DOI 10.1140/epja/i2011-11018-3 Regular Article – Theoretical Physics Eur. Phys. J. A (2011) 47: 18 THE EUROPEAN PHYSICAL JOURNAL A DN interaction from meson exchange J. Haidenbauer1,2,a, G. Krein3, U.-G. Meißner1,2,4, and L. Tolos5,6 1 Institut für Kernphysik and Jülich Center for Hadron Physics, Forschungszentrum Jülich, D-52425 Jülich, Germany 2 Institute for Advanced Simulation, Forschungszentrum Jülich, D-52425 Jülich, Germany 3 Instituto de F́ısica Teórica, Universidade Estadual Paulista, Rua Dr. Bento Teobaldo Ferraz, 271 - 01140-070 São Paulo, SP, Brazil 4 Helmholtz-Institut für Strahlen- und Kernphysik (Theorie) and Bethe Center for Theoretical Physics, Universität Bonn, Nußallee 14-16, D-53115 Bonn, Germany 5 Theory Group, KVI, Zernikelaan 25, NL-9747AA Groningen, The Netherlands 6 Instituto de Ciencias del Espacio (IEEC/CSIC), Campus Universitat Autonoma de Barcelona, Facultat de Ciencies, Torre C5, E-08193 Bellaterra (Barcelona), Spain Received: 23 August 2010 / Revised: 15 November 2010 Published online: 3 February 2011 c© The Author(s) 2011. This article is published with open access at Springerlink.com Communicated by A. Ramos Abstract. A model of the DN interaction is presented which is developed in close analogy to the meson- exchange K̄N potential of the Jülich group utilizing SU(4) symmetry constraints. The main ingredients of the interaction are provided by vector meson (ρ, ω) exchange and higher-order box diagrams involving D∗N , DΔ, and D∗Δ intermediate states. The coupling of DN to the πΛc and πΣc channels is taken into account. The interaction model generates the Λc(2595)-resonance dynamically as a DN quasi-bound state. Results for DN total and differential cross sections are presented and compared with predictions of two interaction models that are based on the leading-order Weinberg-Tomozawa term. Some features of the Λc(2595)-resonance are discussed and the role of the near-by πΣc threshold is emphasized. Selected predictions of the orginal K̄N model are reported too. Specifically, it is pointed out that the model generates two poles in the partial wave corresponding to the Λ(1405)-resonance. 1 Introduction The study of the low-energy interactions of open charm D-mesons with nucleons is challenging for several reasons. The complete lack of experimental data on the interaction makes it very difficult to constrain models. Reliable mod- els are very important for guiding planned experiments by the P̄ANDA [1] and CBM [2] Collaborations at the future FAIR facility at Darmstadt [3]. Estimates for the magnitudes of cross sections are required for the design of detectors and of efficient data acquisition systems. As emphasized in recent publications [4,5], one way to make progress in such a situation is to build models guided by analogies with other similar processes, by the use of sym- metry arguments and of different dynamical degrees of freedom. Physics motivations for studying the interaction of D- mesons with nucleons are abundant. Amongst the most exciting ones is the possibility of studying chiral symmetry in matter. The chiral properties of the light quarks com- posing D-mesons are sensitive to temperature and den- a e-mail: j.haidenbauer@fz-juelich.de sity; one expects to learn about manifestations of such a sensitivity through the detection of changes in the interac- tion of these mesons with nucleons in the medium as com- pared to the corresponding interaction in free space. Also, studies of J/ψ dissociation in matter, since long time [6] suggested as a possible signature for the formation of a quark-gluon plasma, require a good knowledge of the in- teraction of D-mesons with ordinary hadrons [7] in order to differentiate different scenarios and models in this area. Another exciting perspective is the possibility of the for- mation of D-mesic nuclei [8,9] and of exotic nuclear bound states like J/ψ binding to nuclei [10,11]. In this latter case in particular, the interaction of D-mesons with ordinary matter plays a fundamental role in the properties of such exotic states [12,13]. In a recent paper we examined the possibility to ex- tract information about the DN and D̄N interactions from the p̄d → D0D−p reaction [5]. The scattering am- plitudes for DN and D̄N employed in this exploratory study were generated from interaction models that were constructed along the line of the K̄N and KN meson- exchange potentials developed by the Jülich group some time ago [14–16]. While the D̄N interaction has been Page 2 of 16 The European Physical Journal A described in some detail in ref. [4] this has not been done so far for the DN model. With the present paper we want to remedy this situation. The DN interaction is derived in close analogy to the meson-exchange K̄N model of the Jülich group [14], utiliz- ing as a working hypothesis SU(4) symmetry constraints, and by exploiting also the close connection between the DN and D̄N systems due to G-parity conservation. In particular, the DN potential is obtained by substituting the one-boson-exchange (σ, ρ, ω) contributions, but also the box diagrams involving K̄∗N , K̄Δ, and K̄∗Δ inter- mediate states, of the original K̄N model of the Jülich group by the corresponding contributions to the DN in- teraction. Of course, we know that SU(4) symmetry is strongly broken, as is reflected in the mass differences be- tween the strangeness and the charm sectors, for mesons as well as for baryons. However, as already argued in ref. [4], we expect that the dynamics in the DN (D̄N) and K̄N (KN) systems should be governed predominantly by the same “long-range” physics, i.e. by the exchange of ordi- nary (vector and possibly scalar) mesons. Thus, in both systems the dynamics should involve primarily the up- and down-quarks, whereas the heavier quarks (the s and c quarks, respectively) behave more or less like specta- tors and contribute predominantly to the static properties of the mesons and baryons. Therefore, the assumption of SU(4) symmetry for the dynamics seems to be not com- pletely implausible. In any case, invoking SU(4) symmetry for the DN in- teraction involves larger uncertainties than for D̄N . For the former, like in the analogous K̄N system, there are couplings to several other channels which are already open near the DN threshold (πΛc, πΣc) or open not far from the threshold (ηΛc). The coupling to those channels should play an important role for the dynamics of the DN sys- tem —as is the case in the corresponding K̄N system— and, thus, will have an impact on the results, at least on the quantitative level. Specifically, the transitions from DN to those channels involve the exchange of charmed mesons, for example the D∗(2010), where the range argu- ments given above no longer hold, and where the coupling constants and associated vertex form factors, required in any meson-exchange model, are difficult to constrain. Another open issue is the relevance of heavy-quark symmetry (HQS). This symmetry which is an exact sym- metry of the underlying QCD Lagrangian in the limit of large quark masses [17,18], connects the coupling strengths of specific spin doublets such as the D and D∗ mesons or the Σc and Σ∗ c baryons [19,20]. While the cou- pling constants that are used in our DN (and also in our D̄N) model comply with the restrictions imposed by HQS, simply because SU(6) symmetry was used in deriving the Jülich K̄N and KN models, HQS is not implemented fully consistently in our DN and D̄N models. In 1993 a first evidence for the Λc(2595)-resonance was reported by the CLEO Collaboration [21] and sub- sequently confirmed by several other experiments [22–24]. Nowadays it is generally accepted that this resonance is the charmed counterpart of the Λ(1405) [25]. In the Jülich K̄N potential model [14] the latter state is generated dy- namically. It appears as a K̄N quasi-bound state and is produced by the strongly attractive interaction due to the combined effect of ω, ρ and scalar-meson exchanges, which add up coherently in the K̄N channel. When extending the Jülich meson-exchange model to the charm sector via SU(4) symmetry one expects likewise the appearance of a quasi-bound state, namely in the DN channel. Thus, one can actually utilize the experimentally known mass of the Λc(2595)-resonance as an additional constraint for