Threshold J/ψ production in nucleon-nucleon collisions

We analyse J/ψ-production in nucleon-nucleon collisions near threshold in the framework of a general model-independent formalism, which can be applied to any reaction N + N→N + N + V0, where V0 = ω, ϕ or J/ψ. Such reactions show large isotopic effects: a large difference for pp and pn collisions, which is due to the different spin structure of the corresponding matrix elements. The analysis of the spin structure and of the polarization observables is based on symmetry properties of the strong interaction. Using existing experimental data on the different decays of the J/ψ meson, we suggest a model for N + N→N + N + J/ψ, based on t channel η + π exchanges. We predict polarization phenomena for the n + p→n + p + J/ψ reaction and the ratio of cross sections for np and pp collisions. For the processes η(π) + N→N + J/ψ we apply two different approaches: vector meson exchange and local four-particle interaction. In both cases we find larger J/ψ production in np collisions than in pp collisions.


Introduction
It is well known that the J/ψ meson has been observed in two different reactions: in p + Be collisions [1] and e + e − [2].
Since that time experimental and theoretical studies of J/ψ production have been going on. As a result of high statistics and high-resolution experiments, a large amount of information on the properties of the J/ψ meson, on its production processes and on its numerous decays has been collected. From a theoretical point of view, the interpretation of the data, in particular in the confinement regime, is very controversial. As an example, the c-quark mass is too large, if 68.3 applied for q ≤ m c , where q is the J/ψ three-momentum in the reaction centre of mass (CMS). This paper is organized as follows. In section 2 we establish the spin structure of the threshold matrix for the two NN processes: p + p → p + p + J/ψ, p + n → p + n + J/ψ in terms of partial S-wave amplitudes, and calculate the simplest polarization observables in terms of these amplitudes. In section 3 we treat the dynamical issue in terms of t-channel exchange mechanisms by light mesons. We give numerical predictions in the framework of a π + η model. The experimental data about different hadronic decays of the J/ψ meson may give constraints on our predictions.

Spin structure of threshold matrix elements and polarization phenomena
In the general case, the spin structure of the matrix element for the process N+N → N+N+V 0 is described by a set of 48 independent complex amplitudes, which are functions of five kinematical variables [37]. The same reaction, in coplanar kinematics, is described by 24 amplitudes, functions of four variables. In collinear kinematics the number of independent amplitudes is reduced to seven and the description of this reaction is further simplified in the case of threshold V 0 -meson production, where all final particles are in the S state.
Applying the selection rules following from the Pauli principle, the P invariance and the conservation of the total angular momentum, it is possible to prove that the threshold process p + p → p + p + V 0 is characterized by a single partial transition: where S i (S f ) is the total spin of the two protons in the initial (final) states, i is the orbital momentum of the colliding protons, j is the total angular momentum, and P is the parity for the colliding pp system. In the CMS of the considered reaction, the matrix element corresponding to transition (1) can be written as: where χ 1 and χ 2 (χ 3 and χ 4 ) are the two-component spinors of the initial (final) protons; U is the three-vector of the V 0 -meson polarization,ˆ k is the unit vector along the three-momentum of the initial proton; f 10 is the S-wave partial amplitude, describing the triplet-singlet transition of the two-proton system in V 0 -meson production and σ = (σ x , σ y , σ z ) are the standard Pauli matrices.
In the case of np collisions, applying the isotopic invariance for the strong interaction, two threshold partial transitions are allowed: with the following spin structure of the matrix element: complex functions, depending on the energies E, E and E V , where E(E ) and E V are the energies of the initial (final) proton and of the produced V 0 -meson, respectively. Note that f 10 is the common amplitude for pp and np collisions, due to the isotopic invariance of the strong interaction. This explains the presence of the coefficient 2 in equation (2). The parametrizations (2) and (4) are model-independent descriptions of the spin structure for threshold production of any vector meson in NN collisions, N + N → N + N + V 0 , from the light ρ, ω and φ to J/ψ, ψ ,ψ , including the vector bottonium: Υ(1S), Υ(2S) and even the hypothetical exotic vector toponium states. All dynamical information is contained in the partial amplitudes f 01 and f 10 , which are different for the different vector particles. On the other hand, some polarization phenomena have common characteristics, essentially independent of the type of vector meson. For example, vector mesons produced in pp and np threshold collisions are transversally polarized, and the elements of the density matrix ρ are independent of the relative values of the amplitudes f 01 and f 10 : ρ xx = ρ yy = 1 2 , ρ zz = 0. Therefore, the angular distribution shows a sin 2 θ P dependence for the subsequent decay V 0 → P + P (where P is a pseudoscalar meson) and the (1 + cos 2 θ) dependence for the decay V 0 → µ + + µ − , where θ (θ P ) is the angle betweenˆ k and the µ − (P ) momentum (in the rest system of V 0 ). Possible deviations from this behaviour have to be considered as an indication of the presence of higher partial waves in the final state.
All other one-spin polarization observables, related to the polarizations of the initial or final nucleons, identically vanish, for any process of V 0 -meson production.
The dependence of the differential cross section for threshold collisions of polarized nucleons (where the polarization of the final particles is not detected) can be parametrized as follows: where P 1 and P 2 are the axial vectors of the beam and target nucleon polarizations, dω is the element of phase-space for the three-particle final state. The spin correlation coefficients A 1 and A 2 are real and they are different for pp and np collisions: with the following relations −A 1np + A 2np = 1 and 0 ≤ A 2np ≤ 1.
Defining R as the ratio of the total (unpolarized) cross section for np and pp collisions, taking into account the identity of final particles in p + p → p + p + V 0 , we find: So the following relation holds: The polarization transfer from the initial neutron to the final proton ( n + p → n + p + V 0 ), can be parametrized as follows: 68.5 with a simple expression, which relates the real coefficients P 1np and P 2np to the partial amplitudes f 01 and f 10 : where δ is the relative phase of f 01 and f 10 , which is non-zero, in the general case. For the process p + p → p + p + V 0 the relation P 1pp = P 2pp = 0 holds, for any vector meson V 0 .

