Scrutinizing $ππ$ scattering in light of recent lattice phase shifts

Abstract

In this paper, the IJ = 00, 11, 20 partial wave ππ scattering phase shifts determined by the lattice QCD approach are analyzed by using a novel dispersive solution of the S-matrix, i.e. the PKU representation, in which the unitarity and analyticity of scattering amplitudes are automatically satisfied and the phase shifts are conveniently decomposed into the contributions of the cuts and various poles, including bound states, virtual states and resonances. The contribution of the lefthand cut is estimated by the SU (2) chiral perturbation theory to O(p 4 ). The Balanchandran-Nuyts-Roskies relations are considered as constraints to meet the requirements of the crossing symmetry. It is found that the IJ = 00 ππ scattering phase shifts obtained at mπ = 391 MeV by Hadron Spectrum Collaboration (HSC) reveal the presence of both a bound state pole and a virtual state pole below the ππ threshold rather than only one bound state pole for the σ. To reproduce the lattice phase shifts at mπ = 391 MeV, a virtual-state pole in the IJ = 20 channel is found to be necessary in order to balance the left-hand cut effects from the chiral amplitudes. Similar discussions are also carried out for the lattice results with mπ = 236 MeV from HSC. The observed behaviors of the pole positions with respect to the variation of the pion masses can provide deep insights into our understanding of the dynamical origin of σ resonance.

Comment on "Scrutinizing pion-pion scattering in light of recent lattice phase shifts"

“…However, in our recent modelling [10] of f 0 resonances with the seventeen S and D wave coupled channels ππ, K K, ηη, ηη ′ , η ′ η ′ , ρρ, ωω, K ⋆ K⋆ , φφ, f 0 (500)f 0 (500), f 0 (980)f 0 (980), K ⋆ 0 (700) K⋆ 0 (700), and a 0 (980)a 0 (980), we found a dynamically generated pole at (455−i232) MeV on the second Riemann sheet for the standard overall coupling and with m π = 139.57 MeV, Taking a much larger overall coupling we obtain BS and VS poles at 0.16 MeV and 62 MeV below the ππ threshold, respectively. However, for m π = 391 MeV and with the standard overall coupling we find a bound state at 760 MeV, compatible with the lattice [2] result, but contrary to the dispersive analysis [1] for m π = 391 MeV no nearby VS pole.…”

“…We comment on the two-pole description of the σ resonance in S-wave ππ scattering as modelled in Ref. [1], in contrast to the one-pole extraction from the lattice simulation in Ref. [2].…”

“…Figure 7 of Ref. [1] depicts typical S-wave subthreshold pole trajectories in the complexenergy (E) plane. After both poles hit the real axis, they can either end up as representing a pair of virtual bound states or one virtual state (VS) and one true bound state (BS).…”

“…In the one-channel dispersive analysis [1] of lattice [2] S-wave ππ phases for m π = 391 MeV, BS and VS poles are extracted at 1 MeV and 73 MeV below threshold, respectively. However, in our recent modelling [10] of f 0 resonances with the seventeen S and D wave coupled channels ππ, K K, ηη, ηη ′ , η ′ η ′ , ρρ, ωω, K ⋆ K⋆ , φφ, f 0 (500)f 0 (500), f 0 (980)f 0 (980), K ⋆ 0 (700) K⋆ 0 (700), and a 0 (980)a 0 (980), we found a dynamically generated pole at (455−i232) MeV on the second Riemann sheet for the standard overall coupling and with m π = 139.57 MeV, Taking a much larger overall coupling we obtain BS and VS poles at 0.16 MeV and 62 MeV below the ππ threshold, respectively.…”

Reply to comment on `Scrutinizing $ππ$ scattering in light of recent lattice phase shifts'

“…Recently, in our paper [1] we have used the PKU representation of the S matrix to extract the information of the S-matrix poles from the lattice phase shifts of the ππ scattering with m π = 391, 236 MeV by Hadron Spectrum Collaboration (HSC) [2][3][4][5]. In the commenting paper [6], the authors raised doubts about the novelty and the two-pole solution for σ at m π = 391 MeV of our work.…”