Pole-expansion of two-hadron imaginary-time correlation function: A new analysis method for unstable states in lattice QCD

We analyze the pole expansion of the two-hadron imaginary-time correlation function. We first explain the general idea that the imaginary-time correlation function is expressed as a sum of the pole terms, the Mittag-Leffler expansion, in terms of the uniformization variable, which makes the S-matrix single-valued. We then derive explicit expressions of the pole expansion for the single-channel (<a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>ρ</a:mi></a:math> meson) and two-channel [<c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi mathvariant="normal">Λ</c:mi><c:mo stretchy="false">(</c:mo><c:mn>1405</c:mn><c:mo stretchy="false">)</c:mo></c:math>] examples and demonstrate that the pole expansion actually holds employing phenomenological models, the vector-dominance model for the <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" display="inline"><h:mi>ρ</h:mi></h:math> meson and the chiral unitary model for <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline"><j:mi mathvariant="normal">Λ</j:mi><j:mo stretchy="false">(</j:mo><j:mn>1405</j:mn><j:mo stretchy="false">)</j:mo></j:math>. From this observation we propose the pole expansion as a method to extract information of unstable states such as masses and widths from the two-hadron imaginary-time correlation functions obtained by lattice quantum chromodynamics (QCD) simulations.