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Register a new userFour-Body Faddeev-Type Calculation of the $bar{K}NNN$ System: Preliminary Results
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Authors
N. V. Shevchenko
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The paper is devoted to the $bar{K}NNN$ system, consisting of an antikaon and three nucleons. Four-body Faddeev-type AGS equations are being solved in order to find possible quasi-bound state in the system.
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In this paper, we study the relativistic effects in a three-body bound state. For this purpose, the relativistic form of the Faddeev equations is solved in momentum space as a function of the Jacobi momentum vectors without using a partial wave decomposition. The inputs for the three-dimensional Faddeev integral equation are the off-shell boost two-body $t-$matrices, which are calculated directly from the boost two-body interactions by solving the Lippmann-Schwinger equation. The matrix elements of the boost interactions are obtained from the nonrelativistic interactions by solving a nonlinear integral equation using an iterative scheme. The relativistic effects on three-body binding energy are calculated for the Malfliet-Tjon potential. Our calculations show that the relativistic effects lead to a roughly 2% reduction in the three-body binding energy. The contribution of different Faddeev components in the normalization of the relativistic three-body wave function is studied in detail. The accuracy of our numerical solutions is tested by calculation of the expectation value of the three-body mass operator, which shows an excellent agreement with the relativistic energy eigenvalue.
We perform a Faddeev calculation for the three mesons system, $phi K bar{K}$, taking the interaction between two pseudoscalar mesons and between a vector and a pseudoscalar meson from the chiral unitary approach. We obtain a neat resonance peak around a total mass of 2150 MeV and an invariant mass for the $K bar{K}$ system around 970 MeV, very close to the $f_0(980)$ mass. The state appears in I=0 and qualifies as a $phi f_0(980)$ resonance. We enlarge the space of states including $phi pi pi$, since $pi pi$ and $K bar{K}$ build up the $f_0$ (980), and find moderate changes that serve to quantify theoretical uncertainties. No state is seen in I=1. This finding provides a natural explanation for the recent state found at BABAR and BES, the X(2175), which decays into $phi f_0(980)$.
$Kbar N$ interactions are investigated {it via} an effective non-linear chiral meson-baryon Lagrangian. The adjustable parameters are determined by a fitting procedure on the $K^-p$ threshold branching ratios and total cross-section data for $p^{lab}_Kle$ 250 MeV/c. We produce predictions for the $Sigma pi$ mass spectrum, and scattering lenghts $a_{K^-p}$, $a_n(K^-n to K^-n)$, $a_n0(Kbar0 n to Kbar0 n)$, and $a_{ex}(K^-p to Kbar0 n)$. The $Kbar N$ amplitudes thus obtained, as well as those for other two-body channels ($pi N$, $NN$, and $YN$) are used as input to predict the scattering length $A_{K^-d}$, for which we have devised a relativistic version of the three-body Faddeev equations. Results for all two- and three-body coupled channels are reported both in isospin and particle bases. All available $Kbar N$ data are well reproduced and our best results for the $K^-p$ and $K^-d$ scattering lenghts are $a_{K^-p} = (-0.90 + i 0.87) fm$ and $A_{K^-d} = (-1.80 + i 1.55) fm$.
A Bethe-Salpeter-Faddeev (BSF) calculation is performed for the pentaquark $Theta^+$ in the diquark picture of Jaffe and Wilczek in which $Theta^+$ is a diquark-diquark-${bar s}$ three-body system. Nambu-Jona-Lasinio (NJL) model is used to calculate the lowest order diagrams in the two-body scatterings of ${bar s}D$ and $D D$. With the use of coupling constants determined from the meson sector, we find that ${bar s}D$ interaction is attractive while $DD$ interaction is repulsive, and there is no bound $frac 12^+$ pentaquark state. A bound pentaquark $Theta^+$ can only be obtained with unphysically strong vector mesonic coupling constants.
We report on self-consistent calculations of single-K^- nuclear states and multi-Kbar nuclear states in 12C, 16O, 40Ca and 208Pb within the relativistic mean-field (RMF) approach. Gradient terms motivated by the p-wave resonance Sigma(1385) are found to play a secondary role for single-K^- nuclear systems where the mean-field concept is acceptable. Significant contributions from the Kbar N -> pi Lambda conversion mode, and from the nonmesonic Kbar NN -> YN conversion modes which are assumed to follow a rho^2 density dependence, are evaluated for the deep binding-energy range of over 100 MeV where the decay channel Kbar N -> pi Sigma is closed. Altogether we obtain K^- total decay widths of 50-100 MeV for binding energies exceeding 100 MeV in single-K^- nuclei. Multi-Kbar nuclear calculations indicate that the binding energy per Kbar meson saturates upon increasing the number of Kbar mesons embedded in the nuclear medium. The nuclear and Kbar densities increase only moderately and are close to saturation, with no indication of any kaon-condensation precursor.
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