Generalization of the Proca Action

Lavinia Heisenberg,a,b

aPerimeter Institute for Theoretical Physics, 31 Caroline St. N, Waterloo, Ontario, Canada, N2L 2Y5 bDépartment de Physique Théoriqueand Center for Astroparticle Physics, Universitéde Genève, 24 Quai E. Ansermet, CH-1211 Genève, Switzerland E-mails: [email protected]

Abstract. 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. For this we use the degeneracy condition of the Hessian. 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 and purely intrinsic vector interactions. We then extend this analysis to a curved background. The resulting theory is the Horndeski Proca action with second order arXiv:1402.7026v3 [hep-th] 1 Aug 2019 equations of motion on curved space-times. Contents

1 Introduction1

2 The theory of generalized Proca ﬁeld3

3 The propagation of three degrees of freedom6

2 4 Galileon case of the functions f2,3,4,5 = A 10

5 Curved space-times 12

6 Summary and discussion 13

1 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 three propagating physical degrees of freedom (as it also becomes clear from the master formula (2s + 1), where s represents the spin of the particle). 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 and turning the temporal component manifestly non-dynamical. 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.

– 1 – The standard Proca action is given by

Z 1 1 S = d4x − F 2 − m2A2 (1.1) Proca 4 µν 2 where Fµν = ∂µAν −∂νAµ. 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. In the presence of the mass term it represents a second class constraint. 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 Aµ → Aµ +∂µπ. 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 the 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

– 2 – 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. Therefore, 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 backgrounds there only exists one term respecting the gauge symmetry, which is given by the non-minimal coupling between the field strength tensor and the double 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 Aµ → Aµ + ∂µθ, 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.

2 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

5 1 X L = − F 2 + α L , (2.1) gen.Proca 4 µν n n n=2

1A similar no-go theorem has been also studied in [8] in the context of massive graviton.

– 3 – where the self-interactions of the vector ﬁeld are

˜ L2 = f2(Aµ,Fµν, Fµν) 2 L3 = f3(A ) ∂ · A 2 2 σ ρ ˜ 2 2 L4 = f4(A ) (∂ · A) − ∂ρAσ∂ A + c2f4(A )F 2 3 σ ρ γ ρ σ L5 = f5(A ) (∂ · A) − 3(∂ · A)∂ρAσ∂ A + 2∂ρAσ∂ A ∂ Aγ ˜ 2 ˜µα ˜ν +d2f5(A )F Fα ∂µAν 2 ˜αβ ˜µν L6 = e2f6(A )F F ∂αAµ∂βAν , (2.2)

µ with ∂ · A = ∂µA and where the functions f2,3,4,5 are arbitrary functions. Let us first emphasize the dependences of these functions. First of all, they all can depend on A2 = µ AµA . Nevertheless, the function f2 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 f2 can also have dependence on all the possible terms which have U(1) symmetry 2 µν ˜ µν ˜ ˜ like F = F Fµν, F F = F Fµν..etc (where F is the dual of F). Furthermore, the function f2 can depend on terms which does not contain any time derivative applying on the temporal µρ ν component A0 of the vector field like for instance AµAνF Fρ . This is not true for the remaining functions f3,4,5. Thus, we have

˜ 2 f2 = f2(Aµ,Fµν, Fµν) and f3,4,5 = f3,4,5(A ) (2.3)

2 2 2 4 4 4 µρ ν For instance the function f2 can naturally depend on terms like A F , A F , A F , AµAνF Fρ 2 ...etc while the remaining functions f3,4,5 can only depend on A 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 1 2 2 section4. The second Lagrangian L2 naturally contains the mass 2 m A and potential terms 2 V (A ) for the vector field in the function f2. 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 three free parameters c2, d2 and e2. It means that the vector Galileons contain more free parameters then the usual scalar Galileon theory. We would like to mention that the sixth order Lagrangian was first omitted in a previous version of this work, since by then it was imposed that the longitudinal mode should not have any trivial total derivative. Getting rid of this condition permitted the construction of the sixth order Lagrangian in [11], which was confirmed in [12]. An interesting possibility for an infinite series of derivative self-interactions was also discussed in [11], which unfortunately trivializes in four dimensions [12].

