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Excited Platonic Sphalerons in the Presence of a Dilaton Field

We construct sphaleron solutions with discrete symmetries in Yang-Mills-Higgs theory coupled to a dilaton. These platonic sphalerons are related to rational maps of degree N. We demonstrate that, in the presence of a dilaton, for a given rational map excited platonic sphalerons exist beside the fundamental platonic sphalerons. We focus on platonic sphaleron solutions with N=4, which possess cubic symmetry, and construct the two branches of their first excitations. The energy density of these excited platonic sphalerons exhibits a cube within a cube.


Introduction
The non-trivial topology of configuration space of SU(2) Yang-Mills-Higgs (YMH) theory gives rise to a variety of static, finite energy solutions. The simplest classical solutions possess spherical symmetry. In the presence of a triplet Higgs field these are the topologically stable 't Hooft-Polyakov monopoles [1], whereas in the presence of a doublet Higgs field these are the unstable sphalerons [2,3,4].
Multimonopoles and multisphalerons represent YMH solutions with less symmetry. Characterizing multimonopoles by their topological charge N , N ≥ 2, and multisphalerons by their Chern-Simons number N/2, these solutions possess at most axial symmetry [5,6].
For N ≥ 3 multimonopole and multisphaleron solutions with no rotational symmetry appear [7,8]. The symmetries of these solutions are only discrete, and can be identified with the symmetries of platonic solids or crystals. We therefore refer to them as platonic monopoles and platonic sphalerons. Related to certain rational maps of degree N [9], they have many properties in common.
In flat space the static classical solutions associated with a certain rational map of degree N appear to be unique. This has been shown explicitly for the 't Hooft-Polyakov monopole [10] and the Klinkhamer-Manton sphaleron [4], which do not possess radial excitations 1 .
In curved space, in contrast, the absence of radially excited monopole and sphaleron solutions is no longer true. Indeed, when gravity is coupled, both gravitating monopoles and gravitating sphalerons possess a whole sequence of radial excitations [11,12]. These radially excitated Einstein-Yang-Mills-Higgs (EYMH) solutions are related to the Bartnik-McKinnon solutions of pure Einstein-Yang-Mills (EYM) theory [13,14].
The coupling of gravity to Yang-Mills or YMH theory is known to have a very similar effect concerning the existence and the properties of classical spherically symmetric solutions as the coupling of a scalar dilaton [15,11,16]. Dilatonic monopoles possess a sequence of radial excitations analogous to gravitating monopoles [16], and dilatonic sphalerons possess a sequence of radial excitations analogous to gravitating sphalerons, as demonstrated below.
The existence of excited spherically symmetric solutions as well as of excited axially symmetric solutions in the presence of gravity [17] and analogously in the presence of a dilaton [18] suggests, that also excited platonic solutions might exist. We here demonstrate, that this is indeed the case, by explicitly constructing excited platonic sphalerons in the presence of a dilaton. These excited platonic sphalerons represent the first examples of excited platonic solutions.
We focus on platonic Yang-Mills-Higgs-dilaton (YMHD) sphalerons, based on a rational map of degree N = 4, which possess cubic symmetry [8,19]. Like the fundamental platonic YMHD sphalerons [19], excited platonic sphalerons should form two branches of solutions, merging at a maximal value of the coupling constant. The maximal value of the coupling constant is expected to decrease with increasing excitation. We here explicitly construct the two branches of the first excitations of the platonic YMHD sphalerons. We also construct excited spherically and axially symmetric YMHD sphalerons.
We review YMHD theory in section 2. We present the Ansätze and the boundary conditions for axially symmetric and platonic sphalerons in section 3, and discuss our numerical results in section 4.

