PHYSICAL REVIEW D 102, 104066 (2020)

Generalized SU(2) Proca theory reconstructed and beyond

Alexander Gallego Cadavid * Instituto de Física y Astronomía, Universidad de Valparaíso, Avenida Gran Bretaña 1111, Valparaíso 2360102, Chile † Yeinzon Rodríguez Centro de Investigaciones en Ciencias Básicas y Aplicadas, Universidad Antonio Nariño, Cra 3 Este No. 47A-15, Bogotá D.C. 110231, Colombia; Escuela de Física, Universidad Industrial de Santander, Ciudad Universitaria, Bucaramanga 680002, Colombia and Simons Associate at The Abdus Salam International Centre for Theoretical Physics, Strada Costiera 11, I-34151 Trieste, Italy ‡ L. Gabriel Gómez Escuela de Física, Universidad Industrial de Santander, Ciudad Universitaria, Bucaramanga 680002, Colombia

(Received 7 September 2020; accepted 9 October 2020; published 30 November 2020)

As a modified gravity theory that introduces new gravitational degrees of freedom, the generalized SU(2) Proca theory (GSU2P for short) is the non-Abelian version of the well-known generalized Proca theory where the action is invariant under global transformations of the SU(2) group. This theory was formulated for the first time in Phys. Rev. D 94, 084041 (2016), having implemented the required primary constraint-enforcing relation to make the Lagrangian degenerate and remove one degree of freedom from the vector field in accordance with the irreducible representations of the Poincar´e group. It was later shown in Phys. Rev. D 101, 045008 (2020) and 101, 045009 (2020) that a secondary constraint- enforcing relation, which trivializes for the generalized Proca theory but not for the SU(2) version, was needed to close the constraint algebra. It is the purpose of this paper to implement this secondary constraint- enforcing relation in GSU2P and to make the construction of the theory more transparent. Since several terms in the Lagrangian were dismissed in Phys. Rev. D 94, 084041 (2016) via their equivalence to other terms through total derivatives, not all of the latter satisfying the secondary constraint-enforcing relation, the work was not so simple as directly applying this relation to the resultant Lagrangian pieces of the old theory. Thus, we were motivated to reconstruct the theory from scratch. In the process, we found the beyond GSU2P.

DOI: 10.1103/PhysRevD.102.104066

I. INTRODUCTION of gravity is an effective theory [14–16]. The inevitable presence of singularities in general relativity (GR) [17,18], Whether a classical description of the gravitational even assuming the validity of the cosmic censorship interaction is fundamental or effective remains a mystery. conjecture [19–21], points to a breakdown of the theory. What is certain is that, no matter whether the fundamental Should the breakdown take place in the infrared, the new theory of gravity is classical or quantum, and despite its theory that encompasses GR might give us some insight enormous experimental success [1–13], Einstein’s theory about the true nature of the current accelerated expansion of the Universe. The breakdown might take place in the ultraviolet, helping solve the renormalizability problems of *[email protected] † [email protected] GR and illuminating the way to a quantum description of ‡ [email protected] gravity. Of course, the breakdown might take place in both the infrared and the ultraviolet. Another option is at an Published by the American Physical Society under the terms of intermediate scale, in the strong gravity regime, which is the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to particularly interesting because the very young multi- the author(s) and the published article’s title, journal citation, messenger astronomy is giving us, and will continue and DOI. Funded by SCOAP3. doing it, valuable information about the behavior of

2470-0010=2020=102(10)=104066(21) 104066-1 Published by the American Physical Society GALLEGO, RODRÍGUEZ, and GÓMEZ PHYS. REV. D 102, 104066 (2020) gravity at the scales associated to compact objects such as [56], where the nonmetricity is the sole protagonist. Of black holes and neutron stars.1 We might, therefore, be on course, there are more possibilities, some of them with the verge of a scientific crisis and a new revolution in remanent harmless ghosts, they being, therefore, effective physics, in the sense of Kuhn [23]. theories. Over the years, several approaches have been proposed The introduction of new gravitational degrees of freedom to classically extend Einstein’s theory of gravity (see has not been kept only in the realm of a scalar field. Ref. [24] for a review). Perhaps the simplest one, at least Multiple scalar fields have been considered in what are in its conception, is giving mass to the gravitational carrier called the multi-Galileon theories [57–60]. More tensor [25]; nevertheless, starting from the Fierz-Pauli action [26] fields can be considered as well, as in the bimetric theory and arriving to the de Rham-Gabadadze-Tolley (dRGT) [61] which introduces an extra spin-two metric. The intro- ghost-free massive gravity [27], the introduction of a duction of vector fields [62–66] and p-forms [67–69] has massive graviton has shown to be a difficult challenge. also been investigated. Even the mixture of a scalar and a Another possibility is adding space dimensions while vector field, together with gravity, has been explored [70]. preserving the second-order differential structure of the Each one of these proposals has its own motivations, which field equations and keeping untouched the gravitational we will not describe here except for those related to the degrees of freedom; this is the proposal derived from the introduction of vector fields. Lovelock program [28,29], as the only curvature invariant The most frequent question when we speak about vector that satisfies these requirements in four spacetime dimen- fields in gravity and/or cosmology is, why introduce them? sions is the Einstein-Hilbert term. A third alternative is We think the right question is, why not? At the end of the invoking new gravitational degrees of freedom, the simplest day, and being pragmatical, we have observed many more of them being a scalar field; the first proposal in this regard vector fields in nature than fundamental scalar fields. We was the well-known Brans-Dicke theory [30], but this has have to be careful with the problems they can generate: turned out to be just a particular case of a whole family of ghosts, anisotropies in cosmology, etc., but this does not Lagrangians that comprise the, nowadays very famous, preclude their study. In fact, the role of vector fields in Horndeski theory [31–38]. The purpose of preserving the gravitation, astrophysics, and cosmology has attracted a lot second-order differential structure of the field equations is of interest in recent years (see Refs. [24,71–73] for some to remove the Ostrogradski ghost [39–42] that makes the reviews), culminating in the construction and study of what ground state unstable in the presence of interactions. is called the generalized Proca theory [62–66]. This is the Notwithstanding, this is not the only way to remove the Proca theory [74,75], in curved spacetime, devoid of Ostrogradski ghost, although it is the most transparent; the internal gauge symmetries and can be seen as the vector- degeneracy of the kinetic matrix associated to the degrees field version of the Horndeski theory.2 By construction, it is of freedom of the theory can be invoked so that primary plainly degenerate in order to avoid the propagation of a constraints among the phase space variables are generated fourth degree of freedom which clearly disagrees with the [43]—in this way, the unwanted degrees of freedom can be structure of the irreducible representations of the Poincar´e removed [44] even when the differential structure of the group. Its decoupling limit, in contrast, reduces to the field equations is higher order. This idea was put in action Horndeski theory. with the introduction of the beyond Horndeski theory The generalized Proca theory has been well studied in [45,46] and later generalized to what is now known as astrophysics and cosmology [62,78–87]. In the latter, the degenerate higher-order scalar-tensor theory (DHOST) however, special attention has been paid because of the [47–49], where a plethora of Lagrangians rose up to the anisotropies that a vector field produces, inherent to its surface. The application of this idea to the Lovelock nature, both in the expansion of the Universe and in the program has, nonetheless, not been fruitful [50], which cosmological perturbations. Such anisotropies can easily is, paradoxically, very suggestive. A fourth alternative is go beyond the observational constraints, so it is necessary considering other geometric formulations of gravity, i.e., to take some measures such as the rapid oscillations of the considering not only the curvature but also the torsion and vector field [88], the dilution of the vector field by a the nonmetricity as the protagonist geometric objects in the companion scalar field [89], the suppression of the spatial description of the gravitational interaction [24,51–53]. This components of the vector field against its temporal com- has a long history starting from the Einstein-Cartan theory ponent (what is called the temporal gauge setup) [79],or [54,55], which involves curvature and torsion but leaving the implementation of a cosmic triad of vector fields3 that aside the nonmetricity, to the coincident gravity proposal restores the isotropy [90–94]. The latter proposal has been

1At these scales, however, there might be some contributions 2For the U(1) gauge-invariant version of the generalized Proca from the ultraviolet-complete theory. This means that the regime theory in flat spacetime see Ref. [76] and in curved spacetime see of validity of the modified gravity theory must also be ensured Ref. [77]. before applying constraints that belong to other scales or 3The cosmic triad is a set of three vector fields mutually frequencies [22]. orthogonal and of the same norm.

