Chirally motivated K^- nuclear potentials

In-medium subthreshold KbarN scattering amplitudes calculated within a chirally motivated meson-baryon coupled-channel model are used self consistently to confront K^- atom data across the periodic table. Substantially deeper K^- nuclear potentials are obtained compared to the shallow potentials derived in some approaches from threshold amplitudes, with Re V_{chiral} = -(85+/-5) MeV at nuclear matter density. When KbarNN contributions are incorporated phenomenologically, a very deep K^- nuclear potential results, Re V_{chiral+phen.} = -(180+/-5) MeV, in agreement with density dependent potentials obtained in purely phenomenological fits to the data. Self consistent dynamical calculations of K^- nuclear quasibound states are reported and discussed.


Introduction
The interaction of K − mesons with hadronic systems, ranging from K − atoms through K − -nuclear clusters to dense strange hadronic matter realized perhaps in neutron stars, is of topical interest [1]. SU(3) chiral symmetry combined with coupled channel techniques provides a theoretical framework in which low energy meson-baryon observables can be systematically evaluated. In such an approach the Λ(1405) resonance, dominating theKN − πΣ physics at energies near theKN threshold, is generated dynamically. It is The Klein Gordon (KG) dispersion relation satisfied by K − mesons in medium of density ρ is of the form where Π K (ω K , ρ) = 2(Re ω K )V K − is the self energy (SE) operator for a K − meson with momentum p K and energy ω K . A p K momentum dependence of Π K is suppressed since, as is shown below, it may be transformed into density and energy dependence. In terms of the in-medium K − N c.m. forward scattering amplitude F K − N ( √ s, ρ) (here assumed s-wave): where s = (E K + E N ) 2 − ( p K + p N ) 2 is the Lorentz invariant Mandelstam variable s which reduces to the square of the total K − N energy in the twobody c.m. frame and m N is the nucleon mass. In the laboratory frame, E K = ω K . The KG dispersion relation (1) in bound state applications for a K − meson leads to a KG equation satisfied by the K − wavefunction [12]: where µ is the K − -nucleus reduced mass, V c is the Coulomb potential generated by the finite-size nuclear charge distribution, and B K = B K + iΓ K /2 is a complex binding energy, including a strong interaction width Γ K . Finitemedium corrections are applied in our calculations specifically to the coefficient of F K − N in Eq. (3). Wycech [15] and Bardeen and Torigoe [16] suggested long time ago within Λ(1405)-based phenomenological models that subthreshold K − N scattering amplitudes are relevant in kaonic atom studies, where the kaon energy is essentially at threshold. In the present Letter we construct the K − meson SE operator (2) near threshold (ω K ≈ m K ) from in-medium subthreshold K − N scattering amplitudes derived within a chirally motivated mesonbaryon coupled-channel separable-potential model [17]. It is shown for the first time how the energy and density dependence of F K − N ( √ s, ρ) leads to a deep K − -nuclear potential V K − in kaonic atoms. This conclusion holds already at the lowest-order Weinberg-Tomozawa level that provides excellent starting point for modern chiral models [2]. We also report on self-consistent calculations of K − quasibound nuclear states based on in-medium extensions of the free-space model of Ref. [17]. The calculated binding energies and widths are compared with similar entities calculated in Ref. [18].

Subthreshold energy and density dependence
The present calculations are based on the chirally motivated mesonbaryon coupled-channel separable-potential model of Ref. [17]. This freespace model expands consistently and systematically to next-to-leading-order (NLO) within the heavy baryon formulation of chiral perturbation theory. The low-energy constants of the model are fitted to low energy K − p scattering and reaction data. The free-space K − N scattering amplitude F K − N is given then in a separable form: with c.m. momenta k, k ′ . Of the several versions specified in Table 3 of Ref. [17], we chose the one with a range parameter α = 639 MeV. 2 Other chirally motivated potential models that start with zero range, i.e. no momentum dependence, in practical applications often introduce at least implicitly a finite range, e.g., a cutoff momentum q max = 630 MeV [5] or a renormalization scale µ = 630 MeV [20]. The momentum dependence of the amplitude F K − N in the separable-potential model of Ref. [17] is rather weak for the applications discussed in the present work and is secondary to the strong energy dependence generated by the Λ(1405) resonance. A typical resonance-shape energy dependence is shown in Fig. 1 by dashed lines (marked 'free') for the reduced scattering amplitude which is appropriate for the interaction of K − mesons with symmetric nuclear matter. The imaginary part peaks about 25 MeV below theKN threshold, and the real part rapidly varies there from weak attraction above to strong attraction below threshold. While f K − N ( √ s) at and near threshold is constrained by data that serve to determine the parameters of the chiral model, the extrapolation to the subthreshold region suffers from ambiguities and 2 This version, labeled below CS30, produces two I = 0 S-matrix poles in the neighborhood of theKN threshold, the lower one at Re √ s = 1398 MeV evolves from the πΣ channel, the upper one at 1441 MeV -from theKN channel, within the range of values provided by NLO chiral calculations, e.g., Ref. [19].
