QED Radiative Corrections for Precision Measurements of Nucleon Form Factors

Andrei Afanasev, Nucleon’05, Frascati, Oct.12, 2005Operated by the Southeastern Universities Research Association for the U.S. Dept. of Energy

QED Radiative Corrections for Precision Measurements of Nucleon Form Factors

Andrei Afanasev

Jefferson Lab

Nucleon 2005

INFN, Frascati, October 12, 2005


Main problem

. Uncertainties in QED radiative corrections limit interpretability of precision experiments on electron-hadron scattering


Plan of talk

Radiative corrections for electron scattering

. Model-independent and model-dependent; soft and hard photons

Two-photon exchange effects in the process e+p→e+p

. Models for two-photon exchange

. Cross sections

. Polarization transfer

. Single-spin asymmetries

. Parity-violating asymmetry

Refined bremsstrahlung calculations


Elastic Nucleon Form Factors

pm

q

peefi

fifi

utFtFuuueM

MM

))()(( 22121

1

•Based on one-photon exchange approximation

)0(

,

:1

)1(2

:)(

2121

2

220

y

ME

M

E

z

x

z

x

EM

P

FFGFFG

techniqueonPolarizatiG

G

A

A

P

P

techniqueRosenbluthGG

•Two techniques to measure

Latter due to: Akhiezer, Rekalo; Arnold, Carlson, Gross


Do the techniques agree?

. Both early SLAC and Recent JLab experiments on (super)Rosenbluth separations followed Ge/Gm~const

. JLab measurements using polarization transfer technique give different results (Jones’00, Gayou’02)

Radiative corrections, in particular, a short-range part of 2-photon exchange is a likely origin of the discrepancy

SLAC/Rosenbluth

JLab/Polarization

~5% difference in cross-sectionx5 difference in polarization


Basics of QED radiative corrections

(First) Born approximation

Initial-state radiation Final-state radiation

Cross section ~ dω/ω => integral diverges logarithmically: IR catastrophe

Vertex correction => cancels divergent terms; Schwinger (1949)

)}(2

1

36

17)1)(ln

12

13{(ln

2,)1( 2

2

exp

fm

Q

E

E

eBorn

Multiple soft-photon emission: solved by exponentiation,Yennie-Frautschi-Suura (YFS), 1961

e )1(


Complete radiative correction in O(αem )Radiative Corrections:• Electron vertex correction (a)• Vacuum polarization (b)• Electron bremsstrahlung (c,d)• Two-photon exchange (e,f)• Proton vertex and VCS (g,h)• Corrections (e-h) depend on the nucleon structure•Mo&Tsai; Meister&Yennie formalism•Further work by Bardin&Shumeiko; Maximon&Tjon; AA, Akushevich, Merenkov;

•Guichon&Vanderhaeghen’03:Can (e-f) account for the Rosenbluth vs. polarization experimental discrepancy? Look for ~3% ...

Main issue: Corrections dependent on nucleon structureModel calculations: •Blunden, Melnitchouk,Tjon, Phys.Rev.Lett.91:142304,2003•Chen, AA, Brodsky, Carlson, Vanderhaeghen, Phys.Rev.Lett.93:122301,2004


Separating soft 2-photon exchange

. Tsai; Maximon & Tjon (k→0)

. Grammer &Yennie prescription PRD 8, 4332 (1973) (also applied in QCD calculations)

. Shown is the resulting (soft) QED correction to cross section

. Already included in experimental data analysis

. NB: Corresponding effect to polarization transfer and/or asymmetry is zero

ε

δSoftQ2= 6 GeV2


Bethe-Heitler corrections to polarization transfer and cross sectionsAA, Akushevich, Merenkov Phys.Rev.D64:113009,2001;AA, Akushevich, Ilychev, Merenkov, PL B514, 269 (2001)

Pion threshold: um=mπ2

In kinematics of elastic ep-scattering measurements, cross sections are more sensitive to RC


