MODERN HADRONIC RESONANCES THEORY

Transcription

1 JLab, April 2002 MODERN HADRONIC RESONANCES THEORY by Norbert Ligterink Department of Physics and Astronomy University of Pittburgh Pittsburgh

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4 S (535) confusion FIT Γ full (MeV) bf N A p 2 reaction VPI(96) ± 5 N N, γp p Drechsel(99) * 67 γp p Krusche(97) * 20 γp ηp Sauermann(96) ± 20 N N,γp, ηp Pitt-ANL(00) ± 3 All Feuster(99-00) All PDG ± 30 averaging * uses PDG value thanks to Steve Dytman

5 the little page with the big statements we shall overcome... technical... food for mathematicians and philosophers Not really! Extracting microscopic information Unstable states are hard to handle consistently in field theory (arrow-of-time, unitarity) One cannot postulate m+iγ without a microscopic model for the interaction and decay channels

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8 Hamiltonian: two discrete states a and b, one continuum ɛ. H = a m a a + b m b b + dɛ ɛ ɛ ɛ 0 + dɛ g ɛ( ɛ)[ a ɛ + b ɛ + ɛ a + ɛ b ] 0 where ɛ dk[ps] k. Wave function (for energy ω : 0 < ω < ): β = ω = α a a + α b b + ( ω ɛ dɛβ(ɛ) ɛ ) + z(ω)δ(ω ɛ) g ɛ( ɛ)(α a + α b ) Inserting β back gives (ω H) α = 0, hence det[ω H] = 0 yields z: ( ) z(ω) = g 2 + g2 (ω ω ) ω( ω) log ω( ω) ω m b ω m a 2 ω

9 Some properties perturbative definition Γ = a H ɛ 2 = g 2 ɛ( ɛ) The phase shift Scattering amplitude δ r = arctan z(ω) T = z(ω) + i g 0 Some examples: g 2 ω( ω) (ω m a )(ω m b )/(2ω m a m b ) + ig 2 ω( ω)

10 Weak coupling Argand (2X) Real amplitude Imaginary amplitude Strong coupling Argand (2X) Scattering energy -

11 Weak coupling Argand (2X) Real amplitude Imaginary amplitude Strong coupling Argand (2X) Scattering energy -

12 Weak coupling Argand (X) Real amplitude Imaginary amplitude Strong coupling Argand (2X) Scattering energy -

13 T-Matrix / S-Matrix V E H 0 V E H 0 V E H 0 V nothing new Green s Function / Propagator / Resolvent E H 0 V E H 0 V Eigenstates / Möller Operator E H 0 V E H 0 V It all boils down to evaluating: ( N i E H 0 V E H 0 V E H 0 V ) i E H 0 V E H 0 E H 0 V φ 0

14 THE CORE Approximations at the level of the Hamiltonian (state selection) Maintaining unitarity and analyticity Restricting parameters through quantum field theory Renormalization (No fitting with cut-offs)

15 Fano in a nutshell THE HAMILTONIAN (Type I) H = k i= k + i m i i + i= dɛ ɛ ɛ ɛ W i (ɛ)dɛ ( ɛ e iφ i(ɛ) i + i e iφ i(ɛ) ɛ ), THE EIGENSTATE WITH ENERGY ω ω = dɛβ(ω, ɛ) ɛ + k i= α i (ω) i.

16 Fano in a nutshell THE HAMILTONIAN (Type II) H = m + + k a= k a= dɛ ɛ, a ɛ ɛ, a W a (ɛ)dɛ ( ɛ, a e iφ a(ɛ) + e iφ a(ɛ) ɛ, a ), THE EIGENSTATES WITH ENERGY ω ω, b = k a= dɛβ (b) a (ω, ɛ) ɛ, a + α (b) (ω).

17 Summary H I = m W.... m k W k W W k ɛ H II = m W W k W ɛ.... W k ɛ k can be solved in closed form... (Fano)... Many more can be turned into discrete numerical problems with exact (within numerical accuracy) solutions.

18 Fano Type I where the free lunch went for dinner β(ω, ɛ) in terms of the α s: β(ω, ɛ) = ( ω ɛ ) k + z(ω)δ(ω ɛ) α i (ω)w i (ɛ)e iφ i(ɛ) i= For the consistency condition on z(ω) we define: F ji (ξ) = W i (ξ)w j (ξ)e i(φ j(ξ) φ i (ξ)) F ji (η) = dξ F ij(ξ) η ξ F ji is hermitian and yields the shifted, but real, energies of the discrete states: z(ω) = ( W (ω) ((ω ɛ) F(ω)) W(ω) )

19 (the hadronic Lagrangian is not fundamental!) form factors NR formulae renormalization scale low energy constants cut off

20 v(s) OPAL 0 3 0, 3 0 MC corr. perturbative QCD (massless) naïve parton model OPAL (CERN) data τ pions rho meson peak + tail s (GeV 2 )

21 τ ν ρ ρ γ ρ

22 Many intermediate states ω between ρ and ρ a 0 ρ ρ ρ ρ ω ω ρ ρ 770MeV 279MeV ρρ ρρ ρ 540MeV 049MeV ρ ρ ρ ρ ρ ρ ω a 0 922MeV 20MeV 558MeV 837MeV

23 ρ g g ρ ρ g g 2 2 approximated by: (k 2 )k 8 dk/ω 2 ρ

24 COVARIANCE adding the backward diagrams to the real part restores covariance: dɛ f(ɛ2 ) ω ɛ + f(ɛ 2 ) dɛ ω (2ω + ɛ) = dɛ 2 f(ɛ2 ) ω 2 ɛ 2 (Only in the real parts, because threshold > 800 MeV)

25 Problems with multi-loop Feynman diagrams Picking just one: Pseudo-thresholds which turn up at succesive four-momentum integrations (or as singularities in Feynman parameters)

26 CLEO data + my fit log scale 0-4 events/0.025gev energy (GeV)

27 CLEO data + my fit log scale 0-4 events/0.025gev500 only energy (GeV)

28 CLEO data + my fit log scale 0-4 events/0.025gev500 in the presence of energy (GeV)

29 CLEO data + my fit log scale 0-4 events/0.025gev500 in the presence of energy (GeV)

30 CLEO data + my fit log scale 0-4 events/0.025gev500 total and energy (GeV)

31 CLEO data + my fit log scale 0-4 events/0.025gev500 barrier term included energy (GeV)

32 CLEO data + my fit The underlined quantities compare with the data 0-4 events/0.025gev500 log scale 4 suppresses 2 decay total and in the presence of only energy (GeV) in the presence of barrier term included

33 E 83-0M S[GY/.CD-0J E CP9 E -0F>8HGI/ACD->J?K39! =F>L0L0+3=9*= M5UWV & GYX X "!ZI Q$RTSHL B 45CD9,:<CDJ Q\] (from the inclusive data) Events / 25 MeV (2 entries / event) (*),+.- -0/23) ( a )! ( b ) &' "#$ % M (4 ) (GeV) Events / (GeV -2 ) OPAL (c).5 2 Unfolded 3 Tauola s (GeV 2 )

34 This is just the beginning... foundations of modern resonance theory PROJECTS Fano Type 3 V (multiple discrete and continuum states) Three-body states V, t-exchange V, N final states (sic) Systematize renormalization Coupled channel analysis, numerical code

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