Two-photon exchange: hadronic picture

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Two-photon exchange hadronic picture

• Peter Blunden
• University of Manitoba

Trento, May 15, 2008
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Outline

• Introduction Rosenbluth vs polarization
  measurements of GE and GM of nucleon
• puzzle different results extracted for
  GE/GM
• Hadronic model of two-photon exchange (TPE)
• Results for unpolarized and polarized cross
  sections ep! ep (real part of TPE)
• Resonance contributions (?) to elastic scattering
• Low Q2 limit (second Born approximation)
• Parity violating asymmetry APV (?? and ?Z)
• Off-shell effects

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Rosenbluth vs polarization transfer measurements
of GE/GM of proton
SLAC Rosenbluth data
JLab Polarization data
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Rosenbluth separation method
One-photon exchange cross section

• ? is virtual photon polarization
• Forward scattering ?! 1
• Backward scattering ? ! 0
• Extract GE and GM from linear ? dependence
• GE suppressed as Q2 increases
• Sensitive to small corrections linear in ?

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Polarization transfer method

• Look at ratio of transverse (PT) to longitudinal
  (PL) components of recoil proton polarization
  using a longitudinally polarized electron beam
• Doesnt depend on absolute normalization
• Ratio relatively insensitive to radiative
  corrections (e.g. bremsstrahlung corrections
  cancel)

in one-photon exchange approximation
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• Speculation radiative corrections
• d?0 ! d? d?0 (1?RC)
• Missing effect is
• approximately linear in ?
• not strongly Q2 dependent
• Two-photon exchange
• Bremsstrahlung
• SuperRosenbluth(detect proton)

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Comments on radiative corrections

• Radiative corrections different depending on
  whether electron or proton is detected.
• well understood
• Soft bremsstrahlung
• involves long-wavelength photons
• independent of hadronic structure
• Box diagrams (TPEX M??) involve photons of all
  wavelengths
• long wavelength (soft photon) part is
  included in radiative correction (IR divergence
  is cancelled with electron proton bremsstrahlung
  interference)
• also independent of hadronic
  structure (by construction)

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Hadronic approach N, ?, intermediate states
Obeys gauge invariance and crossing
symmetry Crossed box from box by p1!
p3 Consider ? ?2?-?IR(MT) ?IR(MT) is
standard Mo Tsai correction (soft photon
exchange), which is ?-independent IR
divergent IR divergent terms cancel in ?
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Partonic (GPD) calculation of two-photon
exchange contribution(Chen et al.)
handbag
cats ears
valid at large Q2 ?hard handbag diagrams
(one active quark) to reproduce the IR divergent
contribution at nucleon correctly (Low Energy
Theorem) ?soft need cats ears diagrams (two
active quarks)
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Nucleon elastic contribution (BMT)

• Model form factors used as input in calculation

magnetic proton form factorBrash et al. (2002)
electric proton form factor GE/GM of proton
fixed from polarization dataGayou et al. (2002)
Parametrize as sum of monopoles! maintains
analytic form of result Numerical results not
terribly sensitive to model for GE, or to details
of GM
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Corrections to unpolarized cross sections for
Q21 to 6 GeV2
Effect largest at small ? (backward angles) Small
effect as ?! 1? 1 - Q2/(2 E2) Nonlinearity
grows with Q2 JLAB E05-017 (Arrington) will set
limits on nonlinearity
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Dependence on form factor in loop integrals
Effect of LT and PT values for GE/GM on TPEX
correction ! Mostly sensitive to GM
Realistic (solid) vs. dipole (dashed)
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Effect on ratio R
NOT a refit of data Simple model correct
Rosenbluth data assuming TPEX correction is
linear in ? over a certain range
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Effect on SLAC reduced cross sections at
different Q2 (normalized to dipole GD2)
Nonlinearity in ? is displayed here JLAB
proposals to measure nonlinearity
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SuperRosenbluth (JLAB) data
Curves shifted by 1.0 2.64 2.1
3.20 3.0 4.10 (Effect on determination of
GM)
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Effect on ratio of ep to e-p cross sections
(SLAC, Q2 from 0.01 to 5 GeV2)
MBorn opposite sign for ep vs. e-p, so
enhancement instead of suppression as ?!
0 R(ep/e-p) (1-?)/(1?) ¼ 1-2? Curves
are elastic results for Q21, 3, 6 GeV2 Proposed
expts. E04-116 Q2 lt 2 GeV2 VEPP-3 Q21.6 GeV2,
?¼ 0.4
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Phenomenology Generalized form factors
Kinematical invariants
In limit me! 0 (helicity conservation) general
amplitude can be put in form
Generalized (complex) form factors
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Observables including two-photon exchange Real
parts of two-photon amplitudes

