details of ebac-dcc model

Details of EBAC-DCC modelDetails of EBAC-DCC model

Hiroyuki KamanoHiroyuki Kamano

Informal EBAC meeting, May 24-26, 2010(revised version of the talk at 2009 EBAC meeting)

Partial wave (LSJ) amplitude of a b reaction:

Reaction channels:

Potential:

EBAC-DCC model: hadronic part

2-body v potential(no N cut)

2-body v potential(no N cut) bare N* statebare N* state

For details see Matsuyama, Sato, Lee, Phys. Rep. 439,193 (2007)

2-body Z potential(with N cut)

2-body Z potential(with N cut)

e.g.) D13 Total J = 3/2 , Isospin = 1/2 , Parity =

MB( LS )

Meson-baryon Green’s functions

Stable channels = (, N), (, N), (K,), (K,) :

Unstable channels = (, ), (, N), (, N) :

() self-energy in the presence of spectator particle

Meson-baryon Green’s function

Branch point of unstable Green functions (MeV)

Unstable channels = (, ), (, N), (, N) :

Meson-baryon Green’s function

Model 2:

Unstable channels = (, ), (, N), (, N) :

Betz-Lee PRC23 375 (1981)

Simple s-wave (separable) model

Original Model 2

Branch points now have more realistic values.N Green function has two branch points.

2-body “v” potentials (non-strange channels)

5 diagramss-ch Nu-ch Nu-ch t-ch t-ch

2 diagramss-ch Nu-ch N

2 diagramss-ch Nu-ch N

3 diagramss-ch Nu-ch Nt-ch

2 diagramss-ch Nu-ch N

2 diagramss-ch Nu-ch N

4 diagramss-ch Nu-ch Nt-ch t-ch

2 diagramss-ch Nu-ch N

2 diagramss-ch Nu-ch N

2 diagramss-ch Nu-ch N

4 diagramss-ch Nu-ch Nu-ch t-ch

1 diagrams-ch N

1 diagrams-ch N

2 diagramss-ch Nu-ch N

2 diagramss-ch Nt-ch

Total 36 diagrams

2-body “v” potentials (channels with strange hadrons)

3 diagramss-ch Nu-ch t-ch K*

4 diagramss-ch Nu-ch t-ch t-ch

Total 18 diagrams

At present, KY couples to non-strange channelsthrough N channel only.

3 diagramss-ch Nu-ch u-ch t-ch K*

3 diagramss-ch Nu-ch t-ch

5 diagramss-ch Nu-ch t-ch t-ch t-ch

2-body “v” potentialsAll potentials are described in Matsuyama, Sato, Lee, Phys. Rep. 439 193 (2007)

Partial wave decomposition

Plane wave matrix elementin helicity representationPlane wave matrix elementin helicity representation

Used for thecoupled-channelsequation

Unitary transformation (UT) method

e.g.) N N “v” potential (s-channel nucleon)

Independent of total scattering energy s1/2 !!

Potentials agree with usual Feynman diagram at on-shell. The off-shell behavior is uniquely defined within UT method.

UT methodUT method

See e.g., Ann. Phys. 322, 736 (2007)nucl-th/0102037for details of UT method

See e.g., Ann. Phys. 322, 736 (2007)nucl-th/0102037for details of UT method

Rules for attaching cutoff factors

Attach cutoff factors to each vertex (currently dipole form is used)

For s- and u-channel potential, use 3-momentum of external meson

For t-channel potential, use 3-momentum of exchanged meson

For contact potential, use product of two cutoff factors

2-body “Z” potentials

Feshbach projection:

2-body “Z” potentials

Numerical treatment of Z potentials

Z(E)(k,k’;E) potentials have logarithmic singularity.Z(E)(k,k’;E) potentials have logarithmic singularity.

N N, N, KY

Contour-rotation method

N N

Spline method

e.g., Larson et al, PRC9 699 (1974)

e.g., Matsuyama, PLB152 42 (1985); Matsuyama Lee, PRC34 1900 (1986)

Z as a function of kat k’=0.3 GeV, E=1.88 GeVZ as a function of kat k’=0.3 GeV, E=1.88 GeV

Bare N* MB vertex function

e.g.) D13 state ( I = 1/2, J = 3/2, Parity = minus) 18

# of parameters for a bare N* state

# of bare N* stateS11 2, S31 1P11 2, P13 1, P31 1, P33 2D13 1, D15 1, D33 1, D35 0F15 1, F17 0, F35 1, F37 1 total 16 bare N* state (as of today)

# of bare N* stateS11 2, S31 1P11 2, P13 1, P31 1, P33 2D13 1, D15 1, D33 1, D35 0F15 1, F17 0, F35 1, F37 1 total 16 bare N* state (as of today)

