Content uploaded by Alexander Gallego Cadavid
Author content
All content in this area was uploaded by Alexander Gallego Cadavid on Sep 25, 2019
Content may be subject to copyright.
arXiv:1905.10664v2 [hep-th] 19 Sep 2019
PI/UAN-2019-650FT
A systematic procedure to build the beyond generalized Proca field theory
Alexander Gallego Cadavid*
Escuela de Física, Universidad Industrial de Santander,
Ciudad Universitaria, Bucaramanga 680002, Colombia and
Instituto de Física y Astronomía, Universidad de Valparaíso,
Avenida Gran Bretaña 1111, Valparaíso 2360102, Chile
Yeinzon Rodríguez**
Centro de Investigaciones en Ciencias Básicas y Aplicadas, Universidad Antonio Nariño,
Cra 3 Este # 47A-15, Bogotá D.C. 110231, Colombia
Escuela de Física, Universidad Industrial de Santander,
Ciudad Universitaria, Bucaramanga 680002, Colombia and
Simons Associate at The Abdus Salam International Centre for Theoretical Physics,
Strada Costiera 11, I-34151, Trieste, Italy
To date, different alternative theories of gravity, although related, involving Proca fields have
been proposed. Unfortunately, the procedure to obtain the relevant terms in some formulations
has not been systematic enough or exhaustive, thus resulting in some missing terms or ambiguity
in the process carried out. In this paper, we propose a systematic procedure to build the beyond
generalized theory for a Proca field in four dimensions containing only the field itself and its first-
order derivatives. We examine the validity of our procedure at the fourth level of the generalized
Proca theory. In our approach, we employ all the possible Lorentz-invariant Lagrangian pieces made
of the Proca field and its first-order derivatives, including those that violate parity, and find the
relevant combination that propagates only three degrees of freedom and has healthy dynamics for
the longitudinal mode. The key step in our procedure is to retain the flat space-time divergences
of the currents in the theory during the covariantization process. In the curved space-time theory,
some of the retained terms are no longer current divergences so that they induce the new terms that
identify the beyond generalized Proca field theory. The procedure constitutes a systematic method
to build general theories for multiple vector fields with or without internal symmetries.
I. INTRODUCTION
Einstein’s theory of General Relativity is currently the most compelling and simplified theory of classical
gravity. It has survived stringent tests on its validity in different scenarios: the expansion of the universe,
the propagation of gravitational waves, the formation of the large-scale structure, as well as the strong
gravitational field scenarios of neutron stars and black holes [1–13]. Despite its success, General Relativity
is still considered as incomplete since any attempt to produce a quantum theory of gravity (see e.g.
Refs. [14–17]) has shown not to be satisfactory enough. Moreover, when its predictions are compared
with cosmological observations, some authors argue that there exist hints pointing to modifications of the
theory [10–16].
Recently, a plethora of modified gravity theories have been proposed in order to avoid the assumption of
two unknowns constituents of the Standard Cosmological Model (also called ΛCDM), namely, Dark Matter
and Dark Energy [18–21]. Although there exists a large amount of observational data to constrain most
of these modified gravity theories, some of their sectors have only been partially explored, hence their full
cosmological implications are still unknown [10–14]. The general scheme in the formulation of these theories
is the fulfilment of diffeomorphism invariance, unitarity, locality, and the presence of a pseudo-Riemannian
spacetime in the action of the theory [14]. Nonetheless, any attempt to modify General Relativity inevitably
introduces new dynamical degrees of freedom which, depending on the type of modification, could be of
scalar, vector or tensor nature. Unfortunately, such formulation could lead to instabilities or pathologies in
the theory [14,22,23]. A known pathology is the Ostrogradsky’s instability [22–26], where the Hamiltonian
is not bounded from below. The Ostrogradsky’s theorem states that, for a non-degenerate theory1, field
*Electronic address: alexander.gallego@uv.cl
**Electronic address: yeinzon.rodriguez@uan.edu.co
1A non-degenerate theory at nth-order is one in which its Lagrangian fulfil s the condition det ∂2L/∂q(n)
i∂q(n)
j6= 0, where
2
equations higher than second order lead to an unbounded Hamiltonian from below [22–25]. Thus, in order
to formulate a well-behaved fundamental theory, we must build the action in such a way that the field
equations are, at most, second order.
Three relevant formulations of such modified gravity theories correspond to scalar-tensor, vector-tensor,
and scalar-vector-tensor theories, or simply Horndeski, generalized Proca, and scalar-vector-tensor gravity
theories respectively [14,26–40]. These theories satisfy the necessary, but not sufficient, requirement to be
free from the instabilities or pathologies previously mentioned since the actions are built so that the field
equations are second order. Nowadays, extended versions of Horndeski and generalized Proca theories have
been proposed, namely, beyond Horndeski, extended scalar-tensor2, beyond generalized Proca (BGP), and
extended vector-tensor theories3[41–56]. Following similar procedures as those used to build the generalized
Proca theory, the authors in Refs. [57,58] obtained a massive extension of a SU(2) gauge theory, i.e., the
generalized SU(2) Proca theory. This theory is also called the non-Abelian vector Galileon theory since it
considers a non-Abelian vector field Aa
µ, with a= 1,2,3, whose action is invariant under the SU(2) global
symmetry group.
So far the generalized Proca and non-Abelian Proca field theories have been applied extensively to diffe-
rent phenomenological scenarios, which include the construction of inflationary cosmological models [59–
63], the analysis of de Sitter solutions relevant to dark energy models [64], the study of their cosmological
implications in the presence of matter [65–70], the analysis of the strong lensing and time delay effects
around black holes [71], and the construction of static and spherically symmetric solutions for black holes
and neutron stars [71–76].
