一个相对论性的模型无关的三粒子量子化条件
A relativistic, model-independent, three-particle quantization condition
摘要：
本文将Luescher有限体积谱与散射振幅之间的关系推广到三粒子情形。我们考虑一个相对论性的标量场理论，其中的耦合是任意的，除了一个Z2对称性，删除与奇数粒子的顶点。该理论被假定为具有在弹性散射区域中由π/2限定的两粒子相移。我们从奇粒子数有限体积相关器中的极点确定有限体积理论的谱，我们在微扰论中分析到所有阶次。我们表明，它依赖于无限体积的2到2的K-矩阵，以及一个非标准的无限体积的3到3的K-矩阵。我们的结果的一个关键特点是需要减去物理奇点在三到三的振幅，从而处理一个发散自由量。这使得我们的初始，正式的结果被截断到一个有限维行列式方程。目前，三对三K矩阵与相应的散射振幅的关系尚不清楚，尽管以前在非相对论极限下的结果表明存在这样的关系。
英文摘要：
We present a generalization of Luescher's relation between the finite-volume spectrum and scattering amplitudes to the case of three particles. We consider a relativistic scalar field theory in which the couplings are arbitrary aside from a Z2 symmetry that removes vertices with an odd number of particles. The theory is assumed to have two-particle phase shifts that are bounded by π/2 in the regime of elastic scattering. We determine the spectrum of the finite-volume theory from the poles in the odd-particle-number finite-volume correlator, which we analyze to all orders in perturbation theory. We show that it depends on the infinite-volume two-to-two K-matrix as well as a nonstandard infinite-volume three-to-three K-matrix. A key feature of our result is the need to subtract physical singularities in the three-to-three amplitude and thus deal with a divergence-free quantity. This allows our initial, formal result to be truncated to a finite dimensional determinant equation. At present, the relation of the three-to-three K-matrix to the corresponding scattering amplitude is not known, although previous results in the non-relativistic limit suggest that such a relation exists.
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