Generalization of the Proca Action

We consider the Lagrangian of a vector field with derivative self-interactions with a priori arbitrary coefficients. Starting with a flat space-time we show that for a special choice of the coefficients of the self-interactions the ghost-like pathologies disappear. This constitutes the Galileon-type generalization of the Proca action with only three propagating physical degrees of freedom. The longitudinal mode of the vector field is associated to the usual Galileon interactions for a specific choice of the overall functions. In difference to a scalar Galileon theory, the generalized Proca field has more free parameters. We then extend this analysis to a curved background. The resulting theory is the Horndeski Proca action with second order equations of motion on curved space-times.


Introduction
Motivated by the work from de Rham and Gabadadze for the generalization of the Fierz-Pauli action for a massive graviton [1], we investigate here the generalization of the Proca action for a massive vector field with derivative self-interactions. We will be addressing the natural question of what is the Lagrangian for a self-interacting vector field with second order equations of motion yielding three propagating physical degrees of freedom. We will call them the "vector Galileons", since they contain derivative self-interactions for the vector field and the longitudinal mode corresponds to a Galileon [2]. In the standard relativistic quantum field theory we describe particles with local covariant field operators like scalars, vectors, tensors..etc. The finite-dimensional representation of the Lorentz group dictates to us the number of propagating degrees of freedom. For a massless spin-1 field the theory needs to have the gauge symmetry in order to have the Lorentz invariance manifestly built in. The theory describes then a massless spin-1 field with two propagating degrees of freedom h = ±1. On the other hand, for a massive spin-1 field we have (2s + 1), i.e three, propagating physical degrees of freedom. The Proca action is the theory describing a massive vector field, which propagates the corresponding three polarizations (two transverse plus one longitudinal) . The mass term breaks explicitly the U (1) gauge invariance such that the longitudinal mode propagates as well. However, the zero component of the vector field does not propagate. So therefore it is a natural question to investigate also the existence of derivative interactions for the vector field with still only three propagating degrees of freedom. This is exactly what we aim in this paper: we want to find the generalization of the Proca action for a massive vector field with derivative self-interactions. The standard Proca action is given by In this theory, the temporal component of the vector field does not propagate and generates a primary constraint. The consistency condition of this primary constraint generates a secondary constraint, whose Poisson bracket with the primary constraint is proportional to the mass so that only in the massless case it corresponds to a first class constraint generating a gauge symmetry. Gauge invariance is just a redundancy in the description of massless particles describing the same physical state. Therefore, we can use the Stueckelberg trick to restore the gauge invariance. In the case of the standard Proca action we can restore the gauge invariance by adding an additional scalar field via A µ → A µ + ∂ µ π. This trick does not change the number of propagating physical degrees of freedom. We add one additional scalar degree of freedom but we restore the gauge invariance which guaranties the existence of only two physical degrees of freedom for the vector field, making it in total three physical degrees of freedom. When we now take the mass going zero limit m → 0 the Lagrangian results in a theory of a massless scalar field completely decoupled from a massless vector field for a conserved source. This is the reason why there is no vDVZ discontinuity in the case of the Proca field for conserved sources. This is different for the massive graviton. There, the helicity-0 degree of freedom does not decouple and gives rise to an additional fifth force which has to be screened via Vainshtein mechanism. In the case of the vector field, we do not need any Vainshtein mechanism since there is no observational difference between a massless and massive vector fields for conserved sources. In the generalized Proca action that we construct here, the longitudinal mode of the vector field has exactly the same interactions as a Galileon scalar field for a specific choice of the overall functions. The Galileon theory is an important class of infra-red modifications of general relativity . The Galileon interactions were introduced as a natural extension of the decoupling limit of the DGP model [2]. It is constructed as an effective field theory for a scalar field by the restriction of the invariance under internal Galilean and shift transformations and second order equations of motion. This effective action is local and contains higher order derivatives. Nevertheless, these interactions come in a very specific way such that they only give rise to second order equations of motion. The allowed interactions for the Galileon were originally determined order by order by writing down all the possible contractions for the derivative scalar field interactions and finding the proper coefficients giving rise to second order equations of motion. However, de Rham and Tolley could construct an unified class of four dimensional effective theories starting from a higher dimensional setup and show that these effective theories reproduce successfully all the interaction terms of the Galileon in the non-relativistic limit [3]. In a similar way we wonder whether or not one could construct the generalized Proca action that we are proposing here from a higher dimensional set-up which we will investigate in a future work. Naively, we would think, that, starting from a higher dimensional set-up with manifestly covariant Lovelock invariants, one would only construct terms which are gauge invariant after dimensional reduction. There is only one possible non-minimal interaction which fulfills this requirement, namely the contraction of two field strength tensors with the dual Riemann tensor. There-fore, this specific interaction with gauge invariance could be easily constructed from a higher dimensional set-up. However, it would be worth to study, if the other not gauge invariant non-minimal couplings could be constructed by dimensional reduction. The Galileon interactions present a subclass of Horndeski interactions which describe scalar-tensor interactions with at most second order equations of motion on curved backgrounds [4]. Interestingly, a subclass of Horndeski scalar-tensor interactions [5] can also be constructed by covariantizing the decoupling limit of massive gravity [6]. In the literature there has been some attempts to find a theory for vector fields which is equivalent to scalar Galileons, i.e. to find the "vector Galileons" besides the Maxwell kinetic term with second order equations of motion on flat space-times [7]. There, the authors were interested in derivative self-interactions for the vector field with gauge symmetry yielding only two propagating degrees of freedom. They concluded that the Maxwell kinetic term is the only allowed interaction and wrote a no-go theorem for generalized vector Galileons 1 . However, on curved background there only exists one term respecting the gauge symmetry, which is given by the non-minimal coupling between the field strength tensor and the dual Riemann tensor [9,10]. However, if one gives up on the gauge invariance meaning that we allow for terms which are not invariant under A µ → A µ + ∂ µ θ then one can indeed construct "vector Galileons" on flat-spacetimes or Horndeski vector interactions on curved backgrounds giving rise to three propagating physical degrees of freedom with second order equations of motion. We will illustrate this in this work.

