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Nucleon Form Factors
99% of the content of this talk is courtesy of Mark Jones (JLab)
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Overview of nucleon form factor measurements
Review articles C. F. Perdrisat, V. Punjabi, M. VanderhaeghenProg.Part.Nucl.Phys.59:694,2007
J. Arrington, C. D. Roberts, J. M. Zanotti, J.Phys.G34:S23-S52,2007
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Nucleon Form Factors• Nucleon form factors describe the distribution of
charge and magnetization in the nucleon– This is naturally related to the fact that nucleons are
made of quarks • Why measure nucleon form factors?
– Understand structure of the nucleon at short and long distances
– Understand the nature of the strong interaction (Quantum Chromodynoamics) at different distance scales
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Strong Interactions – the right theory
• Strong interactions Quantum Chromodynamics (QCD)
• Protons and neutrons made of quarks (mostly up and down)– Quarks carry “color” charge– Gluons are the mediators of the strong force
• 3 important points about QCD1. We cannot solve QCD “exactly”2. We can solve QCD approximately but only in
certain special circumstances3. We can solve QCD numerically. Eventually. If we
had a lot of computing power. • We need more and faster computers
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QCD and “Asymptotic freedom”• The forces we are familiar with on a day-to-day basis
(gravity, EM) have one thing in common:– They get weaker as things get further apart!
• The strong force (QCD) is not like that!– The force between quarks gets weaker as they get
closer together they are “asymptotically free”– As you pull quarks apart the force gets stronger –
so strong in fact that particles are created out of the vacuum as you pull two quarks apart: you’ll never find a “free quark”
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QCD vs. QED
QED
QCD
Quantum Electrodynamics:Relatively weak coupling lends itself to study using perturbative calculations
Quantum Chromodynamics:Interactions get stronger as you get further away! “confinement”Perturbative techniques only work at small distance scales
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QCD at Short and Long DistancesShort distances Quarks behave as if they are almost unbound
asymptotic freedom Quark-quark interaction relatively weak
perturbative QCD (pQCD)
Long distances Quarks are strongly bound and QCD calculations difficult Effective models often used ”Exact” numerical techniques
Lattice QCD
Running of as
from Particle Data Group
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Electron Scattering and QCDGoal: Understand the transition from confinement (strongly interacting quarks) to the perturbative regime (weakly interacting quarks)
Tool electron scattering Well understood probe (QED!)
e-
e-
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Electron Scattering and QCDGoal: Understand the transition from confinement (strongly interacting quarks) to the perturbative regime (weakly interacting quarks)
Tool electron scattering Well understood probe (QED!)
More powerful tool with development of intense, CW beams in 1990’s
Luminosity:(SLAC, 1978) ~ 8 x 1031 cm-2-s-1
(JLab, 2000) ~ 4 x 1038 cm-2-s-1
e-
e-
Observables:Form factors nucleons and mesons stay intactStructure functions excited, inelastic response
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Elastic Form Factors
Elastic scattering cross section from an extended target:
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objectpoint
)(QFdd
dd
In the example of a heavy, spin-0 nucleus, the form factor is the Fourier transform of the charge distribution
rderQF rqi 32 )()(
Spin 0 particles (p+,K+) have only charge form factor (F)Spin ½ particles (nucleon) have electric (GE) and magnetic (GM) form factors
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Example: Proton GM
Proton magnetic form-factor consistent with dipole form
222 )84.0/1(1
Q 2
2
22
1)(
MQQG
p
SLAC data
Inverse Fourier transform gives
RMeR 0)(
Re 84.0
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Brief history
• 1918, Rutherford discovers the proton• 1932, Chadwick discovers the neutron and
measures the mass as 938 +/- 1.8 MeV• 1933, Frisch and Stern measure the proton’s
magnetic moment = 2.6 +/- 0.3 B = 1 + kp
• 1940, Alvarez and Bloch measure the neutron’s magnetic moment = 1.93 +/- 0.02 B = kn
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Brief history
• 1918, Rutherford discovers the proton• 1932, Chadwick discovers the neutron and
measures the mass as 938 +/- 1.8 MeV• 1933, Frisch and Stern measure the proton’s
magnetic moment = 2.6 +/- 0.3 B = 1 + kp
• 1940, Alvarez and Bloch measure the neutron’s magnetic moment = 1.93 +/- 0.02 B = kn
Proton and neutron have anomalous magnetic moments a finite size.
