Extended Chern-Simons model for a vector multiplet

We consider a gauge theory of vector fields in $3d$ Minkowski space. At the free level, the dynamical variables are subjected to the extended Chern-Simons (ECS) equations with higher derivatives. If the color index takes $n$ values, the third-order model admits a $2n$-parameter series of second-rank conserved tensors, which includes the canonical energy-momentum. Even though the canonical energy is unbounded, the other representatives in the series can have bounded from below $00$-component. The theory admits consistent self-interactions with the Yang-Mills gauge symmetry. The Lagrangian couplings preserve the unbounded from below energy-momentum tensor, and they do not lead to a stable non-linear theory. The non-Lagrangian couplings are consistent with the existence of conserved tensor with a bounded from below $00$-component. These models are stable at the non-linear level. The dynamics of interacting theory admits a constraint Hamiltonian form. The Hamiltonian density is given by the $00$-component of the conserved tensor. In the case of stable interactions, the Poisson bracket and Hamiltonian do not follow from the canonical Ostrogradski construction. The particular attention is paid to the"triply massless"ECS theory, which demonstrates instability already at the free level. It is shown that the introduction of extra scalar field, serving as Higgs, can stabilize dynamics in the vicinity of the local minimum of energy. The equations of motion of stable model are non-Lagrangian, but they admit the Hamiltonian form of dynamics with a bounded from below Hamiltonian.


Introduction
The higher derivative theories are notorious for better convergency properties at classical and quantum level, and a wider symmetry. In most of the instances, these advantages come with a price of dynamics instability, being the typical problem for the models in this class. At the classical level the solutions to the equations of motion demonstrate runaway behavior ("collapse"). At the quantum level the ghost poles appear in the propagator, so the unitarity of dynamics is an issue. These peculiarities follow from a single fact: the canonical energy is unbounded in every non-singular higher derivative theory. For a review of the problem, we cite recent articles [1][2][3] and references therein. The canonical energy of singular models can be bounded. The examples include f (R)-theories of gravity [4][5][6][7].
These models do not demonstrate instability. The stability problem for constrained higher derivative theories is discussed in [8]. In the majority of interesting (constrained or unconstrained) models, the instability has the special form: the classical dynamics is regular (no "collapse"), but the canonical energy is unbounded from below. The Pais-Uhlenbeck oscillator [9], Podolsky [10] and Lee-Wick [11,12] electrodynamics, ECS theory [13], and conformal gravity [14] are examples. For the current studies we mention [15][16][17][18] and references therein. In all these models the quantum theory is expected to be well-defined [2,19], but the application of the canonical quantization procedure based on the Ostrogradski procedure 1 leads to the model with an unbounded from below spectrum of energy. That is why the stability and unitarity of quantum dynamics is the most important issue.
Recently, it has been recognized that the higher derivative dynamics can be stable at the classical and quantum levels even if the canonical energy of the model is unbounded. The articles [25,26] explain the stability of Pais-Uhlenbeck theory form the viewpoint of existence of the Hamiltonian form of dynamics with a bounded (non-canonical) Hamiltonian. The quantization of the model with an alternative Hamiltonian leads to quantum theory with a well-defined vacuum state with the lowest energy. The papers [27,28] use a special PT-symmetry for construction of stable quantum mechanics in the higher derivative oscillator model. The articles [1][2][3]29] tell us that the non-linear higher derivative models may have a well-defined classical dynamics without "collapsing" trajectories with the runaway behavior. In the last articles, the existence of stable classical dynamics serves as a necessary prerequisite for construction of a well-defined quantum theory. For the studies of field theories, we refer to articles [30,31]. All the mentioned in this paragraph models have one common feature: they admit an alternative (different from the canonical energy) bounded from below conserved quantities. The quantum stability is achieved if the model admits Hamiltonian form of dynamics, with the bounded conserved quantity being the Hamiltonian. This means that the stable higher derivative theory is characterized by two ingredients: the bounded conserved quantity, and the Poisson bracket on its phase space that brings the equations of motion to the Hamiltonian form with a bounded from below Hamiltonian.
