Generalized SU(2) Proca theory reconstructed and beyond
Alexander Gallego Cadavid, Yeinzon Rodriguez, L. Gabriel Gomez
aa r X i v : . [ h e p - t h ] N ov PI/UAN-2020-678FT
Generalized SU(2) Proca theory reconstructed and beyond
Alexander Gallego Cadavid ∗ Instituto de F´ısica y Astronom´ıa, Universidad de Valpara´ıso,Avenida Gran Breta˜na 1111, Valpara´ıso 2360102, Chile
Yeinzon Rodr´ıguez † Centro de Investigaciones en Ciencias B´asicas y Aplicadas, Universidad Antonio Nari˜no,Cra 3 Este
L. Gabriel G´omez ‡ Escuela de F´ısica, Universidad Industrial de Santander,Ciudad Universitaria, Bucaramanga 680002, Colombia
As a modified gravity theory that introduces new gravitational degrees of freedom, the generalizedSU(2) Proca theory (GSU2P for short) is the non-Abelian version of the well-known generalizedProca theory where the action is invariant under global transformations of the SU(2) group. Thistheory was formulated for the first time in
Phys. Rev. D (2016) 084041, having implementedthe required primary constraint-enforcing relation to make the Lagrangian degenerate and removeone degree of freedom from the vector field in accordance with the irreducible representations ofthe Poincar´e group. It was later shown in Phys. Rev. D (2020) 045008, ibid 045009, that asecondary constraint-enforcing relation, which trivializes for the generalized Proca theory but notfor the SU(2) version, was needed to close the constraint algebra. It is the purpose of this paper toimplement this secondary constraint-enforcing relation in GSU2P and to make the construction ofthe theory more transparent. Since several terms in the Lagrangian were dismissed in
Phys. Rev.D (2016) 084041 via their equivalence to other terms through total derivatives, not all of thelatter satisfying the secondary constraint-enforcing relation, the work was not so simple as directlyapplying this relation to the resultant Lagrangian pieces of the old theory. Thus, we were motivatedto reconstruct the theory from scratch. In the process, we found the beyond GSU2P. Keywords: modified gravity theories, vector fields
I. INTRODUCTION
Whether a classical description of the gravitationalinteraction is fundamental or effective remains a mys-tery. What is certain is that, no matter whether thefundamental theory of gravity is classical or quantum,and despite its enormous experimental success [1–13],Einstein’s theory of gravity is an effective theory [14–16]. The inevitable presence of singularities in GeneralRelativity (GR) [17, 18], even assuming the validity ofthe cosmic censorship conjecture [19–21], points to abreakdown of the theory. Should the breakdown takeplace in the infrared, the new theory that encompassesGR might give us some insight about the true nature ofthe current accelerated expansion of the universe. Thebreakdown might take place in the ultraviolet, helping ∗ [email protected] † [email protected] ‡ [email protected] solve the renormalizability problems of GR and illumi-nating the way to a quantum description of gravity. Ofcourse, the breakdown might take place in both the in-frared and the ultraviolet. Another option is at an in-termediate scale, in the strong gravity regime, which isparticularly interesting because the very young multi-messenger astronomy is giving us, and will continue do-ing it, valuable information about the behaviour of grav-ity at the scales associated to compact objects such asblack holes and neutron stars . We might, therefore, beon the verge of a scientific crisis and a new revolutionin Physics, in the sense of Kuhn [23].Over the years, several approaches have been pro-posed to classically extend Einstein’s theory of grav-ity (see Ref. [24] for a review). Perhaps the simplest At these scales, however, there might be some contributionsfrom the ultraviolet-complete theory. This means that theregime of validity of the modified gravity theory must also beensured before applying constraints that belong to other scalesor frequencies [22]. one, at least in its conception, is giving mass to thegravitational carrier [25]; nevertheless, starting from theFierz-Pauli action [26] and arriving to the de Rham-Gabadadze-Tolley (dRGT) ghost-free massive gravity[27], the introduction of a massive graviton has shownto be a difficult challenge. Another possibility is addingspace dimensions while preserving the second-order dif-ferential structure of the field equations and keepinguntouched the gravitational degrees of freedom; thisis the proposal derived from the Lovelock programme[28, 29], as the only curvature invariant that satisfiesthese requirements in four space-time dimensions is theEinstein-Hilbert term. A third alternative is invokingnew gravitational degrees of freedom, the simplest ofthem being a scalar field; the first proposal in this re-gard was the well-known Brans-Dicke theory [30], butthis has turned out to be just a particular case of awhole family of Lagrangians that comprise the, nowa-days very famous, Horndeski theory [31–38]. The pur-pose of preserving the second-order differential struc-ture of the field equations is to remove the Ostrogradskighost [39–42] that makes the ground state unstable inthe presence of interactions. Notwithstanding, this isnot the only way to remove the Ostrogradski ghost, al-though it is the most transparent; the degeneracy of thekinetic matrix associated to the degrees of freedom ofthe theory can be invoked so that primary constraintsamong the phase space variables are generated [43] – inthis way, the unwanted degrees of freedom can be re-moved [44] even when the differential structure of thefield equations is higher order. This idea was put inaction with the introduction of the beyond Horndeskitheory [45, 46] and later generalized to what is nowknown as the degenerate higher-order scalar-tensor the-ory (DHOST) [47–49], where a plethora of Lagrangiansrose up to the surface. The application of this ideato the Lovelock programme has, nonetheless, not beenfruitful [50], which is, paradoxically, very suggestive. Afourth alternative is considering other geometric formu-lations of gravity, i.e., considering not only the curva-ture but also the torsion and the non metricity as theprotagonist geometric objects in the description of thegravitational interaction [24, 51–53]. This has a long his-tory starting from the Einstein-Cartan theory [54, 55],which involves curvature and torsion but leaving asidethe non metricity, to the coincident gravity proposal[56], where the non metricity is the sole protagonist. Ofcourse, there are more possibilities, some of them withremanent harmless ghosts, they being, therefore, effec-tive theories.The introduction of new gravitational degrees of free-dom has not been kept only in the realm of a scalar field.Multiple scalar fields have been considered in what arecalled the multi-Galileon theories [57–60]. More tensorfields can be considered as well, as in the bimetric the- ory [61] which introduces an extra spin-two metric. Theintroduction of vector fields [62–66] and p-forms [67–69]has also been investigated. Even the mixture of a scalarand a vector field, together with gravity, has been ex-plored [70]. Each one of these proposals has its ownmotivations, which we will not describe here except forthose related to the introduction of vector fields.The most frequent question when we speak about vec-tor fields in gravity and/or cosmology is: why to intro-duce them? We think the right question is: why not?:at the end of the day, and being pragmatical, we haveobserved many more vector fields in nature than fun-damental scalar fields. We have to be careful with theproblems they can generate: ghosts, anisotropies in cos-mology, etc., but this does not preclude their study. Infact, the role of vector fields in gravitation, astrophysics,and cosmology has attracted a lot of interest in recentyears (see Refs. [24, 71–73] for some reviews), culmi-nating in the construction and study of what is calledthe generalized Proca theory [62–66]. This is the Procatheory [74, 75], in curved spacetime, devoid of internalgauge symmetries and can be seen as the vector-fieldversion of the Horndeski theory . By construction, itis plainly degenerate in order to avoid the propagationof a fourth degree of freedom which clearly disagreeswith the structure of the irreducible representations ofthe Poincar´e group. Its decoupling limit, in contrast,reduces to the Horndeski theory.The generalized Proca theory has been well studiedin astrophysics and cosmology [62, 78–87]. In the lat-ter, however, special attention has been paid because ofthe anisotropies that a vector field produces, inherent toits nature, both in the expansion of the universe and inthe cosmological perturbations. Such anisotropies caneasily go beyond the observational constraints, so it isnecessary to take some measures such as the rapid oscil-lations of the vector field [88], the dilution of the vectorfield by a companion scalar field [89], the suppressionof the spatial components of the vector field against itstemporal component (what is called the temporal gaugesetup) [79], or the implementation of a cosmic triad ofvector fields that restores the isotropy [90–94]. The lat-ter proposal has been investigated in different contextsand finds a natural home in the presence of an internalSU(2) symmetry [95–99]. Indeed, the temporal gaugesetup and the cosmic triad are two of the four possi-ble setups that are compatible with a spatial sphericalsymmetry and that are realized under an internal SU(2) For the U(1) gauge-invariant version of the generalized Procatheory in flat spacetime see Ref. [76] and in curved spacetimesee Ref. [77]. The cosmic triad is a set of three vector fields mutually orthog-onal and of the same norm. symmetry [100–102]. This was the main motivation be-hind the formulation of what was baptized as the gener-alized SU(2) Proca theory (GSU2P for short) [103] (seealso Ref. [104]). The possible setups mentioned abovespontaneously break the internal (global) SU(2) sym-metry along with the Lorentz rotational symmetry andLorentz boosts, leaving, however, a diagonal spatial ro-tation subgroup unbroken. The isotropic expansion ofthe universe can then be naturally modeled with any ofthe four setups or linear combinations of them withoutresorting to fast oscillations or other (scalar) fields. Theprice to pay, however, which is the spontaneous break-ing of the Lorentz invariance, is, anyway, extraordinar-ily reasonable, since this seems to be nature’s strategyto produce all the patterns we see in condensed mat-ter systems (fluids, superfluids, solids, and supersolids;see Ref. [105]). Indeed, according to the pattern clas-sification in Ref. [105], what would be the condensedmatter analogs of the temporal gauge setup and the cos-mic triad in the GSU2P are the, yet unobserved, type-Iand type-II framids, repectively. The application of theGSU2P to dark energy and inflation has been exploredin Refs. [106, 107] and its stability properties in Ref.