1. Overview The Structure of the Proton Quark-Parton Model lecture-1
QCD and the QCD improved parton model
DGLAP resumming LL NLL in ln(Q2)
lecture-2
How to parametrise/determine PDFs
lecture-3
Parton Distribution Functions and their Uncertainties lecture-4
QCD at low-x, low-Q lecture-5
Importance for LHC physics lecture-6
Devenish and Cooper-Sarkar
‘Deep Inelastic Scattering’ OUP 2004

2. PDFs were first investigated in deep inelastic lepton-nucleon scatterning -DIS
Leptonic tensor - calculable 2
Lμν Wμν
dσ ~
Hadronic tensor- constrained by Lorentz invariance E’ Ee Ep
q = k – k’, Q2 = -q2
s= (p + k)2
x = Q2 / (2p.q)
y = (p.q)/(p.k)
Q2 = s x y
This is the scale of the vector boson probe
These are 4-vector invariants
s = 4 Ee Ep
Q2 = 4 Ee E’ sin2θe/2
y = (1 – E’/Ee cos2θe/2)
x = Q2/sy
The kinematic variables are measurable

5. for massless quarks and p2~0 so
Without assumptions as to what goes on in the hadron the double differential cross-section for e± N scattering can be written as
d2(e±N) = [ Y+ F2(x,Q2) - y2 FL(x,Q2) ± Y_xF3(x,Q2)], Y± = 1 ± (1-y)2
dxdy
Leptonic part hadronic part
F2, FL and xF3 are structure functions which express the dependence of the cross-section on the structure of the nucleon (hadron)–
The Quark-Parton Model interprets these structure functions as related to the momentum distributions of point-like quarks or partons within the nucleon AND the measurable kinematic variable x = Q2/(2p.q) is interpreted as the FRACTIONAL momentum of the incoming nucleon taken by the struck quark
We can extract all three structure functions experimentally by looking at the x, y, Q2 dependence of the double differential cross-section- thus we can check out the parton model predictions
(xP+q)2=x2p2+q2+2xp.q ~ 0
for massless quarks and p2~0 so
x = Q2/(2p.q)
The FRACTIONAL momentum of the incoming nucleon taken by the struck quark is the MEASURABLE quantity x

20. Scattering: elastic, inelastic,deep-inelastic
For a charge g from a potential charge Zg through the mediation of a quantum of mass μ.
Assume free waves for r>>1/μ
where
The Matrix Element has the form of a coupling constant squared times a propagator
Then using the Golden Rule
Which can describe Rutherford Scattering as μ→0,
What happens if the scattering centre has finite size?
The form factor for a proton is ~
Such a form is NOT point-like.

21. Now consider Inelastic scattering
q=k-k’, p2=mN2, PX2=MX2
k+p=k’+pX , pX2 = p2 +q2 +2p.q, so
Q2 =-q2= 2p.q + mN2 –mX defines the positive momentum transfer squared.
If the target is at rest and the incident and scattered energy of the lepton probe is E and E’ respectively then
If the final state hadronic mass is known- as for elastic scattering OR the production of
specific hadron resonances like N* and Δ - then E’ and θ are not independent
But if we integrate over final states – ‘inclusive’ cross-sections- then they are and we usually use the two variables Q2 and x=Q2/2p.q to describe the process. (For elastic scattering x=1).
We will then need form factors or ‘structure functions which depend on x and Q2
The figures show inclusive spectra for E’ at low and slightly higher energies.
The peaks correspond to the nucleon resonances. One can see the importance of these decreasing as the energy increases- we are moving into the deep-inelastic region.
What is the origin of region B?
Could it be scattering from constituents of the proton? Wouldn’t this lead to more peaks?
Not if these constituents have momentum~200MeV
-due to their confinement