Internal structure of the resonant $\Lambda$(1405) state in chiral dynamics

The internal structure of the resonant $\Lambda \text{(1405)}$ state is investigated based on meson-baryon coupled-channels chiral dynamics by evaluating density distributions obtained from the form factors of the $\Lambda \text{(1405)}$ state. The form factors are defined as an extension of the ordinary stable particles and are directly evaluated from the current-coupled meson-baryon scattering amplitude, paying attention to the charge conservation of the probe interactions. For the resonant $\Lambda \text{(1405)}$ state we calculate the density distributions in two ways. One is on the pole position of the $\Lambda \text{(1405)}$ in the complex energy plane, which evaluates the resonant $\Lambda \text{(1405)}$ structure without contamination from nonresonant backgrounds, and another is on the real energy axis around the $\Lambda \text{(1405)}$ resonance energy, which may be achieved in experiments. Using several probe interactions and channel decomposition, we separate the various contributions to the internal structure of the $\Lambda \text{(1405)}$. As a result, we find that the resonant $\Lambda \text{(1405)}$ state is composed of widely spread $\mathrm{K̄}$ around $N$, which gives dominant component inside the $\Lambda \text{(1405)}$, with escaping $\pi \Sigma$ component. Furthermore, we consider $\mathrm{K̄}N$ bound state without decay channels, with which we can observe the internal structure of the bound state within real numbers. We also study the dependence of the form factors on the binding energy and meson mass. This verifies that the form factor defined through the current-coupled scattering amplitude serves as a natural generalization of the form factor for the resonance state. The relation between the interaction strength and the meson mass shows that the physical kaon mass appears to be within the suitable range to form a molecular bound state with the nucleon through the chiral SU(3) interaction.

Figure 1

(a) Feynman diagram for meson-baryon scattering amplitudes close to the pole of the excited baryon and (b) Feynamn diagram for meson-baryon scattering amplitudes with the probe current attached to the excited baryon. The double, dashed, solid, and wiggly lines correspond to the excited baryon, the ground-state meson, the ground-state baryon, and the probe current, respectively.

Figure 2

Examples of the Feynman diagrams for the scattering amplitudes $T_{\gamma }^{\mu }$ in which the probe current does not couple to the excited baryon. The double, dashed, solid, and wiggly lines correspond to the excited baryon, the ground-state meson, the ground-state baryon, and the probe current, respectively.

Figure 3

Diagrams for the $T_{\gamma }^{\mu }$ which contain the double-pole terms of the excited baryon [28, 48, 64]. The shaded ellipses represent the BS amplitude. The dashed, solid, and wiggly lines correspond to the ground-state meson, the ground-state baryon, and the probe current, respectively.

Figure 4

Supplemental diagrams for the charge conservation of the $T_{\gamma }^{\mu }$ in addition to the diagrams in Fig. 3 [64]. The shaded ellipses represent the BS amplitude. The dashed, solid, and wiggly lines correspond to the ground-state meson, the ground-state baryon, and the probe current, respectively.

Figure 5

Electromagnetic form factors of the $\Lambda \text{(1405)}$ state on the higher pole position $Z_2$, together with the empirical form factors of the neutron. Left (right) panel shows the electric (magnetic) form factor $F_E$ ($F_M$). The label “w/o CFF” represents the result without inclusion of the common form factor in Eq. (59). The parameter $c$ in the dipole form factor is chosen to be $c=\text{Re}[F_M(Q^2=0)]$, the real part of the magnetic moment of the $\Lambda \text{(1405)}$.

Figure 6

Baryonic ($F_B$) and strangeness ($F_S$) form factors of the $\Lambda \text{(1405)}$ state on the higher pole position $Z_2$. The strangeness form factor is presented with the opposite sign for comparison. The parameter $c$ in the dipole form factor is chosen to be $c=1$.

Figure 7

Normalized distributions $4\pi r^2\rho (r)$ of charge ($P_E$, left) and magnetic moment ($P_M$, right) densities of the $\Lambda \text{(1405)}$ state on the higher pole position $Z_2$. Empirical charge distribution in the neutron is evaluated by Eq. (63). Line denoted as “Dipole” in magnetic moment density is evaluated by Eq. (64) with $c=\text{Re}[F_M(Q^2=0)]$.

Figure 8

Baryonic ($P_B$) and strangeness ($P_S$) density distributions of the $\Lambda \text{(1405)}$ state on the higher pole position $Z_2$. The strangeness density distribution is presented with the opposite sign for comparison. Typical density of nucleon is evaluated by Eq. (64) with $c=1$.

Figure 9

Meson-baryon components of the electromagnetic form factors of the $\Lambda \text{(1405)}$ state on the higher pole position $Z_2$. Components of the electric (magnetic) form factor are shown in the upper panel (middle and lower panels).

Figure 10

Meson-baryon components of the distributions $4\pi r^2\rho (r)$ of charge ($P_E$, upper) and magnetic moment ($P_M$, middle and lower) densities of the $\Lambda \text{(1405)}$ state on the higher pole position $Z_2$.

