Springer Lecture Notes in Physics 716

314 U. C. Täuber
is to find a curve parametrisation µ(l) =µl in the space spanned by the
parameters ˜D, ˜τ, andũ such that
l d ˜D(l)
dl
= ˜D(l) γ D (l) , l d˜τ(l)
dl
= ˜τ(l) γ τ (l) , l dũ(l)
dl
= β u (l) , (7.76)
with initial values D R , τ R ,andu R , respectively at l =1.Thefirst-order ordinary
differential equations (7.76), with γ D (l) =γ D (ũ(l)) etc. define running
couplings that describe how the parameters of the theory change under scale
transformations µ → µl. The formal solutions for ˜D(l) and˜τ(l) read
˜D(l) =D R exp
[ ∫ l
1
γ D (l ′ ) dl′
l ′ ]
, ˜τ(l) =τ R exp
[ ∫ l
1
γ τ (l ′ ) dl′
l ′ ]
. (7.77)
For the function ˆχ(l) =ˆχ R ( ˜D(l), ˜τ(l), ũ(l)), we then obtain another ordinary
differential equation, namely
which is solved by
l dˆχ(l)
dl
ˆχ(l) =ˆχ(1) l 2 exp
=[2+γ S (l)] ˆχ(l) , (7.78)
[ ∫ l
1
γ S (l ′ ) dl′
l ′ ]
. (7.79)
Collecting everything, we finally arrive at
[
χ R (µ, D R ,τ R ,u R , q,ω)=(µl) −2 exp
× ˆχ R
(
−
∫ l
1
γ S (l ′ ) dl′
l ′ ]
˜τ(l), ũ(l), |q|
µl , ω
˜D(l)(µl) 2+a )
. (7.80)
The solution (7.80) of the RG equation (7.75), along with the flow equations
(7.76), (7.77) for the running couplings tell us how the dynamic susceptibility
depends on the (momentum) scale µlat which we consider the theory.
Similar relations can be obtained for arbitrary vertex functions by solving the
associated RG equations (7.65) [13]. The point here is that the right-hand side
of Eq. (7.80) may be evaluated outside the IR-singular regime, by fixing one
of its arguments at a finite value, say |q|/µ l = 1. The function ˆχ R is regular,
and can be calculated by means of perturbation theory. A scale-invariant
regime is characterised by the renormalised nonlinear coupling u R becoming
independent of the scale µl,orũ(l) → u ∗ = const. For an RG fixed point to
be infrared-stable, we thus require
β u (u ∗ )=0, β ′ u(u ∗ ) > 0 , (7.81)