fixing parameters of the DN interaction. Indeed such a strategy was already followed in recent studies of the K̄N and DN interaction within chiral uni- tary (and related) approaches, where likewise the Λ(1405)- resonance but also states in the DN system are generated dynamically [26–31]. In those approaches the strong at- traction is also provided by vector-meson exchange [27, 31], by the Weinberg-Tomazawa (WT) term [26,28,29], or by an extension of the WT interaction to an SU(8) spin-flavor scheme [30]. In refs. [28–33] the authors ar- gued that a state occurring in the S01 channel of the charm C = 1 sector should be identified with the I = 0 resonance Λc(2595). (Throughout we use the standard spectroscop- ical nomenclature LI 2J , with L denoting the orbital an- gular momentum, I the isospin and J the total angular momentum of the two-particle system.) Note, however, that some of those works differ with regard to the nature of the Λc(2595), i.e. in the dominant meson-baryon com- ponent of this state. For example, in the SU(8) spin-flavor scheme of ref. [30] the Λc(2595) becomes predominantly a D∗N quasi-bound state —though still with important additional binding effects from the DN channel. We take the opportunity to present here also some results of the original Jülich K̄N model [14]. Over the last years there has been increasing interest in the prop- erties of the K̄N interaction close to the threshold and the near-by Λ(1405)-resonance, resulting in a vast amount of pertinent publications. Indeed, there is still a contro- versy about the actual value for the strong-interaction en- ergy shift of kaonic hydrogen and there are ongoing de- bates about issues like a possible double-pole structure of the Λ(1405) or deeply-bound kaonic states (for recent overviews, see refs. [34–37]). The initial publication of the Jülich group [14] focussed on a detailed account of the ingredients of the K̄N interaction model and on the de- scription of scattering data available at that time. Results of the Jülich model for quantities that are relevant for the issues mentioned above were not given. This will be reme- died by the present paper. Specifically, we provide here the K̄N scattering length and we determine the pole position corresponding to the Λ(1405)-resonance. For the latter it turns out that also the K̄N model of the Jülich group pre- dicts the existence of two poles in the corresponding S01 K̄N partial wave. The paper is organized in the following way: in sect. 2 a brief description of the ingredients of the DN model is given. The interaction Lagrangians, which are used to de- rive the meson-baryon potential, are summarized in the appendix. In sect. 3 selected results for the Jülich K̄N model are presented. Results for DN scattering are pre- sented in sect. 4. Besides the scattering lengths and the J. Haidenbauer et al.: DN interaction from meson exchange Page 3 of 16 Fig. 1. Meson-exchange contributions included in the direct DN interaction. pole positions corresponding to the Λc(2595)-resonance, also some predictions for total and differential cross sec- tions are given. Furthermore, we compare our results with those obtained from DN interactions that were derived from the leading-order WT contact term, assuming either SU(4) symmetry [28] or even SU(8) symmetry [30]. Con- sidering those results allows us to explore the model de- pendence of the predictions for DN scattering observables. Section 5 is dedicated to the Λc(2595)-resonance and fo- cusses on the consequences of the fact that the position of this resonance coincides practically with the πΣc thresh- old. In particular, we present results for the πΣc invariant- mass spectrum which allows us to discuss the subtle effects of the slightly different thresholds of the π+Σ0 c , π0Σ+ c , and π−Σ++ c channels on the various invariant-mass distribu- tions. Also here a comparison with results based on the SU(4) (and SU(8)) WT approach is made. Furthermore, we discuss similarities and differences between the Λ(1405) and the Λc(2595) as dynamically generated states. The paper ends with a short summary. 2 Coupled-channel DN model in the meson-exchange framework The DN interaction employed in the present study is constructed in close analogy to the meson-exchange K̄N model of the Jülich group [14] utilizing SU(4) symmetry constraints, as well as by exploiting the close connection between the DN and D̄N systems due to G-parity conser- vation. Specifically, we use the latter constraint to fix the contributions to the direct DN interaction potential while the former one provides the transitions to and interactions in channels that can couple to the DN system. The main ingredients of the DN → DN interaction are provided by vector meson (ρ, ω) exchange and higher-order box dia- grams involving D∗N , DΔ, and D∗Δ intermediate states, see fig. 1. The original K̄N and KN models of the Jülich group include besides the exchange of the standard mesons also an additional phenomenological (extremely short-ranged) repulsive contribution, a “σrep”, with a mass of about Fig. 2. Meson-exchange contributions included in the DN → πΛc, πΣc transition potentials and in the πΛc, πΣc → πΛc, πΣc interactions. 1.2GeV [14,16]. This contribution was introduced ad hoc in order to achieve a simultaneous description of the em- pirical KN S- and P -wave phase shifts [15,16]. Evidently, due to its phenomenological nature it remains unclear how that contribution should be treated when going over to the D̄N and DN systems. This difficulty was circumvented in the construction of the D̄N interaction [4] by resorting to a recent study of the KN interaction [38] which provided evidence that a significant part of that short-ranged repul- sion required in the original Jülich model could be due to genuine quark-gluon exchange processes. Indeed, consid- ering such quark-gluon mechanisms together with conven- tional a0(980)-meson exchange instead of that from “σrep” a comparable if not superior description of KN scattering could be achieved [38]. From this model the D̄N inter- action [4] could be derived in a straightforward way un- der the assumption of SU(4) symmetry. Furthermore, the extension to the DN system is possible too. Note, how- ever, that the short-ranged quark-gluon processes, that contribute to the D̄N interaction [4], are absent here be- cause the quark-exchange mechanism cannot occur in the DN interaction due to the different quark structure of the D-meson as compared to D̄. As far as the coupling to other channels is concerned, we follow here the arguments of ref. [14] and take into ac- count only the channels πΛc(2285) and πΣc(2455). Fur- thermore, we restrict ourselves to vector-meson exchange and we do not consider any higher-order diagrams in those channels. Pole diagrams due to the Λc(2285) and Σc(2455) intermediate states are, however, consistently included in all channels. The various contributions to the DN → πΛc, πΣc transition potentials and to the πΛc, πΣc → πΛc, πΣc interaction, taken into account in the present model, are depicted in fig. 2. HQS would require to include further channels in our model. Specifically the DN and D∗N channels and the corresponding interactions should be treated on equal footing. While both channels are indeed taken into ac- count in our model, the direct interaction in the D∗N case has been neglected in analogy to what has been done for K̄∗N in the Jülich K̄N model [14]. Thus, being faced Page 4 of 16 The European Physical Journal A with the decision whether we should maintain the close re- lation with the original Jülich K̄N model or impose fully HQS we chose the former option. Still we would like to point out that the DN interaction and the DN → D∗N transition potential, as mediated by ρ exchange, are con- sistent with HQS due to the SU(6) symmetry that was adopted in the construction of the Jülich K̄N and KN models. The interaction Lagrangians, which are used to de- rive the meson-baryon potentials for the different chan- nels, are summarized in the appendix. Based on the re- sulting interaction potentials Vij (i, j = DN , πΛc, πΣc) the corresponding reaction amplitudes Tij are obtained by solving a coupled-channel Lippmann-Schwinger–type scattering equation within the framework of time-ordered perturbation