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Transcript of Andrei Afanasev, Lepton Scattering… HEP School, January 11-13, 2010, Valparaiso, Chile Lepton...
Andrei Afanasev, Lepton Scattering…
HEP School, January 11-13, 2010, Valparaiso, Chile
Lepton Scattering as a Probe of Hadronic Structure
Andrei Afanasev
Jefferson Lab/Hampton U
USA
Part 2
Andrei Afanasev, Lepton Scattering…
Plan of Lectures
• Basic formulae
• Scattering cross section, spin asymmetries
• Experimental data on nucleon form factors
• Overview of lepton scattering processes, experimental results and future programs
Andrei Afanasev, Lepton Scattering…
Form Factors
Exercise: Calculate the charge distribution ρ(x) is the form factor is described by a dipole formula
F(q2)=1/(1+q2/μ2)2
Andrei Afanasev, Lepton Scattering…
Fourier Transform Excercise
• Magnetic form factor of the proton follows the dipole formula up to Q2~25GeV2:
• Exponential fall-off of magnetic moment distribution with radius
0
32
22
22
2
22
23
22222
22
22
22
)1(
|)||sin(|4(...)cos2
(...)
)1(
]exp[/)(
)frameBreitin0(
12/,71.0,)1(
1/)(
x
B
BBBBBB
BBB
B
BBpMp
B
pMp
eq
xqdqq
xddqq
ddqqq
xqiqdxG
qqqQ
chGeVQ
QG
Andrei Afanasev, Lepton Scattering…
Mean Square Radius
• Defines characteristic size of the charge and magnetic moment distribution
• Believed to be the same for charge and magnetic moment
• But is it really the same?
• JLAB experiments proved otherwise
23322 )81.0()(/)( fmxdxGxdxGxr MpMpMp
Andrei Afanasev, Lepton Scattering…
Elastic Nucleon Form Factors
pm
q
peefi
fifi
uqFqFuuuq
ieM
MM
N))()(( 2
222
12
21
1
• Based on one-photon exchange approximation
)0(
)2
tan21(,4
,
:1
)1(2
:)(
122
2
2
2
2121
2
220
y
elab
N
ME
M
E
z
x
z
x
EM
P
q
q
m
q
FFGFFG
techniqueonPolarizatiG
G
A
A
P
P
techniqueRosenbluthGG
• Two techniques to measure
Latter due to: Akhiezer, Rekalo; Arnold, Carlson, Gross
Andrei Afanasev, Lepton Scattering…
Deriving the Rosenbluth formula
• Matrix element of electron-proton scattering in the first Born approximation:
• For scattering ep->ep, the electromagnetic current is Hermitian and form factors F1,2(q2) are real functions
• Squaring the matrix element, we obtain for the elastic contribution to the hadronic tensor:
)())()()(()()( 12
222
12122
21
1
puqFqFpukukuq
ieM
MM
pm
q
peefi
fifi
N
pp
mpq
pm
FFFmp
qp
m
FFFtrJJW N
NN
Nspin
ˆ
),ˆ()2
()()ˆ()2
()(2
121
22111
221
Andrei Afanasev, Lepton Scattering…
Rosenbluth Formula (cont.)