104066-2 GENERALIZED SU(2) PROCA THEORY RECONSTRUCTED AND … PHYS. REV. D 102, 104066 (2020) investigated in different contexts and finds a natural home relation to the “old” GSU2P and seeing what the result in the presence of an internal SU(2) symmetry [95–99]. would be. However, this turned out to be impractical, since Indeed, the temporal gauge setup and the cosmic triad are many terms had disappeared when employing the total two of the four possible setups that are compatible with a derivatives. Moreover, the total derivatives employed sat- spatial spherical symmetry and that are realized under an isfied the primary constraint-enforcing relation but not internal SU(2) symmetry [100–102]. This was the main necessarily the secondary one, so repairing the old theory motivation behind the formulation of what was baptized as quickly became quite a big deal and, therefore, unworthy. the generalized SU(2) Proca theory (GSU2P for short) The purpose of this paper is to build from scratch the [103] (see also Ref. [104]). The possible setups mentioned GSU2P, paying attention to the two caveats already above spontaneously break the internal (global) SU(2) mentioned and following a style of construction based a symmetry along with the Lorentz rotational symmetry on the decomposition of a first-order derivative ∂μAν of the a and Lorentz boosts, leaving, however, a diagonal spatial vector field Aμ into its symmetric, rotation subgroup unbroken. The isotropic expansion of the a a a Universe can then be naturally modeled with any of the four Sμν ≡∂μAν þ ∂νAμ; ð1Þ setups or linear combinations of them without resorting to fast oscillations or other (scalar) fields. The price to pay, and antisymmetric part, however, which is the spontaneous breaking of the Lorentz a a a invariance, is, anyway, extraordinarily reasonable, since Aμν ≡∂μAν − ∂νAμ: ð2Þ this seems to be nature’s strategy to produce all the patterns we see in condensed matter systems (fluids, superfluids, Employing this decomposition will simplify things and solids, and supersolids; see Ref. [105]). Indeed, according allow us to deal with a lower number of Lagrangian to the pattern classification in Ref. [105], what would be the building blocks as compared with Ref. [103]. In the condensed matter analogs of the temporal gauge setup and process, we will find the beyond GSU2P. the cosmic triad in the GSU2P are the, yet unobserved, The layout of the paper is the following. In Sec. II,we type-I and type-II framids, respectively. The application of will enumerate the requirements for the construction of the the GSU2P to dark energy and inflation has been explored GSU2P. In Sec. III, we will show how an arbitrary function a a in Refs. [106,107] and its stability properties in Ref. [108]. of Aμν and Aμ satisfies both the primary and secondary The GSU2P was built in Ref. [103] (see also Ref. [104]) constraint-enforcing relations, leaving only the work of having in mind the primary constraints required to remove finding the right terms in the action involving at least one a the fourth degree of freedom.4 To that end, a primary Sμν. In Sec. IV, we build the Lagrangian involving one constraint-enforcing relation related to the primary Hessian derivative and two vector fields. Similar procedures are of the system was employed. This was done in flat followed in Secs. V–VIII, where we obtain the Lagrangians spacetime following the standard procedure of later cova- involving one derivative and four vector fields, two riantizing not before having removed redundant terms in derivatives only, two derivatives and two vector fields, the obtained action via total derivatives. Later on, two and three derivatives only, respectively. In all these cases, caveats were recognized. First, the constraint algebra was the number of spacetime indices in the Lagrangian building not closed only with the primary constraints, at least for blocks before contractions with the primitive invariants theories involving more than one vector field5 [109,110];a of the Poincar´e group is less than or equal to six. We secondary constraint was identified that closed the con- prefer to keep the construction of the theory up to this level straint algebra and that, therefore, pointed out to the since, as shown in Ref. [103], the number of Lagrangian existence of ghosts in the GSU2P. Second, the redundant building blocks we have to consider scales very fast terms in flat spacetime turned out to be not necessarily when more spacetime indices are considered. Finally, in “ ” “ ” redundant in curved spacetime, which would lead, for sure, Secs. IX and X, we compare the new or reconstructed to new terms not uncovered in Ref. [103]; indeed, such a GSU2P with the old GSU2P and with the generalized Proca remark led two of us to rediscover the beyond Proca terms theory, respectively. Section XI is devoted to the conclu- in Ref. [111], they being the vector analogous of the sions. Throughout the text, Greek indices are spacetime beyond Horndeski terms, already obtained in Ref. [112]. indices and run from 0 to 3, while Latin indices are internal Reformulating the GSU2P in order to implement the SU(2) group indices and run from 1 to 3. The sign secondary constraint-enforcing relation seemed at first convention is the (þþþ) according to Misner, Thorne, sight very easy, because it was a matter of applying this and Wheeler [113]. II. REQUIREMENTS FOR THE 4Concretely, the temporal component of the vector field. CONSTRUCTION OF THE THEORY 5The constraint algebra of the generalized Proca theory, the latter being a theory that involves just one vector field, turned out The GSU2P must be built having in mind the following to be trivially closed. criteria:

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A A c A (1) The action must be, locally, Lorentz invariant ∂L ∂L ∂Aρσ ∂L i i i δ0 δμ δc (although the symmetry may be nonlinearly _ a ¼ c _ a ¼ c ½ρ σ a ∂Aμ ∂Aρσ ∂Aμ ∂Aρσ realized). ∂LA ∂LA ∂LA (2) The vector field must transform as the adjoint i − i 2 i : 5 ¼ ∂ a ∂ a ¼ ∂ a ð Þ representation of the global transformations belong- Aρμ ρ¼0 Aμσ σ¼0 Aρμ ρ¼0 ing to the SU(2) group [114–116]. Accordingly, the action must be invariant under these transformations. Any possible ambiguity in the second line of the previous 0ν (3) The primary constraint-enforcing relation H ¼ 0, LA ab equation is clarified having in mind that i is always a 8 where written as Aμν contracted with an antisymmetric tensor. Thus, 2 μν ∂ L H ≡ ; ð3Þ ∂2LA ∂2LA ∂ c ab _ a _ b i i Aαβ ∂Aμ∂Aν 2 _ b _ a ¼ a c _ b ∂Aν ∂Aμ ∂Aρμ∂Aαβ ρ 0 ∂Aν ¼ ∂2LA is the “primary” Hessian and a dot means a time 2 i δ0 δν δc ¼ a c ½α β b ∂Aρμ∂Aαβ ρ 0 derivative, must be satisfied in flat spacetime in ¼ order to make the Lagrangian degenerate. This is a ∂2LA ∂2LA 2 i − 2 i necessary condition to remove the unwanted degree ¼ a b a b ∂Aρμ∂Aαν ρ 0 α 0 ∂Aρμ∂Aνβ ρ 0 β 0 of freedom [62,63]. ¼ ; ¼ ¼ ; ¼ ∂2LA (4) The secondary constraint-enforcing relation i 00 ¼ 4 : ð6Þ H˜ 0 ∂ a ∂ b ab ¼ , where Aρμ Aαν ρ¼0;α¼0

2 The primary constraint-enforcing relation is, therefore, μν ∂ L H˜ ≡ ; ð4Þ satisfied: ab ∂ _ ½a∂ b Aμ Aν ∂2LA 0ν i H ¼ 4 ¼ 0 ð7Þ ab ∂ a ∂ b is the “secondary Hessian” and the brackets mean Aρμ Aαν ρ¼0;μ¼0;α¼0 unnormalized antisymmetrization, must be satisfied a in flat spacetime so that the primary constraint holds because of the antisymmetry of Aμν. at all times.6 This condition together with the Regarding the secondary constraint-enforcing relation, preceding one are necessary and sufficient to remove we obtain from Eq. (5) the unwanted degree of freedom in flat spacetime [109,110]. ∂2LA ∂2LA i i ¼ 2 ; ð8Þ (5) The decoupling limit of the theory must be free of b _ a a b ∂Aν ∂Aμ ∂Aρμ∂Aν ρ 0 the Ostrogradski ghost as must happen since the full ¼ theory is free of it. This implies that the scalar limit of GSU2P must belong to the non-Abelian extension which leads to the secondary Hessian of the multi-Galileon theory [57–60,103] or any of ∂2LA its beyond or DHOST versions. H˜ 00 2 i 0; 9 ab ¼ ½a b ¼ ð Þ ∂Aρμ∂Aν ρ¼0;μ¼0;ν¼0 L III. 2 a in view, again, of the antisymmetry of Aμν. LA a a All the Lagrangian pieces i built exclusively from Hence, we can conclude that the L2ðAμν;AμÞ Lagrangian a a contractions of Aμν and Aμ with the primitive invariants of piece satisfies automatically the first and secondary con- 7 the Lorentz group [114–116], collected in a generic straint-enforcing relations necessary to propagate only a a Lagrangian piece called L2ðAμν;AμÞ, satisfy automatically three degrees of freedom. This is the reason why such a both the primary and secondary constraint-enforcing rela- Lagrangian piece is so particular, differing in its structure a tions thanks to the antisymmetry of Aμν. To see it, let us and arbitrariness from the other Lagrangian pieces we are calculate the primary and secondary Hessians. First of all, going to describe in the following. On the other hand, the generalization of L2 to curved spacetime is straightforward. 6This condition bears a great resemblance to that obtained in 8 a Refs. [117,118] for mechanical systems with multiple degrees of Except for the case where no Aμν tensors are involved. freedom. ∂LA i 0 7 However, in such a case, _ a ¼ automatically. They may, of course, either preserve or violate parity. ∂Aμ

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IV. ONE DERIVATIVE AND gμνgρσgαβ; TWO VECTOR FIELDS gμνgραgσβ; Lagrangian building blocks constructed from one deriva- gμνgρβgσα; tive and two vector fields, linearly independent from L2, are μρ νσ αβ terms of the form SμνAρAσ which, as can be seen, involve g g g ; four spacetime indices. Group theory tells us that four gμρgναgσβ; building blocks can be constructed upon contractions of μρ νβ σα SμνAρAσ with the following tensors [114–116]: g g g ; gμσgνρgαβ; gμνgρσ; gμσgναgρβ; gμρgνσ; gμσgνβgρα; gμσgνρ; gμαgνρgσβ; ϵμνρσ; ð10Þ gμαgνσgρβ; μν ϵμνρσ μα νβ ρσ where g is the contravariant Minkowski metric and g g g ; is the Levi-Civita tensor. Thus, the only building blocks μβ νρ σα either different than zero or with the potential of becoming g g g ; different than zero after adding the internal group indices gμβgνσgρα; are the following: gμβgναgρσ; ð13Þ μ SμðA · AÞ; μ ν as well as with the following ten products of a spacetime SμνA A : ð11Þ metric and a Levi-Civita tensor:

The addition of the internal group indices leads to terms of gνρϵμσαβ; the form SaAbAc that involve three internal group indices νσ μραβ and which, from group theory [114–116], can be contracted g ϵ ; ϵ 9 να μρσβ only with the totally antisymmetric tensor abc: g ϵ ; νβ μρσα μa b c g ϵ ; Sμ ðA · A Þϵabc; ρσ μναβ a μb νc g ϵ ; SμνA A ϵabc: ð12Þ gραϵμνσβ; Such terms vanish because of the antisymmetry of ϵ ,so abc ρβϵμνσα we conclude that there do not exist terms in GSU2P, g ; linearly independent of L2, that involve one derivative and gσαϵμνρβ; two vector fields. gσβϵμνρα; V. ONE DERIVATIVE AND FOUR gαβϵμνρσ: ð14Þ VECTOR FIELDS ϵ A. The Lagrangian building blocks Another five contractions of the form g are possible, but they are not linearly independent because of the property Lagrangian building blocks built from one derivative and four vector fields, linearly independent of L2, are terms of μν ρσαβ νρ μσαβ νσ μραβ g ϵ ¼ g ϵ − g ϵ the form SμνAρAσAαAβ that involve six spacetime indices. να μρσβ νβ μρσα Group theory [114–116] tells us that, in this case, the þ g ϵ − g ϵ : ð15Þ building blocks are constructed upon contractions of SμνAρAσAαAβ with the following 15 permutations of the Thus, only three building blocks either are nonvanishing or product of three spacetime metrics: have the potential of becoming different than zero once the internal group indices are added10: 9This tensor represents the structure constants of the SU(2) group. See, in particular, the Misner, Thorne, and Wheeler 10From now on, the starred Lagrangian building blocks and treatise on gravitation [113] for a description of the SU(2) group total derivatives will be those that vanish according to the as a manifold endowed with a metric gab and an orientability form Poincar´e group but that otherwise survive when considering also described by ϵabc. the SU(2) group.

