Application of the uniformized Mittag-Leffler expansion to $\mathrm{\Lambda }\left(1405\right)$

We study the pole properties of $\mathrm{\Lambda }\left(1405\right)$ 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 parametrization which expresses the observable to be single valued, and fitting it to experimental data of the invariant-mass distribution of ${\pi }^{+}{\mathrm{\Sigma }}^{-}, {\pi }^{-}{\mathrm{\Sigma }}^{+}, {\pi }^0{\mathrm{\Sigma }}^0$ final states in the reaction $\gamma p\to K^{+}\pi \mathrm{\Sigma }$ and the elastic and inelastic cross section $K^{-}p\to K^{-}p, {\overline{K}}^{0}n, {\pi }^{+}{\mathrm{\Sigma }}^{-}, {\pi }^{-}{\mathrm{\Sigma }}^{+}$. 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 $\mathrm{\Lambda }\left(1405\right)$. The broad peak structure between the $\pi \mathrm{\Sigma }$ and $\overline{K}N$ thresholds regarded to be $\mathrm{\Lambda }\left(1405\right)$ 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 $\mathrm{\Lambda }\left(1405\right)$. We conclude that the uniformized Mittag-Leffler expansion turns out to be a very powerful and simple method to obtain resonance energy, width, and residues from the near-threshold spectrum.

Figure 1

(a) Schematic figure of the spectral representation, and the uniformized Mittag-Leffler expansion, of the $\mathcal{S}$ matrix in the case of a double-channel two-body system. ${\epsilon }_1, {\epsilon }_2$ are the threshold energies of the two channels. The continuum contribution along the unitary cuts in panel (a), can be decomposed into resonant contributions from poles in the unphysical domain (blue) in panel (b).

Figure 2

A schematic diagram of the process ‘$\gamma p\to K^{+}{Y}^{*}\to K^{+}\pi \mathrm{\Sigma }$’ measured in the CLAS experiment [33].

Figure 3

(a) Analytic structure of the double-channel $\mathcal{S}$ matrix in the $\sqrt{s}$ plane, and (b) the uniformized $z$ plane. In the $\sqrt{s}$ plane, the unitary cuts run along the real axis from ${\epsilon }_1$ to $\infty$ (blue) and from ${\epsilon }_2$ to $\infty$ (green), and the four Riemann sheets of the $\sqrt{s}$ plane correspond to each region in $z$ labeled as $(\pm \pm )$. The red line shows the physical region accessible by experiment.

Figure 4

Results for the invariant-mass distributions of ${\pi }^{+}{\mathrm{\Sigma }}^{-}$ in the reaction $\gamma p\to K^{+}\pi \mathrm{\Sigma }$ in nine bins of center-of-mass energy $W$. The (blue) dots with bars are the experimental data. The (orange) bands represent the $2\sigma$-confidence interval of the fit by the uniformized Mittag-Leffler expansion with $m=3$. The (green) dashed, (red) dot-dashed, and (purple) dotted lines represent the contributions from individual resonant-pole pairs of $1+1^{*}, 2+2^{*}$, and $3+3^{*}$, respectively.

Figure 5

Results for the invariant-mass distributions of ${\pi }^{-}{\mathrm{\Sigma }}^{+}$. Details are the same as in Fig. 4.

Figure 6

Results for the invariant-mass distributions of ${\pi }^0{\mathrm{\Sigma }}^0$. Details are the same as in Fig. 4. In the ${\pi }^0{\mathrm{\Sigma }}^0$ channel the $2\sigma$-confidence interval is wider than that in the ${\pi }^{+}{\mathrm{\Sigma }}^{-}$ and ${\pi }^{-}{\mathrm{\Sigma }}^{+}$ channels due to large experimental errors.

Figure 7

Results for the cross sections $K^{-}p\to K^{-}p, {\overline{K}}^{0}n, {\pi }^{+}{\mathrm{\Sigma }}^{-}, {\pi }^{-}{\mathrm{\Sigma }}^{+}$. Details are the same as in Fig. 4.

Figure 8

Results for the pole positions on the $z$ plane by the uniformized Mittag-Leffler expansion with $m=3$. Let us denote $z_{n^{*}}^{\left(m\right)}=-z_n^{\left(m\right)*}$. Two dotted lines around the poles represent the 70% and 95% confidence intervals of the position of the poles, respectively, from inside to outside. The (red) thick line represents the physical region accessible in the experiment and the labels $\pi \mathrm{\Sigma }$ and $\overline{K}N$ represent the corresponding thresholds.

Figure 9

Results for the pole positions on the $\sqrt{s}$ plane. Details are the same as in Fig. 8.

Figure 10

Results for the invariant-mass distribution of ${\pi }^{+}{\mathrm{\Sigma }}^{-}$ in the reaction $\gamma p\to K^{+}\pi \mathrm{\Sigma }$ by the uniformized Mittag-Leffler expansion with $m=1$, 2, and 3 from left to right.

Figure 11

Results for the pole positions on the $z$ plane by the uniformized Mittag-Leffler expansion with $m=1$, 2, and 3.