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Non-minimal Einstein-Yang-Mills-Higgs theory: Associated, color and color-acoustic metrics for the Wu-Yang monopole model

We discuss a non-minimal Einstein-Yang-Mills-Higgs model with uniaxial anisotropy in the group space associated with the Higgs field. We apply this theory to the problem of propagation of color and color-acoustic waves in the gravitational background related to the non-minimal regular Wu-Yang monopole.

consider in more details the admissible and trapped regions for longitudinal and transversal color and color-acoustic waves.

A. Lagrangian
Consider a modified action functional 1 where the susceptibility tensors R ikmn and ℜ mn are defined as ℜ mn ≡ q 4 Rg mn + q 5 R mn .
Here g = det(g ik ) is the determinant of a metric tensor g ik , R is the Ricci scalar, tensor F (a) ik is the strength of gauge field, the symbol Φ (a) denotes the multiplet of the Higgs scalar fields, 1 2 λ Φ 2 − v 2 2 is a potential of the Higgs field, and Φ 2 ≡ Φ (a) Φ (a) . Latin indices without parentheses run from 0 to 3, (a) and (b) are the group indices. Following [16], Section 4.3, we consider the Yang-Mills field F mn and the Higgs field Φ taking values in the Lie algebra of the gauge group SU (n): Here t (a) are Hermitian traceless generators of SU (n) group, F represents the Yang-Mills field potential), G is a constant of gauge interaction, and the group index (a) runs from 1 to n 2 − 1. The scalar product of the generators t (a) and t (b) is defined via the trace the symmetric tensor G (a)(b) plays a role of a metric in the group space. The representation (4), (5) allows to consider the multiplets of the real fields {F Here ∇ m is a covariant space-time derivative, the symbols f (a) (b)(c) denote the real structure constants of the gauge group SU (n). The gauge invariant derivativeD m Φ (a) is defined according to the formula [16] The definition of the commutator is based on the relation: providing the formula The structure constants f (a)(b)(c) are supposed to be antisymmetric under exchange of any two indices. The quantities X (a)(b) and Y (a)(b) are symmetric with respect to the group indices and depend on the Higgs field Φ (c) ; they are defined below.
B. Higgs field and uniaxial structures in the group space The group space can be considered as isotropic, when it is equipped by the metric G (a)(b) only, and as anisotropic one in all other cases. The presence of the Higgs field multiplet Φ (a) with positive norm G (a)(b) Φ (a) Φ (b) ≡ Φ 2 > 0 allows us to equip the group space by the vector q (a) = Φ (a) /Φ with the unit norm G (a)(b) q (a) q (b) = 1 and by the projector P (a)(b) . The latter is defined as and possesses the evident projector properties: Using these quantities, every tensor A i1i2...is (a)(b) , symmetric with respect to the transposition of the group indices (a) and (b), can be decomposed into the sum where The simplest type of anisotropy in the group space is the uniaxial one. This case assumes that the only direction pointed by q (a) is the selected one, while other orthogonal directions are equivalent, i.e., where S i1i2...is 1 and S i1i2...is 2 are some non-equal tensors without group indices. The conditions (16) mean that Keeping in mind this features, we assume the tensors X (a)(b) and Y (a)(b) in the Lagrangian (1) to have the simplest form with two new coupling constants Q 1 and Q 2 . In principle, we can treat they as quantities, which depend on the vacuum value v of the scalar field. When Q 1 = Q 2 = 1, the tensors X (a)(b) and Y (a)(b) coincide with the metric G (a)(b) in the group space. The introduction of Q 1 = 1 and/or Q 2 = 1 modifies the master equations for the gauge, scalar and gravitational fields. When Q 1 = 0, X (a)(b) = P (a)(b) , and when Q 2 = 0, Y (a)(b) = P (a)(b) . In general case X (a)(b) (and Y (a)(b) , correspondingly) consists of two parts, parallel to the direction pointed by the scalar field, and orthogonal to it, the longitudinal and transversal parts being, respectively In addition, every vector in the group space, say, A (a) k , can be decomposed using longitudinal and transversal components as follows where C. Non-minimal master equations

