arXiv : Three-particle systems with resonant subprocesses in a finite volume

Briceño, Raúl A.; Sharpe, Stephen R.; Hansen, Maxwell T.

doi:10.1103/PhysRevD.99.014516

Article
Report number	arXiv:1810.01429 ; JLAB-THY-18-2819
Title	Three-particle systems with resonant subprocesses in a finite volume
Author(s)	Briceño, Raúl A. (Jefferson Lab ; Old Dominion U.) ; Hansen, Maxwell T. (CERN) ; Sharpe, Stephen R. (Washington U., Seattle)
Publication	2019-01-31
Imprint	2018-10-02
Number of pages	40
Note	46 pages, 9 figures, JLAB-THY-18-2819, CERN-TH-2018-216
In:	Phys. Rev. D 99 (2019) 014516
DOI	10.1103/PhysRevD.99.014516
Subject category	hep-lat ; Particle Physics - Lattice
Abstract	In previous work, we have developed a relativistic, model-independent three-particle quantization condition, but only under the assumption that no poles are present in the two-particle K matrices that appear as scattering subprocesses. Here we lift this restriction, by deriving the quantization condition for identical scalar particles with a G-parity symmetry, in the case that the two-particle K matrix has a pole in the kinematic regime of interest. As in earlier work, our result involves intermediate infinite-volume quantities with no direct physical interpretation, and we show how these are related to the physical three-to-three scattering amplitude by integral equations. This work opens the door to study processes such as $a_2 \to \rho \pi \to \pi \pi \pi$, in which the $\rho$ is rigorously treated as a resonance state.
Copyright/License	preprint: (License: arXiv nonexclusive-distrib 1.0) Publication: © 2019-2025 authors

$Derivation of Eq.~(\ref{eq:symm1}), using the notation of Fig.~\ref{fig:CL0Frecursion_v2}.$ $Derivation of Eq.~(\ref{eq:symmL0}), using the notation of Figs.~\ref{fig:CnL0F} and \ref{fig:CL0Frecursion_v2}. On the left-hand side of the equality, the quantity to the right of the $G$ cut is $\bKdfth^\uu$, with the $u$ above the upper-right dashed line indicating that this is an unsymmetrized quantity. The left-hand cut in all diagrams must be present, but can be either $\bF$ or $\bG$. The box at the left-hand end of each diagram represents whatever lies to the left of the $F/G$ cut, which depends on the context, but for which the details are irrelevant. The first two equalities show how $\bG$ is converted to $\bF$ by adding and subtracting an integral. This method is used extensively in Ref.~\cite{\HSQCa} and is explained in Eqs.~(163)-(165) of that work and the accompanying text. It results in the $u$ superscript on $\bKdfth$ changing to $s+\tilde s$ in the $\bF$ term. In the second step (indicated by the arrow connecting the two boxed diagrams), $\K_2$ is replaced by the pole term, with on-shell projection onto the K-matrix pole, and the smooth part. Since there is now an integral to the right of the $\bG$, rather than a sum, the infinite volume quantity $\bKdfth^\uu$ is extended to the left by the addition of either a $\Gamma$ or $\K'_2$, implicitly defining the integral operators $\mathcal I_{23}$ and $\mathcal I_{33}$, respectively.$ $Derivation of Eq.~(\ref{eq:symm5}), using the notation of Figs.~\ref{fig:CnL0F} and \ref{fig:CL0Frecursion_v2}. The left-hand boxes represent $\bA'_{\ttil}$, aside from the loop that is exposed explicitly to the left of the left-most cut. The steps are similar to those in Fig.~\ref{fig:symm2}: replacing the sum adjacent to the $\bG$ with a sum-integral difference and an integral, the former giving rise to an $\bF$. The difference from Fig.~\ref{fig:symm2} concerns the integral, in which the factor of $\bGr$ can be converted into an $\bFrp$ cut by projecting the entire quantity to the right onto the K-matrix pole onto the $\ttil$ mass shell, leading to the $\mathcal I_{\ttil 3}$ term. The residue (the $\delta_\rho$ term) cancels the K-matrix pole, allowing the sum over the momentum $k$ to be replaced by an integral, so that the implicit $\bA'_{\ttil}$ and the $\bKdfthtw^\u$ are connected by an infinite-volume integral operator denoted $\rho_{\ttil 3}$.$ Show more plots

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