Lorentz Structure of 2-γ amplitude

pAppeee

pm

q

ppeee

pepefi

pm

q

peefi

fififi

uusGuAuuA

uusFusFuVuuV

AAVVM

utFtFuuuM

MMM

55

221

2

2211

21

),(,

)),('),('(,

))()((

Generalized form factors are functions of two Mandelstam invariants;Specific dependence is determined by nucleon structure


New Expressions for Observables

AMEM

AEMEy

AMM

AEME

z

x

AMEM

GGGG

GGGGP

GGG

GGGG

P

P

GGGG

*222

**

*22

**

*2220

Re)1)(1(2||||

)1(21)Im()1(2)Im(

)1(2)Re(1||

)1(21)Re()1(2)Re(

)Re)1)(1(2|||(|

We can formally define ep-scattering observables in terms of the new form factors:

For the target asymmetries: Ax=Px, Ay=Py, Az=-Pz


Calculations using Generalized

Parton Distributions

Hard interaction with a quark

Model schematics: • Hard eq-interaction•GPDs describe quark emission/absorption •Soft/hard separation

•Use Grammer-Yennie prescription

AA, Brodsky, Carlson, Chen, Vanderhaeghen, Phys.Rev.Lett.93:122301,2004; hep-ph/0502013


Short-range effects; on-mass-shell quark(AA, Brodsky, Carlson, Chen,Vanderhaeghen)

Two-photon probe directly interacts with a (massless) quarkEmission/reabsorption of the quark is described by GPDs

su

tii

t

s

t

s

ut

u

s

su

t

s

ui

t

u

s

tf

su

suii

t

s

t

s

ut

u

s

su

t

s

ui

t

u

s

tti

s

uf

uuAuuV

fAAfVVt

eA

A

V

qeqeqe

qeqeqe

Aqe

Vqeemq

eqeq

2)]2))(log(log(

1))((log

1[

4

)(

)]log(1

))(log(1

[2

2)]2))(log(log(

1))((log

1[

4

)(

)]log(1

))(log(1

[2

)log(])[log(2

,

),(2

2

222

2

22

22

222

2

22

2

,5,,

,,,

22

Note the additional effective (axial-vector)2 interaction; absence of mass terms


`Hard’ contributions to generalizedform factors

GPD integrals

Two-photon-exchange form factors from GPDs


Two-Photon Effect for Rosenbluth Cross Sections

. Data shown are from Andivahis et al, PRD 50, 5491 (1994)

. Included GPD calculation of two-photon-exchange effect

. Qualitative agreement with data:

. Discrepancy likely reconciled


Updated Ge/Gm plot

AA, Brodsky, Carlson, Chen, Vanderhaeghen, Phys.Rev.Lett.93:122301,2004; hep-ph/0502013


Full Calculation of Bethe-Heitler Contribution

212

2

1

1

212

2

1

1

152

2

ˆ

2

ˆ

2

ˆ

2

ˆ

)ˆ)(ˆ1()ˆ(2

1

kk

k

kk

k

kk

k

kk

k

kk

k

kk

k

kk

k

kk

k

mkmkTrL er

Radiative leptonic tensor in full formAA et al, PLB 514, 269 (2001)

Additional effect of full soft+hard brem → +1.2% correction to ε-slopeResolves additional ~25% of Rosenbluth/polarization discrepancy!

Additional work by AA et al., using MASCARAD (Phys.Rev.D64:113009,2001) Full calculation including soft and hard bremsstrahlung


Polarization transfer

. Also corrected by two-photon exchange, but with little impact on Gep/Gmp extracted ratio


Charge asymmetry

. Cross sections of electron-proton scattering and positron-proton scattering are equal in one-photon exchange approximation

. Different for two- or more photon exchange

To be measured in JLab Experiment 04-116, Spokepersons W. Brooks et al.


