One-loop divergences in massive gravity theory

0.11 MB·English

by  Ioseph L. Buchbinder

One-loop divergences in massive gravity theory
I. L. Buchbinder,†,‡,1 D. D. Pereira‡,2 and I. L. Shapiro‡,3
† Department of Theoretical Physics, Tomsk State Pedagogical University, 634041,
Tomsk, Russia
‡ Departamento de F´ısica, ICE, Universidade Federal de Juiz de Fora, 36036-330, MG,
2 Brazil
1
0
2

n
a
J Abstract. The one-loop divergences are calculated for the recently pro-

6
posed ghost-free version of massive gravity, where the action depends on both
1

metric and external tensor field f. The non-polynomial structure of the mas-
]
h
sive term is reduced to a more standard form by means of auxiliary tensor
t
-
p field, which is settled on-shell after quantum calculations are performed. As
e
h one should expect, the counterterms do not reproduce the form of the classical
[
action. Moreover, the result has the form of the power series in f.

1
v
5
4
1 1 Introduction
3
.
1
0 It is well known that general relativity is a non-linear dynamical theory of symmetric
2
1 second rank tensor field in curved space-time. In the linear approximation this theory de-
:
v scribes a propagation of massless spin-2 irreducible representation of the Poincare group.
i
X Dynamical theory of massive spin-2 representation of the Poincare group has been con-
r
a structedbyFierzandPauli[1]intermsofsymmetricsecondranktensorfieldinMinkowski
space. It is natural to think that there should exist a non-linear dynamical theory in term
of symmetric second rank tensor field in curved space-time, whose linear limit will be the
Fierz-Pauli theory. However, during a long time such theory has not been constructed.
An evident way to find non-linear generalization of Fierz-Pauli theory is to add some
kind of massive term into Lagrangian of general relativity. It could be a cosmological
constant, but in the linear limit the cosmological term does not provide a true mass term
in Fierz-Pauli theory. As a result the only way to insert a mass term to Lagrangian
1
E-mail: [email protected]
2E-mail: dante.pereira@fisica.ufjf.br
3E-mail: shapiro@fisica.ufjf.br. On leave from Tomsk State PedagogicalUniversity, Tomsk, Russia.
of general relativity is to use, additionally to metric, an extra second rank tensor field
(reference metric) and find a coupling of metric to this extra field in such a way that in
the linear approximation for both metric and Minkowski reference metric, this coupling
should generate a true mass term in Fierz-Pauli theory.
The extension of general relativity described above has been studied in details by
Boulware and Deser [2]. They have shown that inserting the mass term with the help
of additional second rank tensor field can yield, in general, the inconsistent theory with
propagating ghost field (BDghost). However recently therewas a progress inconstructing
a family of the non-linear dynamical theories which are free of problem of BD ghost [3],
[4], [5], [6] (see also the review [7]). As a result, at present one can have the Lagrangians
which satisfy the following set of conditions: (i) describe consistent ghost-free non-linear
dynamics of symmetric second rank tensor field; (ii) have a number of propagating
degrees of freedom exactly corresponding to massive spin-2 field, (iii) depend of reference
metric andonsomemass parameter m, while inthelimit m = 0reproducetheLagrangian
of general relativity without cosmological term; (iv) reproduce the Lagrangian of Fierz-
Pauli theory in linear approximation for dynamical metric and for Minkowski reference
metric.
In this note we study the quantum aspects of the massive gravity theory. To be
more precise, we compute the one loop divergences of a minimal massive theory [4] and
investigate their structure. Of course, the massive gravity theory is non-renormalizable
as well as general relativity since massive gravity Lagrangian includes general relativity
Lagrangian and hence should leat to the quadratic in curvature tensor counter-terms at
one loop. In order to find the one-loop divergences we will use the background field
method and Schwinger-DeWitt proper-time techniques [8].