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Electron as probe of nucleon elastic form factors
Known QED coupling
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Electron as probe of nucleon elastic form factors
Known QED coupling
Unknown g*Ncoupling
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Electron as probe of nucleon elastic form factors
Known QED coupling
Unknown g*Ncoupling
Nucleon vertex:
Elastic Form Factors: F1 is helicity conserving (no spin flip) F2 is helicity non-conserving (spin flip)
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IncidentElectron beam
g*
Qe
Scattered electron
Fixed nucleon target with mass M
Electron-Nucleon Scattering kinematics
Virtual photon kinematicsN
€
Pe = (E e,r k )
€
′ P e = ( ′ E e ,r ′ k )
€
Q2 = −(Pe − ′ P e )2 = 4E e ′ E esin2 ϑ e /2( ) me = 0
€
ν Ee − ′ E e
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IncidentElectron beam
g*
Qe
Scattered electron
Fixed nucleon target with mass M
Electron-Nucleon Scattering kinematics
Virtual photon kinematicsN
€
Pe = (E e,r k )
€
′ P e = ( ′ E e ,r ′ k )
€
Q2 = −(Pe − ′ P e )2 = 4E e ′ E esin2 ϑ e /2( ) me = 0
€
ν Ee − ′ E e
g*N center of mass energy
€
W = M p2 + 2M pν − Q2
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IncidentElectron beam
Qe
Scattered electron
Electron-Nucleon Scattering kinematics
Virtual photon kinematics
€
Pe = (E e,r k )
€
′ P e = ( ′ E e ,r ′ k )
€
Q2 = −(Pe − ′ P e )2 = 4E e ′ E esin2 ϑ e /2( ) me = 0
€
ν Ee − ′ E e
g*N center of mass energy
€
W = M p2 + 2M pν − Q2
Final
States
Elastic scattering
Inelastic scattering
W = M
W > M + mp
W = MR Resonancescattering
W
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Electron-Nucleon Cross SectionSingle photon exchange (Born) approximation
€
dσdΩ
=dσdΩ ⎛ ⎝ ⎜
⎞ ⎠ ⎟Mott
′ E eE e
{F12(Q2)
+ τ κ 2F22(Q2) + 2 F1(Q
2) + κF2(Q2)( )2tan2 θ e
2 ⎡ ⎣ ⎢
⎤ ⎦ ⎥}€
τ 2 = Q2 /4M 2
€
GE (Q2) = F1(Q2) −τκF2(Q2)
€
GM (Q2) = F1(Q2) +κF2(Q2)
Low Q2
€
dσdΩ
=dσdΩ ⎛ ⎝ ⎜
⎞ ⎠ ⎟Mott
′ E eEe
F12(Q2)
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Proton is an extended charge potential
Proton has a radius of 0.80 x 10-13 cm
fm-2 Q2 = 0.5 GeV2
“Dipole” shape
Early Form Factor Measurements
€
(r r )
€
Mott ρ(r r )e ir q ⋅r r d3r r ∫ 2
€
Mott F(r q ) 2
€
(r r ) = 3aea 2 r r
€
F(r q ) = 1+q2
a2
⎛ ⎝ ⎜
⎞ ⎠ ⎟−2
€
/σ Mott
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In center of mass of the eN system (Breit frame), no energy transfer νCM = 0 so
= charge distribution
= magnetization distribution
Sach’s Electric and Magnetic Form Factors
€
qν 2= r q 2
€
(r r )
€
(r r )
€
GE = ρ(r r )e ir q ⋅r r d3r r ∫
€
GM = μ(r r )e ir q ⋅r r d3r r ∫
€
At Q2 = 0 GMp = 2.79 GMn −1.91
GEp =1 GEn = 0
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Electron-Nucleon Cross SectionSingle photon exchange (Born) approximation
€
dσdΩ
=dσdΩ ⎛ ⎝ ⎜
⎞ ⎠ ⎟Mott
′ E eE e
{F12(Q2)
+ τ κ 2F22(Q2) + 2 F1(Q
2) + κF2(Q2)( )2tan2 θ e
2 ⎡ ⎣ ⎢
⎤ ⎦ ⎥€
τ 2 = Q2 /4M 2
€
GE (Q2) = F1(Q2) −τκF2(Q2)
€
GM (Q2) = F1(Q2) +κF2(Q2)
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Slope
Intercept
Elastic cross section in GE and GM
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Proton Form Factors: GMp and GEp
Experiments from the 1960s to 1990s gave a cumulative data set
€
GE /GD ≈ GM /(μ pGD ) ≈1
€
GD = 1+Q2
0.71
⎛ ⎝ ⎜
⎞ ⎠ ⎟2
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Proton Form Factors: GMp and GEp
Experiments from the 1960s to 1990s gave a cumulative data set
€
GE /GD ≈ GM /(μ pGD ) ≈1
€
GD = 1+Q2
0.71
⎛ ⎝ ⎜
⎞ ⎠ ⎟2
At large Q2, GE contribution is smaller so difficult to extract
GE contribution to is small then large error bars
