determining the full meson photo-production amplitude from...

CLAS HadronWG-Sept26’09 cover

Determining the full meson photoDetermining the full meson photo--production production amplitude from “amplitude from “completecomplete” experiments” experiments

A.M. Sandorfi1 and S. Hoblit2

1 Thomas Jefferson National Accelerator Facility 2 University of Virginia

• spin-observables in Jπ=0–production and “complete” experiments

Q: can such data provide total amplitudes ?

• the mechanics of fitting out the amplitudes - potentials and limitations - constraining the phase

• tests with mock data – a work in progress

Ideal steps to resonance search-Sept09

Idealized path to search for N*, Δ* states via Jπ=0– photo-production (1) determine the production amp l i t uamp l i t u d ed e from experiment

- search for resonant structure: Analytic Continuation, Argand circles, phase motion, Speed plots, etc. (2) separate resonance and background components

- determine resonant γN* and decay couplings; contact with LQCD, DSE, Hadron models


Idealized path to search for N*, Δ* states via Jπ=0– photo-production (1) determine the production amp l i t u d eamp l i t u d e from experiment



Ne v e r b e e n d o n eNe v e r b e e n d o n e even after 50 yr of exps


Idealized path to search for N*, Δ* states via Jπ=0– photo-production (1) determine the production amp l i t u d eamp l i t u d e from experiment



Ne v e r b e e n d o n eNe v e r b e e n d o n e even after 50 yr of exps

-without exp Amplitudes models have conjectured resonances and adjusted couplings to compare

with limited data

POL gN-KL plane

Polarized Pseudoscalar meson photo-production:

!! +

!N " K +

!#

φ

PL

!

Pc

!

Pz

T

Px

T

PyT

!

!

P!y

"

P!z

"

P!x

"

0- N Pol table w E&Tz

Polarization observables in J π = 0– meson photo-production :

• single-pol observables measured from double-pol asy • double-pol observables measured from triple-pol asy

Target Recoil Target - Recoil

x' y’ z’ x' x' x' y’ y’ y’ z’ z’ z'

Photon beam

x y z x y z x y z x y z

unpolarized σ0 T P Tx’ Lx’ Σ Tz’ Lz’

linearly Pγ Σ H P G Ox’ T Oz’ Lz’ Cz’ Tz’ E F Lx’ Cx’ Tx’

circular Pγ F E Cx’ Cz’ Oz’ G H Ox’

0- N Pol table w E&Tz

Polarization observables in J π = 0– meson photo-production :



x' y’ z’ x' x' x' y’ y’ y’ z’ z’ z'

Photon beam





E Tz’

Chaing&Tabakin – in practice

Requirements for a complete determination of the amplitude, unique to within a phase and free of ambiguities: ⇔ 8 carefully chosen observables - Barker, Donnachie, Storrow, NP B95(75) - Chiang & Tabakin, PRC55,2054(97) (I) cross section and the three single-polarization observables:

{σ , Σ , P, T} (II) four double-polarization observables:

• 2 + 2 select cases from: Beam-Target, Beam-Recoil, Target-Recoil

• 2 + 1 + 1 select cases from: Beam-Target, Beam-Recoil, Target-Recoil

Chaing&Tabakin – in practice





• in practice, one needs linear and circular beam polarization, at least one orientation of target polarization and recoil polarization

Chaing&Tabakin _practice-principle





• in practice, one needs linear and circular beam polarization, at least one orientation of target polarization and recoil polarization

⇔ by the time you accumulate that in a 4π detector, you have all 16

⇒ ∃ huge redundancy, at least in principle!

gp-K+L JLab tbl

! + p" K+# series of JLab experiments:


x' y’ z’ x' x' x' y’ y’ y’ z’ z’ z'

Photon beam





status CLAS run period

beam target Coordinator/spokesman

complete g1 ! , !!c LH2 Miskimen/Schumacher

complete g8

!!L

LH2 Cole

complete g9a - PTz

!!L

, !!c FROST -

C4

!H9O!H Klein, Pasyuk

2010 g9b - PTx

!!L

, !!c FROST -

C4

!H9O!H Klein, Pasyuk

Full set of 16Full set of 16

gp-K0L g13_HDice tbl

! + n(p)" K0# experiments at JLab with HDice:



x' y’ z’ x' x' x' y’ y’ y’ z’ z’ z'

