ASYMPTOTIC FREEDOM AND THE RENORMALIZATION GROUP 31
The expression for
in (2.60) is exactly the polarization operator for
massless scalar QED multiplied by an isotopic factor:
=
(2.61)
The factor 2 in (2.61) arises from the two physical polarizations of a^.
Computation of
is easy:
]^ (s c ) _______
d V (2p + q)^(2p + q)
1
(2 n r
p \p + q?
2
{2n)*p^
(2.62)
(This equality follows from the conservation of current
= 0. It
ensures cancellation of the quadratic divergence in
Taking the
trace in pv of (2.62) and keeping only the terms proportional to
we
have:
2q^n(q^) =
1 r d"p 1
2 J (2ny
4p^q^ + 4pq + 4p^
i + ^ ’ + i
P
Pi
d“p 1
( ^ 7
q^
2 KpqY
- r - — r- + 4p^-i
(2.63)
Therefore, to within logarithmic accuracy we have the following
diamagnetic part of the coupling renormalization:
1
^ +
1
4 ^
C,(G) log ^ (A < A)
(2.64)
We see that this part of the effect decreases the coupling as we go to
larger wave lengths. The physical interpretation of this in Minkowski
space is that a charge introduced into the vacuum gets screened by
virtual pairs of particles and antiparticles, just as happens in a dielectric
medium. In Euclidean space the explanation of the sign of (2.64) is also
simple. The second order effect in general should decrease the effective
action (or free energy in the language of statistical mechanics), as is seen
from:
^eff= - log J*^-(S(q>) + F(q,))^^
C(2) _
U y ‘w)>(2,65)
-S(