Constraining single energy partial wave analysisPWA 9/ATHOS 4 Bad Honnef (Germany), March 13-17,...
37
Transcript of Constraining single energy partial wave analysisPWA 9/ATHOS 4 Bad Honnef (Germany), March 13-17,...
Constraining single energy partial wave analysis
M. Hadºimehmedovi¢,1R.Omerovi¢,1H. Osmanovi¢,1 J. Stahov,1
V. Kashevarov2 K. Nikonov,2 M. Ostrick, 2 L. Tiator,2
A. varc,3
1University of Tuzla, Faculty of Natural Sciences and Mathematics,Bosnia and Herzegovina,
2Institute of Nuclear Physics, Johannes Gutenberg-Universität Mainz, Germany3Ruer Bo²kovi¢ Institute, Zagreb, Croatia
International Workshop on Partial Wave Analyses and AdvancedTools for Hadron Spectroscopy
PWA 9/ATHOS 4Bad Honnef (Germany), March 13-17, 2017
H. Osmanovi¢, PWA9/ATHOS4 Bad Honnef, March, 2017 Constraining single energy partial wave analysis 1 / 37
Outline
Kinematics of η photoproduction
Problems in the unconstrained single energy partial waveanalysis
Constrained single energy PWA by imposing the xed-tanalyticity
Preliminary results with
Pseudo data MAIDReal data from MAMI and GRAAL
Conclusions
H. Osmanovi¢, PWA9/ATHOS4 Bad Honnef, March, 2017 Constraining single energy partial wave analysis 2 / 37
η photoproduction
pi -four momentum of incoming nucleonpf - four momentum of outgoing nucleonk - four momentum of incident photonq - four momentum of η mesonMadelstam variables:s = w2 = (pi + k)2
t = (q − k)2
u = (pi − q)2
ν = s−u4m
s + t + u = 2m2 + m2η;
m - mass of nucleon,mη - mass of eta meson
H. Osmanovi¢, PWA9/ATHOS4 Bad Honnef, March, 2017 Constraining single energy partial wave analysis 3 / 37
Observables, amplitudes and multipoles in ηphotoproduction
16 observables
Observables are represented by one set of fourcomplex amplitudes:
CGLN amplitudes (Fk(W , cos θ), k = 1, 2, 3, 4)
helicity amplitudes (Hk(W , cos θ), k = 1, 2, 3, 4)
invariant amplitudes (Bk(s, t), k = 1, 2, 6, 8)
Amplitudes are given by expansion in terms ofelectric (E`±) and magnetic (M`±) multipoles(Details in Tiator's talks).
H. Osmanovi¢, PWA9/ATHOS4 Bad Honnef, March, 2017 Constraining single energy partial wave analysis 4 / 37
Observables, amplitudes and multipoles in ηphotoproduction
Example, dierential cross section in terms of helicity amplitudes
dσ
dΩ=
q
2k(|H1|2 + |H2|2 + |H3|2 + |H4|2)
Expansion of CGLN in terms of multipoles (truncated - up to` = Lmax)
F1 =
Lmax∑`≥0
(`M`+ + E`+)P′`+1 + [(`+ 1)M`− + E`−]P
′`−1,
F2 =
Lmax∑`≥1
[(`+ 1)M`+ + `M`−]P′` ,
F3 =
Lmax∑`≥1
[(E`+ −M`+)P′′`+1 + (E`− −M`−)P
′′`−1]
F4 =
Lmax∑`≥2
(M`+ − E`+ −M`− − E`−)P′′`
H. Osmanovi¢, PWA9/ATHOS4 Bad Honnef, March, 2017 Constraining single energy partial wave analysis 5 / 37
Some concepts in analysis of experimental data
Energy dependent (ED)partial waves (multipoles) are parametrized as a function ofenergy (model dependent). In this talk we will use three EDMAID solutions (Details in Kashevarov's talk).
Single energy (SE)multipoles are determined at a single energy.
Amplitude analysis (AA)amplitudes are parametrized as a function of energy in energyrange where data are available.
H. Osmanovi¢, PWA9/ATHOS4 Bad Honnef, March, 2017 Constraining single energy partial wave analysis 6 / 37
Unconstrained SE PWA
Input: pseudo data (relative error 0.1%) created from MAIDsolution - Solution I.We tted 8 observablesdσdΩ ,F
dσdΩ ,T
dσdΩ ,E
dσdΩ ,P
dσdΩ ,G
dσdΩ ,Cx
dσdΩ ,Ox
dσdΩ , (complete set of
observables).Lmax = 5, 40 real parameters in the t. Multipoles with L > 5 setto zero.
