The unpolarized cross section
dσ =  dσ 
dΩ  dΩ 
ε
1  G 2 (Q2 ) + τ G 2 (Q2 ) 

ε M
(1+τ )  E

Mott


= 1+ 2(1+τ ) tan 2


−1
 θ e  ,τ = Q 2
2 +τ G 2
 
σ
=
ε
G
2
R
E
M
4M
 2
Q2 fixed
N. Rosenbluth (1950)





ε
A.Zichichi, S. M. Berman, N. Cabibbo, R. Gatto, Il Nuovo Cimento XXIV, 170 (1962)
→Holds for 1γ exchange only
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Polarization experiments - Jlab
A.I. Akhiezer and M.P. Rekalo, 1967
Gep collaboration: R=µGEp/GMp
1) Large precision/sign
2) Assume "standard" dipole function for GMp
3) Observe linear deviation from dipole for GEp
- QCD scaling not reached
- Zero crossing of Gep?
contradiction between polarized
and unpolarized measurements
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A.J.R. Puckett et al, PRL (2010)
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Model independent statements
In presence of 2γγ exchange :
- Non linearity in the Rosenbluth fit and
Charge asymmetry in the crossed channel
- Non vanishing odd polarization observables
- Different cross section for
electron/positron – proton elastic(inelastic)
scattering
M. P. Rekalo, E. T.-G. , EPJA (2004), Nucl. Phys. A (2003)
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Linearity of the Rosenbluth fit
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Parametrization of 2γ-contribution for e+p
F (Q 2 , ε ) →
f (Q ) =
(a)
2
1 + ε (a) 2
f (Q )
1− ε
C 2 γ GD
[1 + Q 2 [GeV] 2 /ma2 ]2
From the data:
deviation from linearity
<< 1%!
E. T.-G., G. Gakh, Phys. Rev. C 72, 015209 (2005)
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Radiative Corrections
RC to the cross section:
− large (may reach 40%)
− ε and Q2 dependent
− calculated at first order
Q2=1.75 GeV2
Q2=3.75 GeV2
Q2=5 GeV2
May change
the slope of σR
(and even the sign !!!)
E. T.-G., G. Gakh, PRC 72, 015209 (2005)
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12-XII-2008
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GPD (includes inelastic) :
dominated by two-gamma
correction and correction to GE
A.Afanasev,Phys.Rev.D72:013008(2005)
GMp Hadronic (elastic)
dominated by correction to GMP.
Blunden, Phys.Rev.C72:034612(2005)
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Polarization ratio (ε-dependence)
• DATA: No evidence
of ε-dependence at 1%
level
•MODELS: large
correction (opposite
sign) at small ε
•SF method: ε-independent corrections
•Theory: corrections to the Born approximation at Q2= 2.5 GeV2
Y. Bystritskiy, E.A. Kuraev and E.T.-G, Phys.Rev.C75: 015207 (2007)
P. Blunden et al., Phys. Rev. C72:034612 (2005)
A. Afanasev et al., Phys. Rev. D72:013008 (2005)
N.Kivel and M.Vanderhaeghen, Phys. Rev. Lett.103:092004 (2009).
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Annihilation channel
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Unpolarized cross section
Two Photon Exchange:
• Induces four new terms
• Odd function of θ:
• Does not contribute at θ =90°
G. Gakh, E.T-G., Nucl. Phys. A761,120 (2005).
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Symmetry relations
• Differential cross section at complementary angles:
The SUM cancels the 2γ contribution:
The DIFFERENCE enhances the 2γ contribution:
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Radiative Return (ISR)
e+ +e- → p + p + γ
2
E
2
dσ( e + e − → ppγ ) 2m
m
γ
W ( s , x , θ )σ( e + e − → pp )( m ), x =
,
=
= 1−
dm d cos θ
s
s
s
me
α  2 − 2 x + x 2 x 2 
W ( s , x ,θ ) =
, θ >>
.
−
2