depends on the applied model [2]. The free-space reduced scattering amplitude shown in Fig. 1 is quantitatively similar to the corresponding K − N scattering amplitudes in other chirally motivated models exhibited in Fig. 7 of Ref. [20]. Also shown in Fig. 1 is the energy dependence of two in-medium versions in dot-dashed lines (marked 'without SE'), follows Ref. [4] to require Pauli blocking in the intermediateKN states for ρ = 0. The resulting f K − N exhibits now a resonance-like behavior about 20 MeV above threshold, in agreement with Ref. [4]. The other version, in solid lines (marked 'with SE'), follows Ref. [21] to add self consistently meson and baryon self energies in intermediate states, similarly to Refs. [5,22]. The resulting in-medium f K − N is strongly energy dependent below threshold, with a resonance-like behavior about 30 MeV below threshold. Similar results are obtained at full nuclear matter density ρ 0 = 0.17 fm −3 . We note that whereas the two in-medium reduced amplitudes shown in the figure are close to each other far below and far above threshold, they differ substantially at and near threshold. This applies also to the full amplitudes since the form factors g(k) remain intact in the transition from free-space to in-medium separable amplitudes. At threshold, in particular, the real part of the 'with SE' amplitude is about half of that 'without SE', corresponding to a depth −Re V K − (ρ 0 ) ≈ 40−50 MeV, in agreement with Ramos and Oset [5]. The c.m. reduced amplitudes in Fig. 1 are functions of √ s. In the twobody c.m. system p K + p N = 0, but in the K − -nucleus c.m. system (approximately nuclear laboratory system) p K + p N = 0. Averaging over angles . For bound hadrons we expand near threshold, neglecting quadratic terms in the binding energies where . For the square of the relative momenta k, k ′ in form factors g, Eq. (4), we again average on angles, yielding Replacing in Eqs. (6) and (7) the kinetic energy p 2 K /(2m K ) in the local density approximation by and approximating the nucleon kinetic energy p 2 N /(2m N ) in the Fermi gas model by The downward energy shift invoked by Eq. (8), with respect to E th , corresponds to that implied in the impulse approximation when the many-bodyKN amplitude evaluated at the threshold energy E th is replaced by the two-body amplitude at the c.m. energy √ s. Note its density dependence. In most of the subsequent discussion it is used as is, although we also checked the effect of implementing gauge invariance through the substitution √ s → √ s − V c . We note that gauge invariance was not implemented in the solution of the free-space Lippmann-Schwinger equations for the underlying chiral model of Ref. [17], nor in its in-medium extension displayed in Fig. 1.
For completeness, we comment on the range of momenta k, k ′ transcribed by Eq. (7). In naive applications to kaonic atoms where one assumes p K ≈ 0, these momenta are due to Fermi motion, as given by the first term on the r.h.s. of Eq. (7) which is bounded by k(ρ 0 ), k ′ (ρ 0 ) ∼ 72 MeV, quite negligible compared to the momentum dependence scale α = 639 MeV in model CS30. However, for strongly attractive K − nuclear potentials, reaching depths of about 180 MeV in phenomenological studies [12], the second term on the r.h.s. of Eq. (7) dominates, yielding k(ρ 0 ), k ′ (ρ 0 ) ∼ 276 MeV. These momenta are nonnegligible, but they are well within the CS30 momentum dependence scale α = 639 MeV which emerged by fitting to K − p low-energy data.

