A systematic procedure to build the beyond generalized Proca field theory

Physics Letters B 798 (2019) 134958

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Physics Letters B www.elsevier.com/locate/physletb

A systematic procedure to build the beyond generalized Proca field theory Alexander Gallego Cadavid a,b,∗ , Yeinzon Rodríguez a,c,d a

Escuela de Física, Universidad Industrial de Santander, Ciudad Universitaria, Bucaramanga 680002, Colombia Instituto de Física y Astronomía, Universidad de Valparaíso, Avenida Gran Bretaña 1111, Valparaíso 2360102, Chile Centro de Investigaciones en Ciencias Básicas y Aplicadas, Universidad Antonio Nariño, Cra 3 Este # 47A-15, Bogotá D.C. 110231, Colombia d Simons Associate at The Abdus Salam International Centre for Theoretical Physics, Strada Costiera 11, I-34151, Trieste, Italy b c

a r t i c l e

i n f o

Article history: Received 26 May 2019 Received in revised form 18 September 2019 Accepted 19 September 2019 Available online 23 September 2019 Editor: J. Hisano

a b s t r a c t 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 firstorder 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 In the curved space-time theory, some of the retained terms are no longer current divergences so that they induce the new terms that identify the beyond generalized Proca field theory.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. © 2019 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3 .

1. 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].

*

Corresponding author. E-mail addresses: [email protected] (A. Gallego Cadavid), [email protected] (Y. Rodríguez).

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 CDM), 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

https://doi.org/10.1016/j.physletb.2019.134958 0370-2693/© 2019 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3 .

2

A. Gallego Cadavid, Y. Rodríguez / Physics Letters B 798 (2019) 134958

theorem states that, for a non-degenerate theory,1 field 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-vectortensor theories, or simply Horndeski, generalized Proca, and scalarvector-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-tensor,2 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 A aμ , 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 different 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 A μ have 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 LnP+2 in four di-

LnP+2

denotes the Lagrangians containing n ≥ 1 mensions, where first-order derivatives of A μ [26,35–39]. As an exception to the rule, LP2 is defined as the Lagrangian consisting of an arbitrary function of the Faraday tensor F μν ≡ ∂μ A ν − ∂ν A μ , its Hodge dual F˜ μν ≡ μνρσ F ρσ /2, where μνρσ is the Levi-Civita tensor, and A μ only. 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 spacetime theory, hence leading to different field equations for the Lagrangians involved. The layout of the paper is the following. In Section 2, we describe the general procedure to construct the most general Proca

Table 1 Number of Lorentz scalars that can be constructed with multiple copies of A μ . Number of vector fields A μ

1

2

3

4

5

6

Number of Lorentz scalars

0

1

0

4

0

25

Table 2 Number of Lorentz scalars that can be built for a given product of vector fields and vector field derivatives. Number of A ρ A σ Number of ∂ μ A ν

0

1

2

1 2 3

1 4 7

2 10 30

2 11

theory. In Section 3, 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 4, we implement the procedure to obtain the LP4 terms; there we show how to obtain systematically the LP4 terms in the BGP. The conclusions are presented in Section 5. Throughout the paper we use the signature ημν = diag (−, +, +, +) and set A · A ≡ A μ A μ and ∂ · A ≡ ∂μ A μ . We also define the generalized μ1 ...μn− p

[μ

μn− p ]

μ

μn− p

Kronecker delta as δν1 ...νn− p ≡ δν1 1 . . . δνn− p = δ[ν11 . . . δνn− p ] where the brackets mean unnormalized antisymmetrization. 2. 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. 2.1. 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 g μν and the Levi-Civita tensor μνρσ . In Table 1, we show the number of Lorentz scalars that can be constructed with multiple copies of A μ [57], whereas, in Table 2, 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. 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. 2.2. 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 

xi L i ,

(1)

i =1 1

A non-degenerate theory at nth-order is one in which its Lagrangian fulfils the



(n)

(n)

condition det ∂ 2 L/∂ qi ∂ q j



= 0, where q(i n) is the n-th derivative of the gener-

alized coordinate qi of the system. 2 Also called degenerate higher-order scalar-tensor theories (DHOST). 3 Which, by the way, could be called degenerate higher-order vector-tensor theories (DHOVT).

where n is the number of test Lagrangians and xi are constant parameters of the theory. We then calculate the primary Hessian of the test Lagrangian μν

HLtest ≡

∂ 2 Ltest , ∂ A˙ μ ∂ A˙ ν

(2)

A. Gallego Cadavid, Y. Rodríguez / Physics Letters B 798 (2019) 134958

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., μ0

HLtest =

n  μ0 μ0 μ0 μ0 HLi = x1 HL1 + x2 HL2 + · · · + xn HLn = 0 .