Non-minimal extension of the Yang-Mills field equations
The variation of the action functional over the Yang-Mills potential A where the tensor H ik (a) is Thus, the linear response tensor C ikmn the longitudinal and transversal parts being, respectively Mention that C ikmn (trans) can be obtained from C ikmn (long) by the formal replacement Q 1 → 1, and let us use this features below for the simplifications of the formulas. The color permittivity, impermeability and cross-effect tensors are also anisotropic in the group space Here the asterisk denotes the dualization procedure: U i and ∆ im ≡ g im − U i U m are a unit timelike vector (U i U i = 1) and the projection tensor, respectively. The corresponding longitudinal and transversal components of ε im (a)(b) , (µ −1 ) im (a)(b) and ν im (a)(b) can be easily written in analogy with (25) and (26).

Non-minimal extension of the Higgs field equations
The variation of the action S (NMEYMH) over the Higgs scalar field Φ (a) yieldŝ The can also be decomposed into longitudinal and transversal components where The definitionsg introduce two color-acoustic metrics for the colored scalar particles. Again,g ik (trans) can be obtained fromg ik (long) by the formal replacement Q 2 → 1.

Master equations for the gravitational field
The equations for the gravity field related to the action functional S (NMEYMH) are of the form where The first term T is a stress-energy tensor of pure Yang-Mills field, the second term: relates to the standard stress-energy of the Higgs field. The non-minimal contributions enter the last tensor T , which may be represented as a sum of 5 items: The definitions of these five tensors relate to the corresponding coupling constant q 1 , q 2 , . . . , q 5 . The tensors T

III. EXACT SOLUTION OF THE WU-YANG TYPE OF THE NON-MINIMAL EYMH MODEL
A. Ansatz about the fields structure Let the spacetime metric be of the spherically symmetric form Here σ and N are functions depending on the radius r only and satisfying the asymptotic conditions We focus on the gauge field characterized by the special ansatz (see, [17]): The parameter ν is a non-vanishing integer. The generators t (r) , t (θ) and t (ϕ) are the position-dependent ones and are connected with the standard generators of the SU(2) group as follows: The generators satisfy the relations The field strength tensor has only one non-vanishing component: which does not depend on the variable r. Our ansatz for the Higgs field is the following: we consider Φ as a covariantly constant vector in the group space, which contains only one component, i.e., This means that the Yang-Mills field strength tensor F ik and the Higgs field Φ are parallel in the group space [18,19] to the vector q = t (r) . The Higgs equations are satisfied identically, when the field φ is constant and coincides with v, the background value of the scalar field given by Φ (a) Φ (a) = v 2 . Then the Yang-Mills equations (23) are satisfied identically also, and we deal with a new non-minimal Wu-Yang monopole solution, supplemented by the background covariantly constant Higgs field. Finally, the equations for the gravity field are extremely simplified by the ansatz and its particular consequence In fact the equations for the gravity field for the model under discussion coincides with the ones, described in [6], if we replace the coupling parameters q 1 , q 2 , q 3 by the Q 1 q 1 , Q 1 q 2 , Q 1 q 3 , respectively. Thus, we can use the exact solutions of the Wu-Yang monopole type to the non-minimal Einstein-Yang-Mills model, as an exact solutions for the generalized non-minimal EYMH model with the covariantly constant Higgs field. The scalar field in this case does not modify the pressure and the energy of the system as whole, nevertheless, it play an important role, creating the privilege direction in the group space. In particular, the following solution for the non-minimal Wu-Yang monopole with regular center is valid when Q 1 q 1 = −q, Q 1 q 2 = 4q, Q 1 q 3 = −6q, and q is positive.

IV. MULTI-METRIC REPRESENTATION OF THE MATERIAL TENSORS
By analogy with decomposition of material tensor in electrodynamics [20] the tensor C ikmn (a)(b) can be represented as with respect to color associated metrics g , whereμ is some convenient factor [20]. Below we apply the elaborated formalism to the case, when the non-minimal Wu-Yang monopole accompanied by the constant scalar field forms the gravitational background for the moving particles, quasi-particles and waves.