Yang-Mills-Higgs-Dilaton Theory
We consider YMHD theory with Lagrangian and dilaton field φ, where g and κ denote the gauge and dilaton coupling constants, respectively, λ denotes the strength of the Higgs self-interaction, and v the vacuum expectation value of the Higgs field. The Lagrangian (1) is invariant under local SU(2) gauge transformations U , The gauge symmetry is spontaneously broken due to the non-vanishing vacuum expectation value of the Higgs field In the limit of vanishing dilaton coupling constant, the model corresponds to the bosonic sector of Weinberg-Salam theory for vanishing Weinberg angle. In the following we consider only static finite energy solutions, The energy of such solutions is given by The energy E is related to the dilaton charge D, by the simple relation 4πD = κE .
This can be seen by integrating the dilaton equation and using the identity which follows from a Derrick like argument for solutions of the field equations. The Chern-Simons charge N CS of the solutions is given by with For the identification of the baryon number Q B = N CS [2,8], the Chern-Simons number N CS has to be evaluated in a gauge, where asymptotically

Sphaleron Solutions
We here present the Ansätze and the boundary conditions for axially symmetric and platonic sphalerons. The excited sphalerons fall into two classes, distinguished by their boundary conditions and their Chern-Simons charges.
We parametrize the Higgs field as and impose at infinity the boundary conditions where τ a are the Cartesian Pauli matrices and n R is a unit vector to be specified. The boundary conditions for the gauge field at infinity are then obtained from the requirement D i Φ = 0. When the Higgs field is angle-dependent at infinity this yields for the gauge field i.e., the gauge field tends to a pure gauge at infinity, The baryon number of the sphalerons is then obtained from Eq. (9), after performing a gauge transformation with [8] where Ω tends to π/2 at infinity and vanishes at the origin. In contrast, when the Higgs field tends to a constant isospinor, e.g. when Φ a τ a = τ z , the gauge field vanishes identically at infinity, and the baryon number then vanishes as well.
For the dilaton field we require that it vanishes at infinity, φ ∞ = 0, since any finite value of the dilaton field at infinity can always be transformed to zero via φ → φ − φ ∞ , r → re −κφ∞ .

Axially Symmetric Sphalerons
The Ansatz for the axially symmetric YMHD sphalerons corresponds to the Ansatz employed in Weinberg-Salam theory [6] (at vanishing Weinberg angle [20]), and supplemented by the dilaton function φ, where , τ x , τ y and τ z denote the Pauli matrices, and n is an integer. For n = 1 and κ = 0 the Ansatz yields the spherically symmetric Klinkhamer-Manton sphaleron [2]. For n > 1, the functions H 1 -H 4 , Φ 1 , Φ 2 and φ depend on r and θ, only. With this Ansatz the full set of field equations reduces to a system of seven coupled partial differential equations in the independent variables r and θ. A residual U(1) gauge degree of freedom is fixed by the condition r∂ r H 1 − ∂ θ H 2 = 0 [6].
In the presence of gravity or of a dilaton, for each integer n, a whole sequence of regular axially symmetric sphaleron solutions exists, labelled by the node number k of the gauge field functions H 2 and H 4 [13,17,12]. The sphaleron solutions then fall into two classes, odd-k sphalerons with the fundamental sphaleron (k = 1) as their lowest mass member, and even-k sphalerons, with the first excited sphaleron (k = 2) as their lowest mass member.
The boundary conditions differ for the two classes of solutions [13,17,12]. Regularity at the origin requires the boundary conditions where Φ c is a finite constant [12]. Thus the Higgs field vanishes at the origin only for odd-k sphalerons. In contrast the gauge field vanishes at the origin for all sphalerons (in the chosen gauge). Since Φ c is unknown, however, we employ the boundary conditions Regularity on the z-axis requires the boundary conditions The requirement of finite energy implies at infinity the boundary conditions Thus for odd-k sphalerons, the Higgs field is angle-dependent at infinity, Φ a τ a = τ (n) r , and the gauge field then tends to a pure gauge configuration with U ∞ = iτ (n) r . In contrast, for even-k sphalerons, the Higgs field tends to a constant isospinor, Φ a τ a = τ z , and thus the gauge field vanishes at infinity.
The non-gauge transformed Chern-Simons density K 0 vanishes identically for the spherically and axially symmetric sphalerons, because of the ansatz of the gauge potential. After performing the gauge transformation (15), we obtain [21,6,8] N CS = n/2 , odd k , N CS = 0 , even k .
Thus for a given integer n, odd-k sphalerons, and in particular fundamental sphalerons, carry baryon number Q B = n/2, whereas even-k sphalerons have vanishing baryon number.

