P780.02 Spring 2003 L9 Richard Kass

Inelastic ep Scattering and QuarksElastic vs Inelastic electron-proton scattering: In the previous lecture we saw that the scattering reaction:

e-pe-pcould be described by the Rosenbluth formula:

dd

4MpEsin2( / 2)

2E

E2K1(q2)sin2( / 2) K2(q2)cos2( / 2)

The functions K1 and K2 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-po

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&7.5of M&S.

P780.02 Spring 2003 L9 Richard Kass

Inelastic ep Scattering and QuarksSince 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 Ewithin 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.

dd E d

2E sin2( / 2)

2

2W1sin2( / 2)W2 cos2( / 2) simplified version ofM&S eq. 7.53

EE

X

P780.02 Spring 2003 L9 Richard KassInelastic ep Scattering and QuarksThe scattered electron must be described (kinematically) by two variables. A common set of (Lorentz invariant) kinematic variables are:

q2 and qp

qx

2

2

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: q pe p e

Note that for elastic scattering (e-pe-p) we have x=1. In fact x is bounded by 0≤x≤1.

pepe

q

note: q2 <0

It is convenient to use the positive variable Q2:Q2 = -q2 = (pe-pe

)2-(Ee-Ee)2

With this definition Q2 is the square of the invariant mass of the virtual photon.

In the limit Ee& Ee>>me Q2 is:

Q2 = (pe-pe)2-(Ee-Ee

)2 = 2 EeEe- 2pepe

Q2 = 2 EeEe(1-cos)

q = pe-pe = pX-pT

q2 = (pX-pT)2 = (E2X-M)2 - p2

X = M2X + M2 –2MEX

for elastic scattering MX=Mq2 = 2M2 –2MEX = 2M(M –EX)

2qp = 2 (Ee-Ee)M- (pe-pe

)(0)2qp = 2 M(Ee-Ee

)but Ee-Ee

= EX –M2qp = 2 M(EX –M)

x = -q2/(2pq) = -[2M(M –EX)]/[2 M(EX –M)] =1 (elastic scattering)

P780.02 Spring 2003 L9 Richard KassLooking Inside a NucleonWhat if spin ½ point-like objects are inside the proton ?As Q2 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 Q2 and 2pq are large Bjorken predicted that the form factors would only depend on the scaling variable x: x = Q2/2pq

)(),( 12

1 xFxQMW These relationships are a consequence of point-like objects (small compared to the wavelength of the virtual photon)being "inside" the proton or neutron.

)(),(2 2

22

2

xFxQWMx

Q

)2/(cos

),(2)2/(sin

),(2

)2/(sin22

2

222

21

2

2

Q

QxMxF

M

QxF

EdEd

d

Bjorken Scaling

Re-write W1 and W2 in terms of F1 and F2:

s ca l

ing

r egi

on

P780.02 Spring 2003 L9 Richard Kass

Looking Inside a NucleonCallan-Gross Relationship:If the point-like objects are spin 1/2 then Callan and Gross predicted that the two form factors would be related:

2xF1(x)=F2(x)If however the objects had spin 0, then they predicted:

2xF1(x)/F2(x)=0

x

Good agreement withspin 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”

P780.02 Spring 2003 L9 Richard KassAre 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 ofthe 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 ei is:

)(2

)( 22

2

1 xxfeFandxfe

F ii

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:

)()

3

1()()

3

2(

2

1)(

2

1 22

1

21 xdxuxfeF pp

iiip

)()

3

2()()

3

1(

2

1)(

2

1 22

1

21 xuxdxfeF nn

iiin

This simple model does not describe the data!

data

P780.02 Spring 2003 L9 Richard KassAre 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:

up(x) u p(x)dx 0

1 2 dp(x) d p(x)dx

0

1 1 sp(x) s p(x)dx

0

1 0

The generalized structure functions for a proton is:

)()()()()

3

1()()()

3

2(

2

1 221 xsxsxdxdxuxuF ppppppp

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!

0.25F2n

F2 p4

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

P780.02 Spring 2003 L9 Richard KassAre Partons Quarks?

)()

3

1()()

3

2()( 22

1

22 xdxuxxfexF pp

iiip

)()

3

1()()

3

2()( 22

1

22 xdxuxxfexF NN

iiiN

Assume only “valence” quarks in proton and neutron up= pdf for up quark in a proton uN= pdf for up quark in a neutron dp= pdf for down quark in a neutron dN= pdf for down quark in a proton

pdf=probability distribution function

By isospin invariance (up quark= down quark): u=up= dN d=dp= dN

)()(4

)()(4

)()3

1()()

3

2(

)()3

1()()

3

2(

22

22

2

2

xdxu

xuxd

xdxu

xuxd

F

F

p

N

0d(x)if1/4

0u(x)if4

2

2

p

N

F

F

favored as x1

If only “sea” quarks in proton and neutron then: F2N/F2p=1 sea quarks have to be produced in quark anti-quark pairs

P780.02 Spring 2003 L9 Richard KassPartons are QuarksCompare 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 F2 of e(p+n) < F2 of p+n) by the average of the square of the quark charges:

x

eNepvNvp FF 22 5

18

))()(2(9

5

6

1])

3

1(2)

3

2[(])

3

1()

3

2(2[

6

1 2222)(

2 xdxuuddue eNep

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 quarksby integrating the area under the N curve:

quarksthebycarriedmomentumoffraction1

02 dxF eN

Area under the curve is only 0.5!Something else in the nucleon is carrying halfof its momentum! GLUONS

M&S section 7.5

))()(2(3

1])1(2[])1[(

3

1 22)(

2 xdxuude Np