The reaction p̄p → Λ̄ − c Λ + c close to threshold

Abstract Predictions for the charm-production reaction p ¯ p → Λ ¯ c − Λ c + for energies near the threshold are presented. The calculations are performed in a meson-exchange framework in close analogy to our earlier study on p ¯ p → Λ ¯ Λ by connecting the two processes via SU ( 4 ) symmetry. The obtained Λ ¯ c − Λ c + production cross sections are in the order of 1 to 7 μb, i.e. a factor of around 10–70 smaller than the corresponding cross sections for Λ ¯ Λ . However, they are 100 to 1000 times larger than predictions of other model calculations in the literature.


Introduction
The study of the production of charmed hadrons in antiproton-proton (pp) collisions is of importance for the understanding of the strong force in the nonperturbative regime of QCD. The FAIR project at the GSI laboratory has an extensive program aiming at a high-accuracy spectroscopy of charmed hadrons and at an investigation of their interactions with ordinary matter [1]. Presently little is known about such interactions and their knowledge is a prerequisite for investigating issues like in-medium properties of charmed hadrons, e.g. cc-quarkonium dissociation and changes in properties of D mesons due to chiral symmetry restoration effects on the light quarks composing these mesons. Therefore there is an urgent need for theoretical investigations to guide such experiments.
In the present paper we concentrate on the reactionpp →Λ − c Λ + c close to its threshold. Providing predictions or simply estimations for this reaction is very challenging. First of all, the lack of any direct empirical information on this reaction makes it difficult to constrain model parameters. Second, both longdistance and short-distance physics are present in the reaction and this poses the question on the appropriate degrees of freedom to be used in order to describe it, quarks and gluons or mesons and baryons, or a combination of both. To the best of our knowledge, presently there are only four elaborate studies that consider this reaction. The most recent publication is by Goritschnig et al. [2], who employ a quark-gluon description based on a factorization hypothesis of hard and soft processes. This work supersedes an earlier study by that group within a quark-diquark picture, where already concrete predictions for theΛ − c Λ + c production cross section were given [3]. In the study by Kaidalov and Volkovitsky [4] a non-perturbative quark-gluon string model is used, based on secondary Regge pole exchanges including absorptive corrections. On the same lines, there is the more recent publication by Titov and Kämpfer [5].
Our work here builds on the Jülich meson-baryon model [6,7] for the reaction pp →ΛΛ. In that model the hyperon-production reaction is considered within a coupled-channel approach in which the microscopic strangeness production process and the elastic parts of the interactions in the initial (pp) and final (ΛΛ) states are described by meson exchanges, while annihilation processes are accounted for by phenomenological optical potentials. The elastic parts of the initial-and final-state interactions (ISI and FSI) are G-parity transforms of an one-boson-exchange variant of the Bonn NN potential [8] and of the hyperonnucleon model A of Ref. [9], respectively. The model achieved a reasonably good overall description of the wealth ofpp →ΛΛ data collected by the P185 collaboration at LEAR (CERN) on total and differential cross-sections and spin observables [10] -see also the review in Ref. [11].
The extension of the model to the charm sector here follows a strategy similar to our recent work on the DN andDN interactions [12,13], which used the original Jülich meson-exchange model for the KN system [14,15] and improvements from quark-gluon dynamics at short distances [16,17]. Specifically, in the present paper we construct an extension of the meson-exchange model of Refs. [6,7] to the reactionpp →Λ − c Λ + c assuming as a working hypothesis SU(4) symmetry constraints. We examine the sensitivity of the results to changes in the elastic and annihilation parts of the initialpp interaction and inspect the effects of the finalΛ − c Λ + c interaction and of the form factors that enter in thepp →Λ − c Λ + c transition potential described by t-channel D and D * meson exchanges. In addition, we also investigate the effect of replacing this meson-exchange transition by a charm-production potential derived in a quark model. We believe this is important for assessing uncertainties in the model, since one could easily raise the question on the validity of a meson-exchange description of the transition in view of the large masses of the exchanged mesons.
In the next Section we discuss the basic ingredients of the model and fix parameters by fitting differential and total inclusivepp cross sections. In Section 3 we present numerical results for our predictions for differential and total cross sections and compare with the results available in the literature. A summary of our work is presented in Section 4.

The model
We will start discussing the basic ingredients of the original Jülich coupled channel approach [6,7] that we employ here for the reactionpp →Λ − c Λ + c . The transition amplitude is obtained from the solution of a coupled-channel Lippmann-Schwinger equation, where z is the initial energy and q ′ (q) the c.m. (center-of-mass) relative momentum in the initial (final) state. Here V is a 2×2 matrix in channel space containing the interaction potentials (channel 1 = pp, and G 0 (q, z) is the propagator with The diagonal potentials V ii are given by the sum of an elastic part and an annihilation part.