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Proton Form Factors: GMp and GEp
Experiments from the 1960s to 1990s gave a cumulative data set
€
GE /GD ≈ GM /(μ pGD ) ≈1
€
GD = 1+Q2
0.71
⎛ ⎝ ⎜
⎞ ⎠ ⎟2
At large Q2, GE contribution is smaller so difficult to extract
GE > 1 then large error bars and spread in data.
GM measured to Q2 = 30
GE measured well only to Q2= 1
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elastic /Mott drops dramatically
At W = 2 GeVinel/Mott drops less steeply
At W=3 and 3.5 inel/Mott almost constant
As Q2 increases
Point object inside the proton
Q2 dependence of elastic and inelastic cross sections
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Asymptotic freedom to confinement
• “point-like” objects in the nucleon are eventually identified as quarks•Theory of Quantum Chromodynamics (QCD) with gluons mediating the strong force.•At high energies , the quarks are asymptotically free and perturbative QCD approaches can be used.
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Asymptotic freedom to confinement
• “point-like” objects in the nucleon are eventually identified as quarks•Theory of Quantum Chromodynamics (QCD) with gluons mediating the strong force.•At high energies , the quarks are asymptotically free and perturbative QCD approaches can be used.
ConfinementNo free quarks
•The QCD strong coupling increases as the quarks separate from each other• Quantatitive QCD description of nucleon’s properties remains a puzzle•Study of nucleon elastic form factors is a window see how the QCD strong coupling changes
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Elastic FF in perturbative QCDInfinite momentum frameNucleon looks like three massless quarksEnergy shared by two hard gluon exchangesGluon coupling is 1/Q2
g*
u
u
d
u
u
d
gluon
gluon
ProtonProton
F1(Q2) / 1=Q4
F2 requires an helicity flip the spin of the quark. Assuming the L = 0
F2(Q2) / 1=Q6
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Electron as probe of nucleon structure
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IncidentElectron beam
g*
Qe
np
Neutron Form FactorsNo free neutron target - use the deuteron (proton + neutron)
Measure eD eX cross sectionDetect only electron at Ee′ and θe when W = M or Q2 = 2Mν , Quasi-elastic kinematics
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d(e,e’) Inclusive Cross Section
€
r = ε 1+ν 2
Q2
⎛ ⎝ ⎜
⎞ ⎠ ⎟
σσ Mott
= RT +εRL
€
ε 1+ 2(1+ν 2
Q2 )tan2 θ2
⎡ ⎣ ⎢
⎤ ⎦ ⎥−1
RT and RL are the transverse and longitudinal response functions
Assume Plane Wave Impulse Approximation
€
RT ∝ (GMn )2 + (GM
p )2
€
RL ∝ (GEn )2 + (GE
p )2
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Extracting GMn
GM
GE
Measure cross sections at several energiesSeparate RT and RL as function of W2
Solid line is fitμGMn/GD = 0.967 ± 0.03
(GEn/GD)2= 0.164 ± 0.154
Dotted line shows sensitivity to Neutron form factor
Reduce GMn by80%
Set (GEn/GD)=1.5
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Neutron Magnetic Form Factor: GMn
Extract GMn from inclusive d(e,e’) quasielastic scattering cross section data
Difficulties: Subtraction of large proton contribution Sensitive to deuteron model
€
GMn /(μ pGD ) ≈ GM
p /(μ pGD ) ≈ GEp /GD
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• High current continuous-wave electron beams• Double arm detection • Reduces random background so coincidence
quasi-free deuteron experiments are possible• Polarized electron beams
• Recoil polarization from 1H and 2H• Highly polarized, dense 3He, 2H and 1H targets
•Beam-Target Asymmetry • Polarized 3He, 2H as polarized neutron target.
