Appilcation of
Maximum
Likelihood
Method for the
Analysis of Spin
Density Matrix
Elements of
Vector Meson
Production at
A.Borissov
CHEP’06
XV International
Conference on
Computing in
High Energy and
Nuclear Physics
13-17.02.06
Mumbai, India
HERMES Spectrometer
FIELD CLAMPS
TRIGGER HODOSCOPE H1
m
FRONT
MUON
HODO
2
DRIFT CHAMBERS
270 mrad
DRIFT
CHAMBERS
1
FC 1/2
PROP.
CHAMBERS
LUMINOSITY
0
MONITOR
TARGET
CELL
MC 1-3
SILICON
DVC
HODOSCOPE H0
-1
170 mrad
PRESHOWER (H2)
BC 1/2
BC 3/4
STEEL PLATE
CALORIMETER
TRD
IRON WALL
RICH
-2
270 mrad
WIDE ANGLE
MUON HODOSCOPE
MAGNET
0
1
2
3
4
5
6
7
140 mrad
27.5 GeV
e+
140 mrad
170 mrad
MUON HODOSCOPES
8
9
10 m
• Two identical halves of forward spectrometer
having acceptance 40 < Θ < 220 mrad with
momentum resolution ≤ 1%
• Electron identification with efficiency ≥ 98%
at hadron contamination ≤ 1%
• Calorimeter with resolution
p
∆E/E[%] = 1.5±0.5 + 5.1±1.1/ E[GeV ]
• RICH from 1998 with π +− vs K +− and P
separation over all kinematic region
The Target
POLARIZED GAS
e+
ELECTRON BEAM
POLARIMETER
PUMP
Internal storage cell gas target with high density
• polarised: ∼ 1014 nucl/cm2, polarization: ∼
88% for H/D
• unpolarized: ∼ 1015 nucl/cm2
• high density runs at the end of the fill in 2000
with 1H, 2D, 4He and 20Ne gases at ∼ 5 · 1015
nucl/cm2
The Beam
Polarization [%]
Comparison of rise time curves
80
Transverse Polarimeter
Longitudinal Polarimeter
60
40
20
0
6
6.5
7
Longitudinally polarized e+(−) beam with
• P = 27.56 GeV/c,
• current 50...10 mA,
• polarization 40...60%
7.5
8
Time [hours]
Exclusive ρ0 pruduction
e
Q2
e
γ∗
V
W2
p
t
p
+
e +p→e
e
+0
+0
+ p + ρ0(→ π +π −)
and π +π − detected
Entries
Selection of Exclusive ρ0 Events
700
600
DATA
500
DIS MC
400
300
200
100
0
-2
0
∆E =
2
4
6
2 −M 2
MX
p
2Mp
2
with MX
= (p + q − v)2
8
10 12
∆E (GeV)
Measured Φ, φ, Θ Angles of ρ0 production
Photon-Nucleon CMS
Φ
e’
π+
γ*
e
lepton scattering
plane
0
ρ
π-
N
N’
φ
0
ρ decay plane
ρ 0 production
plane
π+
N’
φ
0
θ
ρ
π-
ρ Rest Frame
Maximum Likelihood Method in MINUIT
Binned Likelihood Method with Poisson
distribution for the number of events in each
bin. 8 × 8 × 8 bins of cos(Θ), φ, Φ was used.
Input of data:
3-dimensional matrix of data and
normalized background to be subtracted from the
data before fitting
Monte Carlo Events:
3-dimensional matrix of fully reconstructed MC
events at uniform angular distribution
Extraction procedure: simultaneous fit of 23
SDME, with one set of parameters for negative
and positive beam helicity.
α
Function 1: Fit of 23 SDME rij
W(cos Θ, φ, Φ) = W unpol + W long.pol.
