The Julich Partial-Wave Analysis¨...Deborah Ronchen, PWA 8 / ATHOS 3¨ The J¨ulich Partial-Wave...

The Julich Partial-Wave Analysis

a dynamical coupled-channel analysis of pion- and photon-induced reactions

Deborah Ronchen

HISKP, Bonn University

In collaboration with:

M. Doring, H. Haberzettl, J. Haidenbauer, C. Hanhart, F. Huang, S. Krewald, U.-G. Meißner,

and K. Nakayama

PWA 8 / ATHOS 3April 14, 2015

Motivation and IntroductionData analysis and fit results

Outlook

The excited hadron spectrumThe Julich model

The excited hadron spectrum:Connection between experiment and QCD in the non-perturbative regime

Excited hadron spectrum: testing ground for theories of the strong force at low and medium energy

Experimental study of hadronic reactions

source: ELSA; data: ELSA, JLab, MAMI

⇐⇒

Theoretical predictions of excited hadronse.g. from lattice calculations:(with some limitations)

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

mπ = 396 MeV [Edwards et al., Phys.Rev. D84 (2011)]

⇒ Partial wave decomposition:decompose data with respect to a conservedquantum number:total angular momentum and parity JP

missing resonance problem

→ analyze πN → X and γN → Xsimmultaneously

→ coupled-channel approach

Deborah Ronchen, PWA 8 / ATHOS 3 The Julich Partial-Wave Analysis 2/ 21


Outlook


A dynamical coupled-channel approach: the Julich model

Dynamical coupled-channels (DCC): simultaneous analysis of different reactions

The scattering equation in partial-wave basis

〈L′S′p′|T IJµν |LSp〉 = 〈L′S′p′|V IJ

µν |LSp〉+

∑γ,L′′S′′

∞∫0

dq q2 〈L′S′p′|V IJµγ |L′′S′′q〉

1E − Eγ(q) + iε

〈L′′S′′q|T IJγν |LSp〉

potentials V constructed fromeffective L

s-channel diagrams: T P

genuine resonance states

t- and u-channel: T NP

dynamical generation of polespartial waves strongly correlated



Outlook


The Julich DCC approach

(2-body) unitarity and analyticity respected

Resonances: poles in the T -matrix

Re(E0) = “mass”, -2Im(E0) = “width”

pole position E0 is the same in allchannelsresidues→ branching ratios

opening of inelastic channels

⇒ branch point and new Riemann sheetAnalytic structure (P11)

from PW decomposition, e.g. Vu =1∫

−1

dx P`(x)u −m2

N + iε ,3-body ππN channel:

parameterized effectively as π∆, σN ,ρNπN/ππ subsystems fit the respectivephase shifts

branch points move into complex plane



Outlook


Analysis of pion-induced reactions

calculate observables from T -matrix

fit free parameters of T to data or partial waveamplitudes

σ = 12

4πp2

∑JLS,L′S′ |τ

JL′S′LS |2

with τfi = −π√ρf ρiT fi

ρ: phase factor

s-channel: resonances (T P )

N∗

N π

KΛ

mbare + fπNN∗

t- and u-channel exchange: “background“ (T NP )

K Λ,Σ

π N

K∗

Λ,Σ K

Σ,Σ∗

π N

Λ

Σ K

π N

cut offs Λ in form factors(

Λ2−m2ex

Λ2+~q2

)n

(couplings fixed from SU(3))

⇒ search for poles in the complex energy plane of T

All exchange diagrams data baseDeborah Ronchen, PWA 8 / ATHOS 3 The Julich Partial-Wave Analysis 5/ 21


Outlook


PhotoproductionDifferent approaches

Field theoretical approaches : DMT, ANL-Osaka, Julich-Athens-Washington, ...