• Computing the trace over spinor indices, we obtain structure functions of elastic scattering:
• Differential cross section in terms of structure functions (Rosenbluth formula:
)4
1/()4
(4)(
)()(
)()2
)(2
(1
)()(
2
22
2
2222
2
22221
221
22112
212
NMp
NEpN
el
Mpel
el
N
el
m
qG
m
qGmqW
GqFFqqW
qWq
pq
pm
qWq
qqgW
2
2
1
22
1
22
14
2
0
21
2
1
2
21
22
22
1
2022
2
2
22
22
21
0
22
122
1212
2
4,
21)
21(4
)421(
2/
12
12
tan2
41
4
)2
sin21()(
2cot
2
1
)2
sin21(
1
2sin8
NNN
N
MpMpEp
N
Mpe
N
MpN
Ep
e
N
e
eleel
Ne
N
ee
m
q
mE
mE
qE
q
E
q
mE
qEq
GGG
mE
qG
m
q
Gm
qG
m
Eepep
d
d
WWmE
m
Ed
d
Andrei Afanasev, Lepton Scattering…
Polarization formula
)(2
1
4)(
protons outgoing of Pon vectorPolarizati
tensorleptonicLtarget,protontheofdirectiononpolarizatispinalongvectorunitaisn
2)(2),(2
1
4)(
sectioncrossdependentonPolarizati
2,where,
)(2)(,
)(2)(
:frameBreitinspinorscomponent)(4Dirac
)())()()((),()(
ionapproximatBorninmatrixscatteringofelementMatrix
2
2
1
24
2
212
12212
2
1
24
2)(
012
2
2
21
1
1
12
222
1212
2
2
fftrm
L
E
E
qepep
d
dP
kkiqgkkkkLwherefnftrm
L
E
E
qepep
d
d
GmfGqiffJ
mE
qmEpumE
qmEpu
puqFqFpuJkukuj
Jjq
ieM
Ne
Ne
n
EpNMpBp
NB
BBB
NB
BBB
pm
q
pp
eee
pefi
N
Andrei Afanasev, Lepton Scattering…
Polarization formula: result
)ionapproximatexchangephotononein0(
)2
tan21(,4
,
:1
)1(2
:)(
122
2
2
2
2121
2
22
y
elab
N
ME
M
E
z
x
z
x
EMR
P
q
q
m
q
FFGFFG
formulaonPolarizatiG
G
A
A
P
P
formulaRosenbluthGG
Andrei Afanasev, Lepton Scattering…
The Pre-JLab Era• Stern (1932) measured the proton magnetic moment µp ~ 2.5
µDirac
indicating that the proton was not a point-like particle• Hofstadter (1950’s) provided the first measurement of the
proton’s radius through elastic electron scattering• Subsequent data (≤ 1993) were based on:
Rosenbluth separation for proton, severely limiting the accuracy for GE
p at Q2 > 1 GeV2
• Early interpretation based on Vector-Meson Dominance
• Good description with phenomenological dipole form factor:
GD 2
2 Q2
2
with 0.84GeV
corresponding to r (770 MeV) and w (782 MeV) meson resonances in timelike region and to exponential distribution in coordinate space
Andrei Afanasev, Lepton Scattering…
Global Analysis
P. Bosted et al. PRC 51, 409 (1995)
Three form factors very similar
GEn zero within errors ->
accurate data on GEn early goal
of JLab
First JLab GEp proposal rated B+!
Andrei Afanasev, Lepton Scattering…
Magnetic Form Factor of the Proton at High Momentum Transfers
• In perturbative QCD description, GMp~1/Q4 at large
values of Q2. How large?
• Data from SLAC, (Andivahis et al ,1986) show scaling behavior
Andrei Afanasev, Lepton Scattering…
Modern Era
Akhiezer et al., Sov. Phys. JETP 6 (1958) 588 andArnold, Carlson and Gross, PR C 23 (1981) 363showed that:
accuracy of form-factor measurements can be significantly improved by measuring an interference term GEGM through the beam helicity asymmetry with a polarized target or with recoil polarimetry
Had to wait over 30 years for development of
• Polarized beam with high intensity (~100 µA) and high polarization (>70 %)
(strained GaAs, high-power diode/Ti-Sapphire lasers)• Beam polarimeters with 1-3 % absolute accuracy• Polarized targets with a high polarization or• Ejectile polarimeters with large analyzing powers
Andrei Afanasev, Lepton Scattering…
Pre-Jlab Measurements of GEn
Elastic e-d scattering (Platchkov, Saclay)
dd
{A Btan2(e / 2)} (GEp GE
n )2 [u2(r) w2 (r)] j0 (Qr2
)dr ....