Kaonic atoms
We now apply the method outlined above to the interaction of K − mesons with nuclei close to threshold, as a means of testing the subthreshold chiral amplitude formalism against experimental results. A distinction was made in solving the KG equation (3) between proton and neutron densities, replacing with ρ p and ρ n normalized to Z and N, respectively, and Z + N = A. The reduced amplitudes f K − p and f K − n were evaluated at √ s given by Eq. (8), where the atomic binding energy B K was neglected with respect to B N ≈ 8.5 MeV. A similar approximation was made in Eq. (7) for k 2 , k ′ 2 . B K was also neglected with respect to m K in Eq. (3). The corresponding potential was calculated at each radial point for every target nucleus in the data base. Self consistency was required in solving Eq. (8) with respect to Re V K − , i.e., the value of Re V K − (ρ) in the expression for √ s and in the form factors g had to agree with the resulting Re V K − (ρ). Figure 2 shows, as representative examples, several K − -Ni potentials. The CS30 'with SE' amplitudes, within the self consistent procedure described above and without adjustable parameters, yield the potential marked CS30. For Re V K − , similar depths to within a few MeV are obtained using CS30 'without SE' amplitudes. Re V CS30 K − is twice deeper, −85 MeV with respect to −40 MeV, than the shallow potential (not shown here) used in the kaonic atom calculations of Ref. [23]. That shallow potential followed from a threshold value f K − N (E th , ρ), without going subthreshold. Yet, Re V CS30 K − is not as deep as Re V DD K − , where the density dependent (DD) potential V DD K − , also shown in the figure, represents the best fit with χ 2 = 103 for 65 data points obtained using purely phenomenological DD potentials [12]. The density dependence of the chiral CS30 'with SE' effective scattering amplitude F eff K − N , calculated self consistently for Ni, is shown in Fig. 3. The increase of Re F eff K − N (ρ) with density over the nuclear surface region combined with the decrease of Im F eff K − N (ρ) is the underlying mechanism behind the compression and inflation of successful phenomenological best-fit potentials such as V DD K − [13]. However, the decrease of Im F eff K − N is unreasonably rapid, leading in the lower part of Fig. 2 to the peculiar shape of Im V CS30 K − at the nuclear surface. 3 A substantial K − N energy shift into the subthreshold region is involved in the calculation of F eff K − N , e.g., −40 MeV at 0.5ρ 0 down to −60 MeV at ρ 0 in Ni. For such far subthreshold energies, the reduced amplitude Im f CS30 K − N , as seen in Fig. 1, is too small to provide the absorptivity required by the kaonic atom data. Note that these single-nucleon amplitudes do not account for multi-nucleon absorption which becomes increasingly important at subthreshold energies. 3 No similar bulge at the nuclear surface appears if Im V CS30 K − is derived from CS30 'without SE' amplitudes, and Im V CS30 K − (ρ 0 ) is then about twice that for the 'with SE' case shown in the figure.  is dominated by ρ 2 terms which are required by the fit procedure and which are likely to representKNN absorptive and dispersive contributions, respectively. The emerging phenomenology is similar to that for V π − in pionic atom studies where theoretically motivated single-nucleon contributions are augmented by phenomenological ρ 2 terms representing πNN processes [24]. More work is required to justify microscopically the size of the ρ 2 kaonic atom contributions suggested by the successful V CS30+phen.
Although our study of kaonic atoms focused on K − N s-wave interaction models incorporating the subthreshold Λ(1405) resonance, we also checked whether p-wave contributions from the subthreshold Σ(1385) resonance are sizable and, more importantly, whether they could modify the pattern of V CS30 K − and V CS30+phen. K − established above. To this end we used the Σ(1385)based p-wave potential of Weise and Härtle [18] consisting of resonance and background parts. While detailed account will be given elsewhere, we assert that fits to kaonic atom data within the present self-consistent procedure do not require substantial p-wave contributions, and that V CS30 K − are robust with respect to adding p waves.

K − -nuclear quasibound states
The threshold K − -nuclear potential V CS30 Fig. 2 is sufficiently attractive to generate K − -nuclear quasibound states. However, since the potential is energy dependent, it has to be calculated again self consistently for binding energies B K in the expected range of tens of MeV. Calculations for 16 O and 208 Pb were reported by Weise and Härtle [18] using a subthreshold extrapolation for solving self consistently the K − KG equation within a local potential approximation of a chiral-modelKN amplitude. The present self consistency scheme is based on Eq. (8) for √ s and Eq. (7) for k 2 , as applied here to kaonic atoms, but without neglecting B K for K − nuclear states. We thus solved the KG equation (3), requiring self consistency explicitly for B K (and implicitly for Re V K − ) using Eqs. (7) and (8). Realistic RMF density distributions ρ(r) were employed, within a fully dynamical calculation that allows the nuclear density to get polarized by the K − meson. In this dynamical scheme, following Refs. [14,25], a RMF self consistency cycle is applied in which the K − meson solution serves further as a source in the RMF equations of motion which are solved to produce a modified nuclear density that goes again into the K − meson KG equation. However, the present formulation differs fundamentally from previous RMF calculations: here V K − is generated microscopically from a two-body coupled channel chiral model and, furthermore, energy and