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
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 u d eamp l i t u d 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
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
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 u d eamp l i t u d 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
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 :
• 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’
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
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
• in practice, one needs linear and circular beam polarization, at least one orientation of target polarization and recoil polarization
Chaing&Tabakin _practice-principle
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
• 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:
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’
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:
• 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’
⇑ 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,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.
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[E,M,L3] K0L-m2,10^9Born
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[E,M,L3] K0L-m2,10^9Born
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[E,M,L2] K0L-m2,10^9Born
Re[A]
(mF)
M2-
M2+
E2-
E2+
M2 109+Grad-0.5
0.0
0.5
Re[E,M,L3] K0L-m2,10^9Born
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[E,M,L3] K0L-m2,10^9Born
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
Multipole fits to mock data-summary
- 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
Multipole fits to mock data-summary
- 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)
Multipole fits to mock data-summary
- 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,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)
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{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)
Multipole fits to mock data-summary
- 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;
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-1.0
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0.5
1.0
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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
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
-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)
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
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
-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)
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)
Argand[E2-] EBAC,kMAIDvsFITw M1- phLk
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
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
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-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)
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)
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}
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-1.0
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0.0
0.5
1.0
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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
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-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)
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 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
phase[E0+] M2'vsKMAID EBAC[m2]KaonMAID
phas
e[E0
+]
(deg
)
W (GeV)
γ n → Κο Λ
δ [E0+]
-1.5
-1.0
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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)
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 E0+ of EBAC(m2)
Argand[E2-] EBAC,kMAIDfit w E0+ph
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
phase[E0+] M2'vsKMAID EBAC[m2]KaonMAID
phas
e[E0
+]
(deg
)
W (GeV)
γ n → Κο Λ
δ [E0+]
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-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)
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 E0+ of EBAC(m2)
Argand[E2-] EBAC,kMAIDvsFIT w E0+phLk
fit to expected JLab data, with phase locked to E0+ of KaonMAID
χ2df {fits to mock data} ~ 2-4→ χ2 valleys very narrow !
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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
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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)
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0.0
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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
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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
M1+ multipole of EBAC(M2) with P13(1720) resonance
fit to expected JLab data, with phase locked to E0+ of EBAC(m2)
Argand[M1+] EBAC,kMAID vs fit w E0+ phLk
fit to expected JLab data, with phase locked to E0+ of KaonMAID
χ2df ~ 2-4 for W >2050
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0.0
0.5
-1 -0.5 0 0.5
Argand[M1+]K0L PhE0+ M2st
Κ0Λ - EBAC[m2]
Κ0Λ fit phase-locked to EBAC[E0+]
Re[M1+] (mF)
Im[M1+](mF)
W
W = 1.75 GeV
γ n → Κο Λ
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0.0
0.5
-1 -0.5 0 0.5
Argand[M1+]K0L PhE0+(m2)
Κ0Λ - EBAC[m2]
Κ0Λ fit phase-locked to EBAC[E0+]
Re[M1+] (mF)
Im[M1+](mF)
W
W = 1.75 GeV
γ n → Κο Λ
M1+ multipole of EBAC(M2) with P13(1720) resonance
fit to expected JLab data, with phase locked to E0+ of EBAC(m2)
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 ;
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0.0
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Argand[M1+]K0L 1StepMC pkE0
Κ0Λ - EBAC[m2]
Κ0Λ fit phase-locked to EBAC[E0+]
Re[M1+] (mF)
Im[M1+](mF)
W
W = 1.75 GeV
γ n → Κο Λ
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0.0
0.5
-1 -0.5 0 0.5
Argand[M1+]K0L 2StepMC pkE0
Κ0Λ - EBAC[m2]
Κ0Λ fit phase-locked to EBAC[E0+]
Re[M1+] (mF)
Im[M1+](mF)
W
W = 1.75 GeV
γ n → Κο Λ
M1+ multipole of EBAC(M2) with P13(1720) resonance
fit to expected JLab data, with phase locked to E0+ of EBAC(m2)
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
Partial summary-Mar’09
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
Partial summary-Mar’09
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