A K-Matrix Tutorial

A K-Matrix TutorialCurtis A. MeyerCarnegie Mellon UniversityOctober 23, 20081OutlineWhyThe Formalism.Simple Examples.Recent Analyses.Note: See S. U. Chung, et al., Partial wave analysis in K-matrix formalism, Ann.der Physik 4:404,(1995).2What is the K-matrixIn Partial wave analysis (PWA), resonances are often parameterizedas Breit-Wigners.m Γ0 0BW(m) =2 2m −m −imΓm0 m ρ(m) F (q)0 lΓ(m) ∼ Γ0m ρ(m ) F (q )0 l 0This approximation assumes an isolated resonance with a singlemeasured decay.3What is the K-matrixIf there is more than one resonance in the same partial wave thatstrongly overlap.The Scalar Meson Sector (all couple to ππ final states).f (600) m = 400−1200MeV Γ = 600−1000MeV0f (980) m = 980MeV Γ = 40−100MeV0f (1370) m = 1200−1500MeV Γ = 200−500MeV0f (1500) m = 1507MeV Γ = 109MeV0f (1710) m = 1718MeV Γ = 137MeV0Broadly overlapping states.4What is the K-matrixDecays overlap as well:f (600) → ππ0¯f (980) → ππ,KK0¯f (1370) → ππ,KK,ηη,4π0¯f (1500) → ππ,KK,ηη,ηη′,4π0¯f (1710) → ππ,KK,ηη0Lots of common decay modes.5FormalismStart with a scattering amplitude to connect an initial state to a finalstate.S = < f | S | i >fi†The scattering operator, S, is unitary: SS = I.The transition operator, T, can be defined viaS = I +2iTThis yields an expression: −1† −1T −T = 2iI †−1 −1T +iI = T +iIThis yields a quantity which is Hermitian.6FormalismWe define the K operator in terms of the Hermitian combination:−1 −1K = T ...