– 4 – The interactions can be also expressed in terms of the Levi-Civita tensors

f (A ,F , F˜ ) L = − 2 µ µν µν E µναβE = f (A ,F , F˜ ) 2 24 µναβ 2 µ µν µν f (A2) L = − 3 E µναβE ρ ∂ A = f (A2) ∂ · A 3 6 ναβ µ ρ 3 1 L = − E µνρσE αβ (f (A2)∂ A ∂ A + c f˜ (A2)∂ A ∂ A ) 4 2 ρσ 4 µ α ν β 2 4 µ ν α β 2 2 σ ρ ˜ 2 2 = f4(A ) (∂ · A) − ∂ρAσ∂ A + c2f4(A )Fρσ µνρσ αβδ 2 ˜ 2 L5 = −E E σ(f5(A )∂µAα∂νAβ∂ρAδ + d2f5(A )∂µAν∂ρAα∂βAδ) 2 3 σ ρ γ ρ σ = f5(A ) (∂ · A) − 3(∂ · A)∂ρAσ∂ A + 2∂ρAσ∂ A ∂ Aγ ˜ 2 ˜αµ ˜β + d2f5(A )F F µ∂αAβ µνρσ αβδκ 2 ˜ 2 L6 = −E E (f6(A )∂µAα∂νAβ∂ρAδ∂σAκ + e2f6(A )∂µAν∂αAβ∂ρAδ∂σAκ) 2 β α ν µ µ ν µ β ν = f6(A ) 3∂ A (∂αAβ∂µAν∂ A − 2∂αAν∂ Aβ∂ Aµ) + 8(∂ · A)∂βAν∂ A ∂ Aµ 2 ν 4 −6(∂ · A) ∂µAν∂ Aµ + (∂ · A) ˜ 2 h γ β δ δ δ γ + e2f6(A ) (∂ · A)(2∂ A (∂βAδ∂ Aγ − ∂δAγ∂ Aβ) + (∂ · A)(∂δAγ − ∂γAδ)∂ A ))

β α γ δ γ δ δ γ i + ∂ A (−2∂αAδ∂ Aβ∂ Aβ + ∂γAδ(2∂ Aα∂ Aβ + (∂αAβ − ∂βAα)∂ A )) ˜ ˜αβ ˜µν = e2f6(A)F F ∂αAµ∂βAν . (2.4)

In [11, 12] the first contraction in L6 was then neglected since it corresponds to a total derivative (see also [13]). Motivated by the interesting phenomenology and implications of the generalized Proca theories [14–18], this term was resurrected in [19]. However, as it has been explicitly shown in [20] this term is indeed just a total derivative in four-dimensional flat space-time and hence can be ignored. The Lagrangians L2,3,4,5,6 in (2.2) propagate only three degrees of freedom. Higher order interactions beyond the six order Lagrangian are trivial in four dimensions, hence the series 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 2 consider for example the special choice for the functions f2,3,4,5 = (A ) then in this case, we µ could either consider contractions in the functions as AµA 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 (A2) for example are contracted with the Levi-Civita

– 5 – tensors as well. 1 Lal = − E µναβE ρ A A = (A2) 2 6 ναβ µ ρ 1 Lal = − E µναβE ρσ A A ∂ A = (A2)(∂ · A) − AµAν∂ A 3 2 αβ µ ρ ν σ ν µ al µναβ ρσδ L4 = −E E βAµAρ∂νAσ∂αAδ 2 2 σ ρ µ ν µ ν ρ = (A ) (∂ · A) − ∂ρAσ∂ A − 2A A ∂νAµ(∂ · A) + 2A A ∂νAρ∂ Aµ al µναβ ρσδγ L5 = E E AµAρ∂νAσ∂αAδ∂βAγ 2 3 σ ρ γ ρ σ = (A ) −(∂ · A) + 3(∂ · A)∂ρAσ∂ A − 2∂ρAσ∂ A ∂ Aγ µ ν 2 µ ν ρ µ ν ρ γ +3A A ∂νAµ(∂ · A) − 6A A ∂νAρ∂ Aµ(∂ · A) + 6A A ∂νAρ∂ Aγ∂ Aµ µ ν σ ρ −3A A ∂νAµ∂ρAσ∂ A (2.5)

