meson-baryon interaction in the hadron exchange picture...baker 78 npb 141, 29 knasel 75 pr11,1...

The analytic structure of the scattering amplitudeUnified analysis of different reactions

Meson-baryon interaction in the hadron exchange picture

M. Doring

H. Haberzettl, J. Haidenbauer, C. Hanhart, F. Huang, S. Krewald,

U.-G. Meißner, K. Nakayama, D. Ronchen,Forschungszentrum Julich, Universitat Bonn, University of Georgia, GWU

NSTAR 2011 Newport News, May 17-20, 2011 Julich analysis 1/ 41


The scattering equationAnalytic structure

The Julich model of pion-nucleon interaction

Motivation

Coupled channels πN , ηN , KY ; effective ππN channels σN , ρN , π∆.

Chiral Lagrangian of Wess and Zumino [PR163 (1967), Phys.Rept. 161 (1988)].

Baryonic resonances up to J = 7/2 with derivative couplings.




Scattering equation in the JLS basis

〈L′S

′k

′|T IJµν |LSk〉 = 〈L′

S′k

′|V IJµν |LSk〉

+∑

γ,L′′S′′

∞∫

0

k′′2

dk′′〈L′

S′k

′|V IJµγ |L′′

S′′k

′′〉 1

Z − Eγ(k′′) + iǫ〈L′′

S′′k

′′|T IJγν |LSk〉

Features

Hadron exchange: provides the relevant dynamics.

Full analyticity (dispersive parts).

All partial waves are linked (t-, u-channel processes)

Channels linked (SU(3) symmetry).

Dynamical generation of resonances is possible, but not easy (strongconstraints).




Partial waves in πN → πN (Solution 2002) “Data”: GWU/SAID, PRC74 (2006)

1200 1400 1600 1800

-0.4

-0.2

0

0.2

0.4

1200 1400 1600 1800 1200 1400 1600 1800 1200 1400 1600 1800

-0.4

-0.2

0

0.2

0.4

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

-0.4

-0.2

0

0.2

0.4

-0.4

-0.2

0

0.2

0.4

1200 1400 1600 1800W [MeV]

0

0.2

0.4

0.6

0.8

1

1200 1400 1600 1800W [MeV]

1200 1400 1600 1800W [MeV]

1200 1400 1600 1800W [MeV]

0

0.2

0.4

0.6

0.8

1

Re S11

Im S11

Re P11

Im P11

Re S31

Im S31

Re P31

Im P31

Re P13

Im P13

Re D13

Im D13

Re P33

Im P33

Re D33

Im D33




Structure of the P11 partial wave (Roper) analytic continuation

N (ππ)L=I=0 channel important( chiral σ ); Roper dynamicallygenerated in Julich model.

Roper pole N (1440) + π∆branch point → non-standardresonance shape.

Sub-threshold cuts fromcrossing symmetry.

Where is the 3∗ N(1710)?[S. Ceci, M.D. et al, arXiv 1104.3490]

1200 1400 1600 1800W/MeV

0

0.2

0.4

0.6

Re,

Im

τ

Fit of a model without ρN branch point

(CMB type) [solid lines] to the Julich

amplitude [dashed lines]

CMB fit to JM has pole at1698 − 130 i MeV, simulatesmissing branch point.

→ Inclusion of full analyticstructure important to avoid falsepole signals.

Branch points in γn → ηn




Poles and residues[M.D., C. Hanhart, F. Huang, S. Krewald and U.-G. Meißner, NPA 829 (2009), PLB 681 (2009)]

Re z0 -2 Im z0 |R| θ [deg]

[MeV] [MeV] [MeV] [0]

N∗(1440) P11 1387 147 48 -64ARN 1359 162 38 -98HOE 1385 164 40CUT 1375±30 180±40 52±5 -100±35

N∗(1520) D13 1505 95 32 -18ARN 1515 113 38 -5HOE 1510 120 32 -8CUT 1510±5 114±10 35±2 -12±5

Re z0 -2 Im z0 |R| θ [deg]

[MeV] [MeV] [MeV] [0]

