Feynman Path Integral Formulation

94 3 Gravity in 2 + ε Dimensions
not expect gravity to be screened. On the other hand the infrared growth of the coupling
in the strong coupling phase G > G c can be written equivalently as
[ ( ) m
G(k 2 2 (d−2)/2
]
) ≃ G c 1 + a 0
k 2 + ... , (3.117)
where the dots indicate higher order radiative corrections, and which exhibits a
number of interesting features. Firstly the fractional power suggests new non-trivial
gravitational scaling dimensions, just as in the case of the non-linear σ-model. Furthermore,
the quantum correction involves a new physical, renormalization group
invariant scale ξ = 1/m which cannot be fixed perturbatively, and whose size determines
the scale for the quantum effects. In terms of the bare coupling G(Λ), itis
given by
( ∫ G(Λ)
) dG
′
m = A m ·Λ exp −
β(G ′ , (3.118)
)
which just follows from integrating μ ∂
∂μG = β(G) and then setting as the arbitrary
scale μ → Λ. Conversely, since m is an invariant, one has Λ d m = 0; the running of
G(μ) in accordance with the renormalization group equation of Eq. (3.104) ensures
that the l.h.s. is indeed a renormalization group invariant. The constant A m on the
r.h.s. of Eq. (3.118) cannot be determined perturbatively, it needs to be computed by
non-perturbative (lattice) methods, for example by evaluating invariant correlations
at fixed geodesic distances. It is related to the constant a 0 in Eq. (3.117) by a 0 =
1/(Am 1/ν G c ).
At the fixed point G = G c the theory is scale invariant by definition. In statistical
field theory language the fixed point corresponds to a phase transition, where the
correlation length ξ = 1/m diverges and the theory becomes scale (conformally)
invariant. In general in the vicinity of the fixed point, for which β(G)=0, one can
write
β(G)
If one then defines the exponent ν by
∼
G→Gc
β ′ (G c )(G − G c )+O[(G − G c ) 2 ] . (3.119)
β ′ (G c )=−1/ν , (3.120)
then from Eq. (3.118) one has by integration in the vicinity of the fixed point
m
∼
G→Gc
Λ · A m |G(Λ) − G c | ν , (3.121)
which is why ν is often referred to as the mass gap exponent. Solving the above
equation (with Λ → k) forG(k) one obtains back Eq. (3.117), with the constant a 0
there related to A m in Eq. (3.121) by a 0 = 1/(Am 1/ν G c ) and ν = 1/(d − 2).
That m is a renormalization group invariant is seen from
dΛ