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μ SμðA · AÞðA · AÞ; which can vanish only if μ ν SμνA A ðA · AÞ; ν ϵμσαβ SμνA AσAαAβ :ðÞ ð16Þ x1 ¼ 0; − 0 When adding the internal group indices, these terms x3 x4 ¼ : ð22Þ acquire the form SaAbAcAdAe which can be contracted, according to group theory [114–116], only with the following six products of an internal group metric and Thus, the Lagrangian that satisfies the constraint algebra is the respective structure constants: given by

gabϵcde; 2 3 4 L ¼ x2L3 þ x3ðL3 þ L3Þ: ð23Þ gacϵbde;

gadϵbce;

gbcϵade; C. Total derivatives gbdϵace; ϵ Although the Lagrangian in Eq. (23) satisfies require- gcd abe: ð17Þ ments 1–4inSec.II, some of its Lagrangian pieces might be redundant, compared to L2, via total derivatives. To gϵ Another four contractions of the form are possible, but find it out, we must proceed to build all the possible total they are not linearly independent because of the property derivatives of currents involving five vector fields. To this end, we must follow a path similar to the ones in g ϵ g ϵ − g ϵ g ϵ : 18 ae bcd ¼ ab cde ac bde þ ad bce ð Þ previous sections, i.e., employing group theory. In this way, a term of the form ∂μðAνAρAσAαAβÞ, which involves Therefore, there exist only four linearly independent building blocks in GSU2P that involve one derivative six spacetime indices, must be contracted with all the terms in Eqs. (13) and (14). However, the Lagrangian and four vector fields: 2 3 4 pieces we are interested in, L3 and L3 þ L3, explicitly 1 a μb νc e violate parity. Therefore, only the terms in Eq. (14) are L3 ¼ SμνA A ðAb · A Þϵace; actually needed. This leads to just one term that satisfies L2 a ν c d eϵμσαβϵ 3 ¼ SμνAaAσAαAβ cde; the requirement of either being nonvanishing or having 3 a νb d e μσαβ the potential of being nonvanishing once the internal L3 ¼ SμνAσaA AαAβϵ ϵbde; group indices are added: 4 a νb d e μσαβ L3 ¼ SμνA AσbAαAβϵ ϵade: ð19Þ

μσαβ ∂μ½ðA · AÞAσAαAβϵ :ðÞ ð24Þ B. The Hessian constraints The Lagrangian is, hence, written as a linear combination of the Lagrangian building blocks of Eq. (19): The addition of the internal group indices leads to terms of the form ∂ðAaAbAcAdAeÞ that involve five internal X4 group indices. Therefore, they must be contracted with L Li ¼ xi 3; ð20Þ all the terms in Eq. (17), which results in i¼1 where the x are arbitrary constants. Because only one i ∂ μ ∂ a c d e ϵμσαβϵ derivative has been considered, the primary constraint- μJ1 ¼ μ½ðA · AaÞAσAαAβ cde; enforcing relation is satisfied automatically. Regarding the μ a b d e μσαβ ∂μJ2 ¼ ∂μ½ðA · A ÞAσaAαAβϵ ϵbde: ð25Þ secondary constraint-enforcing relation, the secondary Hessian gives the following result: These total derivatives can be expressed in terms H˜ 00 2 −A0c A · Ae ϵ − A0c A · Ae ϵ ab ¼ ½ ð ½b Þ ace ð c Þ ½abe of Lagrangian building blocks involving one derivative 0 0c 0eϵ 0e 0c 0ϵ þ A½bA A ace þ A A Ae ½ajcjbx1 and four vector fields, which is the key to observe whether some of the two Lagrangian pieces in − 2 d eϵ0σαβϵ − Aσ½ajAαAβ jbdeðx3 x4Þ; ð21Þ Eq. (23) are redundant:

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1 μ a ν c d e where the xi are arbitrary constants. Since this Lagrangian ∂μJ ¼ ½2AμνA AσAαA 1 2 a β involves only vector fields through spacetime derivatives, c d e a μσαβ the secondary constraint-enforcing relation is satisfied þ 3AμσAαAβðA · AaÞϵ ϵcde automatically. Regarding the primary constraint-enforcing L2 þ 3; relation, the primary Hessian gives the following result: 1 ∂ μ a νb d e μJ2 ¼ 2 ½AμνA AσaAαAβ H0ν −8 0ν ab ¼ gabg ðx1 þ x2Þ; ð30Þ a νb d e þ Aν Aμ AσaAαAβ a b d e which vanishes only if þðA · A ÞAμσaAαAβ 2 a b d e ϵμσαβϵ þ ðA · A ÞAσaAμαAβ bde x1 þ x2 ¼ 0: ð31Þ 1 þ ðL3 þ L4Þ: ð26Þ 2 3 3 Thus, the Lagrangian that satisfies the constraint algebra is given by We can see that, even after covariantization, the two Lagrangian pieces in Eq. (23) can be removed, via total 1 2 L ¼ x1ðL4 − L4Þ: ð32Þ derivatives, in favor of terms already contained in L2. Now, from the previous two expressions and the results of Secs. III and VB, we can see that it is legitimate to μ μ employ ∂μJ1 and ∂μJ2, since they satisfy the Hessian C. Total derivatives constraints. Therefore, the conclusion is that there do not Again, it is absolutely necessary to test if the Lagrangian exist terms in GSU2P, linearly independent of L2,that in Eq. (32) is not already included in L2. To this end, it is involve one derivative and four vector fields. necessary to build the total derivatives of currents built with one derivative and one vector field. These terms, being of VI. TWO DERIVATIVES the form ∂μ½Aνð∂ρAσÞ, involve four spacetime indices, so that they are constructed by means of contractions with the A. The Lagrangian building blocks terms in Eq. (10) except for the last one in that equation as When dealing with two derivatives only, the Lagrangian the Lagrangian piece we are interested in, L1 − L2, L 4 4 building blocks, linearly independent of 2, acquire two explicitly preserves parity. In this case, none of the terms possible structures: either AμνSρσ or SμνSρσ. In both cases, vanishes, so we end up with three possible total derivatives: the number of spacetime indices is four, so we have to contract with all the terms in Eq. (10). This results in μ ∂μ½A ð∂ · AÞ; μ ρ μ ν SμSρ; ∂μ½Aνð∂ A Þ; μν ν μ SμνS ; ð27Þ ∂μ½Aνð∂ A Þ: ð33Þ these terms being the only ones that either do not vanish or Since these terms are of the form ∂½Aað∂AbÞ, once the have the potential of being nonvanishing once the internal internal group indices have been added, they can be group indices are added. Indeed, when this is done, these contracted only with a group metric. Thus, the total a b terms acquire the form S S which can be contracted only derivatives we have been looking for are with the group metric gab [114–116]. Thus, the Lagrangian building blocks are μ μa ∂μJ1 ¼ ∂μ½A ð∂ · AaÞ; 1 μa ρ μ a μ ν L4 ¼ Sμ Sρa; ∂μJ2 ¼ ∂μ½Aν ð∂ AaÞ; 2 a μν μ a ν μ L4 ¼ SμνSa : ð28Þ ∂μJ3 ¼ ∂μ½Aν ð∂ AaÞ: ð34Þ

B. The Hessian constraints It is easy to see that these total derivatives, in their actual The Lagrangian is therefore written as a linear combi- form, are anyway useless, because they lead to terms nation of the Lagrangian building blocks of Eq. (28): involving second-order derivatives in addition to the ones we are interested in which involve just two first-order X2 derivatives. The only way to circumvent this situation, at i L ¼ xiL4; ð29Þ least partially but enough, is to construct the linear i¼1 combination

