Lessons from fitting the lowest order energy independent chiral based $\bar{K}N$ potential to experimental data

It is shown, that fitting parameters of a $\bar{K}N$ interaction model to different sets of experimental data can lead to physical conclusions which might provide a deeper insight into the physics of this multichannel system. The available experimental data are divided into three parts: the"classical"set consisting of the low-energy $K^-p$ cross sections and the threshold branching ratios, the SIDDHARTA $1s$ level shift in kaonic hydrogen and the CLAS photoproduction data. We have fitted the parameters of a recently introduced energy-independent chiral based potential to different combinations of these data. We found, that for lowest order potentials the two poles corresponding to the $I=0$ nuclear quasi-bound state ($\Lambda(1405)$) and to the $K^-p$ $1s$ atomic level seem to resist to their simultaneous reproduction at the right place, though a more or less satisfactory compromise can be achieved. Potentials with the $\Lambda(1405)$ pole pinned down close to the PDG value fail to reproduce the classical two-body data with an acceptable accuracy. We think, that the proposed way of fitting potential parameters to experimental data could be useful in the case of other models, too.


I. INTRODUCTION
In the last decade the antikaon nucleon interactions attracted considerable attention.
Several models were proposed and at present it is generally believed, that the correct theoretical background for derivation of this quantity is the chiral perturbation expansion of the SU(3) meson-baryon interaction Lagrangian. The abundant literature on this subject is summarized in several recent review papers, e.g. [1][2][3] and references therein. We do not seek here after repeating this uneasy task. There are two main directions along which these interactions are constructed: a) A possibly complete reproduction of multichannel two-body data in a wide energy range. These approaches use relativistic formulation and in order to obtain good agreement with experimental data, they go beyond the lowest order term in the chiral perturbation expansion. While the resulting interactions provide a good description of meson-baryon two-body data, they are too complicated to be used for the theoretical description of systems involving more than two particles. b) In view of possible existence of kaonic nuclear clusters, the main idea of the other approach is to construct a potential, which can be used for calculation of n > 2 systems. In this case the simplicity of the potential is essential, therefore the usual practice is to keep the form of the lowest order Weinberg-Tomozawa (WT) term of the chiral expansion and try to adjust it to the experimental data. For this purpose data are selected from the vicinity ofKN threshold and in general from channels, open at this energy. Since the n > 2 calculations at present can be performed only in a non-relativistic, potential based framework, for consistence this formalism has to be kept also at the potential fitting level.
Whatever potential model is chosen, at a certain point its parameters have to be fitted to a set of experimental data. This process is usually not reported in detail when a new potential is introduced. We think, however, that the fitting procedure might reveal some important issues providing a deeper insight into the physics of theKN system. In the present paper we try to demonstrate this idea on the example of a recently introducedKN interaction, belonging to the group b) [4,5]. The potential is energy-independent and leads to a one-pole structure of the Λ(1405) resonance, in contrast to the two-pole one, obtained from the commonly used chiral basedKN potentials. 1 Although our potential is a rather special one, we think, that our conclusions concerning the physics of theKN system derived from the data fitting process, have certain relevance for other potential models, too.

II. INPUT: THEORETICAL MODEL AND EXPERIMENTAL DATA
The potential which we shall use to demonstrate the effects of data fitting [4,5], is an energy-independent implementation of the lowest order Weinberg-Tomozawa (WT) term of the chiral SU (3) meson-baryon interaction Lagrangian, designed for non-relativistic calculations in two-and few-body systems. Its actual form is a two-term multichannel separable potential: where the channel indices i(j) = 1, 2, 3, 4, 5 correspond to the channels , respectively. The form factors g iA and g iB are defined as where k i is the relative momentum, m i and µ i are the meson-and the reduced masses in channel i, while the β i are the corresponding range (or cut-off) parameters. The coupling constant λ ij has the form with c ij being the SU (3) Clebsh-Gordan coefficients, while F i (F j ) stand for the meson decay constants in channel i(j). The way, how to calculate two-body observables from this interaction is described in detail in [4,5]. The potential contains 7 adjustable parameters: the two meson decay constants FK and F π and the five range parameters β i in channels i = 1, 2, 3, 4, 5 .
There are three types of data, to whichKN potentials are usually fitted. The "classical" group consists of the six old elastic and inelastic low-energy K − p cross sections with a rather poor accuracy and the three threshold branching ratios γ, R n and R c : with usual experimental values γ = 2.36 ± 0.04; R n = 0.189 ± 0.015; R c = 0.664 ± 0.011 .
In order to avoid overweighting γ in χ 2 we adopt the reasoning of Guo and Oller [7] and take an experimental value with a somewhat increased error: γ = 2.36 ± 0.1.
A "modern" piece of data is the recently measured complex energy shift ∆E of the 1s atomic level in kaonic hydrogen [8]: There is some confusion in the literature how to treat the statistical and systematic errors in the fitting procedure. We have added them quadratically and used And, finally, a somewhat controversial experimental information is provided by the PDG value for the position of the Λ(1405) resonance (Λ * in the following): We call it controversial, since it can be seen only in different reactions involving more than two particles. As a consequence, the position and shape can (and do) depend on the reaction mechanism and details. Therefore, in our opinion, the direct identification of the PDG resonance parameters, deduced from observed line shapes, with a pole position of the two-bodyKN interaction is not justified. This practice leads to potentials with a stronḡ KN attraction, which in turn stimulated far reaching speculations about kaonic nuclear clusters.