In the article [32], the stability is studied in the class of models of derived type. At the free level, the wave operator of the theory is given by a polynomial (characteristic polynomial) in another formally self-adjoint operator of lower order. It has been shown that each derived theory admits a series of conserved tensors, which includes the canonical energy-momentum [33]. The number of entries in the series grows with the order of characteristic polynomial. Even though the canonical energy is unbounded, the other conserved tensors in the series can be bounded from below [33]. The bounded conserved quantity stabilizes the classical dynamics of the theory. The quantum stability is explained by the existence of the Lagrange anchor that relates a bounded conserved quantity with the invariance of the model with respect to the time translations. 2 In the first-order formalism, the Lagrange anchor defines the Poisson bracket [36]. The Hamiltonian is given by the conserved quantity in the series, being expressed in terms of phase-space variables. The linear higher derivative theory is stable if a bounded conserved quantity, being related to the time-translation symmetry, is admitted by the model [32]. The articles [37,38] consider a problem of consistent deformations of symmetries and conserved quantities that preserve the stability of dynamics in the class of derived type models.
In all the cases interaction vertices do not come from the least action principle with higher derivatives, but the equations of motion admit the Hamiltonian form of dynamics. The main difficulty of the cited above procedures is that they do not automatically preserve gauge invariance. This restricts their applications in gauge systems.
The ECS model [13] is the simplest gauge theory of derived type. In the current studies, it is often serves as a prototype of the class of gauge theories with higher derivatives, see eg. [39][40][41]. The stability of the ECS model has been first studied in [33]. It has been observed that the theory of order p admits an p − 1-parameter series of second-rank conserved tensors, which includes the canonical energy-momentum. The canonical energy of the model is always unbounded from below. The other tensors in the series may have a bounded from below 00-component. The stability conditions for the theory are determined by the structure of the roots of characteristic polynomial [33]. The theory is stable if all the nonzero roots of characteristic polynomial are real and different and zero root has multiplicity one or two. The stability of the ECS theory is confirmed in work [42]. The ECS model has been shown to be a multi-Hamiltonian at the free level in [43]. The series of Hamiltonians includes the canonical (Ostrogradski) Hamiltonian, which is unbounded, and the other representatives, being bounded or unbounded depending on the model parameters. If the Hamiltonian is bounded, it ensures the stability of the model at both classical and quantum levels. The model admits inclusion of stable non-Lagrangian interactions with scalar, fermionic and gravitational fields that preserve a selected representative in the series of conserved quantities of free model [38,43,44]. However, the gauge symmetry is abelian in the sector of vector field in all these examples.
The concept of consistency of interaction between the gauge fields has been developed in [45].
The consistent couplings between gauge vectors has been studied in [46,47]. It has been shown that the Yang-Mills vertex is unique in the covariant setting. The Lagrangian self-interactions of the gauge vector multiplet subjected to the ECS equations at free level has been reconsidered in article [42].
The most general consistent non-linear theory is proven to have the Yang-Mills gauge symmetry, and the Lagrangian is given by the covariantization of the free ECS Lagrangian. The interacting model demonstrates the Ostrogradski instability at the non-linear level in all the instances, even though the free theory is stable. This result implies a no-go theorem for stable Lagrangian interactions in the ECS model. The conclusion is not surprising because the Lagrangian couplings preserve the conservation law of canonical energy, being unbounded already at the free level. The inclusion of non-Lagrangian interactions can solve the issue of the dynamics stability at the interacting level, because such couplings preserve bounded conserved quantities. However, the problem of construction of stable consistent non-Lagrangian (self-)couplings (stable or unstable) has not been studied in the ECS model in the literature before.
In the current work, we present a class of stable self-interactions in the theory of vector multiplet.
The free fields are subjected to the ECS equations of third order. The interaction is (in general) non-Lagrangian. The non-linear equations of motion are consistent with the Yang-Mills gauge symmetry.
A selected second-rank conserved tensor of the free model is preserved by the coupling. Depending on the values of coupling constants, it can be bounded or unbounded. The Lagrangian interaction vertex of the article [42] is unstable. The non-Lagrangian coupling can be consistent with the existence of bounded conserved tensor. The equations of motion admit the Hamiltonian form of dynamics. On shell, the Hamiltonian density is given by the 00-component of the conserved tensor. For Lagrangian interactions, the canonical formalism with the unbounded Ostrogradski Hamiltonian is reproduced.
