Radiative Corrections for Elastic Scattering of
Muons and Electrons on a Proton
Andrei Afanasev
The George Washington University, Washington, DC, USA
Andrei Afanasev, Trento workshop, 31 October 2012
Plan of talk
Radiative corrections for elastic charge lepton scattering
. Model-independent and model-dependent; soft and hard photons
. Refined bremsstrahlung calculations
. Role of the masses
Two-photon exchange effects
. Formulation of a problem
. Soft-photon exchange approximation and IR regularization
. Beyond soft-photon approximation
Summary
Andrei Afanasev, Trento workshop, 31 October 2012
Elastic Nucleon Form Factors
•Based on one-photon exchange approximation
M fi  M 1fi
1
fi
M  e ue  ueu p ( F1 (t )  
•Two techniques to measure
2
   q
2m
F2 (t ))u p
   0 (GM 2    GE 2 ) : Rosenbluth technique
GE  2 (1   )
Px
Ax


Pz
Az
GM  1   2
: Polarizati on technique
GE  F1  F2 , GM  F1  F2
( Py  0)
Latter due to: Akhiezer, Rekalo; Arnold, Carlson, Gross
Andrei Afanasev, Trento workshop, 31 October 2012
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)
Assumed Q2/me2>>1
 2
E 13
Q2
17 1
 exp  (1   ) Born,  
{(ln
 )(ln 2  1) 
 f ( )}

E 12
me
36 2
Multiple soft-photon emission: solved by exponentiation,
Yennie-Frautschi-Suura (YFS), 1961
(1   )  e
Andrei Afanasev, Trento workshop, 31 October 2012
Complete radiative correction in O(αem )
Log-enhanced but calculable (a,c,d)
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
•Meister&Yennie; Mo&Tsai
•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 structure
Model calculations:
•Blunden, Melnitchouk,Tjon, Phys.Rev.Lett.91:142304,2003
•Chen, AA, Brodsky, Carlson, Vanderhaeghen, Phys.Rev.Lett.93:122301,2004
Andrei Afanasev, Trento workshop, 31 October 2012
Bremsstrahlung for Relativistic vs Nonrelativistic
Lepton Scattering
.
.
Accelerated charge always radiates, but the magnitude of the effect
depends on kinematics
See Bjorken&Drell (Vol.1, Ch.8):
. For large Q2>>me2 the rad.correction is enhanced by a large
logarithm, log(Q2/me2) ~15 for GeV2 momentum transfers
. For small Q2<<me2, rad.correction suppressed by Q2/me2
. For intermediate Q2~me2, neither enhancement nor suppression,
rad correction of the order 2α/π
Andrei Afanasev, Trento workshop, 31 October 2012
Basic Approaches to QED Corrections
.
.
.
L.W. Mo, Y.S. Tsai, Rev. Mod. Phys. 41, 205 (1969); Y.S. Tsai, Preprint
SLAC-PUB-848 (1971).
. Considered both elastic and inelastic inclusive cases. No polarization.
D.Yu. Bardin, N.M. Shumeiko, Nucl. Phys. B127, 242 (1977).
. Covariant approach to the IR problem. Later extended to inclusive, semiexclusive and exclusive reactions with polarization.
E.A. Kuraev, V.S. Fadin, Yad.Fiz. 41, 7333 (1985); E.A. Kuraev,
N.P.Merenkov, V.S. Fadin, Yad. Fiz. 47, 1593 (1988).
