A K-Matrix Tutorial. Curtis A. Meyer. October 23, Carnegie Mellon University

Transcription

1 A K-Matrix Tutorial Curtis A. Meyer Carnegie Mellon University October 23, 28

2 Outline Why The Formalism. Simple Examples. Recent Analyses. Note: See S. U. Chung, et al., Partial wave analysis in K-matrix formalism, Ann. der Physik 4:44,(995).

3 What is the K-matrix In Partial wave analysis (PWA), resonances are often parameterized as Breit-Wigners. BW(m) = Γ(m) m Γ m 2 m2 imγm ( m ) ( ρ(m) m ρ(m ) )( Fl (q) F l (q ) ) Γ This approximation assumes an isolated resonance with a single measured decay.

4 What is the K-matrix If there is more than one resonance in the same partial wave that strongly overlap. The Scalar Meson Sector (all couple to ππ final states). f (6) m = 4 2 MeV Γ = 6 MeV f (98) m = 98MeV Γ = 4 MeV f (37) m = 2 5 MeV Γ = 2 5MeV f (5) m = 57MeV Γ = 9MeV f (7) m = 78MeV Γ = 37MeV Broadly overlapping states.

5 What is the K-matrix Decays overlap as well: f (6) f (98) f (37) f (5) f (7) ππ ππ, K K ππ, K K, ηη,4π ππ, K K, ηη, ηη, 4π ππ, K K, ηη Lots of common decay modes.

6 Formalism Start with a scattering amplitude to connect an initial state to a final state. S fi = < f S i > The scattering operator, S, is unitary: SS = I. The transition operator, T, can be defined via S = I + 2iT This yields an expression: ( T ) T = 2iI ( T + ii ) = ( T + ii ) This yields a quantity which is Hermitian.

7 Formalism We define the K operator in terms of the Hermitian combination: K = ( T + ii ) such that K is also Hermitian, K = K. Time reversal of S and T leads to K also being symmetric. Thus, the K-operator, or the K-matrix can be chosen to be real and symmetric.

8 Formalism In terms of K, we have that: T = K (I ik) We also have that S is S = (I + ik)(i ik)

9 Formalism We define the K operator in terms of the Hermitian combination: K = ( T + ii ) such that K is also Hermitian, K = K. Time reversal of S and T leads to K also being symmetric. Thus, the K-operator, or the K-matrix can be chosen to be real and symmetric.

10 Formalism The K-matrix can be written as the sum of poles, m α, and decay channels, i and j, K ij = α g αi g αj (m 2 α m 2 ) ρ i ρ j. The decay couplings are given as g 2 αi(m) = m α Γ αi (m) Γ αi (m) = γ 2 αiγ αρ i (BF) 2 and ρ i is the phase space for the specified decay. This yields an matrix whose dimensions is the number of decay modes.

11 Formalism If S = e 2iδ, then T = e iδ sinδ and the S-wave cross section is given as ( ) 4π σ = sin 2 δ The K-matrix can be shown to be: q 2 i K = tan δ

12 Two-pole K-Matrix Consider two resonances in the J PC = ++ channel. We will consider two cases: f (2, Γ = ) f (8, Γ = ) f (35, Γ = 3) f (5, Γ = ) In the first case, the resonances are relatively well isolated. In the second, there is very strong overlap.

13 Two-pole K-Matrix Breit-Wigner parameterization for the mass of the resonances. (left) f (2, Γ = )&f (8, Γ = ) and (right) f (35, Γ = 3)&f (5, Γ = ).

14 Two-pole K-Matrix K ij (m) = m Γ (m) m 2 + m 2Γ 2 (m) m2 m2 2 m2 ( Γ i (m) = Γ mi ) ( ) q i m q i T T = = K ( ik) + m Γ (m) (m 2 m2 ) im Γ (m) i (m2 m2 ) (m 2 2 m2 ) m 2Γ 2 (m) m 2 Γ 2 (m) (m 2 2 m2 ) im 2 Γ 2 (m) i (m2 2 m2 ) (m 2 m2 ) m Γ (m)

15 Two-pole K-Matrix For the case of m and m 2 very far apart, the terms like i (m2 m2 ) (m 2 2 m2 ) m 2Γ 2 (m) are driven to zero far away from the main resonance. Thus, we get that T m Γ (m) (m 2 m2 ) im Γ (m) + m 2 Γ 2 (m) (m 2 2 m2 ) im 2 Γ 2 (m) which looks like two Breit Wigner functions.

