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Suppression of matter couplings with a vector field in generalized Proca theories

We derive the profile of a vector field coupled to matter on a static and spherically symmetric background in the context of generalized Proca theories. The cubic Galileon self-interaction leads to the suppression of a longitudinal vector component due to the operation of the Vainshtein mechanism. For quartic and sixth-order derivative interactions, the solutions consistent with those in the continuous limit of small derivative couplings correspond to the branch with the vanishing longitudinal mode. We compute the corrections to gravitational potentials outside a compact body induced by the vector field in the presence of cubic, quartic, and sixth-order derivative couplings, and show that the models can be consistent with local gravity constraints under mild bounds on the temporal vector component. The quintic Galileon interaction does not allow regular solutions of the longitudinal mode for a rapidly decreasing matter density outside the body.


I. INTRODUCTION
The problems of dark energy and dark matter imply that there may be additional dynamical degrees of freedom (DOFs) beyond those appearing in the Standard Model of particle physics [1]. The most studied case is a scalar field φ with a potential energy V (φ) [2]. Such a new scalar DOF can be responsible for the late-time cosmic acceleration or can mimic the dark matter property, depending on forms of the potential. There are also other scalar-field models with derivative self interactions-like Galileons [3,4] and their extensions [5][6][7]. These derivative interactions can be the source of dark energy [8][9][10] while suppressing the propagation of fifth forces in the Solar System [11][12][13][14][15][16][17][18][19] through the operation of the Vainshtein mechanism [20].
The scalar field is not the only possibility for explaining the dark sector of the Universe; the vector field can also play a similar role [21]. The standard massless Maxwell field, which respects the U (1) gauge symmetry, has two transverse electric and magnetic polarizations. The gauge symmetry is explicitly broken by introducing a mass term or derivative interactions of the vector field, in which case the longitudinal propagation emerges. In Refs. [22][23][24][25], the four-dimensional action of a massive Proca field with nonminimal derivative couplings to gravity was constructed from the requirement of keeping two transverse and longitudinal modes besides two tensor polarizations arising from the gravity sector (see Refs. [26][27][28][29][30][31][32][33] for related works). In such generalized Proca theories, the equations of motion remain of second order, so there are no Ostrogradski instabilities. It is also possible to go beyond the second-order domain without increasing the number of propagating DOFs [34,35].
If we apply generalized Proca theories and their extension to the isotropic and homogenous cosmological background, there exists an interesting de Sitter attractor with a constant temporal vector component [36]. The spatial vector components can be treated as perturbations on such a background, which is consistent with the analysis on the anisotropic cosmological background with time-dependent spatial components [37]. There are dark energy models in which all the stability conditions of perturbations can be consistently satisfied [36]. Moreover, the presence of an intrinsic vector mode offers the possibility for realizing the gravitational interaction weaker than that in general relativity (GR) [38][39][40]. This allows one to distinguish dark energy models in generalized Proca theories from those in GR.
On a static and spherically symmetric background, the existence of hairy black hole solutions was recently studied in the context of generalized Proca theories [41][42][43][44][45][46][47][48][49]. For massless or massive vector fields without derivative interactions the spatial vector components vanish identically [50], so the background geometry is simply described by the Reissner-Nordström or the Schwarzschild space-time. The existence of derivative interactions gives rise to a variety of hairy black hole solutions with nonvanishing temporal and longitudinal vector components [47,48]. This leads to the difference between two metric components around the black hole horizon. The deviation from GR in the nonlinear regime of gravity can be potentially probed by future measurements of gravitational waves around black holes.
Inside the Solar System, the fifth force mediated by the vector field A µ nonminimally coupled to gravity should be screened for the consistency with local gravity experiments [51]. In Ref. [52], the propagation of fifth forces around a spherically symmetric and static body was studied in the presence of cubic and quartic vector Galileon interactions under the approximation of weak gravity. The cubic derivative interaction leads to a suppression of the longitudinal component A 1 thanks to the Vainshtein mechanism. The quartic derivative coupling gives rise to the branch A 1 = 0, so the gravitational potentials are subject to modifications only through the temporal vector component A 0 . For both cubic and quartic interactions, the models can be compatible with local gravity constraints under mild bounds on A 0 . This property also persists even in beyond-generalized Proca theories [53].
