%0 Generic %1 Hansen2014Relativistic %A Hansen, Maxwell T. %A Sharpe, Stephen R. %D 2014 %K resonance %T A relativistic, model-independent, three-particle quantization condition %U http://arxiv.org/abs/1408.5933 %X 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 \pi/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.

@misc{Hansen2014Relativistic, abstract = {{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 \pi/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.}}, added-at = {2019-02-23T22:09:48.000+0100}, archiveprefix = {arXiv}, author = {Hansen, Maxwell T. and Sharpe, Stephen R.}, biburl = {https://www.bibsonomy.org/bibtex/2b85551fa75171b48c5dbb9ee2a8dedc0/cmcneile}, citeulike-article-id = {13466283}, citeulike-linkout-0 = {http://arxiv.org/abs/1408.5933}, citeulike-linkout-1 = {http://arxiv.org/pdf/1408.5933}, day = 19, eprint = {1408.5933}, interhash = {ba5e275cb1afd9a00c23487ac73e343a}, intrahash = {b85551fa75171b48c5dbb9ee2a8dedc0}, keywords = {resonance}, month = nov, posted-at = {2014-12-20 21:33:11}, priority = {2}, timestamp = {2019-02-23T22:15:27.000+0100}, title = {{A relativistic, model-independent, three-particle quantization condition}}, url = {http://arxiv.org/abs/1408.5933}, year = 2014 }