Finite-volume formalism for $Nππ$ at maximal isospin

TL;DR

This paper develops a finite-volume three-particle formalism within the relativistic field theory (RFT) framework for the $N\pi\pi$ system in the maximal-isospin channel $I=5/2$. It derives a quantization condition, $\det\left(1+\widehat{\mathcal{K}}_{\rm df,3} \widehat{F}_3\right)=0$, linking finite-volume spectra to two- and three-particle $K$-matrices, and formulates infinite-volume integral equations that relate these $K$ matrices to physical scattering amplitudes, taking into account nucleon spin and nondegenerate flavor; the formalism reduces to a two-channel ($\{(\pi\pi)_2N, (N\pi)_{3/2}\pi\}$) flavor structure for $I=5/2$ and includes detailed spin-rotation factors via Wigner matrices. A concrete numerical application incorporating a $\Delta$ resonance in the $N\pi$ subchannel demonstrates the viability of the approach, while highlighting the challenges for extending to nonmaximal isospin where $N\pi$–$N\pi\pi$ mixing must be included. The work also identifies subthreshold singularities in $3\to3$ kernels that necessitate modified cutoff functions and outlines a clear path toward a full treatment of all $N\pi\pi \leftrightarrow N\pi$ systems in lattice QCD.

Abstract

We extend the relativistic field theoretic finite-volume formalism to $N ππ$ scattering states at maximal isospin, $I=5/2$. As in previous work using the relativistic field theory approach, we work to all orders in a generic low-energy effective theory, and determine the quantization condition that relates finite-volume energies to intermediate K matrices, and the integral equations connecting the latter to the physical scattering amplitudes. We discuss the parametrization of the K matrices, and explain in detail the new features that arise in implementing the quantization condition due to the spin of the nucleon in combination with the use of non-degenerate particles. As a concrete example, we provide a sample numerical application including the $Δ$ resonance in the $Nπ$ subchannel. The extension to the $I=3/2$ and $1/2$ channels is more involved, due to mixing with $Nπ$ states, and we do not provide a complete formalism for these cases. We explain why $Nπ$ states cannot be included by treating the nucleon as a pole in $p$-wave $Nπ$ scattering, an approach that has been successful in studying $D D^*$ scattering using the three-particle $DDπ$ formalism. We additionally provide results for all isospins under the assumption of no two-to-three mixing, thereby laying the groundwork for a follow-up paper in which all $Nππ\leftrightarrow Nπ$ systems are fully treated. Finally, we study the singularities in $Nππ$ amplitudes arising from $Nπππ$ intermediate states, and find that our subthreshold cutoff functions must be modified to avoid such singularities.

Figures (8)

• Figure 1: Processes leading to left-hand singularities in the $N\pi$ amplitude: (a) $u$-channel nucleon exchange; (b) $s$-channel nucleon pole; (c) $t$-channel two-pion exchange. Nucleon lines are solid, while pion lines are dashed. Vertical dotted lines indicate the cuts leading to the singularities.
• Figure 2: $N\pi\pi$ diagram showing the need for a four-particle intermediate state (shown by the dotted vertical line) to account for the $u$-channel left-hand cut in an $N\pi$ subchannel. Notation as in \ref{['fig:LHC']}.
• Figure 3: Low-lying finite-volume spectrum of $I=5/2$ states with unit baryon number in the $G_{1g}$ irrep for $\boldsymbol P =0$. Results are for the two-particle parameters described in the text and $\mathcal{K}_0 = 0$. Results from the quantization condition are shown as solid, colored lines, with the alternating colors used to distinguish the levels but having no further significance. Noninteracting levels are shown by thin, black solid ($N\pi\pi$) and dashed ($\Delta\pi$) lines. The second $\Delta\pi$ noninteracting level is doubly degenerate.
• Figure 4: Effect of $\mathcal{K}_0$ in the lowest two states in the spectrum. Notation as in \ref{['fig:G1girrep']}.
• Figure 5: Allowed parameter space for the $N\pi\pi$ system with total energy $E$, pion spectator momentum $p = |\bm p|$ (both defined in the $N\pi\pi$ CMF), and $M_\pi/M_N = 0.2$. Solid, dashed and dotted curves correspond to singularities associated with the accompanying Feynman diagram, and are discussed in the text. Solid curves indicate singularities on the physical sheet, while dotted curves correspond to singularities on unphysical sheets. The cyan dashed curve separates the physical region (also shaded in cyan), in which all momenta are real, from the region of subthreshold $N \pi$ kinematics. The latter is shaded (in purple) until the first singularity is encountered, which marks the edge of the physical sheet. The dashed curve is not solid because the associated threshold singularity is absent from the $3\to3$ Bethe-Salpeter kernel and from ${\mathcal{K}_{\mathrm{df},3}}$. The standard RFT cutoff function $H_\pi(\bm p)$ (given by \ref{['eq:cutoff', 'eq:sigmapimin']}) has a lower end of support on the orange curve, and thus includes the upper-right triangular region (to the right of the blue nearly vertical curve) which is not on the physical sheet. The purple dot-dashed curve illustrates a possible choice for the lower end of support for a modified cutoff function, one that ensures that the parameter space is free from subthreshold nonanalyticities (see \ref{['app:newcutoff']}).