0.05
0.2
0.35
-1 -0.5 0 0.5 1
dσ
/dΩ
cos θ
Elab= 1200.00 MeV, W= 1769.80 MeV;
Data
-0.02
0.05
0.12
-1 -0.5 0 0.5 1
F d
σ/d
Ω
cos θ
Fit
0
0.08
0.16
-1 -0.5 0 0.5 1
T d
σ/d
Ω
cos θ
-0.05
0.1
0.25
-1 -0.5 0 0.5 1
Σ d
σ/d
Ω
cos θ
0
0.08
0.16
-1 -0.5 0 0.5 1P
dσ
/dΩ
cos θ
Elab= 1200.00 MeV, W= 1769.80 MeV;
Data
-0.16
-0.08
0
-1 -0.5 0 0.5 1
G d
σ/d
Ω
cos θ
Fit
-0.16
0
0.16
-1 -0.5 0 0.5 1
Cx d
σ/d
Ω
cos θ
Data
-0.16
-0.08
0
0.08
-1 -0.5 0 0.5 1
Ox d
σ/d
Ω
cos θ
Fit
H. Osmanovi¢, PWA9/ATHOS4 Bad Honnef, March, 2017 Constraining single energy partial wave analysis 7 / 37
Unconstrained SE PWA
The starting parameters of the t were randomly selected in a 30%range around the "true" solution.
-15
5
25
1500 1700 1900
E0
+ [
mfm
]
W[MeV]
Real part
-3
0
3
1500 1700 1900
E1
+ [
mfm
]
W[MeV]
Imag part
-3
0
3
1500 1700 1900
M1
+ [
mfm
]
W[MeV]
-6
0
6
1500 1700 1900
M1
- [m
fm]
W[MeV]
-1
0
1
1500 1700 1900
E2
+ [
mfm
]
W[MeV]
Real part
-3
0
3
1500 1700 1900
E2
- [m
fm]
W[MeV]
Imag part
-1.5
0
1.5
1500 1700 1900M
2+
[m
fm]
W[MeV]
-1.5
0
1.5
1500 1700 1900
M2
- [m
fm]
W[MeV]
Problem is more serious - uniqueness problem. How to resolve it?
H. Osmanovi¢, PWA9/ATHOS4 Bad Honnef, March, 2017 Constraining single energy partial wave analysis 8 / 37
Single energy partial wave analysis
One must impose more stringent constraints taking into accountanalyticity of scattering amplitudes.Dispersion relations? Not easy to apply!Important step forward:
In a series of papers E. Pietarinen proposed a substitute fordispersion relations. In his method invariant amplitudes areexpanded in terms of analytic functions having the same analyticstructure.
S. Bowcock, H. Burkhardt, Rep. Prog Phys 38 (1975) 1099
E. Pietarinen: Amplitude analysis using xed-t analyticity of invariant amplitudes
E. Pietarinen, Nuovo Cim. 12 (1972) 522E. Pietarinen, Nucl. Phys. B49 (1972) 315 Discussion of uniquenessproblemE. Pietarinen, Nucl. Phys. 8107 (1976) 21 Discussion of uniquenessproblemJ. Hamilton, J. L. Peterson, New developments in dispersion theory,Vol.1, Nordita, 1975.
H. Osmanovi¢, PWA9/ATHOS4 Bad Honnef, March, 2017 Constraining single energy partial wave analysis 9 / 37
How to impose xed-t analyticity in PWA of scattering data?
In our PWA of η photoproduction data we use the same approachas it was done in KH80 analysis of πN scattering data.
The method consists of two analyses:
Fixed-t amplitude analysis (Fixed-t AA) - determination of theinvariant scattering amplitudes from exp. data at a givenxed-t valueConstrained single energy partial wave analysis - SE PWAFixed-t amplitude analysis and single energy PWA are coupled.Results from one analysis are used as constraint in another inan iterative procedure.
H. Osmanovi¢, PWA9/ATHOS4 Bad Honnef, March, 2017 Constraining single energy partial wave analysis 10 / 37
Imposing the xed-t analyticity in PWA of scattering data
1.5 1.55 1.6 1.65 1.7 1.75 1.8 1.85W [GeV]
-1.5
-1.25
-1
-0.75
-0.5
-0.25
0
t [G
eV2]
Kinematical grid of MAMI dσ/dΩ
Single energy PWA is performed along red lines. Fixed-t amplitudeanalysis is performed along green lines.
H. Osmanovi¢, PWA9/ATHOS4 Bad Honnef, March, 2017 Constraining single energy partial wave analysis 11 / 37
Coupled xed-t amplitude analysis and single energy PWA
Connection between SE PWA and xed-t AAMultipoles obtained from SE PWA at a given set of energies areused to calculate helicity amplitudes which are used as constraint inthe xed-t amplitude analysis.