Photon beam





⇑ simultaneous B-R with HD ⇔

schedule CLAS run period

beam target spokesmen

2006-2007 g13

!!L LD2 Nadel-Turonski

2010-2011 g14

!!L

, !!c

H !!D ice Sandorfi/Klein

Full set of 16Full set of 16

Mechanics I multipoles->observables

The mechanics of inferring amplitudes from data (I) theory ⇒ experiment: eg. generating mock data

d!

d"=

P# ,$ ,KCM

E%cm & N(# ,$,K) F

i % N'

2

, CGLN, PR106(1957)

↑ (Cartesian) ⇔ (Spherical) Hi helicity amplitudes

multipoles ⇒ observables

• E!±

, M!±{ }! P

! , "P

! , ""P

! { } ⇔ functions of W / E

!{ }" functions of #CM{ }

⇒ F1 , F2 , F3 , F4{ }CGLN

⇔ functions of (W / E! , "

CM){ }

⇒ ! 0 , " , T , P , E , ... L#z{ } ⇔ functions of (W / E

! , "

CM , # ){ }

-3.0

-2.0

-1.0

0.0

1.0

2.0

1.6 1.7 1.8 1.9 2 2.1 2.2 2.3

RE[E,M,O,1] K0L-m2soln

Re{E0+] K0L (mF)Re[E1+] K0L (mF)Re[E1-] K0L (mF)Re[M1+] K0L (mF)Re[M1-] K0L (mF)

Re{E/M](mF)

W (GeV)

γ n → Ko Λ

-1.0

-0.5

0.0

0.5

1.0

1.6 1.7 1.8 1.9 2 2.1 2.2 2.3

Im[E,M,O,1] K0L-m2soln

Im{E0+] K0L (mF)Im[E1+] K0L (mF)Im[E1-] K0L (mF)Im[M1+] K0L (mF)Im[M1-] K0L (mF)

Im{E/M](mF)

W (GeV)

Im[E0+] / 5

γ n → Ko Λ

multipoles from: B. Julia-Diaz, T-S. H. Lee - M2 (full) solution -

eg.

-3.0

-2.0

-1.0

0.0

1.0

2.0

1.6 1.7 1.8 1.9 2 2.1 2.2 2.3



Re{E/M](mF)

W (GeV)

γ n → Ko Λ

-1.0

-0.5

0.0

0.5

1.0

1.6 1.7 1.8 1.9 2 2.1 2.2 2.3


Im{E0+] K0L (mF)Im[E1+] K0L (mF)Im[E1-] K0L (mF)Im[M1+] K0L (mF)Im[M1-] K0L (mF)

Im{E/M](mF)

W (GeV)

Im[E0+] / 5

γ n → Ko Λ

multipoles from: B. Julia-Diaz, T-S. H. Lee - M2 (full) solution -

eg.

W = 1.95 GeV W = 1.95 GeV

-1.0

-0.5

0.0

0.5

1.0

-1 -0.5 0 0.5 1

Argand E2- K0LIm[E2-] K0Λ-M2soln (mF)

Im[E2-] K0Λ-M2soln-noD13 (mF)

Im[E2-](mF)

Re[E2-] (mF)

γ n → Κο Λ

W = 1.95 GeV

W

Constructing CGLN Fi amplitudes

Constructing the CGLN Fi amplitudes: F1 (x) = E0+ + M1+ + E1+( ) !P2

" (x)

+ 2M2+ + E2+( ) !P3" (x) + 3M2# + E2#( )

+ 3M3+ + E3+( ) !P4" (x) + 4M3# + E3#( ) !P2

" (x)

+ 4M4+ + E4+( ) !P5" (x) + 5M4# + E4#( ) !P3

" (x)

F3 (x) = + E1+ ! M1+( ) "P2## (x)