Single-Spin Asymmetries in Elastic Scattering

Parity-conserving

. Observed spin-momentum correlation of the type:

where k1,2 are initial and final electron momenta, s is a polarization vector

of a target OR beam

. For elastic scattering asymmetries are due to absorptive part of 2-photon exchange amplitude

Parity-Violating

21 kks

1ks


Quark+Nucleon Contributions to An

. Single-spin asymmetry or polarization normal to the scattering plane

. Handbag mechanism prediction for single-spin asymmetry/polarization of elastic ep-scattering on a polarized proton target

HGPDondependenceNo

BGAGA MER

n

~

)Im(2

1)Im(

1)1(2

Only minor role of quark mass


Radiative Corrections for Weak Processes

Semi-Leptonic processes involving nucleons

. Neutrino-nucleon scattering

Per cent level reached by NuTeV. Radiative corrections for DIS calculated at a partonic level (D. Bardin et al.)

. Neutron beta-decay: Important for Vud measurements; axial-vector coupling gA

Marciano, Sirlin, PRL 56, 22 (1986); Ando et al., Phys.Lett.B595:250-259,2004; Hardy, Towner, PRL94:092502,2005

. Parity-violating DIS: Bardin, Fedorenko, Shumeiko, Sov.J.Nucl.Phys.32:403,1980; J.Phys.G7:1331,1981, up to 10% effect from rad.corrections

. Parity-violating elastic ep (strange quark effects, weak mixing angle)


Parity Violating elastic e-N scattering

Longitudinally polarized electrons,

unpolarized target

2

e e pp

unpol

AMEF

LR

LR AAAQGA

224

2

sRFGQG

GGRGRQGe

AZA

eA

sME

nME

nA

pME

pAW

ZME

)()(

)1()1)(sin41()(2

,,,22

,

45.3

sin41

282

W

MeAWA

MZMM

EZEE

GGA

GGA

GGA

)sin41(

)(

2

Neutral weak form factors contain explicit contributions from strange sea

GZA(0) = 1.2673 ± 0.0035 (from decay)

= Q2/4M2

= [1+2(1+)tan2(/2)]-1

’= [(+1)(1-2)]1/2


Born and Box diagrams for elastic ep-scattering

. (d) Computed by Marciano,Sirlin, Phys.Rev.D29:75,1984, Erratum-ibid.D31:213,1985 for atomic PV

. (c) Presumed small, e.g., M. Ramsey-Musolf, Phys.Rev. C60 (1999) 015501


New Expressions for PV asymmetry

. PV asymmetry in terms of generalized form factors including multi-photon exchange

2/1,2/)sin41(

,)1)(1(2

,2,2

)()(24

2

2

22

2

eAW

eV

MpZA

eV

BornA

EpZMp

eA

BornMEp

ZEp

eA

BornE

EpMp

BornA

BornM

BornEF

LR

LRBornPV

gg

GGgA

GGgAGGgA

GG

AAAQGA

PV-asymmetry, Born Approximation

ApEpMp

AMAMEF

LR

LRPV

GGG

AAAAAQGA

')1(2||||

''

24 22'

2


2γ-exchange Correction to Parity-ViolatingElectron Scattering

. New parity violating terms due to (2gamma)x(Z0) interference should be added:

xγ γ Z0

Electromagnetic Neutral Weak

)Re()1(2'

)Re()1)(1(2''

'2

ApZA

eVA

ApZM

eAM

GGgA

GGgA


GPD Calculation of 2γ×Z-interference. Can be used at higher Q2, but points at a

problem of additional systematic corrections for parity-violating electron scattering. The effect evaluated in GPD formalism is the largest for backward angles:

AA & Carlson, hep-ph/0502128, Phys. Rev. Lett. 94, 212301 (2005): measurements of strange-quark content of the nucleon are affected, Δs may shift by ~10%

Important note: (nonsoft) 2γ-exchange amplitude has no 1/Q2 singularity;1γ-exchange is 1/Q2 singular => At low Q2, 2γ-corrections is suppressed as Q2 P. Blunden used this formalism and evaluated correction of 0.16% for Qweak


