# Presentation: 862977

## High order radiative corrections in Electron-Proton scattering  

*Egle Tomasi-Gustafsson**Saclay, France*

**JLab, August 5, 2008**

*In collaboration with *
*Yu. Bystriskiy, V. Bytev and Prof. E.A. Kuraev*

- Egle TOMASI-GUSTAFSSON

- August 5, 2008

## Cross section of (quasi)elastic ep-scattering

*Elastic *

*Inelastic *

*Higher order inelastic  *

*double brehmstrahlung, *
*pair production..** *

*Classify radiative corrections:**  *

- Egle TOMASI-GUSTAFSSON

- August 5, 2008

## Unpolarized case

_*Unpolarized case*_

- Egle TOMASI-GUSTAFSSON

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## Feynman diagrams, for the scattering amplitude

_*Feynman diagrams, for the scattering amplitude*_

*Elastic scattering*

- Egle TOMASI-GUSTAFSSON

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## Single bremstrahlung

- Egle TOMASI-GUSTAFSSON

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## Double Bremstrahlung and Pair Production

_*Double Bremstrahlung and Pair Production*_

- Lowest order radiative corrections (RC): Mo and Tsai (1969)
- Infrared divergences : « photon mass » , 
- Logarithmic enhancement:

- Egle TOMASI-GUSTAFSSON

- August 5, 2008

## Vacuum Polarization

_*Vacuum Polarization*_

- Main contribution: vacuum polarization due to electron positron pair

- Egle TOMASI-GUSTAFSSON

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## Vertex function

*Vertex function*

- The contribution to F2(Q2) is suppressed by           compared to F1(Q2)

- The  « photon mass » , , is an auxiliary parameter,
- which does not enter in  the final answer for the cross section

- Egle TOMASI-GUSTAFSSON

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## Collinear Emission of Photons

**   ***Contains **logarithmic enhancement*
*   **Suffers from **infrared divergences*

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## Slide 10

- Egle TOMASI-GUSTAFSSON

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## Scattered electron energy

**All orders of PT needed ******** **
**	****	****	****	*****beyond Mo & Tsai approximation******!***

- Initial state emission

- final state emission

- Quasi-elastic scattering

- 3%

- Y0

- Not so small!

- Shift to LOWER Q2

- Egle TOMASI-GUSTAFSSON

- August 5, 2008

## Short history (I)

-    Schwinger: corrections to cross section for electron scattering in external field
- =0(1+)  (1)
- Yennie, Frauchi, Suura: cross section of any pure process (without real photon emission) is zero.
- Kessler, Ericsson, Baier, Fadin, Khoze, Y. Tsai : quasi real electron method. Emission of hard photon is described in terms of a convolution of a radiative function with Born cross section.
-   				    (1)  Not adequate

- Egle TOMASI-GUSTAFSSON

- August 5, 2008

## Short history (II)

- 1977: Altarelli Parisi Gribov Lipatov, Dokshitzer: (DGLAP)
- Asymptotic freedom, evolution equation, Collins factorization
- theorem.
- Drell-Yan picture of hard processed in QED : application of
- QCD ideas to QED: radiative corrections in form of structure
- functions and Drell-Yan picture
- Leading terms:                   and non leading terms
- explicitely taken into account in DGLAP evolution
- equations. In QED known as *Lipatov equations* (1975).

- Egle TOMASI-GUSTAFSSON

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## Structure Function method

- SF method applied to QED processes: calculation of radiative corrections with precision ~ 0.1%.
- Takes into account the dynamics of the process
- Formulated in terms of parton densities (leptons, antileptons, photons)
- Many applications to different processes

*E. A. Kuraev and V.S. Fadin, Sov. J. of Nucl. Phys. 41, 466 (1985)*

*Lipatov equations* (1975)

- Egle TOMASI-GUSTAFSSON

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## Structure Function Method  (Applications)