N∗(1535) S11 1519 129 31 -3ARN 1502 95 16 -16HOE 1487CUT 1510±50 260±80 120±40 +15±45

N∗(1650) S11 1669 136 54 -44ARN 1648 80 14 -69HOE 1670 163 39 -37CUT 1640±20 150±30 60±10 -75±25

N∗(1720) P13 1663 212 14 -82ARN 1666 355 25 -94HOE 1686 187 15CUT 1680±30 120±40 8±12 -160±30

∆(1232) P33 1218 90 47 -37ARN 1211 99 52 -47HOE 1209 100 50 -48CUT 1210±1 100±2 53±2 -47±1

∆∗(1620) S31 1593 72 12 -108ARN 1595 135 15 -92HOE 1608 116 19 -95CUT 1600±15 120±20 15±2 -110±20

∆∗(1700) D33 1637 236 16 -38ARN 1632 253 18 -40HOE 1651 159 10CUT 1675±25 220±40 13±3 -20±25

∆∗(1910) P31 1840 221 12 -153ARN 1771 479 45 +172HOE 1874 283 38CUT 1880±30 200±40 20±4 -90±30

[ARN]: Arndt et al., PRC 74 (2006), [HOE]: Hohler, πN Newsl. 9 (1993), [CUT]: Cutkowski et al., PRD 20 (1979).

Residues to ηN , σN , ρN , π∆. Zeros. Branching ratios to πN , ηN .



Including the KY channels and observablesPhotoproduction & Analysis of lattice data

The reaction π+p → K +Σ

+

Towards a unified analysis of different final states

From the example of the N (1710)P11: different reaction channels providemore information on resonance content than higher precision in onechannel.

SU(3) symmetry provides predictive power.

π+p → K +Σ+: Pure isospin 3/2; relatively few ∆ resonances.

Simultaneous fit to πN → πN partial waves [energy dependent GWU SAID solution, PRC74

(2006)] plus π+p → K +Σ+ observables.

Guiding principle in the fit: Resonances are the last resort (whiletechnically being the easiest way to improve the χ2).

Error analysis on pole positions is crucial.




Inclusion of the KY channelsInclusion via SU(3) symmetry plus s-channel resonance vertices to KY (not shown; 40 parms.).




The reaction π+p → K +Σ

+ Link to full Results & Error analysis

M.D., C. Hanhart, F. Huang, S. Krewald, U.-G. Meißner, D. Ronchen, [NPA851 (2011)]

-1 -0.5 0 0.5cos θ

0

20

40

dσ/d

Ω [

µb

/ sr

]

z = 1813 MeV

-0.5 0 0.5cos θ

z = 2019 MeV

-0.5 0 0.5cos θ

z = 2224 MeV

-1 -0.5 0 0.5cos θ

-0.5

0

0.5

P

z = 1813 MeV

-0.5 0 0.5cos θ

z = 2019 MeV

-0.5 0 0.5cos θ

z = 2224 MeV

Data upper: Candlin 1983, NPB 226 (1983), lower: GWU/SAID, PRC74 (2006)

1200 1400 1600 1800 2000 2200z [MeV]

-0.4

-0.2

0

0.2

0.4

1200 1400 1600 1800 2000 2200z [MeV]

0.2

0.4

0.6

0.8

1Re P33 Im P33

1200 1400 1600 1800 2000z [MeV]

-0.1

0

0.1

0.2

0.3

1200 1400 1600 1800 2000z [MeV]

0.1

0.2

0.3

0.4Im F37Re F37

Linking partial waves and different reactions puts more constraints onresonance content .




Example of other final states [preliminary, no N(1710)P11 needed so far]

0

20

40

60

80

100

120

Baker 78 NPB 141, 29

Knasel 75 PR11,1

Yoder 63, PR 132, 1778

Bertanaz 62, PRL 8, 332

-1 -0.5 0 0.5 10

20

40

60

80

100

120

-1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1

dσ/d

Ω [µ

b/sr

]

cos θ

π−p --> K

0Λ

1633 MeV 1648 MeV 1661 MeV

1671 MeV 1676 MeV 1678 MeV 1682 MeV

z = 1626 MeV




0

20

40

60

80

100

120

140

Knasel 75, PRD 11,1

-1 -0.5 0 0.5 10

20

40

60

80

100

120

140

Baker 78, NPB 141,29

-1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1

π−p --> K

0Λdσ

/dΩ

[µb

/sr]