Although some physical and mathematical motivations to build alternative theories of gravity involving a
Proca field Aµhave been given [34–36], the formulations have not been performed in a systematic enough
or exhaustive way (see however Refs. [26,77,78]). The purpose of this paper is to show a systematic
procedure to build the most general Proca theory LP
n+2 in four dimensions, where LP
n+2 denotes the
Lagrangians containing n≥1 first-order derivatives of Aµ[26,35–39]. As an exception to the rule, LP
2is
defined as the Lagrangian consisting of an arbitrary function of the Faraday tensor Fµν ≡∂µAν−∂νAµ,
its Hodge dual ˜
Fµν ≡ǫµνρσ Fρσ /2, where ǫµνρσ is the Levi-Civita tensor, and Aµonly. As we will show,
the theory thus built is equivalent to the BGP theory since we are able to obtain the Lagrangian LN
4[55]
that identifies it.
In some stages, the procedure is similar to that of Ref. [57]. The difference in our case resides in
retaining the total derivatives of the flat space-time currents. These derivatives lead to some relations
among Lagrangian pieces which, in turn, are used to eliminate some of the pieces since total derivatives do
not contribute to the field equations. However, as we will show below, the convariantized versions of these
relations, in some cases, are no longer total derivatives so they induce new terms in the curved space-time
theory, hence leading to different field equations for the Lagrangians involved.
The layout of the paper is the following. In Section II, we describe the general procedure to construct
the most general Proca theory. In Section III, we discuss the issue of the total derivatives in flat spacetime
and show how these terms are no longer total derivatives, in general, when going to curved spacetime.
Then, in Section IV, we implement the procedure to obtain the LP
4terms; there we show how to obtain
systematically the LP
4terms in the BGP. The conclusions are presented in Section V. Throughout the
paper we use the signature ηµν = diag (−,+,+,+) and set A·A≡AµAµand ∂·A≡∂µAµ. We also
define the generalized Kronecker delta as δµ1...µn−p
ν1...νn−p≡δ[µ1
ν1. . . δµn−p]
νn−p=δµ1
[ν1. . . δµn−p
νn−p]where the brackets
mean unnormalized antisymmetrization.
II. GENERAL PROCEDURE
In this section we describe in detail the procedure to build the most general theory for a Proca field
containing only its first-order derivatives. For most of the description here, we follow the first steps of the
procedure described in Ref. [57] until the consideration of the 4-currents. The procedure is as follows.
q(n)
iis the n-th derivative of the generalized coordinate qiof the system.
2Also called degenerate higher-order scalar-tensor theories (DHOST).
3Which, by the way, could be called degenerate higher-order vector-tensor theories (DHOVT).
3
A. Test Lagrangians
Write down all possible test Lagrangians in a flat spacetime using group theory. The Lorentz-invariant
quantities are constructed out of the metric gµν and the Levi-Civita tensor ǫµνρσ . In Table I, we show the
number of Lorentz scalars that can be constructed with multiple copies of Aµ[57], whereas, in Table II,
we show the number of Lorentz scalars that can be built for a given product of vector fields and vector
field derivatives [57]. These tables are non exhaustive.
number of vector fields Aµ1 2 3 4 5 6
number of Lorentz scalars 0 1 0 4 0 25
Cuadro I: Number of Lorentz scalars that can be constructed with multiple copies of Aµ.
number of ∂µAν
number of AρAσ
0 1 2
1 1 2 2
2 4 10 11
3 7 30
Cuadro II: Number of Lorentz scalars that can be built for a given product of vector fields and vector field
derivatives.
It is worth stressing that, when doing the respective contractions, some Lorentz scalars could be identical
to each other and thus the number of independent terms would be reduced.
Using group theory in this way, we can assure that all possible terms are written down, and that they
are linearly independent.
B. Hessian Conditions
Impose the condition that only three degrees of freedom for the vector field propagate [26,35–39,57,58].
In order to achieve this, we first write down a linear combination of the test Lagrangians in the form
Ltest =
n
X
i=1
xiLi,(1)
where nis the number of test Lagrangians and xiare constant parameters of the theory. We then calculate
the primary Hessian of the test Lagrangian
Hµν
Ltest ≡∂2Ltest
∂˙
Aµ∂˙
Aν
,(2)
where dots indicate derivatives with respect to time. In order to ensure the propagation of only three
degrees of freedom, we impose the vanishing of the determinant of the primary Hessian matrix Hµν [26,35–
39,57,58]. This will guarantee the existence of one primary constraint that will remove the undesired
polarization for the vector field. This condition is equivalent to satisfying Hµ0= 0, i.e.,
Hµ0
Ltest =
n
X
i=1
Hµ0
Li=x1Hµ0
L1+x2Hµ0
L2+···+xnHµ0
Ln= 0 .(3)
Eq. (3) gives a system of algebraic equations for the xiwhose roots impose conditions on the test La-
grangian. For some test Lagrangians their corresponding xiparameters will be zero, thus eliminating
undesirable degrees of freedom. In practice, to calculate the primary Hessian condition in Eq. (3), it turns
out to be easier to separately compute the cases µ= 0 and µ=i.
As shown in Refs. [77,78], the vanishing of the determinant of the primary Hessian matrix is not enough
to guarantee the propagation of the right number of degrees of freedom when multiple vector fields are
present. In this case, an additional condition must be satisfied, namely the vanishing of the secondary
Hessian ( ˜
Hαβ )Ltest :
(˜
Hαβ )Ltest ≡∂2Ltest
∂˙
Aα
0∂Aβ
0
−∂2Ltest
∂˙
Aβ
0∂Aα
0
= 0 ,(4)
4
where the indices αand βdenote the different vector fields involved. Nonetheless, keep in mind that, when
going to a curved spacetime, the Hessian conditions are not sufficient to get rid of the ghost and Laplacian
instabilities that might be present in the theory [35,37,47,55,57]. To this purpose, the positiveness of the
kinetic matrix and squared propagation speeds of the perturbation modes must be imposed respectively
(see, for instance, Refs. [79,80]).