+ E2+ ! M2+( ) "P3## (x)

+ E3+ ! M3+( ) "P4## (x) + E3! + M3!( ) "P2

## (x)

+ E4+ ! M4+( ) "P5## (x) + E4! + M4!( ) "P3

## (x)

F2 (x) = 2M1+ + M1!( ) + 3M2+ + 2M2!( ) "P2

# (x)

+ 4M3+ + 3M3!( ) "P3# (x) + 5M4+ + 4M4!( ) "P4

# (x)

F4 (x) = M2+ ! E2+ ! E2! ! M2!( ) "P2## (x)

+ M3+ ! E3+ ! E3! ! M3!( ) "P3## (x)

+ M4+ ! E4+ ! E4! ! M4!( ) "P4## (x)

P! , P

!! , P

!!! " Legendre functions

x = cos(!CM)

-20.0

-15.0

-10.0

-5.0

0.0

5.0

10.0

15.0

0 30 60 90 120 150 180

Re[CGLN Fi] at Eg 1550

Re[F1] (mF)Re[F2] (mF)Re[F3] (mF)Re[F4] (mF)

Re F(i)(mF)

θCM

(deg)

γ n → Ko Λ

Eγ = 1.55 GeVW = 1.95 GeV

-20.0

-15.0

-10.0

-5.0

0.0

5.0

10.0

15.0

0 30 60 90 120 150 180

Im[CGLN Fi] at Eg 1550

Im[F1] (mF)Im[F2] (mF)Im[F3] (mF)Im[F4] (mF)

Im F(i)(mF)

θCM

(deg)

γ n → Ko Λ

Eγ = 1.55 GeVW = 1.95 GeV

CGLN amplitudes at W = 1.95 GeV- calculated from the multipoles of B. Julia-Diaz, T-S. H. Lee

eg.

hat notation

notation: Aasy = !o" Aasy

eg:

E = !Anti

" !Par

!Anti

+!Par

!o= 1

2!Anti

+!Par( )

# E = 1

2!Anti

" !Par( )

$

%

&&&&

'

&&&&

F1 , F2 , F3 , F4{ }CGLN

⇔ functions of (W / E! , "

CM){ }

⇒ ! 0 , " , T , P , E , ... L

#z{ } ⇔ functions of (W / E!

, "CM

, # ){ }

n γc

Rc

Lc

Λ

Κo

Observables in terms of the CGLN Fi amplitudes, i = 1-4 : ! = P" ,# ,KCM

E$

cm

- Andy & Harry’s signs – confirmed Aug 22’09.

!0=

F1

2

+ F2

2

+ 1

2sin

2 " # F3

2

+ F4

2( )+$e sin2 " # F

2

*F3+ F

1

*F4+ cos" # F

3

*F4( ) % 2cos" # F

1

*F2

&' ()

*

+,

-,

.

/,

0,# 1

! = " 1

2sin

2 # $ F3

2

+ F4

2{ } + sin2 # $%e F2

*F3+ F

1

*F4+ cos# $ F

3

*F4( ){ }&

'()$ *

T = !m sin" F1

*F3# F

2

*F4+ cos" $ F

1

*F4# F

2

*F3( ) # sin2 " $ F

3

*F4

%& '({ } $ )

P = !m sin" #2F1

*F2# F

1

*F3+ F

2

*F4+ cos" $ F

2

*F3# F

1

*F4( ) + sin2 " $ F

3

*F4

%& '({ } $ )

Tx '= !e sin2 " #F

1

*F3# F

2

*F4# F

3

*F4# 1

2cos" $ F

3

2

+ F4

2( )%&

'({ } $ )

Lx '= +!e sin" F

1

2

# F2

2

+ 1

2sin

2 " $ F4

2

# F3

2( ) # F2*F3 + F1*F4 + cos" F1

*F3# F

2

*F4( )%

&'({ } $ )

Tz '= !e sin" #F

2

*F3+ F

1

*F4+ cos" F

1

*F3# F

2

*F4( ) + 1

2sin

2 " $ F4

2

# F3

2( )%&

'({ } $ )

