1. P780.02 Spring 2003 L9Richard Kass Inelastic ep Scattering and Quarks Elastic vs Inelastic electron-proton scattering: In the previous lecture we saw that the scattering reaction: e - p  e - p could be described by the Rosenbluth formula: The functions K 1 and K 2 are form factors and contain information about the structure and size of the proton. This process is an example of an elastic scattering. In an elastic scattering we have the same kind and number of particles in the initial and final state. Since no new particles are created in this type of collision we also satisfy the classical definition of an elastic collision, i.e. initial kinetic energy= final kinetic energy. In an inelastic collision there are "new" particles in the final state. Examples of inelastic e - p scatterings include: e - p  e - p  o e - p  e -  + e - p  e - pK + K - These collisions are mediated by photons for center of mass energies well below the Z-boson mass and hence are classified as electromagnetic interactions. So, all quantum numbers respected by the electromagnetic interaction must be conserved. sections 7.4&7.5 of M&S.

2. P780.02 Spring 2003 L9Richard Kass Inelastic ep Scattering and Quarks Since there are many inelastic final states it is convenient to define a quantity called the inclusive cross section. Here we are interested in the reaction: e - p  e - X + It called an inclusive reaction because we don't measure any of the properties of "X", hence we include all available final states. Experimentally this means that we only measure the 4-vector (energy and angle) of the final state electron. The inclusive cross section (in the lab frame) for e - p  e-X + can be written as: This is similar in form to the Rosenbluth formula. There are some important differences however. First, this is the cross section for the scattered electron to have energy E  within a solid angle d . The Rosenbluth formula does not contain this dependence. This difference comes about because in an inelastic scatter the energy of the scattered electron (E) is not uniquely determined by the scattering angle (  ) as for the elastic case. simplified version of M&S eq. 7.53  E E X

3. P780.02 Spring 2003 L9Richard Kass Inelastic ep Scattering and Quarks The scattered electron must be described (kinematically) by two variables. A common set of (Lorentz invariant) kinematic variables are: q 2 and Here we define p as the 4-vector of the target proton (e.g. (M,0,0,0) in "lab" frame) and q is the difference between the incoming electron (beam) and outgoing (scattered) electron: Note that for elastic scattering (e - p  e - p) we have x=1. In fact x is bounded by 0≤x≤1.  pepe pepe q note: q 2 <0 It is convenient to use the positive variable Q 2 : Q 2 = -q 2 = (p e -p e  ) 2 -(E e -E e  ) 2 With this definition Q 2 is the square of the invariant mass of the virtual photon. In the limit E e & E e  >>m e Q 2 is: Q 2 = (p e -p e  ) 2 -(E e -E e  ) 2 = 2 E e E e  - 2p e  p e Q 2 = 2 E e E e (1-cos  ) q = p e -p e  = p X -p T q 2 = (p X -p T ) 2 = (E 2 X -M) 2 - p 2 X = M 2 X + M 2 –2ME X for elastic scattering M X =M q 2 = 2M 2 –2ME X = 2M(M –E X ) 2qp = 2 (E e -E e  )M- (p e -p e  )(0) 2qp = 2 M(E e -E e  ) but E e -E e  = E X –M 2qp = 2 M(E X –M) x = -q 2 /(2pq) = -[2M(M –E X )]/[2 M(E X –M)] =1 (elastic scattering)

4. P780.02 Spring 2003 L9Richard Kass Looking Inside a Nucleon What if spin ½ point-like objects are inside the proton ? As Q 2 increases, the wavelength of the virtual photon decreases and at some point we should be able to see "inside" the proton. This situation was analyzed by many people in the late 1960's (Bjorken, Feynman, Callan and Gross) and predictions were made for spin 1/2 point-like objects inside the proton (or neutron). These theoretical predictions were quickly verified by a new generation of electron scattering experiments performed at SLAC and elsewhere (DESY, Cornell…). In the limit where Q 2 and 2pq are large Bjorken predicted that the form factors would only depend on the scaling variable x: x = Q 2 /2pq These relationships are a consequence of point-like objects (small compared to the wavelength of the virtual photon) being "inside" the proton or neutron. Bjorken Scaling Re-write W 1 and W 2 in terms of F 1 and F 2 : scaling region

5. P780.02 Spring 2003 L9Richard Kass Looking Inside a Nucleon Callan-Gross Relationship: If the point-like objects are spin 1/2 then Callan and Gross predicted that the two form factors would be related: 2xF 1 (x)=F 2 (x) If however the objects had spin 0, then they predicted: 2xF 1 (x)/F 2 (x)=0 x Good agreement with spin ½ point-like objects inside proton or neutron! But are these objects quarks ? i.e. do they have fractional charge? In the late 1960’s most theorists called these point-like spin ½ objects “partons”

6. P780.02 Spring 2003 L9Richard Kass Are Partons Quarks? Do these objects have fractional electric charge ? 2/3|e| or -1/3|e| First we have to realize that x represents the fraction of the proton’s momentum carried by the struck quark. Next, we take into account how the quarks share the nucleon's momentum. The quarks inside a nucleon now have some probability (P) distribution (f(x)=dP/dx) to have momentum fraction x. The structure function for the inelastic scattering of an electron off of a quark which has momentum fraction x and electric charge e i is: To get the structure function of a proton (or neutron) we must add all the quark contributions. Let u(x) denote the x probability distribution for an up quark and d(x) the x probability for a down quark. Then we can write: This simple model does not describe the data! data

7. P780.02 Spring 2003 L9Richard Kass Are Partons Quarks? We can generalize the structure functions by allowing quark anti-quark pairs to also exist (for a short time) in the nucleon. For the proton we expect: The generalized structure functions for a proton is: We now talk about two types of quarks, valence and sea. The sea quarks are the ones we get from the quark anti-quark pairs, the valence quarks are the ones that we expect to be in the nucleon. Thus a proton has two valence u quarks and one valence d quark. There are many testable relationships between structure functions! If there are only u (d) quarks in the proton (neutron) we expect 0.25 (4) for the ratio. If there are only sea quarks in the nucleon than the ratio is 1. For example for electron nucleon scattering we predict: expect sea quarks to be produced at small x

8. P780.02 Spring 2003 L9Richard Kass Are Partons Quarks? Assume only “valence” quarks in proton and neutron u p = pdf for up quark in a proton u N = pdf for up quark in a neutron d p = pdf for down quark in a neutron d N = pdf for down quark in a proton pdf=probability distribution function By isospin invariance (up quark= down quark): u=u p = d N d=d p = d N favored as x  1 If only “sea” quarks in proton and neutron then: F 2N /F 2p =1 sea quarks have to be produced in quark anti-quark pairs

9. P780.02 Spring 2003 L9Richard Kass Partons are Quarks Compare electron nucleon scattering with neutrino scattering: EM interaction (photon) sensitive to quark electric charge Neutrino scattering via W exchange is blind to quark electric charge But W - only couples to down quark (lepton # conservation) Expect F 2 of e(p+n) < F 2 of  p+n) by the average of the square of the quark charges: x neutrinos electrons Good Agreement with prediction! The parton charges are the quark charges. Can also measure the average fraction of the nucleon momentum carried by the quarks by integrating the area under the N curve: Area under the curve is only  0.5! Something else in the nucleon is carrying half of its momentum! GLUONS M&S section 7.5