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: ,  K, KN,…

4.  scattering 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 2 nd sheet Lattice only get real eigenvalues of Hamiltonian

6. Review of single-channel case Lüscher’s formula Luescher’s Formula A finite volume Exact energy of two hadrons in finite volume 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  scattering length (Sharpe et al, Fukugita et al, CPPACS, JLQCD, C.L,…)  scattering phase (CPPACS) Other hadron scattering processes (KN,  K,  N )

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  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:      f both phase shifts are well-measured, we can compute  from E  f only one phase (say   ) is well-measured, we can get a constraint for  

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.  scattering

19. An illustration of  in  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