Part 2
Lepton Scattering as a Probe of Hadronic Structure
Andrei Afanasev
Jefferson Lab/Hampton U
USA
HEP School, January 11-13, 2010, Valparaiso, Chile
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:
GMp (Q 2 ) /  p 
(1 
1
Q2

2
,  2  0.71GeV 2 , h / 2  c  1
)2


 Q 2  q 2   2  q 2   q B2 (  0 in Breit frame )
 
exp[ iqB x ]
3
GMp ( x) /  p   d qB
  q B2 dq B d B (...) 
2
q
(1  B2 ) 2

4
 2  q dqB d cos B(...) 
x
2
B

 qB dqB
0
sin(| qB || x |)
  2  3e  x
2
q
(1  B2 ) 2

• Exponential fall-off of magnetic moment distribution
with radius
Andrei Afanasev, Lepton Scattering…
Mean Square Radius
• Defines characteristic size of the charge and
magnetic moment distribution
r
2
  x GMp ( x)d x /  GMp ( x)d x  (0.81 fm)
2
Mp
3
3
2
• Believed to be the same for charge and magnetic
moment
• But is it really the same?
• JLAB experiments proved otherwise
Andrei Afanasev, Lepton Scattering…
Elastic Nucleon Form Factors
•Based on one-photon exchange approximation
M fi  M 1fi
M
1
fi
 ie 2
 q
 2 ue  ueu p ( F1 (q 2 )   2m N F2 (q 2 ))u p
q
•Two techniques to measure
   0 (GM 2    GE 2 ) : Rosenbluth technique
G  2 (1   )
Px
A
 x  E
Pz
Az
GM  1   2
: Polarizati on technique
GE  F1  F2 , GM  F1  F2
2
qlab
 q2
2  e 1

,


(
1

2
tan
)
2
2
4m N
q
2
( Py  0)
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:
M fi  M 1fi
M
1
fi
 ie 2
 q
 2 ue (k2 )  ue (k1 )u p ( p2 )( F1 (q 2 )   2m N F2 (q 2 ))u p ( p1 )
q
• 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:

1 
F
q 
F
q 
W  J J   spin  tr ( F1  F2 )   2 ( p1  )  ( pˆ1  mN ) ( F1  F2 )   2 ( p1  )  ( pˆ 2  mN ),
2 
mN
2 
mN
2 

pˆ  p  
Andrei Afanasev, Lepton Scattering…
Rosenbluth Formula (cont.)
•
Computing the trace over spinor indices, we obtain structure functions
of elastic scattering:
W  ( g 
q q
q2
)W1el (q 2 ) 
q
q
1
(
p

)(
p

)W2el (q 2 )
1
1
2
mN
2
2
2
W1el (q 2 )  q 2 ( F1  F2 ) 2   q 2GMp
q2
q2
2
W (q )  4m (G 
GMp ) /(1 
)
4mN2
4mN2
el
2
•
2
2
N
2
Ep
Differential cross section in terms of structure functions (Rosenbluth
formula:
d
2
1

de 8 sin 2  e (1  2 E1 sin 2  e ) E 2 m 2
1
N
2
mN
2
 el 1 2  e el 
W1  2 cot 2 W2 
 2

q2 2
 GEp 

GMp
2
d
0
0
4 mN

2 e
2 
(ep  ep ) 

2
tan
G

Mp
2

E
 
q2
de
2
(1  2 1 sin 2 e )  1  q 2
 1
mN
2 
4 mN
2 E1mN

0 
2 
q2
q2
2
2 

4
E
(
1

)

q
1

 1

q4 
2 E1mN
 2 E1mN

,

 


 G 2  G 2

2
2
 q / 2 E1
Mp
2
 Ep

 GMp
q2
q2 
 1
(1 


2 E1mN 4 E12 ) 

q2
4mN2
Andrei Afanasev, Lepton Scattering…
Polarization formula
Matrix element of scattering matrix in Born approximat ion
 ie 2 e p
M fi  2 j J 
q
je  ue (k2 )  ue (k1 ), J p  u p ( p2 )( F1 (q 2 )  
   q
2mN
F2 (q 2 ))u p ( p1 )
Dirac (4  component) spinors in Breit frame :







   q 2

    q1
, u ( p 2 )  E B  mB 
u ( p1 )  EB  mB 
B
B


 2( E  m )  2 
 2( E  m ) 1 
N
B
N
B







J p   2 f  1 , where f  i  qBGMp , f 0  2mN GEp
Polarizati on  dependent cross section
2

 2  E2  L 1
d ( n )
(ep  ep )  4   2 tr ( f  n f ), where L  2(k1 k2  k2  k1 )  g  q 2  2i  k1 k2 
4q  E1  mN 2
d e

n is a unit vector along spin polarizati on direction of the proton target, L   leptonic tensor

Polarizati on vector P of outgoing protons
2
 d

 2  E2  L 1
(ep  ep )  4   2 tr ( f  f )
P
4q  E1  mN 2
d e
Andrei Afanasev, Lepton Scattering…
Polarization formula: result
   R (GM 2    GE 2 ) : Rosenbluth
GE  2 (1   )
Px
Ax


Pz
Az
GM  1   2
formula
: Polarizati on
formula
GE  F1  F2 , GM  F1  F2
2
2
q
qlab
2  e 1

,


(
1

2
tan
)
2
2
4 mN
q
2
( Py  0 in one  photon  exchange approximat ion )
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 GEp at Q2 > 1 GeV2
• Early interpretation based on Vector-Meson Dominance
• Good description with phenomenological dipole form factor:
2
  