cos θ

z = 1684 MeV 1686 MeV 1687 MeV 1689 MeV

1693.5 MeV 1698 MeV 1701 MeV 1707 MeV




0

20

40

60

80

100

120

140

Baker 78 NPB 141,29

Binford 69 PR 183,1134Knasel 75 PRD 11,1

-1 -0.5 0 0.5 10

20

40

60

80

100

120

140

-1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1

π−p --> K

0Λdσ

/dΩ

[µb/

sr]

cos θ

z = 1724 MeV 1742 MeV 1758 MeV 1792 MeV

1797 MeV 1819 MeV 1825 MeV 1846 MeV




0

20

40

60

80

100

120

Saxon 80, NPB 162,522

Dahl 67, PR 163, 1430

Yoder 63, PRL 132, 1778

-1 -0.5 0 0.5 10

20

40

60

80

100

120

-1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1

Goussu 66, NCA 42, 607

-1 -0.5 0 0.5 1

π−p-->K

0Λdσ

/dΩ

[µb

/sr]

cosθ

z = 1879 MeV 1909 MeV 1930 MeV 1934 MeV

1938 MeV 1966 MeV 1973 MeV 1978 MeV




0

20

40

60

80

100

120

140

Saxon 80 NPB 162, 522

Dahl 67, PR 163, 1430

-1 -0.5 0 0.5 10

20

40

60

80

100

120

140

-1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1

π−p --> K

0Λdσ

/dΩ

[µb

/sr]

cosθ

z = 1999 MeV 2026 MeV 2059 MeV 2097 MeV

2104 MeV 2137 MeV 2159 MeV 2182 MeV




0

20

40

60

80

100

120

Saxon 80, NPB 162, 522

Dahl 67, PR 163, 1430

-1 -0.5 0 0.5 10

20

40

60

80

100

120

-1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1

π−p --> K

0Λdσ

/dΩ

[µb

/sr]

cosθ

z = 2208 MeV 2244 MeV 2259 MeV 2305 MeV

2316 MeV 2405 MeV




Linking reactions: dσ/dΩ and Σγ for γp → π+nF. Huang, K. Nakayama, M.D., et al., in prep. (see also arXiv:1103.2065)

Gauge invariance respected Further results γN → πN

0

8

16

24

0 60 1200

6

12

18

0 60 120 0 60 120

0

8

16

24

0

9

18

27

36

0 60 120 180

d/d

(p

+ n)

(b/

sr)

(600, 1416) (650, 1449) (725, 1497) (775, 1528)

(825, 1558) (875, 1588) (925, 1617) (975, 1646)

(deg.)

(280, 1186)

(320, 1217)

(250, 1162)

(400, 1277) (450, 1313) (500, 1349) (550, 1383)

(360, 1247)

Differential cross section for γp → π+n

-0.8

0.0

0.8

1.6

0 60 120

-0.8

0.0

0.8

1.6

0 60 120 0 60 120

-0.8

0.0

0.8

1.6

-0.8

0.0

0.8

1.6

0 60 120 180

(p

+ n)

(600, 1416) (650, 1449) (688, 1474) (750, 1513)

(800, 1543) (848, 1572) (900, 1603) (952, 1633)

(deg.)

(275, 1182)

(322, 1219)

(244, 1157)

(400, 1277) (450, 1313) (500, 1349) (550, 1383)

(350, 1240)

Photon spin asymmetry for γp → π+nData: CNS Data analysis center [CBELSA/TAPS, JLAB, MAMI,...]




Dynamical coupled channels models in a box Discretization & twisted BC

[M.D., J. Haidenbauer, A. Rusetsky, U.-G. Meißner, E. Oset, in preparation]

Variation of box size L → reconstruction of phase shifts (Luscher)Examples: Λ(1405), σ(600), f0(980) on the lattice.Prediction of lattice levels & including coming lattice-data in analysis.