Although the theory under consideration is non-renormalizable, one can expect it to
have many interesting features in quantum domain. First, the massive term in the action
possesses a complicated structure in tensor indices and its background-quantum splitting
(decomposition of initial field into background and quantum field, which is the element
of the background field method) is not trivial. Second, a functional determinant, defin-
ing the one-loop effective action, has different and in fact more complicated structure
as compared to the massless case of general relativity. Therefore, evaluation of effective
method may require some novelty in the method of calculation. Third, it is interesting
to study whether the one-loop divergences in massive gravity theory vanish on-shell as
it is the case for the general relativity without matter. Fourth, the one-loop divergences
in massive gravity theory are expected to depend on the reference metric. As a result of
computations we obtain the one-loopdivergences in terms of the specific general covariant
functionals depending on reference metric. They can be considered as the possible candi-
2
dates for actions of reference metric if to treat this metric as a dynamical field. Problem
of Lagrangian for reference metric is broadly discussed in the literature (see e.g. [4] and
reference therein). Fifth, any new gravitational model deserves a study of quantum as-
pects in the hope to extend our understanding of a quantum gravity and to shead a light
on a possible role of gravity in quantum domain.
The paper is organized as follows. In Sect. 2 we consider an equivalent representation
of the theory via auxiliary tensor field and the procedure of linearization. The derivation
of bilinear form of the action and the details of background field method are treated in
Sect. 3. In Sect. 4 we present the results of calculating the one-loop divergences. In Sect.
5 we present some discussions of the result and draw our conclusions.
2 Linearization of massive term
Consider the action of a minimal model of massive gravity [4]
S[g ] = S [g ]+S [g ]
µν 0 µν m µν
= d4x√ g R+2Λ + m2tr g−1f . (1)
−
Z h (cid:16)p (cid:17)i
Here m is a mass parameter, f means an external tensor field (reference metric) f ,
µν
expression g−1f in action S means gµαf , all other notations are standard4. According
m αν
to [4] we have to put Λ = 3m2, however it is more convenient to perform all calculations
−
for an arbitrary value of Λ and fix it only in the final result.
To calculate the one-loop divergences of the theory under consideration we should
compute the second variational derivative of action with respect g . Such computation
µν
for the term S is very non-trivial since the matrices gµα and f do not commute5. We
m αν
will avoid this obstacle considering the classically equivalent theory formulated in terms
of dynamical metric g and auxiliary field ϕµ .
µν ν
It is easy to see that the theory, described by the action (1) in terms of dynamical field
g , is classically equivalent to the theory in terms of the fields g and auxiliary field ϕµ
µν µν ν
with the following action
S[g ,ϕµ ] = S [g]+S [g ,ϕµ ], (2)
µν ν 0 m µν ν
where the action S [g ,ϕµ ] is given by
m µν eν e
m2
S e[g ,ϕµ ] = d4x√ g gµνf ϕα +(ϕ−1)µ . (3)
m µν ν να µ µ
2 −
Z
(cid:2) (cid:3)
4The gravitatioenalconstant κ is suppressed
5Thefirstvariationalderivativeis computedsimply enough[4],howeverthesecondderivativecannot
be computed analogously to [4].
3
Indeed, the equation of motion that follows from the variation over ϕµ in (2) has the
ν
form
δS δS
= m = 0,
δϕµ δϕµ
ν ν
e e
The solution to this equation is
ϕµ = g−1f −1/2 µ .
ν ν
(cid:2)(cid:0) (cid:1) (cid:3)
Replacing this solution back into (2), we obtain the action (1). It shows that the two
actions are classically equivalent. Starting from this point we will use the action (2)6.
3 Background field method and bilinear form to ac-
tion
Our main purpose is to develop the background field method ( [8], see also the details in
[9]) to the theory (2) and use it to calculate the one-loop divergences of the theory. The
first step is to obtain the bilinear form of the action (2).