. Developed a method of electron structure functions based on Drell-Yan
representation; currently widely used at e+e- colliders
. Applied for polarized electron-proton scattering by AA et al, JETP 98,
403 (2004).
Andrei Afanasev, Trento workshop, 31 October 2012
Separating soft 2-photon exchange
.
.
.
.
Tsai; Maximon & Tjon (k→0); similar to Coulomb corrections at low Q2
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 for elastic ep
.
Also done for pion electroproduction in AA, Aleksejevs, Barkanova, arXiv:1207.1767
(there we also used Passarino-Veltman parameterization of loop integrals, inclusion of
lepton masses is straightforward)
ε
q1→q
q2→0
δSoft
Q2= 6 GeV2
Lepton mass is not essential for TPE calculation in ultra-relativistic case;
Two-photon effect below 1% for lower energies and Q2<0.1GeV2
Andrei Afanasev, Trento workshop, 31 October 2012
Calculations using Generalized
Parton Distributions
GPD
e-
Model schematics:
• Hard eq-interaction
•GPDs describe quark
emission/absorption
•Soft/hard separation
•Use Grammer-Yennie
prescription
Hard interaction with
a quark
q
AA, Brodsky, Carlson, Chen, Vanderhaeghen,
Phys.Rev.Lett.93:122301,2004; Phys.Rev.D72:013008,2005
Andrei Afanasev, Trento workshop, 31 October 2012
Two-Photon Effect for Rosenbluth Cross Sections
.
.
.
.
Data shown are from Andivahis et al,
PRD 50, 5491 (1994)
Included GPD calculation of twophoton-exchange effect
Qualitative agreement with data:
Discrepancy likely reconciled
Andrei Afanasev, Trento workshop, 31 October 2012
Updated Ge/Gm plot
AA, Brodsky, Carlson, Chen, Vanderhaeghen,
Phys.Rev.Lett.93:122301, 2004; Phys.Rev.D72:013008, 2005
• Significant part of the discrepancy is
removed by the TPE mechanism
• Verification coming soon from VEPP,
OLYMPUS, JLAB CLAS experiments
Andrei Afanasev, Trento workshop, 31 October 2012
Soft TPE for low-energy electron scattering
. Soft TPE correction for ep scattering using Mo-Tsai formalism for
three fixed electron energies: 115 MeV/c (top), 153 MeV/c (middle)
210 MeV/c (bottom curve); no large logs involved, muon correction is
about the same: 0.25% max. Used kinematics for MUSE experiment
proposal at PSI
Andrei Afanasev, Trento workshop, 31 October 2012
TPE for Proton Charge Radius Extraction
.
Electron scattering: Rch increase by +0.0015 fm, according to Blunden
and Sick, Phys. Rev. C 72, 057601 (2005). “The radius after twophoton correction amounts to 0.897+/-0.018 fm”
Andrei Afanasev, Trento workshop, 31 October 2012
Hard Bremsstrahlung
.
Need to include radiative lepton tensor in a complete form:
AA et al, Phys.Rev. D64 (2001) 113009; PLB 514, 269 (2001): terms ~ k
emitted photon momentum) usually neglected in rad.correction
calculations, but can lead to ~1% effect for Rosenbluth slope at high Q2
1
L    Tr (kˆ2  m) (1   5ˆe )( kˆ1  m) 
2
  kˆ    kˆ 
 k1
k 2 
  