2 3

L [mπ-1

]

1300

1400

1500

1600

1700

1800

E [

MeV

]

Uχ (Oset/Ramos, NPA (1997))

Dyn.CC (Muller,..., NPA (1990))

1

2

3 4

0

5

6

Λ(1405)

400

600

800

1000

1200

1400

from chiral-u. (Oller/Oset, NPA 1997)DynCC-approach (Janssen et. al, PRD 1995)

1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6

L [mπ-1

]

800

900

1000

1100

1200

I=L=0 [σ(600), f0(980)]

peri

odic

BC

twis

ted

BC

K_K




Multi-channel dynamicsError propagation from pseudo lattice-data [M.D., U.-G. Meißner, E. Oset, A. Rusetsky, in prep.]

Coupled channels ππ, KK :three unknwons

V (ππ → ππ)V (ππ → KK)V (KK → KK)

How good is the reconstructedphase shift using differentlattice data?

Use pseudo-data generatedfrom hadronic model.

Hadronic input can reduce theerror.

900 1000 1100 1200E [MeV]

50

100

150

200

250

300

δ 00 (π

π-->

ππ)

[de

g]

red: twisted boundary conditions

green: Asymmetric boxes

brown: different levels




Conclusions

Lagrangian based, field theoretical description of meson-baryoninteraction. Unitarity and analyticity are ensured.

Meson and baryon exchange are the relevant degrees of freedom in the 2nd

and 3rd resonance region. Exchange provides constraints, because allpartial waves are linked. With this background, only a few bare states areneeded (cleaner resonance extraction).

⇒ precise determination of model independent resonance parameters(poles).

Branch points in the complex plane/on the real axis may lead to falseresonance signals. Different reaction channels provide more information onresonance content than higher precision in one only.

Coupled channel formalism links different reactions in one combineddescription:

πN → πN , πN → ηN , KY ,..., γN → πN ,....

Dynamical coupled channel models on a momentum lattice (predict levels,error propagation, analyse coming lattice data).


Chiral unitary approach to ππ scattering


Implementation of the chiral σ back


Analytic continuation via Contour deformation...enables access to all Riemann sheets

Πσ(z) =

∞∫

0

q2dq

(vσππ(q))2

z − E1 − E2 + iǫ

z − E1 − E2 = 0 ⇔ q = qc.m.

qc.m. =1

2z

√

[z2 − (m1 − m2)2][z2 − (m1 + m2)2]

Plot qc.m.(z) in the q plane ofintegration (X: Pole positions).

case (a), Im z > 0: straightintegration from q = 0 toq = ∞.

case (b), Im z = 0: Pole is onreal q axis.

case (c), Im z < 0:Deformation gives analyticcontinuation.

Special case: Pole at q = 0⇔ branch point atz = m1 + m2 (= threshold).


Propagator of effective ππN channels σN , ρN , π∆

N

π π

σ σ

1200 1400 1600 1800 2000 2200

-5

-4

-3

-2

-1

0

Re z @MeVD

GΣ

N

gσN(z, k) =

1

z −

√

m2N

+ k2 −

√

(m0σ)2 + k2 − Πσ(zσ(z, k), k)

,

GσN (z) =

∞∫

0

dk k2 F(k) gσN (z, k),

zσ(z, k) = z + m0σ −

√

k2 + (m0σ)2 −

√

k2 + m2N

Re GΣN

Re z=1.75 GeV

-400 0 400

Im z @MeVD

Im GΣN

-400 0 400

Im z @MeVD

Re GΣN

Re z=1.87 GEV

-400 0 400

Im z @MeVD

Im GΣN

-400 0 400

Im z @MeVD

→ Branch point in thecomplex z plane.


Effective ππN channels: Analytic structure back

The cut along Im z = 0 is induced bythe cut of the self energy of theunstable particle.

The poles of the unstable particle (σ)induce branch points in the σN

propagator at

zb2 = mN + z0, zb′2

= mN + z∗0

Three branch points and four sheetsfor each of the σN , ρN , and π∆propagators.


Branch points in coupled channels (γN → ηN ) back

[M.D., K. Nakayama, EPJA43 (2010), PLB683 (2010)]

700 12000

1

2

700 800 900 1000 1100 1200Eγ [MeV]

0

0.5

1

1.5

2

σ n / σ p

[Data: I. Jaegle et al., CBELSA/TAPS, PRL 100 (2008)]

N

γ

B N

π(η)

M

MB = πN , ηN , KΛ, KΣ

Pronounced cusp from dispersive(“real”) part of the loop.