The analysis of Ref. [52] assumed that a direct coupling between the vector field and matter is absent, so the vector-matter interaction arises indirectly from nonminimal gravitational couplings with the vector field. It is not yet clear whether the existence of direct vector-matter interactions leads to the screening of fifth forces at the level of being compatible with local tests of gravity. In this paper, we will address the issue of the screening mechanism in the presence of a matter coupling of the form QA µ A µ T , where Q is the coupling strength and T is the trace of the energy-momentum tensor of matter. The quantum corrections to the generalized Proca action can be generated by matter loops, but the computation of one-loop corrections to the vector-field propagator shows that the theory remains healthy without the appearance of new ghosty DOFs in the domain of the effective theory [54]. We will derive the vector-field profile around a compact body in the presence of cubic, quartic, quintic, and sixth-order-derivative generalized Proca interactions and estimate the corrections to leading-order gravitational potentials in GR.
This paper is organized as follows: In Sec. II we present the full equations of motion on the static and spherically symmetric background in generalized Proca theories with the matter coupling. In Sec. III we derive the profiles of temporal and longitudinal vector components with cubic derivative interactions both inside and outside the body. We compute the corrections to gravitational potentials induced by the vector field and show how the Vainshtein mechanism is efficient to suppress the propagation of fifth forces even with matter couplings. In Secs. IV, V and VI we study the propagation of the vector field in the presence of quartic, quintic, and sixth-order derivative interactions, respectively. While the model with quintic derivative coupling does not possess regular solutions of the longitudinal mode for a rapidly decreasing density profile outside the body, the quartic and sixth-order interactions give rise to solutions with the vanishing longitudinal mode. The latter models can be consistent with local gravity constraints under mild bounds on the temporal vector component. Sec. VII is devoted to conclusions.

II. EQUATIONS OF MOTION
We consider a vector field A µ with the field strength F µν = ∇ µ A ν − ∇ ν A µ , where ∇ µ is the covariant derivative operator. We introduce a matter perfect fluid given by the energy-momentum tensor where g is a determinant of the metric tensor g µν , and L m is the matter Lagrangian density. We assume that the vector field is coupled to the matter sector with the interacting Lagrangian density where Q is a dimensionless coupling constant, M pl is the reduced Planck mass, T is the trace of T µν , and We assume that |Q| is at most of the order 1.
The generalized Proca theories are the second-order vector-tensor theories given by the action [22][23][24][25] with F = −F µν F µν /4, and 8) where G 2,3,4,5,6 and g 4,5 are functions of X with the notation G i,X ≡ ∂G i /∂X, R is the Ricci scalar, and G µν is the Einstein tensor. The tensorsF µν and L µναβ are defined, respectively, bỹ where E µνρσ is the Levi-Cività tensor normalized by E µνρσ E µνρσ = −4!, and R ρσγδ is the Riemann tensor. The theory with constant G 6 (X) corresponds to the U (1) gauge-invariant derivative interaction advocated by Horndeski [55]. The Lagrangians containing the functions g 4 (X) and g 5 (X) correspond to intrinsic vector modes. We consider the static and spherically symmetric background described by the line element where f (r) and h(r) are arbitrary functions of the distance r from the center of symmetry. The vector-field profile compatible with the above background is given by [52] A µ = (A 0 (r), A 1 (r), 0, 0) .