The whole procedure has to be iterated until reaching reasonableagreement in two subsequent iterations
H. Osmanovi¢, PWA9/ATHOS4 Bad Honnef, March, 2017 Constraining single energy partial wave analysis 12 / 37
Fixed-t amplitude analysis - Pietarinen's expansion method
The simplest case-πN elastic scattering at xed-t.Apart from nucleon poles, crossing symmetric invariant amplitudesare analytic function in a complex ν2 plane ν2
th≤ ν2 <∞,
(νth = mπ + t
4m).
Conformal mapping:
z =α−
√ν2th− ν2
α +√ν2th− ν2
Points on the cut in complex ν2 plane is mapped on the circle.H. Osmanovi¢, PWA9/ATHOS4 Bad Honnef, March, 2017 Constraining single energy partial wave analysis 13 / 37
Fixed-t amplitude analysis of η photoproduction data
For a given t invariant amplitudes are represented by two Pietarinenseries :
B = BN +
N1∑i=0
b(1)iz i1 +
N2∑i=0
b(2)iz i2.
B stands for crossing symmetric invariant amplitudes B1,B2,B6
and B8
ν .BN are known nucleon pole contributions. Conformal variables z1and z2 are dened as:
z1 =α−
√ν2th1− ν2
α+√ν2th1− ν2
, z2 =β −
√ν2th2− ν2
β +√ν2th2− ν2
.
H. Osmanovi¢, PWA9/ATHOS4 Bad Honnef, March, 2017 Constraining single energy partial wave analysis 14 / 37
Fixed-t amplitude analysis
Coecients b(1)i, b(2)
i are obtained by minimizing a quadratic
form χ2 = χ2data + χ2PW + Φ
χ2data
contains data at a given t-value.χ2PW
contains as the data the helicity amplitudes from the SEPWA analysis. Φ is Pietarinen's convergence test function.
Φ = λ1Φ1 + λ2Φ2 + λ3Φ3 + λ4Φ4.
Φ1 =N∑n=0
(n + 1)3(c+n )2, . . . ,Φ4 =
N∑n=0
(n + 1)3(b−n )2.
λ1, λ2, λ3 and λ4 are determined in an iterative procedure.
In a rst iteration helicity amplitudes are calculated from initial,already existing PW solution.In subsequent iterations helicity amplitudes are calculated frommultipoles obtained in SE PWA of the same set of experimentaldata.
H. Osmanovi¢, PWA9/ATHOS4 Bad Honnef, March, 2017 Constraining single energy partial wave analysis 15 / 37
Fixed-t amplitude analysis
H. Osmanovi¢, PWA9/ATHOS4 Bad Honnef, March, 2017 Constraining single energy partial wave analysis 16 / 37
How does it work? - testing with pseudo data
Input data:We used pseudo data created from Solution I with relative error0.1%.In present calculations we tted 8 observables (Complete set ofobservables) dσ
dΩ , FdσdΩ , T
dσdΩ , P
dσdΩ , Σ dσ
dΩ , GdσdΩ , Cx
dσdΩ and Ox
dσdΩ .
Observables at t = −0.2GeV 2.
t=-0.200GeV2
0.2
0.8
1.4
1500 1850 2200
dσ
/dΩ
W[MeV]
Data
-0.04
0.02
0.1
1500 1850 2200
F d
σ/d
Ω
W[MeV]
0
0.1
0.2
1500 1850 2200
T d
σ/d
Ω
W[MeV]
0
0.1
0.2
1500 1850 2200
Σ d
σ/d
Ω
W[MeV]
t=-0.200GeV2
-0.15
0
0.15
1500 1850 2200
P d
σ/d
ΩW[MeV]
Data
-0.16
-0.06
0.04
1500 1850 2200
G d
σ/d
Ω
W[MeV]
-0.4
0.4
1.2
1500 1850 2200
Cx d
σ/d
Ω
W[MeV]
-0.25
-0.1
0.05
1500 1850 2200
Ox d
σ/d
Ω
W[MeV]
H. Osmanovi¢, PWA9/ATHOS4 Bad Honnef, March, 2017 Constraining single energy partial wave analysis 17 / 37
Fixed-t amplitude analysis-pseudo data
H. Osmanovi¢, PWA9/ATHOS4 Bad Honnef, March, 2017 Constraining single energy partial wave analysis 18 / 37