But on closer inspection one can see that they give rise to exactly the same interactions once integrations by part are performed. This would just correspond to considering disformal transformations of the metric ηµν → ηµν + AµAν. For instance if we take the cubic interaction ˜ L3 in (2.4) and perform a disformal transformation ηµν → ηµν + f3(X)AµAν, then this will ˜ ˜ µ ν give rise to L3 = f3(X)A A (∂µAν). We do not consider these interactions as genuinely new interactions since they are related to the previous interactions by means of disformal transformations. 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. This will make extensive use of the degeneracy condition of the Hessian and the presence of second class contraints.

3 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 f2 which contains amongst others the mass term and the potential 2 interactions for the vector field f2 ⊃ V (A ), 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 2 ∗ 2 2 2 ∗ ρµ ν f2 ⊃ F + FF + A F + A FF + AµAνF Fρ + ··· . The independent contractions are µ µν µ ν α X = −AµA /2, F = −FµνF /4 and Y = A A Fµ Fνα (see also [21]) and hence we can rewrite the function as L2 = f2(X,F,Y ). The first term that we can have to the next order in the vector field is simply

L3 = f3(X) ∂ · A (3.1)

– 6 – µ with f3 being an arbitrary function of the vector field norm X = −AµA /2. It is a trivial observation that in (3.1) the temporal component of the vector field A0 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 Hµν = 0. L3 Also notice that the presence of the function f3 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

2 ρ σ σ ρ L4 = f4 c1(∂ · A) + c2∂ρAσ∂ A + c3∂ρAσ∂ A (3.2)

2 with a priori free parameters c1, c2 and c3 and f4 an arbitrary function depending on f4(A ). 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   2(c1 + c2 + c3) 0 0 0 2 µν ∂ L4  0 −2c2 0 0  H = = f4(X)   (3.3) L4 ˙ ˙   ∂Aµ∂Aν  0 0 −2c2 0  0 0 0 −2c2

For a vanishing determinant of the Hessian matrix we have two possibilities. First possibility corresponds to choosing c2 = 0. In this case the Hessian matrix contains three vanishing eigenvalues corresponding to three constraints. Therefore, if we choose c2 = 0, only the zero component of the vector ﬁeld propagate while the other three degrees of freedom do not 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 c1 + c2 + c3 = 0. Without loss of generality we can set c1 = 1 and therefore c3 = −(1 + c2). 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:

2 ρ σ σ ρ L4 = f4 (∂ · A) + c2∂ρAσ∂ A − (1 + c2)∂ρAσ∂ A (3.4)

Note that we can write these interactions also as

2 σ ρ 2 L4 = f4 (∂ · A) − ∂ρAσ∂ A + c2Fρσ (3.5) where it becomes immediate that the last term proportional to c2 even with an independent ˜ 2 separate function f4(X) can then simply be absorbed into f2 ⊃ F since the function f2 depends in general on the gauge invariant quantities which do not contain any dynamics for the A0 degree of freedom and therefore will not change the number of propagating degrees of

– 7 – ˜ 2 freedom. One can either include the interaction c2f4(X)Fρσ into the function f2 or leave it at the order of L4, but not both at the same to avoid redundancy. In terms of the Levi-Civita tensors we have the following ways of contracting the indices 1 L = − E µνρσE αβ (f (X)∂ A ∂ A + c f˜ (X)∂ A ∂ A ) 4 2 ρσ 4 µ α ν β 2 4 µ ν α β 2 σ ρ ˜ ρ σ σ ρ = f4 (∂ · A) − ∂ρAσ∂ A + c2f4(∂ρAσ∂ A − ∂ρAσ∂ A ) . (3.6)