The bounded Hamiltonians do not follow from the Ostrograski procedure, so we have non-canonical Hamiltonian formalism in this case. In all the instances, the Poisson bracket is a non-degenerate tensor, so the model admits a Hamiltonian action principle. The quantization of the first-order action with the bounded Hamiltonian leads to a stable quantum theory with a well-defined vacuum state.
In the case of resonance (multiple roots of charlatanistic polynomial), the dynamics of the theory is unstable already at the free level. The inclusion of self-interaction does not improve stability properties of dynamics of this model because the conserved quantities have one and the same structure in the free and non-linear models. In the current article, a model with the third-order zero resonance root is of interest. The free field is subjected to the "triply massless" Chern-Simons (CS) equations.
The wave operator of free equations is given by a cube of the CS operator. To stabilize the dynamics at the non-linear level, we apply the "Higgs-like" mechanism described in [48]. Introducing an extra scalar, we generate a coupling such that the energy of the interacting theory gets a local minimum The article is organized as the following. In Section 2, we consider the third-order ECS model for a vector multiplet. The particular attention is paid to the structure of symmetries and conserved quantities of the theory, and the stability of dynamics. In Section 3, we propose stable self-interactions between the multiplet of vector fields with the Yang-Mills gauge symmetry. The general non-model in the class model preserve a selected conserved tensor of the free theory, which can be bounded or unbounded from below depending on the values of coupling constants. The stable interactions are inevitably non-Lagrangian. In Section 4, we construct the constrained Hamiltonian formalism for the non-linear model. The density of Hamiltonian is given by the 00-component of the conserved tensor, being expressed in terms of the phase-space variables. In Section 5, we consider the issue of stability of "triply massless" ECS theory. We propose the class of consistent couplings with the scalar field stabilizing the dynamics in the vicinity of equilibrium position. The equations of motion are non-Lagrangian, but they admit the constrained Hamiltonian form with a bounded Hamiltonian. The concluding section summarizes the results of the article.

Higher-derivative Chern-Simons model
We consider the most general third-order gauge theory of the vector multiplet A a = A a µ (x)dx µ , µ = 0, 1, 2, a = 1, 2, . . . , n on three-dimensional Minkowski space. The action functional of the model reads The real numbers α 1 , α 2 , α 3 are model parameters. Without loss of generality, we assume that the coefficient at the higher derivative term is nonzero, α 3 = 0. The vectors F a µ , G a µ , K a µ denote the (generalized) field strengthes of the potential A a µ , with the Levi-Civita symbol ǫ µνρ , ǫ 012 = 1 being antisymmetric with respect to indices. All the tensor indices are raised and lowered by the Minkowski metric. We use the mostly positive convention for the signature of metric throughout the paper. The summation is implied over repeated at one level isotopic indices a = 1, . . . , n unless otherwise stated. For α 3 = α 1 = 1, α 2 = 0, n = 1, the action functional (1) has been first proposed in the article [13]. We call the model (1) the ECS theory for a vector multiplet.
The least action principle for the functional (1) brings us to the following Lagrange equations for the field A µ : These equations have the derived form [33] because the wave operator is a polynomial in the CS operator * d. The symbol * denotes the Hodge star operator, and d is the de Rham differential. The structure of the Poincare group representation, being described by the equation (3), is determined by the roots of the characteristic polynomial The polynomial M(α; z) follows from (3) after the formal replacement of the CS operator * d by the complex-valued variable z. The following cases are distinguished in [33].
(1b) α 1 = 0 , α 2 2 − 4α 1 α 3 < 0. The characteristic polynomial has a zero root and two complex roots. The case is similar to (1a), but the masses of vector modes are complex. The Poincare group representation is non-unitary.
(2a) α 2 = 0, α 1 = 0. The characteristic polynomial has a multiplicity two zero root and simple nonzero root. The set of subrepresentations includes a massless spin-1 field and a massive spin-1 mode subjected to the self-duality condition.
As we see, the ECS model describes the unitary representation of the Poincare group if all the nonzero roots of characteristic equation are different, and the zero root has multiplicity one or two.
The dynamical degrees of freedom include the massive spin 1 vector subjected to the self-duality condition and/or spin 1 massless field, which meets the 3d Maxwell equations. In all the instances, the model has two local physical degrees of freedom (four physical polarizations).
