Generalizing the relativistic quantization condition to include all three-pion isospin channels

Abstract

We present a generalization of the relativistic, finite-volume, three-particle quantization condition for non-identical pions in isosymmetric QCD. The resulting formalism allows one to use discrete finite-volume energies, determined using lattice QCD, to constrain scattering amplitudes for all possible values of two-and three-pion isospin. As for the case of identical pions considered previously, the result splits into two steps: the first defines a non-perturbative function with roots equal to the allowed energies, E n (L), in a given cubic volume with side-length L. This function depends on an intermediate three-body quantity, denoted K df,3 , which can thus be constrained from lattice QCD input. The second step is a set of integral equations relating K df,3 to the physical scattering amplitude, M 3. Both of the key relations, E n (L) ↔ K df,3 and K df,3 ↔ M 3 , are shown to be block-diagonal in the basis of definite three-pion isospin, I πππ , so that one in fact recovers four independent relations, corresponding to I πππ = 0, 1, 2, 3. We also provide the generalized threshold expansion of K df,3 for all channels, as well as parameterizations for all three-pion resonances present for I πππ = 0 and I πππ = 1. As an example of the utility of the generalized formalism, we present a toy implementation of the quantization condition for I πππ = 0, focusing on the quantum numbers of the ω and h 1 resonances.

Equivalence of relativistic three-particle quantization conditions

“…The study of resonant three-particle systems using lattice QCD (LQCD) is becoming feasible, due to advances in the underlying theoretical formalism [1][2][3][4][5][6][7][8][9][10][11][12][13] and its practical application [4,[14][15][16], as well as in algorithmic and computational methods necessary to extract three-particle spectra (see, for example, the recent results presented in Refs. [17][18][19]).…”

“…The first is based on an all-orders diagrammatic analysis in a generic relativistic field theory and is usually denoted the RFT approach. It was initially developed for identical scalar particles with a G-parity-like Z 2 symmetry [1,2], and subsequently extended to allow 2 → 3 processes [6], the inclusion of poles in the two-particle K matrix [9,11], and nonidentical but degenerate scalars [12]. In all cases, the formalism allows arbitrary interactions in two-particle subsystems (which we henceforth refer to as "dimers").…”

Alternative derivation of the relativistic three-particle quantization condition

“…This is a complementary generalization to that achieved recently in Ref. [42], which considers degenerate, but potentially distinguishable, spinless particles. We expect it to be profitable to combine the two approaches.…”

“…We stress that these and similar steps have been repeatedly used in previous RFT papers, i.e., in HS1, HS2, and Refs. [39][40][41][42]. As a first step, we define d 23 as m 2 þ m 3 evaluated when c 3 → 0:…”

“…We refer to these papers in the following as HS1 and HS2, respectively. The formalism has been subsequently generalized to allow 2 ↔ 3 transitions [39], K matrix poles [40,41], and nonidentical but degenerate particles [42]. The numerical implementation of the formalism has been studied in Refs.…”