Analyticity is crucial.


Couplings “ g =√

a−1 ” to other channels back

Nπ Nρ(1) (S = 1/2) Nρ(2) (S = 3/2) Nρ(3) (S =N∗(1535) S11 S11 8.1 + 0.5i S11 2.2 − 5.4i − D11 0.5 −N∗(1650) S11 S11 8.6 − 2.8i S11 0.9 − 9.1i − D11 0.3 −N∗(1440) P11 P11 11.2 − 5.0i P11 −1.3 + 3.2i P11 3.6 − 2.6i −∆∗(1620) S31 S31 2.9 − 3.7i S31 0.0 − 0.0i − D31 0.0 +∆∗(1910) P31 P31 1.2 − 3.5i P31 0.2 − 0.4i P31 −0.2 − 0.4i −N∗(1720) P13 P13 3.7 − 2.6i P13 0.1 + 0.8i P13 −1.1 + 0.1i F13 0.1 +N∗(1520) D13 D13 8.4 − 0.8i D13 −0.6 + 0.7i D13 0.9 − 2.0i S13 −2.5 −∆(1232) P33 P33 17.9 − 3.2i P33 −1.3 − 0.8i P33 −0.9 − 3.0i F33 0.0 −∆∗(1700) D33 D33 4.9 − 1.0i D33 −0.2 + 0.9i D33 −0.4 − 0.4i S33 −0.1 −

Nη ∆π(1) ∆π(2) NσN∗(1535) S11 S11 11.9 − 2.3i − D11 −5.9 + 4.8i P11 −1.4 −N∗(1650) S11 S11 −3.0 + 0.5i − D11 4.3 + 0.4i P11 −2.1 −N∗(1440) P11 P11 −0.1 + 0.0i P11 −4.6 − 1.7i − S11 −8.3 −∆∗(1620) S31 − − D31 11.1 − 4.0i −∆∗(1910) P31 − P31 15.0 − 0.3i − −N∗(1720) P13 P13 −7.7 + 5.5i P13 −14.1 + 3.0i F13 0.0 − 0.3i D13 −0.8 +N∗(1520) D13 D13 0.16 − 0.60i D13 0.0 + 0.4i S13 −12.9 − 0.7i P13 −0.6 −∆(1232) P33 − P33 −(4 to 5) + i(0 to 0.5) F33 ∼ 0 −∆∗(1700) D33 − D33 −0.7 − 0.3i S33 −19.7 + 4.5i −

Resonance couplings gi [10−3 MeV−1/2] to the coupled channels i. Also, the LJS type of each

coupling is indicated. For the ρN channels, the total spin S is also indicated.


Zeros and branching ratio to πN , ηN back

first sheet second sheet [FA02]P11 1235 − 0 i S11 1587 − 45 i 1578 − 38 iD33 1396 − 78 i S31 1585 − 17 i 1580 − 36 i

P31 1848 − 83 i 1826 − 197 iP13 1607 − 38 i 1585 − 51 iP33 1702 − 64 i –D13 1702 − 64 i 1759 − 64 i

Position of zeros of the full amplitude T in [MeV]. [FA02]: Arndt et al., PRC 69(2004).

ΓπN /ΓTot [%] ΓηN /ΓTot [%]N∗(1535) S11 48 [33 to 55] 38 [45 to 60]N∗(1650) S11 79 [60 to 95] 6 [3 to 10]N∗(1440) P11 64 [55 to 75] 0 [0 ± 1]∆∗(1620) S31 34 [20 to 30] –∆∗(1910) P31 11 [15 to 30] –N∗(1720) P13 13 [10 to 20] 38 [4 ± 1]N∗(1520) D13 67 [55 to 65] 0.10 [0.23 ± 0.04]∆(1232) P33 100 [100] –∆∗(1700) D33 13 [10 to 20] –

Branching ratios into πN and ηN . The values in brackets are from the PDG,

[Amsler et al., PLB 667 (2008)].


