Abstract
The formalism relating the relativistic three-particle infinite-volume scattering amplitude to the finite-volume spectrum has been developed thus far only for identical or degenerate particles. We provide the generalization to the case of three nondegenerate scalar particles with arbitrary masses. A key quantity in this formalism is the quantization condition, which relates the spectrum to an intermediate K matrix. We derive three versions of this quantization condition, each a natural generalization of the corresponding results for identical particles. In each case we also determine the integral equations relating the intermediate K matrix to the three-particle scattering amplitude, $\mathcal M_3$. The version that is likely to be most practical involves a single Lorentz-invariant intermediate K matrix, $\widetilde{\mathcal K}_{\rm df,3}$. The other versions involve a matrix of K matrices, with elements distinguished by the choice of which initial and final particles are the spectators. Our approach should allow a straightforward generalization of the relativistic approach to all other three-particle systems of interest.
Highlights
The theoretical formalism needed to study three-particle interactions using lattice QCD (LQCD) has advanced considerably in recent years [1,2,3,4,5,6,7,8,9,10,11,12,13,14]
Within the generic relativistic effective field theory (RFT) approach, which we adopt here, the initial development was for identical scalars with a G-parity-like Z2 symmetry [2,3], with the extension to theories without the Z2 symmetry presented in Ref. [4], and that to nonidentical but degenerate scalars given in Ref. [12]
Mirror, at every step, the form of the time-ordered perturbation theory (TOPT) analysis, so that the algebraic simplifications in the latter approach carry over. For this reason the resulting quantization condition is asymmetric. This leads to the major disadvantage of the resulting formalism, namely that it depends on nine intermediate three-particle K matrices, collected in the matrix denoted Kb 0df;3, which are distinguished by the choice of spectator flavors for incoming and outgoing particles
Summary
The theoretical formalism needed to study three-particle interactions using lattice QCD (LQCD) has advanced considerably in recent years [1,2,3,4,5,6,7,8,9,10,11,12,13,14]. Mirror, at every step, the form of the TOPT analysis, so that the algebraic simplifications in the latter approach carry over For this reason the resulting quantization condition is asymmetric. This leads to the major disadvantage of the resulting formalism (shared with the TOPT form), namely that it depends on nine intermediate three-particle K matrices, collected in the matrix denoted Kb 0df;, which are distinguished by the choice of spectator flavors for incoming and outgoing particles. This disadvantage is resolved by the final form of the quantization condition.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
