A K-Matrix Tutorial
28 Pages
English
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A K-Matrix Tutorial

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28 Pages
English

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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 ...

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A
K-Matrix
Tutorial
Curtis A. Meyer
Carnegie Mellon University
October
23,
1
2008
Outline
Why
The Formalism.
Simple Examples.
Recent Analyses.
Note: See S. U. Chung,et der Physik4:404,(1995).
al.,
Partial
wave
2
analysis
in
K-matrix
formalism,
Ann.
What is the K-matrix
In Partial wave analysis (PWA), resonances are often parameterized as Breit-Wigners.
BW(m) =m0Γ0 m02m2imΓm Γ(m)mm0 ρρ((mm0)) FFll((qq0))Γ0
This approximation assumes an isolated resonance with a single measured decay.
3
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 toππfinal states).
f0(600) f0(980) f0(1370) f0(1500) f0(1710)
m= 4001200MeV m= 980MeV m= 12001500MeV m= 1507MeV m= 1718MeV
Broadly overlapping states.
4
Γ = 6001000MeV Γ = 40100MeV Γ = 200500MeV Γ = 109MeV Γ = 137MeV
What is the K-matrix
Decays overlap as well:
f0(600) f0(980) f0(1370) f0(1500) f0(1710)
Lots of common decay modes.
ππ ¯ ππK K ¯ ππK K ηη4π ¯ ππK K ηη ηη4π ¯ ππK K ηη
5
Formalism
Start with a scattering amplitude to connect an initial state to a final state.
S=<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: 1 TT1= 2iI
Thisyielsdauqnati
T1+iI=T1+iI
ythwichisHe6mritian.
Formalism
We define theKoperator in terms of the Hermitian combination: K1=T1+iI
such thatKis also Hermitian,K=K. Time reversal ofSandTleads toK Thus,also being symmetric. the K-operator, or theK-matrixcan be chosen to be real and symmetric.
7
Formalism
In terms ofK, we have
We also have thatSis
S
that:
T
=
=
(I
K(IiK)
1
+iK) (IiK)
8
1
Formalism
We define theKoperator in terms of the Hermitian combination: K1=T1+iI
such thatKis also Hermitian,K=K. Time reversal ofSandTleads toK thealso being symmetric. Thus, K-operator, or theK-matrixcan be chosen to be real and symmetric.
9
Formalism
The K-matrix can be written as the sum of poles,mα, and decay channels, iandj,
Kij=X(m2αgαmig2α)jρiρjα
The decay couplings are given as
gα2i(m) =mαΓαi(m) Γαi(m) =γ2αiΓ0αρi(BF)2
andρi Thisis the phase space for the specified decay. yields an matrix whose dimensions is the number of decay modes.
01
Formalism
IfS=e2iδ, thenT=eiδsinδand theS-wave cross section is given as σ=4qi2πsin2δ
The K-matrix can be shown to be:
K= tanδ
11