GD   2
with   0.84GeV
2 
  Q 
2
corresponding to r (770 MeV) and  (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 and
Arnold, Carlson and Gross, PR C 23 (1981) 363
showed 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
No free neutron target available, early experiments used deuteron
Large systematic errors caused by subtraction of proton contribution
Elastic e-d scattering
(Platchkov, Saclay)
d
Qr
2
p
n 2
2
2
 {A  Btan (e / 2 )}  (GE  GE )  [u (r)  w (r)] j0 ( )dr  ....
d
2
Yellow band represents range of
GEn-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 target
limitations: 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 target
limitations: current on target (12 µA)
target polarization (40 %)
nuclear medium corrections
G  2 (1   )
Px
A
 x  E
Pz
Az
GM  1   2
: Polarizati on technique
GE  F1  F2 , GM  F1  F2
( Py  0)
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 GEp
Andrei Afanasev, Lepton Scattering…
Measuring GnM
Old method: quasi-elastic scattering from 2H
large systematic errors due to subtraction of proton contribution
d 3 (eD  e'n(p))
dE' de ' dn
• Measure (en)/(ep) ratio
RD  3
d  (eD  e' p(n))
Luminosities cancel
Determine neutron detector efficiency
dE'd e' d p
• 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)
cos vT ' RT '  2sin cos vTL' RTL' 
*
A
*
v L RL  v T RT
*
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 d/d at constant Q2
GEp inversely weighted with Q2, increasing the systematic error
above Q2 ~ 1 GeV2
1  E  E, 


 R Q2 ,     1  
 {GMp Q2 }2  {GEp Q2 }2
   E '  Mott


1
1  2(1   ) tan 2 ( / 2)
At 6 GeV2 R changes by only 8%
from =0 to =1 if GEp=GMp/µp
Hence, measurement of Gep with
10% accuracy requires 1.6%
cross-section measurements over
a large range of electron energies
Andrei Afanasev, Lepton Scattering…
Spin Transfer Reaction 1H(e,e’p)
G  2 (1   )
Px
A
 x  E
Pz
Az
GM  1   2
GE  F1  F2 , GM  F1  F2
( Py  0)
: Polarizati on technique
No error contributions from
• analyzing power
• beam polarimetry
These errors cancel in the ratio of
polarizations Px/Pz
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
MC simulations
Andrei Afanasev, Lepton Scattering…
Rosenbluth
Pol Trans
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
2
2

˜
˜

G
G
d

 M    E 2  2  
d
 
G˜ M


Pt
2 
G˜ E  2

 1

Pl
 (1  ) G˜ M  1  

G˜ E
G˜ M


2 
Y2 ( ,Q )



G˜ E
G˜


2 
Y2 ( ,Q )



M
Need ~3% value for Y2 (6% correction to slope), independent of Q2, which yields minor
correction to polarization transfer
Andrei Afanasev, Lepton Scattering…
Two-Photon Contributions (cont.)
Blunden et al. have calculated elastic contribution of TPE
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
Resolves ~50% of discrepancy
Polarization transfer
1+2(hard)
1+2(hard+soft)
Andrei Afanasev, Lepton Scattering…
Experimental Verification of TPE contributions
Experimental verification
• non-linearity in -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 -dependence for
Q2-values up to 2.0 GeV2
Andrei Afanasev, Lepton Scattering…
Reanalysis of SLAC data on GMp
E. Brash et al., PRC submitted,
have reanalyzed SLAC data
with JLab GEp/GMp results
as constraint, using a similar
fit function as Bosted
Reanalysis results in 1.5-3%
increase of GMp data
Andrei Afanasev, Lepton Scattering…
Theory
→ Vector Meson Dominance
•
•
•
Photon couples to nucleon exchanging vector meson (r,,f
Adjust high-Q2 behaviour to pQCD scaling
Include 2π-continuum in finite width of r
Lomon
3 isoscalar, isovector poles, intrinsic core FF
Iachello
2 isoscalar, 1 isovector pole, intrinsic core FF
Hammer
4 isoscalar, 3 isovector poles, no additional FF
→ Relativistic chiral soliton model
•
•
Holzwarth
Goeke
one VM in Lagrangian, boost to Breit frame
NJL Lagrangian, few parameters
→ Lattice QCD (Schierholz, QCDSF)
quenched approximation, box size of 1.6 fm, mπ = 650 MeV
chiral “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 kinematics
wave function and pion cloud adjusted to static parameters
• Cardarelli & Simula
Isgur-Capstick oge potential, light-front kinematics
constituent quark FF in agreement with DIS data
• Wagenbrunn & Plessas
point-form spectator approximation
linear confinement potential, Goldstone-boson exchange
• Giannini et al.
gluon-gluon interaction in hypercentral model
boost 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, hepph/9910565; Guidal et al, PRD 72: 054013 (2005);
Diehl et al, EPJ 39, 1 (2005)
Fits to the data; additional constraints
from parton distributions measured in
deep-inelastic scattering
Andrei Afanasev, Lepton Scattering…
High-Q2 behaviour
Basic pQCD scaling (Bjørken) predicts
F1  1/Q4 ; F2  1/Q6
 F2/F1  1/Q2
Data clearly do not follow this trend
Schlumpf (1994), Miller (1996) and
Ralston (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 obtains
 F2/F1  1/Q
• or equivalently a linear drop off of
GE/GM with Q2
Brodsky argues that in pQCD limit nonzero 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: proton
open: 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
2
2
Q2
 Q 
2
2
˜
GE,M (k)  GE ,M (Q )1   with k 
and  
2M 
1 
• 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
• GEp
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
Andrei Afanasev, Lepton Scattering…