C. Constraints among the Test Lagrangians
Find constraints among the test Lagrangians that involve contractions among the Faraday tensor Fµν ,
its Hodge dual ˜
Fµν , and Aµ. To this end, it is handy to use the identity [38,57,81]
Aµα ˜
Bνα +Bµα ˜
Aνα =1
2(Bαβ ˜
Aαβ )δµ
ν,(5)
valid for all antisymmetric tensors Aand B. In Section IV, we will use this identity to find one constraint,
thus eliminating one of the test Lagrangians. In Ref. [57], a non-Abelian version of this identity was used
to eliminate two test Lagrangians.
D. Flat Space-Time Currents in the Lagrangian
Identify the Lagrangians related by total derivatives of the currents. In the case of Lagrangians involving
two vector-field derivatives, it is useful to use the antisymmetric properties of the generalized Kronecker
delta in order to define currents of the form [57]
Jµ
δ≡f(X)δµµ2
ν1ν2Aν1∂µ2Aν2,(6)
where X≡ −A2/2. We can also use the properties of the Levi-Civita tensor and define the following type
of currents [57]:
Jµ
ǫ≡f(X)ǫµνρσ Aν(∂ρAσ).(7)
Finally, we can define currents involving a divergence-free tensor Dµν [57]:
Jµ
D≡f(X)Dµν Aν.(8)
From Eqs. (6) - (8) we can write algebraic expressions among the test Lagrangians and total derivatives
of the 4-current vectors. In a flat spacetime, we would use these relations to eliminate one or several test
Lagrangians in terms of others since they yield the same field equations. However, in general, when the
derivatives of the flat space-time currents are covariantized, what in flat spacetime are total derivatives,
in curved spacetime are not anymore, so the test Lagrangians that yield the same field equations in flat
spacetime do not yield the same field equations in curved spacetime.
Since this part of the procedure constitutes the main difference with respect to the approach followed
in Refs. [26,35,37,57,58], we will devote Section III to explain this issue further.
E. Covariantization
Covariantize the resulting flat space-time theory. To this purpose, we could simply follow the minimal
coupling principle in which we replace all partial derivatives with covariant ones. One must also include
possible direct coupling terms between the vector field and the curvature tensors [37]. This procedure has
been extensively explained in Refs. [35–37,39,80,82,83] where the authors propose contractions on all
indices with divergence-free tensors built from curvature, such as the Einstein Gµν and the double dual
Riemann Lαβγ δ =−1
2ǫαβµν ǫγ δρσ Rµνρσ tensors.
F. Scalar Limit of the Theory
From Horndeski theories we have learned that, when gravity is turned on, it could excite the temporal
polarization of the vector field, introducing new propagating degrees of freedom [14,35–39,55,57,80,83].
5
This is the reason why, as a final step, we must verify that the field equations for all physical degrees of
freedom, i.e. scalar and vector modes, are at most second order. To this end, we split Aµinto the pure
scalar and vector modes
Aµ=∇µφ+ˆ
Aµ,(9)
where φis the StÃijckelberg field and ˆ
Aµis the divergence-free contribution (∇µˆ
Aµ= 0).
For a theory built out of first-order derivatives in the vector field, the pure vector sector of Aµcannot
lead to any derivative of order higher than two in the field equations. As for the scalar part, derivatives
of order three or more could appear when covariantazing, which can be expressed in terms of derivatives
of some curvature terms and be eliminated, in turn, by adding the appropriate counterterms (arriving
then to the Horndeski or beyond Horndeski theories in curved spacetime); such counterterms can easily
be generalized to the Proca field by employing the StÃijckelberg trick. Care must be taken also with the
mixed pure scalar-pure vector sector, following an identical procedure as the one described lines before4.
It is worth mentioning that some of the built Lagrangians vanish in the scalar limit, indicating that
these interaction terms correspond to purely intrinsic vector modes [37].
III. COVARIANTIZATION OF FLAT SPACE-TIME CURRENTS
As we explained above, from Eqs. (6) - (8), it is possible to write algebraic expression among the test
Lagrangians and the total derivatives of the 4-current vectors. These relations would then be used to
eliminate one or several test Lagrangians in terms of the others since they yield the same field equations
in flat spacetime. However, when the gravity is turned on, the flat space-time current derivatives now
involve curved space-time current derivatives and other curvature terms that arise because second-order
derivatives, being promoted now to space-time covariant derivatives, do not commute anymore. Thus, the
test Lagrangians in the relation do not yield the same field equations. For instance, as we will see more
clearly in the implementation, if we have an expression of the form
∂µJµ=Li+Lj,(10)
which allows us to remove Ljin favour of Lior viceversa, a similar relation holds when promoting this
expression to curved spacetime:
∇µJµ=Li+Lj+F(Aµ,∇µAν),(11)
where Fis a function of the field and the space-time covariant field derivatives. Nonetheless, we can see
from this expression that, in curved spacetime, the field equations for Liand Ljwill no longer be the
same due to the presence of the function F. Anyway, it could also be the case that Fvanishes identically,
or that it is a total derivative, such that we are allowed to replace one of the Lagrangians in terms of the
other since their field equations are the same.
IV. IMPLEMENTATION OF THE PROCEDURE
In this section, we will implement the procedure described in Section II for the case of the LP
4Proca
Lagrangian. Paying attention to the covariantization of the flat space-time currents, discussed in Section
III, we will arrive to the BGP theory whose main characteristic is its equivalence to the beyond Horndeski
theory in the scalar limit.
4This, indeed, is the origin of the counterterm in the LP
6piece of the generalized Proca action.
