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.
%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
}