Alternative derivation of the relativistic three-particle quantization condition
Tyler D. Blanton, Stephen R. Sharpe
Abstract
We present a simplified derivation of the relativistic three-particle quantization condition for identical, spinless particles described by a generic relativistic field theory satisfying a ${\mathbb{Z}}_{2}$ symmetry. The simplification is afforded by using a three-particle quasilocal K matrix that is not fully symmetrized, ${\stackrel{\texttildelow{}}{\mathcal{K}}}_{\mathrm{df},3}^{(u,u)}$, and makes extensive use of time-ordered perturbation theory (TOPT). We obtain a new form of the quantization condition. This new form can then be related algebraically to the standard quantization condition, which depends on a fully symmetric three-particle K matrix, ${\mathcal{K}}_{\mathrm{df},3}$. The new derivation is fully explicit, allowing, for example, a closed-form expression for ${\mathcal{K}}_{\mathrm{df},3}$ to be given in terms of TOPT amplitudes. The new form of the quantization condition is similar in structure to that obtained in the ``finite-volume unitarity'' approach, and in a companion paper we make this connection concrete. Our simplified approach should also allow a more straightforward generalization of the quantization condition to nondegenerate particles, and perhaps also to more than three particles.