Paving to Way for γp→K+K-p: KN Scattering

César Fernández-Ramírez Jefferson Lab

!(Joint Physic Analysis Center)

FDSA Workshop, 20 November 2014, JLab

Table of Contents

Motivation ! γp→K+K-p ! KN Scattering (πN as playground) !

KN Scattering in the Resonance Region ! Poles ! Error-bars

2

Motivation

3

For two kaon photo production Map strangeonia spectrum Hybrids/exotics search

! For KN scattering

Next slides

Dalitz Plot

4

sK+K�

sK�p

fixed s

sK+K�

sK�p

Dalitz Plot

4

sK+K�

sK�p

fixed s

sK+K�

sK�p

Meson

Dalitz Plot

4

sK+K�

sK�p

fixed s

sK+K�

sK�p

Double Regge

Baryon

Meson

Dalitz Plot

5

fixed s

sK+K�

sK�p

Double Regge

Baryon

sK-p

sK+K-

Preliminary  CLAS  g12

C. Salgado, D. Schott

Meson

A Model for γp→K+K-p

6

Low energy fit FR et al.,!

(in preparation)

K�p ! K�p

High energy fit Mathieu et al.,!

(in preparation)

K�p ! K�p

K+

K−

K−

γ

p pω

K+

K− K−

γ

pp

A Model for γp→K+K-p

6

Low energy fit FR et al.,!

(in preparation)

K�p ! K�p

High energy fit Mathieu et al.,!

(in preparation)

K�p ! K�p

K+

K−

K−

γ

p pω

K+

K− K−

γ

pp

Analytical continuation between the two regions via dispersion relations (FESR)

A Model for γp→K+K-p

6

Low energy fit FR et al.,!

(in preparation)

K�p ! K�p

High energy fit Mathieu et al.,!

(in preparation)

K�p ! K�p

K+

K−

K−

γ

p pω

K+

K− K−

γ

pp

Analytical continuation between the two regions via dispersion relations (FESR)

We feed amplitudes to experimentalists and they isolate mesons through PWA

Finite Energy Sum Rules πN (playground)

7

⇡N ! ⇡NDiscuss resonances together with Reggeons !Construct Im(amplitude) from 0 to infinity via FESR !Reconstruct Re(amplitude) from a dispersion relation

ÊÊÊ

Ê

ÊÊÊ

Ê

Ê

Ê

Ê

Ê

ÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊ

Ê

Ê

ÊÊÊÊ

ÊÊÊÊÊ

Ê

Ê

ÊÊÊÊÊÊÊ

ÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊ

ÊÊÊÊÊÊ Ê Ê

Ê ÊÊ Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê

ÊÊÊ

Ê

Ê

Ê

Ê

Ê

ÊÊÊÊÊÊÊÊ

ÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊ

ÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊ Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê

Ê Ê Ê Ê

- t=-0.0 GeV2- t=-0.3 GeV2

0.0 0.5 1.0 1.5 2.0 2.5

-200

-150

-100

-50

0

50

Elab

mb.GeV

2

Re nBH+LHn,tL-- Regge Model-- SAID- New Amplitude

0.0 0.5 1.0 1.5 2.0 2.5 3.0

-150

-100

-50

0

50

100

n HGeVL

mb.GeV

Im nB+Hn, t=0L

Re ⌫B(+)(⌫, t) =g2r2m

2⌫2

⌫2m � ⌫2+

2⌫2

⇡P

Z 1

⌫0

Im B(+)(⌫0, t)

⌫02 � ⌫2d⌫0

in the Resonance Region

8

Coupled channels, analytical and unitary !We can use partial waves !Resonances are incorporated employing relativistic Breit-Wigner !Variation of Zhang et al., PRC 88 (2013) 035205 incorporating analyticity to the amplitudes and adapting for extension to two kaon photo production !Single-energy partial waves from KSU analysis (Manley et al.) are fitted independently !A lot of parameters but also a lot of data points !We can go to the unphysical sheets and get the poles

K̄N ! K̄N,⇡⌃,⇡⇤

Scattering Matrix (p.w.)