• Caution needed about assumptions (generalized
  FFs are not observables)
• Parametrization of amplitude NOT unique
• Axial parametrization GA (???5)(e) (???5)(p)
  instead of F3 (or Y2) term
• shifts some F3 into ?F1 (and hence into ?GE and
  ?GM)

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Real part of elastic results
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SSA in elastic eN scattering
spin of beam OR target OR recoil proton NORMAL to
scattering plane
on-shell intermediate state (MX W)

• Target general formula of order e2
• GPD model allows connection of real and
  imaginary amplitudes
• Hadronic models sensitive to intermediate state
  contributions,
• no reliable theoretical calculations at
  present
• Beam general formula of order me e2 (few ppm)
• Measured in PV experiments (longitudinally
  polarized electrons)
• at SAMPLE and A4 (Mainz)
• Only non-zero result so far for TPEX

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Corrections to PL and PT at Q21, 3, and 6 GeV2

• PT/PL will show some variation with ?, esp. at
  low ?
• JLab data taken at ?0.7
• JLAB expt (Gilman) will measure PT/PL at low ?
• GPD calculation predicts suppression of PT/PL

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Resonance (?) contribution ?(q?) ?(p?) ! N
?N? vertex

• Lorentz covariant form
• Spin ½ decoupled
• Obeys gauge symmetries

3 coupling constants g1, g2, and g3 At ? pole g1
magnetic (g2-g1) electric g3
Coulomb Take dipole FF F?(q2) 1/(1-q2/??2)2
with ?? ¼ 0.84 GeV
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No infrared divergences (since M? gt MN) The ?N?
vertex was used in Dressed K-matrix model
(Kondratyuk and Scholten) to describe pion
photoproduction, ?N scattering, Compton
scattering at low to medium energies g1 and g2
taken from fits to E2/M1 ratio Coulomb
contribution (g3)2 and is small, independent of
sign
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• Smaller than nucleon contribution for reasonable
  range of parameters
• Becomes more important as Q2 increases
• Partially cancels the nucleon only contribution
  at backward angles
• Reduces nonlinear ? dependence somewhat

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• Other resonances
• N (P11), ? (P33) D13, D33, P11, S11, S31
• Parameters from dressed K-matrix model

• Results
• contribution of heavier resonances much smaller
• than N and ?
• D13 next most important (consistent with second
  resonance shape of Compton scattering cross
  section)
• partial cancellation between spin 1/2 and spin
  3/2
• leads to better agreement, especially at high Q2

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• Global Analysis (Arrington et al,
  nucl-ex/0707.1861)
• Incorporate TPE effects directly into analysis of
  Rosenbluth and PT data
• Extract GE and GM over range of Q2
• Input Estimate of Q2 dependence of higher
  resonances from hadronic and GPD calculations
• ?2? 0.01 (?-1) ln Q2/ln 2.2 Q2gt1
  GeV2together with nucleon elastic contribution,
  with 100 uncertainty
• linear in ?
• decreases cross section by 1 at Q2 2.2 GeV2
• Hadronic and GPD agree TPE corrections to PT data
  are small (2), but give opposite signs
• ! Dont include in analysis of PT data

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Effect on ratio R
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Extraction ofGM and GE
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? production e N ! e ?

• Can obtain information on ?N? transition current
  
• Need ??? vertex as well, issues of gauge
  invariance
• JLAB experiment looking for nonlinearities in ?
  over Q2 0.5 to 4 GeV2

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Results for TPEX correction to Born ?
production nucl-th/0601063 No strong
nonlinearities evident, but effect is large
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proton correction at low Q2
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proton correction at Q20.01 GeV2

• Essentially independent of mass (same for muon,
  quarks)
• At high Q2, GM dominates the loop integral
• At low Q2, GE dominates
• neutron correction vanishes at low Q2 (pointlike
  neutron)

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Proton radii and TPEX

• H atom
• 2p-1s transition energy known to 14 digits
• 1s hyperfine structure interval (HFS) known to 12
  digits
• Tests of QED now depend on accuracy of proton
  finite-size corrections,as determined through
  e-p scattering
• 2p-1s needs charge rms radius rrms 0.895
  0.018 fm
• ! calculated and expt. energies agree
• 1s HFS needs Zemach moment (convolution of
  charge and
• magnetization densities)
• r2 1.086 0.012 fm
• ! calculated and expt. energies disagree by 3.6
  ppm
• Partly explained by nuclear polarization
  (1.6 ppm)
• Interpreted using OPE Coulomb distortion (soft
  photons).
• What is role of TPEX?