No free neutron target available, early experiments used deuteronLarge systematic errors caused by subtraction of proton contribution
Yellow band represents range of GE
n-values resulting from the use of different NN-potentials
Andrei Afanasev, Lepton Scattering…
Double Polarization Experiments to Measure GnE
• Study the (e,e’n) reaction from a polarized ND3 targetlimitations: low current (~80 nA) on target
deuteron polarization (~25 %)• Study the (e,e’n) reaction from a LD2 target and
measure the neutron polarization with a polarimeter limitations: Figure of Merit of polarimeter• Study the (e,e’n) reaction from a polarized 3He targetlimitations: current on target (12 µA)
target polarization (40 %)nuclear medium corrections
)0(
,
:1
)1(2
2121
2
y
ME
M
E
z
x
z
x
P
FFGFFG
techniqueonPolarizatiG
G
A
A
P
P
Andrei Afanasev, Lepton Scattering…
Neutron Electric Form Factor GEn
Galster: a parametrization fitted to old (<1971) data set of very limited quality
For Q2 > 1 GeV2
data hint that GEn has
similar Q2-behaviour as GE
p
Andrei Afanasev, Lepton Scattering…
Measuring GnM
• Measure (en)/(ep) ratioLuminosities cancelDetermine neutron detector efficiency
• On-line through e+p->e’+π+(+n) reaction (CLAS)• Off-line with neutron beam (Mainz)
• Measure inclusive quasi-elastic scattering off polarized 3He (Hall A)
RD
d3(eD e 'n(p))
dE ' de 'dn
d3(eD e ' p(n))
dE 'de'd p
Old method: quasi-elastic scattering from 2Hlarge systematic errors due to subtraction of proton contribution
A cos*vT 'RT ' 2sin*cos*vTL'RTL'
vLRL vTRT
RT’ directly sensitive to (GMn)2
Andrei Afanasev, Lepton Scattering…
Measurement of GnM at low Q2
Andrei Afanasev, Lepton Scattering…
New GnM Results from CLAS
J. Lachinet et al, [CLAS Collab.] Phys. Rev. Lett. 102: 192001, 2009
Andrei Afanasev, Lepton Scattering…
Early Measurements of GEp
• relied on Rosenbluth separation• measure ds/dW at constant Q2
• GEp inversely weighted with Q2, increasing the systematic error
above Q2 ~ 1 GeV2
At 6 GeV2 sR changes by only 8% from e=0 to e=1 if GE
p=GMp/µp
Hence, measurement of Gep
with 10% accuracy requires 1.6% cross-section measurements over a large range of electron energies
1
1 2(1 ) tan2 ( / 2)
R Q2 , 11
E
E '
E, Mott
{GMp Q2 }2
{GEp Q2 }2
Andrei Afanasev, Lepton Scattering…
Spin Transfer Reaction 1H(e,e’p)
No error contributions from• analyzing power• beam polarimetry
These errors cancel in the ratio of polarizations Px/Pz
)0(
,
:1
)1(2
2121
2
y
ME
M
E
z
x
z
x
P
FFGFFG
techniqueonPolarizatiG
G
A
A
P
P
Andrei Afanasev, Lepton Scattering…
JLab Polarization-Transfer Data
• E93-027 PRL 84, 1398 (2000)Used both HRS in Hall A with FPP
• E99-007 PRL 88, 092301 (2002)used Pb-glass calorimeter for electron detection to match proton HRS acceptance
• Reanalysis of E93-027 (Pentchev)Using corrected HRS properties
→Clear discrepancy between polarization transfer and Rosenbluth data
→Investigate possible source, first by doing optimized Rosenbluth experiment
Andrei Afanasev, Lepton Scattering…
Charge and Magnetization Densities
• JLAB experiments demonstrated large differences in charge and magnetization densities of a nucleon
Kelly (2002)
Andrei Afanasev, Lepton Scattering…
Super-Rosenbluth (E01-001) 1H(e,p)J. Arrington and R. Segel• Detect recoil protons in HRS-L to
diminish sensitivity to:• Particle momentum and
angle• Data rate