The action functional (1) is preserved by the 2n-parameter series of infinitesimal transformations (no summation in a). The transformation parameters are the constant vector ξ µ and real numbers β a k , a = 0, . . . , n, k = 1, 2. The series (5) includes the space-time translations with the independent parameters for the individual vector A a µ in the multiplet, and higher order transformations, whose value is determined by β a 2 . The Noether theorem associates symmetries (5) with the 2n-parameter series of second-rank conserved tensors, which has the form where (no summation in a). The quantities Θ aµν 1 , a = 1, . . . , n (7) represent the canonical energymomentum tensors of individual fields in the vector multiplet A a µ . The tensors Θ aµν 2 , a = 1, . . . , n (8) are other conserved quantities. The total number of independent conserved tensors in the free theory is 2n because each field in the multiplet admits two different symmetries.
The 00-component of the conserved tensor (6) reads The summation over repeated at one level index i = 1, 2 is implied. The quadratic form (9) can be reduced to the principal axes as follows Here, X a µ = β a 1 F a µ + β a 2 G µ , and the notation is used In Section 4, we see that X a0 X a0 + X ai X ai and F a0 F a0 + F ai F ai depend on different initial data. 4 In this case, the 00-component (9) is bounded from below if the coefficients at squares are positive, We ignore the case of semi-positive quadratic form because the degenerate conserved quantities do not ensure the stability of dynamics. Relations (12) are consistent if and only if the model parameters

Stable interactions
In this section, we present an example of stable self-interactions in the model (1). The interactions are non-Lagrangian. The dynamics of the non-linear theory is determined by the equations of motion.
The interactions are associated with the deformations of free equations of motion that preserve the gauge symmetries and gauge identities of free model. The interaction is consistent if the number of physical degrees of freedom is preserved by coupling. For details of the concept of consistency of interaction in the class of not necessary Lagrangian theories, we refer article [53].
We start construction of non-linear theory by assuming that the dynamical fields take values in the Lie algebra of a semisimple Lie group with the generators t a , a = 1, . . . , n, The covariant analogs of the (generalized) field strength vectors (2) are defined as follows: The vectors F µ , G µ , K µ lie in the Lie algebra of a semisimple Lie group, and D stands for the covariant derivative The vector F µ represents a dual of the standard Yang-Mills strength tensor. The generalized field strengthes G µ , K µ are other covariant quantities that involve second and third derivatives of A µ .
We consider the interactions that are polynomial in the gauge invariants (14) and does not involve the highest derivative in the non-linear part. In this setting, the most general self-consistent non-linear theory is determined by the following equations of motion: We prove the uniqueness of this coupling in Appendix A. The model parameters are the real numbers α k , k = 1, 2, 3, β l , l = 1, 2, and f abc , a, b, c = 1, . . . , n. The constants α 1 , α 2 determine the free limit of the equations (16). The numbers β 1 , β 2 distinguish admissible couplings with one and the same gauge group. Throughout this section and below, we assume that C(β; α) = 0. The equations of motion (16) come from the least action principle if β 1 = 1, β 2 = 0. In this case, the action functional This action functional was first derived in [42]. 5 The same paper tells us that the (17) is the most general form of consistent self-interactions in the gauge theory of vector fields. This means that the most general consistent Lagrangian coupling (17) is included into the model (16). If the parameter β 2 is nonzero, equations (16) do not follow from the least action principle for any functional with higher derivatives. The variational principle with auxiliary fields still exists, even if the higher derivative model is non-Lagrangian. In the last case, the theory (16) admits consequent quantization, and establishing relationship between symmetries and conserved quantities.
The concept of interaction consistency for not necessarily Lagrangian theories has been developed in [53]. This paper tells us that the non-Lagrangian interaction is consistent if the non-linear theory admits the same number of (i) gauge symmetries, (ii) gauge identities, and (iii) physical degrees of freedom as a free model. All these fact are easily verified. At first, the equations of motion (16) are preserved by the Yang-Mills gauge symmetry, with ζ = (ζ a (x), a = 1, . . . , n) being the gauge transformation parameter. The free model (17) is preserved by the standard gradient gauge symmetry, δ ζ A a µ = ∂ µ ζ a . As it is required, the gauge symmetry (18) is given by the deformation of the gradient gauge symmetry of free model. The important difference is that the gauge symmetry (18) is non-abelian. So, the inclusion of interaction 5 The higher-derivative Yang-Mills theory having a similar structure of the Lagrangian has been known long before [54]. leans to the model with non-abelian gauge symmetry. We have an obvious set of gauge identities between the equations (16), Again, the leading term of the gauge identity is given by the free contribution. This agrees with the concept of interaction consistency. At the final step of analysis, we verify that the physical degrees of freedom number is preserved by the coupling. Equation (8) (16) is consistent for the general values of the parameters β, α.