9

S = I + 2i T

T = TB +BT TR B

SR = I + 2i TR

SB = BT B = I + 2i TB

K̄N,⇡⌃,⇡⇤, ⌘⇤, ⌘⌃,⇡⌃(1385),⇡⇤(1520), K̄�(1232), K̄⇤N,�⇤,�⌃

Resonant Part: Single Resonance

10

[Ka(s)]jk = tan �a(s)�`,j(s)�`,k(s)

�a(s)x

ajx

ak

Ta(s) = Ka(s) [I � iKa(s)]�1

[Ta(s)]jk =Ma

M

2a � s� iMa�a(s)

�`,j(s)�`,k(s) xajx

ak

has the angular momentum barrier�`,j(s)

is the total width�a(s) =ncX

j

�

2`,j(s) x

aj

Resonant Part: Two Resonances

11

[Kab]jk = tan �a(s) �`,j(s) �`,k(s) xajx

ak + tan �b(s) �`,j(s) �`,k(s) x

bjx

bk

[Tab]jk = caa(s) �`,j(s) �`,k(s) xajx

ak + cab(s) �`,j(s) �`,k(s) x

ajx

bk

+ cba(s) �`,j(s) �`,k(s) xbjx

ak + cbb(s) �`,j(s) �`,k(s) x

bjx

bk

caa(s) =1

C(s)Ma

M

2a � s� iM�a(s)

cbb(s) =1

C(s)Mb

M

2b � s� iM�b(s)

cab(s) = cba(s) =i"ab(s)

C(s)Ma

M

2a � s� iM�a(s)

Mb

M

2b � s� iM�b(s)

C(s) = 1 + ["ab(s)]2 Ma

M

2a � s� iM�a(s)

Mb

M

2b � s� iM�b(s)

"ab(s) =ncX

j

�

2`,j(s) x

ajx

bj

Energy Dependence

12

sm1

m2

m3

m4

�

2`,j(s) =

q̄k(s,m1,m2)

w0B2`

⇥q̄

2k(s,m1,m2)r

2⇤

q̄k(s,m1,m2) =m1m2

(m1 +m2)2⇥s� (m1 +m2)

2⇤

�

2`,j(s) =

�2

2⇡w0

Z ⇤

m3+m4

dx

q̄k(s,m1, x)B2`

⇥q̄

2k(s,m1, x)r2

⇤

(x�m2)2 + (�2/2)2

Background

13

⇥TBp

⇤jk

=✏p Mp

M2p + s� i✏Mp�p(s)

�`,j(s) �`,k(s) ypj y

pk

TBp = (BT

p Bp � I)/2i

B =Y

j

Bp

TBp = (BT

p Bp � I)/2i

Summary of fits (current version)

14

5

TABLE I. Summary of the fitted single-energy partial waves.Notation: nR: number of resonances; nB : number of back-grounds; nC : number of channels; N : number of fitted single-energy points; dof = N − np: degrees of freedom.

LIJ nR nB nC N nP dof χ2/N χ2/dof

S01 4 2 7 360 41 319 2.09 2.36

P01 4 2 6 358 42 316 4.70 5.33

P03 2 2 8 508 36 472 1.45 1.56

D03 3 1 6 372 28 344 1.83 1.98

D05 2 1 5 302 18 284 0.64 0.68

F05 2 1 8 460 27 433 2.14 2.27

F07 1 1 4 208 10 198 0.08 0.09

G07 1 1 6 350 14 336 0.65 0.68

S11 4 3 10 546 77 469 3.08 3.59

P11 2 3 9 546 50 496 1.48 1.63

P13 2 2 11 722 60 662 0.55 0.60

D13 1 2 13 814 42 772 0.77 0.81

D15 2 1 11 714 36 678 1.52 1.60

F15 2 2 12 782 52 730 0.11 0.12

F17 1 1 11 704 24 680 0.37 0.38

G17 1 1 10 580 22 558 0.09 0.09

III. RESULTS

A. Fits to the Single-Energy Partial Waves

The experimental database with s ∈ [2.19, 4.70] GeV2

[9–27] contains approximately 8000 data points for theK̄N → K̄N reaction, 4500 for the K̄N → πΛ reaction,and 5000 for the K̄N → πΣ reaction. This database wasthoroughly analyzed in [4–6] obtaining the single-energypartial waves up to ℓ = 4 and subsequently employingthem to determine the necessary number of resonances,backgrounds and parameters for the multichannel uni-tary model in [2] that we use as starting point for themodel we presented in Section II. To determine the pa-rameters of our model and to be consistent with the anal-ysis in [2] we have used the same amount of resonances,backgrounds and channels per partial wave and we havefitted the same experimental information employing MI-NUIT [28], namely the single-energy partial waves ob-tained in [4–6]. A summary of the fits is provided inTable I. Due to the large amount of parameters (nP inTable I), many fits have been performed with differentstrategies for each partial wave until a sufficiently sat-isfactory solution was obtained. Also, we have imposedin the fits reasonable values for the resonance poles ofwell-established resonances [29], and allowed more free-dom for those resonances that are not well-known. Theobtained χ2/dof’s are generally good except in two casesS01 and P01 partial waves, which also correspond to thefits that were more difficult to perform. On both casesparameters got easily trapped in local minima and fits

needed additional guidance.The full list of parameters can be found in Tables III