According to the background field method, the fields ϕµ and g are replaced by sums
ν µν
of background and quantum fields as follows
ϕµ ϕµ +ψµ , g g +h . (4)
ν ν ν µν µν µν
→ →
Here ϕµ and g are background fields, while ψµ and h are quantum fields. By means
ν µν ν µν
of a simple algebra one can obtain the bilinear form for the actions S and S in the form
0 m
1
S(2) +S = d4x√ ghµν Jˆ hαβ, e (5)
0 gf 2 − µν,αβ
Z
and
1 1
S (2) = m2 d4x√ g hαβ Gˆ hµν ψα Aˆ β, ν ψµ + Bˆβ ψα . (6)
m αβ,µν β α µ ν α β
− 2 − 2
Z n o
In Eqe. (5) we have introduced the so-called minimal gauge fixing term
1
S = d4x√ g χµχ , (7)
gf µ
−2 −
Z
where
1
χ = hλ h. (8)
µ λ µ µ
∇ − 2∇
6We assumethatthis equivalenceis fulfilled onthe quantumlevelaswell, sincethere isno anysource
for possible anomaly
4
The operators Jˆ , Gˆ , Aˆ β, ν and Bˆβ , which were used in (5) and (6),
µν,αβ αβ,µν α µ α
have the form
1 1 1
Jˆ = K ✷+R +g R (g R +g R ) (R+2Λ)K ,
µν,αβ µν,αβ µανβ νβ µα µν αβ αβ µν µν,αβ
2 − 2 − 2
1 1
Gˆ = K gστf ϕλ +(ϕ−1)σ g f ϕσ +g f ϕσ ,
αβ,µν αβ,µν τλ σ σ αβ νσ µ µβ νσ α
−4 − 2
1 h i
Aˆ β, ν = (ϕ−1)σ (ϕ−1)β (ϕ−1)ν +(ϕ−1)σ (ϕ−1)ν (ϕ−1)β ,
α µ α µ σ µ α σ
−2
1h i
Bˆβ = hβλf , (9)
α λα
−2
where we define
1
K = δ g g (10)
αβ,µν αβ,µν αβ µν
− 2
and use notation
1
δ = (g g +g g )
αβ,µν αµ βν αν βµ
2
for the DeWitt identity matrix in the space of symmetric matrices. In the expressions (9)
one have to assume symmetrization in both couples of indices µν and αβ. Let us note
that the expression for Jˆ in (9) is one for the usual Einstein quantum gravity with
µν,αβ
the cosmological constant and the other terms here are because of the massive terms in
Eq. (1).
It is easy to see that the path integral over the quantum field ψµ can be taken at
ν
once. It is well known that the following identity holds for Hermitian matrices A(y,x):
i
ψexp dy dxψ(y)A(y,x)ψ(x)+i dxB(x)ψ(x)
D − 2
Z n Z Z i Z o
= DetA −1/2 exp dy dxB(y)A−1(y,x)B(x) . (11)
× 2
n Z Z o
(cid:0) (cid:1)
−1/2
Let us note that the quantity DetA corresponds to the determinant of a numerical
matrix. Since we assume dimensional regularization here, this object is irrelevant to the
(cid:0) (cid:1)
analysis of quantum corrections to the effective action and therefore will not be omitted.
Using Eq. (11) in the expression for the generating functional of Green functions, we
present the bilinear form for the action (2) as follows
1
S(2) = S(2) + S + S(2) = d4x√ g hαβ Hˆ hµν ,
0 gf m 2 − αβ,µν
Z
where the opereator Hˆ is given by e
αβ,µν
1
Hˆ = Jˆ +m2Gˆ + m2f (Aˆ−1) λ, σf . (12)
αβ,µν µν,αβ αβ,µν βλ α µ νσ
4
5
In order to obtain the matrix A−1, let us consider the following procedure. The result
(11) is valid for the Hermitian matrix A. Therefore we need to take only symmetric part
of the matrix A. Consider first the matrix A¯ which is not symmetrized. It is an easy
exercise to find its inverse, however one has to work a little bit more to do the same with
the symmetric part of it. One can write A¯ as
A¯ β, ν = (ϕ−1)σ (ϕ−1)β (ϕ−1)ν . (13)
α µ α µ σ
−
The corresponding inverse matrix is given by
(A¯−1) α, ρ = ϕρ ϕτ ϕα . (14)
β σ τ β σ
−
Consider now the following symmetric structure
X α, ρ = ϕρ ϕτ ϕα ϕα ϕτ ϕρ . (15)
β σ τ β σ τ σ β
− −
Then we arrive at the equation
Aˆ ν, β X α, ρ = Z ρ, ν = δρδν +Y ρ, ν, (16)
µ α × β σ µ σ µ σ µ σ
where
1 1
Y ρ, ν = (ϕ−1)ρ ϕν + (ϕ−1)ν ϕρ . (17)
µ σ µ σ σ µ
2 2
Finally, we have the inverse to the symmetrized matrix in the form of the series
(Aˆ−1) ν, β = X ν, σ (Z−1) λ, β, (18)
µ α µ λ × σ α
where the matrix (Z−1) λ, β is given by
σ α
(Z−1) λ, β = δλδβ Y λ, β + Y λ, τ Y ρ, β Y λ, τ Y ρ, δY χ, β + ... . (19)
σ α σ α − σ α σ ρ τ α − σ ρ τ χ δ α
Multiplying the operator Hˆ by the operator 2Kˆ−1 ,αβ, where
αβ,µν λσ
1
Kˆ−1 ,αβ= δ ,αβ g gαβ,
λσ λσ λσ
−2
we arrive at
2Kˆ−1 ,λσHˆ Oˆ = δ ✷+Πˆ , (20)
αβ λσ,µν αβ,µν αβ,µν αβ,µν
≡
where we have
Πˆ = 2R +2g R g R g R RK 2Λδ
αβ,µν αµβν βν αµ αβ µν µν αβ αβ,µν αβ,µν
− − − −
m2 m2
+ f (Aˆ−1) ρ, τf g f (Aˆ−1)σρ, τf +2m2f ϕσ g
ρ(α β) µ ντ αβ σρ µ ντ νσ (β α)µ
2 − 4
m2
δ gστf ϕλ +(ϕ−1)σ . (21)
αβ,µν τλ σ σ
− 2
h i
6
4 Derivation of one-loop divergences
The one-loop quantum corrections to effective action is written by standard way (see e.g.
