JHEP01(2016)069
specify in the next section. Here we assume the conformal mode σ has been Wick rotated
from the Lorentzian action as derived from the functional measure [4] which ensures that
the action is bounded from below. This action depends on two metrics, the dynamical
metric g
µν
, and the non-dynamical background metric ¯g
µν
. The background metric is
needed both to regulate the theory and to implement the gauge fixing. Once we have
inserted this action into the flow equation we shall identify ¯g
µν
= g
µν
in order to determine
the beta functions for the flowing couplings G
k
and Λ
k
. For a discussion of background
field flows in the functional RG see [64]. For later convenience we also identify the wave
function renormalisation of the metric g
µν
and the corresponding anomalous dimension
Z
k
≡
G
N
G
k
, η ≡ ∂
t
ln Z
k
, (2.4)
where G
N
is a constant which can be identified with the the low energy Newton’s constant
G
N
= G
0
for trajectories with a classical limit. From the beta functions we will look for
RG trajectories which emanate from a UV fixed point G
k
→ k
−2
g
∗
and Λ
k
→ k
2
λ
∗
at high
energies k → ∞, while recovering classical k-independent couplings G
0
= G
N
and Λ
0
= Λ
when the regulator is removed in the limit k → 0. Such globally safe trajectories suggest
gravity is a well defined quantum field theory on all length scales.
At a non-gaussian fixed point where g
∗
and λ
∗
are finite the scaling is determined from
the critical exponents θ
n
. These exponents appear in the linear expansion
λ
i
− λ
i
∗
=
X
n
C
n
V
i
n
e
−tθ
n
, (2.5)
where λ
i
is a basis of dimensionless couplings e.g λ
i
= {g, λ} = {k
2
G
k
, k
−2
Λ
k
} and the
range of n is equal to the range of i. Here V
i
n
are the eigen-directions and C
n
are constants.
The exponents −θ
n
(note the minus sign) and the vectors V
i
n
correspond to the eigenvalues
and eigenvectors of the stability matrix
M
i
j
=
∂β
i
∂λ
j
λ
i
=λ
i
∗
, (2.6)
where β
i
= ∂
t
λ
i
are the beta functions which vanish for λ
i
= λ
i
∗
. If θ
n
is positive it cor-
responds to a relevant (UV attractive) direction and supports renormalisable trajectories.
For negative θ
n
the direction is irrelevant and C
n
must be set to zero in order to renormalise
the theory at the fixed point. Including more couplings in the approximation would intro-
duce more directions in theory space. The criteria of asymptotic safety is that the number
of relevant directions should be finite at such a UV fixed point [6]. The fewer number of
relevant directions the more predictive the theory defined at the fixed point will be. High
order polynomial expansions in R suggest there are just three relevant directions [24, 40, 41]
while a general argument for f (R) theories imply that there is a finite number of relevant
directions [65].
3 Physical degrees of freedom
General relativity has just two massless propagating degrees corresponding to the two
polarisations of the graviton. On the other hand conformal fluctuations, which are non-
– 6 –
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