Lz '= !e

2F1

*F2" cos# F

1

2

+ F2

2( ) + sin2 # $ F1

*F3+ F

2

*F4+ F

3

*F4( )

+ 1

2cos# sin2 # $ F

3

2

+ F4

2( )

%

&'

('

)

*'

+'$ ,

E = ! ! F1

2

! F2

2

+"e 2cos# $ F1

*F2( ) ! sin2 # $ F

2

*F3+ F

1

*F4( ){ }%

&'( $ )

G = + sin2 ! " #m F

2

*F3+ F

1

*F4{ } " $

F = sin! "#e F1

*F3$ F

2

*F4$ cos! " F

2

*F3$ F

1

*F4( )%& '( " )

H = ! sin" # $m 2F1

*F2+ F

1

*F3! F

2

*F4+ cos" # F

1

*F4! F

2

*F3( )%& '( # )

Multipole fit to mock data

Simulating the extraction of multipoles (a work in progress):

• create mock data:

- using multipoles from EBAC-Juliá-Días/Lee/Sato “M2”amplitude, generate observables with appropriate energy and angular coverage - Gaussian smear the predictions with the expected statistical widths

Multipole fits to mock data-summary

Simulating the extraction of multipoles (a work in progress):

• create mock data:

- using multipoles from EBAC-Juliá-Días/Lee/Sato “M2”amplitude, generate observables with appropriate energy and angular coverage - Gaussian smear the predictions with the expected statistical widths

• fitting mock data, varying multipoles:

(1) limiting case:limiting case: - generate observables in 10 deg steps from 100 to 1700 CM; assign each an error of 0.1% and Gaussian smear the predictions (Note: no data set will ever be that good !) a) search from a starting point within ~25% of the generating soln ⇒ converges rapidly to EBAC “M2” multipoles ✔ b) as a starting point for the search, use Born values; Monte Carlo sampling of parameter space, then Gradient search

-2.0

-1.0

0.0

1.0

2.0

Re[E,M,O,1] K0L-m2,10^9Born

Re[A]

(mF)M

1-

M1+

E1+E0+

M2 109+Grad

-1.0

0.0

1.0

Re[E,M,L2] K0L-m2,10^9Born

Re[A]

(mF)

M2-

M2+

E2-

E2+

M2 109+Grad-0.5

0.0

0.5


Re[A]

(mF)

M3-

M3+

E3-

E3+

M2 109+GradRe[E,M,L4] K0L-m2,10^9Born

Re[A]

(mF)M

4-M

4+

E4-

E4+

M2 109+Grad

-2.0

-1.0

0.0

1.0

2.0

Im[E,M,O,1] K0L-m2,10^9Born

Im[A]

(mF)

M1-

M1+

E1+

E0+

M2 109+Grad

-1.0

0.0

1.0

Im[E,M,L2] K0L-m2,10^9Born

Im[A]

(mF)

M2-

M2+

E2-

E2+

M2 109+Grad-0.5

0.0

0.5


Im[A]

(mF)M3-

M3+

E3-

E3+

M2 109+GradIm[E,M,L4] K0L-m2,10^9Born

Im[A]

(mF)M

4-M

4+

E4-

E4+

M2 109+Grad

n(γ, ΚοΛ)→ → →

L = 0,1 L = 2 L = 3 L = 4

EBAC M2 vs Born + 109 MC + gradient search

W = 1.95 GeVχ2/df = 0.996 - better than M2 to mock data

M2 vsBorn+10^9MC+Grad freefit

-2.0

-1.0

0.0

1.0

2.0

Re[E,M,O,1] K0L-m2,10^9Born

Re[A]

(mF)M

1-

M1+

E1+E0+

M2 109+Grad

-1.0

0.0

1.0


Re[A]

(mF)

M2-

M2+

E2-

E2+

M2 109+Grad-0.5

0.0

0.5


Re[A]

(mF)

M3-

M3+

E3-

E3+

M2 109+GradRe[E,M,L4] K0L-m2,10^9Born

Re[A]