Fixed-t amplitude analysis-pseudo data
Fit of pseudo data at t = −0.2GeV 2 and t = −0.5GeV 2
t=-0.200GeV2
0.2
0.8
1.4
1500 1850 2200
dσ
/dΩ
W[MeV]
Data
-0.04
0.02
0.1
1500 1850 2200
F d
σ/d
Ω
W[MeV]
Fit
0
0.1
0.2
1500 1850 2200
T d
σ/d
Ω
W[MeV]
0
0.1
0.2
1500 1850 2200
Σ d
σ/d
Ω
W[MeV]
t=-0.200GeV2
-0.15
0
0.15
1500 1850 2200
P d
σ/d
Ω
W[MeV]
Data
-0.16
-0.06
0.04
1500 1850 2200
G d
σ/d
Ω
W[MeV]
Fit
-0.4
0.4
1.2
1500 1850 2200
Cx d
σ/d
Ω
W[MeV]
-0.25
-0.1
0.05
1500 1850 2200
Ox d
σ/d
Ω
W[MeV]
t=-0.500GeV2
0
0.5
1
1850 2200
dσ
/dΩ
W[MeV]
Data
0
0.1
0.2
1850 2200
F d
σ/d
Ω
W[MeV]
Fit
0.06
0.12
0.18
1850 2200
T d
σ/d
Ω
W[MeV]
0
0.1
0.2
1850 2200
Σ d
σ/d
Ω
W[MeV]
t=-0.500GeV2
0
0.1
0.2
1850 2200
P d
σ/d
Ω
W[MeV]
Data
-0.12
-0.05
0.02
1850 2200
G d
σ/d
Ω
W[MeV]
Fit
-0.2
0.3
0.8
1850 2200
Cx d
σ/d
Ω
W[MeV]
-0.2
-0.03
0.15
1850 2200
Ox d
σ/d
Ω
W[MeV]
H. Osmanovi¢, PWA9/ATHOS4 Bad Honnef, March, 2017 Constraining single energy partial wave analysis 19 / 37
Fixed - t amplitude analysis - helicity amplitudes
Obtained helicity amplitudes at t = −0.2GeV 2 and t = −0.5GeV 2.
t=-0.20GeV2
0
2
4
1500 1850 2200
H1 [m
fm]
W[MeV]
Re SE
-25
-8
10
1500 1850 2200
H2 [m
fm]
W[MeV]
Im SE
-4
-1
2
1500 1850 2200
H3 [
mfm
]
W[MeV]
Re FT
-10
5
20
1500 1850 2200
H4 [
mfm
]
W[MeV]
Im FTt=-0.50GeV
2
0
2
4
1850 2200
H1 [m
fm]
W[MeV]
Re SE
-8
1
10
1850 2200
H2 [m
fm]
W[MeV]
Im SE
-5
-1
3
1850 2200
H3 [
mfm
]
W[MeV]
Re FT
-10
5
20
1850 2200
H4 [
mfm
]
W[MeV]
Im FT
Full lines (Re FT and Im FT) are results from 3rd iteration. Dots(Re SE and Im SE) are results from 2nd iteration.
H. Osmanovi¢, PWA9/ATHOS4 Bad Honnef, March, 2017 Constraining single energy partial wave analysis 20 / 37
From xed-t to single energy PWA
In single energy Lmax = 5, 40 real parametrs. Multipoles L > 5 setto zero.
H. Osmanovi¢, PWA9/ATHOS4 Bad Honnef, March, 2017 Constraining single energy partial wave analysis 21 / 37
Constrained SE PWA-pseudo data
Fit to the data at W = 1543.24MeV and W = 1769.8MeV .
1
1.15
1.3
-1 -0.5 0 0.5 1
dσ
/dΩ
cos θ
Elab= 800.00 MeV, W= 1543.24 MeV;
Data
-0.1
0.05
0.2
-1 -0.5 0 0.5 1
F d
σ/d
Ω
cos θ
Fit
0
0.08
0.16
-1 -0.5 0 0.5 1
T d
σ/d
Ω
cos θ
0
0.15
0.3
-1 -0.5 0 0.5 1
Σ d
σ/d
Ω
cos θ
-0.04
0.06
0.16
-1 -0.5 0 0.5 1
P d
σ/d
Ω
cos θ
Elab= 800.00 MeV, W= 1543.24 MeV;
Data
-0.06
-0.03
0
-1 -0.5 0 0.5 1
G d
σ/d
Ω
cos θ
Fit
0
0.7
1.4
-1 -0.5 0 0.5 1
Cx d
σ/d
Ω
cos θ
Data
-0.04
0.04
0.12
-1 -0.5 0 0.5 1
Ox d
σ/d
Ω
cos θ
Fit
0.05
0.2
0.35
-1 -0.5 0 0.5 1
dσ
/dΩ
cos θ
Elab= 1200.00 MeV, W= 1769.80 MeV;
Data
-0.02
0.05
0.12
-1 -0.5 0 0.5 1
F d
σ/d
Ω
cos θ
Fit
0
0.08
0.16
-1 -0.5 0 0.5 1
T d
σ/d
Ω
cos θ
-0.05
0.1
0.25
-1 -0.5 0 0.5 1
Σ d
σ/d
Ω
cos θ
0
0.08
0.16
-1 -0.5 0 0.5 1
P d
σ/d
Ω
cos θ
Elab= 1200.00 MeV, W= 1769.80 MeV;
Data
-0.16
-0.08
0
-1 -0.5 0 0.5 1
G d
σ/d
Ω
cos θ
Fit
-0.16
0
0.16
-1 -0.5 0 0.5 1
Cx d
σ/d
Ω
cos θ
Data
-0.16
-0.08
0
0.08
-1 -0.5 0 0.5 1 O
x d
σ/d
Ω
cos θ
Fit
dσdΩ , F
dσdΩ , T
dσdΩ , P
dσdΩ , Σ dσ
dΩ , GdσdΩ , Cx
dσdΩ and Ox
dσdΩ .
