Inelastic scattering
When the scattering is not elastic (new
particles are produced) the energy and
direction of the scattered electron are
independent variables, unlike the elastic
scattering situation.
W is the mass squared of the produced
hadronic system
From the measurement of the direction 
(solid angle element d) and the energy E'
of the scattered electron, the four
momentum transfer Q2=-q2 can be
calculated.
The differential cross-section is
determined as a function of E' and Q2.
2004,Torino
Aram Kotzinian
e(k',E')

e (k,E)
 (q)
N (P,M)
k'  k  q
W
  E  E'
q 2  4 EE' sin2 2
W 2  (P  q)2 
P 2  2P  q  q 2 
M 2  2M  Q2
1
Electron - proton inelastic scattering
Bloom et al. (SLAC-MIT group) in
1969 performed an experiment with
high-energy electron beams (7-18 GeV).
Scattering of electrons from a hydrogen
target at 60 and 100.
Only electrons are detected in the final
state - inclusive approach.
The data showed peaks when the mass
W of the produced hadronic system
corresponded to the mass of the known
resonances.
2004,Torino
Aram Kotzinian
2
Inelastic scattering cross-section
Similar to the electron-proton elastic scattering, the differential cross-section
of electron-proton inelastic scattering can be written in a general form:
d
 cos 2
2
2
2

W
(

,q
)
W
(

,q
)t
an
'
2
4   2
1
2
ddE 4E sin 2
2
2
The cross-section is double differential because  and E ' are independent
variables.
The expression contains Mott cross-section as a factor and is analogous to the
Rosenbluth formula. It isolates the unknown shape of the nucleon target in two
structure functions W1 and W2, which are the functions of two independent
variables  and q2. The structure functions correspond to the two possible
polarisation states of the virtual photon: longitudinal and transverse.
Longitudinal polarisation exists only because photon is virtual and has a mass.
For elastic scattering, (P+q)2=M2 and the two variables  and Q2 are related
by Q2=2M .
2004,Torino
Aram Kotzinian
3
Scaling
To determine W1 and W2 separately
it is necessary to measure the
differential cross-section at two
values of  and E' that correspond
to the same values of  and Q2.
This is possible by varying the
incident energy E.
SLAC result: the ratio of  /Mott
depends only weakly on Q2 for high
values of W.
For small scattering angles  /Mott
≈ W2 . Thus, the structure function
W2 does not depend on Q2.
2004,Torino
Aram Kotzinian
4
Scaling
Instead, at high values of
W the function W2
depends on the single
variable  = 2M / Q2
(at present the variable
x=1/ is widely used)
This is the so-called
"scaling" behaviour of
the cross-section
(structure function).
It was first proposed by
Bjorken in 1967.
W1,2(,q2)  W1,2(x)
when ,q2  ∞.
2004,Torino
Aram Kotzinian
5
Deep Inelastic Scattering (DIS)
z-axis to be along the incident
lepton beam direction.
2004,Torino
Kinematic Variables
M --The mass of the target hadron.
E -- The energy of the incident lepton.
k -- The momentum of the initial lepton.
 -- The solid angle into which the
outgoing lepton is scattered.
E’ -- The energy of the scattered lepton.
K’ -- The momentum of the scattered
lepton,
K’ =
(E’;E’sinqcosf;E’sinqsinf;E’cos).
P -- The momentum of the target, p = (M;
0; 0; 0), for a fixed target experiment.
q = k-k’ -- the momentum transfer in the
scattering process, i.e. the momentum of
the virtual photon.
Aram Kotzinian
6
Important variables
The invariant mass of the final hadronic system X is
2004,Torino
Aram Kotzinian
7
Some inequalities
The invariant mass of X must be at least that of a nucleon, since baryon
number is conserved in the scattering process.
Since Q2 and n are both positive, x must also be positive.
The lepton energy loss E-E’ must be between zero and E, so the physically
allowed kinematic region is
The value x = 1 corresponds to elastic scattering.
2004,Torino
Aram Kotzinian
8
Any fixed hadron state X with invariant mass M X contributes to the
cross-section at the value of x
2
Q
 , with fixed x
In the DIS limit
So, any hadron state X with fixed invariant mass gets driven to x=1
The experimental measurements give the cross-section as a function
of the final lepton energy and scattering angle. The results are often
presented instead by giving the differential cross-section as a function
of (x, Q 2 ) or (x,y). The Jacobian for converting between these cases is
easily worked out using the definitions of the kinematic variables
2004,Torino
Aram Kotzinian
9
Thus the cross-sections are related by
Q2
Q2 
x