The quantity X can be expressed as X = X 0 + X 1 , where For the matter sector, we consider the perfect fluid with the energy-momentum tensor T µ ν = (ρ m +P m )u µ u ν +P m δ µ ν , where ρ m is the density, P m is the pressure, and u µ = (f −1/2 , 0, 0, 0) is the fluid four-velocity in the rest frame. Since the trace T is given by the interacting Lagrangian density (2.2) does not vanish except for the radiation (ρ m = 3P m ). Varying the action (2.4) with respect to A 0 and A 1 , we obtain the equations of motion for the temporal and longitudinal vector components, respectively, as where a prime represents a derivative with respect to r. To derive the gravitational equations of motion, we write the metric (2.11) in the form ds 2 = −f (r)dt 2 + h −1 (r)dr 2 + r 2 e 2ζ(r) (dθ 2 + sin 2 θ dϕ 2 ). Varying the action (2.4) with respect to f, h, ζ and setting ζ = 0 at the end, it follows that where c i 's (i = 1, 2, . . . , 19) are given in the Appendix, and The continuity equation of the matter sector is given by In the whole analysis of this paper, we take into account the Einstein-Hilbert term, such that where M pl is the reduced Planck mass. If there exist derivative couplings G i (X) and g i (X) with even indices i alone, Eq. (2.16) reduces to The function F consists of the sum of terms written in the forms β iFi (with i = 2, 4, 6) and γ 4G4 , where β i and γ 4 are coupling constants in G i (X) and g 4 (X), respectively, andF i andG 4 are regular functions of A 0 , A 1 , f, h and their derivatives. From Eq. (2.23), the vector component A 1 has two branches satisfying (i) A 1 = 0, or (ii) F = f Qr 2 (ρ m − 3P m )/M 2 pl . In the limit that β i → 0 and γ 4 → 0, the function F vanishes, so branch (ii) does not exist for nonrelativistic matter (P m ≪ ρ m ) with a nonvanishing coupling constant Q. Hence, the branch consistent with the general-relativistic limit corresponds to A 1 = 0. This property does not generally hold for the couplings G i (X) and g i (X) with odd indices i, in which case the nonvanishing branch of A 1 can arise from Eq. (2.16).
We have derived the field equations for general theories given by the action (2.4), but we will focus on the theories with in the following discussion. This reflects the fact that the analysis with the derivative couplings G 3,4,5,6 (X) is sufficiently general to understand basic properties of the screening mechanism in generalized Proca theories. As we will see in subsequent sections, the temporal vector component A 0 is generally dominated by a constant a 0 with a small variation around it. In such cases, it is convenient to express the quantity X 0 in the form where δX 0 characterizes the deviation fromX 0 . Unless a 2 0 is very much smaller than M 2 pl , the last term on the lhs of Eq. (2.21) gives rise to a large modification to the pressure P m relative to the Q = 0 case. By defining it is possible to rewrite Eq. (2.21) without having the contribution fromX 0 . Expressing ρ m and P m in terms of ρ and P and substituting them into Eq. (2.21), it follows that where For nonrelativistic matter characterized by P m ≪ ρ m , the matter-coupling term in Eq. (2.27) does not dominate over P m under the condition which is assumed in the following discussion. As long as the screening mechanism operates to suppress both X 1 and δX 0 , the Q-dependent terms in Eq. (2.28) work as tiny corrections to the continuity equation P ′ + f ′ (ρ + P )/(2f ) = 0 in GR. We also note that the terms on the rhs of Eqs. (2.17)-(2.19) are expressed, respectively, as Unlike Eq. (2.20), the matter-coupling terms in Eq. (2.31) do not containX 0 . For nonrelativistic matter, the condition (2.30) translates to |Qa 2 0 | ≪ M 2 pl f . Then, the couplingQ is approximately equivalent to Q with T * ≃ T in Eq. (2.29). We will exploit the rescaled energy density ρ and the pressure P to discuss the profile of a spherically symmetric body with radius r * . To derive solutions to the vector-field and gravitational potentials, we consider the following matter density profile: where ρ 0 is a constant density, and µ is a dimensionless constant much smaller than 1. In Sec. III we will also numerically obtain solutions to the vector field for a varying matter density. The Schwarzschild radius of the source is defined by Taking the vacuum limit µ → 0 outside the body, it follows that r g ≃ ρ 0 r 3 * /(3M 2 pl ). We introduce the gravitational potentials Ψ and Φ, as and employ the weak gravity approximation which amounts to the condition r g ≪ r * . In GR without the coupling Q, the gravitational potentials Ψ and Φ inside the body (r < r * ) are given by the internal Schwarzschild solution with the pressure Under the approximation (2.36), the ratio between P and ρ 0 can be estimated as P/ρ 0 ≃ Φ * (1 − r 2 /r 2 * )/12 ≪ 1. In such cases, the condition (2.30) translates to In the presence of the vector field coupled to matter, the gravitational potentials (2.37) are subject to modifications due to the existence of theQ-dependent terms in Eq. (2.31). Provided that the two conditions are satisfied, the contributions arising from the matter coupling can be treated as corrections to the leading-order gravitational potentials (2.37). Under the operation of the screening mechanism, we will show that it is possible to satisfy the conditions (2.40).
