1. 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) andfrom G van der Steenhoven(NIKHEF/RuG)

2. Henley & Garcia, Subatomic Physics R. Hofstadter Nobel prize 1961

3. Form factors and charge distributions Henley & Garcia, Subatomic Physics

4. Minima of cross-section are comparable to diffraction minima These kind of data where used as basis for establishing

5.  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 qq p form-factor e -  pp p1p1 p3p3 p2p2 p4p4

6.  e - -p scattering is like e-  scattering if p is point-like  For e-  scattering we obtained: a function of p 1 and p 3 but it could also be p 1 and q Elastic e - -p Scattering Eq. 7.129 qq p form-factor e -  pp p1p1 p3p3 p2p2 p4p4

7.  e - ’s (or  ’s) can be used to “probe” inside the proton  What do we know about K  ?  depend on p 2 =p and q (with q=p 2 -p 4 ) K 3 is reserved to neutrino scattering  As a (virtual)  does the probing, we anticipate two form factors => K 1 (Q 2 ) and K 2 (Q 2 ) Note Q 2 =-q 2 >0 Proton Form-Factor qq p form-factor e -  pp p1p1 p3p3 p2p2 p4p4

8.  Evaluate the cross-section in the lab frame where and we neglect m (<< M)  Traditionally, a different definition of K 1 and K 2 is used. electric (G E ) and magnetic (G M ) form factors are used to obtain the Rosenbluth formula Rosenbluth formula

9. 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’ and  at least three times.  Perform a Rosenbluth separation. Henley & Garcia, Subatomic Physics

10. Effective function for nucleon Form-Factor It turns out that G E p, G M p and G M n have the same functional form (up to a certain Q 2 ) Dipole function for form factors yields an exponential charge distribution

11. Deep Inelastic e-p scattering (DIS)  (electron scattering angle)  ’ (scattered electron energy) Elastic Inelastic 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:

12. 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.

13. DIS cross-section Again just like for elastic scattering where W  can be defined in term of W i W 1 and W 2 are functions of q 2 and q.p (or Q 2 and x) Bjorken scaling variable G E and G M are functions of Q 2 only.

14. 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

15. Looking deep inside the proton  First SLAC experiment (‘69):  expected from proton form factor:  First data show big surprise:  very weak Q 2 -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 ’90 Friedman, Kendall and Taylor

16. 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

17. The Quark-Parton Model  Assumptions (infinite momentum frame) :  Neglect masses and p T’ ’s  Proton constituent = Parton  Impulse Approximation: ignore the binding of quarks between each others  Lets assume: p quark = xP proton  if |x 2 P 2 |=x 2 M 2 <<q 2 it follows: e P parton e’e’ Quasi-elastic scattering off partons  Check limiting case:  Therefore: x = 1: elastic scattering and 0 < x < 1 Definition Bjorken scaling variable

18. Structure Functions F 1, F 2  Instead of W 1 and W 2 use F 1 and F 2 :  Rewrite this in terms of : (elastic e-q scatt.: 2m q = Q 2 )  Experimental data for 2xF 1 (x) / F 2 (x) → quarks have spin 1/2 and are point-like (if bosons: no spin-flip  F 1 (x) = 0) Callan-Gross relation

19. Structure Functions F 1, F 2  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 :

20. In the quark-parton model: [and F 2 = 2xF 1 analogously] Quark momentum distribution Interpretation of F 1 (x) and F 2 (x)

21. Valence quark vs Sea quark

22. Momentum of the proton  Do quark account for the momentum of the proton?  Integrating over F 2 ep (x) and F 2 en (x)  Therefore: Gluons carry about 50% of the proton’s momentum: Indirect evidence for gluons. Momentum sum rule

23. Quarks in protons & neutrons  If q s p (x) = q s n (x) and x  0 :  In the limit x  1: assume isospin symmetry  assume same high-x tail:  assume → u -quark dominance

24. Modern data  First data (1980):  “Scaling violations”:  weak Q 2 dependence  rise at low x  what physics?? PDG 2002 ….. QCD