Feynman Path Integral Formulation

84 3 Gravity in 2 + ε Dimensions
In the weak coupling phase u < u c the fermions stay massless and chiral symmetry
is unbroken, whereas in the strong coupling phase u > u c (which is the only phase
present in d = 2) chiral symmetry is broken, a fermion condensate arises and a nonperturbative
fermion mass is generated. In the vicinity of the ultraviolet fixed point
one has for the mass gap
m(u)
∼
u→uc
Λ (u − u c ) ν , (3.74)
up to a constant of proportionality, with the exponent ν given by
ν −1 ≡−β ′ (u c )=ε −
ε2
¯N − 2 − ( ¯N − 3)π
2( ¯N − 2) 2 ε3 + ... (3.75)
The rest of the analysis proceeds in a way that, at least formally, is virtually identical
to the non-linear σ-model case. It need not be repeated here, as one can just take over
the relevant formulas for the renormalization group behavior of n-point functions,
for the running of the couplings, etc.
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 accordance
with Eq. (3.70). In the weak coupling, small u 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.
Finally, the same model can be solved exactly in the large N limit. There too
one can show that the model is characterized by two phases, a weak coupling phase
where the fermions are massless and a strong coupling phase in which a chiral symmetry
is spontaneously broken.
3.5 The Gravitational Case
In two dimensions the gravitational coupling becomes dimensionless, G ∼ Λ 2−d ,
and the theory appears perturbatively renormalizable. In spite of the fact that the
gravitational action reduces to a topological invariant in two dimensions, it would
seem meaningful to try to construct, in analogy to what was suggested originally for
scalar field theories (Wilson, 1973), the theory perturbatively as a double series in
ε = d − 2 and G.
One first notices though that in pure Einstein gravity, with Lagrangian density
L = − 1
16πG 0
√ gR , (3.76)
the bare coupling G 0 can be completely reabsorbed by a field redefinition
g μν = ω g ′ μν , (3.77)