In the vacuum limit, the gravitational potentials outside the body in GR are given by The presence of the vector field coupled to matter gives rise to modifications to Ψ GR and Φ GR , but they can be again treated as corrections to the leading-order gravitational potentials for the vector components obeying the conditions of Eq. (2.40).
For the comparison with the results derived for Q = 0 in Ref. [52], we will adopt the notations We consider nonrelativistic matter satisfying P ≪ ρ and employ the approximationQ ≃ Q under the condition (2.39).

III. CUBIC VECTOR GALILEONS
Let us begin with the cubic Galileon model given by the functions where m is a constant having a dimension of mass, and β 3 is a dimensionless constant. From Eqs. (2.15) and (2.16), we obtain the following equations of motion: For |m| smaller than the order of the today's Hubble expansion rate H 0 ≃ 10 −33 eV, the term |Qρ/M 2 pl | is much larger than |m 2 | for |Q|ρ ≫ ρ c , where ρ c ≃ 10 −29 g/cm 3 is today's critical density. Unless |Q| is extremely smaller than 1, the condition |Q|ρ ≫ ρ c is well satisfied in the Solar System. Hence, we will ignore the term m 2 relative toQρ/M 2 pl in the whole analysis of this paper. Under the condition (2.39), the couplingQ defined by Eq. (2.32) is at most of the order Q[1 + O(Φ * )]. We neglect the contribution of the order Φ * inQ for the estimations of φ and χ ′ , so that the termQρ/M 2 pl appearing on the rhs of Eqs. (3.2) and (3.3) is approximated as Qρ/M 2 pl . We deal with the general-relativistic gravitational potentials Ψ GR and Φ GR as leading-order contributions to Ψ and Φ, respectively, and substitute them into Eqs. (3.2) and (3.3) to obtain the solutions to φ and χ ′ . We then plug the solutions of vector-field profiles into Eqs. (2.17) and (2.18) to derive corrections to the leading-order gravitational potentials outside the body.
A. r < r * We first derive the field profiles for the distance r smaller than r * . Substituting Eq. (2.37) into Eqs. (3.2) and (3.3), we obtain where we have chosen the branch recovering the solution χ ′ (r) → 0 in the continuous limit β 3 → 0. We will focus on the positive derivative coupling but we will not restrict the signs of Q.

B. r > r *
Let us derive the solutions to φ(r) and χ ′ (r) outside the spherically symmetric body. On using the leading-order gravitational potentials (2.41), Eqs. (3.2) and (3.3) reduce, respectively, to 1 d dr (3.20) We assume that the quantity µ(r), which is defined by Eq. (2.33), is a constant µ much smaller than 1. From Eq. (3.20), the longitudinal mode is expressed as On using the approximation scheme (3.8), we can integrate Eq. (3.19) to give where the integration constant C is fixed to be C = −φ 0 Φ * r * (1 + Q − Qµ)/3 by matching the solution (3.22) with Eq. (3.9) at r = r * . Then, we obtain the following relation: where We first study the case where s β3 ≫ 1. Since ξ(r) ≫ 1 outside the body, we first take the limit ξ(r) → ∞ in Eq. (3.25). The ratio µ is much smaller than 1, so we carry out the expansion of F 2 (ξ(r) → ∞) around µ = 0. This process leads to where we use the approximation ξ(r) ≫ s β3 (i.e., r ≫ r * ) for deriving the third term on the rhs of F 2 (r). For the distance r satisfying Since the second term in the square bracket of Eq. (3.29) rapidly approaches 0 for increasing r, we obtain the approximate integrated solution where we have performed the matching of the solution with Eq. (3.16) at r = r * . This shows that φ(r) is nearly constant around φ 0 . Under the condition (3.28), the second term in the square root of Eq. (3.30) is much larger than 1, so the longitudinal mode reduces to The distance r V can be regarded as the Vainshtein radius, within which the propagation of the longitudinal mode is suppressed due to the existence of cubic Galileon interactions. For |Q| of the order of unity, the Vainshtein radius can be estimated as r V ≃ µ −1/3 r * . The density ρ 0 is related to the Schwarzschild radius r g , as ρ 0 ≃ 3M 2 pl r g /r 3 * . If µρ 0 is of the order of the present cosmological density, it is related to today's Hubble radius, r H ≃ 10 28 cm, as µρ 0 ≃ 3M 2 pl /r 2 H . Then, the Vainshtein radius is of the order of r V ≃ (r g r 2 H ) 1/3 . For the Sun (r g ≃ 10 5 cm), we have r V ≃ 10 20 cm, which is much larger than the Solar-System scale. Thus, the propagation of the longitudinal mode is suppressed inside the Solar System thanks to the Vainshtein mechanism.