theory, Tij = Vij + ∑ k VikG0 kTkj , (1) from which we calculate the observables in the standard way [15]. The assumed SU(4) symmetry and the connection with the K̄N model, respectively, allows us to fix most of the parameters —the coupling constants and the cut- off masses at the vertex form factors of the occurring meson-meson-meson and meson-baryon-baryon vertices, cf. ref. [14]. A list with the specific values of the perti- nent parameters can be found in the appendix. When solving the coupled-channel Lippmann-Schwin- ger equation with this interaction model we observe that two narrow states are generated dynamically below the DN threshold, one in the S01 partial wave and the other one in the S11 phase. In view of the close analogy be- tween our DN model and the corresponding K̄N interac- tion [14] this is not too surprising, because the latter also yields a quasi-bound state in the S01 channel which is as- sociated with the Λ(1405)-resonance. The bound states in both K̄N and DN are generated by the strongly attrac- tive interaction due to the combined effect of ω, ρ and scalar-meson exchanges, which add up coherently in spe- cific channels. Following the arguments in refs. [28–30,32,33] we iden- tify the narrow state occurring in the S01 channel with the I = 0 resonance Λc(2595). Furthermore, we identify the state we get in the S11 channel with the I = 1 resonance Σc(2800) [25]. (For a different scenario, see [39].) In order to make sure that the DN model incorporates the above features also quantitatively, the contributions of the scalar mesons to the DN interaction are fine-tuned so that the position of those states generated by the model coincide with the values given by the Particle Data Group [25]. This could be achieved by a moderate change in the coupling constants of the σ-meson (from 1 to 2.6) and the a0-meson (from −2.6 to −4.8), cf. table 3 in the appendix. We would like to stress that anyhow the application of SU(4) sym- metry (and even SU(3) symmetry) to the scalar-meson sector is problematic, as discussed in ref. [4]. In the present paper we will show results for the latter interaction which we consider as our basic model. However, we present also DN cross sections based on the parameter set that follows directly from the K̄N and KN studies [15,16,38] by as- suming SU(4) symmetry. Those results may be considered as a measure for the uncertainty in our model predictions for DN . With regard to the D∗(2010) exchange that con- tributes to the DN → πΛc and DN → πΣc transition po- tentials (cf. fig. 2) it should be said that the corresponding form factors cannot be inferred from the K̄N model [14] via SU(4) arguments. We fixed the relevant cut-off masses somewhat arbitrarily to be about 1GeV larger than the exchange mass. Anyway, variations in those cut-off masses have very little influence on the results in the DN chan- nel. Moreover, within the spirit of the basic model such variations can be easily compensated by re-adjusting the coupling constants in the scalar sector so that again the masses of the Λc(2595) and Σc(2800) resonances are repro- duced, as discussed above. But the width of those states are certainly sensitive to the values used for those cut-off masses. 3 Selected results of the Jülich K̄N model Before discussing the results of the Jülich DN model in detail, let us first present some selected results of the corre- sponding K̄N model. The general scheme and also the ex- plicit expressions for the various contributions to the K̄N interaction potential of the Jülich group are described in detail in ref. [14]. In the original publication of the model, results for total cross sections of the reaction channels K−p → K−p, K−p → K0n, K−p → π0Λ, K−p → π0Σ0, K−p → π−Σ+, and K−p → π+Σ− were presented and compared with available data. One can see [14] that the model yields a quite satisfactory description of the avail- able experimental information up to K̄ laboratory mo- menta of plab ≈ 300MeV/c. In the present paper we re- frain from showing those results again but we want to fo- cus on additional predictions of the model that have not been included in ref. [14]. First this concerns the behaviour close to the K̄N threshold. The original Jülich potential is derived under the assumption of isospin symmetry and the reaction amplitudes were obtained by solving a Lippmann- Schwinger–type scattering equation (1) in the isospin basis using averaged masses of the involved baryons and mesons. The corresponding S-wave scattering lengths a for I = 0, I = 1, and for K−p (aK−p = (a0 + a1)/2) are summa- rized in table 1. In order to enable a detailed compari- son with available empirical information in the threshold region, now we also performed a calculation in the par- ticle basis. Using the physical masses of the baryons and mesons allows us to take into account the isospin-breaking effects generated by the known mass splittings within the involved isospin multiplets, and specifically between the K− and K̄0 masses. The K−p scattering length calcu- lated in this way is also given in table 1. It differs signifi- cantly from the one obtained in the isospin basis (labelled by (I)). Recently a thorough study of the K−p scattering length and its theoretical uncertainties within chiral J. Haidenbauer et al.: DN interaction from meson exchange Page 5 of 16 Table 1. K̄N results of the Jülich model [14]. The value for the K−p scattering length obtained in the isospin-symmetric calculation is marked with (I). In case of kaonic hydrogen we present the strong-interaction energy shift ΔE and the width Γ of the 1s level. The result for the Jülich model is obtained from the K−p scattering length with the modified Deser-Trueman formula [41]. Jülich model [14] Experiment Scattering lengths [fm] aI=0 −1.21 + i 1.18 aI=1 1.01 + i 0.73 aK−p (I) −0.10 + i 0.96 aK−p −0.36 + i 1.15 Kaonic hydrogen ΔE 217 eV 323 ± 63 ± 11 eV [44,45] 193 ± 37 ± 6 eV [42,43] Γ 849 eV 407 ± 208 ± 100 eV [44,45] 249 ± 111 ± 39 eV [42,43] Threshold ratios – eq. (2) γ 2.30 2.36 ± 0.04 [48,49] Rc 0.65 0.664 ± 0.011 [48,49] Rn 0.22 0.189 ± 0.015 [48,49] Pole positions [MeV] S01 1435.8 + i 25.6 S01 1334.3 + i 62.3 SU(3) unitary approaches was presented [40]. The full calculation which included the WT contact interaction at leading chiral order, the direct and crossed Born terms as well as contact interactions from the Lagrangian of sec- ond chiral order yielded a scattering length of aK−p = −1.05 + i 0.75 fm. Obviously, the result obtained for the Jülich meson-exchange model differs from this value, but it is still within the 1σ confidence region given in ref. [40]. Results for the strong-interaction energy shift ΔE and the width Γ of the 1s level of kaonic hydrogen, deduced from the K−p scattering length with the modified Deser- Trueman formula [41], are also given in table 1. Interest- ingly, the prediction of the Jülich model for ΔE agrees well with the DEAR result [42,43] while Γ is roughly in line with the value found in the KEK experiment [44,45]. Note, however, that the experimental results for kaonic hydrogen were not available at the time when the Jülich K̄N model was constructed and, therefore, not included in the fitting procedure. Since the KEK and DEAR re- sults are not compatible with each other it is important to resolve this discrepancy between the experimental re- sults. Values of higher level of precision are expected to be reached by the ongoing SIDDHARTA experiment at LNF [46,47]. Of interest are also the three measured threshold ra- tios [48,49] of the K−p system, which are defined by γ = Γ (K−p → π+Σ−) Γ (K−p → π−Σ+) , Rc = Γ (K−p → charged particles) Γ (K−p → all) , Rn = Γ (K−p → π0Λ) Γ (K−p → all neutral states) . (2) The corresponding predictions of the Jülich model are given in table 1. As one can see, the values are remarkably close to the experimental ones, specifically when one keeps in mind that these threshold ratios were not included in the fitting procedure when the model was constructed. Re-calculating total cross sections in the particle basis yielded results that do not differ very much from those shown in ref. [14] (obtained in isospin basis) for momenta plab � 100MeV/c where data are available. In particular, the changes are small in comparison to the given exper- imental error