Chiral Breit-Wigner

Chiral symmetry and unitarization are combined into generalized Breit-Wigner expressions describing scalar resonances, which contain free parameters and allow flexible descriptions of masses, widths and pole positions. This theoretical tool is especially designed to be used in analyses of low-energy data.Comment: Talk given at the XI International Conference on Hadron Spectroscopy, Rio de Janeiro, Brazil, August 200

Chiral symmetry: An analytic SU(3)SU(3) unitary matrix

The $SU(2)$ unitary matrix $U$ employed in hadronic low-energy processes has both exponential and analytic representations, related by $U = \exp\left[ i \mathbf{\tau} \cdot \hat{\mathbf{\pi}} \theta\,\right] = \cos\theta I + i \mathbf{\tau} \cdot \hat{\mathbf{\pi}} \sin\theta$. One extends this result to the $SU(3)$ unitary matrix by deriving an analytic expression which, for Gell-Mann matrices $\mathbf{\lambda}$, reads $U= \exp\left[ i \mathbf{v} \cdot \mathbf{\lambda} \right] = \left[ \left( F + \tfrac{2}{3} G \right) I + \left( H \hat{\mathbf{v}} + \tfrac{1}{\sqrt{3}} G \hat{\mathbf{b}} \right) \cdot \mathbf{\lambda} \, \right] + i \left[ \left( Y + \tfrac{2}{3} Z \right) I + \left( X \hat{\mathbf{v}} + \tfrac{1}{\sqrt{3}} Z \hat{\mathbf{b}} \right) \cdot \mathbf{\lambda} \right]$, with $v_i=[\,v_1, \cdots v_8\,]$, $b_i = d_{ijk} \, v_j \, v_k$, and factors $F, \cdots Z$ written in terms of elementary functions depending on $v=|\mathbf{v}|$ and $\eta = 2\, d_{ijk} \, \hat{v}_i \, \hat{v}_j \, \hat{v}_k /3$. This result does not depend on the particular meaning attached to the variable $\mathbf{v}$ and the analytic expression is used to calculate explicitly the associated left and right forms. When $\mathbf{v}$ represents pseudoscalar meson fields, the classical limit corresponds to $\langle 0|\eta|0\rangle \rightarrow \eta \rightarrow 0$ and yields the cyclic structure $U = \left\{ \left[ \tfrac{1}{3} \left( 1 + 2 \cos v \right) I + \tfrac{1}{\sqrt{3}} \left( -1 + \cos v \right) \hat{\mathbf{b}}\cdot \mathbf{\lambda} \right] + i \left( \sin v \right) \hat{\mathbf{v}}\cdot \mathbf{\lambda} \right\}$, which gives rise to a tilted circumference with radius $\sqrt{2/3}$ in the space defined by $I$, $\hat{\mathbf{b}}\cdot \mathbf{\lambda}$, and $\hat{\mathbf{v}}\cdot \mathbf{\lambda}$. The axial transformations of the analytic matrix are also evaluated explicitly

Scalar resonances: scattering and production amplitudes

Scattering and production amplitudes involving scalar resonances are known, according to Watson's theorem, to share the same phase $\delta(s)$. We show that, at low energies, the production amplitude is fully determined by the combination of $\delta(s)$ with another phase $\omega(s)$, which describes intermediate two-meson propagation and is theoretically unambiguous. Our main result is a simple and almost model independent expression, which generalizes the usual $K$-matrix unitarization procedure and is suited to be used in analyses of production data involving scalar resonances.Comment: 10 pages, 4 figures. Minor changes, references added, version to appear in Phys. Rev.