[108].The GSU2P was built in Ref. [103] (see also Ref.[104]) having in mind the primary constraints requiredto remove the fourth degree of freedom . To that end, aprimary constraint-enforcing relation related to the pri-mary Hessian of the system was employed. This wasdone in flat spacetime following the standard procedureof later covariantizing not before having removed redun-dant terms in the obtained action via total derivatives.Later on, two caveats were recognized. First, the con-straint algebra was not closed only with the primaryconstraints, at least for theories involving more thanone vector field [109, 110] ; a secondary constraint wasidentified that closed the constraint algebra and that,therefore, pointed out to the existence of ghosts in theGSU2P. Second, the redundant terms in flat spacetimeturned out to be not necessarily redundant in curvedspacetime, which would lead, for sure, to new terms notuncovered in Ref. [103]; indeed, such a remark led two ofus to rediscover the beyond Proca terms in Ref. [111],they being the vector analogous of the beyond Horn-deski terms, already obtained in Ref. [112]. Reformu-lating the GSU2P in order to implement the secondaryconstraint-enforcing relation seemed at first sight veryeasy, because it was a matter of applying this relationto the “old” GSU2P and seeing what the result would Concretely, the temporal component of the vector field. The constraint algebra of the generalized Proca theory, the lat-ter being a theory that involves just one vector field, turned outto be trivially closed. be. However, this turned out to be impractical, sincemany terms had disappeared when employing the totalderivatives. Moreover, the total derivatives employedsatisfied the primary constraint-enforcing relation butnot necessarily the secondary one, so repairing the oldtheory quickly became quite a big deal and, therefore,unworthy. The purpose of this paper is to build fromscratch the GSU2P, paying attention to the two caveatsalready mentioned and following a style of constructionbased on the decomposition of a first-order derivative ∂ µ A aν of the vector field A aµ into its symmetric, S aµν ≡ ∂ µ A aν + ∂ ν A aµ , (1)and antisymmetric part, A aµν ≡ ∂ µ A aν − ∂ ν A aµ . (2)Employing this decomposition will simplify things andallow us to deal with a lower number of Lagrangianbuilding blocks as compared with Ref. [103]. In theprocess, we will find the beyond GSU2P.The layout of the paper is the following. In the Sec-tion II, we will enumerate the requirements for the con-struction of the GSU2P. In Section III, we will show howan arbitrary function of A aµν and A aµ satisfies both theprimary and secondary constraint-enforcing relations,leaving only the work of finding the right terms in theaction involving at least one S aµν . In section IV, we buildthe Lagrangian involving one derivative and two vectorfields. Similar procedures are followed in Sections V, VI,VII, and VIII, where we obtain the Lagrangians involv-ing one derivative and four vector fields, two derivativesonly, two derivatives and two vector fields, and threederivatives only, respectively. In all these cases, thenumber of space-time indices in the Lagrangian buildingblocks before contractions with the primitive invariantsof the Poincar´e group is less than or equal to six. Weprefer to keep the construction of the theory up to thislevel since, as shown in Ref. [103], the number of La-grangian building blocks we have to consider scales veryfast when more space-time indices are considered. Fi-nally, in Sections IX and X, we compare the “new” or“reconstructed” GSU2P with the old GSU2P and withthe generalized Proca theory, respectively. Section XI isdevoted to the conclusions. Throughout the text, Greekindices are space-time indices and run from 0 to 3, whileLatin indices are internal SU(2) group indices and runfrom 1 to 3. The sign convention is the (+++) accord-ing to Misner, Thorne, and Wheeler [113]. II. REQUIREMENTS FOR THECONSTRUCTION OF THE THEORY
The GSU2P must be built having in mind the follow-ing criteria:1. The action must be, locally, Lorentz invariant (al-though the symmetry may be non-linearly real-ized).2. The vector field must transform as the adjoint rep-resentation of the global transformations belong-ing to the SU(2) group [114–116]. Accordingly,the action must be invariant under these transfor-mations.3. The primary constraint-enforcing relation H νab =0, where H µνab ≡ ∂ L ∂ ˙ A aµ ∂ ˙ A bν , (3)is the “primary” Hessian and a dot means a timederivative, must be satisfied in flat spacetime inorder to make the Lagrangian degenerate. Thisis a necessary condition to remove the unwanteddegree of freedom [62, 63].4. The secondary constraint-enforcing relation˜ H ab = 0, where˜ H µνab ≡ ∂ L ∂ ˙ A [ aµ ∂A b ] ν , (4)is the “secondary Hessian” and the brackets meanunnormalized antisymmetrization, must be satis-fied in flat spacetime so that the primary con-straint holds at all times . This condition togetherwith the preceding one are necessary and sufficientto remove the unwanted degree of freedom in flatspacetime [109, 110].5. The decoupling limit of the theory must be free ofthe Ostrogradski ghost as must happen since thefull theory is free of it. This implies that the scalarlimit of GSU2P must belong to the non-Abelianextension of the multi-Galileon theory [57–60, 103]or any of its beyond or DHOST versions. III. L All the Lagrangian pieces L Ai built exclusively fromcontractions of A aµν and A aµ with the primitive invari-ants of the Lorentz group [114–116], collected in a This condition bears a great resemblance to that obtained inRefs. [117, 118] for mechanical systems with multiple degreesof freedom. They may, of course, either preserve or violate parity. generic Lagrangian piece called L ( A aµν , A aµ ), satisfy au-tomatically both the primary and secondary constraint-enforcing relations thanks to the antisymmetry of A aµν .To see it, let us calculate the primary and secondaryHessians. First of all, ∂ L Ai ∂ ˙ A aµ = ∂ L Ai ∂A cρσ ∂A cρσ ∂ ˙ A aµ = ∂ L Ai ∂A cρσ δ ρ δ µσ ] δ ca = ∂ L Ai ∂A aρµ (cid:12)(cid:12)(cid:12) ρ =0 − ∂ L Ai ∂A aµσ (cid:12)(cid:12)(cid:12) σ =0 = 2 ∂ L Ai ∂A aρµ (cid:12)(cid:12)(cid:12) ρ =0 . (5)Any possible ambiguity in the second line of the pre-vious equation is clarified having in mind that L Ai isalways written as A aµν contracted with an antisymmet-ric tensor . Thus, ∂ L Ai ∂ ˙ A bν ∂ ˙ A aµ = 2 ∂ L Ai ∂A aρµ ∂A cαβ (cid:12)(cid:12)(cid:12) ρ =0 ∂A cαβ ∂ ˙ A bν = 2 ∂ L Ai ∂A aρµ ∂A cαβ (cid:12)(cid:12)(cid:12) ρ =0 δ α δ νβ ] δ cb = 2 ∂ L Ai ∂A aρµ ∂A bαν (cid:12)(cid:12)(cid:12) ρ =0 ,α =0 − ∂ L Ai ∂A aρµ ∂A bνβ (cid:12)(cid:12)(cid:12) ρ =0 ,β =0 = 4 ∂ L Ai ∂A aρµ ∂A bαν (cid:12)(cid:12)(cid:12) ρ =0 ,α =0 . (6)The primary constraint-enforcing relation is, therefore,satisfied: H νab = 4 ∂ L Ai ∂A aρµ ∂A bαν (cid:12)(cid:12)(cid:12) ρ =0 ,µ =0 ,α =0 = 0 , (7)because of the antisymmetry of A aµν .Regarding the secondary constraint-enforcing rela-tion, we obtain from Eq. (5) ∂ L Ai ∂A bν ∂ ˙ A aµ = 2 ∂ L Ai ∂A aρµ ∂A bν (cid:12)(cid:12)(cid:12) ρ =0 , (8)which leads to the secondary Hessian˜ H ab = 2 ∂ L Ai ∂A [ aρµ ∂A b ] ν (cid:12)(cid:12)(cid:12) ρ =0 ,µ =0 ,ν =0 = 0 , (9)in view, again, of the antisymmetry of A aµν .Hence, we can conclude that the L ( A aµν , A aµ ) La-grangian piece satisfies automatically the first and sec-ondary constraint-enforcing relations necessary to prop-agate only three degrees of freedom. This is the rea-son why such a Lagrangian piece is so particular, dif-fering in its structure and arbitrariness from the other Except for the case where no A aµν tensors are involved. How-ever, in such a case, ∂ L Ai ∂ ˙ A aµ = 0 automatically. Lagrangian pieces we are going to describe in the fol-lowing. On the other hand, the generalization of L tocurved spacetime is straightforward. IV. ONE DERIVATIVE AND TWO VECTORFIELDS
Lagrangian building blocks constructed from onederivative and two vector fields, linearly independentfrom L , are terms of the form S µν A ρ A σ which, as canbe seen, involve four space-time indices. Group the-ory tells us that four building blocks can be constructedupon contractions of S µν A ρ A σ with the following ten-sors [114–116]: g µν g ρσ ,g µρ g νσ ,g µσ g νρ ,ǫ µνρσ , (10)where g µν is the contravariant Minkowski metric and ǫ µνρσ is the Levi-Civita tensor. Thus, the only buildingblocks either different than zero or with the potential ofbecoming different than zero after adding the internalgroup indices are the following: S µµ ( A · A ) ,S µν A µ A ν . (11)The addition of the internal group indices leads to termsof the form S a A b A c that involve three internal groupindices and which, from group theory [114–116], can becontracted only with the totally antisymmetric tensor ǫ abc : S µaµ ( A b · A c ) ǫ abc ,S aµν A µb A νc ǫ abc . (12)Such terms vanish because of the antisymmetry of ǫ abc ,so we conclude that there do not exist terms in GSU2P,linearly independent of L , that involve one derivativeand two vector fields. This tensor represents the structure constants of the SU(2)group. See, in particular, the Misner, Thorne, and Wheelertreatise on gravitation [113] for a description of the SU(2) groupas a manifold endowed with a metric g ab and an orientabilityform described by ǫ abc . V. ONE DERIVATIVE AND FOUR VECTORFIELDSA. The Lagrangian building blocks
