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
• Introduction to particle scattering formalism
• Basics
• Scattering cross section, spin asymmetries
• Experimental data on nucleon form factors
• Derivation of basic formulae
• Overview of lepton scattering processes, experimental
results and future programs
Andrei Afanasev, Lepton Scattering…
Some Facts about Quarks, Hadrons
and Nuclei
Andrei Afanasev, Lepton Scattering…
Some facts (cont.)
Andrei Afanasev, Lepton Scattering…
Elementary Particles
Andrei Afanasev, Lepton Scattering…
Hadrons and Interactions
• Quarks carry both electromagnetic charge and color charge
Andrei Afanasev, Lepton Scattering…
Scattering as a Probe of Structure
• Rutherford experiment (1909-1911)
• Beam: alpha-particles from radioactive
214Po
• Target: gold foil
• Result: “Planetary Model” of an atom; beginning of a new era
in modern physics
Compare with a bullet
probing a haystack
Exercise: Run a computer simulation of Rutherford experiment
http://phet.colorado.edu/simulations/sims.php?sim=Rutherford_Scattering
Andrei Afanasev, Lepton Scattering…
Scattering in Quantum Physics
• Described by a Lippmann-Schwinger equation for wave
function of the scattered particles,
 ()   
1
V  () ,
E  H 0  i
p2
H  H0 V , H0 
, H0   E 
2m
x
()
1

(2 )3 / 2
large r
 ikx eikr

e

f
(
k
'
,
k
)


r


• Differential cross section:
d
number of particles scattered into d per unit time
d 

d
number of incident particles crossing unit area per unit time

2
r | jscatt | d

| f (k ' , k ) |2 d
| jincident|
Andrei Afanasev, Lepton Scattering…
Born Approximation
• Scattering amplitude and differential cross section in
the first Born approximation
x'  (  )  x'  

ik x '
e
(2 ) 3 / 2
2m 3 i ( k  k ') x ' 
f (k , k ' )  
d x' e
V (x' )
h 
 

| k  k ' | Q  2k sin ;
2
(1)

2m / q
for sphericall y symmetric potential f ( )  
dr rV (r ) sin( qr )
2 
(h / 2 ) 0
(1)
V0 e  r
2mV0
1
for Yukawa potential V (r ) 
 f (1) ( )  
r
 (h / 2 ) 2 q 2   2
  2  2mV  2
d
1
0

differenti al cross section
 f (k ' , k )  
2
2 
d
  (h / 2 )   2 2 
2
 
 4k sin
2


Andrei Afanasev, Lepton Scattering…
Role of Particle Spin
• Spin is an internal quantum number of a particle
• For charged particles spin relates to their magnetic moments
through a Bohr magneton
• Spin has discrete projection on a selected direction (quantum
phenomenon), demonstrated in Stern-Gerlach experiment
(1922) for a beam of spin-1/2 particles
Exercise: run a computer simulation of a Stern-Gerlach experiment at
http://phet.colorado.edu/simulations/sims.php?sim=SternGerlach_Experiment
Andrei Afanasev, Lepton Scattering…
Electron Scattering
Andrei Afanasev, Lepton Scattering…
Pros and Cons of Electron Scattering
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Elastic Electron Scattering
• Form Factor
response of system to momentum transfer Q,
often normalized to that of point-like system
Examples:
→scattering of photons by bound atoms
→nuclear beta decay
→X-ray scattering from crystal
→electron scattering off nucleon
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…
Nucleon Electro-Magnetic Form Factors
 Fundamental ingredients in “Classical” nuclear theory
• A testing ground for theories constructing nucleons from quarks and gluons
- spatial distribution of charge, magnetization
wavelength of probe can be tuned by selecting momentum transfer Q:
< 0.1 GeV2 integral quantities (charge radius,…)
0.1-10 GeV2 internal structure of nucleon
> 20 GeV2 pQCD scaling
Caveat: If Q is several times the nucleon mass (~Compton wavelength),
dynamical effects due to relativistic boosts are introduced, making physical
interpretation more difficult
 Additional insights can be gained from the measurement of the form factors of
nucleons embedded in the nuclear medium
- implications for binding, equation of state, EMC…
- precursor to QGP
Andrei Afanasev, Lepton Scattering…
Formalism
Sachs Charge and Magnetization Form Factors GE and GM
 GE2   GM2

d
2
2
( E,  )   M 
 2 GM tan  / 2  
d
 1 

 2 E 'cos2  / 2
M 
4E 3 sin 4  / 2
with E (E’) incoming (outgoing) energy,  scattering angle,
k anomalous magnetic moment
In the Breit (centre-of-mass) frame the Sachs FF can be written
as the Fourier transforms of the charge and magnetization
radial density distributions
GE and GM are often alternatively expressed in the Dirac (non-spin-flip) F1
and Pauli (spin-flip) F2 Form Factors
F1  GE   GM
G  GE
F2  M
k (1   )
Q2
=
4 M2
Andrei Afanasev, Lepton Scattering…
Elastic Nucleon Form Factors
•Based on one-photon exchange approximation
M fi  M 1fi
M 1fi  e 2ue  ueu p ( F1 (q 2 )  
   q
F2 (q 2 ))u p
2m
•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…
Do the techniques agree?
SLAC/Rosenbluth
~5% difference in cross-section
x5 difference in polarization
JLab/Polarization
•
Both early SLAC and Recent JLab experiments on (super)Rosenbluth separations
followed Ge/Gm~const
•
JLab measurements using polarization transfer technique give different results
(Jones’00, Gayou’02)
Radiative corrections, in particular, a short-range part of 2-photon
exchange is a likely origin of the discrepancy
Andrei Afanasev, Lepton Scattering…
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)
 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, Lepton Scattering…
Complete radiative correction in O(αem )
Radiative Corrections:
• Electron vertex correction (a)
• Vacuum polarization (b)
• Electron bremsstrahlung (c,d)
Log-enhanced but calculable (a,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, Lepton Scattering…