For the intermediate coupling strength s β3 satisfying the magnitude of the second term in the square root of Eq. (3.35) is much larger than 1 for the distance r * < r < r c . Then, the longitudinal mode reduces to Since ξ(r) ≫ 1 for the distance r c ≪ r ≪ r V , the solutions to φ ′ (r) and χ ′ (r) are described by Eqs. (3.29) and (3.32), respectively.

C. Vector-field profile for varying matter density
While we have derived analytic solutions to the vector field for the constant densities inside and outside the body, we will also study the case of the varying matter density given by where a and µ are constants, with a = O(1) and µ ≪ 1. The density is nearly constant for the distance r ≪ r * , but it starts to decrease rapidly around r = r * to approach the asymptotic value µρ 0 . Numerically, we solve the full equations of motion (2.15)-(2.17), (2.19), and (2.21) by using Eqs. (3.11), (3.14), (2.37), and (2.38) as the boundary conditions of φ, χ ′ , Ψ, Φ, P around the center of the body (r = 10 −3 r * ). We recall that the analytic vector-field profile was derived by employing the leading-order general-relativistic gravitational potentials. To check the consistency of this procedure, we also integrate Eqs. (2.17)-(2.18) by neglecting the contributions of the vector field and solve Eqs. (3.2)-(3.3) for φ(r) and χ ′ (r). We substitute the derived solutions of φ(r) and χ ′ (r) into Eqs. (2.17)-(2.18) and confirm that the corrections to Ψ GR and Φ GR induced by the vector field remain small. These approximate solutions exhibit good agreement with the full numerical results.
We note that, for cubic scalar Galileons, the derivative of the longitudinal scalar χ(r) behaves as χ ′ (r) ∝ r for r < r * and as χ ′ (r) ∝ r −1/2 for r * < r < r V , where r V is the Vainshtein radius [11,[15][16][17]19]. Inside the body, the solution of χ ′ (r) derived above shares the common property to that of cubic scalar Galileons. Outside the body, the solution found for the coupling s β3 ≪ 1 has a similar property [χ ′ (r) ∝ r −1/2 ] to that for cubic scalar Galileons at the distance r * < r < r c . For the coupling s β3 ≫ 1, the exterior solution [χ ′ (r) ∝ r −2 ] decreases faster than that of cubic scalar Galileons.

D. Gravitational potentials outside the body
Let us estimate the corrections to leading-order gravitational potentials (2.41) outside the body induced by the presence of the vector field coupled to matter. In doing so, we express Eqs. (2.17) and (2.18) in the forms where ∆ Φ and ∆ Ψ are corrections to the gravitational equations in GR. Since we are interested in the behavior of Φ and Ψ for r > r * , we set ρ = µρ 0 = µΦ * M 2 pl /r 2 * in Eq. (3.41). For the integrations of Eqs. (3.41) and (3.42), we employ the weak-gravity approximation (2.36) in such a way that the general-relativistic corrections to gravitational potentials higher than second order are neglected. In the following, we will consider the two different cases, (i) s β3 ≫ 1 and (ii) s β3 ≪ 1, separately.
In this case, we exploit the solutions from Eqs. (3.31)-(3.32) and substitute them into Eqs. (2.17)-(2.18). There exists the term of the form ρQφ 2 0 Φ * /M 2 pl , which is at most of the order ρΦ 2 * = µM 2 pl Φ 3 * /r 2 * under the condition (3.38) by identifying a 0 with φ 0 . We neglect such contributions relative to the term of the order φ 2 0 Φ 2 * /r 2 * , which amounts to the condition µΦ * ≪ (φ 0 /M pl ) 2 . Then, the corrections up to the order of Φ 2 * can be estimated as For the integration of Eqs. (3.41)-(3.42), we choose the integration constant in such a way that the solutions Φ = r g /(2r) = r * Φ * /(6r), Ψ = −r g /(2r) = −r * Φ * /(6r) are recovered in the vacuum limit. Then, we obtain the integrated solutions corrected by the presence of the vector field, as The second terms in the square brackets of Eqs. (3.44) and (3.45) induce the difference between the gravitational potentials, as For φ 0 M pl , we have Φ + Ψ U 2 . The quantity β PN may be regarded as the second parametrized post-Newtonian parameter [56]. The experimental bound |β PN | 2.3 × 10 −4 from the Nordtvedt effect [51] can be satisfied for φ 0 10 −2 M pl . We note that the condition (2.39) translates to φ 0 Φ * /|Q|M pl , so this can put a tighter limit of φ 0 for Φ * /|Q| ≪ 10 −4 .

