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We study two particles colliding in a $d$-dimensional finite volume andgeneralize L\"uscher's formula to arbitrary $d$ spatial dimensions. We obtainthe $s$- and $p$-wave approximations of the generalized L\"uscher's formula.For resonant $s$- or $p$-wave interactions, we analytically determine theenergies 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 areproportional 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 threebosons at a three-body resonance in a 2- or higher-dimensional finite volume.

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