(mF)M

4-M

4+

E4-

E4+

M2 109+Grad

-2.0

-1.0

0.0

1.0

2.0

Im[E,M,O,1] K0L-m2,10^9Born

Im[A]

(mF)

M1-

M1+

E1+

E0+

M2 109+Grad

-1.0

0.0

1.0


Im[A]

(mF)

M2-

M2+

E2-

E2+

M2 109+Grad-0.5

0.0

0.5


Im[A]

(mF)M3-

M3+

E3-

E3+

M2 109+GradIm[E,M,L4] K0L-m2,10^9Born

Im[A]

(mF)M

4-M

4+

E4-

E4+

M2 109+Grad

n(γ, ΚοΛ)→ → →

L = 0,1 L = 2 L = 3 L = 4

EBAC M2 vs Born + 109 MC + gradient search

W = 1.95 GeVχ2/df = 0.996 - better than M2 to mock data

new soln = M2 x ei(136.04 deg)

M2 vsBorn+10^9MC+Grad freefit


- χ2 surface has a large number of local minima !

- in general, fitted soln will be rotated from generating amplitude by some angle dependent on the statistical fluctuations in data


- χ2 surface has a large number of local minima !

- in general, fitted soln will be rotated from generating amplitude by some angle dependent on the statistical fluctuations in data

c) fixing the common phase with high L Born :

• S!"

= 1+ 2i T!"

, for reactions ! " #

• T!" = T!"LPL

L

# $ %T!" = ei&!" (w)

T!"LPL

L

#

• Oexp = cT!"#2

$ cei%!"T!"#

2

- vary multipoles for all L that are expected to differ from Born (eg. L=0-3); include higher L multipoles fixed to their Born values

⇒ uniquely finds the generating amplitude !

(but requires sufficient statistics for all non-Born multipoles)


- realistic case: the data will run out of statistical accuracy before the multipoles run out of non-Born components;

(eg. stat sufficient to fit L=0-2, but L=3-4 may be non-Born; fixing L=5-8 to Born DOES NOT determine the phase !)

- alternatively, lock phase of one multipole to something regarded as “known” ⇔ not a trivial choice for γN → ΚΛ

-3.0

-2.0

-1.0

0.0

1.0

2.0

1.6 1.7 1.8 1.9 2 2.1 2.2 2.3



Re{E/M](mF)

γ n → Ko Λ

-1.0

-0.5

0.0

0.5

1.0

1.6 1.7 1.8 1.9 2 2.1 2.2 2.3


Im{E/M](mF)

W (GeV)

Im[E0+] / 5

-5.0

-4.0

-3.0

-2.0

-1.0

0.0

1.0

1600 1700 1800 1900 2000 2100 2200 2300

RE[E,M,O,1] K0L-KMAID

Re[E0+] K0L (mF)Re[E1+] K0L (mF)Re[E1-] K0L (mF)Re[M1+] K0L (mF)Re[M1-] K0L (mF)

Re{E/M](mF)

γ n → Ko Λ

-1.0

-0.5

0.0

0.5

1.0

1600 1700 1800 1900 2000 2100 2200 2300

Im[E,M,O,1] K0L-KMAID

Im{E/M](mF)

W (MeV)

Im[E0+] / 5

Im[M1+] / 3

EBAC(M2) Kaon-MAID

K0L multipoles - KMAID vs EBAC(M2)

-180

-135

-90

-45

0

45

90

135

180

1.6 1.7 1.8 1.9 2 2.1 2.2 2.3

phase[E0+] M2'vsKMAID EBAC[m2]KaonMAID

phas

e[E0

+]

(deg

)

W (GeV)

γ n → Κο Λ

δ [E0+]

-180

-135

-90

-45

0

45

90

135

180

1.6 1.7 1.8 1.9 2 2.1 2.2 2.3

phase[M1-] m2'vsKMAIDEBAC[m2]KaonMAID

phas

e[M

1-]

(de

g)W (GeV)

δ [M1-]

γ n → Κο Λ

K0L phases - KMAIDvsEBAC(m2)


- realistic case: the data will run out of statistical accuracy before the multipoles run out of non-Born components;

(eg. stat sufficient to fit L=0-2, but L=3-4 may be non-Born; fixing L=5-8 to Born DOES NOT determine the phase !)