H. Osmanovi¢, PWA9/ATHOS4 Bad Honnef, March, 2017 Constraining single energy partial wave analysis 22 / 37
Constrained SE PWA-Helicity amplitudes
Helicity amplitudes Hk(W , cos θ) at W = 1543.24MeV andW = 1660.3MeV .
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
-1 -0.5 0 0.5 1
H1 [
mfm
]
cos θ
Elab= 800.00 MeV, Wcm= 1543.24 MeV;
Re FT
-25
-20
-15
-10
-5
0
5
-1 -0.5 0 0.5 1
H2 [
mfm
]
cos θ
Re SE
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
-1 -0.5 0 0.5 1
H3 [
mfm
]
cos θ
Im FT
-5
0
5
10
15
20
25
-1 -0.5 0 0.5 1
H4 [
mfm
]
cos θ
Im SE
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
-1 -0.5 0 0.5 1
H1 [
mfm
]
cos θ
Elab= 1000.00 MeV, Wcm= 1660.39 MeV;
Re FT
-8
-6
-4
-2
0
2
4
6
8
-1 -0.5 0 0.5 1
H2 [
mfm
]
cos θ
Re SE
-4
-3
-2
-1
0
1
2
3
-1 -0.5 0 0.5 1H
3 [
mfm
]
cos θ
Im FT
-10
-8
-6
-4
-2
0
2
4
6
-1 -0.5 0 0.5 1
H4 [
mfm
]
cos θ
Im SE
To be pointed out: Real and imaginary parts of helicity amplitudes(blue and red dots- Re FT and Im FT) are obtained fromindependent xed-t AA at dierent t-values.
H. Osmanovi¢, PWA9/ATHOS4 Bad Honnef, March, 2017 Constraining single energy partial wave analysis 23 / 37
Constrained SE PWA-Multipoles
Red and blues solid lines are multipoles from Solution I. (3rditeration)
-10
-5
0
5
10
15
20
1500 1600 1700 1800 1900 2000 2100 2200
E0
+ [
mfm
]
W[MeV]
Real part
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
1500 1600 1700 1800 1900 2000 2100 2200
E1
+ [
mfm
]
W[MeV]
Imag part
-0.5
0
0.5
1
1.5
2
2.5
1500 1600 1700 1800 1900 2000 2100 2200
M1
+ [
mfm
]
W[MeV]
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
1500 1600 1700 1800 1900 2000 2100 2200
M1
- [m
fm]
W[MeV]
-0.12
-0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
1500 1600 1700 1800 1900 2000 2100 2200
E2
+ [
mfm
]
W[MeV]
Real part
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1500 1600 1700 1800 1900 2000 2100 2200
E2
- [m
fm]
W[MeV]
Imag part
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
1500 1600 1700 1800 1900 2000 2100 2200
M2
+ [
mfm
]
W[MeV]
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1500 1600 1700 1800 1900 2000 2100 2200
M2
- [m
fm]
W[MeV]
-0.001
-0.0005
0
0.0005
0.001
0.0015
0.002
0.0025
0.003
0.0035
0.004
1500 1600 1700 1800 1900 2000 2100 2200
E3
+ [
mfm
]
W[MeV]
Real part
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
1500 1600 1700 1800 1900 2000 2100 2200
E3
- [m
fm]
W[MeV]
Imag part
-0.01
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
1500 1600 1700 1800 1900 2000 2100 2200
M3
+ [
mfm
]
W[MeV]
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
1500 1600 1700 1800 1900 2000 2100 2200
M3
- [m
fm]
W[MeV]
H. Osmanovi¢, PWA9/ATHOS4 Bad Honnef, March, 2017 Constraining single energy partial wave analysis 24 / 37
Real case - η photoproduction data base
Data base consists of following experimental data
A2 Collaboration at MAMIDierential cross section dσ
dΩCBall/MAMI: E.McNicoll et al., PRC 82(2010) 035208
Target asymmetry TC.S. Akondi et al. (A2 Collaboration at MAMI) Phys. Rev. Lett. 113,102001 (2014).
