No hidden physics in resonance pole residue phase

Abstract

In hadron resonant scattering, there are four fundamental resonant parameters: real and imaginary part of the pole position, and the magnitude and the phase of the residue. Out of the four, the last one is the least understood. The search for the residue phase's physical meaning has focused on model-independent phases of the majority of the lowest-mass resonances. Here, we apply a simple mathematical identity to the amplitude in the complex plane to reveal the exact reason for the noticed regularity and show that there is no room for hidden physical variables in the residue phase.

Figures (4)

• Figure 1: Complex phase of the KH80 $\pi N$ scattering amplitude $T$$(I=3/2,J^\pi=3/2^+)$ as a function of complex energy $E$ near the pole of $\Delta(1232)$. The pole is shown as the white circle, the threshold energy as the black circle, and the Breit-Wigner mass, which is here defined as the zero of the real part of the amplitude on the real axis, as the black and white circles. The true residue phase $\theta$ is determined numerically, while phases $\alpha$ and $\beta$ are calculated using Eqs. (\ref{['Eq:alpha']}) and (\ref{['Eq:beta']}). The solid white line shows where $\mathrm{Re} \,T =0$, and the dashed one where $\mathrm{Im} \,T =0$
• Figure 2: The $\pi N$ elastic complex phase of the KH80 scattering amplitude $T$ ($I = 1/2$, $J^\pi=1/2^-$) as a function of complex energy $E$. Black circles are zeros, and white and black is the zero of the real part. We see strongly overlapping $N(1535)$ and $N(1650)$ resonances (white circles). As a result, $\mathrm{Re}\,T=0$ line (solid white line) does not cross the real axis, and without it, Eq. (\ref{['Eq:theta']}) cannot be used for $N(1535)$. A multi-resonant model is needed here
• Figure 3: Complex phase of $T$ near resonances of the type Ib. Zero (black disk) is to the left, zero of the real part (black and white disk) is to the right, and the pole (white disk) is in between, similarly to the type Ia. N(1440) is almost type Ia, but it has a zero on the real axis, rather far from the elastic threshold. The other resonances are fully detached from the real axis, similar to type II. The two "zeros" are not too far from the real axis, and closer to the pole than in the case of Ia, resulting in $\alpha'$ and $\beta'$ being somewhat larger than Ia's $\alpha$ and $\beta$, which leads to $\theta$ typically around $-100^\circ\pm20^\circ$.
• Figure 4: Complex phase of $T$ amplitude near the type II resonances. If the pole (white disk) is to the left of the zero (black disk), we must add or subtract $180^\circ$ according to Eq. (\ref{['Eq:thetaprimeprime']}). N(1895) shows typical type Ib behavior, but it is not the first resonance; it is the third. Two other resonances show arrangements of the three characteristic points rarely seen in Ib type, and impossible for Ia.