Four-body systems are studied using an effective-field theory with two- and three-body contact interactions. A method to systematically address deep trimers (three-body bound states that are more tightly bound than four-body bound or resonant states) in four-body calculations is developed using a diagrammatic approach. Previous calculations were limited by the existence of deep trimers, which this work overcomes. For cold ${}^4\mathrm{He}$ atoms, binding energies of 526.1(5) and 128.517(1) mK are obtained at leading order for the tetramer ground and excited states, respectively, where errors come from the truncation of three-body partial waves. Tetramer binding energies and decay widths are also computed approaching the unitary limit. In the unitary limit, there are two tetramers associated with each trimer of binding energy $B_3^{\left(0\right)}$. The binding energy and decay width for the associated tetramer ground state are $E_4^{\left(0\right)}=4.60\left(1\right)B_3^{\left(0\right)}$ and ${\mathrm{\Gamma }}_4^{\left(0\right)}/2=0.0160\left(1\right)B_3^{\left(0\right)}$, respectively, and for the associated tetramer excited state, $E_4^{\left(1\right)}=1.0022\left(3\right)B_3^{\left(0\right)}$ and ${\mathrm{\Gamma }}_4^{\left(1\right)}/2=2.57\left(2\right)\times {10}^{-4}{B}_3^{\left(0\right)}$, respectively. This calculation is a gateway to higher-order and/or more-body calculations in nuclear and atomic systems.

• Received 8 June 2023
• Accepted 8 January 2024

DOI:https://doi.org/10.1103/PhysRevC.109.024002