Lagrangian building blocks built from one derivativeand four vector fields, linearly independent of L , areterms of the form S µν A ρ A σ A α A β that involve six space-time indices. Group theory [114–116] tells us that, inthis case, the building blocks are constructed upon con-tractions of S µν A ρ A σ A α A β with the following fifteenpermutations of the product of three space-time met-rics: g µν g ρσ g αβ ,g µν g ρα g σβ ,g µν g ρβ g σα ,g µρ g νσ g αβ ,g µρ g να g σβ ,g µρ g νβ g σα ,g µσ g νρ g αβ ,g µσ g να g ρβ ,g µσ g νβ g ρα ,g µα g νρ g σβ ,g µα g νσ g ρβ ,g µα g νβ g ρσ ,g µβ g νρ g σα ,g µβ g νσ g ρα ,g µβ g να g ρσ , (13)as well as with the following ten products of a space-time metric and a Levi-Civita tensor: g νρ ǫ µσαβ ,g νσ ǫ µραβ ,g να ǫ µρσβ ,g νβ ǫ µρσα ,g ρσ ǫ µναβ ,g ρα ǫ µνσβ ,g ρβ ǫ µνσα ,g σα ǫ µνρβ ,g σβ ǫ µνρα ,g αβ ǫ µνρσ . (14)Other five contractions of the form gǫ are possible, butthey are not linearly independent because of the prop-erty: g µν ǫ ρσαβ = g νρ ǫ µσαβ − g νσ ǫ µραβ + g να ǫ µρσβ − g νβ ǫ µρσα . (15)Thus, only three building blocks either are non vanish-ing or have the potential of becoming different than zeroonce the internal group indices are added: S µµ ( A · A )( A · A ) ,S µν A µ A ν ( A · A ) ,S µν A ν A σ A α A β ǫ µσαβ . ( ∗ ) (16)When adding the internal group indices, these terms ac-quire the form S a A b A c A d A e which can be contracted,according to group theory [114–116], only with the fol-lowing six products of an internal group metric and therespective structure constants: g ab ǫ cde ,g ac ǫ bde ,g ad ǫ bce ,g bc ǫ ade ,g bd ǫ ace ,g cd ǫ abe . (17)Other four contractions of the form gǫ are possible, butthey are not linearly independent because of the prop-erty: g ae ǫ bcd = g ab ǫ cde − g ac ǫ bde + g ad ǫ bce . (18)Therefore, there exist only four linearly independentbuilding blocks in GSU2P that involve one derivativeand four vector fields: L = S aµν A µb A νc ( A b · A e ) ǫ ace , L = S aµν A νa A cσ A dα A eβ ǫ µσαβ ǫ cde , L = S aµν A σa A νb A dα A eβ ǫ µσαβ ǫ bde , L = S aµν A νb A σb A dα A eβ ǫ µσαβ ǫ ade . (19) B. The Hessian constraints
The Lagrangian is, hence, written as a linear combi-nation of the Lagrangian building blocks of Eq. (19): L = X i =1 x i L i , (20) From now on, the starred Lagrangian building blocks and totalderivatives will be those that vanish according to the Poincar´egroup but that otherwise survive when considering also theSU(2) group. where the x i are arbitrary constants. Because only onederivative has been considered, the primary constraint-enforcing relation is satisfied automatically. Regard-ing the secondary constraint-enforcing relation, the sec-ondary Hessian gives the following result:˜ H ab = 2[ − A c ( A [ b · A e ) ǫ a ] ce − A c ( A c · A e ) ǫ [ ab ] e + A b A c A e ǫ a ] ce + A e A c A e ǫ [ a | c | b ] ] x − A σ [ a | A dα A eβ ǫ σαβ ǫ | b ] de ( x − x ) , (21)which can vanish only if x = 0 ,x − x = 0 . (22)Thus, the Lagrangian that satisfies the constraint alge-bra is given by L = x L + x ( L + L ) . (23) C. Total derivatives
Although the Lagrangian in Eq. (23) satisfies require-ments 1 to 4 in Section II, some of its Lagrangian piecesmight be redundant, compared to L , via total deriva-tives. To find it out, we must proceed to build all thepossible total derivatives of currents involving five vec-tor fields. To this end, we must follow a path similar tothe ones in previous sections, i.e., employing group the-ory. In this way, a term of the form ∂ µ ( A ν A ρ A σ A α A β ),which involves six space-time indices, must be con-tracted with all the terms in Eqs. (13)-(14). How-ever, the Lagrangian pieces we are interested in, L and L + L , explicitly violate parity. Therefore, only theterms in Eq. (14) are actually needed. This leads to justone term that satisfies the requirement of either beingnon vanishing or having the potential of being non van-ishing once the internal group indices are added: ∂ µ [( A · A ) A σ A α A β ] ǫ µσαβ . ( ∗ ) (24)The addition of the internal group indices leads to termsof the form ∂ ( A a A b A c A d A e ) that involve five internalgroup indices. Therefore, they must be contracted withall the terms in Eq. (17), which results in ∂ µ J µ = ∂ µ [( A a · A a ) A cσ A dα A eβ ] ǫ µσαβ ǫ cde ,∂ µ J µ = ∂ µ [( A a · A b ) A σa A dα A eβ ] ǫ µσαβ ǫ bde . (25)These total derivatives can be expressed in terms of La-grangian building blocks involving one derivative andfour vector fields, which is the key to observe whethersome of the two Lagrangian pieces in Eq. (23) are re-dundant: ∂ µ J µ = 12 [2 A aµν A νa A cσ A dα A eβ +3 A cµσ A dα A eβ ( A a · A a )] ǫ µσαβ ǫ cde + L ,∂ µ J µ = 12 [ A aµν A νb A σa A dα A eβ + A aν A νbµ A σa A dα A eβ +( A a · A b ) A µσa A dα A eβ +2( A a · A b ) A σa A dµα A eβ ] ǫ µσαβ ǫ bde + 12 ( L + L ) . (26)We can see that, even after covariantization, the twoLagrangian pieces in Eq. (23) can be removed, via totalderivatives, in favour of terms already contained in L .Now, from the previous two expressions and the resultsof Sections III and V B, we can see that it is legitimateto employ ∂ µ J µ and ∂ µ J µ , since they satisfy the Hessianconstraints. Therefore, the conclusion is that there donot exist terms in GSU2P, linearly independent of L ,that involve one derivative and four vector fields. VI. TWO DERIVATIVESA. The Lagrangian building blocks
When dealing with two derivatives only, the La-grangian building blocks, linearly independent of L , ac-quire two possible structures: either A µν S ρσ or S µν S ρσ .In both cases, the number of space-time indices is four,so we have to contract with all the terms in Eq. (10).This results in S µµ S ρρ ,S µν S µν , (27)these terms being the only ones that either do not vanishor have the potential of being non vanishing once theinternal group indices are added. Indeed, when this isdone, these terms acquire the form S a S b which can becontracted only with the group metric g ab [114–116].Thus, the Lagrangian building blocks are L = S µaµ S ρρa , L = S aµν S µνa . (28) B. The Hessian constraints
The Lagrangian is therefore written as a linear com-bination of the Lagrangian building blocks of Eq. (28): L = X i =1 x i L i , (29)where the x i are arbitrary constants. Since this La-grangian involves only vector fields through space-time derivatives, the secondary constraint-enforcing re-lation is satisfied automatically. Regarding the primaryconstraint-enforcing relation, the primary Hessian givesthe following result: H νab = − g ab g ν ( x + x ) , (30)which vanishes only if x + x = 0 . (31)Thus, the Lagrangian that satisfies the constraint alge-bra is given by L = x ( L − L ) . (32) C. Total derivatives
Again, it is absolutely necessary to test if the La-grangian in Eq. (32) is not already included in L . Tothis end, it is necessary to build the total derivatives ofcurrents built with one derivative and one vector field.These terms, being of the form ∂ µ [ A ν ( ∂ ρ A σ )], involvefour space-time indices, so that they are constructed bymeans of contractions with the terms in Eq.(10), exceptfor the last one in that equation as the Lagrangian piecewe are interested in, L − L , explicitly preserves parity.In this case, none of the terms vanishes, so we end upwith three possible total derivatives: ∂ µ [ A µ ( ∂ · A )] ,∂ µ [ A ν ( ∂ µ A ν )] ,∂ µ [ A ν ( ∂ ν A µ )] . (33)Since these terms are of the form ∂ [ A a ( ∂A b )], once theinternal group indices have been added, they can becontracted only with a group metric. Thus, the totalderivatives we have been looking for are ∂ µ J µ = ∂ µ [ A µa ( ∂ · A a )] ,∂ µ J µ = ∂ µ [ A aν ( ∂ µ A νa )] ,∂ µ J µ = ∂ µ [ A aν ( ∂ ν A µa )] . (34)It is easy to see that these total derivatives, in theiractual form, are anyway useless, because they lead toterms involving second-order derivatives in addition tothe ones we are interested in which involve just two first-order derivatives. The only way to circumvent this sit-uation, at least partially but enough, is to construct thelinear combination ∂ µ ˜ J µ ≡ ∂ µ J µ − ∂ µ J µ = − A aµν A νµa + 14 ( L − L ) + A µa [ ∂ µ , ∂ ν ] A νa , (35)that removes the second-order derivatives, since thecommutator in the last line trivially vanishes in flatspacetime. Indeed, from this result and the findings inSections III and VI B, we can see that employing ∂ µ ˜ J µ is allowed, since it satisfies the Hessian constraints. TheLagrangian in Eq. (32) is, in consequence, already con-tained in L in flat spacetime up to a total derivative.Things, however, are different in curved spacetime. D. Covariantization
As is usual the case, the covariantization of Eq.(35) implies the replacement of partial derivatives withspace-time covariant derivatives and of the Minkowskimetric with an arbitrary space-time metric. Thus, thecurved spacetime version of Eq. (35) reads ∇ µ ˜ J µ = − A aµν A νµa + 14 ( L − L ) + A µa [ ∇ µ , ∇ ν ] A νa = − A aµν A νµa + 14 ( L − L ) − A µa R µν A νa , (36)where R µν is the Ricci tensor. Then, we can concludethat the Lagrangian in Eq. (32) is actually indepen-dent of L in a non-redundant way in curved spacetime,whereas it is already included in L in flat spacetime.To remind the reader of this fact, we will in the followingdeal with A µa R µν A νa instead of L − L . E. The decoupling limit
The Helmholtz theorem tells us that any vectorfield A µ can be decomposed into its transverse part,a divergence-free vector field A µ , and its longitudinalpart, the gradient of scalar field ∇ µ π : A µ = A µ + ∇ µ π . (37)The decoupling limit of GSU2P, understood as an ef-fective field theory, which corresponds in this case tothe replacement A aµ → ∇ µ π a , must also be a healthytheory; i.e., it must be free of the Ostrogradski insta-bility. Examining the term A µa R µν A νa , we can observethat its decoupling limit ∇ µ π a R µν ∇ ν π a is not healthy, as the field equation resultant of the variation of theaction with respect to π a leads to a term proportionalto ∇ µ R µν , i.e., a higher-order term. To avoid such apathological behaviour (see Ref. [34]), it is necessary toadd − R ( A a · A a ) / R being the Ricciscalar: L , = A µa R µν A νa − R ( A a · A a )= G µν A µa A νa , (38)where G µν is the Einstein tensor. Indeed, this La-grangian is healthy in the decoupling limit becauseof the divergenceless character of G µν . Our conclu-sion, different than the one encountered in Ref. [103],where no term with just two derivatives was found while G µν A µa A νa was just postulated, finds its origin in thefact that the total derivative in Eq. (35) was first co-variantized and later employed (not) to dismiss someterms in favour of others. This way of proceeding wasidentified in Ref. [111], and it is the mechanism to un-cover the beyond SU(2) Proca terms as we will latersee. To finish, the notation L , is introduced in Eq.(38) to label this Lagrangian as one that involves (orcomes from) two derivatives (this is the reason for the4) and no vector fields (this is the reason for the 0). VII. TWO DERIVATIVES AND TWO VECTORFIELDSA. Lagrangian building blocks