Double-polarisation asymmetry FC.S. Akondi et al. (A2 Collaboration at MAMI) Phys. Rev. Lett. 113,102001 (2014).
GRAAL collaboration
Beam asymmetry ΣO. Bartalini et al., EPJ A 33 (2007) 169
H. Osmanovi¢, PWA9/ATHOS4 Bad Honnef, March, 2017 Constraining single energy partial wave analysis 25 / 37
Preparing input data
1.5 1.55 1.6 1.65 1.7 1.75 1.8 1.85W [GeV]
-1.5
-1.25
-1
-0.75
-0.5
-0.25
0t
[GeV
2]
Kinematical grid of MAMI dσ/dΩ
H. Osmanovi¢, PWA9/ATHOS4 Bad Honnef, March, 2017 Constraining single energy partial wave analysis 26 / 37
η photoproduction data base in Fixed-t
Observables at t = −0.2GeV 2 and t = −0.5GeV 2.
t=-0.20GeV2
0
0.5
1
1.5
1500 1650 1800
dσ
/dΩ
W[MeV]
-0.15
0
0.15
1500 1650 1800
F d
σ/d
Ω
W[MeV]
0
0.1
0.2
1500 1650 1800
T d
σ/d
Ω
W[MeV]
0
0.1
0.2
0.3
1500 1650 1800
Σ d
σ/d
Ω
W[MeV]
t=-0.50GeV2
0
0.5
1
1500 1650 1800
dσ
/dΩ
W[MeV]
0
0.1
0.2
0.3
1500 1650 1800
F d
σ/d
Ω
W[MeV]
0
0.1
0.2
0.3
1500 1650 1800T
dσ
/dΩ
W[MeV]
0
0.1
0.2
0.3
1500 1650 1800
Σ d
σ/d
Ω
W[MeV]
H. Osmanovi¢, PWA9/ATHOS4 Bad Honnef, March, 2017 Constraining single energy partial wave analysis 27 / 37
Fixed-t amplitude analysis
H. Osmanovi¢, PWA9/ATHOS4 Bad Honnef, March, 2017 Constraining single energy partial wave analysis 28 / 37
Fixed-t amplitude analysis-real data
In order to explore model dependence of our solution we startedwith two MAID solutions and performed complete analysis(Solution II and Solution III).Invariant amplitudes from initial Solution II and III t = −0.2GeV 2.
-4.5
-2.2
0
1450 1650 1850
B1 [1/G
eV
2]
W[MeV]
Re Solution IIIm Solution II
-6
-2
2
1450 1650 1850
B2 [1/G
eV
2]
W[MeV]
Re Solution IIIIm Solution III
-10
2
1450 1650 1850
B6 [
1/G
eV
3]
W[MeV]
-3
3
9
1450 1650 1850
B8 [
1/G
eV
3]
W[MeV]
H. Osmanovi¢, PWA9/ATHOS4 Bad Honnef, March, 2017 Constraining single energy partial wave analysis 29 / 37
Fixed-t amplitude analysis-real data
Fit to the data at t = −0.2GeV 2 and t = −0.5GeV 2.
t=-0.20GeV2
0
0.5
1
1.5
1500 1650 1800
dσ
/dΩ
W[MeV]
-0.15
0
0.15
1500 1650 1800
F d
σ/d
Ω
W[MeV]
Fit Solution III
0
0.1
0.2
1500 1650 1800
T d
σ/d
Ω
W[MeV]
Fit Solution II
0
0.1
0.2
0.3
1500 1650 1800
Σ d
σ/d
Ω
W[MeV]
t=-0.50GeV2
0
0.5
1
1500 1650 1800
dσ
/dΩ
W[MeV]
0
0.1
0.2
0.3
1500 1650 1800
F d
σ/d
Ω
W[MeV]
Fit Solution III
0
0.1
0.2
0.3
1500 1650 1800T
dσ
/dΩ
W[MeV]
Fit Solution II
0
0.1
0.2
0.3
1500 1650 1800
Σ d
σ/d
Ω
W[MeV]
H. Osmanovi¢, PWA9/ATHOS4 Bad Honnef, March, 2017 Constraining single energy partial wave analysis 30 / 37
Constrained SE PWA-real data
H. Osmanovi¢, PWA9/ATHOS4 Bad Honnef, March, 2017 Constraining single energy partial wave analysis 31 / 37
Constrained SE PWA-real data
Fit to the data at W = 1554.27MeV and W = 1657.62MeV .
