1. 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.
e (k,E)
N (P,M)
e(k',E')
 (q) W 
2004,Torino
Aram Kotzinian

2. 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

3. 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:
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 .
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Aram Kotzinian

4. 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

5. 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

6. Deep Inelastic Scattering (DIS)
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’cosq).
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.
z-axis to be along the incident lepton beam direction.
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Aram Kotzinian

7. Important variables
The invariant mass of the final hadronic system X is
2004,Torino
Aram Kotzinian

8. 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.
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Aram Kotzinian

9. So, any hadron state X with fixed invariant mass gets driven to x=1
Any fixed hadron state X with invariant mass contributes to the cross-section at the value of 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, ) 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

10. Thus the cross-sections are related by
the contours of constant x are straight
lines through the origin with slope x.
the contours of constant angle q are straight lines passing through the point n=E
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Aram Kotzinian

11. For fixed value of x, the maximum allowed value of
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
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Aram Kotzinian

12. Expression for DIS cross section
The scattering amplitude M is given by
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Aram Kotzinian

13. 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
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Aram Kotzinian

14. Translation invariance implies that
Only the first term contributes, since and
Using leptonic and hadronic tensors we have
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Aram Kotzinian

15. Finally, integrating over azimuth, we get
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Aram Kotzinian

16. 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
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Aram Kotzinian

17. For a spin-1/2 particle at rest with spin along the z-axis, the spin
vector is 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

18. 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
Often another structure functions are used in the literature:
2004,Torino
Aram Kotzinian

19. Scaling The hadronic tensor is dimensionless
The structure functions are dimensionless functions of the Lorentz invariant variables
It is conventional to write them as functions of
They can be written as dimensionless functions of the dimensionless variables
In elastic scattering there is a strong dependence on , and the elastic form factors fall o like a power of
Bjorken: in DIS the structure functions only depend on x, and must be independent of
2004,Torino
Aram Kotzinian

20. 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
where is the lepton helicity.
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Aram Kotzinian

21. Longitudinally Polarized Target
A target polarized along the incident beam direction:
where for a target polarized parallel or antiparallel to the
beam 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

22. Transversely Polarized Target
The polarization vector of a transversely polarized target can be
chosen to point along the 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

23. Details of derivation for unpolarized DIS
Consider the general form of inelastic electron-proton scattering
We must construct only from the available 4-vectors, and , and the invariant tensors and Thus we can write the most general structure in terms of the possible 6 tensors and corresponding
form factors
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Aram Kotzinian

24. 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 ,
Hence the coefficients of and in this equation mustseparately vanish
Substituting back into the initial expression we have
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Aram Kotzinian

25. In the laboratory frame we have the following relations between the
kinematic quantities
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Aram Kotzinian

26. Finally the definition of the cross section gives
2004,Torino
Aram Kotzinian