ASYMPTOTIC FREEDOM AND THE RENORMALIZATION GROUP 23
Substitution of (2.16) into (2.15) gives:
e\p) = e\\og{p/pX e \p \ log(A//i))
(2.17)
The first thing renormalizability tells us is that (2.17) actually does not
depend on its last argument, since we can change A and compensate for
it by changing el without changing e^(p). Hence:
e\p) = e\\og(p/fiX
(2.18)
We see that e^(pX being expressed in terms of e^(fiX does not contain
any divergences and does not depend on the structure of the theory at
distances of the order of lattice spacing. But this is not the end of the
story. Actually we have a further constraint on (2.18) which follows
from the fact that the point p was quite arbitrary. Hence, just as it was
with A, it must be possible to compensate for a shift in p by changing
e^{p). We have, therefore, a functional constraint on e^{p) which is quite
easy to solve. Namely, it is clear that:
with
e\p) = /(log(p//z) + g{e\p)))
f{g(x)) = X
(2.19)
(2.20)
The structure (2.19) makes the above property explicit: a shift in
log(p/)u) is obviously compensated by a change of e^{p). The formula
(2.20) follows from the fact that e^{p)\p = ^ =
The relation (2.19)
presents a very strong constraint on the structure of momentum
dependence. It is not fulfilled in a fixed order of perturbation theory and
perm its us to obtain nonperturbative expressions.
It follows from (2.19) that in a theory without dimensional parameters a so-called dimensional transmutation takes place:
e\p) = /(log(p//l))
X = fie
(2.21)
All quantities depend on a universal correlation length 2“ ^ which
should be kept fixed as the lattice spacing A"^ goes to zero. No other
arbitrary parameters enter into the theory. (The last is not generally
true: there are theories with several effective charges, like massless
scalar QED, in which physical quantities depend on the ratios of these
charges). In order to see how (2.19) improves perturbation theory, let us
write it in differential form:
de\p)
= P(e\p))
d \og(p/p)
l^(x) = f'(g(x))
(2.22)
Substitution of (2.16) into (2.15) gives:
e\p) = e\\og{p/pX e \p \ log(A//i))
(2.17)
The first thing renormalizability tells us is that (2.17) actually does not
depend on its last argument, since we can change A and compensate for
it by changing el without changing e^(p). Hence:
e\p) = e\\og(p/fiX
(2.18)
We see that e^(pX being expressed in terms of e^(fiX does not contain
any divergences and does not depend on the structure of the theory at
distances of the order of lattice spacing. But this is not the end of the
story. Actually we have a further constraint on (2.18) which follows
from the fact that the point p was quite arbitrary. Hence, just as it was
with A, it must be possible to compensate for a shift in p by changing
e^{p). We have, therefore, a functional constraint on e^{p) which is quite
easy to solve. Namely, it is clear that:
with
e\p) = /(log(p//z) + g{e\p)))
f{g(x)) = X
(2.19)
(2.20)
The structure (2.19) makes the above property explicit: a shift in
log(p/)u) is obviously compensated by a change of e^{p). The formula
(2.20) follows from the fact that e^{p)\p = ^ =
The relation (2.19)
presents a very strong constraint on the structure of momentum
dependence. It is not fulfilled in a fixed order of perturbation theory and
perm its us to obtain nonperturbative expressions.
It follows from (2.19) that in a theory without dimensional parameters a so-called dimensional transmutation takes place:
e\p) = /(log(p//l))
X = fie
(2.21)
All quantities depend on a universal correlation length 2“ ^ which
should be kept fixed as the lattice spacing A"^ goes to zero. No other
arbitrary parameters enter into the theory. (The last is not generally
true: there are theories with several effective charges, like massless
scalar QED, in which physical quantities depend on the ratios of these
charges). In order to see how (2.19) improves perturbation theory, let us
write it in differential form:
de\p)
= P(e\p))
d \og(p/p)
l^(x) = f'(g(x))
(2.22)