–X. Errors have been computed employing the bootstrapmethod [30] which is straightforward and allows to easilypropagate the errors to observables and pole extraction,although it has the caveat of being computationally de-manding. Figures 1–10 compare the model to the single-energy partial waves for isospin 0 K̄N → K̄N (Figs. 1and 2), isospin 1 K̄N → K̄N (Figs. 3 and 4), isospin 0K̄N → πΣ (Figs. 5 and 6), isospin 1 K̄N → πΣ (Figs.7 and 8)), and K̄N → πΛ (Figs. 9 and 10) channels.Although the agreement with data varies from partialwave to partial wave, and from channel to channel, theoverall description of the single-energy partial waves isfairly good. Only two partial waves have high χ2/dof,S01 and P01, although the general features of them arewell captured by the model.In the case of the S01 partial wave the K̄N → K̄N

channel is better described than the K̄N → Σπ, beingthe latter the main source of the χ2 (GET A NUMBER).Fig. 5 shows where most of the χ2 comes from. The rapidvariation of the partial wave between 2.6 and 3 GeV2.Besides, when a rapid variations of the phase shifts

happens, the single-energy partial-wave extraction be-comes very difficult and systematic errors in the experi-mental data can have a a big impact in the extraction.This problem is also present in the P01 partial wave.This is on of the main candidates to explain the dif-

ficulty in obtaining a fully satisfactory match betweenthe model and the single-energy partial waves. Otherpossible reasons are missing resonances in the descrip-tion of the amplitudes and a non-adequate description ofthe background component of the amplitude. The latestis a model-dependent and might be too rigid to give anadequate description of the data.The S01 and P01 partial waves are a major contribu-

tion to the observables and in the lowest energy regionthe dominant one. For this reason, systematics can bemeaningful, specially It is very difficult to account forsystematics in the single-energy extraction and consti-tutes one of theThe high χ2/dof’s can be due to several reasons such as

missing resonances, inadequate description of the back-ground, or inherent problems related to the single-energyextraction like the rapid variation of the partial wave incertain energy regions or an underestimation of the errorbars.S11 partial wave also has a relatively high χ2/dof (3.59)

which can be understood just by looking into the real partof the partial wave in the K̄N → Σπ channel in Fig. 7K̄N → K̄N and K̄N → πΣ channels contribute

equally to the χ2 of P01 partial wave.The model tries to smear the oscillations of the single-

energy multipolesHence the uncertainty on the number of resonances in

the P01 partial waves [29] remains an open problemTheOur work represent a step forward in the development

Fit single energy partial waves from Kent State University analysis of:

!• ~8000 exp. data for • ~4500 exp. data for • ~5000 exp. data for

K̄N ! K̄N

K̄N ! ⇡⌃K̄N ! ⇡⇤

Some Fits:

15

-0.3-0.2-0.1

0 0.1 0.2 0.3

Re S01

0.4 0.5 0.6 0.7 0.8 0.9

1

Im S01

-0.15-0.1

-0.05 0

0.05 0.1

0.15 0.2

0.25

Re P01

-0.1 0

0.1 0.2 0.3 0.4 0.5 0.6

Im P01

-0.1-0.05

0 0.05

0.1 0.15

0.2 0.25

0.3 0.35

0.4

Re P03

-0.05 0

0.05 0.1

0.15 0.2

0.25 0.3

0.35 0.4

0.45 0.5

Im P03

-0.4-0.3-0.2-0.1

0 0.1 0.2 0.3 0.4

2 2.5 3 3.5 4 4.5 5

s (GeV2)

Re D03

-0.1 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7

2 2.5 3 3.5 4 4.5 5

s (GeV2)