s β 3 ≪ 1
For the coupling s β3 satisfying s β3 ≪ (Qµ) 2 , we employ the solutions from Eqs. (3.34) and (3.37) for the estimations of Φ and Ψ. Up to the order of Φ 2 * , the correction terms ∆ Φ and ∆ Ψ are given, respectively, by The second terms in Eqs.

IV. QUARTIC DERIVATIVE COUPLINGS
We proceed to the case of quartic derivative interactions given by where β 4 is a dimensionless constant, and n is a positive integer. As we already mentioned in Sec. II, we can express Eq. (2.16) in the form (2.23), where which vanishes in the limit that β 4 → 0. Hence, the branch consistent with this limit corresponds to Since there is no longitudinal propagation of the vector field, the temporal component φ alone can affect the solutions to gravitational potentials. Let us consider the matter density profile given by Eq. (2.33). Inside the body, we substitute the leading-order gravitational potentials (2.37) into Eq. (2.15). Then, the temporal vector component obeys Assuming the solution in the form (3.8) and imposing the boundary condition φ ′ (0) = 0, the integrated solution to Eq. (4.4) reads .

(4.6)
Provided that |b 4 | 1, the second term in the square bracket of Eq. (4.5) is at most of the order Φ * . We will assume this condition in the following discussion.
Outside the body, we substitute the leading-order external gravitational potentials (2.41) into Eq. (2.15). This leads to Inside the spherically symmetric body, we substitute the leading-order gravitational potentials of Eq. (2.37) into Eqs. (2.15)-(2.16) and then expand them up to the first order of Φ * . Then, it follows that d dr As in the case of cubic vector Galileons, we search for analytic solutions to φ ′ (r) and χ ′ (r) proportional to r around the center of the body. Provided that φ is nearly constant around φ 0 , the first three terms on the lhs of Eq. (5.3) are in proportion to r 3 , whereas the last term has the dependence 2β 5 Φ * rχ ′4 ∝ r 5 . Neglecting the last term of Eq. (5.3) around r = 0, we can solve Eq. (5.3) for χ ′ as where we have chosen the branch recovering χ ′ → 0 for β 5 → 0. Provided that the longitudinal mode χ ′ is suppressed relative to φ, the β 5 -dependent terms in Eq. (5.2) should work as corrections to the leading-order solution which is nearly constant around φ 0 . Substituting the derivative of Eq. (5.5) into Eq. (5.4), the leading-order longitudinal mode reads For the existence of this solution, we require the condition ε 5 ≤ 1, which translates to so that the coupling β 5 is bounded from above. Let us consider the case in which the condition is satisfied. Assuming that both β 5 and Q are positive, the longitudinal mode (5.6) reduces to which means that χ ′2 leading (r) is suppressed relative to φ 2 0 by the factor (Φ * ε 5 /24)(r/r * ) 2 . By using the solution (5.9), the β 5 -dependent terms in Eq. (5.2) can be estimated as 3Qφ 0 ε 5 Φ * r 2 /[(1 + Q)r 2 * ]. Taking into account this contribution and integrating Eq. (5.2) with respect to r, we obtain and hence χ ′ (r) is well described by the leading-order solution (5.9) for ε 5 ≪ 1.