. Quark-level short-range contributions are substantial (2-3%)

. 2γ-correction to parity-violating asymmetry does not cancel. May reach a few per cent for GeV momentum transfers

. Corrections are angular-dependent, not reducible to re-definition of coupling constants

. Revision of γZ-box contribution and extension of model calculations to lower Q2 is necessary

. Experimentally measurable directly by comparing electrons vs positrons on a spin-0 target- it is difficult => in the meantime need to rely on the studies of 2γ-effect for parity-conserving observables

Two-photon exchange for PV electron-proton scattering


Parity-Conserving Single-Spin Asymmetries in Scattering Processes(early history). N. F. Mott, Proc. R. Soc. (London), A124, 425 (1929), noticed that

polarization and/or asymmetry is due to spin-orbit coupling in the Coulomb scattering of electrons (Extended to high energy ep-scattering by AA et al., 2002).

. Julian Schwinger, Phys. Rev. 69, 681 (1946); ibid., 73, 407 (1948), suggested a method to polarize fast neutrons via spin-orbit interaction in the scattering off nuclei

. Lincoln Wolfeinstein, Phys. Rev. 75, 1664 (1949); A. Simon, T.A.Welton, Phys. Rev. 90, 1036 (1953), formalism of polarization effects in nuclear reactions


Proton Mott Asymmetry at Higher Energies

. Due to absorptive part of two-photon exchange amplitude; shown is elastic contribution

. Nonzero effect observed by SAMPLE Collaboration (S.Wells et al., PRC63:064001,2001) for 200 MeV electrons

. Calculations of Diaconescu, Ramsey-Musolf (2004): low-energy expansion version of hep-ph/0208260

Transverse beam SSA, units are parts per million Figures from AA et al, hep-ph/0208260

Spin-orbit interaction of electron moving in a Coulomb field N.F. Mott, Proc. Roy. Soc. London, Set. A 135, 429 (1932);BNSA for electron-muon scattering: Barut, Fronsdal, Phys.Rev.120, 1871 (1960);BNSA for electron-proton scattering: Afanasev, Akushevich, Merenkov, hep-ph/0208260


Phase Space Contributing to the absorptivepart of 2γ-exchange amplitude

. 2-dimensional integration (Q12, Q2

2) for the elastic intermediate state

. 3-dimensional integration (Q12, Q2

2,W2) for inelastic excitations Examples: MAMI A4

E= 855 MeVΘcm= 57 deg;

SAMPLE, E=200 MeV

`Soft’ intermediate electron;Both photons are hard collinear

One photon is Hard collinear


MAMI data on Mott Asymmetry

. F. Maas et al., Phys.Rev.Lett.94:082001, 2005

. Pasquini, Vanderhaeghen:

Surprising result: Dominance of inelastic intermediate excitations

Elastic intermediatestate

Inelastic excitationsdominate


Beam Normal Asymmetry(AA, Merenkov)

0

ˆ

)ˆ1)(ˆ()ˆ(4

1

)ˆ()ˆ1)(ˆ()ˆ(4

1

2Im

),(

1212

512

512

22

210

3

22

2,

qHqHqHqLqLqL

aa

TMpMpTrH

mkmkmkTrL

QQ

HL

k

kd

QsD

QA

p

eeee

Pen

Gauge invariance essential in cancellation of infra-red singularity for target asymmetry0/0 2

22

1 QorandQifHL

Feature of the normal beam asymmetry: After me is factored out, the remaining expression is singular when virtuality of the photons reach zero in the loop integral!But why are the expressions regular for the target SSA?!