The dynamics of the t channel
The parametrization of the spin structure of the threshold matrix elements given above is based on fundamental symmetry properties. It is therefore model independent and can be applied to any reaction mechanism. Following the standard way of describing the nucleon-nucleon interaction, we will apply t-channel π 0 , η, σ and ρ(ω) meson exchanges to J/ψ production, too. Such an approach has been used to describe the production of light vector mesons such as φ and ω [30]- [34]. The reaction threshold for p + p → p + p + J/ψ in the laboratory system (LAB) is quite large. However, the formalism of Pomeron exchange cannot be applied here, even at such large energies, because the Regge picture is valid when not only the initial energy is large, but also the excitation energy: the quantity (W − W th )/W th (where W is the total energy) has to be essentially larger than unity.
Another important kinematical variable is the momentum transfer squared, t = (p 2 − p 4 ) 2 , where p 2 and p 4 are the four-momenta of the target and of the scattered nucleon. At threshold, one can find that the variable t has only a fixed value, t = −mm V , where m is the nucleon mass and m V is the V-meson mass. So, for J/ψ production this momentum is large: t −3 GeV 2 , therefore all the propagators corresponding to the light mesons are of comparable magnitude In such a situation it is not possible to justify the dominance of a particular exchange mechanism, so we have to conclude that threshold heavy V 0 -meson production in NN collisions is determined by the exchange of the coherent sum of many different mesons, with different masses.
But what about the quantum numbers, J P , of these exchanges (J is the spin and P is the parity of the corresponding meson)?
We can use the parametrizations (2) and (4), which are exact and model-independent results, with definite selection rules, from the point of view of the s channel for N + N → N + N + V 0 to understand the t channel J P picture. Using the Fierz transformations (in two-component form), let us rewrite the general parametrization of the matrix element, see equations (2) and (4), as a t-channel parametrization: Each term in equation (10) has a precise dynamical interpretation, as it corresponds to t-channel meson exchange (figure 1) with a definite spin and parity, J P . The first two terms describe a scalar exchange, where the spin structure σ · ( U * ×ˆ k) (in one vertex) corresponds to the matrix element of the process σ * + N → V 0 + N (σ * is a virtual scalar meson) at its threshold. The other vertex, corresponding to the σNN interaction has a structure of the type χ † Iχ. The last Figure 1. Feynman diagrams for the process n + p → n + p + V 0 , describing t-channel exchanges by neutral σ, η and π 0 mesons. two terms describe the exchange by neutral pseudoscalar mesons (π 0 or η) with the σ ·ˆ U -spin structure of the matrix element for the subprocess π 0 * (η * ) + N → V 0 + N at threshold, and with the σ ·ˆ k structure for the vertex NNπ(η). The same considerations hold for the f 10 partial amplitude. Therefore we can conclude that t-channel exchanges with J P = 0 + (scalar mesons) and J P = 0 − (pseudoscalar mesons) can be considered as the most probable mechanisms to describe the threshold dynamics of J/ψ production in NN collisions. Let us consider these mechanisms in detail.