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∂ ˜ μ ≡∂ μ − ∂ μ 1 μJ1 μJ1 μJ3 μa ν a μa ν L4 0 ¼ A RμνA − RðA · A Þ¼GμνA A ; ð38Þ 1 1 ; a 2 a a − a νμ L1 − L2 μa ∂ ∂ ν ¼ 4 AμνAa þ 4 ð 4 4ÞþA ½ μ; νAa; ð35Þ

where Gμν is the Einstein tensor. Indeed, this Lagrangian is that removes the second-order derivatives, since the com- healthy in the decoupling limit because of the divergence- mutator in the last line trivially vanishes in flat spacetime. less character of Gμν. Our conclusion, different than the one Indeed, from this result and the findings in Secs. III and VI μ encountered in Ref. [103], where no term with just two ∂ ˜ μa ν B, we can see that employing μJ1 is allowed, since it derivatives was found while GμνA Aa was just postulated, satisfies the Hessian constraints. The Lagrangian in finds its origin in the fact that the total derivative in Eq. (35) Eq. (32) is, in consequence, already contained in L2 in was first covariantized and later employed (not) to dismiss flat spacetime up to a total derivative. Things, however, are some terms in favor of others. This way of proceeding was different in curved spacetime. identified in Ref. [111], and it is the mechanism to uncover the beyond SU(2) Proca terms as we will later see. To L D. Covariantization finish, the notation 4;0 is introduced in Eq. (38) to label this Lagrangian as one that involves (or comes from) two As is usual the case, the covariantization of Eq. (35) derivatives (this is the reason for the 4) and no vector fields implies the replacement of partial derivatives with space- (this is the reason for the 0). time covariant derivatives and of the Minkowski metric with an arbitrary spacetime metric. Thus, the curved spacetime version of Eq. (35) reads VII. TWO DERIVATIVES AND TWO VECTOR FIELDS 1 1 ∇ ˜ μ − a νμ L1 − L2 μa ∇ ∇ ν A. Lagrangian building blocks μJ1 ¼ 4 AμνAa þ 4 ð 4 4ÞþA ½ μ; νAa Lagrangian building blocks built from two derivatives 1 νμ 1 − a L1 − L2 − μa ν and two vector fields are terms of the form AμνSρσAαAβ or ¼ 4 AμνAa þ 4 ð 4 4Þ A RμνAa; ð36Þ SμνSρσAαAβ that involve six spacetime indices. In order to uncover them, we must contract with all the terms in where Rμν is the Ricci tensor. Then, we can conclude that Eqs. (13) and (14). As a result, the Lagrangian building L the Lagrangian in Eq. (32) is actually independent of 2 in blocks that either do not vanish or have the potential of a nonredundant way in curved spacetime, whereas it is becoming different than zero once the internal group L already included in 2 in flat spacetime. To remind the indices are added are the following: reader of this fact, we will in the following deal with μa ν 1 2 A RμνA instead of L4 − L4. a μ ν σ AμνSσA A ; ρ μ ν E. The decoupling limit AμνSρA A ; ν μσαβ The Helmholtz theorem tells us that any vector field Aμ AμνSσAαAβϵ ; ðÞ can be decomposed into its transverse part, a divergence- ρ μναβ AμνSρAαAβϵ ; ðÞ free vector field Aμ, and its longitudinal part, the gradient ρ ϵμνσβ of scalar field ∇μπ: AμνSρσA Aβ ; μ ρ SμSρðA · AÞ; Aμ Aμ ∇μπ: μ ρ σ ¼ þ ð37Þ SμSρσA A ; μν SμνS ðA · AÞ; The decoupling limit of GSU2P, understood as an effective μ ν σ field theory, which corresponds in this case to the replace- SμνSσA A ; a a ment Aμ → ∇μπ , must also be a healthy theory; i.e., it ν μσαβ SμνSσAαAβϵ :ðÞ ð39Þ must be free of the Ostrogradski instability. Examining the μa ν term A RμνAa, we can observe that its decoupling limit μ a ν ∇ π Rμν∇ πa is not healthy, as the field equation resultant When the internal indices are added, these terms are of the a a b c d a b c d of the variation of the action with respect to π leads to a form AfgS A A or S S A A ; i.e., they involve four μ term proportional to ∇ Rμν, i.e., a higher-order term. To internal group indices. So, in order to obtain the avoid such a pathological behavior (see Ref. [34]), it is Lagrangian building blocks, and according to group theory a necessary to add −RðA · AaÞ=2 as a counterterm, R being [114–116], we must contract with the following products of the Ricci scalar: two group metrics:

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H0ν −2 0 νc 2 2 gabgcd; ab ¼ AcA gabðx1 þ x12 þ x16Þ 0 0c 0ν gacgbd; − 2AcA g gabðx1 − 2x12 − 2x16Þ − 2 0 ν − 2 gadgbc: ð40Þ AaAbðx2 x4 þ x13 þ x17Þ − 2 0 0 0ν − 2 − 2 − 2 AaAbg ðx2 þ x3 x13 x17 x18Þ This results in the following 19 Lagrangian building blocks − 2 0 ν 2 AbAaðx3 þ x4 þ x13 þ x18Þ linearly independent of L2: 0ναβ − 2ϵ AαbAβaðx5 þ 2x6 − 2x19Þ μ c 0ν 1 a νc σ − 8ðAc · A Þg gabðx10 þ x14Þ L4 ¼ AμνSσaA Ac; − 8 0ν L2 a μb ν σ ðAa · AbÞg ðx11 þ x15Þ; ð43Þ 4 ¼ AμνSσ AaAb; L3 a μb ν σ 4 ¼ AμνSσ AbAa; whereas for the secondary Hessian 4 a ρb μ ν L ¼ AμνSρ A A ; 4 a b ˜ 00 α 0 H −2 1 − 3 − 4 5 a νb μσαβ ab ¼ A½bjA αjaðx x x Þ L4 ¼ AμνSσ AαaAβbϵ ; − 2 α 0 2 − 2 − 2 6 a ρb μναβ A½bjSαjað x12 x13 þ x16 x18Þ L4 ¼ AμνSρ AαaAβbϵ ; − 2ϵβα0σ − 7 a ρc μνσβ Aσ½ajAβαjbðx6 x7 þ x8Þ L4 ¼ AμνSρσaA Aβcϵ ; 0 α ρ 2 4 − 2 2 − 8 a b μνσβ þ A a Sα b ð x10 x11 þ x12 x13Þ L4 ¼ AμνSρσAaAβbϵ ; ½ j j 0 00 L9 a b ρ ϵμνσβ − 4A S ð2x14 − x15 þ x16 − x17Þ: ð44Þ 4 ¼ AμνSρσAbAβa ; ½a b 10 μa ρ L ¼ Sμ Sρ ðAc · A Þ; 4 a c Both expressions vanish, therefore, only when the follow- 11 μa ρb L4 ¼ Sμ Sρ ðAa · AbÞ; ing 11 constraints are satisfied: 12 μa ρc σ L4 ¼ Sμ SρσaA Ac; x1 ¼ 0; L13 μa b ρ σ 4 ¼ Sμ SρσAaAb; x3 ¼ −x2; 14 a μν c L4 ¼ SμνSa ðA · AcÞ; x4 ¼ x2; 15 a μνb L4 ¼ SμνS ðAa · AbÞ; x8 ¼ −x6 þ x7; 16 a μ νc σ L4 ¼ SμνSσaA Ac; x13 ¼ 4x10 − 2x11 þ 2x12; L17 a μb ν σ 4 ¼ SμνSσ AaAb; x14 ¼ −x10; L18 a μb ν σ 4 ¼ SμνSσ AbAa; x15 ¼ −x11; 19 a νb μσαβ L4 ¼ SμνSσ AαaAβbϵ : ð41Þ x16 ¼ −x12;

x17 ¼ −2x10 þ x11 − x12;

x18 ¼ −2x10 þ x11 − x12; B. The Hessian constraints x5 The Lagrangian is written as a linear combination of the x19 ¼ 2 þ x6: ð45Þ Lagrangian building blocks found in the previous section. Thus, Thus, the Lagrangian that satisfies the constraint algebra is given by X19 L Li 19 ¼ xi 4; ð42Þ 2 3 4 5 L4 L x2 L − L L x5 L i¼1 ¼ ð 4 4 þ 4Þþ 4 þ 2 6 8 19 7 8 þ x6ðL4 − L4 þ L4 Þþx7ðL4 þ L4Þ where the x are arbitrary constants. Since the Lagrangian i L9 L10 4L13 − L14 − 2L17 − 2L18 involves two derivatives and two vector fields, none of the þ x9 4 þ x10ð 4 þ 4 4 4 4 Þ 11 13 15 17 18 Hessian constraints is trivially satisfied in this case. þ x11ðL4 − 2L4 − L4 þ L4 þ L4 Þ Performing the calculations, we find for the primary L12 2L13 − L16 − L17 − L18 Hessian þ x12ð 4 þ 4 4 4 4 Þ: ð46Þ

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˜ μ μ μ C. Total derivatives ∂μJ1 ≡∂μJ1 − ∂μJ8 With the purpose of establishing which of the 1 − a νμ c 2 a νμ c ρ Lagrangian pieces in Eq. (46) are redundant, the total ¼ 4 ½AμνAa ðA · AcÞþ Aν Aa AμρAc derivatives of terms involving one derivative and three 1 L10 − L14 2L13 − 2L17 − 2L4 vector fields must be constructed. These derivatives are þ 4 ð 4 4 þ 4 4 4Þ ∂ ∂ terms of the form μ½Aνð ρAσÞAαAβ that involve six μa ν c þ A ½∂μ; ∂νA ðA · A Þ; spacetime indices, so contractions with the terms in a c ˜ μ μ μ Eqs. (13) and (14) must be done. As a result, the only ∂μJ2 ≡∂μJ2 − ∂μJ9 terms that either are different than zero or have the potential 1 a νμb a νμb ρ − AμνA A · A Aν A Aμ Aρ of becoming so after introducing the internal group indices ¼ 4 ½ ð a bÞþ a b are the following: a νμb ρ þ Aν A AμρbAa