Example: Gauge invariant formulation by Haberzettl, Huang and NakayamaPhys. Rev. C56 (1997), Phys. Rrev. C74 (2006), Phys. Rev. C85 (2012)

↓complicated, involved construction/calculation

Focus of the present analysis:

extraction of resonance parameters⇒ flexible, phenomenological parameterization of photo excitation

Advantage: easy to implement, analyze large amounts of dataDisadvantage: no information on microscopic reaction dynamics



Outlook


Photoproduction in a semi-phenomenological approach

Multipole amplitude

M IJµγ = V IJ

µγ +∑κ

T IJµκGκV IJ

κγ

(partial wave basis)

γ

N

π, η

N

Vπγ

γ

N

π, η

N

m

B

VκγG

T

m = π, η , B = N , ∆

Tµκ: Julich hadronic T -matrix → Watson’s theorem fulfilled by construction→ analyticity of T: extraction of resonance parameters

Phenomenological potential:

γ

N

m

B

γ

N

m

B

N ∗,∆∗

PNPµ

PPi

γaµ

+Vµγ =(E, q) =γaµ(q)

mNPNPµ (E) +

∑i

γaµ;i(q)PP

i (E)

E − mbi

γaµ , γa

µ;i : hadronic vertices→ correct threshold behaviour, cancellation of singularity at E = mbi

→ γaµ;i affects pion- and photon-induced production of final state mB

i: resonance number per multipole; µ: channels πN , ηN , π∆ Polynomials



Outlook

Fit resultsResonance parameters

Data analysis and fit results



Outlook


Combined analysis of pion- and photon-induced reactionsSimultaneous fit

Fit parameter:

πN → πNπ−p→ ηn, K 0Λ, K 0Σ0, K+Σ−

π+p→ K+Σ+

s-channel: resonances (T P )

N∗

N π

KΛ

mbare + fπNN∗⇒ 128 free parameters

11 N∗ resonances × (1 mbare+ couplings to πN , ρN , ηN , π∆, KΛ, KΣ))+ 10 ∆ resonances × (1 mbare+ couplings to πN , ρN , π∆, KΣ)

γp→ π0p, π+n, ηp

⇒ up to 456 free parameterscouplings of the polynomials

γ

N

m

B

γ

N

m

B

N ∗,∆∗

PNPµ

PPi

γaµ

+

calculations on the JUROPA supercomputer: parallelization in energy (∼ 300 - 400 processes)



Outlook


Data base simultaneous fit to π- and γ-induced reactions

Fit A Fit B

πN → πN PWA GW-SAID WI08 [Arndt et al., PRC 86 (2012)]

π−p→ ηn dσ/dΩ, P

π−p→ K 0Λ dσ/dΩ, P , β

π−p→ K 0Σ0 dσ/dΩ, P

π−p→ K+Σ− dσ/dΩ

π+p→ K+Σ+ dσ/dΩ, P , β

∼ 6000 data points

γp→ π0p dσ/dΩ, Σ, P , T , ∆σ31, G, H

γp→ π+n dσ/dΩ, Σ, P , T , ∆σ31, G, H

γp→ ηp dσ/dΩ, P , Σ dσ/dΩ, P , Σ, T , F

29,392 data points 29,680 data points

More single/doublepolarization:E , Cx′L , Cz′L ,T ,P,H (ELSA 2014)(γp→ π0p)⇒ predictions

→ T , F : Akondi et al.(A2 at MAMI)PRL 113 10, 102001 (2014)



Outlook


Eta production: π−p → ηn and γp → ηp

Inelastic channels → possibility resolve ”missing resonance” problem

Data quality in π−p→ ηn:

-1 -0.5 0 0.5 1

cosΘ

0

0.05

0.1

0.15

0.2

0.25

dσ

/dΩ

[m

b/s

r]

1507 MeV

-1 -0.5 0 0.5 1

cosΘ

-1

0

1

2

3

P

1793 MeV # data points π−p→ ηn: 732γp→ ηp: 6164

data situation much better in γp→ ηp

⇒ Fix N∗Nη coupling from photoproduction (to a large extent)

γ

N

η

N

N ∗

π

N

η

N

N ∗



Outlook


Fit results γp → ηp selected results, arXiv:1504.01643 [nucl-th]T , F not included T , F included

Differential cross section

0 30 60 90120150

Θ [deg]

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

dσ

/dΩ

[µ

b/s

r]

[1]

1488

0 30 60 90 120150

Θ [deg]

0

0.2

0.4

0.6

0.8

1

[2]

1755

0 30 60 90 120150

Θ [deg]

0

0.1

0.2

0.3

0.4

[3]

[4]