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Comparison with 2nd Born approximation (soft
photon exchange) (Blunden and Sick, PRC 2005)
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Analysis

• Reanalyze world e-p cross section data up to 4
  fm-1 including contribution of TPEX effects
• rrms 0.897 0.018 fm (from 0.895 0.018 fm)
• r2 1.091 0.012 fm (from 1.086 0.012 fm)
• change in r2 is small (40 of error bar), but in
  wrong direction to explain HFS discrepancy
• ! discrepancy cannot be attributed to TPEX

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Alternative recent work on low Q2 TPEX(Borisyuk
Kobushkin, nucl-th/0612104)
total
?0.5
electric
magnetic
Claim significant effects at low Q2, which reduce
proton rms radius Implications for PV?
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Neutron No infrared divergences Positive and
about 2-3 times smaller than proton (dominance of
magnetic form factor?) Some model dependence due
to choice of form factors (blue curve)
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Effect on Parity-violating asymmetry in elastic
ep
Weak radiative corrections interfere with M? (MZ
! MZ M?Z) Electromagnetic radiative corrections
interfere with MZ (M? ! M? M??)
Afanasev and Carlson used generalized form
factors to analyze effect of ?? on A (GPD model)
So for example the proton contribution to the
vector asymmetry is
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Qweak At low Q2, forward angles (?! 1)
A(1 - 4 sin2?W) independent of hadron
structure Bhadronic correction
Qweak aims for a 2 measurement of APV Though
not obvious at first glance, AM and AA are of
order Q2 Our corrections to A vanish as ?! 1 At
Qweak kinematics, TPEX correction is -0.05
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APV vs. ? for Q2 0.1, 0.5, 1.0, 3.0 GeV2
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?Z electroweak as well as TPE Hadronic model, 2
recent PRLs Zhou et al., Tjon Melnitchouk
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?0, ?0 standard EW corrections
At large Q2 where GM dominates, expect 2?
contribution to ??¼ 1-?2?/2, and
??¼ 0.
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• Significant ? dependence of ?Z boxes
• in ?? manifests at low Q2

• Marciano/Sirlin calculation
• Atomic parity violation
• Q2 0 limit (no IR divergences)
• includes Born (nucleon) asymptotic (quark)
  contribution beyond some mass scale M ( 0.5 to
  1.0 GeV)
• Total ?? -0.0037, ?? -0.0053
• Excluded terms in ?Z amplitude corresponding to
  long range Coulomb interaction of bound
  electron with nucleus (already included in
  bound state wavefunction)
• These latter terms are not excluded here (or in
  Zhou et al. Tjon/Melnitchouk)
• Would second Born approximation (soft photon)
  give similar behaviour at low Q2?

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?Z electroweak as well as TPE Hadronic model,
Tjon Melnitchouk (PRL 100, 082003 (2008))
?(? Z)
Z(??)
?(??)
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TPEX using dispersion relations(Borisyuk
Kobushkin, nucl-th/0804.4128)
s dk ?h

• Imaginary part determined by unitarity
• Only on-shell form factors
• Real part determined from dispersion relations
• Numerical differences between naive (solid)
  and dispersion (dashed) analyses are small
• Similar insensitivity seen for ?
• (Blunden, unpublished)

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Outlook

• Theory
• Connect real and imaginary parts of TPEX
  amplitude
• more work needs to be done on hadronic models
• Recent work seems to indicate insensitivity to
  off-shell form factors
• Merge with GPD calculations?
• Firm up understanding of ?Z contributions at low
  Q2
• Resonance contributions to ?Z boxes?
• Experiment
• ep/e-p ratio
• look for nonlinearity in ?
• Collaborators Melnitchouk, Tjon Kondratyuk
  (N?), Kondratyuk (resonances) Scholte (APV)