- alternatively, lock phase of one multipole to something regarded as “known” ⇔ not a trivial choice for γN → ΚΛ (2) Realistic case:Realistic case: - using EBAC(M2) multipoles L=0-4, generate observables for energies of JLab experiments with expected angular coverage; Gaussian smear predictions with the expected statistical widths.

- start multipole search at non-resonant Born values;

- lock phase of M(1-), or E(0+), to EBAC(M2) values;

- 107 Monte Carlo sample of parameter space for L=0-3; Gradient search whenever χ2 is within 104 of best value;

-1.5

-1.0

-0.5

0.0

0.5

1.0

-1 -0.5 0 0.5 1

Argand E2- K0L[M2,nonRes]

K0Λ - EBAC[M2]

K0Λ - nonRes Born

Im[E2-](mF)

Re[E2-] (mF)

γ n → Κο Λ

W = 1.95 GeV

W

E2- multipole of EBAC(m2) with D13(1950) resonance

Argand[E2-] EBAC,kMAIDvsFITw M1- phLk

-1.5

-1.0

-0.5

0.0

0.5

1.0

-1 -0.5 0 0.5 1


K0Λ - EBAC[M2]

K0Λ - nonRes Born

Im[E2-](mF)

Re[E2-] (mF)

γ n → Κο Λ

W = 1.95 GeV

W

-1.5

-1.0

-0.5

0.0

0.5

1.0

-1 -0.5 0 0.5 1

Argand[E2-]K0L PhM1-(EBAC)

K0Λ - EBAC[M2]

K0Λ- fit phase-locked to EBAC[M1-]

Re[E2-] (mF)

γ n → Κο ΛIm[E2-]

(mF)

W = 1.95 GeV

W

E2- multipole of EBAC(M2) with D13(1950) resonance

fit to expected JLab data, with phase locked to M1- of EBAC(m2)


-1.5

-1.0

-0.5

0.0

0.5

1.0

-1 -0.5 0 0.5 1


K0Λ - EBAC[M2]

K0Λ - nonRes Born

Im[E2-](mF)

Re[E2-] (mF)

γ n → Κο Λ

W = 1.95 GeV

W

-1.5

-1.0

-0.5

0.0

0.5

1.0

-1 -0.5 0 0.5 1

Argand[E2-]K0L PhM1-(m2,kM)

Κ0Λ - EBAC[m2]

Κ0Λ fit phase-locked to EBAC[M1-]

Κ0Λ - fit phase-locked to kMAID[M1-]

Re[E2-] (mF)


(mF)

W = 1.95 GeV

W




fit to expected JLab data, with phase locked to M1- of KaonMAID

-1.5

-1.0

-0.5

0.0

0.5

1.0

-1 -0.5 0 0.5 1


K0Λ - EBAC[M2]

K0Λ - nonRes Born

Im[E2-](mF)

Re[E2-] (mF)

γ n → Κο Λ

W = 1.95 GeV

W

-1.5

-1.0

-0.5

0.0

0.5

1.0

-1 -0.5 0 0.5 1

Argand[E2-]K0L PhM1-(m2,kM)

Κ0Λ - EBAC[m2]

Κ0Λ fit phase-locked to EBAC[M1-]

Κ0Λ - fit phase-locked to kMAID[M1-]

Re[E2-] (mF)


(mF)

W = 1.95 GeV

W



Argand[E2-] EBAC,kMAID vs fit w M1- phLk

fit to expected JLab data, with phase locked to M1- of KaonMAID

χ2df{fits to mock data} ~ 1.2

~ 15% less than χ2df{mock data to generating Amp}

-1.5

-1.0

-0.5

0.0

0.5

1.0

-1 -0.5 0 0.5 1


K0Λ - EBAC[M2]

K0Λ - nonRes Born

Im[E2-](mF)

Re[E2-] (mF)