The theory (16) admits a symmetric conserved tensor of second rank of the following form: with α k , k = 1, 2, 3 and β l , l = 1, 2 being the model parameters. The divergence of the quantity Expression (20) is the covariantization of a selected representative in the conserved tensor series (6), As is seen, the model (16) represents the class of deformations of free ECS equations (3) that preserves a selected representative in the series (6) conserved quantities at the non-linear level. It is important to note that the other representatives in the series (20) no longer conserve in the non-linear theory (16).
This happens because the parameters β k , k = 1, 2 in the conserved tensor (20) are unambiguously fixed by the interaction.
The conserved tensor is a bilinear form in (generalized) strengthes F µ , G µ . The bringing of the quadratic form (23) to the principal axes reveals that the model is stable if These conditions can be consistent or inconsistent depending on the values of the model parameters α k , k = 1, 2, 3 and β l , l = 1, 2. The Lagrangian interaction vertex (17) does not meet stability requirements. This confirms the instability of the variational coupling proposed in [42]. The stable interactions in the class of models (16) are inevitably non-Lagrangian. The similar form of stability conditions (3), (16) at the free and interacting cases implies that the linear and non-linear dynamics is stable or unstable simultaneously. In the class of stable at the linear level theories, formula (16) determines a class of non-linear models that preserve a selected bounded conserved quantity in the series (6). Now, we can return to the special case C(β; α) = 0, which is excluded in (16). The conserved quantity (6), (22) is a degenerate quadratic form of the initial data. The 00-component of the free conserved tensor can be bounded from below, but its degeneracy implies existence of zero vector(s).
The motion of the system in the degenerate direction can be infinite, and the corresponding conserved quantity appears to be irrelevant to stability already at the free level. The formula (16) prevents construction of interacting theories that preserve the conserved tensors with a semi-definite 00-component.

Hamiltonian formalism
In this section, we construct constrained Hamiltonian formalism for the higher-derivative theory (16).
The We start the construction of the Hamiltonian formalism from the reduction of order of equations (16). The space components of the field strength F i , i = 1, 2, and generalized field strength of second order G i , i = 1, 2, are chosen as extra fields. By construction, they absorb the first and second time derivatives of space components of the vector field A i , i = 1, 2, Here, ǫ ij , ǫ 12 = 1 is the 2d Levi-Civita symbol. The Latin indices i, j run over the values 1, 2.
The summation over repeated at one and the same level Latin indices is implied. As is seen form In remaining part of the article, we associate the quantities F 0 , G 0 with their expressions in terms of Substituting the extra variables (26) into the system (16), we obtain the first-order equations for The evolutionary equations (28), (29), (30) are supplemented by the constraint The systems (16) and (28) We associate the on-shell Hamiltonian H 0 with the 00-component of the conserved tensor (20).
Here, the functions F 0 , G 0 are defined in (27). The on-shell Hamiltonian H 0 depends on the free model parameters α k , k = 1, 2, 3 and coupling constants β l , l = 1, 2. Off-shell, The Hamiltonian is a sum of on-shell part (32) and a linear combination of constraints. We chose the following ansatz for the total Hamiltonian, The Hamiltonian is well-defined if the parameters β, α subject to the following conditions: These relations have a clear origin. The first condition in this set insures that the on-shell Hamiltonian is a non-degenerate quadratic form of the phase-space variables A i , F i , G i . This requirement is reasonable because the degenerate Hamiltonian cannot generate the evolution of all physical degrees of freedom. The second relation (34) ensures that the numerical factor at the Lagrange multiplier A 0 is non-singular. This is necessary to reproduce the correct gauge transformations for all the dynamical variables. We also note that the obstructions to existence of Hamiltonian remain valid in the free limit. This means that the inclusion of interaction by the scheme of Section 3 does not restrict the class models that admit the Hamiltonian formulation. In the other words, every theory in the class (16) admitting the Hamiltonian formulation with the Hamiltonian (33) in the free limit, is Hamiltonian at the interacting level. Hereinafter, we assume that the relations (34) (28), (29), (30) and equations (25), we obtain the following system of algebraic equations: The sign ≈ means equality modulo constraint (31). The Poisson bracket, being defined by these equations, depends on five independent arguments: the free model parameters α 3 , α 2 , α 1 , and coupling constants β 2 , β 1 . It has the following form: Here, δ (2) (25). In the last case, the Poisson bracket with the Hamiltonian involves higher powers of fields than the right hand side of first-order equations (28), (29), (30).