[10])
i i
Γ(1) = Ln Det(Hˆ) = Tr Ln(Hˆ), (22)
2 2
where the operator Hˆ corresponds to the bilinear part of the action in quantum fields and
Tr means the functional trace.
The divergent part of Tr LnHˆ can be obtained by calculating Tr Ln(Kˆ−1Hˆ) and
then subtracting the Tr LnKˆ−1
Tr LnHˆ = Tr Ln(Kˆ−1Hˆ)+ Tr LnKˆ−1. (23)
− −
However, as far as we are interested in the logarithmic divergent part of the effective
action, the contribution of the last term can be safely omitted.
The computation of (22) can be performed by the use of the Schwinger-DeWitt
proper-time technique [8]. This technique provides the efficient method of evaluating
the Ln DetOˆ, where the operator Oˆ has the form (see e.g. [11], [12])
Oˆ = ˆ1✷+2hˆµ +Πˆ. (24)
µ
∇
Also we introduce the operators
1
Pˆ = Πˆ + Rˆ1 hˆµ hˆµhˆ ,
µ µ
6 −∇ −
Sˆ = = ˆ1[ , ]+ hˆ hˆ +[hˆ ,hˆ ]. (25)
µν µ ν ν µ µ ν ν µ
∇ ∇ ∇ −∇
In our case, exactly as for the Einstein quantum gravity, the hˆ = 0 and this essentially
µ
simplifies the calculations.
In the framework of dimensional regularization, the quantity Γ¯(1) is written as follows
µn−4 ˆ1 1 1
Γ¯(1) = Tr (R2 R2 )+ Pˆ2 + Sˆ2 , (26)
div −(4π)2(n 4) 180 µναβ − µν 2 12 µν
( )
−
where µ is the parameter of dimensional regularization, the operators Pˆ, Sˆ are defined
µν
above and the surface terms are ignored.
In our case the operators Pˆ and Sˆ have the form
µν
5
Pˆ = 2R +2g R g R g R Rδ
αβ,µν αµβν βν αµ αβ µν µν αβ αβ,µν
− − − 6
1 m2
+ Rg g 2Λδ δ gστf ϕλ +(ϕ−1)σ
αβ µν αβ,µν αβ,µν τλ σ σ
2 − − 2
m2 h m2 i
+ 2m2f ϕσ g + f (Aˆ−1) ρ, τf g f (Aˆ−1)σρ, τf ,
νσ (α β)µ ρ(α β) µ ντ αβ σρ µ ντ
2 − 4
Sˆ = [Sˆ ] = 2R g . (27)
λτ λτ µν,αβ − µαλτ νβ
7
Replacing these operators in the expression (26), after some algebra we obtain the ex-
pression for the divergent part of the one-loop effective action,
2µn−4 53 7 1
Γ(d1iv) |Λ=−3m2 = − (4π)2(n 4) dnx√−g 90E + 20Rµ2ν + 120R2
− Z n
m2 m2
+ R(λ|α|σ)βf (Aˆ−1) τ |ρ|f R 948+236 g−1f µ
στ λ (α β)ρ µ
2 − 48
+ +5 g−1f λ (Aˆ−1) ασ g−1f β +5f h(Aˆ−1)βλ,ασ(cid:0)fp 5f(cid:1) (Aˆ−1)αβ,µνf
β αλ, σ αλ βσ αβ µν
−
+ m2(cid:0)Rλα 1(cid:1)2 g−1f (cid:0)+ g−(cid:1)1f β (Aˆ−1) ρ |τ|f +f (Aˆ−1)βρ |τ|f i
4 αλ ρ λ (α β)τ λρ (α β)τ
h (cid:0)p (cid:1) m4 (cid:0) (cid:1)
2f (Aˆ−1)στ ρf + 1440+12 g−1f λ (Aˆ−1) ασ g−1f β
στ α λρ β αλ, σ
− 16
+ 12f (Aˆ−1)βλ,ασfi 12fh (Aˆ−1)αβ(cid:0),µνf (cid:1) 240 g−1f(cid:0) µ +(cid:1)24 g−1f µ