2142

[1] McNicoll 2010 (MAMI), [2] Williams 2009 (JLab), [3] Crede 2009 (ELSA), [4] Crede 2005 (ELSA)

Beam asymmetry

0 30 60 90 120150

Θ [deg]

-1

-0.5

0

0.5

1

Σ

[4]

1496

0 50 100 150

Θ [deg]

-1

-0.5

0

0.5

1

1754

0 50 100 150

Θ [deg]

-1

-0.5

0

0.5

1

[5]1843

[4] Bartalini 2007 (GRAAL), [5] Elsner 2007 (ELSA)

Recoil polarization- only 7 data points in total -

-1

-0.5

0

0.5

1

P

[6] 1561

0 50 100 150

Θ [deg]

-1

-0.5

0

0.5

1

P

1680

[6] Heusch (Caltech),PRL25, 1381 (1970)



Outlook


F and T in γp → ηp arXiv:1504.01643 [nucl-th]

Data: Akondi et al. (A2 at MAMI) PRL 113 10, 102001 (2014)

0

0.5

1497 15161534 1558F

1588 1617

0 30 60 90 120 150-1

-0.5

0

0.5

1

0 30 60 90 120 150

1646 1674

0 30 60 90 120 150 0 30 60 90 120 150 0 30 60 90 120 150 0 30 60 90 120 150

17021743 1796

1847

predictionfit

Polarization:

Beam Target Recoil+1 +x 0−1 +x 0

0

0.5

1497 15161534 1558T

15881617

0 30 60 90 120 150-1

-0.5

0

0.5

1

0 30 60 90 120 150

1646 1674

0 30 60 90 120 150 0 30 60 90 120 150 0 30 60 90 120 150 0 30 60 90 120 150

1702 1743 17961847

Polarization:

Beam Target Recoil0 +y 00 −y 0



Outlook


Pion-induced eta production: π−p → ηn arXiv:1504.01643 [nucl-th]

0.1

0.2

0.3

0.4

dσ

/dΩ

[m

b/s

r]

1509 MeV 1576 MeV 1664 MeV

-1 -0.5 0 0.5 1cosθ

0

0.1

0.2

dσ

/dΩ

[m

b/s

r]

1729 MeV

-1 -0.5 0 0.5 1

cosθ

1805 MeV

-1 -0.5 0 0.5 1

cosθ

2235 MeV

0

0.05

0.1

P x

dσ

/dΩ

[m

b/s

r]

1740 MeV 1872 MeV 1989 MeV

-1 -0.5 0 0.5 1cosθ

0

0.05

0.1

P x

dσ

/dΩ

[m

b/s

r]

2024 MeV

-1 -0.5 0 0.5 1

cosθ

2070 MeV

-1 -0.5 0 0.5 1

cosθ

2235 MeV

fit A: T , F not included fit B: T , F included fit Ahad: Julich2012, only pion-induced data



Outlook


Resonance content: I=1/2 arXiv:1504.01643 [nucl-th]

Pole search on the 2nd sheet ofthe scattering matrix Tµν

Resonance parameter:

”mass” = Re(E0)

”width”= −2Im(E0)

Residues → branchingratios

E0: pole position1000 1200 1400 1600 1800 2000 2200

Re E [MeV]

-300

-250

-200

-150

-100

-50

0

Im E

[M

eV

]

Fit AFit B

1620 1640 1660

-100

-50

Fit AFit B

πN ππN

π∆

KΣ

ρN

N(1720) 3/2+

N(1650) 1/2-

N(1675) 5/2-N(1520) 3/2

-

N(2190) 7/2-

N(1440) 1/2+

N(1680) 5/2+

N(2250) 9/2-

N(1990) 7/2+

N(1710) 1/2+

N(1535) 1/2-

physical axisηN KΛ

N(1750) 1/2+

N(2220) 9/2+

σN

N(1710) 1/2+

N(1675) 5/2-

N(1680) 5/2+

N(1650) 1/2-

x: branch points Notation: N(“name”) J parity



Outlook


Resonance parameters selected results, arXiv:1504.01643 [nucl-th]

