Model independent properties of two photon exchange

1
Model independent properties of two photon
exchange
Egle Tomasi-GustafssonSaclay, France
Collaboration with M.P. Rekalo Presently with
G.I. Gakh and E.A. Kuraev
Frascati, January 20, 2006
2
PLAN

• Introduction
• Generalities on form factors
• Electric proton FF (space-like)
• Two-photon exchange
• History
• Model independent properties
• Observables in time-like region
• Signatures of two-photon exchange
• Search for evidence in the data
• Alternative explanation for the discrepancy of
  FFs ratio
• Perspectives

3
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 2S1 form
  factors.
• Neutron and proton form factors are different.
• Elastic form factors contain information on the
  hadron ground state.
• Playground for theory and experiment.

4
Space-like and time-like regions

• FFs are analytical functions.
• In framework of one photon exchange, FFs are
  functions of the momentum transfer squared of the
  virtual photon, tq2-Q2.

tlt0
tgt0
Scattering
Annihilation
_
_
e- h gt e- h
e e- gt h h
Form factors are real in the space-like region
and complex in
the time-like region.
5
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 ? p1
6
The Rosenbluth separation (1950)

• Elastic ep cross section (1-? exchange)

• point-like particle ? Mott

Linearity of the reduced cross section!
7
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

8
THE RESULTS (gt2000)
Linear deviation from dipole mGEp?GMp
Jlab E93-027 , E99-007SpokepersonsCh.
Perdrisat, V. Punjabi, M. Jones, E. Brash M.
Jones et ql. Phys. Rev. Lett. 84,1398 (2000) O.
Gayou et al. Phys. Rev. Lett. 88092301 (2002)
9
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Electric proton FF

• Different results with different
• experimental methods !!
• New mechanism
• two-photon exchange?

11
Two-Photon exchange

• 1g-2g interference is of the order of
  ae2/4p1/137 (in usual calculations of
  radiative corrections, one photon is hard and
  one is soft)
• In the 70s it was shown J. Gunion and L.
  Stodolsky, V. Franco, F.M. Lev, V.N. Boitsov, L.
  Kondratyuk and V.B. Kopeliovich, R. Blankenbecker
  and J. Gunion that, at large momentum transfer,
  due to the sharp decrease of the FFs, if the
  momentum is shared between the two photons, the
  2g- contribution can become very large.
• The 2g amplitude is expected to be mostly
  imaginary.
• In this case, the 1g-2g interference is more
  important in time-like region, as the Born
  amplitude is complex.

12
Qualitative estimation of 2g exchange
For ed elastic scattering
q/2
q/2
From quark counting rules Fd t-5 and
FNt-2
For t 4 GeV2,
For d, 3He, 4He, 2g effect should appear at 1
GeV2, for protons
10 GeV2
13
Two-Photon exchange

• In 1999 M.P. Rekalo, E. T.-G. and D. Prout
  found
• a model-independent parametrization of the 2g-
• contribution and applied to ed-elastic
  scattering data.
• ? Discrepancy between the results from
• Hall A L.C. Alexa et al. Phys. Rev. Lett.
  82, 1374 (1999)
• and
• Hall C D. Abbott et al. Phys. Rev. Lett.
  82, 1379 (1999).

M. P. Rekalo, E. T-G and D. Prout, Phys. Rev.
C60, 042202 (1999)
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1g-2g interference
M. P. Rekalo, E. T.-G. and D. Prout Phys. Rev. C
(1999)
2g
1g


1g
15
1g-2g interference
D/A
C/A
M. P. Rekalo, E. T-G and D. Prout, Phys. Rev.
C60, 042202 (1999)
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The 1g-2g interference destroys the linearity
of the Rosenbluth plot!
17
Model independent propertiesof two photon
exchange
18
Model independent considerations for

• 4 spin ½ fermions ? 16 amplitudes in the general
  case.
• P- and T-invariance of EM interaction,
• helicity conservation,

1g exchange
1g exchange
Two (complex) EM FFs Functions of one variable (t)
Three (complex) amplitudes Functions of two
variables (s,t)
Crossing symmetry, C-invariance, T-reversal
connect e N ? e N, NN? e e- , and e
e- ? NN
19
The Matrix Element 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)
Assuming P-invariance, and lepton helicity
conservation, the matrix element for 1g2g
exchange is
vector
axial
For 1g -exchange
20
Generalized Form Factors
By analogy with Sachs and Fermi-Dirac FFs
complex functions of 2 variables
Both F1N and F2N contain 1g2g!
Decomposition of the amplitudes
2g-terms
21
Unpolarized cross section
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).
2g-exchange induces three new terms, of the order
of a
22
Symmetry relations

• Odd properties of the 2g amplitudes with respect
  to the transformation

cos ? - cos ? i.e., ? ? ? - ?

• One can remove or single out the 2g contribution
  by doing the sum or the difference of the
  differential cross section at the angles
  connected by this transformation

23
Remove the 2g contribution

• Sum of the differential cross sections

• Total cross section

24
Single out the 2g contribution

• Angular asymmetry

with

• In terms of amplitudes - only interference terms!

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2g contribution

• The sum

is free from 2g contributions!
26
Model independent considerations for e N
scattering
Electron and positron scattering
e N
2 real functions
3 complex functions
8 real functions determine the 6 complexe
amplitudes for e N ?e N

• Relations among the functions!

M. P. Rekalo and E. T-G Eur. Phys. Jour. A
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In the same kinematical conditionsSum and
Difference of e N scattering
Electron and positron scattering
e N
e
e-
Model independent considerations which hold at
O(a2)
M. P. Rekalo and E. T-G Eur. Phys. Jour. A
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Single spin observables

• T-odd observable

• TPE contribution
• Small, of the order of a
• Relative role increases when q2 increases
• Does not vanish, in the general case, for 1g
  exchange

• At 90 expected small (vanishes for 1g exchange)

• At threshold (vanishes for 1g exchange due to
  GEGM)

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Form Factor determination

• C-odd properties of nucleon polarization

with

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

30
Form Factor ratio RGE / GM

• The sum

is free from 2g contributions!

• The Ratio R can be determined by

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Is there any evidence of presence of a
2g-contributionin the existing ep data?
NON
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Parametrization of 2g-contribution for ep

• From the data
• deviation from linearity
• ltlt 1!

E. T.-G., G. Gakh Phys. Rev. C (2005)
33

• Possible explanation
• for the FFs discrepancy

Radiative Corrections
34
Radiative corrections
Mo and Tsai (1969) Schwinger (1949)

• Effects of the order of
• - few percent on polarization observables,
• - up to 40 on cross section!
• Complete calculations in progress

35
Structure function method
Q21 GeV2
Q23 GeV2
Assumes dipole FFs Change the slope !
Q25 GeV2
SF Born
Polarization
RC Born
E.A Kuraev, V.S. Fadin Sov. J. Nucl. Phys. 41,
466 (1985)
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Conclusions

• We have derived model independent formulas for
  all experimental observables in presence of 2g
  exchange, as functions of three complex
  amplitudes for e e- ? NN
• Using symmetry properties one can remove or
  single out 2g contributions
• Crossing symmetry, C-invariance, T-reversal
  connect
• e N ? e N, NN? e e- , and e e- ?
  NN

Novosibirsk-VEPP3

• New data welcome in next future!
• Revise Radiative Corrections!