How to improve FF measurements?
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Theory of electron quasi-free scattering on 3He and 2HDetermine kinematics which reduce sensitivity to nuclear effects Determine which observables are sensitive to form factors Use model to extract form factors
How to improve FF measurements?
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• Detect neutron in coincidence with electron• Detect neutron at energy and angle expected for a “free”
neutron. Sensitive to detection efficiency• In same experimental setup measure d(e,e’p)• Theory predicts that R = (e,e’n)/(e,e’p) is less sensitive to deuteron wavefunction model and final
state interactions compared predictions of (e,e’n)• RPWIA = en/ep = R(1-D) D is calculated from theory
Neutron GM using d(e,e’n) reaction
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Neutron Magnetic Form Factor: GMn
Detect neutron in coincidence
But still sensitive to the deuteron model
Need to know absolute neutron cross section efficiency
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Neutron Magnetic Form Factor: GMn
Measure ratio of quasi-elastic n/p from deuterium
Sensitivity to deuteron model cancels in the ratio
Proton and neutron detected in same detector simultaneously
Need to know absolute neutron detection efficiency
Bonn used p(g,p+)n
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Neutron Magnetic Form Factor: GMn
Measure
Sensitivity to deuteron model cancels in the ratio
Proton and neutron detected in same detector simultaneously
Need to know absolute neutron detection efficiency
NIKHEF and Mainz used p(n,p)nwith tagged neutron beam at PSI
Bonn used p(g,p+)n
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Neutron Magnetic Form Factor: GMn
Measured with CLAS in Hall B at JLab
Simultaneously have 1H and 2H targets
CLAS data from W. Brooks and J. Lachniet, NPA 755 (2005)
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Form Factors from Cross Sections• Focused on cross section measurements to extract
proton and neutron form factors.– Proton GM measured to Q2 = 30 GeV2 – Neutron GM measured to Q2 = 4.5 GeV2
– Discrepancy in neutron GM near Q2 = 1.0 GeV2
• Need new experimental observable to make better measurements of neutron electric form factor and proton electric form factor above Q2 = 1 GeV2
– Spin observables sensitive to GExGM and GM – Get the relative sign of GE and GM
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Form Factors from Spin Observables Polarized beam on polarized nucleon (Beam-Target Asymmetries)
Polarized proton targetPolarized neutron target using polarized 3He and deuteriumElectron storage rings use internal gas target (windowless). Target polarized by
atomic beam source method or spin exchange optical pumping.Linear electron accelerators use external gas target (window)
3He gas targets by spin exchange or metastability optical pumpingSolid polarized deuterium or hydrogen
Polarized beam on unpolarized target Spin of scattered nucleon measured by secondary scattering Linear electron accelerators on high density hydrogen and deuterium targets
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Helicity flipped periodically (rapidly)
h+
h-
Nucleon polarized at q* and f* relative to the momentum transfer
85% longitudinally polarized electron beam
Beam-Target Spin Asymmetry
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Helicity flipped periodically (rapidly)
h+
h-
Nucleon polarized at q* and f* relative to the momentum transfer
85% longitudinally polarized electron beam
€
A = PB PTK1GM
2 cosθ *+K2GEGM sinθ *cosφ *GE
2 + (τ /ε )GM2€
A =N + − N −
N + + N −
Beam-Target Spin Asymmetry
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Polarized 3He as a polarized neutron target