6
A. Test Lagrangians
We start by writing all possible test Lagrangians for LP
4. According to Table II, in the case of two vector
field derivatives only, there exist four terms which turn out to be independent:
L1= (∂·A)2,
L2= (∂µAν)(∂µAν),
L3= (∂µAν)(∂νAµ),
L4=ǫµνρσ (∂µAν)(∂ρAσ).
(12)
In contrast, for two vector fields and two vector field derivatives there exist ten terms, six of them being
independent:
L5= (∂·A)(∂ρAσ)AρAσ,
L6= (∂µAν)(∂µAσ)AνAσ,
L7= (∂µAν)(∂ρAµ)AνAρ,
L8= (∂µAν)(∂ρAν)AµAρ,
L9=ǫµρσβ Aβ(∂νAµ)(∂ρAσ)Aν,
L10 =ǫµρσβ Aβ(∂µAν)(∂ρAσ)Aν,
(13)
and the other four just being the same terms of Eq. (12) multiplied by A2. Regarding the test Lagrangians
formed with four vector fields and two vector field derivatives, there exist eleven terms, four of them being
the same terms of Eq. (12) multiplied by A4, other six being the same terms of Eq. (13) multiplied by A2
and the other one being
L11 = (Aµ(∂µAν)Aν)2.(14)
We can continue looking for test Lagrangians that contract two vector field derivatives with an even number
of vector fields higher than four. However, since the number of space-time indices corresponding to the
two vector field derivatives is already saturated when considering the contractions with four vector fields,
all the possible test Lagrangians that involve more than four vector fields will be exactly the same as the
ones in Eqs. (12) - (14) multiplied by some power of A2. This leads us to conclude that all the possible test
Lagrangians that involve two vector field derivatives are expressed as the ones in Eqs. (12)-(14) multiplied
each one of them by an arbitrary function of A2.
B. Hessian Conditions
Continuing with the procedure, we now write down the linear combination of the terms in Eqs. (12)-(14),
each one of them multiplied by an arbitrary function of A2, to form the test Lagrangian
Ltest =
11
X
i=1
fi(X)Li,(15)
where the fi(X) are the mentioned arbitrary functions (the constants xihave been absorbed into the
fi). It is convenient to calculate first the primary Hessians in Eq. (2) associated with the various test
7
Lagrangians5
Hµν
L1= 2g0µg0ν,
Hµν
L2=−2gµν ,
Hµν
L3= 2g0µg0ν,
Hµν
L4= 0 ,
Hµν
L5=A0Aµg0ν+A0Aνg0µ,
Hµν
L6=−2AµAν,(16)
Hµν
L7=A0Aµg0ν+A0Aνg0µ,
Hµν
L8= 2(A0)2gµν ,
Hµν
L9= 0 ,
Hµν
L10 = 0 ,
Hµν
L11 = 2(A0)2AµAν.
Then imposing the primary Hessian condition in Eq. (3) and considering the cases µ= 0 and µ=i
separately, we obtain
H00 = 2 (f1+f2+f3)−2 (f5+f6+f7+f8) (A0)2+ 2f11(A0)4= 0 ,(17)
H0i=−(f5+ 2f6+f7)A0Ai+ 2f11(A0)3Ai= 0 ,(18)
leading to four independent algebraic equations which we solve for
f1=−f2−f3, f5=−2f6−f7, f6=f8,and f11 = 0 .(19)
Thus, our test Lagrangian in Eq. (15) becomes
Ltest =f2(X)(L2− L1) + f3(X)(L3− L1) + f4(X)L4
+f6(X)(L6−2L5+L8) + f7(X)(L7− L5)
+f9(X)L9+f10(X)L10 .(20)
It is worth emphasizing that the secondary Hessian constraint of Eq. (4) is trivially satisfied in this case
since just one vector field is being considered.
C. Constraints among the Test Lagrangians
In this section, we make use of the identity in Eq. (5) [38,57,81] in order to simplify the test Lagrangians.
Let us consider Aµν =Fµν and Bµν =˜
Fµν . For these tensors, we can write down the relation
Fµα ˜
Fνα AµAν=1
4(A·A)Fαβ ˜
Fαβ .(21)
Now, expanding this expression in terms of the Proca field Aµand its first-order derivatives, we obtain
the following identity relating the Lagrangians in Eq. (13):
L9− L10 =1
2L4(A·A),(22)
which is also valid in curved spacetime. Using this relation we obtain
f9(X)L9=f9(X)L10 − L4Xf9(X).(23)
5The arbitrary functions fi(X) act, for this purpose, as constants since the primary Hessian calculation involves only
derivatives of the test Lagrangians with respect to first-order field derivatives.
8
Therefore, recognizing that
L4=1
2Fαβ ˜
Fαβ ,(24)
which means that it actually belongs to LP
2=LP
2(Aµ, Fµν ,˜
Fµν ) [26,35–39,57,58], we can now write
f9(X)L9in terms of f9(X)L10 and a Lagrangian belonging to LP
2, thus allowing us to remove f9(X)L9
and [f4(X)−Xf9(X)] L4from LP
4.
Another constraint can be found by noticing that
(L2− L1)−(L3− L1) = 1
2Fµν Fµν ,(25)
so that f3(X)(L3− L1) can be removed in favour of f3(X)(L2− L1) and a Lagrangian belonging to LP
2
(which can also be removed).
Thus, our test Lagrangian of Eq. (20) becomes
Ltest = [f2(X) + f3(X)](L2− L1)
+f6(X)(L6−2L5+L8) + f7(X)(L7− L5)
+[f9(X) + f10(X)]L10 .(26)
D. Flat Space-Time Currents in the Lagrangian
This part of the implementation is crucial since, from the flat space-time currents, we can obtain in-
teraction Lagrangians which, before being promoted to curved spacetime, would be discarded in other
methods.