  


2k  k1 2k  k 2
 k  k1 k  k 2 
 k1
k 2 
  kˆ    kˆ 
  
  


2k  k1 2k  k 2
 k  k1 k  k 2 
r
Andrei Afanasev, Trento workshop, 31 October 2012
Electron Structure Functions
. For polarized ep->e’X scattering, AA et al, JETP 98,
403 (2004); elastic ep: AA et al. PRD 64, 113009
(2001).
. Resummation technique for collinear photons
(=peaking approx.)
. Difference <0.5% from previous calculation
including hard brem
.
Bystritskiy, Kuraev, Tomasi-Gustafson (2007)
claimed this approach resolves Rosenbluth vs
polarization discrepancy… but used incorrect
energy cutoff ΔE/E of 3% (instead of e.g. 1.5%)
=>miscalculated rad.correction by ~5% (absolute)
Andrei Afanasev, Trento workshop, 31 October 2012
Rosenbluth vs Super-Rosenbluth
.
.
.
.
Two experimental settings realized for elastic ep-scattering:
. p(e,e’)p: detection of scattered electron, or “Rosenbluth”
. p(e,p’)e: detection of recoil proton, or “super-Rosenbluth”
For GeV energies and momentum transfers bremsstrahlung corrections
are different, but of the same order ~10-20%.
Caution for “super-Rosenbluth”: hard collinear bremsstrahlung from
the scattered electron is allowed, may lead to numerical instabilities in
rad.correction calculations (corresponding large logarithms from soft
and hard collinear photon emission cancel each other)
See AA, Akushevich, Ilychev, PLB 514, 269 (2001).
Third type experiment: JLAB Hall B Positron exp (talk by Brian Raue)
. Detecting final e’ and p’, beam energy reconstructed
Andrei Afanasev, Trento workshop, 31 October 2012
Lepton Mass Effects
.
.
.
.
.
Standard approximations keep the lepton mass in the logarithms but
neglect it in power terms. May be justified in the ultrarelativistic case
and Q2>>(lepton mass)2
Most of analysis codes use exact mass dependence for hard brem, but
use above approximations for the “soft” part of brem correction
Revised approach is required that will NOT result in new theoretical
uncertainties
New rad.correction codes no longer use peaking approximation
(justified for relatively small lepton masses)
Formalism and Monte-Carlo generators can be adapted for this
analysis (ELRADGEN; MASCARAD, etc;
more on www.jlab.org/RC)
Andrei Afanasev, Trento workshop, 31 October 2012
Coulomb and Two-Photon Corrections
.
.
.
.
.
Coulomb correction calculations are well justified at lower energies
and Q2 (see also John Arrington’s slides from this workshop)
Hard two-photon exchange (TPE) contributions cannot be calculated
with the same level of precision as the other contributions.
Two-photon exchange is independent on the lepton mass in an ultrarelativistic case.
Issue: For energies ~ mass TPE amplitude is described by 6
independent generalized form factors; but experimental data on TPE
are for ultrarelativistic electrons, hence independent info on 3 other
form factors will be missing.
Theoretical models show the trend that TPE has a smaller effect at
lower Q2 . The reason is that “hard” TPE amplitudes do not have a
1/Q2 Coulomb singularity, as opposed to the Born amplitude.
Andrei Afanasev, Trento workshop, 31 October 2012
Beam Single-Spin Asymmetry
Parity-conserving
. Observed spin-momentum correlation of the type:
  
s  k1  k2
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 2photon exchange amplitude
. Beam asymmetry (~10-5) measured at Mainz and JLAB in parallel with
parity-violating asymmetries (G0, HAPPEX, PREX, Qweak)
. See most recent data from PREX&HAPPEX on nuclei,
arXiv:1208.6164 (to appear in PRL)
Andrei Afanasev, Trento workshop, 31 October 2012
Special Feature of the Transverse Beam Asymmetry
.
.
.
.
.
Can we use beam asymmetry to constrain TPE in muon scattering
cross section?
Proportional to lepton mass (lepton helicity violation is required for
nonzero asymmetry)
But the asymmetry is enhanced due to hard collinear photon exchange
(at least one of the photons in two-photon exchange mechanism), this
feature is unique for transverse polarization; corresponding mechanism
does not appear in the cross section as a consequence of
electromagnetic current conservation (AA, Merenkov, Phys.Lett. B599
(2004) 48; hep-ph/0407167v2; Phys.Rev. D70 (2004) 073002)
Asymmetry is intedepndent of beam energy at fixed Q2 (AA,
Merenkov’04; Gorchtein’04)
Cannot use the asymmetry to constrain directly lepton helicityviolating TPE contribution to cross section
Andrei Afanasev, Trento workshop, 31 October 2012
Helicity-Flip in TPE; estimate of inelastic contribution
.
Scalar t-channel exchange contributes to TPE if you give up ultrarelativistic limit (no longer setting mlepton to zero!)
X
4 t (1+ t )(1- e 2 )mm M N Fsmm fs NN GEp
ds = 2
2
(t GMp
+ eGEp
)(Q 2 + ms2 )
2g
.
For Q2=10-2 GeV2 ds is about 10-3, lepton helicity-flip dominates,
scales as t , t = Q2 / 4M N2
Can be studied directly in the ratio of μ+ and μ- cross sections
2g
Andrei Afanasev, Trento workshop, 31 October 2012
Conclusions
.
.
.
.
Radiative corrections are standard and well-established in electron
scattering, care must be taken in this experiment that the radiative
correction calculations are correctly implemented without invalid
approximations.
Parts of the radiative corrections are expected to be suppressed for
muons due to the larger muon mass.
Two-photon exchange corrections are generally expected to be small,
helicity-flip mechanism is enhanced for the muons
. Can be studied directly in the ratio of μ+ and μ- cross sections
We evaluate systematic uncertainty to cross sections from radiative
corrections at 0.1% for muon at Q2~0.01 GeV2, due to uncertainty in
two-photon exchange effects
Andrei Afanasev, Trento workshop, 31 October 2012