1. Particle Physics WS 2012/13 (20.11.2012)
Stephanie Hansmann-Menzemer
Physikalisches Institut, INF 226, 3.101

2. Electron Proton Scattering
Electron Proton Scattering are probing the structure of the proton!
In e-p → e-p scattering, the nature of the IA strongly
depends on the wave length of the virtual particle, thus
on the momentum transfer |q|
At very low |q| values λ >> rp:
scattering is equivalent to that from „point-like“ spin-less objects
elastic scattering
At low |q| values λ ~ rp:
Scattering is equivalent to that from an extended charged
object
At high |q| values λ < rp:
The wave length is sufficiently short to resolve sub-structure
→ scattering from constituent quarks
inelastic scattering
At very high |q| values λ << rp
the proton appears to be sea of quarks and gluons

3. Elastic vs. Inelastic Scatteringp3 e p3 e e p1 p1 p4 p2 p2
invariant mass W, p4 p p p
inelastic scattering
exitation of proton
deep inelastic scattering
proton gets split up in many pieces
elastic scattering
need a single variable
to describe scatter process
for given p1,p2
e.g. scatter angle ϴ, E3, q2, …
need two independent variables
to describe scatter process
for given p1,p2

4. Inelastic Scattering e p3 q2 <0 for t-channel process, e
thus define Q2 = -q2 p1 p2
invariant mass W, p4 p
work in lab-frame: p2 = (M,0,0,0)
4 momentum transfer: q = p1 – p3
energy loss: ν = E1-E3= qp2/M
fractional energy loss: y = (E1-E3)/E1 = qp2/p1p2
invariant mass: W2 = (q+p2)2 = q2 + 2qp2 + p22 = q2 + 2qp2 + M2
due to baryon number conservation W2 ≥ M2
(proton is lightest baryon)
= 0 elastic scattering
> 0 inelastic scattering
W2-M2 = q2 +2qp2
= elastic scattering
0 < x < 1 inelastic scattering
Bjorken variable x: x = −𝑞2 2𝑞𝑝2 = 𝑄2 2𝑞𝑝2 =

5. Inelastic Scattering
Need to distinguish carefully between lab frame and CMS and LI form of variables!
Lab frame: proton at rest p2 = (M,0,0,0)
LI form
energy loss of
incoming particle
Bjorken x
fraction energy loss
of incoming particle
4 momentum transfer
square
ν = E1-E ν = qp2/M
x = −𝑞2 2𝑀ν x = 𝑄2 2𝑞𝑝2
y = (E1-E3)/E y = qp2/p1p2
q = (p1-p3) q = (p1-p3)2

6. Inelastic Scattering
Note: most DIS (deep inelastic scattering) results from HERA collider which collides electron
beams on proton beams (proton not at rest before collisions; lab frame ≠ CMS)
example of inelastic
ep scattering in H1 detector
more about the HERA collider and its experiments later in the lecture ….

7. Inelastic Scattering e p3 e p1 p2 invariant mass W, p4 p
Produce excited states
e.g. Δ+ (1232)
deep inelastic scattering
(DIS); proton splits up in
many final state particles

8. Double differential x-section for inelastic scattering
Tow variables used to describe inelastic scattering process
→ need to quote double
differential cross-section,
e.g.
𝑑2σ 𝑑Ω𝑑𝐸3 (in a given lab frame)
or (LI)
𝑑2σ 𝑑𝑥𝑑𝑄2
observation:
inelastic cross-section
drops off significantly less than elastic one as function of q2

9. Q2 dependence of (in)elastic ep cross-section
Rosenbluth formular for elastic scattering process (in lab frame: p2 = (M,0,0,0)):
𝐺𝐸2 𝑞2 + 𝜏𝐺𝑀2 𝑞2 1+τ +2 τ 𝐺𝑀2 𝑞2 𝑡𝑎𝑛2 ϴ/2
𝑑σ 𝑑Ω = α2 4𝐸12𝑠𝑖𝑛4ϴ/2 𝐸3 𝐸1
τ = 𝑄2 4𝑀2
Inelastic scattering
𝑑σ 𝑑Ω Mott
Saw last time from fit to data:
G(q2) = 𝑄2 /0.71 𝐺𝑒𝑉 ~ Q-4
elastic scattering
for Q2 >> M (τ >> 1):
𝑑σ 𝑑Ω / ~ (2τ tan2ϴ/2) GM2 ~ Q-6
𝑑σ 𝑑Ω Mott
Inelastic cross-section almost independent of q2

10. (In)elastic scattering x-section formular
Rosenbluth formular [in lab frame: p2 = (M,0,0,0)]:
𝑑σ 𝑑Ω = α2 4𝐸12𝑠𝑖𝑛4ϴ/2 𝐸3 𝐸1
𝐺𝐸2 𝑞2 + 𝜏𝐺𝑀2 𝑞2 1+τ 𝑐𝑜𝑠2 ϴ 2 +2 τ 𝐺𝑀2(𝑞2)𝑠𝑖𝑛2 ϴ/2
f2(Q2)
f1(Q2)
Rosenbluth formular in LI form:
𝑑σ 𝑑𝑄2 = 4π α2 𝑄4 [ 𝒇𝟐 𝑸𝟐 1−𝑦− 𝑀2𝑦2 𝑄 𝑦2 𝒇𝟏(𝑸𝟐)]
elastic cross section inelastic cross section
Inelastic cross section [in lab frame: p2 = (M,0,0,0)]:
𝑑σ 𝑑𝐸3𝑑Ω = α2 4 𝐸12𝑠𝑖𝑛4ϴ/2 [ 𝑭𝟐 𝒙,𝑸𝟐 ν 𝑐𝑜𝑠2ϴ/2 + 2 𝑀 𝑭𝟏(𝒙,𝑸𝟐)]
Inelastic cross section in LI form:
𝑑σ 𝑑𝑥𝑑𝑄2 = 4π α2 𝑄4 [ 𝑭𝟐 𝒙,𝑸𝟐 𝑥 1−𝑦− 𝑀2𝑦2 𝑄 𝑦2 𝑭𝟏(𝒙,𝑸𝟐)]
Form factors have been replaced by structure functions, which depend on two
paramters, which cannot anymore interpreted as Fourier transforms of electric
charge and magnetic moment distribution.

11. Bjorken Scaling Hypothesis (1967)
„if scattering is caused by point-like constituents (partons) structure functions for
fixed x must be independent of Q2“
Experimental observation: F1(x,Q2) → F1(x) F2(x,Q2) → F2(x) defined as Bjorken saling!
Proton consists out of point-like partons!