σ exchange
The matrix element M σ , corresponding to the two diagrams of figure 1, can be written as: with the following expression for the matrix element M 1σ , corresponding to figure 1(a): where g σN N is the σNN coupling constant, I is the unit 2 × 2 matrix, N = 2m(E + m) = m(m V + 4m) is a normalization factor, which arises from the transformation of the invariant matrix element for the considered process (in terms of the four-component spinors for the initial and final nucleons) to the two-component form, which is better adapted to the description of threshold spin structure in the CMS of the considered process. We used the following formula for the threshold energy of the initial nucleons, E = W/2 = m + m V /2. The complex amplitudes h 1σ and h 2σ describe two possible threshold partial transitions in σ * + N → V 0 + N: where σ is the orbital momentum of the initial σN system. One can find that: where h 1/2 and h 3/2 are the partial amplitudes corresponding to the two possible values of the total angular momentum in σ * + N → V 0 + N.
Comparing the spin structure of the matrix elements for the processes n + p → n + p + V 0 , equation (4) and σ + N → N + V 0 , equation (11), one can see that only the amplitude h 2σ has to be kept, because it generates transversally polarized V 0 mesons. First of all, this means that the 68.7 cross sections of the processes n + p → n + p + V 0 and σ + N → N + V 0 are determined by different combinations of the amplitudes h 1σ and h 2σ . This is a result of the following formulae for the corresponding cross sections: Therefore these cross sections must be independent. Moreover, both amplitudes f 01 and f 10 , for n + p → n + p + V 0 in the case of σ exchange, being proportional to h 2σ , have to satisfy simple relations. In such a case, definite numerical values can be derived for the polarization observables in V-meson production for np collisions. To prove this, let us transform the matrix element (11) into the 'standard' parametrization of equation (4), in terms of definite quantum numbers of the s channel: where A = (ˆ k × U * ). From equation (12) one can see that M 1σ contains not only the structures which are allowed by symmetry selection rules, but also a contribution which corresponds to a triplet-triplet transition in the np system (last term in equation (12)). Such a transition is forbidden by the generalized Pauli principle, following from the isotopic invariance of the strong interaction, and should not appear in the total matrix element M σ . Let us consider in a similar way the matrix element M 2σ : where we applied the following relations: g σpp = g σnn = g σN N , h iσ (σn → nV 0 ) = h iσ (σp → pV 0 ), i = 1, 2, which follow from the isotopic invariance of the strong interaction, in the case of an isoscalar vector meson. Note that here we use the same propagator as in equation (11). This is correct in threshold conditions, because any different propagator will generate higher waves in the initial and final states. Summing the two contributions in the matrix element, M σ , the wrong term corresponding to the triplet-triplet transition in n+p → n+p+V 0 is exactly cancelled: . Therefore, for σ exchange, one finds: Independently of the numerical values of the coupling constant g σN N and of the partial amplitude h 2σ for the process σ + N → N + V 0 , the polarization observables for the process n+p → n+p+V 0 and the ratio R (see equation (7)), take the following values (in the framework of σ exchange): Note that introducing phenomenological form factors in the expression for M σ affects the absolute value of the cross section, but cannot change the relation f

η exchange
Similarly to σ exchange, the η exchange is characterized by two diagrams (figure 1) and the matrix element is the sum of two matrix elements, corresponding to figures 1(a) and (b): where the matrix element M 1η can be written as: where h 1η and h 2η are two independent partial amplitudes, which describe the threshold spin structure for the subprocess η * + N → V 0 + N. These amplitudes correspond to to two allowed threshold partial transitions (in η * + N → V 0 + N): i = 0 → j P = 1/2 − and i = 2 → j P = 3/2 − . The factor k/(E + m) = m V /(m V + 4m) arises from the transformation from the relativistic expression of the ηNN vertex, u(p 2 )γ 5 u(p 1 ), to the twocomponent form in the CMS of the n + p → n + p + V 0 reaction, σ ·ˆ k. Again, one can prove that only the sum M 1η + M 2η generates the correct spin structure for threshold matrix element M η (again taking into account the isotopic invariance for both vertexes of the considered diagrams: ηNN and η + N → N + V 0 , N = p or n): with the following relation for the partial amplitudes for n + p → n + p + η: and definite numerical predictions for the polarization phenomena and for the ratio R: So, only the coefficient P 1np can discriminate between η and σ exchanges.