III. FITTING
For our χ 2 per degree of freedom we take the one used by the majority of papers dealing with this problem, in particular Guo and Oller [7], which tries to establish a certain balance between the weights of discrete observables and cross section data: where K is the number of different measurements included into the fit, n k stands for the number of data points in the kth measurement, n P corresponds to the number of free parameters (7 in our case) and y exp k;i (y th k;i ) represent the ith experimental(theoretical) point in the kth data set with standard error σ k;i . 2 We have performed several fits to different combinations of the three types of experimental data. In general, the fits do not produce sharp minima: small changes in the parameters in the vicinity of the best values lead to other χ 2 local minima, slightly larger, than the "best" ones. To make clearer the contribution of different data to the resulting χ 2 we divided it into two parts: corresponding to the discrete and cross section data. Generally, the χ 2 cs > 0.7 − 0.8 for the best fits and provide the dominant part of the total χ 2 . This is due to the large errors and spreading of the cross section data and cannot be substantially reduced by the fitting procedure.
In our first fit (A) we fitted the potential parameters to the "classical" data set. The results can be seen in the first column of the summary Tables I and II and in part a) of Fig.1.
As for the quality of the fit, the discrete data (branching ratios) are within the experimental errors and the cross sections are also reproduced in a (visually) acceptable way. It can be seen, as we mentioned before, that the dominant part of the χ 2 comes from χ 2 cs . The most remarkable feature of fit A to the "classical" data is, that the position of the pole corresponding to Λ * is while for the 1s level shift in kaonic hydrogen ∆E we have the real part of which is well out of the experimentally allowed range. These numbers suggest, that the simplest (WT) term of a chiral SU (3) basedKN interaction, when fitted to the "classical" data set, supports a Λ * pole at the position (5),while the same potential strongly overestimates the real part of ∆E. The latter observation was already noticed in earlier fits to experimental data using chiral based potentials. Acceptable reproduction of the experimental ∆E could be achieved only by adding next order terms to the lowest order WT one [9].
a) Fit to classical cross sections . It can be seen, that the about 25% worsening of χ 2 comes essentially from the discrete part, signifying, that the WT based potential "has difficulties" in reproducing ∆E. The real part of the level shift ∆E B is close to the experimental value, while its imaginary part is still somewhat outside of the experimentally allowed range. However, the price for bringing the real part close to the experiment is that the Λ * pole moved above theKN threshold: what is probably unacceptable from experimental point of view. This is confirmed by by our fits to the CLAS Λ(1405) data (Fig.2, part b)), which we shall discuss later.
In view of this inability of the simple WT-like potential to produce simultaneously reasonable values both for E(Λ * ) and ∆E, we tried to reach a compromise by adding to the experimental data an artificial E(Λ * c ) somewhere below theKN threshold, say and performed the fitting including both E(Λ * c ) and ∆E. This is our fit C, the results of which are shown in column C of tables I and II, and in part a) of Fig.3. The values of E(Λ * ) C and ∆E C are acceptable as well as the quality of reproduction of "classical" data.
We think, that for practical purposes, that is for calculation of few-body kaonic nuclear systems the choice C is the most acceptable.
In fit D the E(Λ * ) of (7) was combined with the classical data, while in fit E also the experimental ∆E was added. The results are displayed in columns D and E of tables I and II and in part a) of Figs. 4. and 5. In these fits the very small errors in (7) force the pole of the resulting potential to be close to the PDG value (7). The price is a drastic worsening of a) Fit to classical cross sections .   The most recent and accurate experimental information about the Λ(1405) resonance comes from the recent CLAS photoproduction experiment [10]. In our opinion, from the abundance of CLAS data the best candidate for tracing the Λ(1405) is the π 0 Σ 0 missing mass spectrum from the reaction In this case the neutral π and Σ are in pure I = 0 final state corresponding to Λ(1405), while the charged final states contain not easily removable I = 1 contributions, including p-waves from the Σ(1385).
For the reaction (8) Roca and Oset proposed [11] a two-step model corresponding to the diagram on Fig.6. The corresponding amplitude can be written (somewhat schematically) as: For fixed p γ and p K + momentum conservation reduces the above integral to a single one over the relative momentum of particles X, Y : where the non-relativistic propagator G 0 (p K + , P XY , k XY ; E) is Here we introduced the total and relative momenta P XY and k XY of particles X and Y , and P XY is fixed by momentum conservation P XY =p γ − p K + . The last factor in (10) is the half off-shell t-matrix element of ourKN potential.
These unsophisticated formulae were presented to be able to point out an important difference of the present calculation and the original method used in [11]. The basic assumption, which makes the model calculable, is that we can neglect the k XY dependence of the unknown amplitude p γ |Â|p K + , P XY , k XY and and treat it as a complex energy-and (X,Y)-dependent constant to be fitted to experimental data. Apart from this assumption in [11] the remaining integral of the propagator and the t-matrix element was evaluated using the so-called "on-shell factorization", which reduces the effect of propagation between the two steps of the process to a multiplication by an energy-dependent constant instead of integration. While accepting the basic assumption, in the present calculation the remaining integration was performed exactly. In [12] an attempt was made to propose a calculable model for the amplitude p γ |Â|p K + , P XY , k XY of Eq.(10), however reasonable fits to the CLAS data could be achieved only at the price of introducing a number of new adjustable parameters.
Thus finally our fitting procedure to the CLAS π 0 Σ 0 missing mass spectrum for a given γ energy E γ involves two complex constants corresponding to the two possible intermediate states (X, Y ), while for the t-matrix elements we used our previous fits A-E without further adjustment. The actual number of fitted parameters is three, since an overall phase factor of the amplitude cancels out from the cross section. In part b) of the Figs.
[1]- [5] we have shown the fits to the four lowest γ energy bins of the CLAS photoproduction data. The lowest γ-energies and the low-energy part of the missing mass spectrum were chosen due to the non-relativistic nature of our formulation.
Having in mind the approximate nature of the applied reaction model, the quality of the fits seem to be acceptable, confirming the final state interaction nature of the Λ(1405) peak.
The only exception is fit B, where the E(Λ * ) pole position moved above theKN threshold.
This case is unambiguously ruled out by the CLAS data. It is interesting to note, that fits D and E with PDG values for E(Λ * ), which yield poor results for the classical two-body data, still can be adjusted to reproduce the observed peaks. This fact can be misleading if a potential is constructed mainly to fit the observed Λ(1405) line shapes.