μ 1 11 13 12 15 18 16 4 ∂μ A ∂ · A A · A ; L L L − L − L − L L ½ ð Þð Þ þ 4 ð 4 þ 4 þ 4 4 4 4 þ 4Þ μ ρ σ ∂μ½A ð∂ρAσÞA A ; μa νb þ A ½∂μ; ∂νA ðAa · AbÞ; μ ν ∂μ½Aνð∂ A ÞðA · AÞ; ˜ μ μ μ ∂μJ3 ≡∂μJ3 − ∂μJ5 ν μ ∂μ½Aνð∂ A ÞðA · AÞ; 1 μa c ρ σ μa c σ ρ − A AρσAμ A A AρσAμ A ν μσαβ ¼ 4 ½ c a þ a c ∂μ½Aνð∂ AσÞAαAβϵ ; ðÞ 1 ∂ ∂ ν ϵμραβ L12 2L18 − L13 − L16 μ½Aνð ρA ÞAαAβ ; ðÞ þ 4 ð 4 þ 4 4 4 μρσβ ∂μ½ð∂ρAσÞAβðA · AÞϵ ; 17 3 4 2 − L4 − 2L4 − L4 þ 2L4Þ μναβ ∂μ½ð∂ · AÞAνAαAβϵ :ðÞ ð47Þ μa ρc σ þ A ½∂μ; ∂ρAσaA Ac; ð49Þ

These total derivatives become terms of the form while in the following cases the problem is automatically ∂½Aað∂AbÞAcAd once the internal group indices are added. solved thanks to the symmetries of the Levi-Civita tensor: Since they involve four internal group indices, contractions ∂ ˜ μ ≡∂ μ with the terms in Eq. (40) are needed, which results in μJ4 μJ11 1 a νb νb a ¼ ½AμνAρ AαaAβb þ Aρ AμαaAν Aβb μ μa c 4 ∂μJ1 ¼ ∂μ½A ð∂ · AaÞðA · AcÞ; νb a ϵμραβ μ μa b þ Aρ AμβbAν Aαa ∂μJ2 ¼ ∂μ½A ð∂ · A ÞðAa · AbÞ; 1 19 5 8 ∂ μ ∂ μa ∂ ρc σ þ ðL þ 2L − L þ L7Þ μJ3 ¼ μ½A ð ρAσaÞA Ac; 4 4 4 4 4 ∂ μ ∂ μa ∂ b ρ σ 1 μJ4 ¼ μ½A ð ρAσÞAaAb; a νb μραβ þ Aν ½∂μ; ∂ρA AαaAβbϵ ; μ μa b ρ σ 2 ∂μJ5 ¼ ∂μ½A ð∂ρAσÞA A ; b a ∂ ˜ μ ≡∂ μ μ a μ ν c μJ5 μJ12 ∂μJ ¼ ∂μ½Aν ð∂ A ÞðA · A Þ; 6 a c 1 μ a c a c α μρσβ ∂ ∂ a ∂μ νb ¼ ½AρσAμβ ðA · A Þþ2AρσAμαAβ A ϵ μJ7 ¼ μ½Aν ð A ÞðAa · AbÞ; 4 a c a c μ a ν μ c 1 ∂μJ8 ¼ ∂μ½Aν ð∂ AaÞðA · AcÞ; 9 þ L4 μ a ν μb 2 ∂μJ ∂μ Aν ∂ A A · A ; 9 ¼ ½ ð Þð a bÞ 1 μ a c μρσβ ∂ ∂ a ∂ν b ϵμσαβ þ Aβ ½∂μ; ∂ρAσðA · A Þϵ ; μJ10 ¼ μ½Aν ð AσÞAαaAβb ; 2 a c ∂ μ ∂ a ∂ νb ϵμραβ ˜ μ μ μJ11 ¼ μ½Aν ð ρA ÞAαaAβb ; ∂μJ6 ≡∂μJ13 ∂ μ ∂ ∂ a c ϵμρσβ 1 μJ12 ¼ μ½ð ρAσÞAβaðA · AcÞ ; a b a α b ¼ ½AρσAμβðAa · AbÞþAρσAμ aAβAαb μ a b μρσβ 4 ∂μJ13 ¼ ∂μ½ð∂ρAσÞAβðAa · AbÞϵ : ð48Þ a b α μρσβ þ AρσAμαbAβAaϵ

1 7 8 All these total derivatives are useless as long as they þ ðL4 þ L4Þ produce terms with second-order derivatives. Fortunately, 4 1 this circumstance can be redeemed, although not in all the b a μρσβ þ A ½∂μ; ∂ρAσðA · A Þϵ : ð50Þ cases, by building the following linear combinations: 2 β a b

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˜ μ ˜ μ However, even like this, these total derivatives continue to ∂μðJ3 þ 3J2Þ; be useless unless they satisfy the Hessian constraints. ˜ μ ˜ μ ∂μð2J3 − 3J1Þ; Comparison of these expressions with Eqs. (43) and (44) ∂ ˜ μ and with the findings in Sec. III reveals that the following μJ5; ˜ μ linear combinations are the only ones that pass the test: ∂μJ6: ð51Þ

We have now the four total derivatives that will help us remove some redundant terms from Eq. (46). However, covariantization must be performed first.

D. Covariantization The minimal covariantization scheme described in Sec. VI D and applied to the total derivatives of Eq. (51) produces the curved spacetime versions 1 3 ∇ ˜ μ 3˜ μ … ∈ L L2 − L3 L4 L11 − 2L13 − L15 L17 L18 L12 2L13 − L16 − L17 − L18 μðJ3 þ J2Þ¼ð 2Þþ2 ð 4 4 þ 4Þþ4 ð 4 4 4 þ 4 þ 4 Þþð 4 þ 4 4 4 4 Þ μa α ρc σ μa αb þ A R σρμAαaA Ac − 3A RμαA ðAa · AbÞ; 1 3 ∇ 2˜ μ − 3˜ μ … ∈ L L2 − L3 L4 L12 2L13 − L16 − L17 − L18 − L10 4L13 − L14 − 2L17 − 2L18 μð J3 J1Þ¼ð 2Þþð 4 4 þ 4Þþ2 ð 4 þ 4 4 4 4 Þ 4 ð 4 þ 4 4 4 4 Þ μa α ρc σ μa α c þ 2A R σρμAαaA Ac þ 3A RμαAaðA · AcÞ; 1 1 ∇ ˜ μ … ∈ L L9 α a c ϵμρσβ μJ5 ¼ð 2Þþ2 4 þ 2 AβaR σρμAαðA · AcÞ ; 1 1 ∇ ˜ μ … ∈ L L7 L8 b α a ϵμρσβ μJ6 ¼ð 2Þþ4 ð 4 þ 4Þþ2 AβR σρμAαðAa · AbÞ ; ð52Þ α where ð… ∈ L2Þ means terms belonging to L2 and R σρμ is the Riemann tensor. We see, therefore, that some terms in Eq. (46) can be dismissed in flat spacetime but not in curved spacetime. Indeed, to remind the reader of this difference, these terms will be traded by their respective curvature-dependent companions that appear in the total derivatives in Eq. (52):

x5 L 2 − 2 L2 − L3 L4 2L5 L19 L6 − L8 L19 − 2 b α a ϵμρσβ ¼ðx2 þ x11 x12Þð 4 4 þ 4Þþ 2 ð 4 þ 4 Þþx6ð 4 4 þ 4 Þ x7AβR σρμAαðAa · AbÞ 3 4 − α a c ϵμρσβ − 2 L10 4L13 − L14 − 2L17 − 2L18 − μa α ρc σ x9AβaR σρμAαðA · AcÞ þ x10 x11 þ 2 x12 ð 4 þ 4 4 4 4 Þ 3 x11½A R σρμAαaA Ac 8 μa αb μa α ρc σ μa α c − 3A RμαA A · A x11 − 2x12 2A R σρμA A A 3A RμαA A · A : 53 ð a bÞ þ 3 ½ αa c þ að cÞ ð Þ

E. Change of basis There are eight linear independent Lagrangian pieces in Eq. (53) which form a basis set for the construction of the Lagrangian involving two derivatives and two vector fields. For purposes that will be clear in the following section, we will perform a change of basis that will affect the third and sixth to eighth Lagrangian basis elements in Eq. (53):

L6 L8 L6 − L8 L19 → L5 − 4 4 4 4 þ 4 4 2 þ 2 1 1 2L5 L19 − L6 − L8 L19 ¼ 2 ð 4 þ 4 Þ 2 ð 4 4 þ 4 Þ; 1 L10 4L13 − L14 − 2L17 − 2L18 → L10 − L14 2L11 − 2L15 4 þ 4 4 4 4 4 ð 4 4 þ 4 4 Þ 3 … ∈ L L2 − L3 L4 − L10 4L13 − L14 − 2L17 − 2L18 ¼ð 2Þþð 4 4 þ 4Þ 4 ð 4 þ 4 4 4 4 Þ 2 ∇ ˜ μ 3˜ μ − μa α ρc σ − 3 μa αb þ 3 f μðJ3 þ J2Þ ½A R σρμAαaA Ac A RμαA ðAa · AbÞg 4 − ∇ 2˜ μ − 3˜ μ − 2 μa α ρc σ 3 μa α c 3 f μð J3 J1Þ ½ A R σρμAαaA Ac þ A RμαAaðA · AcÞg;

