Electron-nucleon scattering

Rutherford scattering: non relativistic scatters off a nucleus without penetrating in it (no spin involved).

Mott scattering: 2 ultra-relativistic point-like fermion scattering off each others

Most of the figures of this talk are from Henley and Garcia’s book titled “Subatomic Physics”Some of the slides are from B. Meadow (U of Cincinatti) and from G van der Steenhoven(NIKHEF/RuG)

Henley & Garcia, Subatomic Physics

R. HofstadterNobel prize 1961

Form factors and charge distributions

Henley & Garcia, Subatomic Physics

Henley & Garcia, Subatomic Physics

• Minima of cross-section are comparable to diffraction minima

• These kind of data where used as basis for establishing

e--p scattering is like e- scattering if p is point-like

For e- scattering the steps to obtain the cross section are Use the Feynman’s rule for one helicity state (initial and final) Eq 7.106

Apply the Casimir trick to take into account all spin configurations. Eq 7.126

Compute the traces. Eq 7.129

Elastic e-- Scattering

q qp form-factor

e - e -

e - e -

p p

p1 p3

p2 p4

e--p scattering is like e- scattering if p is point-like

For e- scattering we obtained:

a function of p1 and p3 but it could also be p1 and q

Elastic e--p Scattering

Eq. 7.129

q qp form-factor

e - e -

e - e -

p p

p1 p3

p2 p4

e - ’s (or ’s) can be used to “probe” inside the proton

What do we know about K?

depend on p2=p and q (with q=p2-p4)

K3 is reserved to neutrino scattering

As a (virtual) does the probing, we anticipate two form factors => K1(Q2) and K2(Q2) Note Q2=-q2>0

Proton Form-Factor

q qp form-factor

e - e -

e - e -

p p

p1 p3

p2 p4

Evaluate the cross-section in the lab frame where

and we neglect m (<< M)

Traditionally, a different definition of K1 and K2 is used. electric (GE) and magnetic (GM) form factors are used to obtain the Rosenbluth formula

Rosenbluth formula

Strategy to measure nucleon form factors Scatter electron off a hydrogen target Count the number of scattered electron of energy E’ at angle Change E’ andat least three times. Perform a Rosenbluth separation.

Henley & Garcia, Subatomic Physics

Effective function for nucleon Form-FactorIt turns out that GE

p, GMp and GM

n have the same functional form

(up to a certain Q2)

Dipole function for form factors yields an exponential charge distribution

Deep Inelastic e-p scattering (DIS)

(electron scattering angle)

’(sc

att

ere

d e

lect

ron

energ

y)

ElasticInelastic

In inelastic scattering, the energy (E’) of the scattered electron is not uniquely determined by E and .

For a given E

invariant energy of virtual-photon proton system:

DIS cross-section

Start like for the elastic scattering

The cross section is for observing the scattered electron only. Need to integrate over the complete hadronic systems.

DIS cross-sectionAgain just like for elastic scattering where

W can be defined in term of Wi

W1 and W2 are functions of q2 and q.p (or Q2 and x)

Bjorken scaling

variable

GE and GM are functions of Q2 only.

ep cross-section summary

Non relativistic and no spin

Ultra relativistic point like fermions

Point like fermion (one light, one heavy)

ep elastic

DIS ep

Looking deep inside the proton

First SLAC experiment (‘69): expected from proton form factor:

First data show big surprise: very weak Q2-dependence:

scattering off point-like objects?

How to proceed: Find more suitable variable What is the meaning of

As often at such a moment….

…. introduce a clever model!

Nobel prize ’90Friedman, Kendall and Taylor

Looking deep inside the proton

With a larger momentum transfer, the probing wavelength gets smaller and looks “deeper” inside the proton

Therefore :

Consider the case now where the

Electron scatters on quarks/partons

Particles of spin ½

The Quark-Parton Model Assumptions (infinite momentum frame):

Neglect masses and pT ’’s

Proton constituent = Parton

Impulse Approximation:

ignore the binding of quarks between each others

Lets assume: pquark = xPproton

if |x2P2 |=x2M2 <<q2 it follows:

e

P

parton

e’

Quasi-elastic scattering off partons

Check limiting case:

Therefore:

x = 1: elastic scattering

and 0 < x < 1Definition Bjorken scaling variable

Structure Functions F1, F2

Instead of W1 and W2 use F1 and F2:

Rewrite this in terms of : (elastic e-q scatt.: 2mq = Q2 )

Experimental data for 2xF1(x) / F2(x)

→ quarks have spin 1/2 and are point-like

(if bosons: no spin-flip F1(x) = 0)

Callan-Gross relation

Structure Functions F1, F2

From the Callan-Gross relationship:

Introduce the concept of density function

is the number of quark of flavor I that carry a

fractional momentum in the range

Such that :

In the quark-parton model:

[and F2 = 2xF1 analogously]

Quark momentum distribution

Interpretation of F1(x) and F2(x)

Valence quark vs Sea quark

Momentum of the proton Do quark account for the momentum of the proton?

Integrating over F2ep(x) and F2

en(x)

Therefore:

Gluons carry about 50% of the proton’s momentum:Indirect evidence for gluons.

Momentum sum rule

Quarks in protons & neutrons

If qsp(x) = qs

n(x) and x 0:

In the limit x 1:

assume isospin symmetry

assume same high-x tail:

assume → u-quark dominance

Modern data

First data (1980):

“Scaling violations”: weak Q2 dependence rise at low x what physics??

PDG 2002

….. QCD