Re E0 -2Im E0 |rπN | θπN→πNΓ

1/2πN Γ

1/2ηN

ΓtotθπN→ηN

[MeV] [MeV] [MeV] [deg] [%] [deg]

fit

N(1535) 1/2− A 1497 105 23 −48 51 110

B 1499 104 22 −46 51 112

Ahad 1498 74 17 −37 51 120

N(1710) 1/2+ A 1611 140 2.7 −40 6.1 175

B 1651 121 3.2 55 16 −180

Ahad 1637 97 4 −30 24 130

fit A: T , F not included fit B: T , F included fit Ahad: Julich2012, only pion-induced data



Outlook


Photocouplings at the pole selected results, arXiv:1504.01643 [nucl-th]

Ahpole = Ah

poleeiϑh

h = 1/2, 3/2Ah

pole = IF√

qpkp

2π (2J+1)E0mN rπN

Res AhL±

IF : isospin factorqp (kp ): meson (photon) momentum at the poleJ = L ± 1/2 total angular momentumE0: pole positionrπN : elastic πN residue

A1/2pole ϑ1/2

[10−3 GeV−12 ] [deg]

fit

N(1535) 1/2− A 107 4.6

B 106 5.2

2 50+4−4 −14+12

−10

N(1710) 1/2+ A 7.1 −177

B 20 −83

2 28+9−2 103+20

−6

fit A: T , F not included

it B: T , F included

fit 2: Julich2013, onlypion photoproduction

(same pole positions as fit Ahad)



Outlook


Summary

Extraction of the N∗ and ∆ resonance spectrum

from a simultaneous analysis of pion- and photon-induced reactions

DCC analysis of πN → πN , ηN , KΛ and KΣ

The Julich model: lagrangian based, unitarity & analyticity respected→ analysis of over 6000 data points (PWA, dσ/dΩ, P , β)

π and η photoproduction in a semi-phenomenological approach

hadronic final state interaction: Julich DCC analysis→ analysis of more than 30 000 data points for single and double polarization observables

→ extraction of resonance parameters (poles & residues)



Outlook

Outlook



Outlook

Kaon photoproduction: preliminary results for γp → K+Λ

simultaneous fit of γp→ π0p , π+n , ηp, K+Λ

and πN → πN , ηN , KΛ , KΣ

Differential cross section

0 30 60 90 120150

Θ [deg]

0

0.1

0.2

dσ

/dΩ

[µ

b/s

r] [1]

1645 MeV

0 30 60 90 120 150

Θ [deg]

0

0.1

0.2

0.3

0.4

0.5

1845

0 30 60 90 120150

Θ [deg]

0

0.1

0.2

0.3

0.4

0.5

[2]

2206

[1] McCracken et al. 2010 (JLab), [2] Glander et al. 2004 (ELSA)

Recoil polarization

0 30 60 90 120150

Θ [deg]

-1

-0.5

0

0.5

1

P

[3]

1649

0 30 60 90 120150

Θ [deg]

-1

-0.5

0

0.5

1

[1]

1925

0 30 60 90 120150

Θ [deg]

-1

-0.5

0

0.5

1 2315

[3] Lleres et al. 2007 (GRAAL)

Future projects:

More double polarizationobservables in mesonphotoproduction to bepublished in the near futureγN → KΣ



Outlook

Thank you for your attention!


Photon-induced reactions

Pion-induced reactions πN → πN, ηN, KΛ, KΣ arXiv:1504.01643 [nucl-th]

Selected results: Fit A and Fit B

-0.4

-0.2

0

0.2

0.4

1200 1400 1600 1800 2000

E [MeV]

0.2

0.4

0.6

0.8

1

πΝ −> πΝ

Im S11

Re S11

-1 -0.5 0 0.5 1

cosΘ

0

20

40

60

80d

σ/d

Ω [

µb/s

r]1763 MeV

K+Σ

−

-1 -0.5 0 0.5 1

cosΘ

0

0.05

Pxd

σ/d

Ω [

µb

/sr]

2148 MeV

ηN

-1 -0.5 0 0.5 1

cosΘ

-2

0

2

4

6

β [ra

d]

1852 MeV

KΛ



Multipoles for γp → ηp

-16

-8

0

8

Re

(A)

[10

-3 f

m]

-2

0

-0.8

-0.4

0

-2

-1

0

1

-0.6

-0.3

0

0.3

0.6

0

0.2

0.4

0.6

1600 1800 2000 2200

-16

-8

0

Im(A

) [1

0-3

fm

]