density dependencies are introduced directly through the underlying K − N scattering amplitude. Binding energies and widths calculated dynamically for the 1s K − nuclear quasibound state in several nuclei across the periodic table are listed in Table 1. The 'with SE' in-medium version of the CS30 chiral model was used. Similar results are obtained for the 'without SE' version, with slightly larger binding energies and widths. The values of B K listed in the table are in close agreement with binding energies calculated within a dynamical RMF approach for nucleons and antikaons [25] when the K − nuclear interaction is Table 1: Binding energies B K and widths Γ K (in MeV) of 1s K − nuclear quasibound states, calculated self consistently from Eq. (3) using in-medium 'with SE' CS30 amplitudes F K − N ( √ s, ρ) within a dynamical RMF scheme [14,25]. Possible K − N N → Y N ρ 2 contributions are excluded.  Table 1 represent only K − N → πY absorption processes that are accounted for by the coupled channels chiral model without allowing for multinucleon absorption modes. They are relatively small, of order 10 MeV, reflecting the proximity of the πΣ thresholds which suppresses the major decay modes available for a K − meson on a single nucleon. The sensitivity of the width calculation to the precise form of inmedium subthreshold extrapolation of √ s is demonstrated by five separate static calculations of the K − 1s quasibound state in 16 O listed in Table 2. The first three calculations use the CS30 'without SE' version of the in-medium chiral model. The first one uses Eq. (10) for self consistency in the solution of the KG equation, mocking up as much as possible within our model the calculation by Weise and Härtle [18]. The next two calculations use Eq. , is seen to be minor by comparing with the third (B K , Γ K ) column. The last two calculations also use Eq. (8) for self consistency, but the CS30 in-medium version 'without SE' is replaced by 'with SE' which brings the calculated value of Γ K further down to near its value in the dynamical calculation reported in Table 1. The last column includes also a Σ(1385) p-wave contribution given in Ref. [18]. Its effect is found to be rather small since √ s is well below the peak of the Σ(1385) resonance. A more detailed account of K − quasibound state calculations incorporating p waves will be given elsewhere. In addition to the K − N → πY single-nucleon induced widths of K − quasibound states which are calculable from given in-medium chiral models and which were shown above to be quite model dependent, there are also sizable two-nucleon absorption processes K − NN → Y N with considerably lower thresholds that contribute additional widths of order 40 MeV [14,18,25]. These absorption processes could be simulated by adding energy dependent imaginary ρ 2 terms suggested by the (CS30+phen.) potential for kaonic atoms together with the associated dispersive real ρ 2 terms. This requires further consideration.

Conclusion
In conclusion, we have shown within in-medium extensions of the chirally motivated coupled-channel separable-interaction CS30 model [17] how to incorporate the strong energy and density dependence of the K − N scattering amplitude F K − N ( √ s, ρ), at and below threshold, into a self consistent evaluation of the SE operator Π K (ω K , ρ) for kaonic atoms. The procedure adopted here is sufficiently general to be applied within in-medium extensions of other chirally motivated interaction models. Our calculations demonstrate that kaonic atom data probe the subthreshold regime of the in-medium K − N scattering amplitude. The two in-medium extensions of CS30 considered in the present work produce parameter-free K − nuclear potentials for kaonic atoms which are similar to each other, with depths −Re V CS30 K − (ρ 0 ) = (85 ± 5) MeV and −Im V CS30 K − (ρ 0 ) = (20 ± 10) MeV evaluated at the center of a typical medium-weight nucleus such as Ni. Preliminary calculations using the Tomozawa-Weinberg lowest order chiral interaction term, including the new SIDDHARTA 1s level shift and width in kaonic hydrogen [26] produce similar results. The sharp decrease of Im f K − N towards the πΣ threshold is reflected in rapid decrease of Im F eff K − N with increased density. Therefore absorption processes beyond a single nucleon mechanism must be added together with a possible dispersive real term. Indeed to achieve truly low χ 2 values for K − atoms, phenomenological potential terms had to be added, leading to increased V K − depths, i.e., −Re V CS30+phen. K − nuclear quasibound states generated by in-medium extensions of CS30 were also calculated within a self consistent scheme. Potential depths −Re V K − (ρ 0 ) of order 100 MeV were obtained for both extensions used, exceeding somewhat the depths derived for kaonic atoms. This indicates a moderate state, or energy dependence of Π K (ω K , ρ 0 ). The similarity of the results for the two in-medium extensions used here is in striking contrast to the large difference in their Re f K − N values at threshold, as can be judged from Fig. 1. Implementing additional phenomenological term within K − nuclear quasibound calculations is likely to result in B K values higher than 100 MeV. This topic deserves further consideration.
Our calculations provide for the first time a microscopic link between shallow K − nuclear potentials [5,23] obtained from threshold K − N interactions and phenomenological deep ones deduced from kaonic atom data [13,14]. In future work, p-waveKN subthreshold amplitudes associated with the Σ(1385) resonance will be explored in detail, to confirm the secondary role played by these amplitudes in our preliminary extended studies of kaonic atoms 4 and in studies of quasibound K − nuclear states [18,25].