bars. Thus, we do not present the corre- sponding results here. Let us now come to the Λ(1405). As already said above, also in the Jülich model this structure is generated dy- namically. It is predicted “unambiguously” [14] to be a quasi-bound K̄N state. When searching in the region be- tween the πΣ and K̄N threshold we find two poles in the complex plane for the relevant S01 partial wave. The values of the pole positions are listed in table 1. In view of the extensive discussion of the two-pole structure of the Λ(1405) over the last ten years or so [40,50–55] it is certainly not surprising that the Jülich meson-exchange potential of the K̄N interaction generates two such poles as well. But back in 1989 when the model was constructed this was not an issue yet. Thus, no attempt was made to examine the pole structure of the amplitude in detail and, therefore, this feature remained undiscovered. The pole structure predicted by the Jülich model turns out to be quantitatively very similar to the one of the full result of the chiral SU(3) unitary approach by Borasoy et al. [40]. One pole, the K̄N “bound state”, is located fairly close to the K̄N threshold and to the physical real axis while the other one is close to the πΣ threshold and has a significantly larger imaginary part. It was emphasized in refs. [40,54] that only the pole close to the physical axis remains relatively fixed while the other one depends sensitively on the dynamics included in the analysis, on the values chosen for inherent parameters, etc. Indeed, a glance at the published results for the poles predicted by different interactions [50–53,56] reveals that there are significant variations in the values for the positions and widths. For example, the interactions proposed in refs. [50, 53] lead to two poles that are both very close to the phys- ical region. In case of the WT models of refs. [51,52,56] the second pole shows a larger width, more in line with our results and those of refs. [40,54], though the two poles are still fairly close together. Page 6 of 16 The European Physical Journal A 1320 1360 1400 1440 s 1/2 (MeV) 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 |T |2 q (a .u .) π−Σ+ π0Σ0 π+Σ− πΣ −> πΣ 1320 1360 1400 1440 s 1/2 (MeV) 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 |T |2 q (a .u .) K - p -> π−Σ+ K - p -> π0Σ0 K - p -> π+Σ− KN -> πΣ Fig. 3. πΣ invariant-mass spectrum. On the left are results based on the πΣ → πΣ T -matrix and on the right the ones for K−p → πΣ. The histograms are experimental results for the π−Σ+ (dotted line) and π+Σ− (dashed line) channels taken from ref. [57]. In ref. [14] results for the quantity |TπΣ |2 · q were pre- sented, which is commonly associated and compared with the πΣ mass distributions, i.e. dσ dmπΣ ∝ |TπΣ |2 q. (3) Here, q is the center-of-mass momentum of the πΣ sys- tem. Measurements of the πΣ invariant-mass distribu- tion for reactions like K̄N → πππΣ [57] or πN → KπΣ [58] provide the main experimental evidence for the Λ(1405)-resonance. As we realize now, the results pub- lished in ref. [14] were not correct, because of an error in the phase-space factor in the computer code. Moreover, in this reference only the contribution from I = 0 alone was considered. In fig. 3 we present results for different charge chan- nels (π−Σ+, π0Σ0, π+Σ−) in the final state and consider πΣ → πΣ as well as K̄N → πΣ transitions, and com- pare them with the πΣ mass distribution measured in the reaction K−p → πππΣ [57]. We display here curves for the individual T -matrices because we want to illustrate the differences between the various amplitudes. Note that the “true” amplitude to be inserted in eq. (3) will be a coherent sum of transition amplitudes from all allowed in- termediate states to a specific final state, weighted with coefficients that reflect the details of the reaction mecha- nism [50]. As already observed by others in the past [52,59], there is a remarkable difference between the invariant- mass spectrum due to the πΣ → πΣ amplitude (left-hand side of fig. 3) and the one due to the K̄N → πΣ ampli- tude (right-hand side of fig. 3). This is due to the fact that the two poles in the I = 0 S-wave amplitude have different widths and couple also differently to the K̄N and πΣ channels [52]. In case of the Jülich potential the for- mer shows little resemblance with the πΣ invariant-mass spectrum given in ref. [57] while the results based on the K̄N → πΣ amplitude are roughly in line with the empir- ical information. Note that there are also significant differences in the invariant-mass distributions due to the K−p → πΣ ampli- tudes for the different possible charge states of πΣ. Specif- ically, the peak positions for π−Σ+ and π+Σ− differ by almost 30MeV. Interestingly, the experimental invariant- mass distributions for the two charge states, cf. the dot- ted and dashed histograms in fig. 3, respectively, seem to show a similar separation of their peak and both agree roughly with the corresponding predictions of the Jülich K̄N potential. We should mention, however, that the ex- perimental π+Σ− mass spectrum is afflicted by fairly large background contributions [57]. These differences in the invariant-mass distributions are caused primarily by the interference between the I = 0 and I = 1 amplitudes [60] when evaluating the observables in the particle basis. They are not due to isospin-breaking effects. Indeed, we found that an isospin-symmetric calcu- lation based on averaged masses yields very similar results. Note that then the π−Σ+ → π−Σ+ and π+Σ− → π+Σ− results would coincide. In this context let us mention that there have been some recent experimental attempts to achieve a better determination of the πΣ invariant-mass spectrum in the region of the Λ(1405) [61,62]. The π0Σ0 distribution mea- sured in the reaction K−p → π0π0Σ0 by the Crystal Ball J. Haidenbauer et al.: DN interaction from meson exchange Page 7 of 16 Table 2. DN results of the meson-exchange model, the SU(4) WT model of ref. [28] and the SU(8) WT model of ref. [30]. The pole positions of the SU(4) WT model were published in [30]. Meson-exchange model SU(4) DN model [28] SU(8) DN model [30] Scattering lengths [fm] aI=0 −0.41 + i 0.04 −0.57 + i 0.001 0.004 + i 0.002 aI=1 −2.07 + i 0.57 −1.47 + i 0.65 0.33 + i 0.05 Pole positions [MeV] S01 2593.9 + i 2.88 2595.4 + i 1.0 2595.4 + i 0.3 S01 2603.2 + i 63.1 2625.4 + i 51.5 2610.0 + i 35.5 S01 2799.5 + i 0.0 2821.5 + i 0.5 S01 2871.2 + i 45.6 S11 2797.3 + i 5.86 2661.2 + i 18.2 2553.6 + i 0.34 S11 2694.7 + i 76.5 2612.2 + i 89.5 S11 2637.1 + i 40.0 S11 2822.8 + i 17.4 S11 2868.0 + i 19.3 P01 2804.4 + i 2.04 Collaboration [61] has been analyzed in ref. [63] within the chiral unitary approach and, together with corresponding results for the reaction π−p → K0πΣ [64], is considered as evidence of the two-pole nature of the Λ(1405) [63]. The interest in the lineshape of the Λ(1405) has been again revived with the Λ(1405) photoproduction experiment on a proton target at CLAS@JLAB. This experiment is pro- viding the first results on the Λ(1405) photoproduction analyzing all three πΣ decay modes. Preliminary results show differences in the Λ(1405) lineshapes as well as differ- ent Λ(1405) differential cross sections for each πΣ decay mode [65,66]. 