```PHYS 633: Elementary Particle Physics
Spring 2012
Homework #4
Due Date: 10/30/12
1. In the center-of-mass frame for the process AB  CD , show that
dQ 
pf
1
4 4 s
2
d
F  4 pi s
And hence that the differential cross section is
d
1  pf  2
 M

d cm 64 2 s  pi 
2. Show that for very high-energy “spinless” electron-muon scattering,
d
 2  3  cos  



d cm 4s  1  cos  
2
Where θ is the scattering angle and α=e2/4π. Neglect the particle masses.
3. Show that in the reaction A  B  C  D ,
s  t  u  m A2  mB2  mC2  mD2
4. Taking e  e   e  e  to be the s channel process, verify that
s  4( k 2  m 2 )
t  2k 2 (1  cos  )
u  2k 2 (1  cos  )
Where θ is the center-of-mass scattering angle and k  k i  k f , where
k i and k f are, respectively, the momenta of the incident and scattered
electrons in the center-of-mass frame. Show that the process is
physically allowed provided s  4m 2 , t  0, u  0 . The physical region is
shown shaded in the figure drawn in class. Note that t  0(u  0)
correspond to forward (backward) scattering.
5. Show that the invariant amplitude for “spinless” electron-electron
scattering,
 iΜe  e

 2  p A  pC   pD  pB 

p A  pB   pD  pC 
2
 i  e
e
2



 pC  pD 2
p

p
D
B


,


can be written as
 s u t u 
Μe  e  ( s, t , u )  e 2 

.
s 
 t
Comment on the symmetry of M under s ↔ t.
6. In e  e      near threshold, one can obviously not neglect the mass
of the  . Working from
d
1  pf  2
  M and the exact spin
d 64 2 s  pi 
averaged amplitude
M 
2

8e 4
 p  pk  k   k   p k  p  m 2 k   p  M 2 k  p  2m 2 M 2
4
q
k , p : incoming e  , e  four vecto rs
k , p : scattered   ,  four vecto rs
m, M : masses of e,
Show that the total cross-section for  production is given by
4 2

3s
 3  2 
v
 where    .
 
c
 2 

```