Im D03

-0.1-0.05

0 0.05

0.1 0.15

0.2

Re D05

-0.05 0

0.05 0.1

0.15 0.2

0.25 0.3

Im D05

-0.3-0.2-0.1

0 0.1 0.2 0.3 0.4

Re F05

-0.1 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7

Im F05

-0.02-0.01

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

Re F07

-0.02-0.01

0 0.01 0.02 0.03 0.04 0.05

Im F07

-0.05

0

0.05

0.1

0.15

0.2

2 2.5 3 3.5 4 4.5 5

s (GeV2)

Re G07

-0.05 0

0.05 0.1

0.15 0.2

0.25 0.3

0.35

2 2.5 3 3.5 4 4.5 5

s (GeV2)

Im G07

K̄N ! K̄N

Some Fits:

16

K̄N ! ⇡⌃

-0.35-0.3

-0.25-0.2

-0.15-0.1

-0.05 0

0.05 0.1

0.15 0.2

Re S01

-0.15-0.1

-0.05 0

0.05 0.1

0.15 0.2

0.25 0.3

0.35

Im S01

-0.2-0.15

-0.1-0.05

0 0.05

0.1 0.15

0.2

Re P01

-0.4-0.35

-0.3-0.25

-0.2-0.15

-0.1-0.05

0 0.05

0.1

Im P01

-0.05 0

0.05 0.1

0.15 0.2

0.25 0.3

Re P03

-0.05 0

0.05 0.1

0.15 0.2

0.25

Im P03

-0.2

-0.1

0

0.1

0.2

2 2.5 3 3.5 4 4.5 5

s (GeV2)

Re D03

-0.3-0.2-0.1

0 0.1 0.2 0.3 0.4 0.5

2 2.5 3 3.5 4 4.5 5

s (GeV2)

Im D03

-0.1

-0.05

0

0.05

0.1

0.15

Re D05

-0.2

-0.15

-0.1

-0.05

0

0.05

Im D05

-0.2-0.15

-0.1-0.05

0 0.05

0.1 0.15

0.2 0.25

Re F05

-0.35-0.3

-0.25-0.2

-0.15-0.1

-0.05 0

0.05

Im F05

-0.02-0.01

0 0.01 0.02 0.03 0.04 0.05

Re F07

-0.02-0.01

0 0.01 0.02 0.03 0.04 0.05

Im F07

-0.02 0

0.02 0.04 0.06 0.08

0.1 0.12 0.14 0.16 0.18

2 2.5 3 3.5 4 4.5 5

s (GeV2)

Re G07

-0.02 0

0.02 0.04 0.06 0.08

0.1 0.12 0.14 0.16 0.18

2 2.5 3 3.5 4 4.5 5

s (GeV2)

Im G07

Unphysical Sheets

17

IIIII

I

sp

s0p

I

tI� (s)

tII� (s)

tIII� (s)

tI� (s)

�{s}

�{s}

0s1 s2

2N Riemann Sheets

s

s2

s1

I

I

I I

IV

III

Poles, isospin 0

18

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2

Im [s

p1/2 ] (

GeV

)

Re [sp1/2] (GeV)

physical axisKN/Y dR

S01 P01 P03 D03 D05 F05 F07 G07

Poles, isospin 1

19

-0.14

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0

1.2 1.4 1.6 1.8 2 2.2

Im [s

p1/2 ] (

GeV

)

Re [sp1/2] (GeV)

physical axisKN/Y dY/R

S11 P11 P13 D13 D15 F15 F17 G17

Error-bars: Bootstrap

20

Error-bars: Bootstrap

20

1. Randomize the experimental data

Error-bars: Bootstrap

20

1. Randomize the experimental data2. Fit them

Error-bars: Bootstrap

20

1. Randomize the experimental data2. Fit them3. Get a set of parameters

Error-bars: Bootstrap

20

1. Randomize the experimental data2. Fit them3. Get a set of parameters4. Repeat 1→3 until I have enough statistics

Error-bars: Bootstrap

20

1. Randomize the experimental data2. Fit them3. Get a set of parameters4. Repeat 1→3 until I have enough statistics5. Compute observables, poles, …

Error-bars: Bootstrap

20

1. Randomize the experimental data2. Fit them3. Get a set of parameters4. Repeat 1→3 until I have enough statistics5. Compute observables, poles, …6. Apply standard statistical methods

Summary and Future Directions

21

• Reasonable model for KN amplitude • Poles obtained as a by-product (not main objective) !

• Finalize analyticity implementation • Refit single-energy partial waves • Error calculations • Observables • Finite Energy Sum Rules !

• Reggeization of incoming kaon • Photon-Reggeon-Kaon vertex