electromagnetic_form_factors

							Electromagnetic Form Factors and Polarization Phenomena in Space-like and Time-like Regions
Egle Tomasi-Gustafsson
Saclay, France

Bad-Honnef, 24 –VI – 2008
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Hadron Electromagnetic Form factors
– Characterize the internal structure of a particle (≠ point-like) – In a P- and T-invariant theory, the EM structure of a particle of spin S is defined by 2S+1 form factors. – Elastic form factors contain information on the hadron ground state. – Playground for theory and experiment. – New interest due : • to new precise data (polarization method) • new kinematical range accessible (TL)
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PLAN
Towards a unified description of Hadron Form factors to clarify: - zero of GEp - asymptotic properties (Pragmen Lindeloff theorem) - reaction mechanism Underline the essential information which becomes accessible through polarization observables in TL region
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Space-like and Time-like regions
FFs are analytical functions: : real in SL region, complex TL region
In framework of one photon exchange, FFs are functions of the momentum transfer squared of the virtual photon, t = q2 = -Q2.

t<0

t>0

Scatterin g

Annihilation

e- + h => e- + h

_ e+ + e- => h + h

_

Relative phase accessible ONLY through POLARIZATION
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Crossing Symmetry
Scattering and annihilation channels:
- Described

by the same amplitude :

-

function of two kinematical variables, s and t

-

which scan different kinematical regions

k2 → – k2 p2 → – p2

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STATUS on EM Form factors

Space-like region
4) "standard" dipole function for the nucleon magnetic FFs GMp and GMn 2) linear deviation from the dipole function for the electric proton FF GEp 3) contradiction between polarized and unpolarized measurements 4) non vanishing electric neutron FF, GEn.
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The nucleon form factors
E. T.-G., F. Lacroix, Ch. Duterte, G.I. Gakh, EPJA 24, 419 (2005)

Electric proton
VDM : IJL

Magnetic

F. Iachello..PLB 43, 191 (1973)

Extended VDM (G.-K. 92):
E.L.Lomon PRC 66, 045501 2002)

neutron

u o T

d p

te a

so …

a m

y n

ew n

d

a! t a

Hohler
NPB 114, 505 (1976)

Bosted
PRC 51, 409 (1995)

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Proton Form Factors

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The Rosenbluth separation
Linearity of the reduced cross section

The dynamics is contained in FFs → τ, Q2 The kinematics : energies, angles The reaction mechanism?
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→ Holds for 1γ exchange only
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Egle TOMASI-GUSTAFSSON

The polarization method (1967)

The polarization induces a term in the cross section proportional to GE GM Polarized beam and target or polarized beam and recoil proton polarization
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Proton Form Factors Ratio
SLAC Rosenbluth
L. Andivahis PRD50,5491 (1994)

Jlab Super Rosenbluth
I.A. Qattan et al.PRL 94 142301 (2005)

POLARIZATION Exp Jlab E93-027 , E99-007
Spokepersons:
Ch. Perdrisat, V. Punjabi, M. Jones, E. Brash M. Jones et al., Phys. Rev. Lett. 84,1398 (2000) O. Gayou et al., Phys. Rev. Lett. 88,092301 (2002) V. Punjabi et al., Phys. Rev. C 71, 055202 (2005)

Linear deviation from dipole: µ GEp≠GMp
Jlab E04-108/019 , just running !
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Time-like region

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Time-like observables: | GE| 2 and | GM| 2 .

A. Zichichi, S. M. Berman, N. Cabibbo, R. Gatto, Il Nuovo Cimento XXIV, 170 (1962)

B. Bilenkii, C. Giunti, V. Wataghin, Z. Phys. C 59, 475 (1993). G. Gakh, E.T-G., Nucl. Phys. A761,120 (2005).

As in SL region: - Dependence on q2 contained in FFs - Even dependence on cos2θ (1γ exchange) - No dependence on sign of FFs - Enhancement of magnetic term but TL form factors are complex!
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STATUS on EM Form factors Time-like region
3) No individual determination of GE and GM 4) Assume GE=GM (valid only at threshold) VMD or pQCD inspired parametrizations (for p and n):

A GM= 2 2 2 2 s [ π ln  s/ Λ  ]

A(p) = 56.3 GeV4 A(n) = 77.15 GeV4

3) TL nucleon FFs are twice larger than SL FFs 4) Recent data from Babar (radiative return) : • interesting structures in the Q2 dependence of GM(=GE) • GM≠GE.
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Λ=0.3 GeV is the QCD scale parameter

Time-Like Region
proton
VDM : IJL
F. Iachello..PLB43 191 (1973)

Extended VDM (G.-K. 92):
E.L.Lomon PRC66 045501(2002)

neutron

‘QCD inspired’

E. T-G., F. Lacroix, C. Duterte, G.I. Gakh, EPJA 24, 419 (2005)
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Spin Observables(1γ-exchange)
Analyzing power, A

Double spin observables

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Models in T.L. Region (polarization)
Ay Axx Ayy
VDM : IJL Ext. VDM ‘QCD inspired’

Axz R Azz

E. T-G., F. Lacroix, C. Duterte, G.I. Gakh, EPJA 24, 419(2005)
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Asymptotics

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Phragmèn-Lindelöf theorem Asymptotic properties for analytical functions
If f(z) →a as z→∞ along a straight line, and f(z) →b as z→∞ along another straight line, and f(z) is regular and bounded in the angle between, then a=b and f(z) →a uniformly in the angle.