Couplings and dressed vertices back

Residue a−1 vs. dressed vertex Γ vs. bare vertex γ.

Γ(†)D

=

γ(†)B

+

Gγ(†)B TNP

Σ

=

Gγ(†)B ΓD

SD

=

SB

+

SB Σ SD

TP

=

ΓD SD Γ(†)D

a−1 =Γd Γ

(†)d

1 − ∂∂Z

Σ

g =√

a−1

r = |(ΓD − γB)/ΓD|,r

′ = |1 −√

1 − Σ′|,

Dressed Γ depends on T NP.√

a−1 6= Γ 6= γ

γC ΓC r [%] r ′ [%]N∗(1520) D13 6.4 − 0.6i 13.2 + 1.2i 53 61N∗(1720) P13 −0.1 + 5.4i 0.9 + 4.8i 24 45∆(1232) P33 1.3 + 13.0i −2.8 + 22.2i 45 40∆∗(1620) S31 0.1 + 14.3i 5.0 + 5.7i 130 66∆∗(1700) D33 5.4 − 0.8i 6.7 + 1.0i 33 54∆∗(1910) P31 9.4 + 0.3i 1.9 − 3.2i 222 22


Pole repulsion in P33 back

-0.4

-0.2

0

0.2

0.4

0.6

Re

P 33

1100 1200 1300 1400 1500z [MeV]

0

0.2

0.4

0.6

0.8

1

Im P

33

a0 + a

-1/(z-z

0)

a0 + a

-1/(z-z

0)

TNP

TNP

Poles in T NP may occur ⇒ pole repulsion inT = T NP + T P!


Observables back

gfi und hfi in JLS-Basis:

gfi =1

2√

kf ki

∑

j

(2j + 1)d j12

12

(θ)[

τ j(j− 12

) 12 + τ j(j+ 1

2) 1

2

]

cosθ

2

+1

2√

kf ki

∑

j

(2j + 1)d j

− 12

12

(θ)[

τ j(j− 12

) 12 − τ j(j+ 1

2) 1

2

]

sinθ

2

hfi =−i

2√

kf ki

∑

j

(2j + 1)d j12

12

(θ)[

τ j(j− 12

) 12 + τ j(j+ 1

2) 1

2

]

sinθ

2

+i

2√

kf ki

∑

j

(2j + 1)d j

− 12

12

(θ)[

τ j(j− 12

) 12 − τ j(j+ 1

2) 1

2

]

cosθ

2


Observables back

dσ

dΩ=

kf

ki

(|gfi |2 + |hfi|2)

=1

2k2i

1

2·(

∣

∣

∣

∣

∣

∑

j

(2j + 1)(τ j(j− 12

) 12 + τ j(j+ 1

2) 1

2 ) · dj12

12

(Θ)

∣

∣

∣

∣

∣

2

+

∣

∣

∣

∣

∣

∑

j

(2j + 1)(τ j(j− 12

) 12 − τ j(j+ 1

2) 1

2 ) · dj

− 12

12

(Θ)

∣

∣

∣

∣

∣

2)

~Pf =2Re(gfih∗

fi)

|gfi|2 + |hfi |2· n

β = arctan

(

2Im(h∗figfi)

|gfi|2 − |hfi |2)


Differential cross section of π+p → K +Σ

+ back

20

40

60

-1 -0.5 0 0.5 10

20

40

60

-0.5 0 0.5 1 -0.5 0 0.5 1 -0.5 0 0.5 1

dσ/d

Ω [

µb

/ sr

]

z = 1729 MeV 1732 MeV 1757 MeV 1764 MeV

1783 MeV 1789 MeV 1790 MeV 1813 MeV

cos θ



+ back

20

40

60

80

100

-1 -0.5 0 0.5 10

20

40

60

80

100

-0.5 0 0.5 1 -0.5 0 0.5 1 -0.5 0 0.5 1

dσ/d

Ω [

µb

/ sr

]

cosθ

z = 1822 MeV 1845 MeV 1870 MeV 1891 MeV

1926 MeV 1939 MeV 1970 MeV 1985 MeV



+ back

20

40

60

80

100

120

-1 -0.5 0 0.5 10

20

40

60

80

100

120

-0.5 0 0.5 1 -0.5 0 0.5 1 -0.5 0 0.5 1

dσ/d

Ω [

µb

/ sr

]