1600 1800 2000 2200

-2

-1

0

1

1600 1800 2000 2200

-0.6

-0.4

-0.2

0

1600 1800 2000 2200

-0.2

0

0.2

1600 1800 2000 2200

-0.6

-0.3

0

1600 1800 2000 2200

-0.6

-0.4

-0.2

0

0.2

-0.2

0

0.2

Re

(A)

[10

-3 f

m]

-0.2

0

0.2

-0.2

0

0.2

-0.1

0

0.1

0.2

-0.03

0

0.03

-0.1

0

1600 1800 2000 2200

-0.4

-0.2

0

Im(A

) [1

0-3

fm

]

1600 1800 2000 2200

-0.2

-0.1

0

0.1

0.2

0.3

1600 1800 2000 2200

-0.2

-0.1

0

0.1

1600 1800 2000 2200

-0.2

-0.1

0

0.1

0.2

1600 1800 2000 2200

-0.08

-0.04

0

1600 1800 2000 2200

-0.05

0

0.05

-0.15

-0.1

-0.05

0

0.05

Re

(A)

[10

-3 f

m]

0

0.1

0.2

-0.04

-0.02

0

0.02

0.04

-0.04

0

0.04

0.08

0

0.02

0.04

0

0.02

1600 1800 2000 2200

-0.03

-0.02

-0.01

0

0.01

Im(A

) [1

0-3

fm

]

1600 1800 2000 2200

-0.1

0

1600 1800 2000 2200

-0.01

0

0.01

0.02

1600 1800 2000 2200

-0.02

0

0.02

0.04

1600 1800 2000 2200

-0.01

-0.005

0

0.005

1600 1800 2000 2200

-0.02

-0.01

0

E0+

M1-

E1+ M

1+E

2-M

2-

E2+

M2+ E

3-M

3- E3+

E4-

M3+

M4-

E4+ M

4+E

5-M

5-

E [MeV]

Figure : (Black) dash-dotted line: BG 2014-02 solution. Dashed (blue) line: fit A; solid(red) line: fit B.



Pion photoproduction: selected fit results arXiv:1504.01643 [nucl-th]

γp→ π0p

0 30 60 90 120150

Θ [deg]

0

0.01

dσ

/dΩ

[µ

b/s

r] [1]

1074 MeV

0 30 60 90 120150

Θ [deg]

-1

-0.5

0

0.5

1

Σ

[2][3][4]

1785

0 30 60 90120150

Θ [deg]

-1

-0.5

0

0.5

1

G

[5]1660

0 30 60 90 120 150 180

Θ [deg]

-4

-2

0

2

-∆σ

31

[6] 1469

[1] Schmidt 2001 (MAMI)[2] Elsner 2009 (ELSA)[3] Sparks 2010 (ELSA)[4] Bartalini 2005 (GRAAL)[5] Thiel 2012 (ELSA)[6] Ahrens 2002 (MAMI)

γp→ π+n

0 50 100 150 200

Θ [deg]

5

10

15

20

dσ

/dΩ

[µ

b/s

r] [7]

1162

0 50 100 150

Θ [deg]

-1

-0.5

0

0.5

1

Σ

[8][9]

1660

0 50 100 150

Θ [deg]

-1

-0.5

0

0.5

1

G

[10]

1232

0 50 100 150

Θ [deg]

-5

0

5

10

-∆σ

31

[11]

1364

[7] Ahrens 2004 (MAMI)[8] Bartalini 2002 (GRAAL)[9] Ajaka 2000 (GRAAL)[10] Ahrens 2005 (MAMI)[11] Ahrens 2006 (MAMI)



Double polarization in γp → π0p selected results, arXiv:1504.01643 [nucl-th]Data NOT included in fit

[1] Gottschall et al. 2013 (ELSA) PRL. 112 1, 012003

Polarization

Beam Target Recoil+1 −z 0−1 −z 0

0 30 60 90 120 150

-1

-0.5

0

0.5

1

E

[1]