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 Πµ = ∂L4 . The zero component of the conjugate momentum is given by L4 ∂A˙ µ

0 ΠL4 = −2f4 ∇A . (3.7)

As one can see, the zero component of the conjugate momentum does not contain any time derivative yielding the constraint equation

0 C1 = ΠL4 + 2f4 ∇A. (3.8)

This constraint equation will generate a secondary constraint given by

∂H ∂C1 ∂H ∂C1 {H, C1} = µ − µ (3.9) ∂Aµ ∂Π ∂Π ∂Aµ or equivalently one can obtain the secondary constraint by calculating the time derivative of ˙ µ ∂H ˙ µ ∂H ˙ the conjugate momentum Π and use the Hamiltonian equations = −Π and µ = A . ∂Aµ ∂Π µ We have checked explicitly the existence of the secondary constraint and therefore the La- grangian L4 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:

3 ρ σ σ ρ L5 = f5 d1(∂ · A) − 3d2(∂ · A)∂ρAσ∂ A − 3d3(∂ · A)∂ρAσ∂ A γ ρ σ γ ρ σ +2d4∂ρAσ∂ A ∂ Aγ + 2d5∂ρAσ∂ A ∂γA ] (3.10) with a priori the arbitrary parameters d1, d2, d3, d4 and d5 and function f5 depending only on 2 γ ρ σ A . In this quintic Lagrangian (3.10) the additional possible term ∂σAρ∂ A ∂γA is actually γ ρ σ γ ρ σ equal to ∂ρAσ∂ A ∂γA since ∂ A ∂γA is symmetric under the exchange of ρ and σ. The

– 8 – Hessian matrix for this quintic Lagrangian is giving by

00 ˙ HL5 = −6(d1 − d2 − d3)(∇A) + 6(d1 − 3d2 − 3d3 + 2(d4 + d5))At 0i i0 HL5 = HL5 = (6d3 − 2(3d4 + d5))At,i + 2(3d2 − 2d4)Ai,t 11 2 HL5 = −6d2Aα(Az,z + Ay,y) − 2(3d2 − 2d4)(Ax,x − At,t) 12 21 HL5 = HL5 = 2d5(Ax,y + Ay,x) 13 31 HL5 = HL5 = 2d5(Ax,z + Az,x) 22 HL5 = 2(−3d2Az,z + (−3d2 + 2d5)Ay,y − 3d2Ax,x + (3d2 − 2d5)At,t) 23 32 HL5 = HL5 = 2d5(Ay,z + Az,y) 33 HL5 = (−6d2 + 4d5)Az,z − 6d2(Ay,y + Ax,x) + 2(3d2 − 2d5)At,t (3.11)

In order to have only three propagating degrees of freedom the parameters need to fulﬁll the following conditions

d1 − d2 − d3 = 0, d1 − 3d2 − 3d3 + 2(d4 + d5) = 0,

3d3 − 3d4 − d5 = 0, 3d2 − 2d5 = 0 (3.12) which are fulﬁlled by choosing (again without loss of generality we can choose d1 = 1) 3d 3d d = 1 − d , d = 1 − 2 , d = 2 (3.13) 3 2 4 2 5 2 Hence, the quintic Lagrangian with only three propagating physical degrees of freedom is given by

3 ρ σ σ ρ L5 = f5 (∂ · A) − 3d2(∂ · A)∂ρAσ∂ A − 3(1 − d2)(∂ · A)∂ρAσ∂ A 3d 3d +2 1 − 2 ∂ A ∂γAρ∂σA + 2 2 ∂ A ∂γAρ∂ Aσ (3.14) 2 ρ σ γ 2 ρ σ γ