(3)

i =1

Eq. (3) gives a system of algebraic equations for the xi whose roots impose conditions on the test Lagrangian. For some test Lagrangians their corresponding xi parameters 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 (H˜ α β )Ltest :

(H˜ α β )Ltest ≡

∂ 2 Ltest ∂ 2 Ltest − = 0, β β ∂ A˙ α0 ∂ A 0 ∂ A˙ 0 ∂ A α0

(4)

where the indices α and β denote 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]). 2.3. Constraints among the test Lagrangians

1 2

( B α β A˜ α β )δν , μ

(5)

valid for all antisymmetric tensors A and B. In Section 4, 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. 2.4. 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] μ

μμ

J δ ≡ f ( X )δν1 ν22 A ν1 ∂μ2 A ν2 ,

(6)

where X ≡ − A 2 /2. We can also use the properties of the LeviCivita tensor and define the following type of currents [57]: μ

J  ≡ f ( X ) μνρσ A ν (∂ρ A σ ) .

(7)

Finally, we can define currents involving a divergence-free tensor D μν [57]:

(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 3 to explain this issue further. 2.5. 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 G μν and the double dual Riemann L α β γ δ = − 12  α β μν  γ δ ρσ R μνρσ tensors. 2.6. 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]. 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 A μ into the pure scalar and vector modes

ˆμ, Aμ = ∇ μφ + A

Find constraints among the test Lagrangians that involve contractions among the Faraday tensor F μν , its Hodge dual F˜ μν , and A μ . To this end, it is handy to use the identity [38,57,81]

˜ να = A μα B˜ να + B μα A

μ

J D ≡ f ( X ) D μν A ν .

3

(9)

ˆ μ is the divergence-free conwhere φ is the Stückelberg field and A μ ˆ tribution (∇μ A = 0). For a theory built out of first-order derivatives in the vector field, the pure vector sector of A μ cannot 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 Stückelberg trick. Care must be taken also with the mixed pure scalar-pure vector sector, following an identical procedure as the one described lines before.4 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]. 3. 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 4 This, indeed, is the origin of the counterterm in the LP6 piece of the generalized Proca action.

4

A. Gallego Cadavid, Y. Rodríguez / Physics Letters B 798 (2019) 134958

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

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 A 2 . 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 A 2 .

∂μ J μ = L i + L j ,

4.2. Hessian conditions

(10)

which allows us to remove L j in favour of Li or viceversa, a similar relation holds when promoting this expression to curved spacetime:

∇μ J μ = Li + L j + F ( A μ , ∇ μ A ν ) ,

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 A 2 , to form the test Lagrangian

(11)

where F is 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 Li and L j will no longer be the same due to the presence of the function F . Anyway, it could also be the case that F vanishes 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.

Ltest =

11 

f i ( X )Li ,

(15)

i =1

where the f i ( X ) are the mentioned arbitrary functions (the constants xi have been absorbed into the f i ). It is convenient to calculate first the primary Hessians in Eq. (2) associated with the various test Lagrangians5 μν

HL1 = 2g 0μ g 0ν , μν

HL2 = −2g μν ,

4. Implementation of the procedure

μν

In this section, we will implement the procedure described in Section 2 for the case of the LP4 Proca Lagrangian. Paying attention to the covariantization of the flat space-time currents, discussed in Section 3, we will arrive to the BGP theory whose main characteristic is its equivalence to the beyond Horndeski theory in the scalar limit.