• Use HRS-R as luminosity monitor• Very careful survey• Careful analysis of background
RosenbluthPol TransMC simulations
Andrei Afanasev, Lepton Scattering…
Rosenbluth Compared to Polarization Transfer
• John Arrington performed detailed reanalysis of SLAC data• Hall C Rosenbluth data (E94-110, Christy) in agreement with SLAC data• No reason to doubt quality of either Rosenbluth or polarization transfer
data→Investigate possible theoretical sources for discrepancy
Andrei Afanasev, Lepton Scattering…
Two-photon Contributions
Guichon and Vanderhaeghen (PRL 91 (2003) 142303) estimated the size of two-photon effects (TPE) necessary to reconcile the Rosenbluth and polarization transfer data
dd
˜ G M
2
˜ G E2
˜ G M2 2
˜ G E˜ G M
Y2 (,Q2)
Pt
Pl
2
(1 )
˜ G E˜ G M
12
1
˜ G E˜ G M
Y2 (,Q2 )
Need ~3% value for Y2g (6% correction to e-slope), independent of Q2, which yields minor correction to polarization transfer
Andrei Afanasev, Lepton Scattering…
Two-Photon Contributions (cont.)
Afanasev, Brodsky, Carlson et al,
PRD 72:013008 (2005); PRL 93:122301 (2004)
Model schematics: • Hard eq-interaction• GPDs describe quark
emission/absorption • Soft/hard separation• Assume factorization
Polarization transfer1g+2g(hard)1g+2g(hard+soft)
Blunden et al. have calculated elastic contribution of TPE
Resolves ~50% of discrepancy
Andrei Afanasev, Lepton Scattering…
Experimental Verification of TPE contributions
Experimental verification• non-linearity in e-dependence
(test of model calculations)• transverse single-spin asymmetry
(imaginary part of two-photon amplitude)
• ratio of e+p and e-p cross section (direct measurement of two-photon contributions)
CLAS experiment PR04-116 aims at a measurement of the e-dependence for Q2-values up to 2.0 GeV2
Andrei Afanasev, Lepton Scattering…
Reanalysis of SLAC data on GM
p
E. Brash et al., PRC submitted,have reanalyzed SLAC datawith JLab GE
p/GMp results
as constraint, using a similarfit function as BostedReanalysis results in 1.5-3% increase of GM
p data
Andrei Afanasev, Lepton Scattering…
Theory
→Vector Meson DominancePhoton couples to nucleon exchanging vector meson ( , , )r w fAdjust high-Q2 behaviour to pQCD scalingInclude 2π-continuum in finite width of r
• Lomon 3 isoscalar, isovector poles, intrinsic core FF• Iachello2 isoscalar, 1 isovector pole, intrinsic core FF• Hammer 4 isoscalar, 3 isovector poles, no additional FF
→Relativistic chiral soliton model • Holzwarth one VM in Lagrangian, boost to Breit frame• Goeke NJL Lagrangian, few parameters
→Lattice QCD (Schierholz, QCDSF)quenched approximation, box size of 1.6 fm, mπ = 650 MeVchiral “unquenching” and extrapolation to mπ = 140 MeV (Adelaide)
Andrei Afanasev, Lepton Scattering…
Vector-Meson Dominance Model
charge magnetization
proton
neutron
Andrei Afanasev, Lepton Scattering…
Constituent quark model predictions
• Relativistic Constituent Quark Models: |p>=|uud>, |n>=|udd>Variety of q-q potentials (harmonic oscillator, hypercentral, linear)Non-relativistic treatment of quark dynamics, relativistic EM currents
• Miller: extension of cloudy bag model, light-front kinematicswave function and pion cloud adjusted to static parameters
• Cardarelli & SimulaIsgur-Capstick oge potential, light-front kinematicsconstituent quark FF in agreement with DIS data