- e+e- hadrons( J/ width)
*        **E. A. KURAEV and V.S. FADIN, Sov. J. of Nucl. Phys. 41, 466 (1985)*
- ep  e’X (elastic and inelastic scattering)
*	**E. A. KURAEV ;N.P. MERENKOV and V.S. FADIN, Sov. J. of Nucl. Phys. 47,1009  (1988)*
- Decay width of mesons (FSI)
*	**E. A. KURAEV, **JETP Lett.65, 127 (1997) *
- Radiative corrections for LEP beam (small angle BHABHA scattering)
*	**A.B.Arbuzov, E.A.Kuraev  et al, Phys. Lett.B 399, 312 (1997) *
- Compton and double Compton scattering
*	**A.N.Ilyichev, E.A. Kuraev, V.Bytev and Y. P. Peresun'ko, J. Exp. Theor. Phys.100 31 (2005)*
- Structure function method applied to polarized and unpolarized electron-proton scattering: A solution of the GE(p)/GM(p) discrepancy.
*        **Y. Bystricky, E.A.Kuraev, E. Tomasi-Gustafsson, Phys. Rev. C75, 015207 (2007)**.*
-  Radiative corrections to DVCS electron tensor.* V.Bytev, E.A.Kuraev, E. Tomasi-Gustafsson, Phys. Rev. C (2008)*

- Egle TOMASI-GUSTAFSSON

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## The Structure Function Method

- Distinguish:
- 	 -leading contributions of higher order
- 	 -non leading ones

*E. A. K. and V.S. FADIN, Sov. J. of Nucl. Phys. 41, 466 (1985)*

- The SF method is based on:
-  Renormalization group evolution equation
-  Drell-Yan parton picture of the cross section in QCD

**Electron SF: ****probability to ‘find’ electron in the **
**initial electron, with energy fraction**** *****x *****and virtuality up to**** Q****2**

- Egle TOMASI-GUSTAFSSON

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## LSF: ‰ precision

*E. A. K. and V.S. FADIN, Sov. J. of Nucl. Phys. 41, 466 (1985)*

- LLA (Leading Logarithm Approximation)

- Precision of LLA

*Even when corrections in first order PT are**  ****~100%, **the accuracy of higher order RC (LSF)  is** ****1% !** *

- Including K-factor

- Egle TOMASI-GUSTAFSSON

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## The LSF cross section (for ep )

- If the electron is detected in a calorimeter: the cross section
- is integrated over the scattered electron energy fraction:

- The K-factor includes all non leading contributions:

- Egle TOMASI-GUSTAFSSON

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## Results

*Results*

- Both calculations assume dipole FFs
*The slope changes *
*(due to different RC) *

- Q2=1 GeV2

- Q2=5 GeV2

- Q2=3 GeV2

- SF Born

- Polarization

- RC Born

- .........

* **E.T-G,** **Phys. Part. Nucl. Lett. ****4****, 281-288 (2007).*

- Egle TOMASI-GUSTAFSSON

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## Unpolarized Cross section

- Born +dipole FFs
- (=unpolarized experiment+Mo&Tsai)
- SF (with dipole FFs)
- SF+2 exchange

- Q2=3 GeV2

- Q2=5 GeV2

*SF: change the slope!*

- Q2=1 GeV2

*2**** exchange very small!*

- Egle TOMASI-GUSTAFSSON

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## Polarization ratio

*Polarization ratio*

- Born
- SF
- SF+2 exchange

-  =60°

*2**** destroys linearity!*

*2**** exchange very small!*

-  =80°

-  =20°

*Yu. Bystricky, E.A.Kuraev, E. T.-G,  Phys. Rev. C 75, 015207 (2007)*

- Egle TOMASI-GUSTAFSSON

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## Radiative Corrections (SF method)

- Polarization data

- JLab data

- SLAC data

*Yu. Bystricky, E.A.Kuraev, E. Tomasi-Gustafsson,  Phys. Rev. C75, 015207 (2007)*

*SF corrected*

*SF corrected*

- Rosenbluth parameters
-       highly correlated!