∆=0.05, 0.1

E. T-G. and G. Gakh, Eur. Phys. J. A 26, 265 (2005)
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Phragmèn-Lindelöf theorem
GM (TL) Connection with QCD asymptotics? Applies to NN and NN Interaction (Pomeranchuk theorem ): t=0 : not a QCD regime!

GM (SL)

GE (SL)

E. T-G. and M. P. Rekalo, Phys. Lett. B 504, 291 (2001)
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Reaction mechanism 1γ-2γ interference

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Two-Photon exchange •The 2γ amplitude is expected to be mostly imaginary. •In this case, the 1γ-2γ interference is more important in time-like region, as the Born amplitude is complex.

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Model independent considerations for
4 spin ½ fermions → 16 amplitudes in the general case. P- and T-invariance of EM interaction, helicity conservation,

•For one-photon exchange:
•Two (complex) EM form factors •Functions of one variable (t)

•For two-photon exchange:
•Three (complexe) amplitudes •Functions of two variables (s,t)

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Model independent considerations for
M. L. Goldberger, Y. Nambu and R. Oehme, Ann. Phys 2, 226 (1957) M.P. Rekalo and E. Tomasi-Gustafsson, EPJA 22, 331 (2004)

The hadronic current:

Decomposition of the amplitudes:

For 1γ -exchange:

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Unpolarized cross section
G. Gakh, E.T-G., Nucl. Phys. A761,120 (2005).

p + p → e+ +e-

2γ−term
•Induces four new terms •Odd function of θ: •Does not contribute at θ =90°

Destroys the linearity of the Rosenbluth fit in SL region!
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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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Angular distribution
Mpp=1.877-1.9

Mpp=2.4-3

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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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Single spin asymmetry
•T-odd observable

•TPE contribution:

•Does not vanish, in the general case (1γ exchange) •At 90° (vanishes for 1γ exchange) : •At threshold (vanishes for 1γ exchange due to GE=GM) :

•Small, of the order of α •Relative role increases when q2 increases

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Single spin asymmetry
•Symmetry properties can be applied to the polarization observables as, for example, the single spin asymmetry. Let us introduce:

•This difference can be written as:

• δ is the phase difference of the form factors GM and GE

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Double spin observables

with specific properties, at threshold, or at 90°…
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Conclusions…
Towards a unified description in TL and SL region - Symmetry properties of fundamental interaction, kinematical constraints - Relations SL/TL, reaction mechanism, asymptotics… - Think nucleon models in all kinematical region

Model independent properties Lessons from QED
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The Pauli and Dirac Form Factors



The electromagnetic current in terms of the Pauli and Dirac FFs: Normalization F1p(0)=1, F2p(0)= κp GEp(0)=1, GMp(0)=μp=2.79



Related to the Sachs FFs :

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Interference of 1γ ⊗2γ exchange

•Explicit calculation for structureless proton – The contribution is small, for unpolarized and polarized ep scattering – Does not contain the enhancement factor L – The relevant contribution to K is ~ 1
E.A.Kuraev, V. Bytev, Yu. Bystricky, E.T-G, Phys. Rev. D74, 013003 (2006)
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Rosenbluth separation
Contribution of the electric term

ε=0.8
Before …to be compared to the absolute value of the error on σ and to the size and ε dependence of RC
E.T-G, Phys. Part. Nucl. Lett. 4, 281 (2007)
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ε=0.2

ε=0.5
After

Radiative Return (ISR) e+ +e- → p + p + γ

2 2E γ dσ e e−  p p γ  2m  m = W  s , x ,θ σe e−  p  m, x= p =1− , dm d cos θ s s s me α 2−2x x2 x 2 W  s , x ,θ = − , θ >> . 2 πx sin θ 2 s





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Two photon exchange
•The calculation of the box amplitude requires the description of intermediate nucleon excitation and of their FFs at any Q2 •Different calculations give quantitatively different results

Theory not enough constrained!
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The nucleon form factors and Il Nuovo Cimento
W. Wataghin, 1968

T. Massam and A. Zichichi, 1966 - one parameter fit! - Time-like region!
Iachello, Jackson, A. Lande, 1973 (PLB)
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Time-Like Region: GE versus GM

| GM| 2 Cross section at 900

GE=0 GE=GM

Asym

GE=GD
E. T-G. and M. P. Rekalo, Phys. Lett. B 504, 291 (2001)
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1γ-2γ interference
M. P. Rekalo, E. T.-G. and D. Prout, Phys. Rev. C60, 042202 (1999)

1γ

2γ

{
1{g

{

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Two-photon exchange
Different results with different experimental methods !! - Both methods based on the same formalism - Experiments repeated New mechanism? •1γ-2γ ~ α=e2/4π=1/137 •1970’s: Gunion, Lev…

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