The formulation of the Minkowskian and Euclidean homogeneous Faddeev-Bethe-Salpeter (FBS) equations for a three-boson system with a contact interaction in an infinite four-dimensional (4D) volume are reviewed, and the Euclidean one in a finite 4D volume is derived and discussed. In an Euclidean 4D box the periodic boundary conditions leading to quantized momentum turns the continuum homogeneous FBS equation into a discretized form, with solutions determining the spectrum running above the scattering threshold. A truncated set of the discretized FBS equation is solved for three-pions in the maximum isospin state, with scattering length, pion mass and 4D volume from recent lattice QCD calculations, and the results of the approximate FBS equation for the ground state mass are in qualitative agreement with the lattice result. The approximate dependence of the ground state mass on the pion-pion scattering length, pion mass and 4D volume is also suggested.

回顾了在无限四维（4D）体积中具有接触相互作用的三玻色子系统的Minkowskian和Euclidean均匀Faddeev-Bethe-Salpeter（FBS）方程的公式，而在有限4D体积中的Euclidean方程为派生和讨论。在欧几里得4D框中，导致量化动量的周期性边界条件使连续统一FBS方程变为离散形式，其解决方案确定了在散射阈值以上运行的光谱。对于最近最大晶格QCD计算中具有散射长度，pion质量和4D体积的最大同位旋态的三离子，求解了一个离散FBS方程的截断集，并且对于基态质量的近似FBS方程的结果在定性与晶格结果一致。