Provided that the last term of Eq. (5.13) is much smaller than the other terms, Eq. (5.13) can be explicitly solved as (5.14) In the vacuum limit (µ → 0), the solution to Eq. (5.12) derived by neglecting the contributions of χ ′ and χ ′′ reads φ ′ (r) ∝ 1/r 2 and φ(r) ≃ constant. Then, the second term in the square root of Eq. (5.14), which is always positive, exhibits the divergence for µ → 0. If we consider a rapidly decreasing density profile like Eq. (3.40), the solution to Eq. (5.14) becomes imaginary above the distance r d at which the second term in the square root of Eq. (5.14) is equivalent to 1. Unless we consider slowly varying density profiles like ρ(r) = ρ 0 (r * /r) p with p < 9/2, it is not possible to realize the existence of regular solutions for the distance r > r d . The solution in Eq. (5.14) has been derived by neglecting the contribution of the last term of Eq. (5.13). As we estimated in Eq. (5.11), this term works as a tiny correction to the leading-order longitudinal mode (5.6) around r = r * . For the density profile (3.40), we numerically integrate the vector-field equations of motion coupled to the gravitational equations and find that the last term of Eq. (5.13) remains small relative to the other terms for the distance r * < r < r d . For r > r d , there are no real solutions to the longitudinal mode. Thus, the quintic vector Galileon does not allow the existence of consistent solutions to χ ′ (r) for realistic density profiles that rapidly decrease for r > r * .
We also studied the model of the linear coupling G 5 (X) = β 5 X/M 2 pl and found that the solution to χ ′ has a similar property to that of the model (5.1). In this case, the χ ′4 term is absent unlike Eq. (5.13), so we have the exact solution to χ ′ (r) analogous to Eq. (5.14). Hence, it is not possible to realize the solution of χ ′ (r) regular throughout the region r > r * for the rapidly decreasing matter density. Thus, the models with quintic derivative couplings are generally plagued by the absence of regular external solutions operated by the Vainshtein mechanism.

VI. SIXTH-ORDER DERIVATIVE COUPLINGS
Finally, we study the model with sixth-order derivative interactions given by where β 6 is a dimensionless constant, and n is a positive integer of order 1. According to the discussion given in Sec. II, the branch consistent with the limit β 6 → 0 corresponds to Let us consider the matter density profile (2.33). Inside the body, we substitute Eq. (2.37) into Eq. (2.15) and expand it up to second order in Φ * for the terms containing β 6 . Then, the temporal vector component obeys d dr (6.4) In the limit that b 6 → 0, we obtain the leading-order solution φ leading (r) = φ 0 [1 − Φ * (1 + Q)r 2 /(6r 2 * )], so φ(r) is nearly frozen around φ 0 . Assuming that |b 6 Φ * | ≪ 1, the terms containing b 6 in Eq. (6.3) can be regarded as the constant mode a 0 in A 0 from the matter continuity equation, we defined the density ρ and the pressure P in the forms (2.26) and (2.27), respectively. The matter-coupling term induced by the constant mode of A 0 is smaller than the intrinsic pressure under the condition (2.30), which translates to the inequality (2.39).
In Sec. III, we derived the analytic vector-field profile both inside and outside a spherically symmetric body in the presence of cubic Galileon interactions by employing the general-relativistic gravitational potentials (2.37) and (2.41). This procedure can be justified under the conditions of Eq. (2.40), whose consistency was checked after deriving solutions to the vector field. We showed that both the longitudinal and temporal vector components are sufficiently suppressed due to the operation of the Vainshtein mechanism. This result was also numerically confirmed for the decreasing density profile (3.40). We computed corrections to the general-relativistic gravitational potentials outside the body and found that the model can be consistent with local gravity constraints even for the coupling Q of order unity.
For the derivative couplings G i (X) with even indices i, the solution to the longitudinal mode, which is consistent with the continuous limit of small couplings, corresponds to χ ′ = 0. In Sec. IV, we obtained solutions to the temporal vector component and the gravitational potentials for quartic power-law couplings given by Eq. (4.1). This model is compatible with Solar-System constraints under the bound (4.16). In Sec. VI, we also carried out the similar analysis for sixth-order power-law couplings (6.1) and showed the compatibility of solutions with local gravity tests.
In Sec. V, we studied the vector-field profile in the presence of quintic Galileon interactions. We found that there are no consistent solutions of the longitudinal mode in the vacuum limit outside the body. This fact does not allow the existence of regular solutions for a realistic compact object whose density rapidly decreases outside the body. Thus, the quintic vector Galileon is the special case in which the screening mechanism for the longitudinal mode does not work. It remains to be seen whether this property also holds for general quintic derivative interactions with matter couplings other than Eq. (2.2).