Scattering phase shifts for two particles of different mass and nonzero total momentum in lattice QCD

Abstract

We derive the relation between the scattering phase shift and the two-particle energy in the finite box, which is relevant for extracting the strong phase shifts in lattice QCD. We consider elastic scattering of two particles with different mass and with non-zero total momentum in the lattice frame. This is a generalization of the Lüscher formula, which considers zero total momentum, and the generalization of Rummukainen-Gottlieb's formula, which considers degenerate particles with non-zero total momentum. We focus on the most relevant total momenta in practice, i.e. P = (2π/L) e z and P = (2π/L) (e x + e y ) including their multiples and permutations. We find that the P -wave phase shift can be reliably extracted from the two-particle energy if the phase shifts for l ≥ 2 can be neglected, and we present the corresponding relations. The reliable extraction of S-wave phase shift is much more challenging since δ l=0 is always accompanied by δ l=1 in the phase shift relations, and we propose strategies for estimating δ l=0 . We also propose the quark-antiquark and mesonmeson interpolators that transform according the considered irreducible representations.

Coupled-channel Dπ, Dη and D s K ¯ $$ {D}_s\overline{K} $$ scattering from lattice QCD

“…In particular, the Lüscher method and its extensions for relating finite-volume spectra to scattering amplitudes are now well established for elastic [5][6][7][8][9][10][11][12][13][14] and coupled-channel [15][16][17][18][19] hadron-hadron scattering. These methods have been demonstrated in a number of applications, notably for the ρ-resonance seen in P -wave ππ scattering [20][21][22][23][24][25][26][27][28][29], and for the σ resonance seen in S-wave ππ scattering [30].…”

“…Having determined the finite-volume spectra, we relate these to infinite-volume scattering amplitudes using the Lüscher method [5,6] and its extensions to moving frames [7,10,13,51] and coupled-channels [16][17][18]52]. In this approach, the dependence of the spectra on finite volume is used as a tool but exponentially-suppressed corrections in the volume are neglected -typically the leading such corrections fall off as e −mπL and, since our volumes have m π L ∼ 4 to 6, we can safely neglect these.…”

S-waveKK*interactions in a finite volume and thefGeng1, Ren2, Zhou3 et al. 2015Phys. Rev. D292330View full textAdd to dashboardCite

“…The problem has been thoroughly studied in Ref. [18] and it is particularly relevant when one performs lattice simulations for particles in a moving frame [22,[35][36][37][38][39][40][41][42][43][44]. The formulation for moving frames along the lines of Ref.…”

Multi-Hadron Observables from Lattice Quantum Chromodynamics

“…[129]. The result is [143,144,145,146] 36) where γ = E/E * , n runs over integer vectors, and r is obtained from n by r = γ −1 [n −c n P ]…”

“…where γ = P 0,M /P * 0,M , the sum is performed over 144,145], andγ −1 x ≡ γ −1 x || +x ⊥ , with x || (x ⊥ ) denoting the x component that is parallel(perpendicular) to the total momentum, P. In Appendix C we show the generalization of this for asymmetric volumes with twisted boundary conditions.…”