Analogously, we can write these interactions also as

3 σ ρ γ ρ σ L5 = f5 (∂ · A) − 3(∂ · A)∂ρAσ∂ A + 2∂ρAσ∂ A ∂ Aγ 3d − 2 (∂ · A)F 2 + 3d ∂ A F σF ργ . (3.15) 2 ρσ 2 σ γ ρ

The Hessian matrix with this chosen parameters then becomes   0 0 0 0

µν 2 0 −6d2(Az,z + Ay,y) 3d2(Ax,y + Ay,x) 3d2(Ax,z + Az,x)  H = f5(A )   (3.16) L5   0 3d2(Ax,y + Ay,x) −6d2(Az,z + Ax,x) 3d2(Ay,z + Az,y)  0 3d2(Ax,z + Az,x) 3d2(Ay,z + Az,y) −6d2(Ay,y + Ax,x)

– 9 – with a vanishing determinant det (Hµν) = 0. As required, the Hessian matrix only contains L5 one vanishing eingenvalue and hence only one propagating constraint which is again given by the corresponding zero component of the conjugate momentum Πµ = ∂L5 L5 ∂A˙ µ 0 2 2 2 2 2 2 ΠL5 = −3f5(A ) d2(Ax,z + Ay,z + Ax,y) − 2Az,zAy,z − 2(−1 + d2)Ay,zAz,y + d2Az,y + d2Az,x 2 −2(Az,z + Ay,y)Ax,x + 2Ax,yAy,x − 2d2Ax,yAy,x + d2Ay,x − 2(−1 + d2)Ax,zAz,x .(3.17) As you can see, there is no time derivatives appearing in the expression of the zero component of the conjugate momentum, representing the constraint equation. Associated to this constraint, there will be a secondary constraint guarenting the propagation of the constraint equation and removing the unphysical degree of freedom. Expressed with the Levi-Civita tensors, the distinctive nature of the interactions become transparent µνρσ αβδ ˜ L5 = −E E σ(f5(X)∂µAα∂νAβ∂ρAδ + d2f5(X)∂µAν∂ρAα∂βAδ) 3 σ ρ γ ρ σ = f5(X) (∂ · A) − 3(∂ · A)∂ρAσ∂ A + 2∂ρAσ∂ A ∂ Aγ ˜ ˜αµ ˜β + d2f5(X)F F µ∂αAβ . (3.18) Note that the longitudinal part of the vector field belongs to the Galileon scalar interactions. Imposing the condition that the longitudinal mode should not have any trivially vanishing interactions would make that the series for the longitudinal mode would stop here. However, since the vector field contains two transverse modes besides the longitudinal mode, one can construct the interactions forth order in derivates of the vector field, that would give rise to a non-trivial mixing between the transverse and the longitudinal modes [11, 12] as it was the ˜ ˜αµ ˜β case for the d2f5(X)F F µ∂αAβ interaction in L5. The sixth order Lagrangian contains µνρσ αβδκ ˜ L6 = − E E (f6(X)∂µAα∂νAβ∂ρAδ∂σAκ + e2f6(X)∂µAν∂αAβ∂ρAδ∂σAκ) ˜ ˜αβ ˜µν = e2f6(X)F F ∂αAµ∂βAν . (3.19) The first contraction corresponds to just a total derivative [13, 20] and can be neglected.

2 4 Galileon case of the functions f2,3,4,5 = A

In this section we will pay attention to the special case where the arbitrary functions are 2 chosen to be f2,3,4,5 = A in the Lagrangians L2,3,4,5. Considering only those Lagrangians in 2.2 simpliﬁes to

2 L2 = A 2 L3 = A (∂ · A) 2 2 ρ σ σ ρ L4 = A (∂ · A) + c2∂ρAσ∂ A − (1 + c2)∂ρAσ∂ A 2 3 ρ σ σ ρ L5 = A (∂ · A) − 3d2(∂ · A)∂ρAσ∂ A − 3(1 − d2)(∂ · A)∂ρAσ∂ A 3d 3d +2 1 − 2 ∂ A ∂γAρ∂σA + 2 2 ∂ A ∂γAρ∂ Aσ . (4.1) 2 ρ σ γ 2 ρ σ γ

