d-dimensional L\"uscher's formula and the near-threshold three-body states in a finite volume

We study two particles colliding in a $d$-dimensional finite volume and generalize L\"uscher's formula to arbitrary $d$ spatial dimensions. We obtain the $s$- and $p$-wave approximations of the generalized L\"uscher's formula. For resonant $s$- or $p$-wave interactions, we analytically determine the energies of the low-lying states at large box size $L$. At $s$-wave resonance, we discover two low-lying states with nearly opposite energies, which are proportional to $\pm 1/L^{d/2}$ for $d\ge 5$, or $\pm 1/L^{2}\sqrt{\ln L}$ for $d=4$. This provides important insights into the near-threshold states of three bosons at a three-body resonance in a 2- or higher-dimensional finite volume.