z = 2019 MeV 2031 MeV 2059 MeV 2074 MeV

2106 MeV 2118 MeV 2147 MeV 2158 MeV

cos θ



+ back

20

40

60

80

100

-1 -0.5 0 0.5 10

20

40

60

80

100

-0.5 0 0.5 1 -0.5 0 0.5 1 -0.5 0 0.5 1

dσ/d

Ω [

µb

/ sr

]

cosθ

z = 2188 MeV 2202 MeV 2224 MeV 2243 MeV

2261 MeV 2282 MeV 2304 MeV 2318 MeV


Polarization of π+p → K +Σ

+

-1.5

-1

-0.5

0

0.5

1

1.5

-1 -0.5 0 0.5 1-1.5

-1

-0.5

0

0.5

1

1.5

-0.5 0 0.5 1 -0.5 0 0.5 1-1 -0.5 0 0.5 1

P

z = 1729 MeV 1732 MeV 1757 MeV 1764 MeV

1783 MeV 1789 MeV 1790 MeV 1813 MeV

cos θ



+

-1.5

-1

-0.5

0

0.5

1

1.5

-1 -0.5 0 0.5 1-1.5

-1

-0.5

0

0.5

1

1.5

-0.5 0 0.5 1 -0.5 0 0.5 1-1 -0.5 0 0.5 1

P

cosθ

z = 1822 MeV 1845 MeV 1870 MeV 1891 MeV

1926 MeV 1939 MeV 1970 MeV 1985 MeV



+

-1.5

-1

-0.5

0

0.5

1

1.5

-1 -0.5 0 0.5 1-1.5

-1

-0.5

0

0.5

1

1.5

-0.5 0 0.5 1 -0.5 0 0.5 1-1 -0.5 0 0.5 1

P

z = 2019 MeV 2031 MeV 2059 MeV 2074 MeV

2106 MeV 2118 MeV 2147 MeV 2158 MeV

cos θ



+

-1.5

-1

-0.5

0

0.5

1

1.5

-1 -0.5 0 0.5 1-1.5

-1

-0.5

0

0.5

1

1.5

-0.5 0 0.5 1 -0.5 0 0.5 1-1 -0.5 0 0.5 1

P

cosθ

z = 2188 MeV 2202 MeV 2224 MeV 2243 MeV

2261 MeV 2282 MeV 2304 MeV 2318 MeV


Error analysis

Determination of the non-linear parameter errorχ2 + 1 criterion.Varying 39 of 40 parameters to get parameter error.

Get error on derived quantities like pole positions and residues.

So far, simplified consideration (error from πN not available, becauseenergy dependent GWU/SAID solution is fitted [PRC74 (2006)]).

1600 1800 2000z [MeV]

-0.1

0

1600 1800 2000z [MeV]

0

0.1

Re F35 Im F35

1740 1760 1780Re z

0 [MeV]

-140

-120

-100

Im z

0 [M

eV]


Error estimates for parameters and derived quantities

Table: Error estimates of bare mass mb and bare coupling f for the ∆(1905)F35

resonance.

mb [MeV] πN ρN π∆ ΣK

2258+44−43 0.0500+0.0011

−0.0012 −1.62+1.29−1.61 −1.15+0.030

−0.022 0.120+0.0065−0.0059

Table: Error estimates of pole position and residues for the ∆(1905)F35 resonance.

πN → πN πN → KΣ

Re z0 [MeV] 1764+18−20 |r | [MeV] 11+1.7

−1.4 1.4+0.24−0.21

Im z0 [MeV] −109+13−12 θ [0] −45+3.8

−11 −313+4.2−10


Resonance content from πN → πN plus π+p → K +Σ

+ back

Re z0 [MeV] |r| [MeV] (Γ1/2

πNΓ

1/2

KΣ)/Γtot [%]

-2 Im z0 [MeV] θ [0] This study Candlin (1983) Gießen (2004)