γ n → Κο Λ

W = 1.95 GeV

W

-1.5

-1.0

-0.5

0.0

0.5

1.0

-1 -0.5 0 0.5 1

Argand[E2-]K0L PhE0+(m2,kM)

ΚοΛ - EBAC[m2]

ΚοΛ - fit phase-locked to EBAC[E0+]

ΚοΛ - fit phase-locked to kMAID[E0+]

Re[E2-] (mF)


(mF)

W = 1.95 GeV

W


fit to expected JLab data, with phase locked to E0+ of EBAC(m2)

Argand[E2-] EBAC,kMAIDvsFIT w E0+phLk

fit to expected JLab data, with phase locked to E0+ of KaonMAID

-180

-135

-90

-45

0

45

90

135

180

1.6 1.7 1.8 1.9 2 2.1 2.2 2.3


phas

e[E0

+]

(deg

)

W (GeV)

γ n → Κο Λ

δ [E0+]

-1.5

-1.0

-0.5

0.0

0.5

1.0

-1 -0.5 0 0.5 1


ΚοΛ - EBAC[m2]



Re[E2-] (mF)


(mF)

W = 1.95 GeV

W



Argand[E2-] EBAC,kMAIDfit w E0+ph


-180

-135

-90

-45

0

45

90

135

180

1.6 1.7 1.8 1.9 2 2.1 2.2 2.3


phas

e[E0

+]

(deg

)

W (GeV)

γ n → Κο Λ

δ [E0+]

-1.5

-1.0

-0.5

0.0

0.5

1.0

-1 -0.5 0 0.5 1


ΚοΛ - EBAC[m2]



Re[E2-] (mF)


(mF)

W = 1.95 GeV

W



Argand[E2-] EBAC,kMAIDvsFIT w E0+phLk


χ2df {fits to mock data} ~ 2-4→ χ2 valleys very narrow !

-0.5

0.0

0.5

-1 -0.5 0 0.5

Argand M1+ K0L[M2,nonRes] Κ0Λ - EBAC[m2]

Κ0Λ - Born

Im[M1+](mF)

Re[M1+] (mF)

W = 1.72 GeV

W

γ n → Κο Λ

-0.5

0.0

0.5

-1 -0.5 0 0.5

Argand M1+ K0L[M2,Res]

Κ0Λ - EBAC[m2, resonant part]

Im[M1+](mF)

Re[M1+] (mF)

γ n → Κο Λ

W = 1.72 GeV

W

M1+ multipole of EBAC(M2) with P13(1720) resonance

Argand[M1+] EBAC(m2)

- **** P13 with small resonant part and large background;

- EBAC PRC73 (06); not seen in latest EBAC analysis (09)

-0.5

0.0

0.5

-1 -0.5 0 0.5

Argand M1+ K0L[M2,nonRes] Κ0Λ - EBAC[m2]

Κ0Λ - Born

Im[M1+](mF)

Re[M1+] (mF)

W = 1.72 GeV

W

γ n → Κο Λ

-0.5

0.0

0.5

-1 -0.5 0 0.5

Argand[M1+]K0L PhE0+(m2,kM)

Κ0Λ - EBAC[m2]

Κ0Λ fit phase-locked to EBAC[E0+]

Κ0Λ - fit phase-locked to kMAID[E0+]

Re[M1+] (mF)

Im[M1+](mF)

W

W = 1.75 GeV

γ n → Κο Λ



Argand[M1+] EBAC,kMAID vs fit w E0+ phLk


χ2df ~ 2-4 for W >2050

-0.5

0.0

0.5

-1 -0.5 0 0.5

Argand[M1+]K0L PhE0+ M2st

Κ0Λ - EBAC[m2]


Re[M1+] (mF)

Im[M1+](mF)

W

W = 1.75 GeV

γ n → Κο Λ

-0.5

0.0

0.5

-1 -0.5 0 0.5

Argand[M1+]K0L PhE0+(m2)

Κ0Λ - EBAC[m2]


Re[M1+] (mF)

Im[M1+](mF)

