P780.02 Spring 2003 L9
Richard 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:
sections 7.4&7.5
of M&S.
2
 E 
d 

2
2
2
2
 

2K1(q )sin ( / 2)  K2 (q )cos ( / 2)
2
d 4M p Esin ( / 2)  E


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-ppo
e-p  e-D+
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.
P780.02 Spring 2003 L9
Richard 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:
2

d


 2W1 sin 2 ( / 2)  W2 cos2 ( / 2)
 
2
dE d 2E sin ( / 2)


simplified version of
M&S eq. 7.53
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.
E
E
g
X
P780.02 Spring 2003 L9
Inelastic ep Scattering and Quarks
Richard Kass
The scattered electron must be described (kinematically) by two variables.
A common set of (Lorentz invariant) kinematic variables are:
q2
and
q2
x
2qp
note: q2 <0
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:
It is convenient to use the positive variable Q2:
q  pe  pe
Q2 = -q2 = (pe-pe )2-(Ee-Ee )2
With this definition Q2 is the square of the invariant mass of the virtual photon.
pe
g
pe
q
In the limit Ee& Ee >>me Q2 is:
Q2 = (pe-pe )2-(Ee-Ee )2 = 2 EeEe - 2pepe
Q2 = 2 EeEe(1-cos)
Note that for elastic scattering (e-pe-p) we have x=1. In fact x is bounded by 0≤x≤1.
q = pe-pe = pX-pT
q2 = (pX-pT)2 = (E2X-M)2 - p2X = M2X + M2 –2MEX
for elastic scattering MX=M
q2 = 2M2 –2MEX = 2M(M –EX)
2qp = 2 (Ee-Ee )M- (pe-pe )(0)
2qp = 2 M(Ee-Ee )
but Ee-Ee = EX –M
2qp = 2 M(EX –M)
x = -q2/(2pq) = -[2M(M –EX)]/[2 M(EX –M)] =1 (elastic scattering)
Looking Inside a Nucleon
P780.02 Spring 2003 L9
Richard Kass
What 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…).
Re-write W1 and W2 in terms of F1 and F2:
2


  2 F1 ( x, Q 2 ) 2
d

2 MxF2 ( x, Q 2 )
2



sin
(

/
2
)

cos
(

/
2
)


dE d  2 E sin 2 ( / 2)  
M
Q2

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
2
Q
Bjorken Scaling
W2 (Q 2 , x)  F2 ( x)
2 Mx
MW1 (Q 2 , x)  F1 ( x)
These relationships are a consequence of
point-like objects (small compared to
the wavelength of the virtual photon)
being "inside" the proton or neutron.
P780.02 Spring 2003 L9
Richard 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:
2xF1(x)=F2(x)
If however the objects had spin 0, then they predicted:
2xF1(x)/F2(x)=0
Good agreement with
spin ½ point-like objects
inside proton or neutron!
x
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
Are Partons Quarks?
Richard Kass
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 ei
is:
ei2 f ( x)
F1 
and F2  ei2 xf ( x)
2
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:
F1 p 
1
1 2 2
1 2

2
 ei fi ( x)  ( ) u p ( x)  ( ) d p ( x)
2 i 1
2 3
3

F1n 
1
1  1 2
2 2

2
 ei fi ( x)  ( ) d n ( x)  ( ) un ( x)
2 i 1
2 3
3

This simple model does not describe the data!
data
Are Partons Quarks?
P780.02 Spring 2003 L9
Richard Kass
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:
1


 up (x)  up (x)dx  2
0
1


 d p (x)  d p (x)dx  1
0
1


 sp (x)  sp (x)dx  0
0
The generalized structure functions for a proton is:
expect sea quarks to be
produced at small x
1 2
1

F1 p  ( )2 u p ( x)  u p ( x)  ( )2 d p ( x)  d p ( x)  s p ( x)  s p ( x) 
2 3
3





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!
For example for electron nucleon scattering we predict:
F
0.25  2n  4
F2 p
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.
P780.02 Spring 2003 L9
Are Partons Quarks?
Richard Kass
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
1
 2

F2 p  x  ei2 fi ( x) x ( )2 u p ( x)  ( )2 d p ( x)
3
i 1
 3

1
 2

F2 N  x  ei2 fi ( x) x ( )2 u N ( x)  ( )2 d N ( x)
3
i 1
 3

By isospin invariance (up quark= down quark):
u=up= dN d=dp= dN
F2 N
F2 p
2
1
( ) 2 d ( x)  ( ) 2 u ( x)
4d ( x )  u ( x )
3
 3

2
1
( ) 2 u ( x)  ( ) 2 d ( x) 4u ( x)  d ( x)
3
3
F2 N  4 if u(x)  0 


F2 p 1/4 if d(x)  0
favored as x1
If only “sea” quarks in proton and neutron then: F2N/F2p=1
sea quarks have to be produced in quark anti-quark pairs
Partons are Quarks
P780.02 Spring 2003 L9
Richard Kass
Compare electron nucleon scattering with neutrino scattering:
M&S section 7.5
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 n(p+n) by the average of the square of the quark charges:
1 2
1
2
1
 1 5
 e2 (ep  eN )  [2( )2 u  ( )2 d ]  [( )2 d  2( )2 u ]    (2u ( x)  d ( x))
6 3
3
3
3
 6 9
1
1
 e2 (np nN )  [(1)2 d ]  [2(1)2 u ]   (2u ( x)  d ( x))
3
3

F2vp vN

18 ep  eN Good Agreement with prediction!
 F2
The parton charges are the quark charges.
5
Can also measure the average fraction of the
nucleon momentum carried by the quarks
by integrating the area under the nN curve:
neutrinos
1
 F2eN dx  fraction of momentum carried by the quarks
0
Area under the curve is only 0.5!
Something else in the nucleon is carrying half
of its momentum! GLUONS
electrons
x