The theory of generalized Proca field
Now we want to generalize the Proca action 1.1 to include derivative self-interactions of the vector field, but without changing the number of propagating degrees of freedom. In order to obtain such interactions, we will analyze all the possible Lorentz invariant terms that can be built at each order and constrain the interactions to remove the ghost-instabilities. The Lagrangian for the generalized Proca vector field with derivative self-interactions is given by where the self-interactions of the vector field are with ∂ · A = ∂ µ A µ and where the functions f 2,3,4,5 are arbitrary functions. Let us first emphasize the dependences of these functions. First of all, they all can depend on Nevertheless, the function f 2 is special in the sense that it is the only function which is not multiplied by any term with derivatives acting on the vector field. Therefore, this function f 2 can also have dependence on all the possible terms which have U (1) symmetry Furthermore, the function f 2 can depend on terms which does not contain any time derivative applying on the temporal component A 0 of the vector field like for instance A µ A ν F µρ F ν ρ . This is not true for the remaining functions f 3,4,5 .
where A · F encodes all the possible contractions between A's and F 's. For instance the function f 2 can naturally depend on terms like .etc while the remaining functions f 3,4,5 can only depend on A 2 since these functions are multiplied by terms which contain derivatives acting on the vector field. These functions do not change the number of propagating physical degrees of freedom since they do not contain any dynamics for the temporal component of the vector field. We will comment more on that in section 4. The second Lagrangian L 2 naturally contains the mass 1 2 m 2 A 2 and potential terms V (A 2 ) for the vector field in the function f 2 . In the next section we will illustrate order by order why these interactions give rise to only three propagating degrees of freedom and illustrate the absence of ghost instabilities. Note also the appearance of the two free parameters c 2 and d 2 . It means that the "vector Galileons" contain more free parameters then the usual scalar Galileon theory. The interactions can be also expressed in terms of the Levi-Civita tensors (2.2) propagate only three degrees of freedom. Higher order interactions beyond the quintic order are trivial in four dimensions, being just total derivatives, hence the serie stops here. Expressed in terms of the Levi-Civita tensors this means that we run out of the indices. When we wrote the derivative self-interactions in terms of the Levi-Civita tensors, the indices of the potential interactions were always contracted with each other. Without loss of generality consider for example the special choice for the functions f 2,3,4,5 = (A 2 ) and the special choice for the parameters c 2 = 1 and d 2 = 1, then in this case, we could either consider contractions in the functions as A µ A µ or contract the indices of these two vector fields with the Levi-Civita tensor as well. One might wonder, if it yields different interactions once the indices of the term (A 2 ) for example are contracted with the Levi-Civita tensors as well.
But on closer inspection one can see that they give rise to exactly the same interactions once integrations by part are performed. In the following we will start with the general interactions order by order with arbitrary coefficients and demonstrate that imposing the absence of the unphsical degree of freedom gives rise to the above Lagrangian with only three propagating physical degrees of freedom.