Lagrangian building blocks built from two derivativesand two vector fields are terms of the form A µν S ρσ A α A β or S µν S ρσ A α A β that involve six space-time indices. Inorder to uncover them, we must contract with all theterms in Eqs. (13) and (14). As a result, the Lagrangianbuilding blocks that either do not vanish or have the po-tential of becoming different than zero once the internalgroup indices are added are the following: A µν S µσ A ν A σ ,A µν S ρρ A µ A ν ,A µν S νσ A α A β ǫ µσαβ , ( ∗ ) A µν S ρρ A α A β ǫ µναβ , ( ∗ ) A µν S ρσ A ρ A β ǫ µνσβ ,S µµ S ρρ ( A · A ) ,S µµ S ρσ A ρ A σ ,S µν S µν ( A · A ) ,S µν S µσ A ν A σ ,S µν S νσ A α A β ǫ µσαβ . ( ∗ ) (39)When the internal indices are added, these terms areof the form A a {} S b A c A d or S a S b A c A d ; i.e., they involvefour internal group indices. So, in order to obtain theLagrangian building blocks, and according to group the-ory [114–116], we must contract with the following prod-ucts of two group metrics: g ab g cd ,g ac g bd ,g ad g bc . (40)This results in the following nineteen Lagrangian build-ing blocks linearly independent of L : L = A aµν S µσa A νc A σc , L = A aµν S µbσ A νa A σb , L = A aµν S µbσ A νb A σa , L = A aµν S ρbρ A µa A νb , L = A aµν S νbσ A αa A βb ǫ µσαβ , L = A aµν S ρbρ A αa A βb ǫ µναβ , L = A aµν S ρσa A ρc A βc ǫ µνσβ , L = A aµν S bρσ A ρa A βb ǫ µνσβ , L = A aµν S bρσ A ρb A βa ǫ µνσβ , L = S µaµ S ρρa ( A c · A c ) , L = S µaµ S ρbρ ( A a · A b ) , L = S µaµ S ρσa A ρc A σc , L = S µaµ S bρσ A ρa A σb , L = S aµν S µνa ( A c · A c ) , L = S aµν S µνb ( A a · A b ) , L = S aµν S µσa A νc A σc , L = S aµν S µbσ A νa A σb , L = S aµν S µbσ A νb A σa , L = S aµν S νbσ A αa A βb ǫ µσαβ . (41) B. The Hessian constraints
The Lagrangian is written as a linear combination ofthe Lagrangian building blocks found in the previoussection. Thus, L = X i =1 x i L i , (42)where the x i are arbitrary constants. Since the La-grangian involves two derivatives and two vector fields,none of the Hessian constraints is trivially satisfied in this case. Performing the calculations, we find for theprimary Hessian: H νab = − A c A νc g ab ( x + 2 x + 2 x ) − A c A c g ν g ab ( x − x − x ) − A a A νb ( x − x + x + 2 x ) − A a A b g ν ( x + x − x − x − x ) − A b A νa ( x + x + x + 2 x ) − ǫ ναβ A αb A βa ( x + 2 x − x ) − A c · A c ) g ν g ab ( x + x ) − A a · A b ) g ν ( x + x ) , (43)whereas for the secondary Hessian:˜ H ab = − A α [ b | A α | a ] ( x − x − x ) − A α [ b | S α | a ] (2 x − x + 2 x − x ) − ǫ βα σ A σ [ a | A βα | b ] ( x − x + x )+2 A a | S αα | b ] (4 x − x + 2 x − x ) − A a S b ] (2 x − x + x − x ) . (44)Both expressions vanish, therefore, only when the fol-lowing eleven constraints are satisfied: x = 0 ,x = − x ,x = x ,x = − x + x ,x = 4 x − x + 2 x ,x = − x ,x = − x ,x = − x ,x = − x + x − x ,x = − x + x − x ,x = x x . (45)Thus, the Lagrangian that satisfies the constraint alge-bra is given by L = x ( L − L + L ) + x (cid:18) L + L (cid:19) + x ( L − L + L ) + x ( L + L )+ x L + x ( L + 4 L − L − L − L )+ x ( L − L − L + L + L )+ x ( L + 2 L − L − L − L ) . (46) C. Total derivatives
With the purpose of establishing which of the La-grangian pieces in Eq. (46) are redundant, the total0derivatives of terms involving one derivative and threevector fields must be constructed. These derivatives areterms of the form ∂ µ [ A ν ( ∂ ρ A σ ) A α A β ] that involve sixspace-time indices, so contractions with the terms inEqs. (13) and (14) must be done. As a result, the onlyterms that either are different than zero or have thepotential of becoming so after introducing the internalgroup indices are the following: ∂ µ [ A µ ( ∂ · A )( A · A )] ,∂ µ [ A µ ( ∂ ρ A σ ) A ρ A σ ] ,∂ µ [ A ν ( ∂ µ A ν )( A · A )] ,∂ µ [ A ν ( ∂ ν A µ )( A · A )] ,∂ µ [ A ν ( ∂ ν A σ ) A α A β ] ǫ µσαβ , ( ∗ ) ∂ µ [ A ν ( ∂ ρ A ν ) A α A β ] ǫ µραβ , ( ∗ ) ∂ µ [( ∂ ρ A σ ) A β ( A · A )] ǫ µρσβ ,∂ µ [( ∂ · A ) A ν A α A β ] ǫ µναβ . ( ∗ ) (47)These total derivatives become terms of the form ∂ [ A a ( ∂A b ) A c A d ] once the internal group indices areadded. Since they involve four internal group indices,contractions with the terms in Eq. (40) are needed,which results in ∂ µ J µ = ∂ µ [ A µa ( ∂ · A a )( A c · A c )] ,∂ µ J µ = ∂ µ [ A µa ( ∂ · A b )( A a · A b )] ,∂ µ J µ = ∂ µ [ A µa ( ∂ ρ A σa ) A ρc A σc ] ,∂ µ J µ = ∂ µ [ A µa ( ∂ ρ A bσ ) A ρa A σb ] ,∂ µ J µ = ∂ µ [ A µa ( ∂ ρ A bσ ) A ρb A σa ] ,∂ µ J µ = ∂ µ [ A aν ( ∂ µ A νa )( A c · A c )] ,∂ µ J µ = ∂ µ [ A aν ( ∂ µ A νb )( A a · A b )] ,∂ µ J µ = ∂ µ [ A aν ( ∂ ν A µa )( A c · A c )] ,∂ µ J µ = ∂ µ [ A aν ( ∂ ν A µb )( A a · A b )] ,∂ µ J µ = ∂ µ [ A aν ( ∂ ν A bσ ) A αa A βb ] ǫ µσαβ ,∂ µ J µ = ∂ µ [ A aν ( ∂ ρ A νb ) A αa A βb ] ǫ µραβ ,∂ µ J µ = ∂ µ [( ∂ ρ A aσ ) A βa ( A c · A c )] ǫ µρσβ ,∂ µ J µ = ∂ µ [( ∂ ρ A aσ ) A bβ ( A a · A b )] ǫ µρσβ . (48)All these total derivatives are useless as long as they pro-duce terms with second-order derivatives. Fortunately,this circumstance can be redeemed, although not in all the cases, by building the following linear combinations: ∂ µ ˜ J µ ≡ ∂ µ J µ − ∂ µ J µ = −
14 [ A aµν A νµa ( A c · A c ) + 2 A aν A νµa A cµρ A ρc ]+ 14 ( L − L + 2 L − L − L )+ A µa [ ∂ µ , ∂ ν ] A νa ( A c · A c ) ,∂ µ ˜ J µ ≡ ∂ µ J µ − ∂ µ J µ = −
14 [ A aµν A νµb ( A a · A b ) + A aν A νµb A ρµ a A ρb + A aν A νµb A µρb A ρa ]+ 14 ( L + L + L − L − L − L + L )+ A µa [ ∂ µ , ∂ ν ] A νb ( A a · A b ) ,∂ µ ˜ J µ ≡ ∂ µ J µ − ∂ µ J µ = −
14 [ A µa A cρσ A ρµ c A σa + A µa A cρσ A σµ a A ρc ]+ 14 ( L + 2 L − L − L −L − L − L + 2 L )+ A µa [ ∂ µ , ∂ ρ ] A σa A ρc A σc , (49)while in the following cases the problem is automaticallysolved thanks to the symmetries of the Levi-Civita ten-sor: ∂ µ ˜ J µ ≡ ∂ µ J µ =14 [ A aµν A νbρ A αa A βb + A νbρ A µαa A aν A βb + A νbρ A µβb A aν A αa ] ǫ µραβ + 14 ( L + 2 L − L + L )+ 12 A aν [ ∂ µ , ∂ ρ ] A νb A αa A βb ǫ µραβ ,∂ µ ˜ J µ ≡ ∂ µ J µ =14 [ A aρσ A µβa ( A c · A c ) + 2 A aρσ A cµα A βa A αc ] ǫ µρσβ + 12 L + 12 A βa [ ∂ µ , ∂ ρ ] A aσ ( A c · A c ) ǫ µρσβ ,∂ µ ˜ J µ ≡ ∂ µ J µ =14 [ A aρσ A bµβ ( A a · A b ) + A aρσ A αµ a A bβ A αb + A aρσ A µαb A bβ A αa ] ǫ µρσβ + 14 ( L + L )+ 12 A bβ [ ∂ µ , ∂ ρ ] A aσ ( A a · A b ) ǫ µρσβ . (50)1However, even like this, these total derivatives continueto be useless unless they satisfy the Hessian constraints.Comparison of these expressions with Eqs. (43)-(44)and with the findings in Section III reveals that thefollowing linear combinations are the only ones that passthe test: ∂ µ ( ˜ J µ + 3 ˜ J µ ) ,∂ µ (2 ˜ J µ − J µ ) ,∂ µ ˜ J µ ,∂ µ ˜ J µ . (51)We have now the four total derivatives that will help usremove some redundant terms from Eq. (46). However,covariantization must be performed first. D. Covariantization
The minimal covariantization scheme described inSection VI D and applied to the total derivatives of Eq.(51) produces the curved space-time versions ∇ µ ( ˜ J µ + 3 ˜ J µ ) =( ... ∈ L )+ 12 ( L − L + L )+ 34 ( L − L − L + L + L )+( L + 2 L − L − L − L )+ A µa R α σρµ A αa A ρc A σc − A µa R µα A αb ( A a · A b ) , ∇ µ (2 ˜ J µ − J µ ) =( ... ∈ L )+( L − L + L )+ 12 ( L + 2 L − L − L − L ) −
34 ( L + 4 L − L − L − L )+2 A µa R α σρµ A αa A ρc A σc + 3 A µa R µα A αa ( A c · A c ) , ∇ µ ˜ J µ =( ... ∈ L )+ 12 L + 12 A βa R α σρµ A aα ( A c · A c ) ǫ µρσβ , ∇ µ ˜ J µ =( ... ∈ L )+ 14 ( L + L )+ 12 A bβ R α σρµ A aα ( A a · A b ) ǫ µρσβ , (52) where ( ... ∈ L ) means terms belonging to L and R α σρµ is the Riemann tensor. We see, therefore, thatsome terms in Eq. (46) can be dismissed in flat space-time but not in curved spacetime. Indeed, to remind thereader of this difference, these terms will be traded bytheir respective curvature-dependent companions thatappear in the total derivatives in Eq. (52): L = ( x + 2 x − x )( L − L + L )+ x L + L )+ x ( L − L + L ) − x A bβ R α σρµ A aα ( A a · A b ) ǫ µρσβ − x A βa R α σρµ A aα ( A c · A c ) ǫ µρσβ + (cid:18) x − x + 32 x (cid:19) ( L + 4 L − L − L − L ) − x [ A µa R α σρµ A αa A ρc A σc − A µa R µα A αb ( A a · A b )]+ (cid:18) x − x (cid:19) [2 A µa R α σρµ A αa A ρc A σc +3 A µa R µα A αa ( A c · A c )] . (53) E. Change of basis
There are eight linear independent Lagrangian piecesin Eq. (53) which form a basis set for the construction ofthe Lagrangian involving two derivatives and two vectorfields. For purposes that will be clear in the followingsection, we will perform a change of basis that will affectthe third and sixth to eighth Lagrangian basis elementsin Eq. (53): L − L + L → L − L L
2= 12 (2 L + L ) −
12 ( L − L + L ) , L + 4 L − L − L − L →
14 ( L − L + 2 L − L )= ( ... ∈ L )+( L − L + L ) −
34 ( L + 4 L − L − L − L )+ 23 n ∇ µ ( ˜ J µ + 3 ˜ J µ ) − [ A µa R α σρµ A αa A ρc A σc − A µa R µα A αb ( A a · A b )] o − n ∇ µ (2 ˜ J µ − J µ ) − [2 A µa R α σρµ A αa A ρc A σc +3 A µa R µα A αa ( A c · A c )] o ,A µa R α σρµ A αa A ρc A σc − A µa R µα A αb ( A a · A b ) → A µa R α σρµ A αa A ρc A σc − A µa R µα A αb ( A a · A b )+˜ a (cid:20)
14 ( L − L + 2 L − L ) (cid:21) , A µa R α σρµ A αa A ρc A σc + 3 A µa R µα A αa ( A c · A c ) → A µa R α σρµ A αa A ρc A σc + 3 A µa R µα A αa ( A c · A c )+˜ b (cid:20)
14 ( L − L + 2 L − L ) (cid:21) , (54)where ˜ a and ˜ b are arbitrary constants. Thus, the La-grangian involving two derivatives and two vector fieldsis written as follows: L = X i =1 ˆ α i ˆ L i , (55)withˆ L = 14 ( L − L + 2 L − L ) , ˆ L = L − L + L , ˆ L = A µa R α σρµ A αa A ρc A σc − A µa R µα A αb ( A a · A b )+˜ a (cid:20)
14 ( L − L + 2 L − L ) (cid:21) , ˆ L = 2 A µa R α σρµ A αa A ρc A σc + 3 A µa R µα A αa ( A c · A c )+˜ b (cid:20)
14 ( L − L + 2 L − L ) (cid:21) , ˆ L = 2 L + L , ˆ L = L − L L , ˆ L = A bβ R α σρµ A aα ( A a · A b ) ǫ µρσβ , ˆ L = A βa R α σρµ A aα ( A c · A c ) ǫ µρσβ , (56) where the ˆ α i are arbitrary constants. We have delib-erately ordered the Lagrangian pieces this way so thatthe first four are the ones that preserve parity while thelast four, in contrast, are the ones that do not preserveit. F. The decoupling limit
