Dispersive determination of resonances from $ππ$ scattering data

Abstract

We provide a precise, model- and parametrization-independent dispersive determination of the $f_0(500)$, $ρ(770)$, $f_0(980)$, $f_2(1270)$, $f_0(1370)$, $ρ(1450)$, $f_0(1500)$, and $ρ_3(1690)$ resonance pole parameters. They are obtained from the analytic continuation, by means of continued fractions, of forward dispersion relations, whose input is a recent global dispersive analysis of $ππ$ scattering data. From this dispersive study, we find no indications of other resonant poles below 1.7 GeV. Beyond this energy, we also provide resonance pole parameters from the direct analytic continuation of Global Fits to the three existing incompatible data sets. Depending on the data set we find poles for the $ρ(1700)$, $f_0(1710)$, $ρ(1900)$, $f_2(1950)$, and $f_0(2020)$ resonances. We also present the Argand diagrams of these Global Fits and illustrate that each resonance does not necessarily have to trace a full circle in the diagram.

Figures (16)

• Figure 1: Analytic continuation of the $F^{0+}$ FDR output, for the central values of the Global Fit I input, and a representative $N$. The amplitude is continued from the intervals [0.65, 0.85] GeV and [1.35, 1.55] GeV in the elastic and inelastic regions, respectively. Each interval gives access to different contiguous sheets, which is illustrated with a discontinuity attached to the $\pi\omega$ threshold, $\sim$0.922 GeV. Note the presence of the $\rho(770)$, $\rho(1450)$, and $\rho_3(1690)$ resonance poles.
• Figure 2: Pole masses (top) and half-widths (bottom) for the $\rho(770)$ and $\rho_3(1690)$ resonances, obtained from the analytic continuations of the $F^{0+}$ FDR output using continued fractions $C_N$. Results are remarkably stable against variations of $N$.
• Figure 3: Pole masses (top) and half-widths (bottom) for the $\rho(1450)$ resonance, obtained from the analytic continuations of the $F^{0+}$ FDR output using continued fractions $C_N$. Results are remarkably stable against variations of $N$.
• Figure 4: Reliability of the continued fraction analytic continuation method in the upper half $\sqrt s$-plane near the $\rho(770)$ region. The first Riemann sheet value of the $F^{0+}$ FDR output, $F^{0+}_{FDR}$, lies well within the estimated uncertainty of the continuation made with continued fractions $F^{0+}_{C_N}$. We show the absolute value of the difference for the real parts (left) and for the imaginary parts (right), divided by the uncertainty of the $C_N$ calculation. For reference, we show the conjugate position of the $\rho(770)$ pole (red cross) and the continued segment (red line).
• Figure 5: The $\rho(1450)$ poles obtained from the analytic continuation of the $F^{0+}$ FDR for the three Global Fits. We have provided a single range covering the results from Global Fits II and III, as they are compatible. For comparison, we also show the RPP educated guess (green area) and $\pi\pi$ mode masses and widths collected there ParticleDataGroup:2022pth.