We start by writing all possible test Lagrangians for According to Table 2, in the case of two vector field derivatives only, there exist four terms which turn out to be independent:

(12)

In contrast, for two vector fields and two vector field derivatives there exist ten terms, six of them being independent:

μν

HL6 = −2 A μ A ν , μν

(16)

HL 7 = A A μ g 0 ν + A 0 A ν g 0 μ , μν H = 2( A 0 )2 g μν , 0

HL 9 = 0 , μν

HL10 = 0 , μν

HL11 = 2( A 0 )2 A μ A ν . Then imposing the primary Hessian condition in Eq. (3) and considering the cases μ = 0 and μ = i separately, we obtain

H00 = 2 ( f 1 + f 2 + f 3 ) − 2 ( f 5 + f 6 + f 7 + f 8 ) ( A 0 )2

+ 2 f 11 ( A 0 )4 = 0 , 0i



0

H = − ( f 5 + 2 f 6 + f 7) A A

i



(17) 0 3

i

+ 2 f 11 ( A ) A = 0 ,

(18)

leading to four independent algebraic equations which we solve for

(13)

f1 = − f2 − f3 ,

f 5 = −2 f 6 − f 7 ,

f 6 = f 8 , and

f 11 = 0 . (19)

Thus, our test Lagrangian in Eq. (15) becomes

and the other four just being the same terms of Eq. (12) multiplied by A 2 . 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 A 4 , other six being the same terms of Eq. (13) multiplied by A 2 and the other one being

L11 = ( A μ (∂μ A ν ) A ν )2 .

μν

HL 5 = A 0 A μ g 0 ν + A 0 A ν g 0 μ ,

μν

LP4 .

L5 = (∂ · A )(∂ρ A σ ) A ρ A σ , L6 = (∂μ A ν )(∂ μ A σ ) A ν A σ , L7 = (∂μ A ν )(∂ρ A μ ) A ν A ρ , L8 = (∂μ A ν )(∂ρ A ν ) A μ A ρ , L9 =  μρσ β A β (∂ν A μ )(∂ρ A σ ) A ν , L10 =  μρσ β A β (∂μ A ν )(∂ρ A σ ) A ν ,

μν

HL 4 = 0 ,

L8

4.1. Test Lagrangians

L1 = (∂ · A )2 , L2 = (∂μ A ν )(∂ μ A ν ) , L3 = (∂μ A ν )(∂ ν A μ ) , L4 =  μνρσ (∂μ A ν )(∂ρ A σ ) .

HL3 = 2g 0μ g 0ν ,

(14)

Ltest = f 2 ( X )(L2 − L1 ) + f 3 ( X )(L3 − L1 ) + f 4 ( X )L4

+ f 6 ( X )(L6 − 2L5 + L8 ) + f 7 ( X )(L7 − L5 ) + f 9 ( X )L9 + f 10 ( X )L10 .

(20)

5 The arbitrary functions f i ( 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.

A. Gallego Cadavid, Y. Rodríguez / Physics Letters B 798 (2019) 134958

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. 4.3. 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 A μν = F μν and B μν = F˜ μν . For these tensors, we can write down the relation

F μα F˜ να A μ A ν =

1 4

( A · A ) F α β F˜ α β .

(21)

Now, expanding this expression in terms of the Proca field A μ and its first-order derivatives, we obtain the following identity relating the Lagrangians in Eq. (13):

1

A μ [∂μ , ∂ν ] A ν ≡ A μ ∂μ ∂ν A ν − A μ ∂ν ∂μ A ν = 0 ,

f 7 ( X )(L7 − L5 )



=

f 9 ( X )L9 = f 9 ( X )L10 − L4 X f 9 ( X ) .

(23)

Therefore, recognizing that

L4 =

1 αβ F F˜ α β , 2

(24)

which means that it actually belongs to LP2 = LP2 ( A μ , F μν , F˜ μν ) [26,35–39,57,58], we can now write f 9 ( X )L9 in terms of f 9 ( X )L10 and a Lagrangian belonging to LP2 , thus allowing us to remove f 9 ( X )L9 and [ f 4 ( X ) − X f 9 ( X )] L4 from LP4 . Another constraint can be found by noticing that

(L2 − L1 ) − (L3 − L1 ) =

1 2

F μν F μν ,

(25)

so that f 3 ( X )(L3 − L1 ) can be removed in favour of f 3 ( X )(L2 − L1 ) and a Lagrangian belonging to LP2 (which can also be removed). Thus, our test Lagrangian of Eq. (20) becomes

Ltest = [ f 2 ( X ) + f 3 ( X )](L2 − L1 )

+ f 6 ( X )(L6 − 2L5 + L8 ) + f 7 ( X )(L7 − L5 ) +[ f 9 ( X ) + f 10 ( X )]L10 .

