From pole parameters to line shapes and branching ratios

Abstract

Resonances are uniquely characterized by their complex pole locations and the corresponding residues. In practice, however, resonances are typically identified experimentally as structures in invariant mass distributions, with branching fractions of resonances determined as ratios of count rates. To make contact between these quantities it is necessary to connect line shapes and resonance parameters. In this work we propose such a connection and illustrate the formalism with detailed studies of the $\rho \left(770\right)$ and $f_0\left(500\right)$ resonances. Based on the line shapes inferred from the resonance parameters along these lines, expressions for partial widths and branching ratios are derived and compared to other approaches in the literature.

Fig. 1

The left (right) figure shows the absolute value (phase) of the $\pi \pi$ scattering amplitude, in both cases for the various analyses presented here: (i), (ii), (iii), and (iv) are shown as the blue, orange, green, and red line or band, respectively. In both figures we also show for comparison the results from Ref. [6]

Fig. 2

The pion vector form factor compared to data derived from ${\tau }^{-}\to {\pi }^{-}{\pi }^0{\overline{\nu }}_{\tau }$ [53] on a linear scale (left) and on a logarithmic scale (right). Legend as in Fig. 1

Fig. 3

Comparison of the phase shifts (left) and the absolute value of the scattering amplitude (right) that result for the scalar–isoscalar $\pi \pi$ channel, once the pole parameters are fixed via the different variants of the model: (i), (ii), (iii), and (iv) are shown as the blue, orange, green, and red line or band, respectively. The black dashed line shows the phase shift and the related absolute value of the scattering amplitude (between $\pi \pi$ and $\overline{K}K$ threshold) from Ref. [4] for comparison. The dots show the phase shifts extracted from $K_{e4}$ decays [59]. The first and second perpendicular lines show the locations of the Adler zero and the $\pi \pi$ threshold, respectively

Fig. 4

Two-pion production amplitude that results for the different fits. The color code agrees with that of Fig. 3, only that now the dashed line is the non-strange scalar pion form factor of Ref. [21], derived from the phase shifts of Ref. [4]