Let us consider the current defined in Eq. (6):
Jµ
δ≡f(X)δµµ2
ν1ν2Aν1∂µ2Aν2
=f(X) [Aµ(∂·A)−Aν∂νAµ],(27)
whose total derivative results in
∂µJµ
δ=f(X)(∂·A)2+Aµ∂µ(∂·A)−∂µAν∂νAµ−Aν∂µ∂νAµ
−fX(X)Aν∂µAν[Aµ(∂·A)−Aρ∂ρAµ]
=−f(X)(L3− L1) + fX(X)(L7− L5),(28)
where fX(X)≡∂f (X)/∂X , and we have used
Aµ[∂µ, ∂ν]Aν≡Aµ∂µ∂νAν−Aµ∂ν∂µAν= 0 ,(29)
since, in flat spacetime, the partial derivatives of the Proca field commute. As we said before, this part of
the calculation is crucial since, in a curved spacetime, the covariant derivatives of the Proca field do not
commute. We see from Eq. (28) that a term of the form f7(X)(L7− L5) can be fully removed from LP
4,
only in flat spacetime, since it gives the same field equations as a term of the form (Rf7(X)dX)(L3− L1):
f7(X)(L7− L5) = Zf7(X)dX(L3− L1) + ∂µJµ
δ.(30)
The actual expression, without removing the commutator of the partial derivatives is:
f7(X)(L7− L5) = Zf7(X)dX(L3− L1−Aµ[∂µ, ∂ν]Aν) + ∂µJµ
δ,
=Zf7(X)dX(L2− L1−Aµ[∂µ, ∂ν]Aν) + ∂µJµ
δ,(31)
where in the last line we have replaced L3− L1in terms of L2− L1plus a term belonging to LP
2, using
Eq. (25). The latter term has been removed.
9
A similar procedure follows when considering the current defined in Eq. (7):
∂µJµ
ǫ=f(X) [ǫµνρσ (∂µAν)(∂ρAσ) + ǫµνρσ Aν(∂µ∂ρAσ)]
−fX(X)Aα∂µAα[ǫµνρσ Aν(∂ρAσ)]
=f(X)L4−fX(X)L10 ,(32)
showing that [f9(X) + f10(X)] L10 can be removed from LP
4since it gives the same field equations as a
term belonging to LP
2:
[f9(X) + f10(X)] L10 =Z[f9(X) + f10(X)] dXL4−∂µJµ
ǫ.(33)
This last formula is valid even in curved spacetime because the commutation of partial second-order
derivatives has not been invoked.
Finally, using Eq. (8), and by virtue of the divergence-free properties of ˜
Fµν , we introduce the current
Jµ
F≡f(X)˜
Fµν Aν,(34)
but this is nothing else than Jµ
ǫ, which leads us to the same results of Eq. (33).
The test Lagrangian of Eq. (26), after removing the redundant pieces, looks like
Ltest =f2(X) + f3(X) + Zf7(X)dX(L2− L1)
+f6(X)(L6−2L5+L8)−Zf7(X)dXAµ[∂µ, ∂ν]Aν
=f2(X) + f3(X) + Z[2f6(X) + f7(X)] dX(L2− L1)
+f6(X)(L6−2L7+L8)−Z[2f6(X) + f7(X)] dXAµ[∂µ, ∂ν]Aν,(35)
where we have used Eq. (31) in the last line. We now notice that
(L6−2L7+L8) = AµAνFµ
αFνα ,(36)
so that it can be removed in favour of a Lagrangian belonging to LP
2. Thus, our test Lagrangian becomes
Ltest =f2(X) + f3(X) + Z[2f6(X) + f7(X)] dX(L2− L1)
−Z[2f6(X) + f7(X)] dXAµ[∂µ, ∂ν]Aν.(37)
E. Covariantization
In this section, we will covariantize our theory and show that it contains the BGP theory of Ref. [55] at
the level of the LP
4sector of the Proca theory. We will also show how the LP
4sector of the BGP theory is
induced by promoting the flat space-time currents into currents in curved spacetime.
By promoting all the partial derivatives to covariant ones in the test Lagrangian of Eq. (37), the different
pieces that it is made of can now be written as
f2(X) + f3(X) + Z[2f6(X) + f7(X)] dX(L2− L1) = −F4(X)δµ1µ2
ν1ν2(∇µ1Aν1)(∇ν2Aµ2),
−Z[2f6(X) + f7(X)] dXAµ(∇µ∇ν− ∇ν∇µ)Aν=GN(X)RµνAµAν,(38)
where Rµν is the Ricci tensor and F4(X) and GN(X) are arbitrary functions of X. Therefore, our test
Lagrangian can be written as
Ltest =−F4(X)δµ1µ2
ν1ν2(∇µ1Aν1)(∇ν2Aµ2) + GN(X)RµνAµAν.(39)
10
The existence of the second term in the previous expression had not been recognized before, in Refs. [26,35–
37,39,57,58], because the covariantization was performed over the final flat space-time Lagrangian, i.e.,
the one obtained after removing all the equivalent terms up to four-current divergences. Nobody had paid
attention to the fact that new terms could be generated in curved spacetime, terms that simply vanish in
flat spacetime.