4 Results for DN As already stated above, the SU(4) extension of the Jülich K̄N model to the DN interaction generates narrow states in the S01 and S11 partial waves which we identify with the experimentally observed Λc(2595) and Σc(2800) res- onances, respectively. Not surprisingly, we find an addi- tional pole in the S01 partial wave, located close to the other one. Similar to the corresponding state in the K̄N sector, the second pole has a much larger width, cf. ta- ble 2. Interestingly, our model even generates a further state, namely in the P01 partial wave, which, after fine- tuning (cf. sect. 2), lies at 2804MeV, i.e. just below the DN threshold. We are tempted to identify this state with the Λc(2765)-resonance, whose quantum numbers are not yet established [25]. Though we do not reproduce the resonance energy quantitatively, we believe that further refinements in the DN model, specifically the inclusion of the ππΛc channel in terms of an effective σΛc chan- nel, can provide sufficient additional attraction for obtain- ing also quantitative agreement. The mechanism could be the same as in the case of the Roper (N∗(1440)) reso- nance, which is generated dynamically in the Jülich πN model [67,68]. Here the required strong attraction is pro- duced via the coupling of the πN P -wave (where the Roper occurs) to the S-wave in the σN system, facilitated by the different parities of the π and σ mesons. Besides shifting the resonance position, the coupling to an effective σΛc would certainly also increase the width significantly, as is required for a reproduction of the experimental in- formation [25]. For comparison we include here some predictions of the DN interactions presented in refs. [28,30] which are based on the leading-order WT contact term. One of them [28] is derived in an SU(4) framework compara- ble to our meson-exchange model, while the other one is based on an extended SU(8) spin-flavor scheme [30]. Both these DN interactions have also been adjusted to repro- duce the Λc(2595)-resonance. However, compared to the Jülich meson-exchange model, the total number of poles and their energies are different, cf. table 2. Three S01 and two S11 states appear up to 2900MeV in the SU(4) DN WT model, as reported in ref. [30]. Among them, there is a S11 state at 2694MeV with a width of Γ = 153MeV that strongly couples to the DN channel. Thus, it is observed as a structure in the real axis close to the DN threshold with similar effects as the S11-resonance at 2797MeV of our model. The SU(8) DN WT model [30] predicts even more states in this energy region, as indicated in table 2. But it also yields a Λc(2595)-resonance with a significantly smaller width than the other two interaction models. This has consequences for the πΣc invariant-mass spectrum, to be discussed below. Some results for the DN scattering are presented in figs. 4 and 5. Specifically, we show DN cross sections for I = 0 and I = 1 based on the parameter set that reproduces the positions of the Λc(2595) and Σc(2800) of the Particle Data Group (solid line) together with re- Page 8 of 16 The European Physical Journal A 0 25 50 75 100 125 150 s 1/2 -mN-mD (MeV) 0 100 200 300 400 500 σ to t ( m b) I=0 0 25 50 75 100 125 150 s 1/2 -mN-mD (MeV) 0 100 200 300 400 500 σ to t ( m b) I=1 Fig. 4. DN cross sections for the isospin I = 0 (top) and I = 1 (bottom) channels as a function of ε = √ s − mN − mD. The solid lines are the results of our meson-exchange model. The dash-dotted lines are based on parameters taken directly from earlier K̄N and KN potentials [15,16,38], cf. text, while the dashed lines are predictions of the SU(4) WT model of, ref. [28] and the dash–double-dotted lines are predictions of the SU(8) WT model of ref. [30]. The dotted lines are the corresponding results for K̄N of the Jülich meson-exchange model [14]. sults that follow directly from the K̄N and KN studies of refs. [15,16,38] by assuming SU(4) symmetry (dash- dotted line). The DN cross sections of the SU(4) WT model of ref. [28] (dashed line) and of the SU(8) WT model of ref. [30] (dash–double-dotted line) are also dis- played. Finally, we include the K̄N cross sections of the original Jülich model [14] for reference (dotted line). Obviously, the DN cross sections show a significant momentum dependence. Furthermore, they are substan- tially larger than those we obtain for D̄N [4]. In particular, the cross section in the isospin I = 1 channel amounts to almost 600mb at threshold. This is not too surprising in -1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1 cos θ 0 5 10 15 20 dσ /d Ω ( m b/ sr ) I=0 -1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1 cos θ 0 5 10 15 20 dσ /d Ω ( m b/ sr ) I=1 Fig. 5. Differential cross sections for DN and K̄N for the isospin I = 0 (top) and I = 1 (bottom) channels at ε = 25 MeV. The solid lines are the results of our meson-exchange model. The dash-dotted lines are based on parameters taken directly from earlier K̄N and KN potentials [15,16,38], cf. text, while the dotted lines are the corresponding results for K̄N of the Jülich meson-exchange model [14]. view of the near-by S11 quasi-bound state. The structure of the cross section in the isospin I = 0 channel is strongly influenced by the P01 partial wave where our model pro- duces a near-threshold quasi-bound state or, in case of the parameter set directly fixed by SU(4) constraints, a resonance state around 25MeV above threshold. The DN cross section of the SU(4) WT approach of ref. [28] shows a similar behaviour as the one of the Jülich model for the I = 1 channel. As already discussed above, this model gen- erates likewise poles in the S11 partial wave, though not as close to the DN threshold as the Jülich model. In case of the I = 0 channel there are pronounced differences. But this is not surprising because the SU(4) WT model yields only S-wave contributions while the results of the Jülich J. Haidenbauer et al.: DN interaction from meson exchange Page 9 of 16 model are dominated by the P -wave. Overall larger differ- ences are seen in comparison to the results for the SU(8) WT model. Specifically, for I = 0 it predicts cross sections close to zero in the near-threshold region and one sees a clear resonance signal from the narrow state generated at around 2821MeV, cf. table 2. Please note that the SU(8) WT model of [30] is not just a straight extension of the SU(4) WT model [28]. For example, the regularization procedures are different: in the SU(4) WT model a subtraction at a certain scale is im- plemented while for the SU(8) WT model the subtraction is performed slightly below the lowest threshold. Further- more, the SU(4) symmetry is broken in different ways. In the SU(4) WT model the symmetry is broken primarily by the used physical masses and by a fixed reduction factor in the (non-diagonal) transitions where a heavy (charmed) meson can be exchanged. In the SU(8) WT model the symmetry is broken by using physical masses but also due to the different weak decay constants used, cf. the dis- cussion in [30]. In the Jülich model the D∗N channel is already explicitly included as required by HQS. However, unlike in the SU(8) WT model the direct interaction in the D∗N system is ignored. Nonetheless, one should not expect that an extension of the Jülich model with full implementation of HQS, which is certainly an interesting project for the future, will lead to results similar to the SU(8) WT model. Realizing that there is a significant can- cellation between the π and ρ exchanges (that strongly re- duces the net effect of their important tensor components) it has been the strategy of the Bonn and Jülich groups to include always both contributions. This philosophy was also followed in the strangeness sector [14–16] and is fol- lowed now in the charm case too. As a consequence, the DN → D∗N transition potential due to the combined π and ρ exchange is fairly weak —and the same would happen for the corresponding (direct) D∗N → D∗N in- teraction. In the SU(8) WT model of [30] only the WT term is included, corresponding to vector-meson (ρ) ex- change, for D∗N → D∗N as well as for DN → D∗N suggesting that those interactions should be significantly stronger than what we would get in the meson-exchange picture. Let us now come to the scattering lengths. The dis- cussed quasi-bound state in the S11 channel of the meson- exchange model is also reflected in the pertinent results, cf. table 2, namely by the rather large value of the real part of aI=1. The same situation is observed in the SU(4) WT approach of [28]. In fact, the S-wave scattering lengths predicted by our model and by the WT approach turn out to be very similar qualitatively for the I = 1 as well as for the I = 0 channel, as can be seen in table 2. This cannot be said about the SU(8) WT model of [30] which predicts radically different scattering lengths in both isospin chan- nels. (Note that there is a typo in the real part of the I = 0 scattering length in ref. [69], where the results were first published.) For completeness, let us mention here that the scattering lengths of the DN interaction of Hofmann and Lutz [27], reported in ref. [29], amount to about −0.4 fm for both isospin channels. In agreement with that work we find that the imaginary part in the Jülich model is negli- gibly small for I = 0. However, the imaginary part in the I = 1 channel for our DN model is not negligible, con- trary to ref. [29]. This is due to the fact that in ref. [29] there is no quasi-bound state close to DN threshold, but lies 180MeV below. Angular distributions for the reaction DN → DN are shown in fig. 5. Obviously, in the I = 0 case there is a strong anisotropy already at fairly low momenta. It is due to significant contributions in the P01 partial wave in this momentum region induced by the near-threshold quasi- bound state or resonance, respectively, produced by our model, as discussed above. For higher momenta, the dif- ferential cross section becomes forward peaked, similar to the predictions of our model for the D̄N system [4]. Predictions for DN scattering observables in the par- ticle basis (D0n → D0n, D0p → D0p, D0p → D+n) can be found in ref. [5]. 