Dynamical Model Analysis of Hadron Resonances (IV) T.- S. Harry Lee Argonne Na1onal Laboratory

Transcription

1 Dynamical Model Analysis of Hadron Resonances (IV) T.- S. Harry Lee Argonne Na1onal Laboratory

2 Lecture IV: 1. Dynamical model for extrac3ng heavy/exo3c mesons from 3- mesons produc3on reac3ons 2. Summary _

3 Most of current analysis : Isobar model π M* (D, a 1(1260), π 2 (1670) ) R (f 0, ρ, f 2) π π

4 Three- Body Unitarity Condi3on M* R + Z (a) (b) (c) πππ cut f 0,ρ,f 2 π π π f 0, ρ, f 2

5 What are the correc3ons on isobar- model analysis?? Need to inves3gate within a unitary model Consider a model Hamiltonian (Kamano, Nakamura, Lee, Sato, Phys. Rev. C (2011)) H = H 0 + H, H = [Γ cr,m + Γ cr,m ]+H, cr M * = heavy/exo3c mesons R, R = f 0, ρ, f 2, a,b,c =π, or K H = c R,cR v c R,cR + R [f ab,r + f ab,r ], ab

6 First step : Fit ππ - > ππ, KK scaaering to determine H int = f ππ, R + f kk,r R = bare f 0, ρ, f 2 Solve T(E)= H int + H int 1 E- H+iε H int

7 π π scaaering amplitude T LI ππ,ππ(q,q; E) = R,R f LI ππ,r (q )τ LI R,R(E) f LI R,ππ(q), [(τ LI ) 1 (E)] R R =(E m R )δ R,R Σ LI R,R(E), f LI LI Σ LI R,R(E) = q 2 R dq,ab(q) f ab,r (q) E E a (q) E b (q)+iɛ, ab=ππ,k K f L ab,i ab ab,r (q) = R ππ,k K interaction

8 _ Coupled- channel model for ππ and K K scaaering Fits of ππ amplitudes (degree) (degree) (degree) W (MeV) W (MeV) W (MeV)

9 _ Coupled- channel model for ππ and K K scaaering Bare masses R (L, I) M R1 (MeV) M R2 (MeV) f 0 (0, 0) ρ (1, 1) f 2 (2, 0) 1607 L I Pole Pole positions posi3on (GeV) [Riemann sheet] i [II] i [II] i [III] i [II] i [III] i [III] II : uu III :up

10 Inves3gate 3- π decays of heavy mesons e + e - - > D D - > D + πππ πν, γν - > M* N f 0, a 1,.. Exo3c - > πππ

11 M * - > 3- mesons amplitude T (ab)c,m (E) =T Isobar (ab)c,m (E)+T FSI (ab)c,m (E), T Isobar (ab)c,m (E) = R c R ab f ab,r G cr,c R (E)Γ c R,M M, T FSI (ab)c,m (E) = R c R c R,c R ab f ab,r G cr,c R (E)T c R,c R (E) G c R,c R (E)Γ c R,M M. M* R R a b + T R R a b c c

12 f 0, ρ, f 2 π Use Spline method π f 0, ρ, f 2 T c R,cR(E) = Z c R,cR(E) + Z c R,c R (E)G c R,c R (E)T c R,cR(E). c R,c R Z c R,cR(E) = c f R,cc 1 E E c E c E c + iɛ f c c,r. [G 1 (E)] c R,cR = δ c,c [(E E c E R )δ R,R Σ R,R(E E c )]. Σ R,R(w) = ab R f R,ab B ab w E a E b + iɛ f ab,r R,

13 Effect of three- body unitarity on pole posi3ons M* R + Z (a) (b) (c) TABLE I. Pole masses of a 1 (1260), π 2 (1670) and π 2 (2100). Here with Z denotestheresultsof the full unitary model, while w/o Z denotes the results in which the Z-diagrams are turned off from the full unitary model. Pole masses (MeV) a 1 (1260) π 2 (1670) π 2 (2100) with Z i i i w/o Z No Z i i i Δ Im(M R )=

14 M* R R a b + T R R a b c c d 2 / d M d M 2-0 (arbitrary) M = 0.3 GeV 2 T = M 2-0 (GeV 2 ) D - >3π

15 Dalitz ploa of D- >3π σ(full)/σ(no T ) M 2-0 (GeV 2 ) M 2-0 (GeV 2 ) M (GeV 2 ) M (GeV 2 )

16 M* R R a b R + T R a b Fit with T =0 c T =0 c d 2 / d M d M 2-0 (GeV -3 ) M = 0.60 GeV 2 a 1 (1260) d 2 / d M d M 2-0 (GeV -3 ) (1670) M = 0.61 GeV M 2-0 (GeV 2 ) M 2-0 (GeV 2 )

17 M a 1 (1260) π 2 (1670) π 2 (2100) Pole mass Pole mass Pole mass (MeV) (MeV) (MeV) Full i i i Isobar fit i i i Δ Im(M R )= Note: Re(M R) is kept the same in the fits fit with T =0

18 Experiment at Jlab γ p γ ρ π q M R p c p a p b N N p N p N c a b M* = a 1 (1230), a 2 (1320) π 1 (1600), π 2 (1670) a 1 (1700), a 2 (1700), π 2 (1800)

19 Bare paremeters of M* decays are taken from 3 P 0 model of Barnes, Close, Page, Swanson (Phys. Rev. D 55, 4157 (1997))

20 Dalitz plot of γp - > M* n - >π + π + π - n W= 1.0 GeV W=1.7 GeV

21 d 4 / dt dw d d (!b / GeV 3 sr) Overlapping resonances W (GeV) a 1 (1230), a 2 (1320) π 1 (1600), π 2 (1670) a 1 (1700), a 2 (1700), π 2 (1800)

22 Isobar- fit to pseudo- data from unitary model: 1. Set T = 0 2. Complex M* - > πf 0, π ρ, πf 2 coupling constant 3. Add compex W- dependent background constants Similar to the usual isobar- model fit

23 Accurate fit to Dalitz plot data can be obtained. But the interpreta3ons of the data are very different: M* can decay into more par3al waves than those included in genera3ng the Dalitz plot data

24 a 1 (1260) Unitary model Isobar-fit model MM C([πf 0 (1)] L=1 ) i C([πf 0 (1)] L=2 ) i C([πρ(1)] L=0 ) i C([πρ(1)] L=2 ) - - C([πρ(2)] L=0 ) i C([πρ(2)] L=2 ) i C([πf 2 (1)] L=1 ) i C([πf 2 (1)] L=3 ) i Coupling with M* Leaking decay channels

25 Final state interac3ons due to three- body unitarity must be accounted for in extrac3ng structure informa3on from future experiments for tes3ng hadron models of heavy/exo3c mesons.

26 Current works and plans: 1. Include meson- meson interac3ons calculated from tree diagrams of Chiral Lagrangian 2. Analyze 3- meson decay data from KEK to determine CP viola3on phases 3. Plan to analyze CLAS data of γν - > πππ N (presented at St. Petersburg, Florida) when become accessible

27 Summary 1. Informa3on on excited hadrons are from hadron resoances extracted from experiments 2. Hadron resonances contain informa3on a. Structure of excited hadrons b. Hadron- hadron Interac3ons 3. Resonances correspond to solu3ons of full theory (QCD)

28 Lessons from recent analysis of N* and M* : Current hadron calcula3ons from Cons3tuent quark models, Dyson- Schwinger- Equa3on models. can be properly tested only when the reac3on mechanisms have been accounted for within a dynamical reac3on model