– 10 – We can restore the U(1) gauge symmetry using the Stueckelberg trick by adding an additional scalar ﬁeld via Aµ → Aµ + ∂µπ. To zeroth order in Aµ we extract out only the longitudinal mode of the vector ﬁeld and recover exactly the Galileon interactions

2 L2 = (∂π) 2 L3 = (∂π) 2π 2 2 2 L4 = (∂π) (2π) − (∂µ∂νπ) 2 3 2 3 L5 = (∂π) (2π) − 32π(∂µ∂νπ) + 2(∂µ∂νπ) . (4.2)

Note that after introducing the gauge symmetry the dependence of the free parameters c2 and d2 disappears in the purely longitudinal sector as expected. Similarly, to ﬁrst order in Aµ we obtain the following scalar-vector interactions

µ L2 = 2A ∂µπ 2 µ L3 = (∂π) (∂ · A) + 22π∂µπA 2 2 µ 2 ν µ 2 ρ L4 = 2(∂π) 2π(∂ · A) + 2(2π) ∂µπA − 2(∂π) ∂µ∂νπ∂ A − 2(∂µ∂νπ) ∂ρπA 2 σ β α α 3 2 2 β α 2 L5 = 6(∂π) ∂σ∂βπ∂α∂ π∂ A + 2A ∂απ(2π) + 3(∂ · A)(∂π) (2π) − 6∂α∂βπ∂ A (∂π) 2π α ρ β σ α 2 2 + 4A ∂απ∂ ∂ π∂σ∂βπ∂ ∂ρπ − 3 2A ∂απ2π + (∂ · A)(∂π) (∂ρ∂σπ) (4.3)

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 ρ σ σ ρ vector field ∂ρAσ∂ A and ∂ρAσ∂ A are not the same, but when we replace Aµ = ∂µπ, they are since the derivatives acting on the scalar field commute ∂µ∂νπ = ∂ν∂µπ 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 ∂µπ → Aµ we would have been missing some of the interactions which also yield three propagating degrees of freedom. The vector interactions have three more free parameters (namely what we called c2, d2 and e2 in (2.2)). 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 L4, no additional constraints arise from the mixing terms, since we obtain terms of the µν general form K (Aµ)∂µπ∂νπ, which automatically leads to second order contributions for π. αβγδ However, for L5 we obtain terms like K (Aµ)∂α∂βπ∂γ∂δπ so we need to impose the tensor αβγδ K (Aµ) to have the correct structure. This Stueckelberg analysis in the decoupling limit and the conditions for the K tensors of the allowed mixing between the Stueckelberg ﬁeld and the transverse ﬁelds have been analyzed in detail in [12].

– 11 – It is also worth to emphasize one more time that the arbitrary functions f2,3,4,5 appearing in our generalized Proca action have been chosen to be A2 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 f2 could actually depend also on the combinations F 2 or FF ∗. It can naturally also depend on any µα ν possible contraction between Aµ and Fµν as well, in a way like for example AµAνF Fα ..etc. From the vector field perspective, these terms do not contain time derivatives of A0, so that it will not spoil the existence of the constraints. Indeed, if you look at the interactions in L4 which are proportional to the parameter c2 then you trivially recognize that these terms are 2 2 just c2F . Since the function f2 also depends on F then the term for instance in L4 could be absorbed into f2(X,F,Y ). 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

E2 = 2Aµ ν E3 = 2Aµ(∂ · A) − 2A ∂µAν 2 σ ρ σ ρ 2 ν E4 = 2 Aµ (∂ · A) − (1 + c2)∂ρAσ∂ A + c2∂ρAσ∂ A + c2A (−2Aµ + ∂ν∂µA )