(26)

This part of the implementation is crucial since, from the flat space-time currents, we can obtain interaction Lagrangians which, before being promoted to curved spacetime, would be discarded in other methods. Let us consider the current defined in Eq. (6): μ

(27)

whose total derivative results in

  μ ∂μ J δ = f ( X ) (∂ · A )2 + A μ ∂μ (∂ · A ) − ∂μ A ν ∂ν A μ − A ν ∂μ ∂ν A μ   − f X ( X ) A ν ∂μ A ν A μ (∂ · A ) − A ρ ∂ρ A μ = − f ( X )(L3 − L1 ) + f X ( X )(L7 − L5 ) , where f X ( X ) ≡ ∂ f ( X )/∂ X , and we have used

(30)





μ f 7 ( X ) d X (L3 − L1 − A μ [∂μ , ∂ν ] A ν ) + ∂μ J δ ,



μ f 7 ( X ) d X (L2 − L1 − A μ [∂μ , ∂ν ] A ν ) + ∂μ J δ ,

(31)

where in the last line we have replaced L3 − L1 in terms of L2 − L1 plus a term belonging to LP2 , using Eq. (25). The latter term has been removed. A similar procedure follows when considering the current defined in Eq. (7):

  μ ∂μ J  = f ( X )  μνρσ (∂μ A ν )(∂ρ A σ ) +  μνρσ A ν (∂μ ∂ρ A σ )   − f X ( X ) A α ∂μ A α  μνρσ A ν (∂ρ A σ ) = f ( X )L4 − f X ( X )L10 ,

(32)

showing that [ f 9 ( X ) + f 10 ( X )] L10 can be removed from LP4 it gives the same field equations as a term belonging to LP2 :





[ f 9 ( X ) + f 10 ( X )] L10 =

since

μ

[ f 9 ( X ) + f 10 ( X )] d X L4 − ∂μ J  . (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 F˜ μν , we introduce the current

J F ≡ f ( X ) F˜ μν A ν , μ

(34)

μ but this is nothing else than J  , which leads us to the same results

4.4. Flat space-time currents in the Lagrangian

μμ J δ ≡ f ( X )δν1 ν22 A ν1 ∂μ2 A ν2   = f ( X ) A μ (∂ · A ) − A ν ∂ν A μ ,

μ

f 7 ( X ) d X (L3 − L1 ) + ∂μ J δ .

The actual expression, without removing the commutator of the partial derivatives is:

which is also valid in curved spacetime. Using this relation we obtain

2





f 7 ( X )(L7 − L5 ) =

=

L4 ( A · A ) ,

(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 f 7 ( X )(L7 − L5 ) can be fully removed from LP4 , only in flat spacetime, since it gives the same field equations as a term of the form ( f 7 ( X ) d X )(L3 − L1 ):

(22)

L9 − L10 =

5

(28)

of Eq. (33). The test Lagrangian of Eq. (26), after removing the redundant pieces, looks like

Ltest =





f 2( X ) + f 3( X ) +

f 7 ( X ) d X (L2 − L1 )



 f 7 ( X ) d X A μ [∂μ , ∂ν ] A ν + f 6 ( X )(L6 − 2L5 + L8 ) − 

 = f 2 ( X ) + f 3 ( X ) + [2 f 6 ( X ) + f 7 ( X )] d X (L2 − L1 ) + f 6 ( X )(L6 − 2L7 + L8 ) 

 − [2 f 6 ( X ) + f 7 ( X )] d X A μ [∂μ , ∂ν ] A ν ,

(35)

where we have used Eq. (31) in the last line. We now notice that μ

(L6 − 2L7 + L8 ) = A μ A ν F α F να ,

(36)

6

A. Gallego Cadavid, Y. Rodríguez / Physics Letters B 798 (2019) 134958

μ

so that it can be removed in favour of a Lagrangian belonging to LP2 . Thus, our test Lagrangian becomes

Ltest =





f 2( X ) + f 3( X ) +

[2 f 6 ( X ) + f 7 ( X )] d X (L2 − L1 )

 −

 [2 f 6 ( X ) + f 7 ( X )] d X

A μ [∂μ , ∂ν ] A ν .

(37)

 GN(X) ≡

μ μ

Thus, comparing Eqs. (39), (40), and (42), we may conclude that our theory is equivalent to the BGP theory in the case of the LP4 Proca sector:

G 4, X ≡ F 4 ( X ) − 2 X G N , X ( X ) − G N ( X ) . (38)

where R μν is the Ricci tensor and F 4 ( X ) and G N ( X ) are arbitrary functions of X . Therefore, our test Lagrangian can be written as μ μ

Ltest = − F 4 ( X )δν11ν2 2 (∇μ1 A ν1 )(∇ ν2 A μ2 ) + G N ( X ) R μν A μ A ν . (39) 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. 4.5.1. 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]: β β β γ

LN4 = f 4N ( X )δα11 α22 α33 γ44 A α1 A β1 ∇ α2 A β2 ∇ α3 A β3 .

