region where D > 0. Explicit form of i∆Z and Q 1 are given in Appendix B. The angular integration can be performed using the form of the phase volume given above and the set of integrals given in Appendix B.
As we are interested in the energy region close to the mass of resonance, we use some trick to restore the real part of Z by means of dispersion relations. For this aim we do a replacement We use the Cauchy theorem (un-substracted dispersion relation) to obtain the real part: The quantity H(β) as a function of β is shown in Fig. 3.

The quarkonium ψ(3770) contribution: D 0 mesons loop vertex
It is known that the main contribution to the decay width of ψ(3770) arise from the OZI non-violating channels ψ(3770) →DD [5]. However the contribution ofDD state as an intermediate state converting to protonantiproton is expected to be small. The main reason for this is the absence of charmed quarks inside a proton. In this section we will estimate the contribution of D mesons loop to the process of our interest by using only  (18) and (24)) as a function of β.
D 0D0 loop in the vertex ψ → pp (see Fig. 2(b)). The amplitude of the process (1) with the ψ intermediate state which converts via D 0D0 loop into proton-antiproton we write in the form similar to (6): where current J µ D has a form: where g ψDD is the constant for vertex ψD 0D0 which can be estimated from the decay width Γ ψ→D 0D0 = 0.26 keV [5] which gives The loop integral in (26) diverges in case of point-like particles. Usually one uses some formfactor to cut this divergency [7,8]. Following this tradition we use formfactors for the vertex D 0 pΛ + c in the form [9]: where

Discussion
In order to see the relative contribution of different mechanisms to the phase we will consider first the contribution of three gluons in the intermediate state. The total cross section then has a form where and the quantities P and φ are defined as The function f (y, φ) is shown in Fig. 4. At the point of ψ(3770) resonance, β = β 0 = 0.86, we have both quantities F and H negative and thus the phase φ is equal to The ratio of the B(β 0 ) to P is It is known that the main contribution to the width of ψ(3770) arise from the OZI non-violating channels ψ(3770) →DD [5]. However the contribution ofDD state as an intermediate state converting to proton-antiproton is small. Main reason of it is the absence of charm quarks inside a proton. In order to demonstrate this we add the D-loop contribution δσ D from (30) to the cross section in (34), i.e.: and then, to calculate the phase φ, we need to use complete expressions for the amplitudes, i.e. S 3g (s) from (17) and S D (s) from (31). This gives the following result for the phase: and thus we conclude that D-meson loop contribution to the phase is rather small and the main contribution to the phase goes from three gluon intermediate state.
We should also notice that we did not evaluate the contribution of a square of amplitude with ψ(3770) intermediate state. It is small compared with the contribution of interference of Born amplitude with the one with ψ(3770) meson and will be estimated elsewhere. It does not exceed ten percents.
The quantities for phase φ in (37) and in (40) are in good agreement with recent experimental data for phase at BES III collaboration [2].
with e(k i ) and ǫ(q) are the polarization vectors of photons and the orthoopositronium respectively. The quantity A includes the information on the wave function of ortho-positronium. Operator Comparing this value with the known result Γ Ops = (2m e /(9π))(π 2 − 9)α 6 we conclude that Here we used the following formulae For the case of decay of ψ(3770) to three gluons with the subsequent turning them to hadrons we define the amplitude in the form similar to (A.2) (see Fig. A.5): with q = 2p and ε(q) are the momentum and the polarization vector of ψ(3770). The decay width then reads as: And comparing this result with the known one [13,14]: we conclude that if one assumes that α s ≈ 0.3. Note that both A and R are real.