If the respective Hamiltonian is bounded, the Poisson bracket cannot follow from the Ostrogradski construction. So, the Hamiltonian formalism for the stable non-linear theories does not follow from the canonical formulation.
In the conclusion of this section, we present the Poisson brackets between the constraints (38), (39), (40), (41), (42), The Hamiltonian is gauge-invariant, As is seen, the first-order model (43)  . So, the model has to be considered as unstable. In this section, we demonstrate that the dynamics of the theory with resonance can be stabilized by means of inclusion of interaction with extra dynamical field. We apply the "Higgs-like" mechanism, which has been first proposed in the context of study of the "doubly massless" generalized Podolsky electrodynamics in the paper [48]. Here, we use it in the theory with non-abelian gauge symmetry for the first time. We mostly consider the model (16)  We start construction of interaction by extending the set of dynamical variables by a real scalar field φ(x). The non-linear theory of vector multiplet A µ (x) and scalar field φ(x) is determined by the equations of motion, which have the following form: Here, the vectors F µ , G µ are defined in (2), and the abbrevialtion γ 2 = γ 2 β 2 /β 1 is used. The constants β 2 , β 1 , γ, m are model parameters, being real numbers. Throughout the section we assume that m, γ > 0 and β 1 = 0. The option β 1 = 0 is not admissible because the self-coupling (16) between the vector fields becomes inconsistent in this case. The value β 2 can be arbitrary real number (positive, negative or zero). Only positive values of β 2 , β 1 lead to stable couplings. This justifies notation γ 2 for the quantity γ 2 β 2 /β 1 . Without loss of generality, we put a unit coefficient at φ 3 -term. An overall factor at the φ 3 -vertex can be absorbed by the scaling of scalar field, φ → λφ with the appropriate Equations (48) have a clear meaning. The first line of the system (48) describes the motion of the vector multiplet. The linear in the fields term in equations corresponds to the "triply massless" ECS theory. The coupling includes self-interaction term (16), and extra contribution involving scalar field. It is convenient to think that (48) follows from (16) after the formal redefinition of the model parameters The characteristic polynomial (4) for the model reads If the scalar field is set to nonzero constant from outset, we obtain theory (16) with the second-order resonance for zero root (case (2a) of classification on pages 4-5). As it has been explained above, the corresponding model describes a massless spin-1 field, and a massive spin 1 vector subjected to the self-duality condition. It is stable at the classical and quantum levels. The second equation in the system (48) describes the motion of scalar field. This equation includes φ 3 -coupling, which ensures the existence of non-zero stationary solution for A µ = 0. This means that φ serves as the Higgs field in the model (48).
The interaction, being defined by equations (48), is consistent. The equations are preserved by the standard Yang-Mills gauge symmetry (18). The scalar field is preserved by the gauge transformation.
The gauge identity has a slightly different form, We note that the gauge generator involves the scalar field explicitly. The model has the same number of physical degrees of freedom as the free theory because the orders of equations of motion (48), gauge identities (51), and gauge symmetries (18) are preserved by coupling. Equations (48) do not follow from the least action principle from the least action principle for a functional with higher derivatives unless β 2 = 0. In the case of Lagrangian coupling, the action principle reads The dynamics of the Lagragian theory is unstable. In the case of non-Lagragian couplings, the dynamics of the model can be stable. In the remaining part of the section, we address an issue of construction of the constrained Hamiltonian formalism with a bounded Hamiltonian for the higher derivative equations (48). Existence of such a formalism implies classical and quantum stability of the model. To avoid "tautology" we outline to the most crucial steps of the construction, while the details are provided above.