Following the general description of Section VI E,the decoupling limit of the theory described by Eqs.(55) and (56), obtained by making the replacement A aµ → ∇ µ π a , must be free of the Ostrogradski insta-bility. This is easy to verify for ˆ L and ˆ L whose de-coupling limits vanish thanks to the antisymmetry of A aµν . It is also easy to verify for ˆ L and ˆ L having inmind their relation to ∇ µ ˜ J µ and ∇ µ ˜ J µ , respectively,as shown in Eq. (52), and, again, the antisymmetry of A aµν . Now, regarding ˆ L , its decoupling limit leads tohigher-order field equations, because, contrary to par-tial derivatives, covariant derivatives do not commute.This can be redeemed by adding a specific countertermso that the healthy version of ˆ L becomes:ˆ L ,h = 14 ( A b · A b )[ S µaµ S ννa − S µaν S νµa − R ( A a · A a )]+ 12 ( A a · A b )[ S µaµ S νbν − S µaν S νbµ − R ( A a · A b )] . (57)In contrast, although the decoupling limit of ˆ L , specif-ically the term L , leads as well to higher-order fieldequations, it turned out impossible to find out the re-quired counterterm . This leaves us with two possibil-ities: either we must discard ˆ L , as it is pathological inthe decoupling limit, or we must keep it, because its de-coupling limit is degenerate and this property might, inprinciple, remove the ghostly degree of freedom [42, 43].We will not know which possibility is the right one untila proper and dedicated analysis of the degeneracy con-ditions in the decoupling limit is performed . Finally,ˆ L and ˆ L are the non-Abelian versions of a term inthe generalized Proca theory identified unequivocally inRef. [111] as the beyond Proca term [112]. We con-jecture then that ˆ L and ˆ L are the beyond generalizedSU(2) Proca terms whose decoupling limits must satisfyall the conditions required to remove the Ostrogradski The isolation of L in just one Lagrangian piece is motivatedby the impossibility of finding out a counterterm, and it is thereason of the first change in basis elements shown in the previ-ous section. This seems quite non trivial, so we rather leave it for futurework. a and ˜ b constants, but, since thenon-Abelian extension of the beyond multi-Galileon the-ory has not been constructed yet, the actual values of˜ a and ˜ b are unknown to us. To circumvent this lack ofknowledge, we can take advantage of the fact that, al-though the Abelian and non-Abelian vector-tensor theo-ries are different despite sharing many aspects in theirconstruction, the non-Abelian theory stripped of the in-ternal group indices must be contained in the Abeliantheory. Thus, once ˆ L is stripped of the internal groupindices, it becomesˆ L → − A µ R µα A α A + ˜ a A ( S µµ S νν − S µν S νµ ) , (58)which must be compared with Eq. (42) in Ref. [111]: L BP = G N ( X ) R µν A µ A ν − [2 XG N,X ( X ) + G N ( X )] 14 ( S µµ S νν − S µν S νµ ) , (59)where X = − A / G N ( X ) is an arbitrary function of X , and G N,X ( X ) is the derivative of G N ( X ) with re-spect to X . We see that these two Lagrangian pieces areequivalent for G N ( X ) = 6 X and ˜ a = 3. Similarly, onceˆ L is stripped of the internal group indices, it becomesˆ L → A µ R µα A α A + ˜ b A ( S µµ S νν − S µν S νµ ) , (60)which is equivalent to the Lagrangian piece in Eq. (59)for G N ( X ) = − X and ˜ b = − G. A new change of basis
Having found the actual values for ˜ a and ˜ b in theprevious section, ˆ L and ˆ L acquire the formˆ L = A µa R α σρµ A αa A ρc A σc − A µa R µα A αb ( A a · A b )+3 h
14 ( A b · A b )( S µaµ S ννa − S µaν S νµa )+ 12 ( A a · A b )( S µaµ S νbν − S µaν S νbµ ) i , ˆ L = 2 A µa R α σρµ A αa A ρc A σc + 3 A µa R µα A αa ( A c · A c ) − h
14 ( A b · A b )( S µaµ S ννa − S µaν S νµa )+ 12 ( A a · A b )( S µaµ S νbν − S µaν S νbµ ) i , (61) Abelian theories display some terms whose non-Abelian ver-sions do not exist and vice versa. This is the reason of the third and fourth changes in basis ele-ments shown in the previous section. which can be replaced byˆ L → A µa R α σρµ A αa A ρc A σc + 34 ( A b · A b )( A a · A a ) R = ˆ L − L ,h + 3 G µν A µa A νb ( A a · A b ) , ˆ L → A µa R α σρµ A αa A ρc A σc + 34 [( A b · A b )( A a · A a ) − A a · A b )( A a · A b )] R = ˆ L + 3 ˆ L ,h − G µν A µa A νa ( A b · A b ) , (62)where we have added and subtracted, respectively,the Lagrangian pieces G µν A µa A νb ( A a · A b ) and G µν A µa A νa ( A b · A b ) that exist only in curved spacetimeand whose decoupling limit is healthy, since G µν is di-vergenceless. Furthermore, we can replace the secondLagrangian piece in the previous expression as follows:ˆ L + 3 ˆ L ,h − G µν A µa A νa ( A b · A b ) → −
34 [( A b · A b )( A a · A a ) + 2( A a · A b )( A a · A b )] R = ˆ L + 3 ˆ L ,h − G µν A µa A νa ( A b · A b ) −
2[ ˆ L − L ,h + 3 G µν A µa A νb ( A a · A b )] , (63)which is indeed very interesting, because now ˆ L ,h canbe replaced byˆ L ,h → n ( A b · A b )[ S µaµ S ννa − S µaν S νµa ]+2( A a · A b )[ S µaµ S νbν − S µaν S νbµ ] o = 3 ˆ L ,h + 34 [( A b · A b )( A a · A a )+2( A a · A b )( A a · A b )] R , (64)this just being the original L , i.e., without its respectivecounterterm.All together, we can formulate the reconstructedGSU2P Lagrangian composed of two derivatives andtwo vector fields as follows: L , = X i =1 α i m P L i , + X i =1 ˜ α i m P ˜ L i , , (65)4where L , =( A b · A b )[ S µaµ S ννa − S µaν S νµa ]+ 2( A a · A b )[ S µaµ S νbν − S µaν S νbµ ] , L , = A aµν S µbσ A νa A σb − A aµν S µbσ A νb A σa + A aµν S ρbρ A µa A νb , L , = A µa R α σρµ A αa A ρb A σb + 34 ( A b · A b )( A a · A a ) R , L , =[( A b · A b )( A a · A a ) + 2( A a · A b )( A a · A b )] R , L , = G µν A µa A νa ( A b · A b ) , L , = G µν A µa A νb ( A a · A b ) , (66)˜ L , = − A aµν S µbσ A αa A βb ǫ νσαβ + S aµν S νbσ A αa A βb ǫ µσαβ , ˜ L , = A aµν S µbσ A αa A βb ǫ νσαβ − ˜ A αβa S bρα A ρa A βb + ˜ A αβa S ρρb A aα A bβ , ˜ L , = A bβ R α σρµ A aα ( A a · A b ) ǫ µρσβ , ˜ L , = A βa R α σρµ A aα ( A b · A b ) ǫ µρσβ , (67)the α i and ˜ α i being arbitrary dimensionless constants, m P being the reduced Planck mass, ˜ A µνa ≡ ǫ µνρσ A ρσa being the Hodge dual of A aµν , and the Lagrangian pieceshaving been deliberately split into those that preserveparity (the ones without a tilde) and those that do notpreserve it (the ones with a tilde). It is worthwhilementioning that the subscripts 4,2 have been introducedto remind the reader that two derivatives and two vec-tor fields have been employed to build the different La-grangian pieces. VIII. THREE DERIVATIVESA. Lagrangian building blocks
Terms of the form A µν A ρσ S αβ , A µν S ρσ S αβ , and S µν S ρσ S αβ , that involve six space-time indices, are theones that become the Lagrangian building blocks of aLagrangian built with just three derivatives once theyare contracted with the terms in Eqs. (13) and (14).Upon the contractions, the only blocks that either donot vanish or have the potential of becoming non van-ishing once the internal group indices are introduced arethe following: A µν A µν S αα ,A µν A µ σ S νσ ,A µν A ρσ S νβ ǫ µρσβ ,A µν A ρσ S αα ǫ µνρσ ,A µν S µσ S νσ , ( ∗ ) A µν S ρσ S ρβ ǫ µνσβ , ( ∗ ) S µµ S ρρ S αα ,S µµ S ρσ S ρσ ,S µν S µσ S νσ . (68)The introduction of the internal group indices makesthese terms become of the form A a {} A b {} S c , A a {} S b S c , or S a S b S c , involving three internal group indices, whichlead to group-invariant Lagrangian building blocks uponcontractions with ǫ abc . Most of these blocks, however,vanish because of the antisymmetric nature of ǫ abc , theonly survivals being L = A aµν A bρσ S νcβ ǫ µρσβ ǫ abc , L = A aµν S µbσ S νσc ǫ abc , L = A aµν S bρσ S ρcβ ǫ µνσβ ǫ abc . (69) B. The Hessian constraints
The linear combination L = X i =1 x i L i , (70)where the x i are arbitrary constants and the L i arethe ones in Eq. (69), makes the GSU2P Lagrangianbuilt with just three derivatives. Because no singlevector field appears in this Lagrangian, the secondaryconstraint-enforcing relation is trivially satisfied. Re-garding the primary constraint-enforcing relation, theprimary Hessian gives the following result: H νab = 2 A cρσ ǫ νρσ ǫ bca ( x + 2 x )+4( S νc + g ν S c − A νc ) ǫ abc x , (71)5which vanishes only if x + 2 x = 0 ,x = 0 . (72)The Lagrangian that satisfies the constraint algebra is,therefore, L = x ( − L + L ) . (73) C. Total derivatives
As with the other Lagrangians involving a differentnumber of derivatives and/or vector fields, we mustbe sure that the Lagrangian in Eq. (73) is not re-dundant compared with terms in L . To this end, wemust construct total derivatives of terms involving twoderivatives and one vector field, i.e., total derivatives ofthe form ∂ µ [ A ν ( ∂ ρ A σ )( ∂ α A β )]. These terms involve sixspace-time indices, so that they must be contracted withthose terms in Eq. (13) and (14). However, since theLagrangian in Eq. (73) does not preserve parity, it willbe enough to contract with the terms in Eq. (14). Thus,the only terms that either are non vanishing or can be-come non vanishing once the internal group indices areadded are the following: ∂ µ [ A ν ( ∂ ν A σ )( ∂ α A β )] ǫ µσαβ ,∂ µ [ A ν ( ∂ ρ A ν )( ∂ α A β )] ǫ µραβ ,∂ µ [ A ν ( ∂ · A )( ∂ α A β )] ǫ µναβ ,∂ µ [ A ν ( ∂ ρ A σ )( ∂ ρ A β )] ǫ µνσβ , ( ∗ ) ∂ µ [ A ν ( ∂ ρ A σ )( ∂ α A ρ )] ǫ µνσα ,∂ µ [ A ν ( ∂ ρ A σ )( ∂ α A σ )] ǫ µνρα , ( ∗ ) (74)which, in turn, can be contracted only with ǫ abc afteradding the internal group indices, since the total deriva-tives acquire the form ∂ [ A a ( ∂A b )( ∂A c )]: ∂ µ J µ = ∂ µ [ A aν ( ∂ ν A bσ )( ∂ α A cβ )] ǫ µσαβ ǫ abc ,∂ µ J µ = ∂ µ [ A aν ( ∂ ρ A νb )( ∂ α A cβ )] ǫ µραβ ǫ abc ,∂ µ J µ = ∂ µ [ A aν ( ∂ · A b )( ∂ α A cβ )] ǫ µναβ ǫ abc ,∂ µ J µ = ∂ µ [ A aν ( ∂ ρ A bσ )( ∂ ρ A cβ )] ǫ µνσβ ǫ abc ,∂ µ J µ = ∂ µ [ A aν ( ∂ ρ A bσ )( ∂ α A ρc )] ǫ µνσα ǫ abc ,∂ µ J µ = ∂ µ [ A aν ( ∂ ρ A bσ )( ∂ α A σc )] ǫ µνρα ǫ abc . (75)As the reader has already learned, these total derivativesare completely useless unless the second derivatives theyproduce may be canceled out. After a careful observa-tion of these terms, only two are able by themselves toget rid of the second derivatives in flat spacetime thanks to the antisymmetry of the Levi-Civita tensor: ∂ µ ˜ J µ ≡ ∂ µ J µ =18 A aµν A νbρ A cαβ ǫ µραβ ǫ abc + 18 ( − L + L )+ 14 { A aν [ ∂ µ , ∂ ρ ] A νb A cαβ + A aν A νbρ [ ∂ µ , ∂ α ] A cβ + A aν S νbρ [ ∂ µ , ∂ α ] A cβ } ǫ µραβ ǫ abc ,∂ µ ˜ J µ ≡ ∂ µ J µ =18 A aµν A bρσ A σcα ǫ µνρα ǫ abc + 18 ( − L + L )+ 12 { A aν [ ∂ µ , ∂ ρ ] A bσ A σcα + A aν [ ∂ µ , ∂ ρ ] A bσ S σcα } ǫ µνρα ǫ abc . (76)Indeed, from this result and the findings in SectionsIII and VIII B, we can see that employing either ∂ µ ˜ J µ or ∂ µ ˜ J µ is allowed, since they satisfy the Hessian con-straints. The conclusion is that the Lagrangian in Eq.(73) is already contained in L in flat spacetime, upto a total derivative, so that, in this framework, theGSU2P does not contain terms built exclusively withthree derivatives that are linearly independent of L .The conclusion is, nonetheless, completely different incurved spacetime. D. Covariantization
