A K-Matrix Tutorial

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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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• Language: English

Exrait

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) = m 0 Γ 0 m 02 −m 2 −imΓm Γ(m)∼  m m 0   ρ ρ ( ( m m 0 ) )   F F l l ( ( q q 0 ) )  Γ 0

This approximation assumes an isolated resonance with a single measured decay.

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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).

f 0 (600) f 0 (980) f 0 (1370) f 0 (1500) f 0 (1710)

m= 400−1200MeV m= 980MeV m= 1200−1500MeV m= 1507MeV m= 1718MeV

Broadly overlapping states.

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Γ = 600−1000MeV Γ = 40−100MeV Γ = 200−500MeV Γ = 109MeV Γ = 137MeV

What is the K-matrix

Decays overlap as well:

f 0 (600) f 0 (980) f 0 (1370) f 0 (1500) f 0 (1710)

→ → → → →

Lots of common decay modes.

ππ ¯ ππK K ¯ ππK K ηη4π ¯ ππK K ηη ηη′4π ¯ ππK K ηη

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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: 1  T †  − −T −1 = 2iI

Thisyielsdauqnati

 T −1 +iI  † =  T −1 +iI 

ythwichisHe 6 mritian.

Formalism

We define theKoperator in terms of the Hermitian combination: K −1 =  T −1 +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.

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Formalism

In terms ofK, we have

We also have thatSis

S

that:

T

=

=

(I

K(I−iK)

−1

+iK) (I−iK)

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−1

Formalism

We define theKoperator in terms of the Hermitian combination: K −1 =  T −1 +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.

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Formalism

The K-matrix can be written as the sum of poles,m α , and decay channels, iandj,

K ij = X (m 2 α − g α m i g 2 α ) j √ ρ i ρ j  α

The decay couplings are given as

g α 2 i (m) =m α Γ αi (m) Γ αi (m) =γ 2 αi Γ 0 α ρ i (BF) 2

andρ i  Thisis the phase space for the specified decay. yields an matrix whose dimensions is the number of decay modes.

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Formalism

IfS=e 2iδ , thenT=e iδ sinδand theS-wave cross section is given as σ=  4 q i 2 π  sin 2 δ

The K-matrix can be shown to be:

K= tanδ

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