ρ ν ρ ρ ν −2c2A ∂νAρ∂ Aµ − 2(∂ · A)A ∂µAρ + 2(1 + c2)A ∂νAρ∂µA 3 σ ρ ρ σ γ ρ σ E5 = 2Aµ (∂ · A) + 3(−1 + d2)(∂ · A)∂ρAσ∂ A − 3d2(∂ · A)∂ρAσ∂ A + (2 − 3d2)∂ρAσ∂ A ∂ Aγ σ ρ γ ρ ν ν ν σ +3d2∂ρA ∂ A ∂σAγ] − 3A − d2(4∂νAρ∂ Aµ(∂ · A) − 2(∂νAµ∂ Aσ + ∂ Aµ∂σAν)∂ Aρ ν σ σ σ ν +Aρ(∂ Aµ(∂σ∂νA − 2Aν) + 2(∂ · A)(2Aµ − ∂σ∂µA ) + (∂ν∂µAσ − 2∂σ∂νAµ + ∂σ∂µAν)∂ A )) 2 σ ν +2((∂ · A) + ((−1 + d2)∂νAσ − d2∂σAν)∂ A )∂µAρ + (4(−1 + d2)∂νAρ(∂ · A) σ σ ν +d2Aρ(−∂σ∂νA + 2Aν) + 2((2 − 3d2)∂νAσ + d2∂σAν)∂ Aρ)∂µA (4.4)

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 Aµ = ∂µπ.

5 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

– 12 – 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

5 1 X Lcurved = − F 2 + β L (5.1) gen.Proca 4 µν n n n=2 where now the self-interactions are encoded in the following Lagrangians

˜ L2 = G2(Aµ,Fµν, Fµν) µ L3 = G3(X)∇µA µ 2 σ ρ L4 = G4(X)R + G4,X (∇µA ) − ∇ρAσ∇ A 1 h L = G (X)G ∇µAν − G (∇ · A)3 5 5 µν 6 5,X γ ρ σ σ ρi + 2∇ρAσ∇ A ∇ Aγ − 3(∇ · A)∇ρAσ∇ A ˜αµ ˜β − g5(X)F F µ∇αAβ G L = G (X)Lµναβ∇ A ∇ A + 6,X F˜αβF˜µν∇ A ∇ A (5.2) 6 6 µ ν α β 2 α µ β ν

˜αµ ˜β with ∇ denoting the covariant derivative. The fifth order interaction F F µ∇αAβ does not need any non-minimal coupling on curved backgrounds in order to compensate higher ˜αβ ˜µν order equations of motion [11, 12], unless the sixth order Lagrangian F F ∇αAµ∇βAν [12], which requires the relative tuning with the non-minimal coupling between the field strength tensors and the dual Riemann tensor considered in [10]. These interactions give rise to the standard scalar Horndeski interactions for the longitudinal mode of the vector field. Terms µν like G AµAν, which does not contain any dynamics for the temporal component of the vector field, are already contained in the above interactions after integrations by parts. All these interactions give only rise to three propagating degrees of freedom in curved background.

6 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 ghost- like instabilities. The resulting theory is simple and constitutes five 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.

– 13 – These are the vector Galileons with three propagating degrees of freedom. At each order the Lagrangian has an overall function which depends on A2 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 FF ∗..etc. Similarly this function can also depend on any contractions µρ ν between the vector field and the field strength tensor AµAνF Fρ which does not contain any time derivative applied on the temporal component of the vector field. The dependence of the function f2 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.

Acknowledgments

We would like to thank Claudia de Rham for useful discussions. Specially, we would like to express our inﬁnite gratitude to Jose Beltran Jimenez for his collaboration on this subject. This work is supported by the Swiss National Science Foundation.

You are the sun, so radiant and warm. The dawn cannot compete against you, the clouds cannot cover you. I am the moon trying to reﬂect your rays. If I can succeed this even slightly, so I’m happy.

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