1. Beyond generalized Proca theory
We will now show that the theory composed of the Lagrangians in Eq. (39) is the usual generalized
Proca theory, before adding the required counterterms, plus the new BGP terms. To this end, we write
the Lagrangian for two fields and two field derivatives unveiled in Ref. [55]:
LN
4=fN
4(X)δβ1β2β3γ4
α1α2α3γ4Aα1Aβ1∇α2Aβ2∇α3Aβ3.(40)
Using the properties of the generalized Kronecker delta function, Eq. (40) can be written as
LN
4=fN
4(X)−2Xδµ1µ2
ν1ν2(∇µ1Aν1)(∇µ2Aν2) + (∇µAν)(∇ρAµ)AνAρ−(∇ · A)(∇µAρ)AµAρ
=fN
4(X) [−2X(L1− L3)−(L5− L7)]
=−2XfN
4(X)−ZfN
4(X)dX(L1− L3) + ∇µJµ
δ+ZfN
4(X)dXRµν AµAν
=−2XfN
4(X)−ZfN
4(X)dX1
2Fµν Fµν −(L2− L1)+∇µJµ
δ+ZfN
4(X)dXRµν AµAν,
(41)
where the covariantized versions of Eqs. (25) and (31) have been used. Therefore, after removing the total
derivative and the term belonging to LP
2,LN
4turns out to be
LN
4=−[2XGN,X (X) + GN(X)] δµ1µ2
ν1ν2(∇µ1Aν1)(∇ν2Aµ2) + GN(X)RµνAµAν,(42)
where
GN(X)≡ZfN
4(X)dX . (43)
Thus, comparing Eqs. (39), (40), and (42), we may conclude that our theory is equivalent to the BGP
theory in the case of the LP
4Proca sector:
Ltest =−G4,X (X)δµ1µ2
ν1ν2(∇µ1Aν1)(∇ν2Aµ2) + LN
4,(44)
where
G4,X ≡F4(X)−2XGN,X (X)−GN(X).(45)
F. Scalar Limit of the Theory
We will now verify that the longitudinal mode φof the Proca field yields the correct scalar-tensor theory.
We will show that our theory reduces to the beyond Horndeski theory [41,42] in the scalar limit Aµ→ ∇µφ.
In order to show this, we first write the Horndeski LH
4and beyond Horndeski LBH
4Lagrangians given in
Refs. [41,42]:
LH
4=G4(φ, X)R−G4,X (φ, X )(φ)2−φµνφµν ,(46)
LBH
4=fN
4(φ, X)ǫµνρ
σǫµ′ν′ρ′σφµφµ′φνν′φρρ′
=fN
4(φ, X)hX(φ)2−φµνφµν +2φµφν(φµα φν
α−φ φµν )i,(47)
where X≡ −∇µφ∇µφ/2, φµ≡ ∇µφ,φµν ≡ ∇µ∇νφ,Ris the Ricci scalar, and G4and fN
4are arbitrary
functions of φand X.
11
In the scalar limit Aµ→ ∇µφ, our test Lagrangian in Eq. (44) takes the form
Ltest → −G4,X (X)(φ)2−φµν φµν+fN
4(X)hX(φ)2−φµν φµν+2φµφν(φµα φν
α−φ φµν )i,(48)
such that it reduces to the Horndeski and beyond Horndeski theories in Eqs. (46) and (47) respectively,
except for the term proportional to the Ricci scalar. This means that our final LP
4Lagrangian is our test
Lagrangian in Eq. (44) supplemented with a term G4(X)R:
LP
4=G4(X)R−G4,X (X)δµ1µ2
ν1ν2(∇µ1Aν1)(∇ν2Aµ2) + fN
4(X)δβ1β2β3γ4
α1α2α3γ4Aα1Aβ1∇α2Aβ2∇α3Aβ3.(49)
V. CONCLUSIONS
The generalized Proca theory is the vector field version of the Horndeski theory and, as such, satisfies a
necessary condition required to avoid the Ostrogradsky’s instability. The original way to build it [35,39]
consisted in finding out all the possible contractions of first-order vector field derivatives with a couple of
Levi-Civita tensors, it being an extrapolation of the method employed in the construction of the scalar
Galileon action which, in turn, lies on a formal demonstration given in Ref. [30]. This method is very
appropriate for the vector field case [35,39,58], even for the BGP theory [55], but it is incomplete since it
does not generate parity-violating terms that we know exist in the theory [26,37,38,57]; a very similar and
formally proved methodology, which does generate the parity-violating terms, has been recently presented
in Refs. [77,78].
A more lengthy procedure was followed in Refs. [37,38,57] with the advantage that all the terms,
including those that violate parity, can be produced. This procedure does not rely on unproved hypothesis
and, therefore, becomes a trustworthy way of building the generalized Proca theory.
Despite the methodology employed, however, earlier attempts did not take into account that what are
total derivatives in flat spacetime may no longer be total derivatives in curved spacetime. Thus, a few
terms were ignored that we, in this paper, have unveiled, finding out that they produce the BGP terms.
Before finishing, let us discuss a bit about what the BGP theory is. The BGP theory is a non-degenerate
theory built from first-order space-time derivatives of the vector field and the field itself. As such, its
field equations are second order so that it satisfies the necessary requirement to avoid the Ostrogradsky’s
instability. It satisfies the conditions for the propagation of the right number of degrees of freedom, at
least in flat spacetime, and reduces to the beyond Horndeski theory in the scalar limit. However, although
this scalar limit corresponds to a degenerate theory, the full vector version, as we mentioned above, is not.
Having followed a lengthy but exhaustive procedure to build the generalized Proca theory, there was no
reason at all not to find the BGP theory. This was not the case in earlier attempts but the BGP theory
should be there, hidden in some way. We have discovered in this work that, in fact, the BGP theory at
the LP
4level was hidden in those terms that look as total derivatives in the Lagrangian but that only are
in flat spacetime. We are then in the position to conclude that the BGP theory at the levels of LP
5and
LP
6can be obtained following the systematic procedure described in this paper. The method can also be
applied to extensions of the generalized Proca theory, such as the scalar-vector-tensor theory developed in
Ref. [40] or the generalized SU(2) Proca theory of Refs. [57,58]. Indeed, the construction of the beyond
generalized SU(2) Proca theory will be discussed in a forthcoming paper [84].