πx
2
s
 sin θ

B. Aubert ( BABAR Collaboration) Phys Rev. D73, 012005 (2006)
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1.877÷1.950
1.950÷2.025
2.025÷2.100
2.100÷2.200
2.200÷2.400
2.400÷3.000
Events/0.2 vs. cos θ
Angular distribution
2γ−exchange?
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Mpp=1.877-1.9
A=0.01±0.02
Mpp=2.4-3
E. T.-G., E.A. Kuraev, S. Bakmaev, S. Pacetti, Phys. Lett. B (2008)
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60000
N
N
N=a0+a2cosθ sin2θ +a1 cos2θ, a2~2γ
1γ
60000
55000
55000
50000
50000
45000
0
0.2
0.4
0.6
0.8
1
45000
2γ
0
0.02
0.2
0.4
0.6
0.8
cos2θ
60000
cos2θ
N
N
q2=5.4 GeV2
2γ
60000
0.05
55000
55000
50000
50000
45000
0
0.2
0.4
0.6
0.8
1
45000
2γ
0
0.2
0.2
0.4
0.6
0.8
cos2θ
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1
cos2θ
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Electron/Positron scattering
• What about data: electron positron scattering
(elastic or inelastic) in the same experimental
conditions?
• If R ≠1 is there any other source of asymmetry?
• Any evidence of two photon exchange (real part
of hard box diagram)?
Recent analysis of existing data:
J. Arrington PRC69 (2004) 032201: evidence of TPE
W.M. Alberico, S.M. Bilenky, C. Giunti,K.M. Graczyk : small TPE
D.Y Chen, H.Q. Zhou and Y.B. Dong, PRC 78 (2008) 045208
depending on data selection, TPE parametrization, Radiative
Corrections..
….controversial,
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Electron/Positron scattering
M± = ±M
2
2
Born approximation
1γ
=M
1γ
2
2γ exchange:
M
± 2
= ± M 1γ + M 2γ
Asymmetry
2
= M 1γ ±2 Re M 1γ M 2γ + M 2γ
2
*
2
2 Re M 1 γ M 2 γ
σ (e + p) − σ (e − p)
A =
=
2
σ (e + p ) + σ (e − p )
M
1γ
The effect is enhanced in the ratio
4 Re( M 1γ M
σ (e + p ) 1 + A
R=
=
≅ 1+
2
−
σ (e p ) 1 − A
M
*
2γ
)
1γ
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World data e+e- scattering
1.4
Yount 62
Browman 65
Anderson 66
Cassiday
Bartel
Mar 68
Anderson 68
Bouquet
Camilleri
Jostlein 74
Hartwig
Rochester
Fancher
1.3
R(e+/e−)
1.2
1.1
1
0.9
0.80
20
40
60
80 100 120 140
N
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Elastic scattering
1.3
R(e+/e−)
1.2
Mar 68
Yount 62
Browman 65
Anderson 68
Anderson 66
Cassiday
Bartel
Bouquet
1.4
q2 =1, 3, 5 GeV2
1.3
1.2
1.1
1.1
0.80
ε=0.2, 0.5, 0.8
1
1
0.9
Mar 68
Yount 62
Browman 65
Anderson 68
Anderson 66
Cassiday
Bartel
Bouquet
Hartwig 79
Hartwig 75
Camilleri
R(e+/e−)
1.4
0.9
Hartwig 79
Hartwig 75
Camilleri
0.2
0.8
0.4
∈
0.6
0.8
1
10−2
10−12
1
q [GeV2]
10
R=(-0.071 ± 0.016 )ε +(1.058 ±0.014)
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Dipole Approximation and pQCD
Dimensional scaling
– Fn (Q2)= Cn [1/( 1+Q2/mn) n-1],
• mn=nβ2 , <quark momentum squared>
• n is the number of constituent quarks
– Setting β2 =(0.471±.010) GeV2 (fitting pion data)
•
pion: Fπ (Q2)= Cπ [1/ (1+Q2/0.471 GeV2)1],
•
nucleon: FN (Q2)= CN [1/( 1+Q2/0.71 GeV2)2],
deuteron: Fd (Q2)= Cd [1/( 1+Q2/1.41GeV2)5]
•
V. A. Matveev, R. M. Muradian, and A. N. Tavkhelidze (1973), Brodsky and Farrar
(1973), Politzer (1974), Chernyak & Zhitnisky (1984), Efremov & Radyuskin (1980)…
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Systematics
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Space-like and Time-like
E.T-G, Phys. Part. Nucl. Lett. 4, 281 (2007)
FM
1
AM
1
FE=εG2E/σred
10%
ε=0.8
10−1
AE
10%
10−1
ε=0.5
ε=0.2
10−20
FE
0
5
10
15
20
2
[GeV ]
25
30
q2
1
2
3
Q [GeV2]
4
5
6
2
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