"
3 1 (1 − r 04 ) + 1 (3r 04 − 1) cos2 Θ
Wunpol (cos Θ, φ, Φ) = 4π
00
00
2
2
-
√
04 ) sin 2Θ cos φ − r 04 sin2 Θ cos 2φ
2Re(r10
1−1
2 Θ cos 2φ
1 sin2 Θ + r 1 cos2 Θ − r 1
sin
- cos 2Φ r11
1−1
00
- sin 2Φ
√
2 ) sin2 Θ sin φ + Im(r 2 ) sin 2Θ sin 2φ
2Im(r10
1−1
p
+
2(1 + ) cos Φ
5
5 cos2 Θ − √2Rer 5 sin 2Θ cos φ − r 5
2 Θcos2φ
r11 sin2 Θ + r00
sin
10
1−1
p
+ 2(1 + ) sin Φ
#
√
2
6
6
2Im(r10 ) sin 2Θ sin φ + Im(r1−1 ) sin Θ sin 2φ
3 P
Wlong.pol. (cos Θ, φ, Φ) = 4π
beam
p
1 − 2
"
√
3
3
2
2Im(r10 ) sin 2Θ sin φ + Im(r1−1 ) sin Θ sin 2φ
p
+
√ 2(1 − ) cos Φ
7 ) sin 2Θ sin φ + Im(r 7 ) sin2 Θ sin 2φ
2Im(r10
1−1
p
+ 2(1 − ) sin Φ
#
√
2
8 sin2 Θ+r 8 cos2 Θ− 2Re(r 8 ) sin 2Θ cos φ−r 8
r11
00
1−1 sin Θ cos 2φ
10
(1)
Data, Initial and Fitted Angular Distributions
HERMES PRELIMINARY
Diffractive ρ Electroproduction ( H)
0
1
Fitted Monte Carlo
Events
Isotropic Monte Carlo
-1
-0.5
0
0.5
1 -1
-0.5
0
1
cosθ
Events
cosθ
0.5
0
-2
0
-2
0
2
-2
0
2
-2
0
2
-2
0
φ (rad)
2
φ (rad)
Events
-2
Φ (rad)
Events
Φ (rad)
2
Ψ (rad)
2
Ψ (rad)
Result: 23 Spin Density Matrix Elements
SDMEs in Terms of the Helicity Amplitudes
1
R
1
|T01 |2 + |T0−1 |2 +
|T00 |2
1 + R 2NT
NL
1
R
1
04
∗
∗
∗
Re r10
=
T11 T01
Re
+ T1−1 T0−1
T10 T00
+
1 + R
2NT
NL
1
1
R
04
∗
∗
∗
r1−1 =
T11 T−11 + T1−1 T−1−1 +
Re
T10 T−10
1 + R
2NT
NL
1
1
∗
∗
1
T0−1 T01 + T01 T0−1
r00 =
1 + R 2NT
1
1
1
∗
∗
r11
=
T1−1 T11
+ T11 T1−1
1 + R 2NT
1
1
∗
∗
1
Re T1−1 T01
+ T11 T0−1
Re r10
=
1 + R 2NT
1
1
1
∗
∗
=
r1−1
+ T11 T−1−1
T1−1 T−11
1 + R 2NT
1
1
∗
∗
2
Re T1−1 T01
− T11 T0−1
Im r10
=
1 + R 2NT
1
1
∗
∗
2
Re T1−1 T−11
− T11 T−1−1
Im r1−1
=
1 + R 2NT
1
1
3
∗
∗
Im r10
=
Im T11 T01
− T1−1 T0−1
1 + R 2NT
1
1
∗
∗
3
Im T11 T−11
− T1−1 T−1−1
Im r1−1
=
1 + R 2NT
√
R
1
1
5
∗
∗
∗
∗
r00
=
√
T00 T01
+ T01 T00
− T00 T0−1
− T0−1 T00
1 + R 2NT NL 2
√
R
1
1
∗
∗
∗
∗
5
√
T10 T11
+ T11 T10
− T10 T1−1
− T1−1 T10
r11
=
1 + R 2NT NL 2
√
R
1
1
∗
∗
∗
∗
5
√
Re T10 T01
+ T11 T00
− T10 T0−1
− T1−1 T00
Re r10
=
1 + R 2NT NL 2
√
R
1
1
∗
∗
∗
∗
5
√
T10 T−11
+ T11 T−10
− T10 T−1−1
− T1−1 T−10
r1−1
=
1 + R 2NT NL 2
√
1
R
1
∗
∗
∗
∗
6
√
Re T10 T01
− T11 T00
+ T10 T0−1
− T1−1 T00
Im r10 =
1 + R 2NT NL 2
√
R
1
1
∗
∗
∗
∗
6
√
Re T10 T−11
− T11 T−10
+ T10 T−1−1
− T1−1 T−10
Im r1−1
=
1 + R 2NT NL 2
√
1
R
1
∗
∗
∗
∗
7
√
Im T10 T01
+ T11 T00
+ T10 T0−1
+ T1−1 T00
Im r10
=
1 + R 2NT NL 2
√
R
1
1
∗
∗
∗
∗
7
√
Im T10 T−11
+ T11 T−10
+ T10 T−1−1
+ T1−1 T−10
Im r1−1
=
1 + R 2NT NL 2
√
i
1
R
∗
∗
∗
∗
8
√
T00 T01
− T01 T00
− T00 T0−1
+ T0−1 T00
r00
=
1 + R 2NT NL 2
√
R
i
1
8
∗
∗
∗
∗
√
r11
=
T10 T11
− T11 T10
− T10 T1−1
+ T1−1 T10
1 + R 2NT NL 2
√
R
1
1
∗
∗
∗
∗
8
√
Im T10 T01
− T11 T00
− T10 T0−1
+ T1−1 T00
Re r10
=−
1 + R 2NT NL 2
√
R
i
1
∗
∗
∗
∗
8
√
T10 T−11
− T11 T−10
− T10 T−1−1
+ T1−1 T−10
r1−1
=
1 + R 2NT NL 2
04
r00
=
(18)
(19)
(20)
(21)
(22)
(23)
(24)
(25)
(26)
(27)
(28)
(29)
(30)
(31)
(32)
(33)
(34)
(35)
(36)
(37)
(38)
(39)
(40)
The ratio R of the longitudinal to transverse γ ∗ p cross section and the two normalization
factors NL and NT are given by
R=
NL
,
NT
(41)
NL = |T00 |2 + |T10 |2 + |T−10 |2 ,
1
|T11 |2 + |T−1−1 |2 + |T01 |2 + |T0−1 |2 + |T1−1 |2 + |T−11 |2 .