0.85
1.1
1.35
-1 0 1
dσ
/dΩ
cos θ
Elab= 818.20 MeV, W= 1554.27 MeV;
-0.1
0.1
0.3
-1 0 1
F d
σ/d
Ω
cos θ
-0.05
0.15
0.3
-1 0 1
T d
σ/d
Ω
cos θ
Fit Solution III
0
0.2
0.4
-1 0 1
Σ d
σ/d
Ω
cos θ
Fit Solution II
0.05
0.2
0.4
-1 0 1
dσ
/dΩ
cos θ
Elab= 995.10 MeV, W= 1657.62 MeV;
-0.02
0.1
0.2
-1 0 1
F d
σ/d
Ω
cos θ
0
0.07
0.14
-1 0 1
T d
σ/d
Ω
cos θ
Fit Solution III
0.02
0.09
0.16
-1 0 1
Σ d
σ/d
Ω
cos θ
Fit Solution II
Fit to the data at W = 1765.4MeV and W = 1840.09MeV .
0.05
0.2
0.35
-1 0 1
dσ
/dΩ
cos θ
Elab= 1191.70 MeV, W= 1765.40 MeV;
-0.1
0.05
0.2
-1 0 1
F d
σ/d
Ω
cos θ
0
0.09
0.18
-1 0 1
T d
σ/d
Ω
cos θ
Fit Solution III
0
0.15
0.3
-1 0 1
Σ d
σ/d
Ω
cos θ
Fit Solution II
0.05
0.15
0.35
-1 0 1
dσ
/dΩ
cos θ
Elab= 1335.20 MeV, W= 1840.09 MeV;
-0.06
0.04
0.14
-1 0 1
F d
σ/d
Ω
cos θ
-0.04
0.03
0.1
-1 0 1
T d
σ/d
Ω
cos θ
Fit Solution III
-0.05
0.1
0.25
-1 0 1Σ
dσ
/dΩ
cos θ
Fit Solution II
H. Osmanovi¢, PWA9/ATHOS4 Bad Honnef, March, 2017 Constraining single energy partial wave analysis 32 / 37
Constrained SE PWA-helicity amplitudes
Helicity amplitudes at W = 1602.01MeV .
-1
0
1
2
-1 -0.5 0 0.5 1
H1 [m
fm]
cos θ
Elab= 898.50 MeV, Wcm= 1602.01 MeV;
Re FT
-6
0
6
12
-1 -0.5 0 0.5 1
H2 [m
fm]
cos θ
Re SE
-4
-2
0
2
-1 -0.5 0 0.5 1
H3 [m
fm]
cos θ
Im FT
-20
-10
0
10
-1 -0.5 0 0.5 1
H4 [m
fm]
cos θ
Im SE
(a) Helicity amplitudes obtained using
Solution II as constraint.
-1
0
1
2
-1 -0.5 0 0.5 1
H1 [m
fm]
cos θ
Elab= 898.50 MeV, Wcm= 1602.01 MeV;
Re FT
-6
0
6
12
-1 -0.5 0 0.5 1
H2 [m
fm]
cos θ
Re SE
-4
-2
0
2
-1 -0.5 0 0.5 1
H3 [m
fm]
cos θ
Im FT
-20
-10
0
10
-1 -0.5 0 0.5 1
H4 [m
fm]
cos θ
Im SE
(b) Helicity amplitudes obtained using
Solution III as constraint.
To be pointed out: Real and imaginary parts of helicity amplitudes(blue and red dots- Re FT and Im FT) are obtained fromindependent xed-t AA at dierent t-values.
H. Osmanovi¢, PWA9/ATHOS4 Bad Honnef, March, 2017 Constraining single energy partial wave analysis 33 / 37
Constrained SE PWA-helicity amplitudes
Helicity amplitudes at W = 1602.01MeV .
-2
-1
0
1
2
-1 -0.5 0 0.5 1
H1 [m
fm]
cos θ
Elab= 898.50 MeV, W= 1602.01 MeV;Re Solution IIIm Solution II
-6
0
6
12
-1 -0.5 0 0.5 1
H2 [m
fm]
cos θ
-6
-3
0
3
-1 -0.5 0 0.5 1
H3 [m
fm]
cos θ
Re Solution IIIIm Solution III
-20
-10
0
10
-1 -0.5 0 0.5 1
H4 [m
fm]
cos θ
(c) Initial helicity amplitudes- solutions II
and III.