(40)

Using the properties of the generalized Kronecker delta function, Eq. (40) can be written as

+(∇ μ A ν )(∇ρ A μ ) A ν A ρ − (∇ · A )(∇μ A ρ ) A μ A ρ



= f 4N ( X ) [−2 X (L1 − L3 ) − (L5 − L7 )]    μ = −2 X f 4N ( X ) − f 4N ( X ) d X (L1 − L3 ) + ∇μ J δ

 + f 4N ( X ) d X R μν A μ A ν     1 μν N N = −2 X f 4 ( X ) − f4 (X) d X F F μν − (L2 − L1 ) 2

(43)

(44)

where

Aμ Aν ,

 μ μ LN4 = f 4N ( X ) −2 X δν11ν2 2 (∇μ1 A ν1 )(∇μ2 A ν2 )

f 4N ( X ) d X .

μ μ

= − F 4 ( X )δν11ν2 2 (∇μ1 A ν1 )(∇ ν2 A μ2 ) , 

 − [2 f 6 ( X ) + f 7 ( X )] d X A μ (∇μ ∇ν − ∇ν ∇μ ) A ν

(42)

where

Ltest = −G 4, X ( X )δν11ν2 2 (∇μ1 A ν1 )(∇ ν2 A μ2 ) + LN4 ,

[2 f 6 ( X ) + f 7 ( X )] d X (L2 − L1 )

(41)

  μμ LN4 = − 2 X G N , X ( X ) + G N ( X ) δν11ν2 2 (∇μ1 A ν1 )(∇ ν2 A μ2 )





= G N ( X ) R μν

R μν A μ A ν ,

+ G N ( X ) R μν A μ A ν ,

In this section, we will covariantize our theory and show that it contains the BGP theory of Ref. [55] at the level of the LP4 sector of the Proca theory. We will also show how the LP4 sector 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

f 2( X ) + f 3( X ) +

f 4N ( X ) d X

where the covariantized versions of Eqs. (25) and (31) have been used. Therefore, after removing the total derivative and the term belonging to LP2 , LN 4 turns out to be

4.5. Covariantization





+ ∇μ J δ +

(45)

4.6. Scalar limit of the theory We will now verify that the longitudinal mode φ of 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 A μ → ∇μ φ . In order to show this, we first write the BH Horndeski LH Lagrangians given in 4 and beyond Horndeski L4 Refs. [41,42]:

  L4H = G 4 (φ, X ) R − G 4, X (φ, X ) (2φ)2 − φμν φ μν ,

(46)

  

L4B H = f 4N (φ, X ) μνρ σ  μ ν ρ σ φμ φμ φνν  φρρ     = f 4N (φ, X ) X (2φ)2 − φμν φ μν   + 2φμ φν φ μα φαν − 2φ φ μν ,

(47)

where X ≡ −∇μ φ ∇ μ φ/2, φμ ≡ ∇μ φ , φμν ≡ ∇μ ∇ν φ , R is the Ricci scalar, and G 4 and f 4N are arbitrary functions of φ and X . In the scalar limit A μ → ∇μ φ , our test Lagrangian in Eq. (44) takes the form

  Ltest → −G 4, X ( X ) (2φ)2 − φμν φ μν    + f 4N ( X ) X (2φ)2 − φμν φ μν   + 2φμ φν φ μα φαν − 2φ φ μν ,

(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 LP4 Lagrangian is our test Lagrangian in Eq. (44) supplemented with a term G 4 ( X ) R: μ μ

LP4 = G 4 ( X ) R − G 4, X ( X )δν11ν2 2 (∇μ1 A ν1 )(∇ ν2 A μ2 ) β β β γ

+ f 4N ( X )δα11 α22 α33 γ44 A α1 A β1 ∇ α2 A β2 ∇ α3 A β3 .

(49)

A. Gallego Cadavid, Y. Rodríguez / Physics Letters B 798 (2019) 134958

5. 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 LeviCivita 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 LP4 level 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 LP5 and LP6 can 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]. Acknowledgements 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 REXE 315/3692 20219 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).

7

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