5 Discussion of the Λc(2595) The excited charmed baryon Λc(2595) was first observed by the CLEO Collaboration [21] and its existence was later confirmed in experiments by the E687 [22] and AR- GUS [23] Collaborations. In all these experiments the reso- nance appears as a pronounced peak in the invariant-mass distribution of the π+π−Λ+ c channel. It is now generally accepted that the Λc(2595) is the charmed counterpart of the Λ(1405) [25]. Therefore, it is natural that interaction models of the K̄N system in which the Λ(1405) appears as a dynamically generated state, as is the case in chiral unitary approaches as well as in the traditional meson-exchange picture, likewise gener- ate the Λc(2595) dynamically, provided that SU(4) sym- metry is assumed when extending the interactions from the strangeness to the charm sector. In this context it is important to realize that there are also drastic differences between the strangeness and the charm case. In particu- lar, the Λ(1405) is located fairly close to the K̄N thresh- old, which is around 1430MeV, while the Λc(2595) co- incides practically with the πΣc threshold, which is at around 2593MeV. Furthermore, the πΣ and K̄N thresh- olds are roughly 100MeV apart, while there are more than 200MeV between the πΣc and DN thresholds. Finally, the ππΛc channel —where the Λc(2595) is experimentally observed— opens 35MeV below the resonance, while the corresponding ππΛ channel is barely open at the location of the Λ(1405). Note that the ππ channel must be in the IG(JP ) = 0+(0+) (“σ”) state when ππΛc couples to the Λc(2595). And, due to parity conservation, it is the P01 partial wave of the ππΛc (ππΛ) system which couples to the S01 DN (K̄N) channel. In view of the mentioned kinematical differences, it is certainly not surprising that the models do not really predict the Λc(2595) at exactly the position where it was found in the experiment. A fine-tuning of inherent param- eters such as subtraction constants or coupling constants is required to shift the resonance to the observed energy. Specifically, in case of the Jülich DN model the coupling constants of the scalar mesons were adjusted in such a way Page 10 of 16 The European Physical Journal A 2580 2600 2620 2640 s 1/2 (MeV) 0 2 4 6 8 10 |T |2 q (M eV fm 2 ) π−Σc ++ π0Σc + π+Σc 0 πΣc 2580 2600 2620 2640 s 1/2 (MeV) 0 2 4 6 8 10 |T |2 q (M eV fm 2 ) π−Σc ++ π0Σc + π+Σc 0 πΣc 2580 2600 2620 2640 s 1/2 (MeV) 0 2 4 6 8 10 |T |2 q (M eV fm 2 ) π−Σc ++ π0Σc + π+Σc 0 πΣc Fig. 6. πΣc invariant-mass spectrum in an isospin-symmetric calculation. In the top-left panel are results based on our meson- exchange model. Those in the top-right panel are for the SU(4) WT model of ref. [28] while those at the bottom are for the SU(8) WT model of ref. [30]. The lower curves are based on the πΣc → πΣc T -matrix while the upper ones correspond to D0p → πΣc. that the πΣc S-wave phase shift in the I = 0 channel goes through 90◦ at 2595MeV, i.e. at the nominal Λc(2595) mass as listed in the PDG [25]. It should be said, how- ever, that for an investigation of the DN interaction, as performed in ref. [5], the precise position of the Λc(2595)- resonance does not play a role. The experimental papers report uniformly that the Λc(2595) decays dominantly into the π+Σ0 c and π−Σ++ c channels [22–24]. In the latter reference one can even read that Λc(2595) decays almost 100% to πΣc. At first sight this seems not unreasonable. Considering the reported mass difference M(Λc(2595)) − M(Λc) of 307.5 ± 0.5 ± 1.2MeV (CLEO ‘95), 309.7 ± 0.9 ± 0.4MeV (E687 ‘96), 309.2 ± 0.7 ± 0.3MeV (ARGUS ‘97), J. Haidenbauer et al.: DN interaction from meson exchange Page 11 of 16 respectively, and the corresponding threshold values for the πΣc channels M(π−) + M(Σ++ c ) − M(Λc) = 307.13 ± 0.18MeV, M(π0) + M(Σ+ c ) − M(Λc) = 301.42 ± 0.4MeV, M(π+) + M(Σ0 c ) − M(Λc) = 306.87 ± 0.18MeV, where we use the latest values from the PDG [25], there is some phase space for the Λc(2595) → πΣc decay. However, the new CLEO measurement with improved statistics and with better momentum resolution [24] sug- gests a mass difference of only 305.3 ± 0.4 ± 0.6MeV (CLEO ‘99). (4) A very similar mass difference (305.6 ± 0.3MeV) was ob- tained in an independent analysis of the new CLEO data by Blechman et al. [70]. Such a value reduces the phase space for the decay of the Λc(2595) into the π+Σ0 c and π−Σ++ c channels significantly. Indeed the decay is only possible due to the finite widths of the involved particles. But, since the widths are only in the order of 2MeV (Σ++ c , Σ0 c ) to 4MeV Λc(2595) [25] and the detector resolution is 1.28MeV [24], it is still surprising that the Λc(2595) should decay domaninatly into those two channels as sug- gested by the experiment. We will come back to this issue at the end of this section. In the following we present predictions of the Jülich DN model and of the SU(4) and SU(8) WT models of refs. [28,30] for the invariant-mass distributions, i.e. for the quantity |T |2 · q, in the relevant πΣc channels. This allows us to explore whether there are any quantitative or even qualitative differences in the predictions of those models. Furthermore, we can illustrate the subtle effects of the slightly different thresholds of the π+Σ0 c , π0Σ+ c , and π−Σ++ c channels on the various invariant-mass dis- tributions due to the presense of a near-by pole. But first, let us show results for the isospin-symmetric calculation (based on averaged masses), cf. fig. 6. The upper curves correspond to the DN → πΣc T -matrix while the lower ones are for πΣc → πΣc. In this figure the relative nor- malization of the πΣc to the DN → πΣc channel is kept as predicted by the models but all T -matrices are multi- plied with the mass factor MπMΣc /(Mπ + MΣc ) in order to obtain convenient units for plotting. The results shown in fig. 6 make clear that in case of our meson-exchange model the amplitudes of the DN -πΣc systems are completely dominated by the I = 0 contribu- tion in the region of the Λc(2595). As a consequence, the predictions for all charge states practically coincide. As mentioned above, this is not the case for K̄N -πΣ where the interference between the I = 0 and I = 1 amplitudes is significant. For example, there the invariant-mass dis- tribution for π±Σ∓ → π±Σ∓ differs by roughly a factor of two from that of π0Σ0 → π0Σ0, cf. fig. 3. Furthermore, it can be seen from fig. 6 that the shape of the invariant- mass distribution obtained from the πΣc and DN → πΣc T -matrices are fairly similar. In fact, a comparison of the results shown in the left and right panels of fig. 6 reveals that there is even hardly any difference between the re- sults of the meson-exchange model and the SU(4) WT interaction. Even the absolute magnitudes are rather simi- lar. This might be surprising but is certainly a reflection of the specific situation with the Λc(2595) being located very close to the πΣc threshold. In such cases one expects to see features that are practically model independent [71–73]. On the other hand, the results for the SU(8) WT model differ drastically. This is due to the fact that the Λc(2595) generated by this model is much narrower, cf. table 2, and the near-by second state in the I = 0 sector is also narrower than in the other two models considered. Moreover, there is even a near-by state in the I = 1 chan- nel (at 2612MeV) so that the I = 1 amplitude is size- able —in contradistinction