104066-11 GALLEGO, RODRÍGUEZ, and GÓMEZ PHYS. REV. D 102, 104066 (2020) 1 μa α ρc σ −3 μa αb → μa α ρc σ −3 μa αb ˜ L10 −L14 2L11 −2L15 A R σρμAαaA Ac A RμαA ðAa ·AbÞ A R σρμAαaA Ac A RμαA ðAa ·AbÞþa 4ð 4 4 þ 4 4 Þ ; 1 2 μa α ρc σ 3 μa α c → 2 μa α ρc σ 3 μa α c ˜ L10 −L14 2L11 −2L15 A R σρμAαaA Ac þ A RμαAaðA ·AcÞ A R σρμAαaA Ac þ A RμαAaðA ·AcÞþb 4ð 4 4 þ 4 4 Þ ; ð54Þ where a˜ and b˜ are arbitrary constants. Thus, the Lagrangian This can be redeemed by adding a specific counterterm so ˆ 1 involving two derivatives and two vector fields is written as that the healthy version of L4 becomes follows: 1 ˆ 1;h b μa ν μa ν a X8 L4 ¼ ðAb · A Þ½Sμ Sνa − Sν Sμa − RðA · AaÞ ˆ i 4 L ¼ αˆ iL4; ð55Þ 1 μa ν μa ν i¼1 b − b − a b þ 2 ðAa · AbÞ½Sμ Sν Sν Sμ RðA · A Þ: ð57Þ with ˆ 5 In contrast, although the decoupling limit of L4, specifi- 1 L19 ˆ 1 10 14 11 15 cally the term 4 , leads as well to higher-order field L4 L − L 2L − 2L ; ¼ 4 ð 4 4 þ 4 4 Þ equations, it turned out impossible to find out the required 11 Lˆ 2 L2 − L3 L4 counterterm. This leaves us with two possibilities: Either 4 ¼ 4 4 þ 4; ˆ 5 we must discard L4, as it is pathological in the decoupling ˆ 3 μa α ρc σ μa αb L4 ¼ A R σρμAα A A − 3A RμαA ðA · A Þ a c a b limit, or we must keep it, because its decoupling limit is 1 degenerate and this property might, in principle, remove the ˜ L10 − L14 2L11 − 2L15 þ a 4 ð 4 4 þ 4 4 Þ ; ghostly degree of freedom [42,43]. We will not know which possibility is the right one until a proper and dedicated ˆ 4 μa α ρc σ μa α c L4 ¼ 2A R σρμAα A A þ 3A RμαA ðA · A Þ a c a c analysis of the degeneracy conditions in the decoupling 1 12 Lˆ 3 Lˆ 4 ˜ L10 − L14 2L11 − 2L15 limit is performed. Finally, 4 and 4 are the non-Abelian þ b 4 ð 4 4 þ 4 4 Þ ; versions of a term in the generalized Proca theory identified ˆ 5 5 19 unequivocally in Ref. [111] as the beyond Proca term L4 ¼ 2L4 þ L4 ; ˆ 3 ˆ 4 [112]. We conjecture then that L4 and L4 are the beyond L6 L8 ˆ 6 5 4 4 generalized SU(2) Proca terms whose decoupling limits L4 ¼ L − þ ; 4 2 2 must satisfy all the conditions required to remove the ˜ ˆ 7 b α a μρσβ Ostrogradski ghosts. This fixes the a˜ and b constants, but, L4 ¼ AβR σρμAαðAa · AbÞϵ ; since the non-Abelian extension of the beyond multi- Lˆ 8 α a c ϵμρσβ 4 ¼ AβaR σρμAαðA · AcÞ ; ð56Þ Galileon theory has not been constructed yet, the actual values of a˜ and b˜ are unknown to us. To circumvent this αˆ where the i are arbitrary constants. We have deliberately lack of knowledge, we can take advantage of the fact that, ordered the Lagrangian pieces this way so that the first four although the Abelian and non-Abelian vector-tensor theo- are the ones that preserve parity while the last four, in ries are different13 despite sharing many aspects in their contrast, are the ones that do not preserve it. construction, the non-Abelian theory stripped of the inter- nal group indices must be contained in the Abelian theory. ˆ 3 F. The decoupling limit Thus, once L4 is stripped of the internal group indices, it Following the general description of Sec. VI E, the becomes decoupling limit of the theory described by Eqs. (55) 3 a → ∇ πa ˆ 3 μ α 2 2 μ ν μ ν and (56), obtained by making the replacement Aμ μ , L4 → −3A RμαA A þ a˜ A ðSμSν − SνSμÞ; ð58Þ must be free of the Ostrogradski instability. This is easy to 4 ˆ 2 ˆ 6 verify for L4 and L4 whose decoupling limits vanish thanks a ˆ 7 11 L19 to the antisymmetry of Aμν. It is also easy to verify for L4 The isolation of 4 in just one Lagrangian piece is Lˆ 8 ∇ ˜ μ ∇ ˜ μ motivated by the impossibility of finding out a counterterm, and 4 having in mind their relation to μJ6 and μJ5, and it is the reason of the first change in basis elements shown in respectively, as shown in Eq. (52), and, again, the anti- the previous section. 12 a ˆ 1 This seems quite nontrivial, so we rather leave it for future symmetry of Aμν. Now, regarding L4, its decoupling limit work. leads to higher-order field equations, because, contrary to 13Abelian theories display some terms whose non-Abelian partial derivatives, covariant derivatives do not commute. versions do not exist and vice versa.

104066-12 GENERALIZED SU(2) PROCA THEORY RECONSTRUCTED AND … PHYS. REV. D 102, 104066 (2020) which must be compared with Eq. (42) in Ref. [111]14: Furthermore, we can replace the second Lagrangian piece in the previous expression as follows: BP μ ν L4 ¼ GNðXÞRμνA A ˆ 4 ˆ 1;h μa ν b 1 L4 þ 3L − 3GμνA A ðA · A Þ − 2 μ ν − μ ν 4 a b ½ XGN;XðXÞþGNðXÞ 4 ðSμSν SνSμÞ; ð59Þ 3 → − b a 2 a b 4 ½ðAb · A ÞðA · AaÞþ ðAa · AbÞðA · A ÞR − 2 2 where X ¼ A = , GNðXÞ is an arbitrary function of X, ˆ 4 ˆ 1;h μa ν b ¼ L4 þ 3L4 − 3GμνA AaðA · AbÞ and GN;XðXÞ is the derivative of GNðXÞ with respect to X. ˆ 3 ˆ 1;h μa νb We see that these two Lagrangian pieces are equivalent for − 2½L4 − 3L4 þ 3GμνA A ðAa · AbÞ; ð63Þ ˆ 4 GNðXÞ¼6X and a˜ ¼ 3. Similarly, once L4 is stripped of ˆ 1;h the internal group indices, it becomes which is indeed very interesting, because now L4 can be replaced by 3 ˆ 4 μ α 2 ˜ 2 μ ν μ ν L4 → 3A RμαA A b A SμSν − SνSμ ; 60 þ 4 ð Þ ð Þ 1 Lˆ 1;h → b μa ν − μa ν 4 4 fðAb · A Þ½Sμ Sνa Sν Sμa which is equivalent to the Lagrangian piece in Eq. (59) for μa νb μa νb ˜ þ 2ðAa · AbÞ½Sμ Sν − Sν Sμ g GNðXÞ¼−6X and b ¼ −3. 3 ˆ 1;h b a ¼ 3L4 þ ½ðAb · A ÞðA · AaÞ G. A new change of basis 4 2 A A Aa Ab R; Having found the actual values for a˜ and b˜ in the þ ð a · bÞð · Þ ð64Þ ˆ 3 ˆ 4 previous section, L4 and L4 acquire the form 1 this just being the original L4, i.e., without its respective ˆ 3 μa α ρc σ μa αb counterterm. L4 ¼ A R σρμAα A A − 3A RμαA ðA · A Þ a c a b All together, we can formulate the reconstructed GSU2P 1 b μa ν μa ν Lagrangian composed of two derivatives and two vector þ 3 ðA · A ÞðSμ Sν − Sν Sμ Þ 4 b a a fields as follows: 1 μ μ a νb a νb 6 4 þ ðAa · AbÞðSμ Sν − Sν Sμ Þ ; X α X α˜ 2 L i Li i L˜ i 4;2 ¼ 2 4;2 þ 2 4;2; ð65Þ ˆ 4 μa α ρc σ μa α c 1 mP 1 mP L4 ¼ 2A R σρμAα A A þ 3A RμαA ðA · A Þ i¼ i¼ a c a c 1 − 3 b μa ν − μa ν where 4 ðAb · A ÞðSμ Sνa Sν SμaÞ 1 μa ν μa ν 1 L b μ − ν μa νb μa νb 4;2 ¼ðAb ·A Þ½S Sνa S Sμa þ ðAa · AbÞðSμ Sν − Sν Sμ Þ ; ð61Þ 2 μa νb μa νb þ2ðAa ·AbÞ½Sμ Sν −Sν Sμ ; L2 a μb ν σ − a μb ν σ a ρb μ ν which can be replaced by 4;2 ¼ AμνSσ AaAb AμνSσ AbAa þAμνSρ AaAb; L3 μa α ρb σ 3 b a 3 4;2 ¼ A R σρμAαaA Ab þ 4ðAb ·A ÞðA ·AaÞR; ð66Þ Lˆ 3 → μa α ρc σ b a 4 A R σρμAαaA Ac þ ðAb · A ÞðA · AaÞR L4 b a 2 a b 4 4;2 ¼½ðAb ·A ÞðA ·AaÞþ ðAa ·AbÞðA ·A ÞR; Lˆ 3 − 3Lˆ 1;h 3 μa νb L5 μa ν b ¼ 4 4 þ GμνA A ðAa · AbÞ; 4;2 ¼ GμνA AaðA ·AbÞ; Lˆ 4 → 2 μa α ρc σ L6 μa νb 4 A R σρμAαaA Ac 4;2 ¼ GμνA A ðAa ·AbÞ; 3 b a a b A · A A · A − 2 A · A A · A R μ þ 4 ½ð b Þð aÞ ð a bÞð Þ L˜ 1 −2 a b ϵνσαβ a νb ϵμσαβ 4;2 ¼ AμνSσ AαaAβb þSμνSσ AαaAβb ; 4 1 ˆ ˆ ;h μa ν b 2 μ αβ ¼ L4 þ 3L4 − 3GμνA AaðA · AbÞ; ð62Þ L˜ a b ϵνσαβ − ˜ b ρa 4;2 ¼AμνSσ AαaAβb Aa SραA Aβb ˜ αβ ρ a b where we have added and subtracted, respectively, the þAa SρbAαAβ; ð67Þ μa νb μa ν b Lagrangian pieces GμνA A ðAa · AbÞ and GμνA AaðA · L˜ 3 b α a ϵμρσβ 4;2 ¼AβR σρμAαðAa ·AbÞ ; AbÞ that exist only in curved spacetime and whose L˜ 4 α a b ϵμρσβ decoupling limit is healthy, since Gμν is divergenceless. 4;2 ¼AβaR σρμAαðA ·AbÞ ;