0 30 60 90 120 150 0 30 60 90 120 150

E=1456 -1543 MeV

1638 - 1716 1848 - 1898

0 30 60 90 120 150 0 30 60 90 120 150 0 30 60 90 120 150

1898 - 1947 1947 - 1995 2109 - 2280

Θ [deg][2] Hartmann et al. 2014 (ELSA) PRL 113, 062001

-1

-0.5

0

0.5

1

[2]

0 30 60 90 120 150-1

-0.5

0

0.5

1

0 30 60 90 120 150 0 30 60 90 120 150 0 30 60 90 120 150

H

1471 MeV 1491 1513 1533

1553 15741594 1613

1491

Θ [deg]

Polarization

Beam Target Recoil⊥′ x 0‖′ x 0

new ELSA T,P data



T , P in γp → π0pData NOT included in the Fit

Data: J. Hartmann et al. 2014 (ELSA)

-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1

0 30 60 90 120 150-1

-0.5

0

0.5

1

0 30 60 90 120 150 0 30 60 90 120 150 0 30 60 90 120 150

T

P

1471 MeV 1491 1513 1533

15531574

15941613

1471 MeV

1491

1491 1513 1533

1613159415741553

arXiv:1504.01643[nucl-th] back



Partial wave contribution in F and T in γp → ηp arXiv:1504.01643 [nucl-th]

0 30 60 90 120 150-1

-0.5

0

0.5

1

0 30 60 90 120 150

θ [deg]

Fit B, full solution

S11

, P13

, D13

0 30 60 90 120 150

15341646

1847

F

0 30 60 90 120 150-1

-0.5

0

0.5

1

0 30 60 90 120 150

θ [deg]


S11

, P13

, D13

0 30 60 90 120 150

15341646 1847

T



Partial wave contribution in F and T in γp → ηp arXiv:1504.01643 [nucl-th]

0 30 60 90 120 150-1

-0.5

0

0.5

1

0 30 60 90 120 150

θ [deg]


S11

, P13

, D13

all s-, p-, d-, f-waves

0 30 60 90 120 150

15341646

1847

F

0 30 60 90 120 150-1

-0.5

0

0.5

1

0 30 60 90 120 150

θ [deg]


S11

, P13

, D13

all s-, p-, d-, f-waves

0 30 60 90 120 150

15341646 1847

T



Details of the formalism

Polynomials:

PPi (E) =

n∑j=1

gPi,j

(E − E0

mN

)je−gP

i,n+1(E−E0)

PNPµ (E) =

n∑j=0

gNPµ,j

(E − E0

mN

)je−gNP

µ,n+1(E−E0)