Acknowledgments
A.G.C. dedicates this work to his mother María Libia Cadavid Carmona who is fighting cancer. A.G.C.
thanks L. Gabriel Gómez for useful comments on the manuscript. A.G.C. was supported by Programa
de Estancias Postdoctorales VIE - UIS 2019000052 and Beca de Inicio Postdoctoral 2019 UV. This work
was supported by the following grants: Colciencias-DAAD - 110278258747 RC-774-2017, VCTI - UAN -
2017239, DIEF de Ciencias - UIS - 2460, and Centro de Investigaciones - USTA - 1952392. Some calculations
were cross-checked with the Mathematica package xAct (www.xact.es).
[1] Supernova Search Team, A. G. Riess et al., Astron. J. 116, 1009 (1998), arXiv:astro-ph/9805201.
[2] Supernova Cosmology Project, S. Perlmutter et al., Astrophys. J. 517, 565 (1999), arXiv:astro-ph/9812133.
[3] SDSS, M. Ata et al., Mon. Not. Roy. Astron. Soc. 473, 4773 (2018), arXiv:1705.06373.
12
[4] WMAP, C. L. Bennett et al., Astrophys. J. Suppl. 208, 20 (2013), arXiv:1212.5225.
[5] Planck, Y. Akrami et al., (2018), arXiv:1807.06205.
[6] Virgo, LIGO Scientific, B. P. Abbott et al., Phys. Rev. Lett. 116, 061102 (2016), arXiv:1602.03837.
[7] C. Schimd, J.-P. Uzan, and A. Riazuelo, Phys. Rev. D71, 083512 (2005), arXiv:astro-ph/0412120.
[8] B. Jain and J. Khoury, Annals Phys. 325, 1479 (2010), arXiv:1004.3294.
[9] G.-B. Zhao et al., Phys. Rev. D85, 123546 (2012), arXiv:1109.1846.
[10] T. Clifton, P. G. Ferreira, A. Padilla, and C. Skordis, Phys. Rept. 513, 1 (2012), arXiv:1106.2476.
[11] K. Koyama, Rept. Prog. Phys. 79, 046902 (2016), arXiv:1504.04623.
[12] J. M. Ezquiaga and M. Zumalacárregui, Front. Astron. Space Sci. 5, 44 (2018), arXiv:1807.09241.
[13] M. Ishak, Living Rev. Rel. 22, 1 (2019), arXiv:1806.10122.
[14] L. Heisenberg, Phys. Rept. 796, 1 (2019), arXiv:1807.01725.
[15] I. Agullo and A. Ashtekar, Phys. Rev. D91, 124010 (2015), arXiv:1503.03407.
[16] I. Agullo, A. Ashtekar, and W. Nelson, Class. Quant. Grav. 30, 085014 (2013), arXiv:1302.0254.
[17] C. Rovelli, Quantum gravity, Cambridge Monographs on Mathematical Physics (Univ. Pr., Cambridge, UK,
2004).
[18] S. Weinberg, Cosmology (Univ. Pr., Oxford, UK, 2008).
[19] G. Ellis, R. Maartens, and M. MacCallum, Relativistic Cosmology (Univ. Pr., Cambridge, UK, 2012).
[20] P. Peter and J.-P. Uzan, Primordial Cosmology, Oxford Graduate Texts (Univ. Pr., Oxford, UK, 2013).
[21] L. Amendola and S. Tsujikawa, Dark Energy (Univ. Pr., Cambridge, UK, 2015).
[22] R. P. Woodard, Lect. Notes Phys. 720, 403 (2007), arXiv:astro-ph/0601672.
[23] R. P. Woodard, Scholarpedia 10, 32243 (2015), arXiv:1506.02210.
[24] M. Ostrogradsky, Mem. Acad. St. Petersbourg 6, 385 (1850).
[25] H. Motohashi and T. Suyama, Phys. Rev. D91, 085009 (2015), arXiv:1411.3721.
[26] Y. Rodríguez and A. A. Navarro, J. Phys. Conf. Ser. 831, 012004 (2017), arXiv:1703.01884.
[27] G. W. Horndeski, Int. J. Theor. Phys. 10, 363 (1974).
[28] T. Kobayashi, Rept. Prog. Phys. 82, 086901 (2019), arXiv:1901.07183.
[29] C. Deffayet and D. A. Steer, Class. Quant. Grav. 30, 214006 (2013), arXiv:1307.2450.
[30] C. Deffayet, X. Gao, D. A. Steer, and G. Zahariade, Phys. Rev. D84, 064039 (2011), arXiv:1103.3260.
[31] C. Deffayet, G. Esposito-Farese, and A. Vikman, Phys. Rev. D79, 084003 (2009), arXiv:0901.1314.
[32] C. Deffayet, S. Deser, and G. Esposito-Farese, Phys. Rev. D80, 064015 (2009), arXiv:0906.1967.
[33] A. Nicolis, R. Rattazzi, and E. Trincherini, Phys. Rev. D79, 064036 (2009), arXiv:0811.2197.
[34] G. W. Horndeski, J. Math. Phys. 17, 1980 (1976).
[35] L. Heisenberg, JCAP 1405, 015 (2014), arXiv:1402.7026.
[36] G. Tasinato, JHEP 1404, 067 (2014), arXiv:1402.6450.
[37] E. Allys, P. Peter, and Y. Rodríguez, JCAP 1602, 004 (2016), arXiv:1511.03101.
[38] E. Allys, J. P. Beltrán Almeida, P. Peter, and Y. Rodríguez, JCAP 1609, 026 (2016), arXiv:1605.08355.
[39] J. Beltrán Jiménez and L. Heisenberg, Phys. Lett. B757, 405 (2016), arXiv:1602.03410.
[40] L. Heisenberg, JCAP 1810, 054 (2018), arXiv:1801.01523.
[41] J. Gleyzes, D. Langlois, F. Piazza, and F. Vernizzi, Phys. Rev. Lett. 114, 211101 (2015), arXiv:1404.6495.