We introduce a special notation for the linear combination of generalized strengths F µ , G µ enter-ing the free part equations (48), We chose the space components of the vectors F i , W i , i = 1, 2 (26) as variables absorbing first and second derivatives of the original dynamical field A µ . In the sector of scalar field, we introduce the canonical momentum π =φ. We consider W 0 as a special notation for the combination (53) of the derivatives of phase-space variables A µ , F i . In terms of the variables φ, π A µ , F i , G i , the first-order equations eventually readȦ The evolutionary equations (54), (55), (56), (57), (58) are supplemented by the constraint The constraint conserves on-shell if the identity (51) is taken into account. The higher derivative system (48) follows from equations (54) The model (48) admits a second-rank symmetric conserved tensor, The Hamiltonian is given by a sum of the 00-component of conserved tensor, being expressed in terms of variables φ, π, A µ , F i , W i (26), (53), and a constraint term, Here, the quantities F 0 , W 0 denote abbreviations (27), (53), and the numbers m, γ are model param-

eters. The Hamiltonian is on-shell bounded if
These conditions involve the scalar field φ. Once the initial value of φ is a Cauchy data for equations (48), the Hamiltonian cannot be globally bounded. However, the Hamiltonian is given by a positive definite quadratic form in the variables F , G in the range | γφ| > 1 of values of scalar field. This corresponds to the case of stability island.
The Poisson bracket between the fields φ, π, A µ , F i , G i is determined by the following system of equations: φ, H(y)dy ≈ π.
All the equalities are considered modulo terms that vanish modulo constraint (59). The solution for the equations (64), (65), (66), (67), (68) reads All the Poisson brackets between φ, π and A i , F i , G i vanish. The relations (69), (70), (71) can be obtained from (38), (39), (40), (41) after substitution (49). We note that the Poisson bracket between the fields A i , F i , W i is constant, even though the theory involves extra fields π, φ. Now, we can demonstrate the phenomenon of dynamics stabilization by means of "Higgs-like" mechanism proposed in [48]. Equations (48) admit a nonzero stationary solution Introducing notation φ * (x) = φ(x) ∓ m and expanding equations (48) in the vicinity of this vacua, we obtain the linearized equations for the fields The dots denote the terms that are at least quadratic in the fields. As we see, the dynamics of the field A is described by the higher derivative equation (3) with α 3 = m 2 γ 2 − 1, α 2 = m 2 γ 2 , α 1 = 0. In this cases, the system has a second order resonance for zero root, which does not affect the stability of dynamics, at least at the free level. The on-shell Hamiltonian reads The dots denote the terms that are at least cubic in the fields. The quadratic part of the Hamiltonian is on-shell bounded if m γ > 1. This means that the dynamics of small fluctuations in the vicinity of vacua (73) is stable. Once local stability is sufficient for construction of stable quantum theory, the "Higgs-like" mechanism can stabilize the dynamics of the non-linear theory (16) with the third-order resonance.

Conclusion
In this article, we have studied the issue of stability of the most general gauge ECS theory of third order for a vector multiplet. We have shown that the free theory with n dynamical fields admits a has been previously proposed in [42], leads to unstable dynamics. The couplings that preserve the conserved tensors with bounded 00-component are non-Lagrangian. The non-linear dynamics admits a Hamiltonian form. On-shell, the Hamiltonian is given by the 00-component of the conserved tensor.
In case of Lagrangian interacting theory, the Hamiltonian follows from the Ostrogradski procedure.
In the case of non-Lagrangian interactions, we have constructed an alternative Hamiltonian formalism for a system of higher-derivative equations. The alternative Hamiltonian is on-shell bounded if a conserved tensor with the bounded 00-component is preserved at the interacting level. This observation demonstrates that the stability of higher derivative dynamics can be consistent with non-abelian gauge symmetry, even though the canonical energy of the model is unbounded. To our knowledge the proposed theory (16) is a first stable higher derivative model with non-abelian gauge symmetry.
