Richard Kass
Physics 780.02 Homework Set 3
Due 5/14/2003
1)The square of the matrix element for elastic electron proton scattering can be written as:
M
2
(
( p1 p)( p3 p) q 2 
2g 2 2 
2
)
K
[
p
p

2
(
mc
)
]

K
[
 ]

1
1 3
2
4 
q2 
( Mc) 2
with: p1= 4-vector of initial state electron (mass=m), p3= 4-vector of final state electron, p= 4vector of target proton (mass=M), and q=p1- p3. Show that the above expression can be re-written
in the lab frame (p=(M, 0, 0, 0)) as:
g 4c2
| M |2  (
) 2 K1 sin 2 ( / 2)  K 2 cos 2 ( / 2)
4 EE  sin 4 ( / 2)


In the above equation E is the energy of the initial state electron, E’ the energy of the final state
electron, and  is the angle between the two electrons.
2) For a neutron evaluate the quark distributions:
1
 [u( x )  u ( x )]dx
0
1
 [d ( x )  d ( x )]dx
0
1
 [ s( x )  s ( x )]dx
0
Repeat the problem for an – instead of a neutron.
Hint: Look at the proton example given in the notes.
3) Draw the “spectator” diagram for the following decays:
n  pe  e
D 0  K  

0 
D  K  
D 0  K  

  pe e
D 0    
What is the Cabibbo factor for each of these decays (i.e. factors of sinc and/or cosc)?
Note: there are many examples of spectator decay diagrams in M&S ch8 (e.g. Figs. 8.19, 8.25).
4) Show that the CKM matrix, given in the notes in terms of 12, 13, 23, and , is unitary.
Suppose the CKM matrix was diagonal, with all diagonal elements equal to one. What are the
implications for the decay of mesons containing an s, c, or b quark? Give an example (or two) of
the possible decay modes of the K0, D0 and B0 mesons.
5) As discussed in lecture and in your book (p248) it is possible for neutral mesons to “mix” with
their anti-particles. Particle anti-particle mixing has been observed with Ko’s and B0’s (= anti-b
quark and a d quark). Mixing is also expected to occur with neutral D mesons (charm and antiup quark) and the Bs meson (anti-b and strange quark). In analogy with Fig. 10.8 (p249) draw
the “box” diagrams (that’s the official term for graph in Fig. 10.8) for the D, B, and Bs systems.
Note: each of these systems has more than one box diagram. Include the CKM factors for each
diagram. Which diagrams are largest and smallest according to their CKM factors?