17

×

9

Use same cutoff

Meson-exchange amplitude:

Dressed N* propagating amplitude:

EBAC-DCC model: electromagnetic partFor details see Matsuyama, Sato, Lee, Phys. Rep. 439,193 (2007)

= hadronic process

gamma N MB potentials

7 diagramss-ch Nu-ch Nu-ch t-ch t-ch t-ch contact

2 diagramss-ch Nu-ch N

2 diagramss-ch Nu-ch N

4 diagramss-ch Nu-ch Nt-ch contact

Total 32 diagrams

5 diagramss-ch Nu-ch Nu-ch t-ch contact

6 diagramss-ch Nu-ch u-ch t-ch Kt-ch K*contact

6 diagramss-ch Nu-ch u-ch t-ch Kt-ch K*contact

gamma N MB potentialsAll potentials are described in Matsuyama, Sato, Lee, Phys. Rep. 439 193 (2007)

gamma N MB potentials

helicity-JLS mixed representation(angular projection of N part is not needed for our purpose)

gamma N N* bare vertex function

Bare N* helicity amplitude:

for transverse photon

for longitudinal photon

where

Parameterizations of bare helicity amplitudes

Introduce appropriate threshold behavior + dipole form factorIntroduce appropriate threshold behavior + dipole form factor

Plan for EBAC-DCC analysis in 2010

Full combined analysis (global fit) of:

N N (W < 2 GeV)

N N (W < 2 GeV)

N N (W < 1.6 GeV 2 GeV)

N N (W < 2 GeV)

N KY (W < 2 GeV)

N N (W < 2 GeV)

N N (W < 1.5 GeV 2 GeV)

EBAC second generation modelEBAC second generation model

~ End of 2010

2010 ~ 2011

pi- p K0 Lambda

Julia-Diaz, Saghai, Lee, Tabakin PRC73 055204

PreliminaryPreliminary

EBAC-DCC

gamma p K Lambda

PreliminaryPreliminary

gamma p K Lambda

PreliminaryPreliminary

Coupling effect of KY channels on piN PWA

PreliminaryPreliminary

Add KY channels5ch calc.SAID-EDS

P11 Re

P11 Im P13 Im

D13 Im

D15 Im F15 Im

F17 Im

S11 Re

P13 Re

D13 Re

F15 Re

F17 Re

S11 Im

D15 Re

Coupling effect of piN, pipiN, etaN channels on KY observables

Meson-exchange amplitudesMeson-exchange amplitudes

Amplitude with Dressed N*Amplitude with Dressed N*

Coupling effect of piN, pipiN, etaN channels on KY observables

Couplings to N, N, N channels off

Current EBAC-DCC result

PreliminaryPreliminary

(At least) about 20% reduction except backward angles is observed.(At least) about 20% reduction except backward angles is observed.

Back upBack up

pi N pi N PWAs

PreliminaryPreliminary

Add KY channels5ch calc.SAID-EDS

P11 Re

P11 Im P13 Im

D13 Im

D15 Im F15 Im

F17 Im

S11 Re

P13 Re

D13 Re

F15 Re

F17 Re

S11 Im

D15 Re

gamma p K LambdaKamano, Nakamura, Lee, Sato in preparation

Effect of N channels on p KEffect of N channels on p K

Couplings to N (D,N,N) channels are turned off

Current EBAC-DCC result

PreliminaryPreliminary

pi N pi N @ W=1232 MeV

V

Re(T)

Im(T)

EBAC

Juelich

Output is T,

not

pi N pi N @ W=1600 MeV

V

Re(T)

Im(T)

EBAC

Juelich

pi N pi @ W=1232 MeV

V

Re(T)

Im(T)

pi N pi @ W=1600 MeV

V

Re(T)

Im(T)

Meson-exchange amplitude:

Dressed N* amplitude:

EBAC-DCC model: hadronic partFor details see Matsuyama, Sato, Lee, Phys. Rep. 439,193 (2007)

Parameters

29 + 247 (15 bare N*) = 271

nonresonantpotentialnonresonantpotential

N* parametersN* parameters

Hadronic part:

Electromagnetic part:

N* barehelicity amps.N* barehelicity amps.

2 + 39 (15 bare N*) = 41

& couplings & couplings

(roughly 20 parameters for each partial wave)

(roughly 3 parameters for each partial wave)

Pion-nucleon elastic scatteringJulia-Diaz, Lee, Matsuyama, Sato, PRC76 065201 (2007)

　　　　　　　　　　　　　　　　　　　　　 coupled-

channels is considered.

Fitted to the SAID N partial wave amplitudes up to 2GeV.

MINUIT library is employed for the numerical minimization.

Unitarity is satisfied in ~ 1 % !!Unitarity is satisfied in ~ 1 % !!

Re(T) with I = 1/2