- E0 = 1077 MeV

- gPi,j , gNP

µ,j : fit parameter

- e−g(E−E0): appropriatehigh energy behavior

- n = 3

back



Data basesimultaneous fit to πN → πN, ηN, KΛ, KΣ

World data base on ηN , KΛ, KΣ

PWA σtotdσdΩ

P β

πN → πN GWU/SAID 2006

up to J=9/2

π−p → ηn 62 data points 38 energy points 12 energy points

z=1489 to 2235 MeV 1740 to 2235 MeV

π−p → K 0Λ 66 data points 46 energy points 27 energy points 7 energy points

1626 to 1405 MeV 1633 to 2208 MeV 1852 to 2262 MeV

π−p → K 0Σ0 16 data points 29 energy points 19 energy points

1694 to 2405 MeV 1694 to 2316 MeV

π−p → K+Σ− 14 data points 15 energy points

1739 to 2405 MeV

π+p → K+Σ+ 18 data points 32 energy points 32 energy points 2 energy pionts

1729 to 2318 MeV 1729 to 2318 MeV 2021 and 2107 MeV

∼ 6000 data pointsback



back

NN

N N

NN

N

N

N

N

N

N

N N

N N

N N

N N

N N

N

π

π π π

π π

π π

π π π π π π

π

N

N

Δ

Δ

σ ρ

ρ

ρ ρ ρ ρ

N

N

π

ρ

π

N

ω a1

a0

η η

N N N N N Nπ π π π π π

Δ Δ Δ

ΔN ρ

π π π

N

σN σ

π

N

K*

K Λ



back

N N N N N N

N N

π π π π π π

π π

KΛ ΛK KΛ

Σ κ Σ* K*

ΣK

Σ

Σ K Σ K

Λ

κ

Σ ΣK K

Σ*

N

N

N N

N N

N N

N

N N

ρ

ρρ

ρ

ρ

ρ ρ

ρ

ρ

ρ ρ

π π

Δ

π

Δ Δ

N

N

N N

N

η

η

η

η

f0

N N

Δ

Δ

π

π π

π Δ

Δ

ρρ



back

Δ

Δ

Δ

π

π Δπ

π

NNσ σ

σ σ

σ

σ

N

N N

N Λ

Λ

Λ

Λ

f0

K

K

K

K

ω

K

K

Λ

Λ

N

φ

ΛK

K Σ

ρ f0

K

K

Σ

Σ

ω φ ρ

K K K

KKK Σ

Σ Σ

Σ Σ

Σ

Nη

K*

ΛK

N

Λ K

η

Λ K*

K Σ

Nη N

K

η

Σ

Σ

N

K

Σ*

η

Σ Λ K

Ξ

K Λ



Error analysis

χ2 + 1 criterion: determination of the non-linear parameter errorerror of parameter pi determined by range of pi such that χ2

min rises by lessthan 1

⇒ error on pole positions and residues.

1600 1800 2000z [MeV]

-0.1

0

1600 1800 2000z [MeV]

0

0.1

Re F35 Im F35

1740 1760 1780

Re z0

[MeV]

-140

-120

-100

Im z

0 [

MeV

]

NPBA 851, 58 (2011)

BUT: numerically not possible with ≥ 500 free parameters

Work in progress: Developing of techniques to apply Monte-Carlo error propagation usingbootstrap method (M. Doring et al.)



Matching to latticePrediction & analysis of lattice data [M. Doring et al., EPJ A47, 163 (2011)]

Scattering equation:

T (q′′, q′) = V (q′′, q′) +

∞∫0

dq q2 V (q′′, q)1

z − E1(q)− E2(q) + iεT (q, q′)

Discretization of momenta in the scattering equation:∫ ~d 3q(2π)3 f (|~q |2) →

1L3

∑~ni

f (|~qi|2), ~qi =2πL~ni, ~ni ∈ Z3

T (q′′, q′) = V (q′′, q′) +2π2

L3

∞∑i=0

ϑ(i)V (q′′, qi)1

z − E1(qi)− E2(qi)T (qi, q′),

ϑ(P)(i) seriesStudy finite-volume effectsPredict lattice spectra



πN → πN partial wave amplitudes selected results, arXiv:1504.01643 [nucl-th]

Fit A and Fit B

-0.4

-0.2

0

0.2

0.4

0.2

0.4

0.6

0.8

1

1200 1400 1600 1800 2000

E [MeV]

-0.4

-0.2

0

0.2

0.4

1200 1400 1600 1800 2000

E [MeV]

0.2

0.4

0.6

0.8Im P

11

Im S11

Re S11

Re P11

-0.4

-0.2

0

0.2

0.4

0.2

0.4

0.6

0.8

1

1200 1400 1600 1800 2000

E [MeV]

-0.2

-0.1

0

1200 1400 1600 1800 2000

E [MeV]

0.05

0.1

0.15

0.2

0.25

Re P33

Re D33

Im P33

Im D33

Notation: L2I 2J

Input to fit: energy-dependent partial wave analysis, GWU/SAID 2006up to J = 9/2 ( ∼ H39)



πN → ηN,KΛ selected results, arXiv:1504.01643 [nucl-th]

π−p→ ηN

-1 -0.5 0 0.5 1

cosΘ

0

0.05

0.1

0.15

0.2

0.251496 MeV

dσ

/dΩ

[ m

b/s

r ]

-1 -0.5 0 0.5 1

cosΘ

1507 MeV

-1 -0.5 0 0.5 1

cosΘ

1897 MeV

-1 -0.5 0 0.5 1

cosΘ

-0.05

0

0.05

0.1

P x

dσ

/dΩ

[m

b/s

r]

1793 MeV

-1 -0.5 0 0.5 1

cosΘ

2070 MeV

-1 -0.5 0 0.5 1

cosΘ

0

0.05

Pxd

σ/d

Ω [

µb/s

r]