∆(1905)F35 1764 1.4 1.23 1.5(3) <1

5/2+ **** 218 -313

∆(1910)P31 1721 5.5 2.98 <3 1.1

1/2+ **** 323 -6

∆(1920)P33 1884 5.9 5.07 5.2(2) 2.1(3)

3/2+ *** 229 -38

∆(1930)D35 1865 1.6 2.14 <1.5

5/2− *** 147 -43

∆(1950)F37 1873 2.7 2.54 5.3(5) —

7/2+ **** 206 -255


Coupled channels and gauge invarianceHaberzettl, PRC56 (1997), Haberzettl, Nakayama, Krewald, PRC74 (2006), back


dσ/dΩ and Σγ for γn → π−ppreliminary back

0 60 1200

6

12

18

0 60 120 0 60 120 180

0

9

18

27

0

12

24

36

48

0

9

18

27

0 60 120

(700, 1483) (748, 1513)

(800, 1545) (850, 1575)

(900, 1604) (950, 1633)

(540, 1378)

(640, 1444)

d/d

(n

- p)

(b/

sr)

(270, 1179)

(313, 1213)

(240, 1155)

(390, 1271) (436, 1305) (501, 1351)

(600, 1418)

(360, 1249)

(deg.)

Differential cross section for γn → π−p

-0.8

0.0

0.8

1.6

0 60 120

-0.8

0.0

0.8

1.6

0 60 120 0 60 120

-0.8

0.0

0.8

1.6

-0.8

0.0

0.8

1.6

0 60 120 180

(n

- p)

(670, 1463) (710, 1489) (750, 1514) (790, 1539)

(819, 1557) (850, 1575) (900, 1604) (947, 1632)

(deg.)

(300, 1203)

(330, 1226)

(250, 1163)

(400, 1278) (450, 1315) (500, 1350) (630, 1438)

(350, 1241)

Photon spin asymmetry for γn → π−p


dσ/dΩ and Σγ for γp → π0ppreliminary back

0

2

4

6

0 60 1200

2

4

6

0 60 120 0 60 120

0

7

14

21

0

13

26

39

52

0 60 120 180

(597, 1414) (652, 1450) (705, 1484) (758, 1517)

(808, 1548) (859, 1579) (909, 1608) (957, 1636)

(deg.)

d/d

(p

0 p)

(b/

sr)

(278, 1184)

(320, 1217)

(245, 1158)

(400, 1277) (450, 1313) (510, 1356) (555, 1386)

(360, 1247)

Differential cross section for γp → π0p

-0.8

0.0

0.8

1.6

0 60 120

-0.8

0.0

0.8

1.6

0 60 120 0 60 120

-0.8

0.0

0.8

1.6

-0.8

0.0

0.8

1.6

0 60 120 180

(p

0 p)

(615, 1426) (666, 1459) (703, 1483) (769, 1524)

(801, 1544) (862, 1580) (895, 1600) (961, 1638)

(deg.)

(275, 1182)

(322, 1219)

(244, 1157)

(400, 1277) (450, 1313) (500, 1349) (551, 1384)

(350, 1240)

Photon spin asymmetry for γp → π0p


Dynamical coupled channels models in a box back

Prediction & analysis of lattice data [M.D., J. Haidenbauer, A. Rusetsky, U.-G. Meißner, E. Oset, in preparation]

Discretization of momenta in the scattering equation:

T (q′′, q′) = V (q′′, q

′) +

∞∫

0

dq q2

V (q′′, q)1

z − E1(q) − E2(q) + iǫT (q, q

′)

∫

~d 3q

(2π)3f (|~q |2) → 1

L3

∑

~ni

f (|~qi |2), ~qi =2π

L~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

′),

Can be also expressed in terms of the Luscher Z00 function up to e−L

relativistic corrections.

Takes into account discretization effects of the potentials themselves.

Twisted boundary conditions, e.g.

u(x + Lei) = u(x), d(x + Lei) = d(x), s(x + Lei) = eiθ

s(x),

especially suited for coupled-channels problem (enables to movethresholds) [V. Bernard, M. Lage, U.-G. Meißner, A. Rusetsky, JHEP (2011)].


meson-baryon interaction in the hadron exchange picture...baker 78 npb 141, 29 knasel 75 pr11,1...

Documents