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In 3He polarization of the neutron is larger than the proton
Pn >> Pp
In PWIA
€
Set θ* = 0o then AT ' = −ν T RT ' /(ν T RT +ν LRL )
AT in quasi-elastic 3He(e,e’)
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Meson Exchange Coupling
AT %
Coupling to correlated nucleon pairs
Coupling to in-flight mesons
u ( MeV)
u ( MeV)
Extracting GMn from AT in 3He(e,e)
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Neutron Magnetic Form Factor: GMn
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Neutron Magnetic Form Factor: GMn
• New results using the BLAST detector at MIT-Bates
•Electron ring and internal gas target
•Use inclusivewhich is sensitive to GMn/GMp
€
2r H (r e ,e')
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GEn from quasielastic
€
3r H e(r e ,e'n)
€
A = PB PTK1GM
2 cosθ * +K2GEGM sinθ *cosφ *GE
2 + (τ /ε )GM2
q*90o
€
A⊥∝ PB PTGE /GM
q*0o
€
A|| ∝ PB PT
€
GE /GM ∝ A⊥/ A||
€
D (r e ,e'n)
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NIKHEF used electron storage ring with internal gas target.
JLab used solid 15ND3
targetMeasured to Q2 = 1
In PWIA with Q* = 90o
MIT-Bates used internal gas target and large acceptance BLAST detector
Quasi-free Helium-3 and Deuterium
Neutron Electric Form Factor: GEn
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One-photon exchange (Born) approximation: PN = 0
Recoil Polarization
Components of
€
S
€
In k × k' planePL : along r p PT : transverse to r p ⊥ to k × k' planePN : transverse r p
€
S
€
I0 = GE2 +
τε
GM2
€
PT = −2 τ (1+τ )GEGM tan θ e2 /I0
€
PL = E +E 'M τ (1+τ )GM
2 tan2 θ e2 /I0
€
GE
GM
= −PT
PL
E + E '2M
tan θ e2
€
′ k
€
k
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Measure proton spin by secondary scattering on a nucleus ( 12C, CH2, H2 …)An asymmetry in the scattering is caused by the spin-orbit part of the nucleon-nucleon potential
Only components of S that perpendicular to the momentum vector produce an asymmetry
A is the analyzer power of the scattering material.
Polarimetry basics
€
V (r r ) ∝r L ⋅
r S
€
L =
r R × r p
€
ΣUD = PT A =Nup − Ndown
Nup + Ndown
€
ΣLR = PN A =N left − Nright
N left + Nright
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Top view
Target
PT
Magnetic field into page
AnalyzerSpin precesses relative to momentum
PL
P polT
Neutron traveling through a magnet field
Spin rotation in magnetic field
€
χ2κβ
r B ∫ dl€
χ
€
PTpol = PL sinχ + PT cosχ
€
ΣUV = PTpol A =
Nup − Ndown
Nup + Ndown
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Σ %
Measure asymmetry Σ at different precession angles χ by varying magnet field.
Extracting Neutron GE/GM
€
Σ PB A(PL sinχ + PT cosχ)
€
Σ A0 sin(χ − χ0)
€
tanχ0 =PB APT
PB APL
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At Mainz, Q2 = 0.15 to 0.8
At JLab, Q2= 0.45, 1.13,
1.45
In PWIA
Recoil Polarization
Quasi-free
€
D(r e ,e' r n )
€
GE
GM
= −PT
PL
E + E '2M
tan θ e2
Neutron Electric Form Factor: GEn
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Neutron Form Factors
GEn
GMn
Additional polarized target data at large Q2 (Hall A at JLab)
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Recoil Polarization and Proton
Recall: proton GE not terribly well measured at large Q2 using cross sections
Like the neutron – polarization techniques can be used to increase precision
In particular, recoil polarization techniques pursued in the case of the proton
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Proton GE/GM
€
GE
GM
= −PT
PL
E + E '2M
tan θ e2
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Proton GE/GM
€
GE
GM
= −PT
PL
E + E '2M
tan θ e2
Cross section measurements
Recoil polarization measurements
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Proton GE/GM
Why the discrepancy between cross section extraction and recoil polarization?