A. Associated metrics
In the spherically symmetric model under consideration the material tensor C ikmn (a)(b) reduces to where C ikmn (long) and C ikmn (trans) are defined by (25) and (26), respectively. The reconstruction of the tensor C ikmn (long) can be done using two longitudinal color metrics g ik(A) (long) and g ik(B) (long) as follows (see [20] for details) where g im(A) Mention that g ik(B) (long) can be obtained from g ik(A) (long) by a formal replacement of the symbol ε by the symbol µ. We used the following notations A new parameter a with dimensionality of length is defined as follows: a 4 = κq, where κ = 8πν 2 /G 2 . To obtain the decomposition for C ikmn (trans) one can simply replace the mark (long) by (trans) and then replace Q 1 by one. Clearly, this simple model involves the following non-trivial components of the tensor G with (α), (β) = (A), (B), which satisfy the linear relations Thus, in order to reconstruct the linear response tensor C ikmn (a)(b) for the model under discussion one needs four associated metrics g are satisfied in the WKB-approximation for A and B-waves, respectively, when the Yang-Mills potential has a form A (r) k t (r) . Second, the equalities hold, when the Yang-Mills potential is of the form A k t (ϕ) . Indeed, let us consider the Yang-Mills equations in the leading order WKB-approximation: with the tensor C ikmn (a)(b) given by (57), (25), (26). When the group index (a) coincides with (r), one obtains When (a) = (θ) or (a) = (ϕ), the system (72) gives two equations for the transversal components of the Yang-Mills potential A (θ) In the WKB approximation the gauge potentials A kl can be extrapolated as follows Mention that the nonlinear terms in (6) give the values of the next order in WKB approximation, thus, such a model of gauge field is effectively Abelian. In the leading order approximation the Yang-Mills equations reduce to For the sake of simplicity and in this subsection only we omit the marks (long) and (trans) in the associated metrics and in the parameters µ and γ, since the results are similar for the longitudinal and transversal cases. Substitution C ikmn (a)(b) from (58) with g ik(A) from (59) and g ik(B) from (60) withμ and γ given by (63) yields where X i {1} , X i {2} are unit spacelike vectors, orthogonal to U i , X i {r} , and each other. Projection of this equations onto the velocity four-vector U i gives the scalar ratio which is satisfied if, for instance, we use the Landau gauge U n A (a) n = 0 and the condition of orthogonality of the wave four-vector and amplitude four-vector in the first associated metric, i.e., g kn p k A (a) n = 0. Projections onto the axes, given by where A (a) one can conclude that nontrivial solution of (79) exists, when Thus, the associated metrics g km(A) and g km(B) are the color metrics, which indicate both longitudinal and transversal ones.

C. Color-acoustic metrics
Tensor C ik (a)(b) (32) can be associated with two metricsg ik (long) andg ik (trans) : In the leading order of the WKB approximation, the propagation of the scalar particle with the color index (a) = (r), described by the potential Φ (r) t (r) , is equivalent to the motion in the effective spacetime with the metricg ik (long) . As for the particle, described the potential Φ (θ) t (θ) or Φ (ϕ) t (ϕ) , the corresponding effective spacetime has a metricg ik (trans) . We obtain color birefringence for scalar waves (quasi-particles), since the velocity of the wave depends on its color. Taking into account the formulas we can reconstruct the color-acoustic metric for arbitrary parameters Q 2 , M , q 4 and q 5 . Nevertheless, to complete the illustration of the possibilities of the effective metric approach we restrict ourselves by the model with M = 0. Then the longitudinal color-acoustic metric is where the conformal factor A 2 is and the angular factor B 2 is given by Here we introduced a new convenient dimensionless variable ξ = r 4 /κq, which clearly takes non-negative values only. Again, the transversal metric can be obtained from this formulas after the formal replacement Q 2 → 1.