W

W = 1.75 GeV

γ n → Κο Λ



Argand[M1+] fitVSstart w E0+phLk

χ2df ~ 1

- Gradient search- start at EBAC(m2)

- 107 Monte Carlo + Gradient - start at Born

small errors → very narrow minima

(2) start at soln(1); vary L=0-4 ;

-0.5

0.0

0.5

-1 -0.5 0 0.5

Argand[M1+]K0L 1StepMC pkE0

Κ0Λ - EBAC[m2]


Re[M1+] (mF)

Im[M1+](mF)

W

W = 1.75 GeV

γ n → Κο Λ

-0.5

0.0

0.5

-1 -0.5 0 0.5

Argand[M1+]K0L 2StepMC pkE0

Κ0Λ - EBAC[m2]


Re[M1+] (mF)

Im[M1+](mF)

W

W = 1.75 GeV

γ n → Κο Λ



Argand[M1+] 2StepMC wE0+phLk

χ2df ~ 2-4 for W >2050

- 107 Monte Carlo + wide Gradient (1) start at Born(L=0-4); vary L=0-3;

- 107 Monte Carlo + narrow Gradient

χ2df ~ 1 everywhere

CC constraints on phase

Coupled-channel constraints on the phase:

• S!"

= 1+ 2i T!"

, for reactions ! " #

• Oexp = cT!"#2

$ cei%!"T!"#

2

• S+S = 1 ! i T"# $ T"#

+%& '( = T"n+Tn#

n

)

• phase rotation ⇒ i !T"# $ !T"#+%& '( = !T"n

+ !Tn#

n

)

but ei!"#i T"# $ T"#

+%& '( ! e" i(#$n "#n% )

T$n+Tn%

n

&

With coupled channels, arbitrary phase may leave Oexp invariant, but this does not generally satisfy Unitarity. (Mark Paris, Alfred Svarc)

⇒ complete experiments in two channels could constrain phases - obvious choice is γ N → π N; challenge is recoil polarization !

Partial summary-Mar’09

Pa r t i a l S umma r yPa r t i a l S umma r y – of a work in progress

Complete sets of spin-observables will soon be available for γ+N → ΚΛ (and possibly γ n → π–

p) from g9 and g14 Prospects for directly fitting out the f u l lf u l l (Res+Bgk+cc+…) multipoles:

• 16 observables X N angles fitted to 24 parameters (L=0-3) at each W; χ2 space has a large number of valleys (even when the Chiang-Tabakin requirements are met)

• in principle, phase can be constrained by one of the following: - the high L Born limit (which requires sufficient statistics in ALL non-Born partial waves),

- by CC-Unitarity (which requires complete data in at least 2 channels),

- or by locking phase to that of one partial wave (if models can agree on one with minimal uncertainty - eg. E0+ ?)

• from a combination of Monte Carlo sampling and Gradient searches, with the phase of one multipole determined, it will be possible to deduce the f u l lf u l l amplitude, which will reveal the resonant structure in the multipoles


CaviatsCaviats: - fits to the current round of experiments will probably be limited to L=0-3; fits to L=4 improve χ2 but produce meaningless results ⇒ such analyses should provide reliable L=0-2 multipoles.

- below W=2000, χ2{to “smeared data”} < χ2{“data” to unsmeared pts}; ⇒ “better” solutions than the generating amplitude, but still reasonably close;

⇒ statistical errors, coupled with many local minima, lead to systematic uncertainties in multipoles; statistics could help and some observables may be more useful than others - still to come

- for W > 2050, present χ2df >2 and valleys are very narrow;

which will certainly complicate searches for potentially new states


Next stepsNext steps:

(a) expand MC search:

MC iterations net samples/parameter 107 2 109 2.4 1011 3 1024 10

(b) test effects of systematic errors on data ⇔ will lead to systematic uncertainties in deduced multipoles - asymmetry ratios can cancel many exp. systematics, BUT, observables no longer bi-linear combinations of multipoles ⇒ greatly increased number of local minima

- potentially larger in γp → Κ+Λ since BR cannot be measured simultaneously with BT and TR

– more to come

Extras

determining the full meson photo-production amplitude from...

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