Though the Jülich group has developed rather sophisticated models of thē N N interaction [18,19] these potentials cannot be used anymore at such high energies. Indeed, already in the study ofpp →ΛΛ the elastic part of thē N N potential was deduced (via G-parity transform) from a simple, energyindependent one-boson-exchange NN potential (OBEPF) and a phenomenological spin-, isospin-, and energy-independent optical potential of Gaussian form, and Berglund et al. [23].
was added in order to take into account annihilation. Now, at even much higher energies, any NN potential has to be considered as being purely phenomenological. Still we keep the longest ranged (and model-independent) part of the elasticpp interaction, namely one-pion exchange. To it we add again an optical potential of the form given in Eq. (4) and determine the parameters (U 0 , W 0 , r 0 , cf. Table 1) by a fit toNN data in the energy range relevant for the reactionpp →Λ − c Λ + c . Fortunately, there are total cross sections [20,21,22] and even differential cross sections [21,23] around p lab = 10 GeV/c, i.e. fairly close to theΛ − c Λ + c threshold which is at 10.162 GeV/c. Thispp potential is called model A in the following. As can be seen from Table 2 and Fig. 1 the integrated cross sections as well as thepp different cross section are fairly well reproduced.
Since one knows from studies onpp →ΛΛ that the magnitude of the cross sections depends very sensitively on the ISI [6,7,24,25] we consider here several variants of theNN potential. First, we take the full G-parity transformed OBEPF interaction and add again the optical potential (model D). Obviously in this case, only the total cross section can be still brought in line with the experiment by adjusting the parameters of the optical potential, while all other cross sections are strongly overestimated. Still we consider this potential too with the intention that it serves as an illustration for the uncertainties in our predictions due to the usedpp interaction. Realizing that the overestimation of the cross sections is primarily caused by the vector-meson exchange (ρ, ω) contributions [26] to OBEPF, we prepare two more potentials where we (i) leave out those vector mesons altogether (B) or (ii) reduce the elastic part (except for one-pion exchange) to 10% (C). Within both scenarios a rather satisfying description of theN N data around 10 GeV/c can be obtained, cf. Table 2 and Fig. 1. In particular, not only the slope but even the shoulder in the differential cross section is reproduced quantitatively by these two interactions. Table 1 Parameters of the phenomenological optical potential in thepp channel for the four different models described in the text.
0.56 fm 0.56 fm 0.58 fm 0.41 fm Table 2 Integratedpp cross sections at p lab = 10.1 GeV/c for the four models considered. The experimental value for the charge-exchange channelpp →nn (cex) is for p lab = 9 GeV/c [22]. The interaction in the finalΛ − c Λ + c system is assumed to be the same as the one inΛΛ. Specifically, this means that the elastic part of the interaction is given by the isospin-zero σ and ω exchanges with coupling constants taken from the hyperon-nucleon model A of Ref. [9], while the annihilation part is again parameterized by an optical potential which contains, however, spin-orbit and tensor components in addition to a central component [6]: The free parameters in the opticalΛΛ potential were determined in Ref. [6] by a fit to data on total and differential cross sections, and analyzing power for pp →ΛΛ. Clearly, there are no reasons to believe that theΛ − c Λ + c interaction will be the same on a quantitative level. But we expect that at least the bulk properties are similar. Specifically, in both cases near threshold the interactions will be govered by strong annihilation processes. We will investigate the role played by the FSI for our results by simply switching it off. We use the parameters as given in Table II of Ref. [7].
In principle, the parameters of Vp p→pp opt can only be determined together with those forΛ − c Λ + c because we consider a coupled-channels problem. However, like inΛΛ the branching ratiopp →Λ − c Λ + c is so small that the effect of thē Λ − c Λ + c channel on the diagonalpp T -matrix can be savely neglected.
The transition potential frompp toΛ − c Λ + c is given by t-channel D and D * exchanges and their explicit expressions are the same as for K and K * as given in Ref. [9]. They are of the generic form where g N ΛcM are coupling constants and F 2 N ΛcM (t) are form factors. Under the assumption of SU(4) symmetry the coupling constants are the same as in the corresponding exchanges inpp →ΛΛ. However, the cutoff masses in the vertex form factors cannot be taken over, because the masses of the exchanged particles are now much larger. We use here a monopole form factor with a cutoff mass Λ of 3 GeV, at the NΛ c D as well as at the NΛ c D * vertex but we will explore the sensitivity of the results to variations of the cutoff mass.
