Four-boson system with short-range interactions

We consider the nonrelativistic four-boson system with short-range forces and large scattering length in an effective quantum mechanics approach. We construct the effective interaction potential at leading order in the large scattering length and compute the four-body binding energies using the Yakubovsky equations. Cutoff independence of the four-body binding energies does not require the introduction of a four-body force. This suggests that two- and three-body interactions are sufficient to renormalize the four-body system. We apply the equations to ${}^4\mathrm{H}\mathrm{e}$ atoms and calculate the binding energy of the ${}^4\mathrm{H}\mathrm{e}$ tetramer. We observe a correlation between the trimer and tetramer binding energies similar to the Tjon line in nuclear physics. Over the range of binding energies relevant to ${}^4\mathrm{H}\mathrm{e}$ atoms, the correlation is approximately linear.

Figure 1

The shallowest three-body binding energies indicated by the solid, dashed, and dash-dotted lines as a function of the momentum cutoff $\Lambda$. The vertical dashed line indicates the cutoff range in which the three-body system has exactly two bound states. The horizontal solid line shows the energy at which the shallowest three-body state is fixed.

Figure 2

The three-body coupling constant ${\lambda }_3$ as a function of the cutoff parameter $\Lambda$. The solid blue line shows a fit of Eq. (26) with $c=0.076$ and $L_3=23.3\sqrt{B_2}$ to the points for $\Lambda ⩾245\sqrt{B_2}$.

Figure 3

Binding energies of the three- and four-body system as a function of the cutoff $\Lambda$. $B_n^{\left(0\right)}$ and $B_n^{\left(1\right)}$ denote the ground and first excited state of the $n$-body system.

Figure 4

The correlations between the ground- and excited-state energies of the ${}^4\mathrm{He}$ trimer and tetramer. First and third figure from the top: the four-body excited-state energy $B_4^{\left(1\right)}$ and ground-state energy $B_4^{\left(0\right)}$ as a function of the three-body excited-state energy $B_3^{\left(1\right)}$. The second and fourth figure from the top: the same quantities as a function of three-body ground-state energy $B_3^{\left(0\right)}$. The solid line shows the leading-order effective theory result and the cross denotes the calculation for the LM2M2 potential by Blume and Greene [21]. The triangles show the results for the TTY, HFD-B, and HFDHE2 potentials [17, 20].