Application of the Uniformized Mittag-Leffler Expansion to $\Lambda(1405)$

TL;DR

This paper introduces a model-independent method using the Uniformized Mittag-Leffler expansion to analyze the pole structure of the $ ext{Lambda}(1405)$ resonance, providing precise resonance parameters from experimental data.

Contribution

The study applies the Uniformized Mittag-Leffler expansion to extract resonance properties of $ ext{Lambda}(1405)$ in a model-independent way, demonstrating its effectiveness near thresholds.

Findings

The expansion converges with up to three pole pairs near $ ext{Lambda}(1405)$.

The resonance is characterized by a pole at 1420 $ ext{MeV}$ with a width of 48 $ ext{MeV}$.

The method accurately reproduces the resonance parameters consistent with the single-pole picture.

Abstract

We study the pole properties of $\Lambda(1405)$ in a model-independent manner by applying the Uniformized Mittag-Leffler expansion proposed in our previous paper. The resonant energy, width and residues are determined by expanding the observable as a sum of resonant-pole pairs under an appropriate parameterization which expresses the observable to be single-valued, and fitting it to experimental data of the invariant-mass distribution of $\pi^+\Sigma^-$, $\pi^-\Sigma^+$, $\pi^0\Sigma^0$ final states in the reaction, $\gamma p \rightarrow K^+ \pi \Sigma$, and the elastic and inelastic cross section, $K^-p\to K^-p$, $\bar{K}^0n$, $\pi^+\Sigma^-$, $\pi^-\Sigma^+$. As we gradually increase the number of pairs from one to three, the first pair converges while the second and third pairs emerge further and further away from the first pair, implying that the Uniformized Mittag-Leffler expansion with three pairs is almost convergent in the vicinity of the $\Lambda(1405)$. The broad peak structure between the $\pi\Sigma$ and $\bar{K} N$ thresholds regarded to be $\Lambda(1405)$ is explained by a single pair with a resonant energy of 1420 $\pm$ 1 MeV, and a half width of 48 $\pm$ 2 MeV, which is consistent with the single-pole picture of $\Lambda(1405)$. We conclude that the Uniformized Mittag-Leffler expansion turns out to be a very powerful method to obtain resonance energy, width and residues from the near-threshold spectrum.