Threshold expansion of the three-particle quantization condition

Abstract

We recently derived a quantization condition for the energy of three relativistic particles in a cubic box [1,2]. Here we use this condition to study the energy level closest to the three-particle threshold when the total three-momentum vanishes. We expand this energy in powers of 1/L, where L is the linear extent of the finite volume. The expansion begins at O(1/L 3 ), and we determine the coefficients of the terms through O(1/L 6 ). As is also the case for the two-particle threshold energy, the 1/L 3 , 1/L 4 and 1/L 5 coefficients depend only on the two-particle scattering length a. These can be compared to previous results obtained using nonrelativistic quantum mechanics [3-5] and we find complete agreement. The 1/L 6 coefficients depend additionally on the two-particle effective range r (just as in the two-particle case) and on a suitably defined threshold three-particle scattering amplitude (a new feature for three particles). A second new feature in the three particle case is that logarithmic dependence on L appears at O(1/L 6 ). Relativistic effects enter at this order, and the only comparison possible with the nonrelativistic result is for the coefficient of the logarithm, where we again find agreement. For a more thorough check of the 1/L 6 result, and thus of the quantization condition, we also compare to a perturbative calculation of the threshold energy in relativistic λφ 4 theory, which we have recently presented in Ref. [6]. Here all terms can be compared and we find full agreement.

Three-particle quantization condition in a finite volume: 2. General formalism and the analysis of data

“…The first formal investigation dates back to 2012, when it was rigorously shown that the three-body spectrum in a finite volume is determined solely by the three-body S-matrix elements in the infinite volume [26]. In the following years, further important aspects of the three-body problem in a finite volume have been addressed [27][28][29][30][31][32][33][34][35][36]. In particular, the relativistic threeparticle quantization condition in a finite volume has been obtained in refs.…”

Progress in three-particle scattering from LQCD

“…[21], in which we reproduce the finite-volume shift to a spin-zero three-particle bound state in a system with two-particle interactions at unitarity (divergent scattering length). This system was studied by Meißner, Ríos and Rusetsky (MRR) using non-relativistic quantum mechanics.…”

“…[21] we also generalize the expression for the finite-volume shift to the case where the bound state carries nonzero momentum, P, in the finitevolume frame. Given our derivation, the generalization to moving frames turns out to be a straightforward kinematic extension.…”

“…This relation is not particularly intuitive but can be derived by studying our quantization condition in the vicinity of the three-particle bound-state pole, as we describe in detail in Ref. [21]. Second, we derive the functional form of the two-to-two scattering amplitude, by substituting the definition of E * 2,k…”

“…This is necessary because the derivation is complicated and because such checks also shed light on the utility of the quantization condition. Recently, two of us performed two such checks, studying the quantization condition in the limit of very weak [20] and very strong [21] interactions. In both cases we were able to reproduce and extend existing results derived in non-relativistic quantum mechanics.…”