Recoil polarization data from different experiments, different experimental halls (A and C) technique robust and consistent
Cross section technique checked with dedicated experiment in Hall A
?
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Radiative contributions to ep elastic cross sections
Electron side contributions
a) Born term, Single virtual photon exchange
b) Virtual photon exchange between beam and scattered electron
c) Virtual photon self energyd) Virtual photon loop in beam or
scattered electron e) Internal radiation of a real
photon by the beam or scattered electron
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Radiative contributions to ep elastic cross sections
Proton side contributions
a) Internal real photon emissionb) Virtual photon exchange
between incoming and outgoing proton
c) Virtual photon loop in incoming or outgoing proton
d) Two virtual or multi-photon exchange between electron and proton
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Radiative contributions to ep elastic cross sections
Born after radiative corrections forQ2 = 1.75, 3.25 and 5 GeV2
Measured before radiative corrections forQ2 = 1.75, 3.25 and 5 GeV2
Possible missing element: 2-photon exchange
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Why the discrepancy in GE/GM?
Neglected 2g contributions to the ep elastic cross section could explain the difference
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Calculations of 2g effects
• Effects predicted to change cross sections but small effect on polarization observables:
» Hadronic Model of Blunden, Melnitchouk and Tjon
» GPD model of Chen, Afanasev, Brodsky, Carlson and Vanderhaeghen
•But no significant 2g effect predicted in calculation of Y. Bystritskiy, E. Kureav and E. Tomasi-Gustafsson
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Measuring Two Photon Exchange
Measure GE/GM as function of ε using polarization transfer technique
Look for deviations from a constant
Experiment completed in Jan 08M. Meziane et al. PRL 106, 132501 (2011)
No evidence for any deviation from 1-photon exchange picture within experimental uncertainties
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Measuring Two Photon Exchange
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Compare positron and electron elastic scattering: e+-p vs. e--p elastic scattering
Three new e+/e- experiments
BINP Novosibirsk – internal target
JLab – mixed e+/e- beam, CLAS
DESY (OLYMPUS) - internal target
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Measuring Two Photon Exchange
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Rosenbluth data with TPE correction
Polarization transfer
IF TPE corrections fully explain the discrepancy, THEN they are constrained well enough that they
do not limit our extractions of the form factor
Still awaiting experimental verification…
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Nucleon Form Factors: 15 years agoProton Neutron
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Nucleon Form Factors: Present statusProton Neutron
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What have we learned from new form factor data?
New information on proton structure– GE(Q2) ≠ GM(Q2) different charge, magnetization
distributions
Model-dependent extraction of charge, magnetization distribution of proton:J. Kelly, Phys. Rev. C 66, 065203 (2002)
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Quark Orbital Angular MomentumMany calculations able to reproduce the falloff in GE/GM
– Descriptions differ in details, but nearly all were directly or indirectly related to quark angular momentum
S. Boffi, et al.
C. Perdrisat, V. Punjabi, and M. Vanderhaeghen, PPNP 59 (2007)
G.A. Miller, M. Frank
F. Gross, P. AgbakpeP. Chung, F. CoesterF. Cardarelli, et al.
What have we learned from new form factor data?
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What have we learned from new form factor data?
Transverse Spatial Distributions
Simple picture: Fourier transform of the spatial distribution– Relativistic case: model dependent
“boost” corrections Model-independent relation found
between form factors and transverse spatial distribution
proton
neutron(b,x) = ∑ eq ∫ dx q(x,b) = transverse density distribution in infinite momentum frame (IMF) for quarks with momentum x
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Slide from G. Cates
Q4F2q/k
Q4 F1q
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Future form factor measurements
Proton Neutron JLab 12 GeV Upgrade will allow us to extend form factor measurements to even larger Q2
8.5 14.5 GeV2
4.5 13.5+ GeV2
3.5 10.5 GeV2