V. DISCUSSION
The formulas (59) and (60) present the associated metrics for the non-minimal EYMH model under discussion. As well, in the WKB approximation the Yang-Mills equations are satisfied when (70) or (71) are valid. Thus, these associated metrics can be interpreted as color ones. This means, in particular, that the propagation of a test Yang-Mills wave in the non-minimally active spacetime can be described by two dispersion relations where When the wave propagates in the radial direction, i.e., p ⊥ = 0, or in the transverse one, i.e., p || = 0, the corresponding phase velocities can be easily found as V || = ω/p || , V ⊥ = ω/p ⊥ For A-wave and B-wave they are, respectively, i.e., both the longitudinal (Q 1 = 1) and transversal (Q 1 = 1) color waves, propagating radially, have the phase velocity equal to the speed of light in the minimal vacuum.
As an illustration, consider now a specific particular case of the exact solution (55) characterized by M = 0. This solution does not admit singularities of the horizon type, since N > 1 everywhere. It is convenient to present the basic formulas for the velocities of the waves propagating in the direction, orthogonal to the radius, as follows For these formulas one can obtain the transversal color wave velocity when Q 1 = 1, as well as, for longitudinal one, say, when Q 1 = 2. When do the phase velocities coincide with the speed of light in minimal vacuum, i.e., when do they equal to one? The following facts clarify this question. depend on the value of the parameter Q 1 . Remember, that the value Q 1 = 1 relates to the transversal color waves, and Q 1 = 1 relates to the longitudinal one. In order to clarify the principal moments of analysis we put further Q 1 = 2 for the longitudinal case. Let us attract the attention to two details. The first one is connected with the nulls of the functions under consideration, the second relates to the infinite value. Both points relate to the singularities of the color metrics. When the functions (91) and (92) become negative, we deal with inaccessibility zones.
for Q 1 = 2 is negative for the ranges √ 65 − 8 < ξ < 1 and 3.283 < ξ < 21.731. In this ranges of radial variable ξ = r 4 /κq the propagation of the color waves of this type is impossible. At r = r 1 ≡ (κq( √ 65−8)) 1/4 and r = r 2 ≡ (κq) 1/4 the phase velocity vanishes and the wave stops. At r = 3.283 and r = 21.731 the phase velocity is infinite. Admissible ranges of ξ for the wave propagation are, therefore, 0 < ξ < √ 65 − 8, 1 < ξ < 3.283 and 21.731 < ξ < ∞. The first admissible range is of finite length, the corresponding part of the space can be indicated as first inner admissible region or trapped region. This trapped region is arranged between two spheres of the zeroth radius and of the radius r 1 . On the left edge of this region (r = 0) the phase velocity is equal to one relating to the absence of classical gravity force (N = 1, N ′ = 0). With increasing of radius the phase velocity decreases monotonically and vanishes at the right edge of the trapped region, thus, the transversally propagating wave stops at this surface. The second (inner) admissible region can also be indicated as trapped one, nevertheless, the phase velocity at the left edge, ξ = 1, is vanishing, whereas the right edge is characterized by the infinite phase velocity, passing the value equal to one at ξ = 11/5. The third admissible range is infinite. The phase velocity is infinite at the left edge of this region and tends monotonically to one at infinity. The non-minimal radius r = a indicates the boundary between the first inaccessibility zone and second admissible region. The second inaccessibility zone 3.283 < ξ < 21.731 is characterized by the infinite phase velocity barriers at the left and right edges.
Longitudinal B-wave When Q 1 = 2 the formula (92) reduces to i.e., the denominator and numerator are presented by the polynomials quadratic in ξ. Now this function is discontinuous at ξ = √ 65 − 8, this point being the right edge of the first admissible region, 0 < ξ < √ 65 − 8, which is the trapped region of the second type. When the radius grows the phase velocity increases from one to infinity, and a wave does not stop at the boundary, but is, on the contrary, infinite. The range √ 65 − 8 < ξ < √ 65 + 8 relates to the inaccessible zone, since V is negative. The second admissible region is √ 65 + 8 < ξ < ∞, the phase velocity is equal to zero at ξ = √ 65 + 8 and tends to one asymptotically at ξ → ∞.
Transversal A-wave When Q 1 = 1, the curve V has no discontinuity. The trapped region of the first type is situated at 0 < ξ < 0.191; the second admissible region is arranged at 0.539 < ξ < ∞; curve passes one at ξ = 11/5, then reaches the maximal value and tends to one at ξ → ∞. The inaccessibility zone is 0.191 < ξ < 0.539.