To examine further the uncertainties, as an alternative we consider here also a charm-production potential inspired by quark-gluon dynamics. There is a large literature associated with strange-hadron production in pp reactions, the best known works are those of Kohno and Weise [24], Furui and Faessler [27], Burkardt and Dillig [28], Roberts [29] and Alberg, Henley and Wilets [25]a more complete list of references can be found in the review of Ref. [11]. In the present study we adopt the interaction proposed by Kohno and Weise, derived in the so-called 3 S 1 mechanism of a constituent quark model. In this model the ss pair in the final state is created from an initial uū or dd pair via s-channel gluon exchange. After quark degrees-of-freedom are integrated out the potential has the form [24]: Here α/m 2 G is an effective (quark-gluon) coupling strength, r 2 is the mean square radius associated with the quark distribution in the p or Λ, and S and T are the total spin and isospin in thepp system. This simple potential has actually a very modern form, namely that of a contact term, though smeared out by the quark distribution. The effective coupling strength is practically a free parameter that was fixed by a fit to thepp →ΛΛ data [7]. But it depends implicitly on the effective gluon propagator, i.e. on the square of the energy transfer from initial to final quark pair, cf. Refs. [27,28,30]. Heuristically this energy transfer corresponds roughly to the masses of the produced constituent quarks, i.e. m G ≈ 2m q , we expect the effective coupling strength α/m 2 G for charm production to be reduced by the ratio of the constituent quark masses of the strange and the charmed quark squared, (m s /m c ) 2 ≈ (550 MeV / 1600 MeV) 2 ≈ 1/9 as compared to the one forpp →ΛΛ. Note that a different suppression factor arises for the 3 P 0 quark-antiquark annihilation vertex, considered by Alberg et al. in their study ofpp →ΛΛ [25]. In this case the amplitude scales with 1/m q so that the effective strength of the transition potential is reduced by m s /m c ≈ 1/3 only when going from strangeness to charm production.

Results
Our predictions for the total reaction cross section for pp →Λ − c Λ + c are presented in Fig. 2. The cross section is shown as a function of the excess energy ǫ = √ s − m Λc − mΛ c so that we can compare it with the one for pp →ΛΛ at the corresponding ǫ. The curve in Fig. 2 correspond to thepp model C, for which we obtained the largest results.
Obviously, and as expected, the cross section forΛ − c Λ + c production is smaller than the one forΛΛ. But the difference is about one order of magnitude only. Indeed, one can get a very rough estimation for the reduction just by considering the following: When going fromΛΛ toΛ − c Λ + c within the mesonexchange picture the main change occurs in the meson propagators of the transition potentials where the masses of the strange mesons (K, K * ) are replaced by those of the charmed mesons (D, D * ), m Ms → m Mc , cf. Eq. (6). All coupling constants remain the same because of the assumed SU(4) symmetry. Thus, the ratio of the transition potentials is then given roughly by Ms /m 2 Mc ≈ 1/4 so that one expects the cross section to be smaller by a factor of around 16. Our explicit calculation is pretty much in line with this admittedly rather qualitative estimation.
Let us now discuss the sensitivity of the results to the various ingredients of our model calculation. The most crucial component is certainly thepp interaction in the initial state. Without it, and specifically in Born approximation, we would get cross sections that are more than a factor 100 larger. This is no surprise because the very same gross overestimation in the Born approximation occurs forpp →ΛΛ [6,7,24,25] but also for other exclusivepp annihilation channels likepp → ππ,pp → KK, etc. [18,19], if one does not take into account the ISI. However, with the ISI included, it is reassuring to see that the variations of the predictedpp →Λ − c Λ + c cross section for the differentpp potentials we have prepared is fairly small. To be concrete, the cross sections at ǫ = 100 MeV are 5.8, 6.3, and 7.3 µb for the interactions A, B, and C, respectively. Only the result based on the full G-parity transformed OBEPF potential differs more significantly. Here we get a cross section that is with 0.8 µb a factor 10 smaller than the other values. The suppression for thispp model   is presumably due to the fact that it yields app elastic cross section that is much larger than the ones of the other potentials considered, and actually in disagreement with experimental information. Still, the variation of one order of magnitude may be considered as a realistic estimation of the uncertainty due to the ISI.
Once the ISI is included our results turned out to be rather insensitive to the finalΛ − c Λ + c interaction. Even when we leave it out altogether the cross sections do not change (decrease) by more than 10-15 %. We also considered variations of the cutoff mass at the NΛ c D and NΛ c D * vertices. For the results discussed above a value of Λ = 3 GeV has been used. When reducing this value to 2.5 GeV the cross sections at ǫ = 100 MeV drop by roughly a factor 3. Of course, employing even smaller cutoff masses would further decrease the cross sections. However, since the exchanged mesons have a mass of around 1.9 to 2 GeV we consider values below 2.5 GeV as being not really realistic.