In the final part of the paper, we have considered the theory with the third-order resonance, which can be called the "triply massless" ECS model. This theory admits non-abelian interactions with Yang-Mills gauge symmetry, but the 00-component of the conserved quantity is unbounded in all instances. To solve the problem, we apply the "Higgs-like" mechanism of the paper [48]. An extra scalar field with the φ 3 (at the level of equations of motion) self-coupling is introduced. The dynamics of the model admits the Hamiltonian form, while the Hamiltonian has a local minimum for nonzero value of the Higgs field. The dynamics of small fluctuations in the vicinity of energy minimum is stable.
The dynamics of theory is determined by a system of partial differential equations (equations of motion) imposed onto the dynamical fields ϕ I (x), Here, T (0) , T (1) , and T (2) are linear, quadratic, and cubic in the dynamical variables. Throughout the section, the system (76) (or, equivalently, (77)) is supposed to be involutive. The concept of involution implies that equations (76) have no differential consequences of lower order (hidden integrability conditions). The ECS equations (16) are involutive.
The defining relations for gauge symmetries and gauge identifies in the system of partial derivative equations (76) read Here R I α , L a A are certain differential operators. For non-Lagrangian equations, the gauge symmetries and gauge identities are not related to each other, so the multi-indices A, α are different. In the perturbative setting, the gauge symmetry and gauge identity generators are supposed to be polynomial in fields Here, R (0) , L (0 , R (1) , L (1) , and R (2) , L (2) are field-independent, linear, and quadratic in the dynamical Relations (78), (79) imply the following consistency conditions for T (k) , R (k) , L (k) , k = 0, 1, 2, . . .: (the symbol ∂ I denotes a variational derivative with respect to the field ϕ I ); The equations (81) . . . allows a complete classification of consistent interactions in a given field theory. An important subtlety of this procedure is that some lower-order couplings can be inconsistent at the higher orders of perturbation theory. The first critical step is the extension of the first-order (quadratic) interaction vertex at the second-order of perturbation theory.
Relation (3) determines the left hand side of the free ECS equations T (0) , The gauge symmetries and gauge identities are defined by the gradient and divergence operators, The free gauge identity (81) and free gauge transformation read where ζ's are gauge transformation parameters. We consider the Poincare-covariant interactions such that are expressed in terms of gauge covariants F µ , G µ , K µ (14) with no higher-derivative terms being included into coupling. In this case, the equations of motion are automatically preserved by the Yang-Mills gauge symmetry (equations (81), (82), (83) are satisfied). Consistent interaction vertices of first and second orders are selected by the conditions (85), (86). We elaborate on this problem below.
We assume that the equations of motion are polynomial in gauge covariants F µ , G µ , K µ . The linear term is given by the covariantization of the free equations (3). The most general covariant first-order interaction vertex without higher-derivatives reads where k l , l = 1, 2, 3 are constants. The covariant divergence of equations of motion reads The gauge identity is satisfied in the first-order approximation (85) if the coefficients k l , l = 1, 2, 3 satisfy relation The general solution to this equations reads k 1 = − β 1 2 α 3 2 C(β; α) + α 1 β 3 , k 2 = − β 2 β 1 α 3 2 C(β; α) k 3 = − β 2 2 α 3 2 C(β; α) where β 1 , β 2 , β 3 are coupling parameters. The parameters β 1 , β 2 determine the coupling vertex (16) (two constants determine a single coupling because the ratio β 1 /β 2 is relevant). The constant β 3 is responsible for another interaction vertex, which is consistent at the first order of perturbation theory. The interaction vertex (16) is self-consistent with no higher-order corrections required for the equations of motion. The other coupling needs cubic in the fields corrections to the equations of motion. To prove the uniqueness of interaction (16) we should demonstrate that the ansatz (90), (93) is inconsistent at the second order of perturbation theory for β 3 = 0.
The most general second-order covariant interaction vertex reads where l p , p = 1, 6 are constants. The covariant divergence of equations of motion reads (only cubic terms are written out) (A20) Here, the notation is used, The interaction is consistent at the second order of perturbation theory if the right hand side of this expression vanish modulo free equations (3). The critical observation is that the expressions of the , where Y µ , Z µ , Y µ = F µ or G µ represent on-shell independent combinations of fields and their derivatives. Once they have to vanish, we conclude k 2 2 − k 3 k 1 = 0 , l p = 0 , p = 1, 6 .
The general solution to these equations has the form (93) with β 3 = 0. With account of this fact, the interaction (16) is unique in the class of covariant couplings without higher derivatives.