to what happens for the Jülich and the SU(4) WT interactions. In this context let us mention that the DN → πΣc results of the Jülich model are calculated from the half-off-shell transition T -matrix assuming the DN momentum to be zero, while for the WT results [28,30] the on-shell amplitude is used. In the latter case the corresponding DN momentum is purely imaginary. Results based on the physical masses are presented in fig. 7. Here the invariant-mass distributions are shown in arbitrary units and normalized in such a way that one can easily compare the results based on the πΣc → πΣc (left panel) and DN → πΣc (right panel) T -matrices. But we keep the relative normalization between the different charge channels as predicted by the model. Obviously, there are drastic effects due to the different thresholds. The threshold of the π0Σ+ c channel is about 6MeV lower than those of the other two charge channels and, as a consequence, the predicted invariant-mass distribution is about twice as large. Moreover, there is a clear cusp in π0Σ+ c at the opening of the π+Σ0 c channel. The threshold of π−Σ++ c is just about 0.3MeV above the one for π+Σ0 c . It produces a noticeable kink in the π0Σ+ c invariant-mass distribution. On the other hand, and as already antici- pated from the comparison of the isospin-symmetry cal- culation above, there are only rather subtle differences be- tween the lines shapes predicted from the πΣc → πΣc and from the corresponding DN → πΣc amplitudes. The only more qualitative difference consists in the stronger fall- off with increasing invariant mass exhibited by the results based on the πΣc → πΣc T -matrix. Finally, let us come to the experimental π+π−Λ+ c mass distribution where the signal for the Λ(2595) was found. The corresponding data [23,24] are also displayed in fig. 7. The results for the D0p → π0Σ+ c as well as for the π0Σ+ c → π0Σ+ c channels resemble indeed very much the measured signal and one can imagine that smearing out our results by the width of the Σ+ c , which is roughly 4MeV [24,25], would yield a fairly good fit to the data. However, experimentally it was found that the Λc(2595) decays predominantly into the π+Σ0 c and π−Σ++ c chan- nels with a branching fraction in the range of 66% [23] to close to 100% [24]. Smearing out the corresponding results with the significantly smaller and better known widths of the Σ0 c and Σ++ c , of just 2MeV [25], would still leave many of the events found below the nominal π+Σ0 c and π−Σ++ c threshold unexplained, especially for the CLEO Page 12 of 16 The European Physical Journal A 2580 2590 2600 2610 s 1/2 (MeV) 0 1 2 3 4 |T |2 q ( a. u. ) π−Σ++ π0Σ+ π+Σ0 πΣc -> πΣc 2580 2590 2600 2610 s 1/2 (MeV) 0 1 2 3 4 |T |2 q ( a. u. ) D 0 p -> π−Σc ++ D 0 p -> π0Σc + D 0 p -> π+Σc 0 DN -> πΣc Fig. 7. πΣc invariant-mass spectrum predicted by our DN meson-exchange model. On the left are results based on the πΣc → πΣc T -matrix and on the right the ones for D0p → πΣc. For illustration purposes we show also data for the π+π−Λ+ c invariant-mass distribution taken from [23] (squares) and [24] (circles). experiment [24]. Thus, it seems to us that the position of the Λc(2595)-resonance being so close to or even be- low the nominal π+Σ0 c /π−Σ++ c thresholds and the found large branching ratios into those channels are difficult to reconcile. Independently of that, we would like to say also a word of caution concerning our own results. In view of the fact that the signal for the Λc(2595)-resonance is seen in the π+π−Λ+ c channel, any more rigorous model analysis would definitely require the explicit inclusion of this channel. In principle, the presence of the ππΛ+ c channel could be simu- lated within our model by adding a phenomenological σΛ+ c channel, analogous to the treatment of the ππN channel in our πN model [67,68]. But then many new parameters would have to be introduced that can no longer be fixed by SU(4) arguments in a reasonable way. In any case, first it would be important to confirm the new CLEO data by independent measurements of the ππΛ+ c and πΣc mass spectra in the region of the Λc(2595). Specifically, it would be essential to establish unambigously that there is a large decay rate of that res- onance into the πΣc channels. A precise determination of the pole position of the Λc(2595) could then be done along model-independent approaches such as the ones suggested in refs. [70,71]. Besides being much better suited for per- forming a fit to data and for deducing uncertainties, these approaches allow one to incorporate also finite widths ef- fects appropriately which is very difficult to achieve in models like the ones discussed in the present paper. In view of the experimental situation discussed above such finite widths effects might play a crucial role. Note that the predictions of the SU(4) WT model for the quantities |TπΣc→πΣc |2 · q and |TD0p→πΣc |2 · q are very similar to the ones of the meson-exchange model and, therefore, we do not show the corresponding curves here. On the other hand, those for the SU(8) WT interaction are characterized by the very narrow width of the Λc(2595) predicted by this model and, thus, do not resemble at all the measured πΣc invariant-mass spectrum. We refrain from showing those results here. In order to understand the differences in the mass spec- tra for the strangeness and charm sectors it is instructive to take a look at the phase shifts of the πΣ and πΣc channels in the S01 partial wave where the poles corre- sponding to the Λ(1405) and Λc(2595) are located. Cor- responding results are presented in fig. 8. The standard relation of the (partial-wave–projected) T -matrix to the phase shift is T (q) = − exp(iδ(q)) sin(δ(q))/q. The quan- tity sin2(δ(q)) has its maximum where δ(q) passes through 90◦. The maximum of the corresponding invariant-mass distribution, |T |2·q = sin2(δ(q))/q, will occur at somewhat smaller invariant masses, due to the additional 1/q factor and depending on the slope with which the phase shift passes through 90◦. The invariant-mass distribution will be zero where the phase shift passes through 180◦. These are indeed the features of the πΣ → πΣ invariant-mass spectrum, shown in fig. 3 on the left side. On the other hand, the K̄N → πΣ invariant-mass spectrum has its maximum close to the position of the pole. This corre- sponds to the region where the πΣ phase shift exhibits the strongest variation with energy, which is around 1420 to 1430MeV, cf. fig. 8 (left). J. Haidenbauer et al.: DN interaction from meson exchange Page 13 of 16 1320 1360 1400 1440 s 1/2 (MeV) 0 20 40 60 80 100 120 140 160 180 200 220 240 δ ( de gr ee s) πΣ 2600 2620 2640 2660 s 1/2 (MeV) 0 20 40 60 80 100 120 140 160 180 200 220 240 δ ( de gr ee s) πΣc Fig. 8. S01 πΣ and πΣc phase shifts. Solid lines represent results for the Jülich K̄N and DN models while the dashed lines are those for the corresponding WT interactions [28,56]. The dash–double-dotted line is for the SU(8) WT interaction of [30]. In this context we would like to draw attention to the fact that the behavior of the πΣ S01 phase shift and the pertinent invariant-mass distribution is very similar to the one of the ππ δ00 partial wave. In the latter case the phase shift shows a broad shoulder at lower energies, passing slowly through 90◦, a behavior usually associated with the σ-meson (or f0(600)), while finally rising quicky through 180◦ around 1GeV at the location of the f0(980)- resonance. The corresponding mass spectrum consists in a broad bump on which the f0(980) appears as a dip struc- ture, cf. ref. [74]. Also in the case of the K̄N -πΣ sys- tem the pole at around 1436MeV with the smaller width, which one might associate with the Λ(1405), produces a peak in the K̄N → πΣ invariant-mass spectrum but a dip in the one computed from the πΣ → πΣ T -matrix. The behavior of the corresponding phase shift for the πΣc system is very much different, cf. fig. 8 (right). Here the strongest variation with energy occurs already very close to the threshold and in the same region the phase also goes through 90◦. The phase shift predicted by the SU(4) WT model [28, 56] is shown by the dashed line in fig. 8. Results for πΣc for the SU(8) WT model [30] (dash–double-dotted line) are included too. Obviously, for the DN -πΣc system the SU(4) WT result is very similar to the prediction of the meson-exchange model. Thus, it is not surprising that also the corresponding invariant-mass distributions, presented in fig. 7, are very similar to those of the Jülich potential. For K̄N -πΣ there are noticeable quantitative differences. In particular, the slope of the phase shift is significantly larger where it passes through 90◦, reflecting the fact that the two poles of the WT model [56] are much closer to each other [52] than in case of the Jülich model, and, as a consequence, the maxima of the invariant-mass spectra based on the K̄N → πΣ and πΣ → πΣ amplitudes are also closer to each other [52]. The πΣc phase shift for the SU(8) WT model clearly reflects the much narrower width of the Λc(2595) (and of the close-by S01 2610MeV resonance) predicted by this interaction. 