-2
-1
0
1
2
-1 -0.5 0 0.5 1
H1 [m
fm]
cos θ
Elab= 898.50 MeV, W= 1602.01 MeV;Re Solution IIIm Solution II
-6
0
6
12
-1 -0.5 0 0.5 1
H2 [m
fm]
cos θ
-6
-3
0
3
-1 -0.5 0 0.5 1
H3 [m
fm]
cos θ
Re Solution IIIIm Solution III
-20
-10
0
10
-1 -0.5 0 0.5 1
H4 [m
fm]
cos θ
(d) Helicity amplitudes after three
iterations using solutions II and III as a
constraint.
H. Osmanovi¢, PWA9/ATHOS4 Bad Honnef, March, 2017 Constraining single energy partial wave analysis 34 / 37
Constrained SE PWA-multipoles
s, p, d waves
-12
-4
4
1450 1650 1850
Re (
E0+
)
Solution II
-5
10
25
1450 1650 1850
Im (
E0
+)
Solution III
-0.6
0
0.6
1450 1650 1850
Re
(E
1+
)
W (MeV)
-0.3
0.15
0.6
1450 1650 1850
Im (
E1+
)
W (MeV)
0
0.7
1.4
1450 1650 1850
Re (
M1+
)
W (MeV)
Solution II
-0.2
0.6
1.4
1450 1650 1850
Im (
M1+
)
W (MeV)
Solution III
-0.5
1
2.5
1450 1650 1850
Re (
M1-)
W (MeV)
-1.5
0.25
1.5
1450 1650 1850
Im (
M1-)
W (MeV)
-0.15
0
0.15
1450 1650 1850
Re (
E2+
)
W (MeV)
Solution II
-0.1
0.05
0.2
1450 1650 1850
Im (
E2+
)
W (MeV)
Solution III
-1
-0.4
0.2
1450 1650 1850
Re (
E2-)
W (MeV)
-0.4
0.4
1.2
1450 1650 1850
Im (
E2
-)
W (MeV)
-0.2
0.15
0.5
1450 1650 1850
Re (
M2+
)
W (MeV)
Solution II
-0.4
-0.05
0.3
1450 1650 1850
Im (
M2+
)
W (MeV)
Solution III
-0.7
-0.2
0.3
1450 1650 1850
Re (
M2-)
W (MeV)
-0.4
0.1
0.6
1450 1650 1850
Im (
M2
-)W (MeV)
H. Osmanovi¢, PWA9/ATHOS4 Bad Honnef, March, 2017 Constraining single energy partial wave analysis 35 / 37
Conclusions
Multipoles obtained in our coupled xed-t-single energy PWAare:
model independent - almostconsistent with xed-s and xed-t analyticityconsistent with crossing symmetry
H. Osmanovi¢, PWA9/ATHOS4 Bad Honnef, March, 2017 Constraining single energy partial wave analysis 36 / 37
Conclusions
Unconstrained single energy PWA
-15
5
25
1500 1700 1900
E0+
[m
fm]
W[MeV]
Real part
-3
0
3
1500 1700 1900
E1+
[m
fm]
W[MeV]
Imag part
-3
0
3
1500 1700 1900
M1+
[m
fm]
W[MeV]
-6
0
6
1500 1700 1900
M1-
[mfm
]
W[MeV]
-1
0
1
1500 1700 1900
E2+
[m
fm]
W[MeV]
Real part
-3
0
3
1500 1700 1900
E2-
[mfm
]
W[MeV]
Imag part
-1.5
0
1.5
1500 1700 1900
M2+
[m
fm]
W[MeV]
-1.5
0
1.5
1500 1700 1900
M2-
[mfm
]
W[MeV]
SE PWA with xed-t constraint ⇓
-12
-4
4
1450 1650 1850
Re (
E0+
)
Solution II
-5
10
25
1450 1650 1850
Im (
E0+
)
Solution III
-0.6
0
0.6
1450 1650 1850
Re (
E1+
)
W (MeV)
-0.3
0.15
0.6
1450 1650 1850
Im (
E1+
)
W (MeV)
0
0.7
1.4
1450 1650 1850
Re (
M1+
)
W (MeV)
Solution II
-0.2
0.6
1.4
1450 1650 1850
Im (
M1+
)
W (MeV)
Solution III
-0.5
1
2.5
1450 1650 1850
Re
(M
1-)
W (MeV)
-1.5
0.25
1.5
1450 1650 1850Im
(M
1-)
W (MeV)
H. Osmanovi¢, PWA9/ATHOS4 Bad Honnef, March, 2017 Constraining single energy partial wave analysis 37 / 37