The propagation of three degrees of freedom
The simplest modification of the Proca action 1.1 is of course promoting the mass term to an arbitrary function f 2 which contains amongst others the mass term and the potential interactions for the vector field f 2 ⊃ V (A 2 ), since this trivially does not modify the number of degrees of freedom. As we already emphasized, this function can also contain gauge invariant interactions which are invariant under the U (1) transformations and terms which do not contain any dynamics for the temporal component of the vector field, i.e. terms of the form The first term that we can have to the next order in the vector field is simply with f 3 an arbitrary function of the vector field norm f 3 (A 2 ). It is a trivial observation that in (3.1) the temporal component of the vector field A 0 does not propagate, even if we include the Maxwell kinetic term, and it acts as a lagrange multiplier. The easiest way to see it is by computing the corresponding Hessian, which vanishes trivially. Also notice that the presence of the function f 3 is crucial since if it was simply a constant, that term would be a total divergence and, thus, with no contribution to the field equations. To next order, the independent interaction terms that we can have are given by with a priori free parameters c 1 , c 2 and c 3 and f 4 an arbitrary function depending on f 4 (A 2 ). Now, we need to fix the parameters such that only three physical degrees of freedom propagate, i.e., such that we still have a second class constraint. In order to eliminate one propagating degree of freedom, we need a constraint equation, which is guaranteed if the determinant of the Hessian matrix vanishes. The Hessian matrix for (3.2) is given by propagate. This is not what we are looking for, therefore we disregard this choice. The other possibility for a vanishing determinant of the Hessian matrix corresponds to c 1 + c 2 + c 3 = 0. Without loss of generality we can set c 1 = 1 and therefore c 3 = −(1 + c 2 ). In this case the Hessian matrix only contains one vanishing eingenvalue and hence only one propagating constraint. This case corresponds to three propagating degrees of freedom with the Lagrangian at this order given by: Note that we can write these interactions also as which then in principal could be absorbed into f 2 ⊃ F 2 since the function f 2 depends in general on the gauge invariant quantities which do not contain any dynamics for the A 0 degree of freedom and therefore will not change the number of propagating degrees of freedom. One can either include the interaction c 2 F 2 ρσ into the function f 2 or leave it at the order of L 4 , but not both at the same to avoid redundancy. The vanishing of the determinant of the Hessian matrix guaranties the existence of a constraint. To find the expression for the constraint, we have to compute the conjugate momentum Π µ L 4 = ∂L 4 ∂Ȧµ . The zero component of the conjugate momentum is given by As one can see, the zero component of the conjugate momentum does not contain any time derivative yielding the constraint equation This constraint equation will generate a secondary constraint given by or equivalently one can obtain the secondary constraint by calculating the time derivative of the conjugate momentumΠ µ and use the Hamiltonian equations ∂H ∂Aµ = −Π µ and ∂H ∂Π µ =Ȧ µ . We have checked explicitly the existence of the secondary constraint and therefore the Lagrangian L 4 possesses only three propagating degrees of freedom. For the next order interactions we write down all the possible contractions between the derivative self-interactions which gives: with a priori the arbitrary parameters d 1 , d 2 , d 3 , d 4 and d 5 and function f 5 depending only on A 2 . In this quintic Lagrangian (3.9) the additional possible term symmetric under the exchange of ρ and σ. The Hessian matrix for this quintic Lagrangian is giving by In order to have only three propagating degrees of freedom the parameters need to fulfill the following conditions which are fulfilled by choosing (again without loss of generality we can choose d 1 = 1) Hence, the quintic Lagrangian with only three propagating physical degrees of freedom is given by Analogously, we can write these interactions also as (3.14) The Hessian matrix with this chosen parameters then becomes with a vanishing determinant det (H µν L 5 ) = 0. As required, the Hessian matrix only contains one vanishing eingenvalue and hence only one propagating constraint which is again given by the corresponding zero component of the conjugate momentum Π µ In this section we will pay attention to the special case where the arbitrary functions are chosen to be f 2,3,4,5 = A 2 . In this case the four Lagrangians L 2,3,4,5 in 2.2 simply become We can restore the U (1) gauge symmetry using the Stueckelberg trick by adding an additional scalar field via A µ → A µ + ∂ µ π. To zeroth order in A µ we extract out only the longitudinal mode of the vector field and recover exactly the Galileon interactions Note that after introducing the gauge symmetry the dependence of the free parameters c 2 and d 2 disappears. Similarly, to first order in A µ we obtain the following scalar-vector interactions This is another way of observing that the interactions we found for the vector field indeed only propagate three degrees of freedom, since when we plug in the longitudinal mode, we obtain the Galileon interaction with at most second order equations of motion. The terms for the vector field ∂ ρ A σ ∂ ρ A σ and ∂ ρ A σ ∂ σ A ρ are not the same, but when we replace A µ = ∂ µ π, they are since the derivatives acting on the scalar field commute ∂ µ ∂ ν π = ∂ ν ∂ µ π on flat space-time. This has a huge concequence: the interactions for the vector field have more free parameters than the Galileon interactions. It means that if we had started with the Galileon interactions and performed the replacement ∂ µ π → A µ we would have been missing some of the interactions