The minimal covariantization scheme applied to thesuitable combination ∂ µ (2 ˜ J µ + ˜ J µ ) of terms in Eq. (76)leads to ∇ µ (2 ˜ J µ + ˜ J µ ) = ( ... ∈ L )+ 38 ( − L + L )+ 12 A νa R σ νρµ A bσ A cαβ ǫ µραβ ǫ abc . (77)The Lagrangian in Eq. (73) is, therefore, not redun-dant against L in curved spacetime. As a remainderof this fact, we will dismiss − L + L in favour of A νa R σ νρµ A bσ A cαβ ǫ µραβ ǫ abc . We conclude then that thereconstructed GSU2P exhibits the following Lagrangianbuilt from just three derivatives:˜ L , = A νa R σ νρµ A bσ ˜ A µρc ǫ abc . (78)6 E. The decoupling limit
Since the Lagrangian given in the previous expressionvanishes in the decoupling limit A aµ → ∇ µ π a , because ofthe antisymmetry of ˜ A aµν , it is free of the Ostrogradskiinstability. IX. COMPARISON WITH THE “OLD” GSU2P
The old GSU2P, formulated in Ref. [103], is describedby the following Lagrangian: L old = L old2 + X i =1 α i L i, old4 + X i = i β i L i, oldCurv , (79)where the α i and β i are dimensionful arbitrary con-stants, L old2 ≡ L old2 ( A aµν , A aµ ) is an arbitrary functionof A aµν and A aµ , and L , old4 = ( A b · A b )[ S µaµ S ννa − S µaν S νµa − R ( A a · A a )]+2( A a · A b )[ S µaµ S νbν − S µaν S νbµ − R ( A a · A b )] , L , old4 = ( A a · A b )[ S µaµ S νbν − S µaν S νbµ − R ( A a · A b )]+ A aµ A bν [ S µαa S ναb − S µαb S ναa +2 A µαa S ναb − A µαb S ναa + 2 A ρa A σb R µνρσ ] , L , old4 = A µa ˜ A bµσ S σνa A νb , L , oldCurv = G µν A µa A νa , L , oldCurv = L µνρσ A µνa A ρσa , L , oldCurv = L µνρσ A µνa A ρb A σc ǫ abc , L , oldCurv = L µνρσ A µa A νb A ρa A σb , (80)where L µνρσ ≡ ǫ µναβ ǫ ρσγδ R αβγδ is the double dual ofthe Riemann tensor. This old theory was built followingthe same steps that we followed here except for threeaspects:1. All the Lagrangian building blocks were con-structed employing the full ∂ µ A aν instead of split-ting it into its symmetric S aµν and antisymmetric A aµν parts. This, of course, produced a lot moreblocks (and a lot more work) than needed, manylinear combinations of them already included in L .2. Only the primary constraint-enforcing relationwas considered. As was shown in Refs. [109, 110],this is not enough to remove the Ostrogradskighost.3. Many terms were dismissed by employing totalderivatives already at the flat spacetime level,which led to a loss of several terms that exists only in curved spacetime, including the beyondSU(2) Proca ones. Moreover, most of the totalderivatives employed do not satisfy the secondaryHessian constraint.The application of this theory to inflation and dark en-ergy was investigated in Refs. [106, 107], and the stabil-ity analysis of the same was performed in Ref. [108], sowe wonder how the results of these works could changein the light of the new theory presented in this paper.As can be seen, our ˆ L ,h in Eq. (57) is identical to L , old4 , this being one of the reasons of the second changein basis elements in Section VII E. Examined from theviewpoint of the reconstructed GSU2P [see Eqs. (65)-(66)], L , old4 can also be written as L , old4 = L , − L , . (81)Thus, we conclude that L , old4 is free of the Ostrogradskighost (at least in flat spacetime).Now, L , old4 was shown in Ref. [110] not to satisfy thesecondary Hessian constraint and, so, as an exampleof the ghost instabilities that plagued the old GSU2P.Nevertheless, a bit of algebra shows us that L , old4 − ∂ µ ˜ J µ = ( ... ∈ L )+ 112 L , − L , − ∂ µ ( ˜ J µ + 3 ˜ J µ ) + 19 ∂ µ (2 ˜ J µ − J µ ) , (82)at the flat space-time level, where the quantities in thisexpression, except for L , old4 , are those of Section VII.Thus, although neither L , old4 is healthy, nor ∂ µ ˜ J µ is,the combination L , old4 − ∂ µ ˜ J µ satisfies the secondaryconstraint-enforcing relation, and, therefore, all thephysics extracted from the unhealthy curved space-timeversion of L , old4 , for instance, in Ref. [108], is equivalentto that extracted from the healthy L , old4 − ∇ µ ˜ J µ .Something similar occurs for L , old4 : L , old4 + 2 ∂ µ ˜ J µ = ( ... ∈ L )+ 12 ( ˜ L , + 4 ∂ µ ˜ J µ ) , (83)at the flat space-time level, so although neither L , old4 nor ∂ µ ˜ J µ are healthy, the combination L , old4 + 2 ∂ µ ˜ J µ is, and, therefore, all the physics extracted from the un-healthy curved space-time version of L , old4 is equivalentto that extracted from the healthy L , old4 + 2 ∇ µ ˜ J µ .Now, as can be seen in Eq. (67), there exist onlytwo parity-violating terms in flat spacetime in the recon-structed GSU2P. Then, why is it that in the old GSU2P7there exists only one? Leaving aside the fact that ˜ L , might be unhealthy in its decoupling limit, the reasonlies in a small mistake in the conditions of Eq. (37)in Ref. [103] to make the primary constraint-enforcingrelation vanish that prevented the authors of that workfrom finding a second parity-violating Lagrangian piece.Finally, among the L oldCurv of Eq. (80), the only onethat appears in the reconstructed theory is L , oldCurv , whichis exactly the same as our L , of Eq. (38). Theother L oldCurv were just postulated, as they are obviouslyhealthy because of the divergenceless nature of L µνρσ .We could have postulated them as well in the recon-structed GSU2P, but we would rather not do it. Thisis because we expect them to naturally appear in thetheory when more than six space-time indices are con-sidered in the Lagrangian building blocks without con-tractions. X. COMPARISON WITH THE GENERALIZEDPROCA THEORY
Finding the beyond GSU2P in Section VII requireddetermining the values of the constants ˜ a and ˜ b in Eq.(56). We could have followed the standard procedureof finding out the kinetic matrix of its decoupling limitand making it degenerate [42, 43]. However, we fol-lowed an alternative route based on the fact that theGSU2P stripped of the internal group indices must becontained in the generalized Proca theory. Indeed, theother reason why we performed the second change inbasis elements in Section VII E is that ˆ L ,h stripped ofthe internal group indices is nothing else than L of thegeneralized Proca theory (see Ref. [65]): L = G ( X ) R + G ,X ( X )4 ( S µµ S νν − S µν S νµ ) , (84)for G ( X ) = − X . Then, what about the other La-grangian pieces that make L , and L , ? First of all, L , stripped of the internal group indices is just L ,up to a total derivative, with G ( X ) = X . In con-trast, L , and ˜ L , reduce to zero when stripped ofthe internal group indices. Regarding L , and L , without internal group indices, they are just healthy ex-tensions of G µν A µ A ν that were not recognized in Ref.[103]. Finally, ˜ L , , ˜ L , , and ˜ L , , stripped of theirinternal group indices, reduce, up to total derivatives,to A β ˜ A βα S αρ A ρ , which was shown in Refs. [106, 119]to be part of L up to a total derivative. To end up,the only parity-violating terms in the generalized Procatheory belong to L [66], so ˜ L , stripped of its internalgroup indices should be either zero, a total derivative,or contained in L ; in fact, observing Eq. (78), the firstalternative is the correct one. XI. CONCLUSIONS
GSU2P and beyond GSU2P are described by the La-grangians in Eq. (38), Eqs. (65)-(67), and Eq. (78).The theory has been written so as to make it explicitwhich Lagrangian pieces exist only in curved spacetimeand which ones exist even in flat spacetime; indeed, fromthe twelve Lagrangian pieces that compose the theory,only four, L , , L , , ˜ L , , and ˜ L , , survive in flat space-time. The nature of some of the Lagrangian pieces ispurely non-Abelian – i.e., they vanish when stripped oftheir internal group indices – specifically, L , , ˜ L , , and˜ L , belong to this subset. It is worthwhile mentioningthat ˜ L , is the parity-violating version of L , as can beeasily observed. On the other hand, the theory is diffeo-morphism invariant, so that the energy and momentumare locally conserved [113].Much remains to be done in the exploration of thistheory as a candidate of an effective theory for the grav-itational interaction. First of all, it is not clear yetwhether the decoupling limits of the beyond GSU2Pterms as well as that of ˜ L , are actually healthy .Other self-consistency issues must be addressed, suchas the possible existence of ghosts (other than the Os-trogradski one) and Laplacian instabilities, as a follow-up of the work in Ref. [108], the generalization of theconstraint algebra to curved spacetime [120, 121], theanalysis of the causal structure [122], and the calcula-tion of the cutoff scale of the theory and its comparisonwith the GW170817 event frequency [22] (to see whetherthe bound on the gravitational waves speed applies toGSU2P ). We might as well construct an extended ver-sion of this theory, considering all the possibilities to de-generate the kinetic matrix in curved spacetime, as wasdone for the generalized Proca theory in Refs. [119, 124].The theory must, of course, be put under test againstobservations; in this regard, determining whether thereexists a screening mechanism at Solar System scales, aswas studied in Ref. [78] for the generalized Proca theory,is a crucial aspect. Of course, the cosmological and as-trophysical implications must be properly studied bothat the background (see, for instance, Ref. [107]) andat the perturbative level (see, for instance, Refs. [125–127]). We finish this paper by reminding the readersand ourselves of one important message given to us by It is unlikely that the decoupling limits of the beyond GSU2Pterms are unhealthy: there is actually no reason to believe thathealthy beyond extensions do exist for the Horndeski theoryand the generalized Proca theory but do not for the GSU2P. Incontrast, there is no clue regarding the healthiness of ˜ L , , thisterm being of a purely non-Abelian nature. How the gravitational wave speed bound affects the generalizedProca theory was investigated in Ref. [123]. “To be complete, a theory of grav-ity must be capable of analyzing from ‘first principles’the outcome of every experiment of interest. It musttherefore mesh with and incorporate a consistent set oflaws for electromagnetism, quantum mechanics, and allother physics.”
There is a long road in this directionahead of us that we hope we will travel.