We display here also the results based on an adaption of the 3 S 1 quark-gluon transition mechanism of Ref. [24]. We scale the effective coupling strength α/m 2 G = 0.25 fm 2 , fixed in our study ofpp →ΛΛ [7], with (m c /m s ) 2 using the constituent quark masses m s = 550 MeV and m c = 1600 MeV, i.e. the same values as employed in our previous works in Refs. [12,13]. As expected, we obtain cross sections that are of the same magnitude as those predicted in the meson-exchange picture though roughly a factor three smaller, cf. the dashed line in Fig. 2. In principle the factor r 2 in Eq. (7) also changes when replacing m s by m c , since it gives the size of the quark distribution of the overlapping hadrons -r 2 decreases as m s is replaced by m c . We have not changed this factor in the present work, since we expect this change to be much smaller than the change in the square of the energy transfer m 2 G , since hadron sizes do not scale as simple powers of quark masses and so the overlap between hadron wave functions is not expected to change drastically from Λ to Λ c -for explicit numbers for the size parameters of harmonic oscillator wave functions for N and Λ c see Ref. [31]. The size parameter we use is r 2 1/2 = 0.55 fm.
Predictions for differential cross sections, the singlet fraction F s , the depolarization D nn and the spin-correlation parameter C yy (cf. Ref. [7] for definitions) at ǫ = 40 MeV are presented in Fig. 3 for all four ISI considered. Note that the singlet fraction is defined by F s = 1 4 (1 − σ 1 · σ 2 ) = 1 4 (1 + C xx − C yy + C zz ) [6]. It is zero if theΛ − c Λ + c pair is produced purely in a triplet state. Indeed in case ofpp →ΛΛ it was found experimentally that the singlet fraction is close to zero [10] and this feature was reproduced by meson-exchange potentials [6] as well as by transition interactions based on quark-gluon dynamics [7,24,25]. In the former it is due to a strong tensor force generated by the combined K +K * exchange which leads to a dominance of transitions of the form LΛ Λ = Lp p − 2 where L refers to the orbital angular momentum. As can be seen from Fig. 3, the very same feature is predicted also for the reactionpp →Λ − c Λ + c . The differential cross section exhibits a peak in forward direction, similar to what has been also observed forpp →ΛΛ [6,7,10]. Let us now compare our predictions with those by other groups. This is done in Fig. 4. Goritschnig et al. [2] as well as Kaidalov and Volkovitsky [4] have presented explicit results in their publications and we reproduce them in Fig. 4 to facilitate a comparison. Our results are shown as dark (red) shaded band (and grid) in order to reflect the variation of the predictions when the four different ISI are used. It is remarkable that our results differ drastically from those of the preceeding works. Specifically, our cross sections are a factor 1000 larger than those given by Goritschnig et al. and they are still about 100 times larger than the ones by Kaidalov and Volkovitsky. Thus, even when considering the variation of about a factor ten due to the ISI that we see in our results and the uncertainties due to the unconstrained FSI and form factors in the transition potential that amount to roughly a factor three, we are faced with an impressive qualitative difference.

Summary
In this paper we presented predictions for the charm-production reactionpp → Λ − c Λ + c . The calculations were performed in the meson-exchange framework in close analogy to our earlier study onpp →ΛΛ by connecting the two processes via SU(4) symmetry. The interaction in the initalpp interaction, which plays a crucial role for the quantiative predictions and which is now needed at a much higher energy, is re-adjusted so that availablepp scattering data in the relevant energy range are reproduced.
The obtainedΛ − c Λ + c production cross sections are in the order of 1 to 7 µb. Thus, they are just about a factor 10 smaller than the corresponding cross sections forΛΛ. The reduction is in line with a naive estimation based on the mass ratio of strange (K, K * ) versus charmed (D, D * ) mesons. The exchange of those mesons governs the range of the forces that are responsible for producing a Λ or a Λ c , respectively, in the meson-exchange picture.
Interestingly, other model calculations in the literature predictΛ − c Λ + c production cross sections that are 100 to 1000 times smaller than what we found. Since the approaches used in those calculations are rather different from ours it is practically impossible to say where this drastic difference in the predicted cross sections comes from. Hence, a discrimination between those scenarios appears to be possible only when the reactionpp →Λ − c Λ + c will be examined in a future experiment at FAIR. Of course, if our predictions turn out to be closer to reality than the others, it will be much easier to perform a pertinent experiment at FAIR. Moreover, Λ c 's andΛ c 's can then be produced more copiously so that also any other experiment that requires charmed baryons as a probe will be much more promising.
The model presented here leaves room for improvements in several directions. In particular, it would be important to extend the simple picture of s-channel qq annihilation followed by cc creation for the transition amplitude in the quark model as used here. An interesting possibility is the one pursued in Ref. [29] forpp →ΛΛ where interacting many-quark intermediate states are considered in the process.