Feynman Path Integral Formulation

76 3 Gravity in 2 + ε Dimensions
For our purposes it will sufficient to look, in the zero-field case h = 0, at the
β-function of Eq. (3.14) which incorporates, as should already be clear from the
result of Eq. (3.33), a tremendous amount of information about the model. Herein
lies the power of the renormalization group: the knowledge of a handful of functions
[β(g),ζ (g)] is sufficient to completely determine the momentum dependence of all
n-point functions Γ (n) (p i ,g,h,Λ).
One can integrate the β-function equation in Eq. (3.14) to obtain the renormalization
group invariant quantity
( ∫ g
ξ −1 dg ′ )
(g)=m(g)=const. Λ exp −
β(g ′ , (3.34)
)
which is identified with the correlation length appearing, for example, in Eq. (3.25).
The multiplicative constant in front of the expression on the r.h.s. arises as an integration
constant, and cannot be determined from perturbation theory in g. Conversely,
it is easy to verify that ξ is indeed a renormalization group invariant,
ξ [Λ,g(Λ)] = 0, as stated previously in Eq. (3.26).
In the vicinity of the fixed point at g c one can do the integral in Eq. (3.34), using
Eq. (3.15) and the resulting linearized expression for the β-function in the vicinity
of the non-trivial ultraviolet fixed point,
Λ
dΛ d
and one finds
β(g)
∼
g→gc
β ′ (g c )(g − g c )+... (3.35)
ξ −1 (g)=m(g) ∝ Λ |g − g c | ν , (3.36)
with a correlation length exponent ν = −1/β ′ (g c ) ∼ 1/(d − 2)+.... Thus the correlation
length ξ (g) diverges as one approaches the fixed point at g c .
In general the existence of a non-trivial ultraviolet fixed point implies that the
large momentum behavior above two dimensions is not given by naive perturbation
theory; it is given instead by the critical behavior of the renormalized theory. In
the weak coupling, small g phase the scale m can be regarded as a crossover scale
between the free field behavior at large distance scales and the critical behavior
which sets in at large momenta.
In the non-linear σ-model another quantity of physical interest is the function
M 0 (g),
[ ∫ g
M 0 (g)=exp − 1 2
dg ′ ζ ]
(g′ )
0 β(g ′ , (3.37)
)
which is proportional to the order parameter (the magnetization) of the non-linear
σ-model. To one-loop order one finds ζ (g)=
2π 1 (N − 1)g + ... and therefore
M 0 (g)=const.(g c − g) β , (3.38)
with β = 1 2 ν(d −2+η) and η = ζ (g c)−ε. To leading order in the ε expansion one
has for the anomalous dimension of the π field η = ε/(N − 2)+O(ε 2 ). In gauge