Figure 11

Meson-baryon components of baryonic ($P_B$) and strangeness ($P_S$) density distributions of the $\Lambda \text{(1405)}$ state on the higher pole position $Z_2$. The strangeness density distribution is presented with opposite sign for comparison. Here $\mathrm{K̄}N$ represents the sum of the contributions from $K^{-}p$ and ${\mathrm{K̄}}^{0}n$, whereas $\pi \Sigma$ represents the sum of the contributions from $\pi {}^0\Sigma {}^0$, $\pi {}^{+}\Sigma {}^{-}$, and $\pi {}^{-}\Sigma {}^{+}$.

Figure 12

Effective baryon number of the resonance, $F_B^{\text{eff}}(Q^2=0)$, as a function of the meson-baryon center-of-mass energy $\sqrt{s}$. Calculations are performed in the $\mathrm{K̄}N(I=0)\gamma {}^{*}\to \mathrm{K̄}N(I=0)$ process on the real energy axis.

Figure 13

Real part of the effective electromagnetic form factors ($F_E^{\text{eff}}$ and $F_M^{\text{eff}}$) on the real energy axis, together with the empirical form factors of the neutron. Calculations are performed with the center-of-mass energy $\sqrt{s}=1410$, $1420$, and $1430\text{MeV}$. The parameter $c$ in the dipole form factor is chosen to be $c=\text{Re}[F_M^{\text{eff}}(Q^2=0;\sqrt{s}=1430\text{MeV})]$; the real part of the magnetic moment at $\sqrt{s}=1430\text{MeV}$.

Figure 14

Real part of the effective charge ($P_E^{\text{eff}}$, left) and magnetic moment ($P_M^{\text{eff}}$, right) density distributions on the real energy axis. Calculations are performed with the center-of-mass energy $\sqrt{s}=1410$, $1420$, and $1430\text{MeV}$. Empirical charge distribution in the neutron is evaluated by Eq. (63). Line denoted as “Dipole” in magnetic moment density is evaluated by Eq. (64) with $c=\text{Re}[F_M^{\text{eff}}(Q^2=0{\text{GeV}}^2;\sqrt{s}=1430\text{MeV})]$.

Figure 15

The $\mathrm{K̄}$, $N$ (left), and electric (right) form factors of the $\mathrm{K̄}N$ bound state with mass $1424\text{MeV}$, together with the empirical form factors of the neutron. The parameter $c$ in the dipole form factor is chosen to be $c=1$.

Figure 16

The $\mathrm{K̄}$, $N$ (left), and charge (right) density distributions of the $\mathrm{K̄}N$ bound state with mass $1424\text{MeV}$. Typical density of nucleon is evaluated by Eq. (64) with $c=1$. Empirical charge distribution in the neutron is evaluated by Eq. (63).

Figure 17

Interaction strength needed to generate a bound state with different binding energies, $C_B$. In the figure, $M_{\text{bound}}$ represents the mass of the bound state.

Figure 18

Mean-squared radii of $\mathrm{K̄}N$ bound state with different binding energies. In the figure, $M_{\text{bound}}$ represents the mass of the bound state.

Figure 19

Mean-squared distance between $\mathrm{K̄}$ and $N$ in the bound system in our approach (solid line), together with that obtained from the nonrelativistic wave function $\psi (x)$ (dashed-dotted line, denoted as “NR”). In the figure, $M_{\text{bound}}$ represents the mass of the bound state.

Figure 20

Interaction strength $C_B$ as a function of the meson mass $m$ to generate a bound state with the condition (75).

Figure 21

Charge density distribution of meson-nucleon bound state with different meson masses, $200$, $500$, $1000$, $1500$, and $2000\text{MeV}$.

Figure 22

Electric, baryonic, and strangeness mean-squared radii of meson-nucleon bound state with different meson masses. We also plot the mean-squared distance between meson and nucleon, $\langle x^2\rangle$.

Figure 23

The real part (left) and the imaginary (right) parts of the loop integrals $D_M^0+D_B^0$ of the $K^{-}p$ channel (upper) and the $\pi {}^{+}\Sigma {}^{-}$ channel (lower) as functions of $\text{Im}[z]$ and $Q^2$. The real part of the energy is chosen to be $\text{Re}[z]=1420\text{MeV}$.

Figure 24

The real part (left) and the imaginary (right) parts of the $I_{k=\pi \Sigma }^{\text{div}}$ (B7) as functions of $\text{Im}[z]$ and $Q^2$. The real part of the energy and ${\mathrm{m̄}}_k$ are chosen to be $\text{Re}[z]=1420\text{MeV}$ and ${\mathrm{m̄}}_k=M_{\Sigma }$, respectively.

Figure 25

Meson-baryon components of the effective electric form factors ($F_E^{\text{eff}}$, left) and effective electric density distribution (${\text{P}}_E^{\text{eff}}$, right) on the real energy axis. Calculations are performed with the center-of-mass energy $\sqrt{s}=1420\text{MeV}$.