*E.T-G, **Phys. Part. Nucl. Lett. ****4,**** 281 (2007)*

- Egle TOMASI-GUSTAFSSON

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*Interference*

*DVCS*

*Bethe-Heitler*

- Egle TOMASI-GUSTAFSSON

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## Charge  Asymmetry

*RC (LSF)*

*RC(1st order)*

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## HELICITY Asymmetry

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## Conclusions

*High precision experiments need **highly  precise*
*  **Radiative Corrections*
*Higher order** **corrections** become *
* **more and more important** at large Q**2*
*The lepton structure function method can *
* **be applied to different electromagnetic*
* **processes with **permille precision*
*Higher order corrections depend on the relevant*
* **kinematical variables*

- Egle TOMASI-GUSTAFSSON

- August 5, 2008

## Radiative Corrections (first order)

*The cross section:*

*The correction ( in powers of Z):*
*	**Z0: electron emission and vacuum polarization*
*	**Z1: interference 1****-2****exchange*
***Z2: target emission*

*L.C. Maximon and J.A Tjon, ****PRC 62, 054320(2000****) *

- Egle TOMASI-GUSTAFSSON

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## LSF Corrections (High orders included)

*The cross section:*

*The correction ( Leading Logarithm Approximation):*

*The vacuum polarization:*

*The K-factor:*

- Egle TOMASI-GUSTAFSSON

- August 5, 2008

## Radiative Corrections 

*Radiative Corrections** *

*Yu. Bystricky, E.A.Kuraev, E. T-G,  Phys. Rev. C75, 015207 (2007)*

*Q**2**=1 GeV**2*

*L.C. Maximon and J.A Tjon, *
***PRC 62, 054320(2000****) *

- Egle TOMASI-GUSTAFSSON

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## Radiative Corrections 

*LSF electron (not LLA)*

*MT (Z0) electron*

*MT  total*

*LSF LLA*

*MT (Z) two photon*

*LSF proton*

*MT (Z2) proton*

*LSF total*

*Radiative Corrections*

- Egle TOMASI-GUSTAFSSON

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*LSF total*

*LSF LLA*

*MT (Z0) total*

*MT (Z0) electron*

*MT (Z0) two photon*

*LSF proton*

*MT proton*

*LSF electron (not LLA)*

- Egle TOMASI-GUSTAFSSON

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- Egle TOMASI-GUSTAFSSON

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## LSF correction (SLAC data) point by point

*Q**2**=5 GeV**2*

- Egle TOMASI-GUSTAFSSON

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## The Pauli and Dirac Form Factors

-       Normalization
- F1p(0)=1,  F2p(0)= κp

- GEp(0)=1, GMp(0)=μp=2.79

-     The electromagnetic current in terms of the Pauli and Dirac FFs:

-     Related to the Sachs FFs :

- Egle TOMASI-GUSTAFSSON

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## Analytical properties of Compton amplitude

-  Elastic form factors and inelastic channels
-    are not independent  SUM RULES

- Egle TOMASI-GUSTAFSSON

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## Analytical properties of Compton amplitude

- Neglect left contribution and close contour on the right side (*10% accuracy*):

*Cancellation** of strong interaction effects  *
*in FFs and inelastic channels**!*

- Cancellation proved exactly in QED: *the L**2** ***** -contribution to FFs is cancelled by soft photon emission*

- Egle TOMASI-GUSTAFSSON

- August 5, 2008

## resonance (example of inelastic channels)

- Small  contribution ~0.5%
- Opposite sign with respect to proton intermediate state

*Cancellation of contributions in *
*elastic and inelastic channels **	**	*

- Egle TOMASI-GUSTAFSSON

- August 5, 2008

## Interference of 1 2 exchange

*Interference of 1**** ****2**** exchange*

- Enhancement due to the fast decreasing of form factors (transferred momentum equally shared between the two photons).
- Dipole approximation of FFs: Q02=0.71GeV2

- Egle TOMASI-GUSTAFSSON

- August 5, 2008

## Interference of 1 2 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 (1076)*

- Egle TOMASI-GUSTAFSSON

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## Two Photon Exchange

- No exact calculation for ep scattering
- ( inelastic intermediate states..)
- but
- electron-muon scattering
- 				constitutes an upper limit!

- Egle TOMASI-GUSTAFSSON

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## QED versus QCD

- Imaginary part of the 2 amplitude

*electron*

*proton*

- Egle TOMASI-GUSTAFSSON

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## QED versus QCD

- Q2=0.05 GeV2

- Q2=1.2 GeV2

- Q2=2 GeV2

- Egle TOMASI-GUSTAFSSON

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