• Wagenbrunn & Plessaspoint-form spectator approximationlinear confinement potential, Goldstone-boson exchange
• Giannini et al.gluon-gluon interaction in hypercentral modelboost to Breit frame
• Metsch et al.solve Bethe-Salpeter equation, linear confinement potential
Andrei Afanasev, Lepton Scattering…
Relativistic Constituent Quark Model
charge magnetization
proton
neutron
Andrei Afanasev, Lepton Scattering…
Form Factors and Generalized Parton Distributions
• Radyushkin, PRD58:114008 (1998); AA, hep- ph/9910565; Guidal et al, PRD 72: 054013 (2005); Diehl et al, EPJ 39, 1 (2005)
Fits to the data; additional constraintsfrom parton distributions measured in deep-inelastic scattering
Andrei Afanasev, Lepton Scattering…
High-Q2 behaviour
Basic pQCD scaling (Bjørken) predicts F1 µ1/Q4 ; F2 µ1/Q6 Þ F2/F1 µ1/Q2
Data clearly do not follow this trend
Schlumpf (1994), Miller (1996) andRalston (2002) agree that by• freeing the pT=0 pQCD condition• applying a (Melosh) transformation to
a relativistic (light-front) system• an orbital angular momentum
component is introduced in the proton wf (giving up helicity conservation) and one obtainsÞ F2/F1 µ1/Q
• or equivalently a linear drop off of GE/GM with Q2
Brodsky argues that in pQCD limit non-zero OAM contributes to F1 and F2
Andrei Afanasev, Lepton Scattering…
High-Q2 Behaviour (cont)
Belitsky et al. have included logarithmic corrections in pQCD limit
Solid: protonopen: neutron
They warn that the observed scaling could very well be precocious
Andrei Afanasev, Lepton Scattering…
Low-Q2 Behaviour
All EMFF allow shallow minimum (max for GEn) at Q ~ 0.5 GeV
Andrei Afanasev, Lepton Scattering…
Pion Cloud of a Nucleon
• Kelly has performed simultaneous fit to all four EMFF in coordinate space using Laguerre-Gaussian expansion and first-order approximation for Lorentz contraction of local Breit frame
˜ G E,M (k) GE ,M (Q2) 1 2 with k 2
Q2
1 and
Q
2M
2
• Friedrich and Walcher have performed a similar analysis using a sum of dipole FF for valence quarks but neglecting the Lorentz contraction
• Both observe a structure in the proton and neutron densities at ~0.9 fm which they assign to a pion cloud
• Hammer et al. have extracted the pion cloud assigned to the NN2π component which they find to peak at ~ 0.4 fm
_
Andrei Afanasev, Lepton Scattering…
Summary
• Very successful experimental program at JLab on nucleon form factors thanks to development of polarized beam (> 100 µA, > 75 %), polarized targets and polarimeters with large analyzing powers
• GEn 3 successful experiments, precise data up to Q2 = 1.5
GeV2
• GMn Q2 < 1 GeV2 data from 3He(e,e’) in Hall A
Q2 < 5 GeV2 data from 2H(e,e’n)/2H(e,e’p) in CLAS• GE
p Precise polarization-transfer data set up to Q2 =5.6 GeV2
New Rosenbluth data from Halls A and C confirm SLAC data
• The experimental discrepancy between polarization transfer and Rosenbluth data may be resolved with two-photon exchange contributions; further experimental tests coming up
• JLab at 12 GeV will make further extensions to even higher Q2 possible
Andrei Afanasev, Lepton Scattering…
Coming up next
• Electroweak form factors and parity-violating electron scattering
• Form factors of mesons
• Excitation of baryon resonances
• Virtual Compton scattering on a nucleon
• Primakoff production of mesons
• Search for mesons with exotic quantum numbers