Transversal B-wave
For this wave one has two trapped regions, two inaccessibility zones and one infinite admissible region. The first trapped region is characterized by the inequality 0 < ξ < 0.191, the phase velocity starts from one at the left edge and reaches infinity at the right edge. The second one is at 0.539 < ξ < 1.853, the phase velocity starts from infinity at the left edge and vanished at the right one. The first inaccessibility zone is arranged at 0.191 < ξ < 0.539, the phase velocity is infinite at both edges. The second one is characterized by 1.853 < ξ < 5.249, the phase velocity being vanishing at both edges. The infinite admissible region is 5.249 < ξ < ∞. The phase velocity starts from zero at the left edge and tends monotonically to one at ξ → ∞.

Color-acoustic waves
In order to complete our illustration let us consider a particular model with the following parameters: Our choice relates to the idea that non-minimal radius a = (κq) 1/4 , appeared in the equations for color gauge waves, is the same one for the color-acoustic waves, i.e., there is no additional non-minimal Constant of Nature (see, e.g., [21]). With (94) the functions A 2 (ξ) and B 2 (ξ) are, respectively, (see formulas (86) and (87)). The toy models with Q 2 = 2 (longitudinal case) and Q 2 = 1 (transversal case) are non-singular. Indeed, in both cases the cubic polynomials in the numerators and denominators of (95) and (96) have no real positive roots. This means that both functions, A 2 (ξ) and B 2 (ξ) are continuous and positive for ξ > 0. In addition, both functions, start from one at ξ = 0, pass through one at ξ = 7 and tend to one at ξ → ∞. As well, the function B 2 (ξ) reaches the maximum at ξ = 8 − √ 57 and minimum at ξ = 8 + √ 57. Thus, in contrast to the case of color waves the color-acoustic waves in this toy model have neither trapped regions, nor inaccessible zones. The phase velocities of the corresponding color-acoustic waves are non-monotonic functions of the radius, nevertheless, they can reach neither zero value, nor infinity.

Remark
In general case the propagation direction of the color or color-acoustic wave is not pure longitudinal or pure transversal with respect to the radial direction. In general case the trajectory of the corresponding quasi-particle becomes very sophisticated depending on the value of the impact parameter. This requires a detailed numerical analysis, and we hope to discuss that results in a separate paper.

Alternative description
Particle dynamics can be alternatively described using the equation of motion with effective force. Instead of equation of null geodesics in the effective spacetime with metric g ik(α) (h) , where the upper index (α) indicates the A and B cases in the classification of effective metrics, the lower index (h) is treated as (long) and (trans), one can write the equation kl(h) , the difference of the Christoffel symbols, symmetric with respect to indices k and l, is known to be a tensor, thus, the quantity F i(α) (h) is a vector. Since we consider the interaction of Yang-Mills and Higgs field with spacetime curvature, we can indicate this force as a tidal one. The tidal force F i(α) (h) is quadratic in the particle four-velocity and is predetermined by the structure of the tensor Π The non-vanishing components of the tensor Π i(B) kl(long) can be obtained from (99) by the formal replacement of the symbol ε by the symbol µ. As for Π i(A) kl(trans) and Π i(B) kl(trans) , they can be obtained from the corresponding longitudinal components if we put Q 1 → 1. When the particle moves radially, i.e., dθ dτ = dϕ dτ = 0, the tidal force vanishes. This agrees with the result obtained above that the radially propagating color waves have the phase velocity equal to the speed of light in the standard (minimal) vacuum.