6 Summary In this paper we presented a model for the interaction in the coupled systems DN , πΛc, and πΣc, which was developed in close analogy to the meson-exchange K̄N interaction of the Jülich group [14], utilizing SU(4) sym- metry constraints. The main ingredients of the DN in- teraction are provided by vector meson (ρ, ω) exchange but higher-order box diagrams involving D∗N , DΔ, and D∗Δ intermediate states, are also included. The coupling of the DN system to the πΛc and πΣc channels is facil- itated by D∗(2010) exchange and by nucleon u-channel pole diagrams. The interaction model generates several states dy- namically. The narrow DN quasi-bound state found in the S01 partial wave is identified with the (I = 0) Λc(2595)-resonance. Narrow states were also found in the S11 and P01 partial waves. We identify the former with the I = 1 resonance Σc(2800) and the latter with the Λc(2765)-resonance, whose quantum numbers are not yet established [25]. Results for DN total and differential cross sections were presented and compared with predictions of two Page 14 of 16 The European Physical Journal A interaction models that are based on the leading-order Weinberg-Tomozawa term using SU(4) [28] or SU(8) [30] symmetry. While the predictions of the Jülich and SU(4) WT models for the I = 1 channel are fairly similar, in magnitude as well as with regard to the energy depen- dence, this is not the case for the I = 0 amplitude. Here the possible presence of a P -wave resonance near the threshold, i.e. the Λc(2765), has a dramatic influence on the shape and the energy dependence of the cross sec- tion. The SU(8) WT results are drastically different alto- gether, reflecting the different resonant structure of this interaction. Finally, we discussed the Λc(2595)-resonance and the role of the near-by πΣc threshold. In particular, we pre- sented results for the πΣc invariant-mass spectrum in the particle basis which illustrate the subtle effects of the slightly different thresholds of the π+Σ0 c , π0Σ+ c , and π−Σ++ c channels on the various invariant-mass distri- butions. We also pointed out that there seems to be a contradiction between the observation that the nar- row Λc(2595)-resonance decays almost exclusively into the π+Σ0 c and π−Σ++ c channels and the latest values of its mass, which place the resonance about 2MeV below the thresholds of those channels [24,70]. Indeed with a mass of 2592.06±0.3MeV, as determined in ref. [70], the Λc(2595) would lie just between the π0Σ+ c and the π+Σ0 c /π−Σ++ c thresholds, which surely would be an interesting scenario. J.H. acknowledges instructive discussions with M. Döring about the determination of poles in the complex plane. This work is partially supported by the Helmholtz Association through funds provided to the virtual institute “Spin and strong QCD” (VH-VI-231), by the EU Integrated Infrastruc- ture Initiative HadronPhysics2 Project (WP4 QCDnet), by BMBF (06BN9006) and by DFG (SFB/TR 16, “Subnuclear Structure of Matter”). G.K. acknowledges financial support from CAPES, CNPq and FAPESP (Brazilian agencies). L.T. wishes to acknowledge support from the Rosalind Franklin Pro- gramme of the University of Groningen (The Netherlands) and the Helmholtz International Center for FAIR within the frame- work of the LOEWE program by the State of Hesse (Germany). Appendix A. The interaction Lagrangians In this appendix we list the specific interaction La- grangians which are used to derived the meson-exchange D̄N interaction. The baryon-baryon-meson couplings are given by [14] LBBS = gBBSΨ̄B(x)ΨB(x)ΦS(x), LBBP = gBBP Ψ̄B(x)iγ5ΨB(x)ΦP (x), LBBV = gBBV Ψ̄B(x)γμΨB(x)Φμ V (x) + fBBV 4mN Ψ̄B(x)σμνΨB(x)(∂μΦν V (x) − ∂νΦμ V (x)), LBΔP = fBΔP mP Ψ̄Δμ(x)ΨB(x)∂μΦP (x) + H.c., LBΔV = fBΔV mV i(Ψ̄Δμ(x)γ5γμΨB(x) −Ψ̄B(x)γ5γμΨΔμ(x))(∂μΦν V (x) − ∂νΦμ V (x)). (A.1) Here, ΨB and ΨΔμ are the nucleon (or hyperon) and Δ field operators and ΦS , ΦP , and Φμ V are the field operators for scalar, pseudoscalar and vector mesons, respectively. The employed three-meson couplings are LPPS = gPPSmP ΦP (x)ΦP (x)ΦS(x), LPPV = gPPV ΦP (x)∂μΦP (x)Φμ V (x), LV V P = gV V P mV iεμντδ∂ μΦν V (x)∂τΦδ V (x)ΦP (x), (A.2) where εμντδ is the totally antisymmetric tensor in four dimensions with ε0123 = 1. Details on the derivation of the meson-baryon interaction potential from those La- grangians can be found in ref. [14] together with explicit expressions for those potentials. The SU(4) flavour struc- ture that leads to the characteristic relations between the coupling constants is discussed in sect. II of [4]. All ver- tices are supplemented with form factors in order to sup- press the meson-exchange contributions for high-momen- tum transfer and guarantee convergence when solving the Lippmann-Schwinger equation. For all (t-channel) ex- change diagrams those vertex form factors are parameter- ized in a conventional monopole form [14] Fα(q2) = ( Λ2 α − m2 exch Λ2 α + q2 )nα , (A.3) where q2 is the square of the three-momentum transfer. Here nα = 1 is used for all vertices except for the NΔρ vertex where nα = 2 [14,75]. In case of (s and u channels) pole contributions a slightly different form is used to avoid problems of convergence and singularity, viz. Fβ(q2) = ( Λ4 β + m4 exch Λ4 β + (q2)2 ) , (A.4) where q2 is the square of the four-momentum transfer [14]. Note that both forms are normalized in such a way that F ≡ 1 when the exchanged particle is on its mass shell. The values for the vertex parameters (coupling constants and cut-off masses) that are used in our meson-exchange model of the DN interaction are summarized for conve- nience in table 3. The following averaged masses are used for evaluating the interaction potential and when solving the Lippmann-Schwinger equation in isospin basis: MN = 938.926MeV, MΛc = 2286.5MeV, MΣc = 2455.0MeV, Mπ = 138.03MeV, MD = 1866.9MeV. For the calcula- tion in the particle basis we use the masses as given in the PDG listing [25]. Open Access This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited. J. Haidenbauer et al.: DN interaction from meson exchange Page 15 of 16 Table 3. Vertex parameters used in the meson-exchange model of the DN interaction. mexch is the mass of the exchanged particle. gM and gB/fB (ΛM and ΛB) refer to the coupling constants (cut-off masses) used at the meson-meson-meson and baryon-baryon-meson (for pole and baryon-exchange graphs: upper and lower) vertex, respectively. The σ and a0 coupling constants in brackets are those that follow from our D̄N model [4] under the assumption of SU(4) symmetry. Process Exch. part. mexch gMgB/(4π) gMfB/(4π) ΛM ΛB [MeV] [GeV] [GeV] DN → DN ρ 769 0.773 4.713 1.4 1.6 ω 782.6 −2.318 0.0 1.5 1.5 σ 600 2.60 [1.00] – 1.7 1.2 a0 980 −4.80 [−2.60] – 1.5 1.5 Λc 2286.5 15.55 – 1.4 1.4 Σc 2455 0.576 – 1.4 1.4 DN → D∗N π 138.03 3.197 – 1.3 0.8 ρ 769 −0.773 −4.713 1.4 1.0 DN → D∗Δ π 138.03 0.506 – 1.2 0.8 ρ 769 −4.839 – 1.3 1.0 DN → DΔ ρ 769 4.839 – 1.3 1.6 DN → πΛc D∗ 2009 −1.339 −4.365 3.1 3.1 N 938.926 −14.967 – 2.5 1.4 Σc 2455 1.995 – 3.5 1.4 DN → πΣc D∗ 2009 −0.773 1.871 3.1 3.1 N 938.926 2.88 – 2.5 1.4 Λc 2286.5 −10.368 – 2.8 1.4 Σc 2455 2.304 – 3.5 1.4 πΛc → πΛc Σc 2455 6.912 – 3.5 3.5 πΛc → πΣc ρ 769 0.0 7.605 2.0 1.35 Σc 2455 7.891 – 3.5 3.5 πΣc → πΣc ρ 769 3.092 5.689 2.0 1.16 Λc 2286.5 6.912 – 2.8 2.8 Σc 2455 9.216 – 3.5 3.5 References 1. PANDA Collaboration (I. Lehmann), arXiv:0909.4237 [hep-ex]. 2. 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