α α˜ 14This is the reason of the third and fourth changes in basis the i and i being arbitrary dimensionless constants, mP ˜ μν 1 μνρσ elements shown in the previous section. being the reduced Planck mass, Aa ≡ 2 ϵ Aρσa being the

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a i Hodge dual of Aμν, and the Lagrangian pieces having been where the xi are arbitrary constants and the L5 are the ones deliberately split into those that preserve parity (the ones in Eq. (69), makes the GSU2P Lagrangian built with just without a tilde) and those that do not preserve it (the ones three derivatives. Because no single vector field appears in with a tilde). It is worthwhile mentioning that the subscripts this Lagrangian, the secondary constraint-enforcing rela- 4,2 have been introduced to remind the reader that two tion is trivially satisfied. Regarding the primary constraint- derivatives and two vector fields have been employed to enforcing relation, the primary Hessian gives the following build the different Lagrangian pieces. result:

H0ν 2 c ϵνρσ0ϵ 2 VIII. THREE DERIVATIVES ab ¼ Aρσ bcaðx1 þ x3Þ 0νc 0ν 00c 0νc A. Lagrangian building blocks þ 4ðS þ g S − A Þϵabcx2; ð71Þ Terms of the form AμνAρσSαβ, AμνSρσSαβ, and SμνSρσSαβ, that involve six spacetime indices, are the ones that become which vanishes only if the Lagrangian building blocks of a Lagrangian built with just three derivatives once they are contracted with the 2 0 terms in Eqs. (13) and (14). Upon the contractions, the only x1 þ x3 ¼ ; blocks that either do not vanish or have the potential of x2 ¼ 0: ð72Þ becoming nonvanishing once the internal group indices are introduced are the following: The Lagrangian that satisfies the constraint algebra is, μν α therefore, AμνA Sα; μ νσ AμνA σS ; 1 3 L ¼ x3ð−2L þ L Þ: ð73Þ ν μρσβ 5 5 AμνAρσSβϵ ; α μνρσ AμνAρσSαϵ ; μ νσ AμνSσS ; ðÞ C. Total derivatives ρ μνσβ AμνSρσSβϵ ; ðÞ As with the other Lagrangians involving a different μ ρ α number of derivatives and/or vector fields, we must be sure SμSρSα; that the Lagrangian in Eq. (73) is not redundant compared μ ρσ SμSρσS ; with terms in L2. To this end, we must construct total μν μ νσ derivatives of terms involving two derivatives and one S SσS : 68 ð Þ vector field, i.e., total derivatives of the form ∂μ½Aνð∂ρAσÞð∂αAβÞ. These terms involve six spacetime The introduction of the internal group indices makes indices, so that they must be contracted with those terms in these terms become of the form Aa Ab Sc, Aa SbSc,or fg fg fg Eq. (13) and (14). However, since the Lagrangian in a b c S S S , involving three internal group indices, which lead Eq. (73) does not preserve parity, it will be enough to to group-invariant Lagrangian building blocks upon con- contract with the terms in Eq. (14). Thus, the only terms ϵ tractions with abc. Most of these blocks, however, vanish that either are nonvanishing or can become nonvanishing ϵ because of the antisymmetric nature of abc, the only once the internal group indices are added are the following: survivals being

ν μσαβ 1 a b νc μρσβ ∂μ½Aνð∂ AσÞð∂αAβÞϵ ; L5 ¼ AμνAρσSβ ϵ ϵabc; ν μραβ 2 a μb νσc ∂μ½Aνð∂ρA Þð∂αAβÞϵ ; L5 ¼ AμνSσ S ϵabc; μναβ 3 a b ρc μνσβ ∂μ½Aνð∂ · AÞð∂αAβÞϵ ; L5 ¼ AμνSρσSβ ϵ ϵabc: ð69Þ ρ μνσβ ∂μ½Aνð∂ρAσÞð∂ AβÞϵ ; ðÞ ρ μνσα ∂μ½Aνð∂ρAσÞð∂αA Þϵ ; B. The Hessian constraints σ μνρα ∂μ½Aνð∂ρAσÞð∂αA Þϵ ; ðÞ ð74Þ The linear combination

X3 which, in turn, can be contracted only with ϵabc after adding i L ¼ xiL5; ð70Þ the internal group indices, since the total derivatives acquire i¼1 the form ∂½Aað∂AbÞð∂AcÞ:

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μ a ν b c μσαβ ˜ μ ˜ μ ∂μJ1 ¼ ∂μ½Aν ð∂ AσÞð∂αAβÞϵ ϵabc; ∇μð2J1 þ J2Þ μ a νb c μραβ … ∈ L ∂μJ2 ¼ ∂μ½Aν ð∂ρA Þð∂αAβÞϵ ϵabc; ¼ð 2Þ μ a b c μναβ 3 1 3 ∂μJ3 ¼ ∂μ½Aν ð∂ · A Þð∂αAβÞϵ ϵabc; −2L L þ 8 ð 5 þ 5Þ ∂ μ ∂ a ∂ b ∂ρ c ϵμνσβϵ μJ4 ¼ μ½Aν ð ρAσÞð AβÞ abc; 1 νa σ b c ϵμραβϵ μ a b ρc μνσα þ A R νρμAσAαβ abc: ð77Þ ∂μJ5 ¼ ∂μ½Aν ð∂ρAσÞð∂αA Þϵ ϵabc; 2 ∂ μ ∂ a ∂ b ∂ σc ϵμνραϵ μJ6 ¼ μ½Aν ð ρAσÞð αA Þ abc: ð75Þ The Lagrangian in Eq. (73) is, therefore, not redundant against L2 in curved spacetime. As a remainder of 1 3 this fact, we will dismiss −2L5 þ L5 in favor of 1 νa σ b c μραβ As the reader has already learned, these total derivatives are 2 A R νρμAσAαβϵ ϵabc. We conclude then that the recon- completely useless unless the second derivatives they structed GSU2P exhibits the following Lagrangian built produce may be canceled out. After a careful observation from just three derivatives: of these terms, only two are able by themselves to get rid of the second derivatives in flat spacetime thanks to the ˜ νa σ b ˜ μρc L5 0 ¼ A R νρμAσA ϵ : ð78Þ antisymmetry of the Levi-Civita tensor: ; abc E. The decoupling limit ˜ μ μ ∂μJ ≡∂μJ Since the Lagrangian given in the previous expression 1 2 a → ∇ πa 1 vanishes in the decoupling limit Aμ μ , because of the a νb c ϵμραβϵ ˜ a ¼ 8 AμνAρ Aαβ abc antisymmetry of Aμν, it is free of the Ostrogradski 1 instability. −2L1 L3 þ 8 ð 5 þ 5Þ 1 IX. COMPARISON WITH THE “OLD” GSU2P a νb c a νb c þ fAν ½∂μ; ∂ρA A þ Aν Aρ ½∂μ; ∂αA 4 αβ β The old GSU2P, formulated in Ref. [103], is described a νb c μραβ by the following Lagrangian: þ Aν Sρ ½∂μ; ∂αAβgϵ ϵabc; μ μ ˜ 3 4 ∂μJ2 ≡∂μJ6 X X Lold Lold α Li;old β Li;old 1 ¼ 2 þ i 4 þ i Curv; ð79Þ a b σcϵμνραϵ 1 ¼ 8 AμνAρσAα abc i¼ i¼i 1 α β −2L1 L3 where the i and i are dimensionful arbitrary constants, þ ð 5 þ 5Þ old old a a a a 8 L2 ≡ L2 ðAμν;AμÞ is an arbitrary function of Aμν and Aμ, 1 a ∂ ∂ b σc and þ 2 fAν ½ μ; ρAσAα 1;old b μa ν μa ν a a b σc μνρα L4 ¼ðA · A Þ½Sμ Sν − Sν Sμ − RðA · A Þ þ Aν ½∂μ; ∂ρAσSα gϵ ϵabc: ð76Þ b a a a μa νb μa νb a b þ 2ðAa · AbÞ½Sμ Sν − Sν Sμ − RðA · A Þ; 2;old μa νb μa νb a b L ¼ðA · A Þ½Sμ Sν − Sν Sμ − RðA · A Þ Indeed, from this result and the findings in Secs. III and 4 a b ˜ μ ˜ μ a b μα ν − μα ν VIII B, we can see that employing either ∂μJ1 or ∂μJ2 is þ AμAν ½Sa Sαb Sb Sαa allowed, since they satisfy the Hessian constraints. The 2 μα ν − 2 μα ν 2 μνρσ þ Aa Sαb Ab Sαa þ AρaAσbR ; conclusion is that the Lagrangian in Eq. (73) is already L3;old μ ˜ b σνa contained in L2 in flat spacetime, up to a total derivative, so 4 ¼ AaAμσS Aνb; that, in this framework, the GSU2P does not contain terms L1;old μa ν Curv ¼ GμνA Aa; built exclusively with three derivatives that are linearly L2;old μνa ρσ independent of L2. The conclusion is, nonetheless, com- Curv ¼ LμνρσA Aa ; pletely different in curved spacetime. L3;old μνa ρb σcϵ Curv ¼ LμνρσA A A abc; 4;old μa νb ρ σ L ¼ LμνρσA A AaA ; ð80Þ D. Covariantization Curv b 1 αβγδ The minimal covariantization scheme applied to the where Lμνρσ ≡ ϵμναβϵρσγδR is the double dual of the ˜ μ ˜ μ 2 suitable combination ∂μð2J1 þ J2Þ of terms in Eq. (76) Riemann tensor. This old theory was built following the leads to same steps that we followed here except for three aspects:

