34
3 Scalar-Tensor Gravities
L 1 = π,
L 2 = −
1
2
∂π · ∂π;
L 3 = −
1
2
[]∂π · ∂π;
L 4 = −
1
4
[]
2 ∂π · ∂π − 2[]∂π · · ∂π − [
2 ]∂π · ∂π + 2∂π ·
2 · ∂π
;
L 5 = −
1
5
[]
3 ∂π · ∂π − 3[]
2 ∂π · · ∂π − 3[][
2 ]∂π · ∂π +
+ 6[]∂π ·
2 · ∂π + 2[]
3 ∂π · ∂π + 3[
2 ]∂π · · ∂π − 6∂π ·
3 · ∂π
. (3.47)
The complete Lagrangian of π is a linear combination of these terms: L = c 1 L 1 +
c 2 L 2 + c 3 L 3 + c 4 L 4 + c 5 L 5 . Clearly, the next step consists in coupling of these
Lagrangians to gravity. But let us first describe some perturbative effects of these
couplings.
One of the interesting effects is that these galileon terms L i are not renormalized
under quantum corrections! The reasons are as follows [64]. First, the galileon is
massless, so, its propagator is 1/k
2 . Then, all galileon couplings c 3 , c 4 , c 5 have negative mass dimensions, therefore the contributions to these terms possess quadratic
and even higher divergences. After integration of subloops, the leading divergence
is proportional to
d
4 k(k
2
)
n , with n ≥ −1, and this integral vanishes within dimensional regularization. Finally, the subleading contributions to galileon vertices vanish
as well (this proof is more sophisticated being based on analysis of symmetries). In
principle, such conclusions are natural for a massless theory with derivative couplings. Other divergent contributions in the galileons theory in the flat space, which
do not match the form of the classical action, in particular, involve more derivatives
(f.e.
2 terms), are discussed in [65].
Clearly, the next step is the coupling of the scalar π to the gravity. One of the first
ideas consists in coupling of galileons to the curvature, so we have terms like [66,
67]:
δS 4 =
d
4 x
√ −g(π μ π
μ
)(π ν G
νρ
π ρ ),
(3.48)
where π μ ≡ ∇ μ π, etc., or the higher terms like π μ π
μν
π
ρ G νρ , or the simplest terms
π
μ
π
ν G μν (the last term is the example of the John term, see below). So, effectively
we have a gravity-coupled scalar field with strongly nonlinear dynamics involving
derivative depending couplings. As it has been claimed in [67], these terms are of
special interest within the cosmological context, where it has been explicitly shown
that the solutions with constant H =
˙
a
a
are consistent for the presence of galileons,
therefore de Sitter-like exponential expansion is possible in this case, with neither
potential term for the scalar nor cosmological constant are employed, therefore the
galileons theory is a sound candidate for the role of the dark energy. In [68], it was
argued that only the minimal scalar-gravity couplings must be considered, as a result,
3 Scalar-Tensor Gravities
L 1 = π,
L 2 = −
1
2
∂π · ∂π;
L 3 = −
1
2
[]∂π · ∂π;
L 4 = −
1
4
[]
2 ∂π · ∂π − 2[]∂π · · ∂π − [
2 ]∂π · ∂π + 2∂π ·
2 · ∂π
;
L 5 = −
1
5
[]
3 ∂π · ∂π − 3[]
2 ∂π · · ∂π − 3[][
2 ]∂π · ∂π +
+ 6[]∂π ·
2 · ∂π + 2[]
3 ∂π · ∂π + 3[
2 ]∂π · · ∂π − 6∂π ·
3 · ∂π
. (3.47)
The complete Lagrangian of π is a linear combination of these terms: L = c 1 L 1 +
c 2 L 2 + c 3 L 3 + c 4 L 4 + c 5 L 5 . Clearly, the next step consists in coupling of these
Lagrangians to gravity. But let us first describe some perturbative effects of these
couplings.
One of the interesting effects is that these galileon terms L i are not renormalized
under quantum corrections! The reasons are as follows [64]. First, the galileon is
massless, so, its propagator is 1/k
2 . Then, all galileon couplings c 3 , c 4 , c 5 have negative mass dimensions, therefore the contributions to these terms possess quadratic
and even higher divergences. After integration of subloops, the leading divergence
is proportional to
d
4 k(k
2
)
n , with n ≥ −1, and this integral vanishes within dimensional regularization. Finally, the subleading contributions to galileon vertices vanish
as well (this proof is more sophisticated being based on analysis of symmetries). In
principle, such conclusions are natural for a massless theory with derivative couplings. Other divergent contributions in the galileons theory in the flat space, which
do not match the form of the classical action, in particular, involve more derivatives
(f.e.
2 terms), are discussed in [65].
Clearly, the next step is the coupling of the scalar π to the gravity. One of the first
ideas consists in coupling of galileons to the curvature, so we have terms like [66,
67]:
δS 4 =
d
4 x
√ −g(π μ π
μ
)(π ν G
νρ
π ρ ),
(3.48)
where π μ ≡ ∇ μ π, etc., or the higher terms like π μ π
μν
π
ρ G νρ , or the simplest terms
π
μ
π
ν G μν (the last term is the example of the John term, see below). So, effectively
we have a gravity-coupled scalar field with strongly nonlinear dynamics involving
derivative depending couplings. As it has been claimed in [67], these terms are of
special interest within the cosmological context, where it has been explicitly shown
that the solutions with constant H =
˙
a
a
are consistent for the presence of galileons,
therefore de Sitter-like exponential expansion is possible in this case, with neither
potential term for the scalar nor cosmological constant are employed, therefore the
galileons theory is a sound candidate for the role of the dark energy. In [68], it was
argued that only the minimal scalar-gravity couplings must be considered, as a result,