Universal physics of the few-body system of two neutrons and one flavored meson

We investigate the $s$-wave three-body system of two neutrons and one flavored meson with total spin-isospin $J=0,I=3/2$. The meson-neutron scattering length can become infinitely large when extrapolated to an unphysical region of the quark mass between strangeness and charm in the so-called zero coupling limit. Using a low-energy cluster effective field theory, we demonstrate that the Efimov effect is formally manifest in the three-body system when the meson-neutron scattering length approaches the unitary limit of the two-body interaction. We thereby discuss the consequence of remnant universal physics in the physical $K^{-}nn$ and $D^{0}nn$ systems.

Figure 1

Complex meson-neutron scattering length in the coupled-channel contact interaction model as a function of the extrapolation parameter $x$. Solid (dashed) line represents the real (imaginary) part.

Figure 2

Trajectory of the pole of the scattering amplitude in the coupled-channel contact interaction model. The energy is measured with respect to the threshold of channel 1, i.e., $E_h=W-M_1\left(x\right)-m_1\left(x\right)$.

Figure 3

Meson-neutron inverse scattering length in the absence of decay channels in the zero coupling limit as a function of the flavor extrapolation parameter $x$.

Figure 4

Trajectory of the pole of the scattering amplitude in the zero coupling limit. The energy is measured with respect to the threshold of channel 1 $[E_h=W-M_1\left(x\right)-m_1\left(x\right)]$. Dashed (solid) line represents the pole on the physical (unphysical) Riemann sheet.

Figure 5

Deviation of the eigenmomentum from the universal prediction $|k_h-k_a|/m_{\pi }$ as a function of the flavor extrapolation parameter $x$ in the coupled-channel contact interaction model.

Figure 6

Deviation of the eigenmomentum from the universal prediction $|k_h-k_a|/m_{\pi }$ in the zero coupling limit as a function of the flavor extrapolation parameter $x$.

Figure 7

Feynman diagrams for the three-body coupled integral equations where the three-body contact terms are omitted for brevity. The solid and dashed lines represent the bare propagators of $n$ and $K\left(x\right)$, respectively, and the double and zigzag lines stand for the dressed dihadron propagators of $s_{\left(nn\right)}$ and $d_{\left(nK\right)}$, respectively.

Figure 8

Flavored meson mass dependence on the asymptotic parameter $s_0^{\infty }$ for $m_{K^{-}}<m<m_{D^0}$.

Figure 9

RG limit cycle for the three-body coupling $g_s$ as a function of the sharp momentum cutoff $\mathrm{\Lambda }$. The pertinent integral equations are solved with the input two-body scattering lengths from Eq. (36) in the zero coupling limit for fixed trimer eigenenergies, namely, $B_d=0.1$ MeV for $K^{-}nn$, measured with respect to the three-particle breakup threshold, and $B_T=B_d-E_D=0.1$ MeV for $D^{0}nn$, measured with respect to the particle-dimer breakup threshold energy of $E_D=1.82$ MeV.

Figure 10

Binding energies for the ground ($n=0$) and first ($n=1$) excited level states as a function of the sharp momentum cutoff $\mathrm{\Lambda }$, excluding the three-body force. The pertinent integral equations are solved with the input two-body scattering lengths from Eq. (36) in the zero coupling limit. For the $K^{-}nn$ system, the trimer binding energy $B_d$, is measured with respect to the three-particle breakup threshold, while for the $D^{0}nn$ system the binding energy, $B_T=B_d-E_D$, is measured with respect to the particle-dimer breakup threshold energy, $E_D=1.82$ MeV.

Figure 11

Extrapolation of the critical cutoff ${\mathrm{\Lambda }}_c^{\left(0\right)}$, associated with the ground state ($n=0$) for fixed binding energies ($B_d$ or $B_T$) of the meson-neutron-neutron bound state, measured along the three-particle or particle-dimer breakup threshold. The double dashed vertical line at $x=0.615$ denotes the unitary limit. The plot corresponds to the zero coupling limit with input scattering lengths, as given in Eq. (36). The three-body force is excluded in these results.

Figure 12

$\mathrm{Det}[X-I]$ vs energy, $-E=(E_{D^{0}nn}-E_D)$, plotted for the $D^{0}nn$ system for different sharp momentum cutoffs $\mathrm{\Lambda }$. $E_D=1.82$ MeV is the threshold energy for breakup into a $D^{0}n$ dimer and a neutron, obtained using the $D^{0}nn$ scattering length in Eq. (36), as the input. The nodes (zeros) correspond to eigenenergies of the trimer level states. Here solutions up to the first three bound states are displayed.

Figure 13

Convergence of the three-body parameter $s_0$ for a fixed eigenenergy, $B_d=B_T=0.1$ MeV, for the $K^{-}nn$ and $D^{0}nn$ trimers. The points corresponding to equal values of the two-body (inverse) scattering lengths (in ${\mathrm{fm}}^{-1}$) are connected by straight lines in order to guide the eyes. The long-dashed (red) lines that correspond to true two-body unitarity shows the quickest asymptotic convergence. The other lines correspond to varying the meson-neutron scattering lengths with the neutron-neutron scattering length fixed at the physical value, $a_{s\left(nn\right)}=-18.63$ fm. To achieve reasonable convergence up to the $n=7\text{th}$ excited state, the number of quadrature points used is $N=300$.