ACKNOWLEDGMENTS
We appreciate all the discussions and exchange ofideas we had with Juan Camilo Garnica Aguirre andCarlos Mauricio Nieto Guerrero which helped a lot in the development of this work. The work presented herewas supported by the following grants: Colciencias-Deutscher Akademischer Austauschdienst Grant No.110278258747 RC-774-2017, Vicerrector´ıa de Ciencia,Tecnolog´ıa, e Innovaci´on - Universidad Antonio Nari˜noGrant No. 2019248, and Direcci´on de Investigaci´on yExtensi´on de la Facultad de Ciencias - Universidad In-dustrial de Santander Grant No. 2460. A.G.C. wassupported by Beca de Inicio Postdoctoral REXE RANo. 315-3269-2020 Universidad de Valpara´ıso. L.G.G.was supported by the postdoctoral scholarship No.2020000102 Vicerrector´ıa de Investigaci´on y Extensi´on -Universidad Industrial de Santander. Some calculationswere cross-checked with the
Mathematica [1] C. M. Will,
Theory and Experiment in GravitationalPhysics (Cambridge University Press, 2018).[2] D. Psaltis et al. (Event Horizon Telescope),“Gravitational Test Beyond the First Post-Newtonian Order with the Shadow of the M87Black Hole,” Phys. Rev. Lett. , 141104 (2020),arXiv:2010.01055 [gr-qc].[3] K. Akiyama et al. (Event Horizon Tele-scope), “First M87 Event Horizon TelescopeResults. I. The Shadow of the Supermas-sive Black Hole,” Astrophys. J. , L1 (2019),arXiv:1906.11238 [astro-ph.GA].[4] B. P. Abbott et al. (LIGO Scientific, Virgo),“GW170817: Observation of GravitationalWaves from a Binary Neutron Star Inspi-ral,” Phys. Rev. Lett. , 161101 (2017),arXiv:1710.05832 [gr-qc].[5] B. P. Abbott et al. (LIGO Scientific, Virgo, FermiGBM, INTEGRAL, IceCube, AstroSat CadmiumZinc Telluride Imager Team, IPN, Insight-Hxmt,ANTARES, Swift, AGILE Team, 1M2H Team, DarkEnergy Camera GW-EM, DES, DLT40, GRAWITA,Fermi-LAT, ATCA, ASKAP, Las Cumbres Observa-tory Group, OzGrav, DWF (Deeper Wider FasterProgram), AST3, CAASTRO, VINROUGE, MAS-TER, J-GEM, GROWTH, JAGWAR, CaltechNRAO,TTU-NRAO, NuSTAR, Pan-STARRS, MAXI Team,TZAC Consortium, KU, Nordic Optical Telescope,ePESSTO, GROND, Texas Tech University, SALTGroup, TOROS, BOOTES, MWA, CALET, IKI-GW Follow-up, H.E.S.S., LOFAR, LWA, HAWC,Pierre Auger, ALMA, Euro VLBI Team, Pi of Sky,Chandra Team at McGill University, DFN, AT-LAS Telescopes, High Time Resolution Universe Sur-vey, RIMAS, RATIR, SKA South Africa/MeerKAT),“Multi-messenger Observations of a Binary NeutronStar Merger,” Astrophys. J. Lett. , L12 (2017),arXiv:1710.05833 [astro-ph.HE].[6] A. Goldstein et al. , “An Ordinary ShortGamma-Ray Burst with Extraordinary Im- plications: Fermi-GBM Detection of GRB170817A,” Astrophys. J. Lett. , L14 (2017),arXiv:1710.05446 [astro-ph.HE].[7] R. Abuter et al. (GRAVITY), “Detectionof the gravitational redshift in the orbit ofthe star S2 near the Galactic centre massiveblack hole,” Astron. Astrophys. , L15 (2018),arXiv:1807.09409 [astro-ph.GA].[8] T. E. Collett et al. , “A precise extragalactictest of General Relativity,” Science , 1342 (2018),arXiv:1806.08300 [astro-ph.CO].[9] J. M. Ezquiaga and M. Zumalac´arregui, “Dark En-ergy in light of Multi-Messenger Gravitational-Waveastronomy,” Front. Astron. Space Sci. , 44 (2018),arXiv:1807.09241 [astro-ph.CO].[10] J.-h. He, L. Guzzo, B. Li, and C. M.Baugh, “No evidence for modifications ofgravity from galaxy motions on cosmologi-cal scales,” Nature Astron. , 967–972 (2018),arXiv:1809.09019 [astro-ph.CO].[11] T. Do et al. , “Relativistic redshift of the starS0-2 orbiting the Galactic center supermas-sive black hole,” Science , 664–668 (2019),arXiv:1907.10731 [astro-ph.GA].[12] B. P. Abbott et al. (LIGO Scientific,Virgo), “Tests of General Relativity withGW170817,” Phys. Rev. Lett. , 011102 (2019),arXiv:1811.00364 [gr-qc].[13] M. Ishak, “Testing General Relativity inCosmology,” Living Rev. Rel. , 1 (2019),arXiv:1806.10122 [astro-ph.CO].[14] A. Kosteleck´y and Z. Li, “Backgrounds ingravitational effective field theory,” (2020),arXiv:2008.12206 [gr-qc].[15] C. P. Burgess, “Quantum gravity in everyday life:General relativity as an effective field theory,”Living Rev. Rel. , 5–56 (2004), arXiv:gr-qc/0311082.[16] J. F. Donoghue, “General relativity as an ef-fective field theory: The leading quantum cor-rections,” Phys. Rev. D , 3874–3888 (1994), arXiv:gr-qc/9405057.[17] R. Penrose, “Gravitational collapse and space-time sin-gularities,” Phys. Rev. Lett. , 57–59 (1965).[18] S. W. Hawking and R. Penrose, “The Singu-larities of gravitational collapse and cosmology,”Proc. Roy. Soc. Lond. A , 529–548 (1970).[19] R. Penrose, “Gravitational collapse: The role of gen-eral relativity,” Riv. Nuovo Cim. , 252–276 (1969).[20] R. Penrose, “Singularities and time-asymmetry,” in General Relativity: An Einstein Centenary Survey,(Hawking and Israel, editors) (1979) pp. 581–638.[21] R. Penrose, “The Question of Cosmic Censorship,”in
Black Holes and Relativistic Stars, (Wald, editor) (1979) pp. 581–638.[22] C. de Rham and S. Melville, “GravitationalRainbows: LIGO and Dark Energy at itsCutoff,” Phys. Rev. Lett. , 221101 (2018),arXiv:1806.09417 [hep-th].[23] T. S. Kuhn,
The structure of scientific revolutions (University of Chicago Press; (50th anniversary ed edi-tion - 2012), 1962).[24] L. Heisenberg, “A systematic approach to gener-alisations of General Relativity and their cosmo-logical implications,” Phys. Rept. , 1–113 (2019),arXiv:1807.01725 [gr-qc].[25] C. de Rham, “Massive Grav-ity,” Living Rev. Rel. , 7 (2014),arXiv:1401.4173 [hep-th].[26] M. Fierz and W. Pauli, “On relativistic wave equationsfor particles of arbitrary spin in an electromagneticfield,” Proc. Roy. Soc. Lond. A , 211–232 (1939).[27] C. de Rham, G. Gabadadze, and A. J.Tolley, “Resummation of Massive Grav-ity,” Phys. Rev. Lett. , 231101 (2011),arXiv:1011.1232 [hep-th].[28] D. Lovelock, “The Einstein tensor and its generaliza-tions,” J. Math. Phys. , 498–501 (1971).[29] D. Lovelock, “The four-dimensionality of space and theEinstein tensor,” J. Math. Phys. , 874–876 (1972).[30] C. Brans and R. H. Dicke, “Mach’s princi-ple and a relativistic theory of gravitation,”Phys. Rev. , 925–935 (1961).[31] G. W. Horndeski, “Second-order scalar-tensorfield equations in a four-dimensional space,”Int. J. Theor. Phys. , 363–384 (1974).[32] A. Nicolis, R. Rattazzi, and E. Trincherini,“The Galileon as a local modification ofgravity,” Phys. Rev. D , 064036 (2009),arXiv:0811.2197 [hep-th].[33] C. Deffayet, S. Deser, and G. Esposito-Far`ese, “Generalized Galileons: All scalarmodels whose curved background extensionsmaintain second-order field equations andstress-tensors,” Phys. Rev. D , 064015 (2009),arXiv:0906.1967 [gr-qc].[34] C. Deffayet, G. Esposito-Far`ese, and A. Vikman, “Co-variant Galileon,” Phys. Rev. D , 084003 (2009),arXiv:0901.1314 [hep-th].[35] C. Deffayet, X. Gao, D. A. Steer, andG. Zahariade, “From k-essence to generalised Galileons,” Phys. Rev. D , 064039 (2011),arXiv:1103.3260 [hep-th].[36] T. Kobayashi, M. Yamaguchi, and J. Yokoyama,“Generalized G-inflation: Inflation withthe most general second-order field equa-tions,” Prog. Theor. Phys. , 511–529 (2011),arXiv:1105.5723 [hep-th].[37] C. Deffayet and D. A. Steer, “A formal introductionto Horndeski and Galileon theories and their gen-eralizations,” Class. Quant. Grav. , 214006 (2013),arXiv:1307.2450 [hep-th].[38] T. Kobayashi, “Horndeski theory and beyond:a review,” Rept. Prog. Phys. , 086901 (2019),arXiv:1901.07183 [gr-qc].[39] M. Ostrogradski, “M´emoires sur les ´equationsdiff´erentielles, relatives au probl`eme desisop´erim`etres,” Mem. Acad. St. Petersbourg ,385–517 (1850).[40] R. P. Woodard, “Avoiding dark en-ergy with 1/r modifications of grav-ity,” Lect. Notes Phys. , 403–433 (2007),arXiv:astro-ph/0601672.[41] R. P. Woodard, “Ostrogradsky’s theorem on Hamil-tonian instability,” Scholarpedia , 32243 (2015),arXiv:1506.02210 [hep-th].[42] A. Ganz and K. Noui, “Reconsidering the Ostrograd-sky theorem: Higher-derivatives Lagrangians, Ghostsand Degeneracy,” (2020), arXiv:2007.01063 [hep-th].[43] D. Langlois and K. Noui, “Degenerate higherderivative theories beyond Horndeski: evadingthe Ostrogradski instability,” JCAP , 034 (2016),arXiv:1510.06930 [gr-qc].[44] D. Langlois and K. Noui, “Hamiltonian anal-ysis of higher derivative scalar-tensor theories,”JCAP , 016 (2016), arXiv:1512.06820 [gr-qc].[45] J. Gleyzes, D. Langlois, F. Piazza, andF. Vernizzi, “Healthy theories beyond Horn-deski,” Phys. Rev. Lett. , 211101 (2015),arXiv:1404.6495 [hep-th].[46] M. Zumalac´arregui and J. Garc´ıa-Bellido, “Transform-ing gravity: from derivative couplings to matter tosecond-order scalar-tensor theories beyond the Horn-deski Lagrangian,” Phys. Rev. D , 064046 (2014),arXiv:1308.4685 [gr-qc].[47] J. Ben Achour, D. Langlois, and K. Noui,“Degenerate higher order scalar-tensor theo-ries beyond Horndeski and disformal trans-formations,” Phys. Rev. D , 124005 (2016),arXiv:1602.08398 [gr-qc].[48] J. Ben Achour et al. , “Degenerate higher order scalar-tensor theories beyond Horndeski up to cubic order,”JHEP , 100 (2016), arXiv:1608.08135 [hep-th].[49] M. Crisostomi, K. Koyama, and G. Tasi-nato, “Extended Scalar-Tensor Theories of Gravity,”JCAP , 044 (2016), arXiv:1602.03119 [hep-th].[50] M. Crisostomi, K. Noui, C. Charmousis, and D. Lan-glois, “Beyond Lovelock gravity: Higher deriva-tive metric theories,” Phys. Rev. D , 044034 (2018),arXiv:1710.04531 [hep-th]. [51] M. Blagojevi´c, Gravitation and gauge symmetries (In-stitute of Physics Publishing, 2002).[52] M. Blagojevi´c and F. W. Hehl, eds.,
