9.7.4. The Running Coupling Constant
The modified Coulomb potential V  r  
 r
can be interpreted as the screening of
r
the bare charge by a cloud of virtual ee+ pairs. In the momentum space, eq(9.90)
may be taken as defining a running coupling constant
   p2  

1    I  p 2   I  0 
(9.92)
There are several important theoretical issues associated with    p 2  . First among
all is its relation with the renormalization process. Thus, instead of eq(9.89), we
could in principle make use of another coupling constant defined by

0
1  0 I    2 
so that
   p2  

1    I  p   I    2 
2

, we have
1   I  0

 2    



I    2 
1   I  0   1 
 1   I  0

Setting  p 2   2 and using 0 


1    I    2   I  0 
(9.93)
This aspect of the renormalization can be further explored and developed into the
renormalization group machinery [see Chapter 11].
The existence of a running coupling constant can be interpreted as an energy
dependence of the effective strength of EM interactions. This dependence is
appreciable only when  p 2 m2 whereupon
  p
2


  p 2 
 1  ln   2 
 3  m 
1
(9.94)
Thus, even at the maximum energies (of order TeV ) that are accessible at present-day
particle accelerators,    p 2  can increase by only 2% from its value ate zero
energy. On the other hand,    p 2    at  p 2
m2e3 / 
m 2 e 411 .
Although
energy of this magnitude can never be reached in any conceivable experiment, there is
cause for concern on theoretical grounds. In analogy of the pole of (9.90) at p 2  0
that indicates a real photon of zero mass, the pole in (9.94) suggests the existence of a
particle of imaginary mass M  i me3 / 2 , known as the Landau ghost. This
M2
 1 . Since the existence of such
E2
particles is generally not acceptable, there seems to indicate some fundamental flaw in
would be a tachyon with velocity v  1 
QED.
A related problem is that the infinite constant I  0  in (9.89) is positive. Thus,
there is no positive values of 0 for which the renormalized  is nonzero. This in
turn means that the theory is basically non-interacting and thereby trivial. Now, this
problem may not have physical significance since 0 has no direct physical meaning
and the arguments were based on all sorts of approximations. Nonetheless, most
people expect QED to break down at sufficiently high energies.