V. CONCLUSIONS
We have confronted a certain potential from class b) (see Introduction) with different pieces of experimental data. In spite of the special choice of the potential we think, that the physical conclusions can be valid for other potentials from the same class.
Comparison with classical data set yields a good agreement with minimal χ 2 . From the unfitted quantities E(Λ * ) shows a modest binding (≈ 10 M eV ) and a reasonable width (≈ 40 M eV ). On the other hand, the 1s level shift ∆E is far from its experimental value, its real part is strongly overestimated, while the imaginary part is just below the experimentally allowed region.
When the experimental ∆E is added to the classical data, the resulting potential produces a ∆E with a real part close to the experimental value, while the imaginary part is slightly above the allowed region. The price of bringing the real part to the right place is, however, that E(Λ * ) moved above theKN threshold, what contradicts to the Λ(1405) experiments.
This sharp contradiction between the E(Λ * ) position and experimental ∆E, probably characteristic for all class b) potentials, can be smoothed by adding to the fit a requirement of having an E(Λ * ) "somewhere below theKN threshold", say at E(Λ * c ) of (6). The potential from the resulting fit C can be considered as an acceptable compromise between the two extremes.
Finally, fits with E(Λ * ) pinned down close to the PDG value produce unacceptable results both for the classical data set and ∆E, while they are able to reproduce the peaks in the photoproduction experiment.