[42] J. Gleyzes, D. Langlois, F. Piazza, and F. Vernizzi, JCAP 1502, 018 (2015), arXiv:1408.1952.
[43] J. Ben Achour, D. Langlois, and K. Noui, Phys. Rev. D93, 124005 (2016), arXiv:1602.08398.
[44] M. Crisostomi, M. Hull, K. Koyama, and G. Tasinato, JCAP 1603, 038 (2016), arXiv:1601.04658.
[45] M. Crisostomi, R. Klein, and D. Roest, JHEP 1706, 124 (2017), arXiv:1703.01623.
[46] M. Crisostomi, K. Koyama, and G. Tasinato, JCAP 1604, 044 (2016), arXiv:1602.03119.
[47] C. Deffayet, G. Esposito-Farese, and D. A. Steer, Phys. Rev. D92, 084013 (2015), arXiv:1506.01974.
[48] X. Gao, Phys. Rev. D90, 104033 (2014), arXiv:1409.6708.
[49] D. Langlois and K. Noui, JCAP 1602, 034 (2016), arXiv:1510.06930.
[50] D. Langlois and K. Noui, JCAP 1607, 016 (2016), arXiv:1512.06820.
[51] C. Lin, S. Mukohyama, R. Namba, and R. Saitou, JCAP 1410, 071 (2014), arXiv:1408.0670.
[52] M. Zumalacárregui and J. García-Bellido, Phys. Rev. D89, 064046 (2014), arXiv:1308.4685.
[53] H. Motohashi et al., JCAP 1607, 033 (2016), arXiv:1603.09355.
[54] J. Ben Achour et al., JHEP 1612, 100 (2016), arXiv:1608.08135.
[55] L. Heisenberg, R. Kase, and S. Tsujikawa, Phys. Lett. B760, 617 (2016), arXiv:1605.05565.
[56] R. Kimura, A. Naruko, and D. Yoshida, JCAP 1701, 002 (2017), arXiv:1608.07066.
[57] E. Allys, P. Peter, and Y. Rodríguez, Phys. Rev. D94, 084041 (2016), arXiv:1609.05870.
[58] J. Beltrán Jiménez and L. Heisenberg, Phys. Lett. B770, 16 (2017), arXiv:1610.08960.
[59] R. Emami, S. Mukohyama, R. Namba, and Y.-l. Zhang, JCAP 1703, 058 (2017), arXiv:1612.09581.
[60] A. Maleknejad and M. M. Sheikh-Jabbari, Phys. Lett. B723, 224 (2013), arXiv:1102.1513.
[61] A. Maleknejad and M. M. Sheikh-Jabbari, Phys. Rev. D84, 043515 (2011), arXiv:1102.1932.
[62] C. M. Nieto and Y. Rodríguez, Mod. Phys. Lett. A31, 1640005 (2016), arXiv:1602.07197.
[63] A. Oliveros and M. A. Jaraba, Int. J. Mod. Phys. D28, 1950064 (2019), arXiv:1903.06005.
[64] Y. Rodríguez and A. A. Navarro, Phys. Dark Univ. 19, 129 (2018), arXiv:1711.01935.
[65] R. Kase and S. Tsujikawa, JCAP 1811, 024 (2018), arXiv:1805.11919.
[66] L. Heisenberg, R. Kase, and S. Tsujikawa, Phys. Rev. D98, 024038 (2018), arXiv:1805.01066.
[67] R. Kase and S. Tsujikawa, Phys. Rev. D97, 103501 (2018), arXiv:1802.02728.
13
[68] A. De Felice et al., JCAP 1606, 048 (2016), arXiv:1603.05806.
[69] A. De Felice et al., Phys. Rev. D94, 044024 (2016), arXiv:1605.05066.
[70] S. Nakamura, R. Kase, and S. Tsujikawa, Phys. Rev. D95, 104001 (2017), arXiv:1702.08610.
[71] M. Rahman and A. A. Sen, Phys. Rev. D99, 024052 (2019), arXiv:1810.09200.
[72] L. Heisenberg, R. Kase, M. Minamitsuji, and S. Tsujikawa, JCAP 1708, 024 (2017), arXiv:1706.05115.
[73] E. Babichev, C. Charmousis, and A. Lehébel, Class. Quant. Grav. 33, 154002 (2016), arXiv:1604.06402.
[74] J. Chagoya, G. Niz, and G. Tasinato, Class. Quant. Grav. 34, 165002 (2017), arXiv:1703.09555.
[75] R. Kase, M. Minamitsuji, and S. Tsujikawa, Phys. Lett. B782, 541 (2018), arXiv:1803.06335.
[76] R. Kase, M. Minamitsuji, and S. Tsujikawa, Phys. Rev. D97, 084009 (2018), arXiv:1711.08713.
[77] V. Errasti Díez, B. Gording, J. A. Méndez-Zavaleta, and A. Schmidt-May, (2019), arXiv:1905.06968.
[78] V. Errasti Díez, B. Gording, J. A. Méndez-Zavaleta, and A. Schmidt-May, (2019), arXiv:1905.06967.
[79] L. G. Gómez and Y. Rodríguez, (2019), arXiv:1907.07961.
[80] J. Beltrán Jiménez, R. Durrer, L. Heisenberg, and M. Thorsrud, JCAP 1310, 064 (2013), arXiv:1308.1867.
[81] P. Fleury, J. P. Beltrán Almeida, C. Pitrou, and J.-P. Uzan, JCAP 1411, 043 (2014), arXiv:1406.6254.
[82] M. Hull, K. Koyama, and G. Tasinato, Phys. Rev. D93, 064012 (2016), arXiv:1510.07029.
[83] C. de Rham and L. Heisenberg, Phys. Rev. D84, 043503 (2011), arXiv:1106.3312.
[84] A. Gallego Cadavid and Y. Rodríguez, work in progress (2019).