2148 MeV

ηN

π−p→ K 0Λ

-1 -0.5 0 0.5 1

cosΘ

0

40

80

120

dσ

/dΩ

[µ

b/s

r]

z = 1626 MeV

-1 -0.5 0 0.5 1

cosΘ

1707 MeV

-1 -0.5 0 0.5 1

cosΘ

2316 MeV

-1 -0.5 0 0.5 1

cosΘ

-1

-0.5

0

0.5

1

1.5

2

P

1661 MeV

-1 -0.5 0 0.5 1

cosΘ

2159 MeV

-1 -0.5 0 0.5 1

cosΘ

-2

0

2

4

6

β [ra

d]

1852 MeV

KΛ



πN → KΣ selected results, arXiv:1504.01643 [nucl-th]

π−p→ K 0Σ0

-1 -0.5 0 0.5 1

cosΘ

0

20

40

60

80

dσ

/dΩ

[µ

b/s

r]

z = 1694 MeV

-1 -0.5 0 0.5 1

cosΘ

2026 MeV

-1 -0.5 0 0.5 1

cosΘ

-2

-1

0

1

2

P

1724.5 MeV

-1 -0.5 0 0.5 1

cosΘ

2208 MeV

π−p→ K+Σ−

-1 -0.5 0 0.5 1

cosΘ

0

20

40

60

80

dσ

/dΩ

[µ

b/s

r]

1763 MeV

K+Σ

−

-1 -0.5 0 0.5 1

cosΘ

2305 MeV

No polarization data!

π+p→ K+Σ+

-1 -0.5 0 0.5 1

cosΘ

20

40

60

80

100

dσ

/dΩ

[µ

b/s

r]

z = 1729 MeV

-1 -0.5 0 0.5 1

cosΘ

2261 MeV

-1 -0.5 0 0.5 1

cosΘ

2031 MeV

-1 -0.5 0 0.5 1

cosΘ

-2

0

2

4

6

β [ra

d]

z = 2021 MeV



Partial wave contribution to F in γp → ηp arXiv:1504.01643 [nucl-th]

Switch off different PWs in Fit B

0

0.5

-1

-0.5

0

0.5

1

0 30 60 90 120 150-1

-0.5

0

0.5

1

E0+

switched off

0 30 60 90 120 150 0 30 60 90 120 150 0 30 60 90 120 150

1497 15161534 1558

15881617

1646 1674

1702 1743 17961847

F





0

0.5

-1

-0.5

0

0.5

1

0 30 60 90 120 150-1

-0.5

0

0.5

1

M1-

switched off

0 30 60 90 120 150 0 30 60 90 120 150 0 30 60 90 120 150

1497 15161534 1558

15881617

1646 1674

1702 1743 17961847

F





0

0.5

-1

-0.5

0

0.5

1

0 30 60 90 120 150-1

-0.5

0

0.5

1

E1+

, M1+

switched off

0 30 60 90 120 150 0 30 60 90 120 150 0 30 60 90 120 150

1497 15161534 1558

15881617

1646 1674

1702 1743 17961847

F



Photoproduction of pseudoscalar meson

Photocouplings of resonanceshigh precision data from ELSA, MAMI, JLab...→ resolve questionable/find new states

Photoproduction amplitude of pseudoscalar mesons:Chew, Goldberger, Low, and Nambu, Phys. Rev. 106, 1345 (1957)

M = F1~σ · ~ε+ iF2~ε · (k × q) + F3~σ · k q · ~ε+ F4~σ · qq · ~ε

~q: meson momentum~k (~ε): photon momentum

(polarization)

Fi : complex functions of the scattering angle, constructed from multipole amplitudes M IJµγ

⇒ 16 polarization observables:asymmetries composed of beam, target and/or recoil polarization measurements

⇒ Complete Experiment: unambiguous determination of the amplitude

8 carefully selected observables, including Chiang and Tabakin, PRC 55, 2054 (1997)

single and double polarization observbalesmeasurement of beam, target and recoil polarization

easier to realize in K than in π or η photoproduction

→ Caveat: in reality more observables needed (data uncertainties)Deborah Ronchen, PWA 8 / ATHOS 3 The Julich Partial-Wave Analysis 16/ 16

The Julich Partial-Wave Analysis¨...Deborah Ronchen, PWA 8 / ATHOS 3¨ The J¨ulich Partial-Wave...

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