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3;old (1) All the Lagrangian building blocks were constructed Something similar occurs for L4 : a employing the full ∂μAν instead of splitting it into its a a 3;old ˜ μ symmetric Sμν and antisymmetric Aμν parts. This, of L4 þ 2∂μJ4 ¼ð… ∈ L2Þ course, produced a lot more blocks (and a lot more 1 μ work) than needed, many linear combinations of L˜ 1 4∂ ˜ þ 2 ð 4;2 þ μJ6Þ; ð83Þ them already included in L2. (2) Only the primary constraint-enforcing relation was 3 at the flat spacetime level, so although neither L ;old nor considered. As was shown in Refs. [109,110], this is 4 ∂ ˜ μ L3;old 2∂ ˜ μ not enough to remove the Ostrogradski ghost. μJ4 are healthy, the combination 4 þ μJ4 is, and, (3) Many terms were dismissed by employing total therefore, all the physics extracted from the unhealthy 3;old derivatives already at the flat spacetime level, which curved spacetime version of L4 is equivalent to that 3;old ˜ μ led to a loss of several terms that exist only in curved extracted from the healthy L4 þ 2∇μJ4. spacetime, including the beyond SU(2) Proca ones. Now, as can be seen in Eq. (67), there exist only two Moreover, most of the total derivatives employed do parity-violating terms in flat spacetime in the reconstructed not satisfy the secondary Hessian constraint. GSU2P. Then, why is it that in the old GSU2P there exists The application of this theory to inflation and dark energy L˜ 1 only one? Leaving aside the fact that 4;2 might be was investigated in Refs. [106,107], and the stability unhealthy in its decoupling limit, the reason lies in a small analysis of the same was performed in Ref. [108],sowe mistake in the conditions of Eq. (37) in Ref. [103] to make wonder how the results of these works could change in the the primary constraint-enforcing relation vanish that pre- light of the new theory presented in this paper. vented the authors of that work from finding a second Lˆ 1;h L1;old As can be seen, our 4 in Eq. (57) is identical to 4 , parity-violating Lagrangian piece. this being one of the reasons of the second change in basis Lold Finally, among the Curv of Eq. (80), the only one that elements in Sec. VII E. Examined from the viewpoint of the 1 appears in the reconstructed theory is L ;old, which is L1;old Curv reconstructed GSU2P [see Eqs. (65) and (66)], 4 can L Lold exactly the same as our 4;0 of Eq. (38). The other Curv also be written as were just postulated, as they are obviously healthy because of the divergenceless nature of Lμνρσ. We could have postulated them as well in the reconstructed GSU2P, but L1;old L1 − L4 : 4 ¼ 4;2 4;2 ð81Þ we would rather not do it. This is because we expect them to naturally appear in the theory when more than six spacetime indices are considered in the Lagrangian build- L1;old Thus, we conclude that 4 is free of the Ostrogradski ing blocks without contractions. ghost (at least in flat spacetime). 2 Now, L ;old was shown in Ref. [110] not to satisfy the 4 X. COMPARISON WITH THE GENERALIZED secondary Hessian constraint and, so, as an example of the PROCA THEORY ghost instabilities that plagued the old GSU2P. Never- theless, a bit of algebra shows us that Finding the beyond GSU2P in Sec. VII required deter- mining the values of the constants a˜ and b˜ in Eq. (56).We could have followed the standard procedure of finding out 2;old ˜ μ L4 − 2∂μJ2 ¼ð… ∈ L2Þ the kinetic matrix of its decoupling limit and making it 1 1 degenerate [42,43]. However, we followed an alternative L1 − L2 þ 12 4;2 3 4;2 route based on the fact that the GSU2P stripped of the 5 1 internal group indices must be contained in the generalized − ∂ ˜ μ 3˜ μ ∂ 2˜ μ − 3˜ μ 9 μðJ3 þ J2Þþ9 μð J3 J1Þ; ð82Þ Proca theory. Indeed, the other reason why we performed the second change in basis elements in Sec. VII E is that ˆ 1;h L4 stripped of the internal group indices is nothing else at the flat spacetime level, where the quantities in this than L4 of the generalized Proca theory (see Ref. [65]): 2;old expression, except for L4 , are those of Sec. VII. Thus, 2;old ˜ μ G4;XðXÞ μ ν μ ν although neither L is healthy, nor ∂μJ is, the combi- L μ − ν 4 2 4 ¼ G4ðXÞR þ 4 ðS Sν S SμÞ; ð84Þ 2;old ˜ μ nation L4 − 2∂μJ2 satisfies the secondary constraint- 2 enforcing relation, and, therefore, all the physics extracted for G4ðXÞ¼−3X . Then, what about the other Lagrangian 2;old from the unhealthy curved spacetime version of L4 , for pieces that make L4;0 and L4;2? First of all, L4;0 stripped of instance, in Ref. [108], is equivalent to that extracted from the internal group indices is just L4, up to a total derivative, L2;old − 2∇ ˜ μ L2 L˜ 1 the healthy 4 μJ2. with G4ðXÞ¼X. In contrast, 4;2 and 4;2 reduce to zero

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L5 comparison with the GW170817 event frequency [22] when stripped of the internal group indices. Regarding 4;2 L6 (to see whether the bound on the gravitational waves speed and 4;2 without internal group indices, they are just healthy 16 μ ν applies to GSU2P ). We might as well construct an extensions of GμνA A that were not recognized in extended version of this theory, considering all the pos- L˜ 2 L˜ 3 L˜ 4 Ref. [103]. Finally, 4;2, 4;2,and 4;2,strippedoftheir sibilities to degenerate the kinetic matrix in curved space- internal group indices, reduce, up to total derivatives, to time, as was done for the generalized Proca theory in ˜ βα ρ AβA SαρA , which was shown in Refs. [106,119] to be part Ref. [119,120]. The theory must, of course, be put under of L2 up to a total derivative. To end up, the only parity- test against observations; in this regard, determining violating terms in the generalized Proca theory belong to L2 whether there exists a screening mechanism at Solar ˜ [66],soL5;0 stripped of its internal group indices should be System scales, as was studied in Ref. [78] for the either zero, a total derivative, or contained in L2;infact, generalized Proca theory, is a crucial aspect. Of course, observing Eq. (78), the first alternative is the correct one. the cosmological and astrophysical implications must be properly studied both at the background (see, for instance, XI. CONCLUSIONS Ref. [107]) and at the perturbative level (see, for instance, Refs. [125–127]). We finish this paper by reminding the GSU2P and beyond GSU2P are described by the readers and ourselves of one important message given to us – Lagrangians in Eqs. (38), (65) (67), and (78). The theory by Misner, Thorne, and Wheeler in their marvelous treatise has been written so as to make it explicit which Lagrangian on gravitation [113]: “To be complete, a theory of gravity pieces exist only in curved spacetime and which ones exist must be capable of analyzing from ‘first principles’ the even in flat spacetime; indeed, from the 12 Lagrangian outcome of every experiment of interest. It must therefore L1 L2 L˜ 1 pieces that compose the theory, only four, 4;2, 4;2, 4;2, mesh with and incorporate a consistent set of laws for L˜ 2 and 4;2, survive in flat spacetime. The nature of some of electromagnetism, quantum mechanics, and all other phys- the Lagrangian pieces is purely non-Abelian—i.e., they ics.” There is a long road in this direction ahead of us that we vanish when stripped of their internal group indices— hope we will travel. L2 L˜ 1 L˜ specifically, 4;2; 4;2, and 5;0 belong to this subset. It is L˜ 2 ACKNOWLEDGMENTS worthwhile mentioning that 4;2 is the parity-violating L2 version of 4;2 as can be easily observed. On the other We appreciate all the discussions and exchange of hand, the theory is diffeomorphism invariant, so that the ideas we had with Juan Camilo Garnica Aguirre and energy and momentum are locally conserved [113]. Carlos Mauricio Nieto Guerrero which helped a lot in the Much remains to be done in the exploration of this theory development of this work. The work presented here was as a candidate of an effective theory for the gravitational supported by the following grants: Colciencias-Deutscher interaction. First of all, it is not clear yet whether the Akademischer Austauschdienst Grant No. 110278258747 decoupling limits of the beyond GSU2P terms as well as RC-774-2017, Vicerrectoría de Ciencia, Tecnología, ˜ 1 15 e Innovación—Universidad Antonio Nariño Grant that of L4 2 are actually healthy. Other self-consistency ; No. 2019248, and Dirección de Investigación y Extensión issues must be addressed, such as the possible existence of — ghosts (other than the Ostrogradski one) and Laplacian de la Facultad de Ciencias Universidad Industrial instabilities, as a follow-up of the work in Ref. [108], the de Santander Grant No. 2460. A. G. C. was supported by generalization of the constraint algebra to curved spacetime Beca de Inicio Postdoctoral REXE RA No. 315-3269-2020 [121,122], the analysis of the causal structure [123], and the Universidad de Valparaíso. L. G. G. was supported by the calculation of the cutoff scale of the theory and its postdoctoral scholarship No. 2020000102 Vicerrectoría de Investigación y Extensión - Universidad Industrial de Santander. Some calculations were cross-checked with 15 It is unlikely that the decoupling limits of the beyond GSU2P the Mathematica package xAct. terms are unhealthy: There is actually no reason to believe that healthy beyond extensions do exist for the Horndeski theory and the generalized Proca theory but do not for the GSU2P. In L˜ 1 16 contrast, there is no clue regarding the healthiness of 4;2, this How the gravitational wave speed bound affects the gener- term being of a purely non-Abelian nature. alized Proca theory was investigated in Ref. [124].

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