Two particle states in a finite volume and the multi-channel S-matrix elements

1. Two particle states in a finite volume and the multi-channel S-matrix elements Chuan Liu in collaboration with S. He, X. Feng Institute of Theoretical Physics School of Physic, PKU

2. Outline • Motivations • Review of single-channel case (elastic scattering) • Lüscher’s formula • Relevant lattice calculations • Generalization to multi-channel scattering • Quantum Mechanical Model • Possible generalization to QFT • Summary

3. Motivations • Experimentally • Scattering is the most common method in studying particle-particle interactions • Hadron-hadron scattering at low energies is important • By partial wave analysis (PWA), one measures the S-matrix parameters • At low energies, hadron-hadron scattering phases have been measured experimentally • Examples:pp, pK, KN,…

4. ppscattering phase shifts for I=0 and I=2 I=0 I=2

5. Motivations • Theoretically • Hadron-hadron scattering is non-perturbative in nature at low energies • Better be handled by non-perturbative methods, like lattice QCD • How to get indications of resonances from the lattice? Resonace complex pole of S-matrix on the 2nd sheet Lattice only get real eigenvalues of Hamiltonian

6. Review of single-channel case A finite volume • Lüscher’s formula Exact energy of two hadrons in finite volume Luescher’s Formula Elastic scattering phase of two hadrons in infinite volume

7. The Formula for Scattering Phases

8. Review of single-channel case • Lattice calculations using Lüscher’s formula • pp scattering length (Sharpe et al, Fukugita et al, CPPACS, JLQCD, C.L,…) • pp scattering phase (CPPACS) • Other hadron scattering processes (KN,pK, pN)

9. Generalization to multi-channel case • Quantum Mechanical Model

10. Multi-channel scattering in infinite volume • Lippman-Schwinger wave functions:

11. Multi-channel scattering in infinite volume • Radial wave-functions:

12. Multi-channel scattering in infinite volume • Structure of solutions Theorem: Under certain conditions, the radial Schrödinger equation has 2 linearly independent, regular solutions near origin, which may be chosen such that:

13. Influence of a cubic box • Singular Periodic Solutions (SPS) of the Helmholtz equation

14. Influence of a cubic box • Symmetry group of the box

15. The formula • Let G be a irrep of O(Z),

16. A special case • Only s-wave, neglecting g-wave contaminations

17. Usefulness • Given E from lattice calculations, we establish a non-perturbative relation between E and three physical parameters of S -matrix elements:h, d1, d2 • If both phase shifts are well-measured, we can compute h from E • If only one phase (say d1) is well-measured, we can get a constraint for h, d2

18. Possible extension to massive quantum field theories • Like in the single channel case, we expect such a relation to be valid also in massive quantum field theories, apart from corrections which are exponentially small in the large volume limit. However, a tight proof is still lacking. • If this were true, our formula provides a way to study the coupled channel hadron scattering processes, e.g.pp-KK scattering

19. An illustration ofhinpp-KK scattering

20. Summary • A formula is derived which relates the exact energy of two (interacting) particles in a finite volume with the S-matrix parameters of the two-particle scattering in the infinite volume • It is a generalization of the well-known Lüscher’s formula to the multi-channel case • Opens a possibility of calculating multi-channel S -matrix elements in inelastic hadron-hadron scattering using lattice QCD