Universal physics of three bosons with isospin

We show that there exist two types of universal phenomena for three-boson systems with isospin degrees of freedom. In the isospin symmetric limit, there is only one universal three-boson bound state with the total isospin one, whose binding energy is proportional to that of the two-boson bound state. With large isospin symmetry breaking, the standard Efimov states of three identical bosons appear at low energies. Both phenomena can be realized by three pions with the pion mass appropriately tuned in lattice QCD simulations, or by spin-one bosons in cold atom experiments. Implication to the in-medium softening of multi-pion states is also discussed.

Figure 1

Energies of the bound and virtual states in the isospin symmetric limit with $\overline{E}=\mathrm{sgn}\left(E\right){\left|mE\right|}^4$ and $\overline{a}=\mathrm{sgn}\left(a\right){\left|a\right|}^4$ in arbitrary units. The thick (thin) solid line represents the energy of the three-pion (two-pion) bound state, and the thick (thin) dashed line the energy of the three-pion (two-pion) virtual state.

Figure 2

Energies of the bound states of the three ${\pi }_0$ system with the large isospin breaking. The solid lines represent the energies of the bound states, and the dotted (dashed) lines represent the real parts (imaginary parts with the opposite sign) of the resonance poles. Both the axes are normalized to be dimensionless by the Efimov parameter ${\kappa }_{*}$ as $E^{\prime }=\mathrm{sgn}\left(E\right){|mE/{\kappa }_{*}^2|}^{1/4}$ and $a^{\prime }=\mathrm{sgn}\left(a\right){\left|a{\kappa }_{*}\right|}^{1/4}$. The inset shows the trajectories of the pole positions in the complex energy plane for a period corresponding to $-0.663293\ge 1/\left(a{\kappa }_{*}\right)\ge -15.0530$. The solid (dashed) line stands for the resonance (anti-resonance) state. As the inverse scattering length is decreased, the pole moves to the direction indicated by the arrows.

Figure 3

Binding energies from the interpolation model with the mass ratio $m_{\pm }/m_0=1.03403$. The filled circle at origin expresses the accumulation of the Efimov bound states. The axes are given by the dimensionless quantities $\underline{E}=\mathrm{sgn}\left(E\right){|E/\Delta |}^{1/2}$ and $1/\underline{a}=\mathrm{sgn}\left(a\right){|1/a\sqrt{m_0\Delta }|}^{1/2}$.