π exchange
Due to the isotopic invariance of the strong interaction, it is necessary to consider four Feynman diagrams, corresponding to the exchange of neutral and charged pions in n + p → n + p + V 0 (see figure 2). Taking into account the isotopic relations between different coupling constants in NNπ vertexes and different amplitudes for the processes π + N → N + V 0 , one can find the following expressions for the amplitudes f (π) 10 and f (π) 01 : and the single amplitude f (π) 10 for the process p + p → p + p + V 0 is equal to 2f (π) 10 (np → npV 0 ). Note that the spin structure for the processes π + N → N + V 0 and η + N → N + V 0 has to be similar.
which are very different from the previous cases of pure σ or η exchanges.

'Realistic model': π + η exchange
Based on the previous results for t-channel exchanges, we can build a more realistic model, combining the contributions of different mesons. As an example, let us consider the case of π + η exchange, with the following expressions for the allowed threshold partial amplitudes f 10 and f 01 of the process n + p → n + p + V 0 : where characterizes the relative role of η and π exchanges in n + p → n + p + V 0 . Therefore, we can find the following results for the polarization observables in n + p → n + p + V 0 and for the ratio R of the total cross section for n + p and p + p collisions: A 1np = − 9 + 6Re r + |r| 2 2(5 + 2Re r + |r| 2 ) , This primarily means that, in the framework of the considered model, two independent parameters Rer and |r| 2 enter into the definition of three observables. Therefore a simultaneous measurement of P 1np and A 1np can determine uniquely Re r and |r| 2 (with the evident condition |Re r| < |r|): The situation is simplified if the ratio r is a real parameter. This is the case in the framework of the effective Lagrangian approach for the processes η(π) + N → N + V 0 , near threshold, where the corresponding pole Feynman diagrams originate the real amplitudes h 1η and h 1π . It is also the case for the s channel N * contribution, which is common to η +N and π +N interactions. For J/ψ production the first case seems to be the most probable.
For a real value of r, the following quadratic relation between polarization observables P 1np and A 1np holds: After measuring the ratio R (of total cross sections for np and pp interactions) it should be possible to find two different values for r: Knowing r, it is straightforward to predict any polarization observable for the process n + p → n + p + V 0 . The behaviour of R, A 1np and P 1np as functions of r, (when r is real) are shown in figures 3-5. One can see a strong dependence of the ratio R on r in the region −1 ≤ r ≤ 3, where R > 1, i.e. with strong isotopic effects. Only for r < −1 do we have R < 1, with a weak dependence on the parameter r. The coefficients A 1np and P 1np show particular sensitivity to r.

Attempts to estimate r
The previous analysis is based on the most general properties of threshold vector meson production in NN collisions and t-channel exchanges. In the last case, we used only properties related to the quantum numbers of the corresponding t-channel mesons: spin, parity and isotopic spin. All previous results are valid for any isoscalar vector meson production, ω, φ or J/ψ. The properties, which are specific to a definite V 0 -meson reaction, appear first of all in the kinematics, in particular in the different V 0 -meson masses, and in the values of the partial amplitudes for the binary subprocess, M * + N → N + V 0 , where M * = σ, η, π is a virtual meson (with space-like four-momentum). So, for the considered (η + π) model, all dynamical information, which is necessary for the calculations of such observables as R, A 1np and P 1np , is contained in one complex parameter r, which characterizes the ratio of the threshold amplitudes for η(π) + N → N + V 0 processes. Therefore the identity of the V 0 production is also contained in the ratio r.
Let us estimate this ratio in the case of J/ψ production. To do this, we refer to the existing experimental information about the different hadronic decays of the J/ψ meson. For example, the following branching ratios [38]: allow us to determine r in the framework of a simple vector exchange model for the process η(π) + N → N + N+J/ψ (see figure 6). The nice property of this model is that all the coupling constants are known. The ρ-exchange mechanism for J/ψ production in πN collisions has been considered earlier [39,40]. The corresponding matrix element can be written in the following form: where F 1 and F 2 are the Dirac and Pauli form factors of the V NN vertex of the considered diagram. At the reaction threshold, the matrix element, equations (26) and (27), can be simplified as: Taking into account VDM predictions for the (F 1 +F 2 ) term, which in the case of the ωNN(ρNN) vertex, is proportional to the isoscalar, µ p + µ n , (isovector, µ p − µ n ,) magnetic moment of the nucleon, one can find:

68.13
where g (Γ) is the corresponding coupling constant (decay width). All existing analysis of threshold η-meson photoproduction on protons [41] indicates that g ηN N g πN N . Therefore, in the framework of such a model, we have |r| 1. From these considerations we cannot determine the sign of the ratio r, but the small value of |r| indicates that the np cross section can be essentially larger than pp cross section, in the case of J/ψ production.
However, conclusions could be different if one takes another set of experimental data about J/ψ decays [38]: Evidently both these decays can be considered as crossed channels of the processes η(π) + N → N + J/ψ. Generally, each decay J/ψ → N + N + P (P is the pseudoscalar meson, P = π or η) is characterized by a complicated spin structure-with six independent scalar (and complex) amplitudes, which are functions of two independent kinematical variables. It is not possible to restore the full spin structure, from a knowledge of the branching ratio alone (with unpolarized particles). Moreover, there is the delicate problem of the extrapolation from the decay region of the kinematical variables (of the process J/ψ → N + N + P) to the scattering region (of the process P + N → N + J/ψ). To overcome this problem, let us consider the oversimplified assumption that the two types of processes J/ψ → N + N + P and P + N → N + J/ψ are driven by an effective contact four-particle interaction, with a single coupling constant. The exact spin structure of this interaction is not important for our considerations. In such an approximation the ratio r can be estimated from the following formula: Again the sign of r cannot be determined by such considerations, but, again the value of r which has been derived is in the range where the ratio R is very sensitive to the value of r. Note that the dependence of the polarization observables A 1np and P 1np is also quite large, in this region of r.
In the previous considerations, the effects of the final-state interaction in the produced NNJ/ψ three-particle system were not taken into account. In the near-threshold region, the NN interaction is well known, in terms of the corresponding scattering energies and effective radius. It is not the case for the J/ψN interaction. Note in this connection, that the total J/ψN cross section is not well known at present [42]. For example, photoproduction data give values of 3-4 mb [43]- [45], while the analysis of charmonium absorption on nucleons (at relatively high momentum) in p + A and A + A reactions suggests larger values, 6-7 mb [46,47]. These larger values can be explained in the framework of effective Lagrangian approaches [48]- [50]. Different methods have been suggested for a direct measurement of this quantity, through the processes π + d → J/ψ + p + p [39,40], p + d → J/ψ + n [51] and charmed meson production in p + A collisions [52].
Note that, in the general case, it is necessary to define two different J/ψ cross sections corresponding to transversal and longitudinal J/ψ polarization. For example, the data about γ +N → J/ψ +N are sensitive to σ T (J/ψN) cross sections, with transversal J/ψ polarization, if the VDM hypothesis is correct. Another possible method is to determine the average σ Av (J/ψN) cross section, obtained by averaging over the J/ψ polarization. In the case of interest here, in the near-threshold J/ψ production in nucleon-nucleon collisions, N + N → N + N + J/ψ, the 68.14 possible effects of the J/ψN interaction are given by the σ T (J/ψN) cross section alone, because we showed that the kinematical conditions are such that the J/ψ is produced with transversal polarization, only. To avoid double counting in the calculation of the J/ψN final interaction, one has to take into account only the J/ψN interaction with a nucleon spectator, produced in the vertex πNN.
Note that the ρ-exchange model for σ(πN → J/ψ) gives a cross section one order of magnitude smaller than other possible theoretical approaches [53]- [55]. One possibility is to explain the value of σ exp (pp → ppJ/ψ). Another possibility is to take [F (t J/ψ )/F (t φ )] 2 10, which could be plausible, because the J/ψ = cc system must have a smaller size than φ = ss. This can be realized by the following form factor:

Conclusions
Let us summarize the main results concerning the theoretical analysis of J/ψ production in nucleon-nucleon collisions in the threshold regime.
• We established the spin structure of the threshold matrix element in terms of a limited number of partial transitions, corresponding to S-wave production of final particles in the process N + N → N + N + J/ψ. • We proved the essential role of isotopic effects for J/ψ production in pp and np collisions.
The two reactions present very different characteristics concerning: * the number of independent partial transitions, * the spin structure of the threshold matrix elements, * the value of the absolute cross sections, * the polarization phenomena.
Note that all these differences are generated by a common mechanism: the origin of the essential difference has to be found in the different role of the Pauli principle for pp and pn collisions in the near-threshold region.

68.15
• This model-independent analysis shows the universality of theoretical considerations of threshold production of different vector mesons in nucleon-nucleon collisions, starting from light ρ, ω, φ to charmed mesons. • Only one polarization observable, the J/ψ polarization, is identical for pp and pn collisions: the J/ψ meson is transversally polarized-even in collisions of unpolarized nucleons. The experimental determination of the ratio of the total cross sections for np and pp collisions is important for the identification of the reaction mechanism. Polarization phenomena, which are trivial for threshold pp collisions, will be very useful for np collisions. The polarization transfer coefficients show the largest sensitivity to the nature (quantum numbers) of tchannel exchanges.