which also yield three propagating degrees of freedom. The vector interaction have two more free parameters (namely what we called c 2 and d 2 in (4.1)). In fact, an alternative way of finding our generalized Proca action is by restoring the U (1) gauge invariance and imposing that the Stueckelberg field propagates only one degree of freedom, i.e., it satisfies second order field equations. One must be careful though, since in addition to the pure Stueckelberg sector, it is also necessarily to analyse the terms mixing the Stueckelberg field and the vector field. For L 4 , no additional constraints arise from the mixing terms, since we obtain terms of the general form K µν (A µ )∂ µ π∂ ν π, which automatically leads to second order contributions for π. However, for L 5 we obtain terms like K αβγδ (A µ )∂ α ∂ β π∂ γ ∂ δ π so we need to impose the tensor K αβγδ (A µ ) to have the correct structure. It is also worth to emphasize one more time that the arbitrary functions f 2,3,4,5 appearing in our generalized Proca action have been chosen to be A 2 in this section to be able to relate them to the Galileon interactions. In the Stueckelberg language, this is so in order to guarantee the second order nature of the field equations with respect to π. There are however additional contributions upon which the functions might depend without altering the number of degrees of freedom. Such terms are those for which the Stueckelberg field give a trivial contribution, i.e., those which are U (1) gauge invariant. Therefore, the function f 2 could actually depend also on the combinations F 2 or F F * . It can naturally also depend on any possible contraction between A µ and F µν as well, in a way like for example A µ A ν F µα F ν α ..etc. From the vector field perspective, these terms do not contain time derivatives of A 0 , so that it will not spoil the existence of the constraints. Indeed, if you look at the interactions in L 4 which are proportional to the parameter c 2 then you trivially recognize that these terms are just c 2 F 2 . Since function f 2 also depends on F 2 then the term for instance in L 4 could be absorbed into f 2 (A 2 , F 2 ). One must be cautious however, since arbitrary functions of such invariants typically give rise to violations of the hyperbolicity of the field equations and hence to superluminal propagation, which we do not discuss in this work.
The equations of motion for the Lagrangian of the derivative self-interacting vector field (4.1) on top of the Maxwell kinetic term are given by Note also that the equations of motion for the vector field does reproduce the equations of motion of the Galileon field if we take the divergence of it and replace A µ = ∂ µ π.

Curved space-times
In the flat space-time the derivatives applied on the vector field were simply partial derivatives which commute. When we consider a general non-flat background the derivatives become covariant derivatives and therefore we have to add non-minimal couplings to the graviton in order to maintain second order equations of motion and healthy propagating degrees of freedom. When we generalize the derivative self-interactions in 5.1 on a curved space-time, the Lagrangian for the generalized Proca field becomes L curved gen.Proca = − where now the self-interactions are encoded in the following Lagrangians with the short cut X = − 1 2 A 2 µ . These interactions give rise to the standard scalar Horndeski interactions for the longitudinal mode of the vector field. The non-minimal coupling between the field strength tensors and the dual Riemann tensor considered in [10] is already incorporated in the above interactions. Similarly the function G 2 does not need to depend only on X but can also depend on terms like G µν A µ A ν which does not contain any dynamics for the temporal component of the vector field. Note again the appearance of the two additional free parameters c 2 and d 2 as in flat space-time case. All these interactions give only rise to three propagating degrees of freedom in curved background.

Summary and discussion
In this paper we have constructed the generalized Proca action for a vector field with derivative self-interactions with only three propagating degrees of freedom. We started our analysis with the case of a flat Minkowksi spacetime. We successfully showed that for appropriate choices of the coefficients of the derivative self-interactions that generalize the Proca action, one can construct a consistent and local theory of massive vector field without the presence of ghostlike instabilities. The resulting theory is simple and constitutes four Lagrangians for the self-interactions of the vector field. We were able to show that the constrained coefficients yield the necessary propagating constraint in order to remove the unphysical degree of freedom. These are the "vector Galileons" with three propagating degrees of freedom. At each order the Lagrangian has an overall function which depends on A 2 and the function of the quadratic Lagrangian can also depend on all the possible terms invariant under U (1) symmetry like for instance F 2 and F F * ..etc. Similarly this function can also depend on any contractions between the vector field and the field strength tensor A µ A ν F µρ F ν ρ which does not contain any time derivative applied on the temporal component of the vector field. The dependence of the function f 2 on the gauge invariant terms or terms in which the zero component of the vector field does not have any dynamics, do not alter the number of propagating degrees of freedom. We have also shown, that these interactions have more free parameters than the corresponding scalar Galileon interactions. We then generalized our results to the case of curved space-time and obtained the corresponding "Horndeski vector" interactions. In the very latest stage of this work we became aware that a similar idea has been explored in [11] even though the interactions we constructed here are more general.