Finite-Volume Spectrum of π+π+ and π+π+

Abstract

The ab initio understanding of hadronic three-body systems above threshold, such as exotic resonances or the baryon spectrum, requires the mapping of the finite-volume eigenvalue spectrum, produced in lattice QCD calculations, to the infinite volume. We present the first application of such a formalism to a physical system in form of three interacting positively charged pions. The results for the ground state energies agree with the available lattice QCD results by the NPLQCD collaboration at unphysical pion masses. Extrapolations to physical pion masses are performed using input from effective field theory. The excited energy spectrum is predicted. This demonstrates the feasibility to determine three-body amplitudes above threshold from lattice QCD, including resonance properties of axial mesons, exotics, and excited baryons.

Three-body unitarity versus finite-volume π+π+π+ spectrum from lattice QCD

“…1. There, solid lines indicate the spectator π + and double lines represent the isospin I = 2 isobar; note that any two-body amplitude, as for example the repulsive I = 2, = 0 can be mapped to the isobar picture [26,77]. In the present scheme, three-body forces arise naturally as real parts that can be added to the interaction without destroying unitarity.…”

“…In Ref. [26], for the first time, a three-body formalism was compared to LQCD data of an actual physical system, π + π + π + , including a fit of the three-body force. In summary, the only missing ingredient for the prediction of the new LQCD data consists in the development of a finite-volume formalism allowing for three-body systems in moving frames.…”

“…This function is chosen as in Ref. [26], where it is shown that the dependence on the particular choice of the cutoff is very weak. The quantity s represents the square of the total four momentum of the three-body system, such that the isobar-spectator amplitudeT reads [4] T nm (s) = τ n (s)T nm (s)τ m (s) − 2E n L 3 τ n (s)δ nm , (2)…”

“…Following the discussion of Ref. [26], we choose f (Q 2 ) = 1/(1 + e −(Λ/2−1) 2 +(Q/mπ) 2 /4 ) with Λ = 42. The dependence of the results on Λ has been checked thoroughly in Ref.…”

“…The dependence of the results on Λ has been checked thoroughly in Ref. [26] and was found to be very mild. Similarly, various analytic forms of the form factor have been evaluated and compared in the same publication.…”