NT =
2
47
(42)
(43)
Function 2: Fit of Amplitudes Tλρ0
λγ ∗
Input: 23 (15) SDME
Output: Helicity Transfer Amplitudes of γ ∗ → ρ0
2
2
2
1
R
+
|T01 | + |T0−1 |
|T |
2NT
NL 00
04
∗
∗
∗
1
1
R
Re r10 =
Re
T11 T01 + T1−1 T0−1 +
T T
1+R
2NT
NL 10 00
04
∗
∗
∗
1
1
R
r1−1 =
Re
T11 T−11 + T1−1 T−1−1 +
T T
1+R
2NT
NL 10 −10
∗ + T T∗
1 =
1
1
T0−1 T01
r00
01 0−1
1+R 2NT 1 =
∗ + T T∗
1
1
r11
T1−1 T11
11 1−1
1+R 2NT
∗ + T T∗
1 =
1 Re T
1
T
Re r10
1−1 01
11 0−1
1+R 2NT
∗
∗
1
1
1
T1−1 T−11 + T11 T−1−1
r1−1 =
1+R 2NT
2
∗
∗
1
1
Im r10 =
Re T1−1 T01 − T11 T0−1
1+R 2NT
∗
∗
2
1 Re T
1
T
−
T
T
Im r1−1
=
1−1 −11
11 −1−1
1+R 2NT ∗
3 =
1 Im T T ∗ − T
1
T
Im r10
11 01
1−1 0−1
1+R 2NT
3
∗
1
1 Im T T ∗
=
Im r1−1
−
T
T
1−1 −1−1
11 −11
1+R 2NT
√
∗
5 =
R p 1
1 T T∗ + T T∗ − T T∗
−
T
T
r00
0−1 00
00 0−1
01 00
00 01
1+R 2N N 2
T L √
5 =
∗
R p 1
1 T T∗ + T T∗ − T T∗
r11
−
T
T
10 11
11 10
10 1−1
1−1 10
1+R 2N N 2
T L
√
5 =
∗
R p 1
1 Re T T ∗ + T T ∗ − T T ∗
Re r10
−
T
T
10 01
11 00
10 0−1
1−1 00
1+R 2N N 2
T L √
∗
∗
∗
R p 1
5
1 T T∗
−
T
T
−
T
T
+
T
T
=
r1−1
1−1 −10
10 −1−1
11 −10
10 −11
1+R 2N N 2
T L
(2)
1
r04
00 = 1+R
where
R= NL /NT
NL = |T00 |2 + |T10 |2 + |T−10 |2
NT = 0.5 |T11 |2 + |T−1−1 |2 + |T01 |2 + |T0−1 |2 + |T1−1 |2 + |T−11 |2 (3)
Result: Spin Transfer Amplitudes
2.5
2.25
2
HERMES
H1
ZEUS
1.75
1.5
|T01|/|T00|
|T11|/|T00|
HERMES PRELIMINARY
0.35
0.3
Ivanov et al.
Royen et al.
0.25
0.2
1.25
1
0.15
0.75
0.1
0.5
0.05
0.25
0
5
0
10
2
2
Q (GeV )
0.25
φ11-φ00 (deg)
|T10|/|T00|
0
0.2
0.15
0
5
10
2
2
Q (GeV )
0
5
10
2
2
Q (GeV )
70
60
50
40
0.1
30
0.05
20
10
0
0
-0.05
-0.1
-10
0
5
10
2
2
Q (GeV )
-20