αλ βσ αβ µν µ µ
− −
+ 4 g−1f µµ g−1f νν + g−1f λ(ρ(Aˆ−1)τ)λµσ (cid:0)gp−1f νσ(cid:1)g−1f ρθ(cid:0)(Aˆ−1)(cid:1)τθµα g−1f αν
+ g(cid:0)p−1f λ(ρ(cid:1)(Aˆ−(cid:0)1p)τ)λµσ (cid:1)g−1f (cid:0)σν g−(cid:1)1f ρθ(Aˆ−1)τθνα(cid:0) g−1f(cid:1) αµ(cid:0) (cid:1) (cid:0) (cid:1)
4(cid:0) g−(cid:1)1f λλ g−1f β(cid:0)ϕfθ(α((cid:1)Aˆ−(cid:0)1)β)θαϕ(cid:1)+2 g−1f(cid:0) λλfα(cid:1)β(Aˆ−1)αβ,µνfµν
−
λ λ
4(cid:0)fp g−(cid:1)1f(cid:0) f(cid:1) (Aˆ−1)ρθαϕ +2f f (cid:0)p g−1f(cid:1) (Aˆ−1)ρθαϕ
ρθ (α ϕ)λ αθ λϕ ρ
−
+ 2f (cid:16)gp−1f β(cid:17) g−1f λ (Aˆ−1) θαϕ +2 g(cid:16)−p1f θ g(cid:17)−1f λ g−1f α (Aˆ−1)ρ ϕ
βλ ϕ θ α λ ϕ ρ θα
+ 2 g−(cid:0)1f τ f(cid:1) (cid:0)pg−1f(cid:1)α (Aˆ−1)λθ ϕ . (cid:0) (cid:1) (cid:0) (cid:1) (cid:0)p (cid:1) (28)
θ λϕ τ α
(cid:0) (cid:1) (cid:0)p (cid:1) io
where we included the mass-independent ghost contribution and used the special value
Λ = 3m2. In the Eq. (28) E = R2 4R2 +R2 is the integrand of the Gauss-Bonnet
− µναβ− µν
topological term and the expression (Aˆ−1)αβµν has been defined in Eq. (16). One has to
note that the matrix (Aˆ)−1 is an infinite power series on the external field f and hence
the divergences (28) have essentially non-polynomial structure in this field too.
Let us note that before the use of the condition Λ = 3m2 the divergences represent
−
the corresponding expression for Einstein quantum gravity [11] with the contribution of
the cosmological term and the rest of the expression is due to additional mass dependent
term in the action. The reason for such a result is that we performed calculations is the
situation when the diffeomorphism symmetry is unbroken. This means we treat f as
µν
external tensor field which does not violate general covariance of the theory.
An interesting observation concerning the Eq. (28) is that there is an explicit simple
hierarchy of the terms, for example the ones with higher derivatives do not depend on
mass and/or on the field f. At the same time, if we consider the classical action with the
algebraic structures presented in (28), we note that there are no derivatives acting on f
there. However, despite there are no such derivatives of f, this field will be dynamical in
action (28) because of the mixture with Ricci tensor and scalar curvature which emerge
8
in the third line of the expression.