16 Two-pole K-matrix Breit-Wigner parameterization (red) and K-Matrix (blue) for the mass of the resonances, (left) f (2, Γ = )&f (8, Γ = ) and (right) f (35, Γ = 3)&f (5, Γ = ).

17 Two-pole K-Matrix Breit-Wigner parameterization (red) and K-Matrix (blue) for the Argand diagrams for the two resonances, (left) f (2, Γ = )&f (8, Γ = ) and (right) f (35, Γ = 3)&f (5, Γ = ).

18 One-pole, 2-decay K-Matrix Consider the a (98), a J PC = ++ resonance that couples to both ηπ and K K. The K K occurs near the peak of the resonance, (987.3 MeV). with γ 2 + γ2 2 =. K = γ2 m Γ m 2 m2 K 22 = γ2 2 m Γ m 2 m2 K 2 = γ γ 2 m Γ m 2 m2

19 One-pole, 2-decay K-Matrix T = ( ) m Γ γ 2 m 2 ( m2 im Γ ρ γ 2 + ρ ) γ γ 2 2γ2 2 γ γ 2 γ2 2

20 One-pole, 2-decay K-Matrix Im(T) T Blue χπ Red KK x Re(T) m K-matrix mass (.98) and width (.8) with γ 2 ηπ =.8.

21 One-pole, 2-decay K-Matrix Im(T) T Blue χπ Red KK x Re(T) m K-matrix mass (.98) and width (.8) with γ 2 ηπ =.5.

22 One-pole, 2-decay K-Matrix Im(T) T Blue χπ Red KK x Re(T) m K-matrix mass (.98) and width (.8) with γ 2 ηπ =.2.

23 One-pole, 2-decay K-Matrix Im(T) T Blue χπ Red KK x Re(T) m K-matrix mass (.98) and width (.3) with γ 2 ηπ =.5.

24 Crystal Barrel Analysis The scalar mesons, f (37) and f (5) are strongly produced in pp annihilation at rest. These can be searched for in three-pseudoscalar final states: pp pp pp pp pp (ππ)π (ππ)η (ηη)π (K K)π (ηη )π The CRYSTAL BARREL Collaboration (C. Amsler et al.), Coupled channel analysis of antiproton proton annihilation into π π π, ηηπ and ηπ π, Phys. Lett. B355, 425, (995).

25 Crystal Barrel Analysis M = (39 ± 3)MeV M = (5 ± )MeV ; Γ = (38 ± 8)MeV ; Γ = (54 ± 3)MeV

26 CLAS Photo-production Analysis The reaction γp pω from threshold up to s 2.8 GeV. Partial wave analysis carried out which includes both t-channel and s-channel processes. γ ω γ ω π,η N p p p p The CLAS Collaboration (M. Williams et al.), Partial wave analysis of the reaction γp pω and the search for nucleon resonances, to be submitted to Phys. Rev. D (28).

27 CLAS Photo-production Analysis φ (radians) Pole K-Matrix[F (68+2)]/G (29) F 5 (95)/G (29) 7 F 5 (68)/G (29) W (MeV) Phase difference between the J P = ( 5 2 )+ and the J P = ( 7 2 ) waves. A reasonable fit to the amplitude and phase differences is obtained by using a two-k-matrix description for the ( 5 2 )+ wave.

28 Summary The K-matrix formalism is derived for S-matrix scattering and provides a method to build a unitary T-matrix. The formalism accommodates multiple (overlapping) resonances in the same partial wave. The formalism allows one to couple data on different final states of the same resonances. This is important when one is trying to measure branching fractions. While not mentioned, the extensions to broad daughter particles adds complications to the formalism. In particular, the handling of thresholds and phase-space factors become nebulous.