2M 2ME E
the contours of constant x are straight
lines through the origin with slope x.
Q2
1
 ( E  )(1  cos  )
2ME M
the contours of constant angle q
are straight lines passing through the
point n=E
2004,Torino
Aram Kotzinian
10
For fixed value of x, the maximum allowed value of Q 2
It is useful to have formulae for the different components of q as a
function of x and y.
This expressions are valid in the Lab frame with z-axis along lepton
momentum
2004,Torino
Aram Kotzinian
11
Expression for DIS cross section
The scattering amplitude M is given by
sl - lepton polarizati on
 - target polarizati on
2004,Torino
Aram Kotzinian
12
It is conventional to define the leptonic tensor
The definition of the hadronic tensor is slightly more complicated.
Inserting a complete set of states gives
where the sum on X is a sum over the allowed phase space for the final state X
2004,Torino
Aram Kotzinian
13
Translation invariance implies that
Only the first term contributes, since p X0  p 0 and q 0  0
Using leptonic and hadronic tensors we have
2004,Torino
Aram Kotzinian
14
Finally, integrating over azimuth, we get
2004,Torino
Aram Kotzinian
15
Leptonic tensor
The polarization of a spin 1/2 particle can be described by a spin vector
defined in the rest frame of the particle by
The spin vector in arbitrary frame is obtained by Lorentz boost
2004,Torino
Aram Kotzinian
16
For a spin-1/2 particle at rest with spin along the z-axis, the spin
vector is s  mzˆ . This differs from the conventional normalization of s
by a factor of the fermion mass m. Here we use the relativistic spinors
normalized to 2E. In the extreme relativistic limit have s=Hk, where k is
the lepton momentum and H is the lepton helicity.
Using trace theorems we obtain for leptonic tensor
Unpolarized lepton beam probes only the symmetric part of hadronic
tensor
2004,Torino
Aram Kotzinian
17
The Hadronic Tensor for Spin-1/2 Targets
Using parity, time-reversal invariance, hermiticity and current
conservation one can show that
Where the structure functions F1 , F2 , g1 and g 2 depend on Q 2 and  .
Often another structure functions are used in the literature:
2004,Torino
Aram Kotzinian
18
Scaling
The hadronic tensor is dimensionless
The structure functions are dimensionless functions of the Lorentz
2
2
2
invariant variables p  M , pq and Q
It is conventional to write them as functions of x  Q 2 2 pq and Q 2
They can be written as dimensionless functions of the dimensionless
2
variables x and Q 2 M
2
In elastic scattering there is a strong dependence on Q 2 M , and the
2
elastic form factors fall o like a power of Q 2 M .
Bjorken: in DIS the structure functions only depend on x, and must be
independent of Q 2 .
2004,Torino
Aram Kotzinian
19
The Cross-Section for Spin-1/2 Targets
Useful relation:
Contracting hadronic and leptonic tensors we get:
For a longitudinally polarized lepton beam, the polarization is sl   k
where    1 is the lepton helicity.
2004,Torino
Aram Kotzinian
20
Longitudinally Polarized Target

A target polarized along the incident beam direction: s h  M z
where    1 for a target polarized parallel or antiparallel to the
beam. sh   p in the evaluation of the cross-section
Where the azimuthal angle f has been integrated over since the
cross-section is independent on f.
2004,Torino
Aram Kotzinian
21
Transversely Polarized Target
The polarization vector of a transversely polarized target can be

x
chosen to point along the  axis in the Lab frame. So, the
azimuthal angle of scattered lepton is counted from that direction.
Then, in this case we get:
The structure functions g1 and g2 are equally important for a
transversely polarized target, and so an experiment with a
transversely polarized target can be used to determine g2, once g1
has been measured using a longitudinally polarized target.
2004,Torino
Aram Kotzinian
22
Details of derivation for unpolarized DIS
Consider the general form of inelastic electron-proton scattering
We must construct W only from the available 4-vectors, p and q ,
and the invariant tensors g  and   . Thus we can write the most
general structure in terms of the possible 6 tensors and corresponding
form factors
2004,Torino
Aram Kotzinian
23
Leptonic tensor is symmetric and we can ignore the antisymmetric
terms in hadronic tensor for unpolarized DIS. Then conservation of the
neutral current requires that
, or, for arbitrary p  , q
Hence the coefficients of p  and q in this equation mustseparately
vanish
Substituting back into the initial expression we have
2004,Torino
Aram Kotzinian
24
In the laboratory frame we have the following relations between the
kinematic quantities
2004,Torino
Aram Kotzinian
25
Finally the definition of the cross section gives
2004,Torino
Aram Kotzinian
26