Also calculations by Vanderhaeghen, Pasquini (2004); Gorschtein, hep-ph/0505022Confirm quasi-real photon exchange enhancement

2

2

2

222

22

1 log,log~0/e

e

e

eem

Qm

m

QmAQorandQifconstmHL


Peaking Approximation

. Dominance of collinear-photon exchange =>

. Can replace 3-dimensional integral over (Q12,Q2

2,W) with one-dimensional integral along the line (Q1

2≈0; Q22=Q2(s-W2)/(s-M2))

. Save computing time

. Avoid uncertainties associated with (unknown) double-virtual Compton amplitude

. Provides more direct connection to VCS and RCS observables


Special property of normal beam asymmetry

. Reason for the unexpected behavior: hard collinear quasi-real photons

. Intermediate photon is collinear to the parent electron

. It generates a dynamical pole and logarithmic enhancement of inelastic excitations of the intermediate hadronic state

. For s>>-t and above the resonance region, the asymmetry is given by:

)2)(log(8

)(2

2

22

21

212

2

e

ep

en m

Q

FF

FFQmA

Also suppressed by a standard diffractive factor exp(-BQ2), where B=3.5-4 GeV-2 Compare with no-structure asymmetry at small θ:

AA, Merenkov, Phys.Lett.B599:48,2004, Phys.Rev.D70:073002,2004;

+Erratum (2005)

3s

mA ee

n


Input parameters

. Used σγp from parameterization by N. Bianchi at al., Phys.Rev.C54 (1996)1688

(resonance region) and Block&Halzen, Phys.Rev. D70 (2004) 091901


Predictions for normal beam asymmetry

Use fit to experimental data on σγp and exact 3-dimensional integration over phase space of intermediate 2 photons

HAPPEX

Data from HAPPEX More to come from G0

2γ-exchange is in diffractive regime


No suppression for beam asymmetry with energyat fixed Q2

x10-6 x10-9

Parts-per-million vs. parts-per billion scales: a consequence ofsoft Pomeron exchange, and hard collinear photon exchange

SLAC E158 kinematics


Beam SSA in the resonance region


Normal Beam Asymmetry on Nuclei

. Important systematic correction for parity-violation experiments (HAPPEX on 4He, PREX on Pb)

. For diffractive 2γ-exchange, scales as ~A/Z~2

Five orders of magnitude enhancement for (HAPPEX) due to excitation of inelastic intermediate states in 2γ-exchange (AA, Merenkov, to be published)


Summary on SSA in Elastic ep-Scattering

. Collinear photon exchange present in (light particle) beam SSA

. (Electromagnetic) gauge invariance of is essential for cancellation of collinear-photon exchange contribution for a (heavy) target SSA

. Unitarity is crucial in reducing model dependence of calculations at small scattering angles, in particular for beam asymmetry

. Strong-interaction dynamics for BNSA small-angle ep-scattering above the resonance region is soft diffraction

. For the diffractive mechanism An is a) not supressed with beam energy and b) does not grow with Z

. If confirmed experimentally → first observation of diffractive component in elastic electron-hadron scattering


. Quark-level short-range contributions are substantial (3-4%) ; correspond to J=0 fixed pole (Brodsky-Close-Gunion, PRD 5, 1384 (1972)).

. Structure-dependent radiative corrections calculated using GPDs bring into agreement the results of polarization transfer and Rosenbluth techniques for Gep measurements

. Full treatment of brem corrections removed ~25% of R/P discrepancy in addition to 2γ

. Collinear photon exchange + inelastic excitations dominance for beam asymmetry

. Experimental tests of multi-photon exchange. Comparison between electron and positron elastic scattering (JLab E04-116). Measurement of nonlinearity of Rosenbluth plot (JLab E05-017). Search for deviation of angular dependence of polarization and/or asymmetries

from Born behavior at fixed Q2 (JLab E04-019)

. Elastic single-spin asymmetry or induced polarization (JLab E05-015)

. 2γ additions for parity-violating measurements (HAPPEX, G0)

Through active theoretical support emerged a research program of

Testing precision of the electromagnetic probeDouble-virtual VCS studies with two space-like photons

Two-photon exchange for electron-proton scattering

QED Radiative Corrections for Precision Measurements of Nucleon Form Factors

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