The next problem is to see what happens with the result (28) on-shell. For this end we
have to derive the classical equations of motion and replace them into (28). The equation
of motion for the theory (1) with Λ = 3m2 have the form
−
1 1 α 1 νµ
Rµν Rgµν = 3m2gµν + m2gµν g−1f m2 g−1f . (29)
α
− 2 − 2 − 2
(cid:16)p (cid:17) (cid:16)p (cid:17)
After using this relation in Eq.(28), we arrive at the following on-shell result
µn−4 53
Γ(1) = dnx√ g E +m2R(λ|α|σ)βf (Aˆ−1) τ |ρ|f
div |onshell − (4π)2(n 4) − 45 στ λ (α β)ρ
− Z h i
m4µn−4
dnx√ g 1.5 32 111 8f (Aˆ−1)βλ,ασf +0.4 77 g−1f µ
αλ βσ µ
− 8(4π)2(n 4) − · · − ·
7 − Z h (cid:0)p (cid:1)
+ g−1f µ +1.5 13 29 g−1f µ g−1f ν
µ µ ν
5 · ·
+ g(cid:0)−1f λ(cid:1) (Aˆ−1) σ g−1(cid:0)fpν g−1(cid:1)f (cid:0)ρp(Aˆ−1)τ(cid:1)θµ g−1f α
(ρ τ)λµ σ θ α ν
+ (cid:0)g−1f(cid:1)λ(ρ(Aˆ−1)τ)λµσ(cid:0)g−1f(cid:1)σν(cid:0)g−1f(cid:1)ρθ(Aˆ−1)τθνα(cid:0)g−1f(cid:1)αµ
3
4(cid:0) g−(cid:1)1f λ g−1f β(cid:0)f ((cid:1)Aˆ−(cid:0)1) θαϕ(cid:1)+ g−1(cid:0)f λ (cid:1)+16 f (Aˆ−1)αβ,µνf
λ ϕ θ(α β) λ αβ µν
− 2
4f(cid:0)p g−(cid:1)1f(cid:0)λ f(cid:1) (Aˆ−1)ρθαϕ +2f fh (cid:0)pg−1f (cid:1)λ (Aˆ−1)iρθαϕ
ρθ (α ϕ)λ αθ λϕ ρ
−
+ 2f (cid:16)pg−1f β(cid:17) g−1f λ (Aˆ−1) θαϕ +2 g(cid:16)−p1f θ g(cid:17)−1f λ g−1f α (Aˆ−1)ρ ϕ
βλ ϕ θ α λ ϕ ρ θα
1 α
2 g−(cid:0)1f τ f(cid:1) (cid:0)pg−1f(cid:1)α (Aˆ−1)λθ ϕ (cid:0) g(cid:1)−1f(cid:0) µ +(cid:1)6 (cid:0)pg−1f (cid:1)(Aˆ−1)βρ |τ|f
θ λϕ τ α µ ρ (α β)τ
− − 2
+ 2(cid:0) g−(cid:1)1f α (cid:0)gp−1f λ(cid:1)(Aˆ−1)βρ |τ|f h+(cid:0)2p g−1f(cid:1) α g−i(cid:16)1f β ((cid:17)Aˆ−1)λρ |τ|f
λ ρ (α β)τ λ ρ (α β)τ
+ 3(cid:0)p2 5 (cid:1) g−(cid:16)1f µ (cid:17) g−1f λ (Aˆ−1) ασ (cid:0)gp−1f β (cid:1). (cid:16) (cid:17) (30)
µ β αλ σ
− 2
h (cid:0)p (cid:1) i(cid:16) (cid:17) (cid:16) (cid:17) i
It is easy to see that the on-shell result does not vanish as it was for the massless theory
[11]. Moreover, in the first line one can see the term which explicitly depends on the
Riemann tensor.
5 Conclusion
We have developed the background field method and calculated the one-loop divergences
for minimal massive gravity models suggested in [4]. The divergences are formulated in
terms of geometrical invariants constructed from metric and reference metric and contain
the inverse matrix (16) which is an infinite power series in the reference metric f . There
µν
are no doubts that the divergences for the non-minimal, more complicated actions of [4]
will have qualitatively the same structure. The final expression (28) shows that the UV
9
completion of the massive gravity theory would be essentially more complicated than
the one of Einstein quantum gravity. Along with the usual fourth-derivative metric-
dependent terms such completion should include also dependence on the reference metric
f . Furthermore, this field gains dynamics due to the mixture with curvature tensor
µν
components. Therefore the counterterm (28) can be considered as the action functional
defying dynamics of reference metric.
Acknowledgments
I.B. is grateful to CAPES for supporting his visit to Juiz de Fora, where the main part
of this work has been done. Also he acknowledges to RFBR grant, project No 12-02-
00121, RFBR-Ukraine grant, project No 11-02-90445 and grant for LRSS, project No
224.2012.2. D.P. thanks FAPEMIG for supporting his PhD project. I.Sh. is grateful to
CNPq, CAPES and FAPEMIG for partial support.
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