Gauge Theoriesof Gravitation: A Reader with Commentaries (WorldScientific, Singapore, 2013).[53] J. Beltr´an Jim´enez, L. Heisenberg, and T. S.Koivisto, “The Geometrical Trinity of Gravity,”Universe , 173 (2019), arXiv:1903.06830 [hep-th].[54] E. Cartan, “Sur les vari´et´es `a connexion affine et lath´eorie de la relativit´e g´en´eralis´ee. (premi`ere partie),”Annales Sci. Ecole Norm. Sup. , 325–412 (1923).[55] E. Cartan, “Sur les vari´et´es `a connexion affine et lath´eorie de la relativit´e g´en´eralis´ee. (Suite).” AnnalesSci. Ecole Norm. Sup. , 1–25 (1924).[56] J. Beltr´an Jim´enez, L. Heisenberg, andT. Koivisto, “Coincident General Rel-ativity,” Phys. Rev. D , 044048 (2018),arXiv:1710.03116 [gr-qc].[57] A. Padilla and V. Sivanesan, “Covariantmulti-galileons and their generalisation,”JHEP , 032 (2013), arXiv:1210.4026 [gr-qc].[58] V. Sivanesan, “Generalized multiple-scalar fieldtheory in Minkowski space-time free of Ostro-gradski ghosts,” Phys. Rev. D , 104006 (2014),arXiv:1307.8081 [gr-qc].[59] A. Padilla, P. M. Saffin, and S.-Y. Zhou,“Multi-galileons, solitons and Derrick’stheorem,” Phys. Rev. D , 045009 (2011),arXiv:1008.0745 [hep-th].[60] E. Allys, “New terms for scalar multi-Galileonmodels and application to SO(N) and SU(N) grouprepresentations,” Phys. Rev. D , 064051 (2017),arXiv:1612.01972 [hep-th].[61] S. F. Hassan and R. A. Rosen, “Bimetric Gravity fromGhost-free Massive Gravity,” JHEP , 126 (2012),arXiv:1109.3515 [hep-th].[62] G. Tasinato, “Cosmic Acceleration from AbelianSymmetry Breaking,” JHEP , 067 (2014),arXiv:1402.6450 [hep-th].[63] L. Heisenberg, “Generalization of the Proca Action,”JCAP , 015 (2014), arXiv:1402.7026 [hep-th].[64] E. Allys, P. Peter, and Y. Rodr´ıguez, “Gener-alized Proca action for an Abelian vector field,”JCAP , 004 (2016), arXiv:1511.03101 [hep-th].[65] J. Beltr´an Jim´enez and L. Heisenberg, “Deriva-tive self-interactions for a massive vectorfield,” Phys. Lett. B , 405–411 (2016),arXiv:1602.03410 [hep-th].[66] E. Allys, J. P. Beltr´an Almeida, P. Peter, andY. Rodr´ıguez, “On the 4D generalized Proca actionfor an Abelian vector field,” JCAP , 026 (2016),arXiv:1605.08355 [hep-th].[67] C. Deffayet, S. Deser, andG. Esposito-Far`ese, “Arbitrary p -formGalileons,” Phys. Rev. D , 061501 (2010),arXiv:1007.5278 [gr-qc].[68] J. P. Beltr´an Almeida, A. Guarnizo,and C. A. Valenzuela-Toledo, “Arbitrar-ily coupled p − forms in cosmological back-grounds,” Class. Quant. Grav. , 035001 (2020), arXiv:1810.05301 [astro-ph.CO].[69] J. P. Beltr´an Almeida et al. , “Topological mass genera-tion and 2 − forms,” Phys. Rev. D , 063521 (2020),arXiv:2003.11736 [hep-th].[70] L. Heisenberg, “Scalar-Vector-Tensor Gravity Theo-ries,” JCAP , 054 (2018), arXiv:1801.01523 [gr-qc].[71] K. Dimopoulos, “Statistical Anisotropyand the Vector Curvaton Paradigm,”Int. J. Mod. Phys. D , 1250023 (2012), [Er-ratum: Int.J.Mod.Phys.D 21, 1292003 (2012)],arXiv:1107.2779 [hep-ph].[72] A. Maleknejad, M. M. Sheikh-Jabbari,and J. Soda, “Gauge Fields and Infla-tion,” Phys. Rept. , 161–261 (2013),arXiv:1212.2921 [hep-th].[73] J. Soda, “Statistical Anisotropy from AnisotropicInflation,” Class. Quant. Grav. , 083001 (2012),arXiv:1201.6434 [hep-th].[74] A. Proca, “Sur la th´eorie ondula-toire des electrons positifs et negatifs,”J. Phys. Radium , 347–353 (1936).[75] A. Proca, “Th´eorie non relativiste des particules `a spinentier,” J. Phys. Radium , 61–66 (1938).[76] C. Deffayet, A. E. G¨umr¨uk¸c¨uouglu, S. Mukohyama,and Y. Wang, “A no-go theorem for generalized vec-tor Galileons on flat spacetime,” JHEP , 082 (2014),arXiv:1312.6690 [hep-th].[77] G. W. Horndeski, “Conservation of Chargeand the Einstein-Maxwell Field Equations,”J. Math. Phys. , 1980–1987 (1976).[78] A. De Felice et al. , “Screening fifth forces in general-ized Proca theories,” Phys. Rev. D , 104016 (2016),arXiv:1602.00371 [gr-qc].[79] A. De Felice et al. , “Cosmology in general-ized Proca theories,” JCAP , 048 (2016),arXiv:1603.05806 [gr-qc].[80] A. De Felice et al. , “Effective gravitational cou-plings for cosmological perturbations in general-ized Proca theories,” Phys. Rev. D , 044024 (2016),arXiv:1605.05066 [gr-qc].[81] L. Heisenberg, R. Kase, and S. Tsujikawa,“Anisotropic cosmological solutions in mas-sive vector theories,” JCAP , 008 (2016),arXiv:1607.03175 [gr-qc].[82] A. De Felice, L. Heisenberg, and S. Tsu-jikawa, “Observational constraints on generalizedProca theories,” Phys. Rev. D , 123540 (2017),arXiv:1703.09573 [astro-ph.CO].[83] L. Heisenberg and H. Villarrubia-Rojo, “Proca in thesky,” (2020), arXiv:2010.00513 [astro-ph.CO].[84] L. Heisenberg, R. Kase, M. Minamitsuji, andS. Tsujikawa, “Black holes in vector-tensor theories,”JCAP , 024 (2017), arXiv:1706.05115 [gr-qc].[85] R. Kase, M. Minamitsuji, and S. Tsu-jikawa, “Relativistic stars in vector-tensortheories,” Phys. Rev. D , 084009 (2018),arXiv:1711.08713 [gr-qc].[86] R. Kase, M. Minamitsuji, S. Tsujikawa, and Y.-L.Zhang, “Black hole perturbations in vector-tensor the-ories: The odd-mode analysis,” JCAP , 048 (2018), arXiv:1801.01787 [gr-qc].[87] R. Kase, M. Minamitsuji, and S. Tsujikawa, “Neutronstars with a generalized Proca hair and spontaneousvectorization,” Phys. Rev. D , 024067 (2020),arXiv:2001.10701 [gr-qc].[88] J. A. R. Cembranos, C. Hallabrin, A. L. Maroto,and S. J. N´u˜nez Jare˜no, “Isotropy theorem for cosmo-logical vector fields,” Phys. Rev. D , 021301 (2012),arXiv:1203.6221 [astro-ph.CO].[89] M.-a. Watanabe, S. Kanno, and J. Soda,“Inflationary Universe with AnisotropicHair,” Phys. Rev. Lett. , 191302 (2009),arXiv:0902.2833 [hep-th].[90] C. Armend´ariz-Pic´on, “Could dark energy be vector-like?” JCAP , 007 (2004), arXiv:astro-ph/0405267.[91] A. Golovnev, V. Mukhanov, and V. Vanchurin,“Vector Inflation,” JCAP , 009 (2008),arXiv:0802.2068 [astro-ph].[92] R. Emami, S. Mukohyama, R. Namba, andY.-l. Zhang, “Stable solutions of inflationdriven by vector fields,” JCAP , 058 (2017),arXiv:1612.09581 [hep-th].[93] M. ´Alvarez, J. B. Orjuela-Quintana,Y. Rodr´ıguez, and C. A. Valenzuela-Toledo,“Einstein Yang–Mills Higgs dark energy re-visited,” Class. Quant. Grav. , 195004 (2019),arXiv:1901.04624 [gr-qc].[94] L. G. G´omez and Y. Rodr´ıguez, “CoupledMulti-Proca Vector Dark Energy,” (2020),arXiv:2004.06466 [gr-qc].[95] A. Maleknejad and M. M. Sheikh-Jabbari,“Gauge-flation: Inflation From Non-AbelianGauge Fields,” Phys. Lett. B , 224–228 (2013),arXiv:1102.1513 [hep-ph].[96] P. Adshead and M. Wyman, “Chromo-NaturalInflation: Natural inflation on a steep po-tential with classical non-Abelian gaugefields,” Phys. Rev. Lett. , 261302 (2012),arXiv:1202.2366 [hep-th].[97] C. M. Nieto and Y. Rodr´ıguez, “Massive Gauge-flation,” Mod. Phys. Lett. A , 1640005 (2016),arXiv:1602.07197 [gr-qc].[98] P. Adshead and E. I. Sfakianakis, “Hig-gsed Gauge-flation,” JHEP , 130 (2017),arXiv:1705.03024 [hep-th].[99] A. Guarnizo, J. B. Orjuela-Quintana, andC. A. Valenzuela-Toledo, “Dynamical analysisof cosmological models with non-Abelian gaugevector fields,” Phys. Rev. D , 083507 (2020),arXiv:2007.12964 [gr-qc].[100] E. Witten, “Some Exact Multipseudoparti-cle Solutions of Classical Yang-Mills theory,”Phys. Rev. Lett. , 121–124 (1977).[101] D. W. Sivers, “Variational Approach to ClassicalSU(2) Gauge Theory With Spherical Symmetry,”Phys. Rev. D , 1141 (1986).[102] P. Forgacs and N. S. Manton, “Space-Time Symmetries in Gauge Theories,”Commun. Math. Phys. , 15 (1980). [103] E. Allys, P. Peter, and Y. Rodr´ıguez,“Generalized SU(2) Proca The-ory,” Phys. Rev. D , 084041 (2016),arXiv:1609.05870 [hep-th].[104] J. Beltr´an Jim´enez and L. Heisenberg, “Generalizedmulti-Proca fields,” Phys. Lett. B , 16–26 (2017),arXiv:1610.08960 [hep-th].[105] A. Nicolis, R. Penco, F. Piazza, and R. Rattazzi,“Zoology of condensed matter: Framids, ordinarystuff, extra-ordinary stuff,” JHEP , 155 (2015),arXiv:1501.03845 [hep-th].[106] Y. Rodr´ıguez and A. A. Navarro, “Scalar and vectorGalileons,” J. Phys. Conf. Ser. , 012004 (2017),arXiv:1703.01884 [hep-th].[107] Y. Rodr´ıguez and A. A. Navarro, “Non-Abelian S -term dark energy and infla-tion,” Phys. Dark Univ. , 129–136 (2018),arXiv:1711.01935 [gr-qc].[108] L. G. G´omez and Y. Rodr´ıguez, “StabilityConditions in the Generalized SU(2) ProcaTheory,” Phys. Rev. D , 084048 (2019),arXiv:1907.07961 [gr-qc].[109] V. Errasti D´ıez, B. Gording, J. A. M´endez-Zavaleta,and A. Schmidt-May, “Complete theory of Maxwelland Proca fields,” Phys. Rev. D , 045008 (2020),arXiv:1905.06967 [hep-th].[110] V. Errasti D´ıez, B. Gording, J. A. M´endez-Zavaleta, and A. Schmidt-May, “Maxwell-Proca theory: Definition and construc-tion,” Phys. Rev. D , 045009 (2020),arXiv:1905.06968 [hep-th].[111] A. Gallego Cadavid and Y. Rodr´ıguez, “A sys-tematic procedure to build the beyond generalizedProca field theory,” Phys. Lett. B , 134958 (2019),arXiv:1905.10664 [hep-th].[112] L. Heisenberg, R. Kase, and S. Tsu-jikawa, “Beyond generalized Proca the-ories,” Phys. Lett. B , 617–626 (2016),arXiv:1605.05565 [hep-th].[113] C. W. Misner, K.S. Thorne, and J.A. Wheeler, Grav-itation (W. H. Freeman, San Francisco, 1973).[114] J. Fuchs and C. Schweigert,
Symmetries, Lie algebrasand representations: A graduate course for physicists (Cambridge University Press, 2003).[115] P. Ramond,
Group theory: A physicist’s survey (Cam-bridge University Press, 2010).[116] R. Feger and T. W. Kephart, “LieART—A Mathemat-ica application for Lie algebras and representation the-ory,” Comput. Phys. Commun. , 166–195 (2015),arXiv:1206.6379 [math-ph].[117] H. Motohashi et al. , “Healthy degenerate theo-ries with higher derivatives,” JCAP , 033 (2016),arXiv:1603.09355 [hep-th].[118] R. Klein and D. Roest, “Exorcising the Ostrograd-sky ghost in coupled systems,” JHEP , 130 (2016),arXiv:1604.01719 [hep-th].[119] R. Kimura, A. Naruko, and D. Yoshida, “Ex-tended vector-tensor theories,” JCAP , 002 (2017),arXiv:1608.07066 [gr-qc]. [120] V. Errasti D´ıez, M. Maier, J. A. M´endez-Zavaleta,and M. Taslimi Tehrani, “Lagrangian constraint anal-ysis of first-order classical field theories with an appli-cation to gravity,” Phys. Rev. D , 065015 (2020),arXiv:2007.11020 [hep-th].[121] M. J. Heidari and A. Shirzad, “Structure ofConstrained Systems in Lagrangian Formal-ism and Degree of Freedom Count,” (2020),arXiv:2003.13269 [physics.class-ph].[122] S. W. Hawking and G. F. R. Ellis, The Large Scale Structure of Space-Time , CambridgeMonographs on Mathematical Physics (CambridgeUniversity Press, 1973).[123] T. Baker et al. , “Strong constraints on cos-mological gravity from GW170817 and GRB170817A,” Phys. Rev. Lett. , 251301 (2017),arXiv:1710.06394 [astro-ph.CO].[124] C. de Rham and V. Pozsgay, “New class of Procainteractions,” Phys. Rev. D , 083508 (2020), arXiv:2003.13773 [hep-th].[125] K. Dimopoulos, M. Karˇciauskas, D. H. Lyth, andY. Rodr´ıguez, “Statistical anisotropy of the curva-ture perturbation from vector field perturbations,”JCAP , 013 (2009), arXiv:0809.1055 [astro-ph].[126] L. G. G´omez and Y. Rodr´ıguez, “StatisticalAnisotropy in Inflationary Models with ManyVector Fields and/or Prolonged Anisotropic Ex-pansion,” AIP Conf. Proc. , 270–276 (2013),arXiv:1306.1150 [astro-ph.CO].[127] J. P. Beltr´an Almeida, Y. Rodr´ıguez, andC. A. Valenzuela-Toledo, “Scale and shape depen-dent non-Gaussianity in the presence of inflation-ary vector fields,” Phys. Rev. D90