Platonic Sphalerons
To obtain YMHD solutions with discrete symmetry we make use of rational maps, i.e. holomorphic functions from S 2 → S 2 [9]. Treating each S 2 as a Riemann sphere, the first having coordinate ξ, a rational map of degree N is a function R : and p and q are polynomials of degree at most N , where at least one of p and q must have degree precisely N , and p and q must have no common factors [9]. We recall that via stereographic projection, the complex coordinate ξ on a sphere can be identified with conventional polar coordinates by ξ = tan(θ/2)e iϕ [9]. Thus the point ξ corresponds to the unit vector and the value of the rational map R(ξ) is associated with the unit vector We here consider platonic YMHD solutions obtained from the map R 4 , Note, that the map R 4 (ξ) = ξ 4 yields the axially symmetric sphalerons for n = 4 in a different gauge.
The rational map Eq. (30) leads to the unit vector (29) The boundary conditions for platonic sphalerons are, in principle, only needed at infinity. Specifying the unit vector n R thus completely determines the boundary conditions for the Higgs field (13) and for the gauge field (14). For fundamental N = 4 platonic sphalerons with cubic symmetry n R = n R4 . Thus at infinity the Higgs field and the gauge field exhibit the angledependence of the rational map, since Φ a τ a = τ R4 and U ∞ = iτ R4 . In contrast, for the first excited N = 4 platonic sphalerons an adequate choice for the Higgs field at infinity is again Φ a τ a = τ z , just as for the first excited rotationally symmetric sphalerons. Consequently the gauge field vanishes at infinity, and the discrete symmetry of the first excited platonic sphalerons does not enter via the boundary conditions at infinity.
For platonic sphalerons the non-gauge transformed Chern-Simons density K 0 is non-trivial, but as checked numerically it does not contribute to their baryon number [8]. Thus the only contribution arises again from the gauge transformation U , Eq. (15), yielding the Chern-Simons number N CS = N/2 fundamental , N CS = 0 first excitation .
The general set of field equations involves the dilaton function φ(x, y, z), three functions for the Higgs field, Φ a (x, y, z), and nine functions for the gauge field, V a i (x, y, z), while the remaining Higgs and gauge field functions vanish consistently, Φ 0 = V a 0 = 0. Subject to the boundary conditions at infinity, and to the gauge condition one can then solve this set of field equations in the full space R 3 .
To obtain better numerical accuracy, however, one can make use of the discrete symmetries of the platonic sphalerons, to restrict the region of numerical integration. After specifying the reflection symmetries of the fields with respect to the xy-, xz-and yz-plane, it is then sufficient to solve the field equations for x ≥ 0 and y ≥ 0 and z ≥ 0 only.
The boundary conditions at infinity must then be supplemented by conditions at the other boundaries of the integration region. These boundaries consist of the origin, the xy-, xz-and yz-plane, and the positive z-axis. (Together with the boundary at infinity these boundaries are implemented numerically as the boundaries of a cube [22].) At the origin r = 0, the Higgs field vanishes for the fundamental platonic sphalerons, while it is finite for the first excited platonic sphalerons, with boundary conditions Φ 1 = Φ 2 = 0 and ∂ r Φ 3 = 0. The gauge field always vanishes at the origin, and the dilaton field satisfies ∂ r φ = 0.
The boundary conditions in the xz-plane (ϕ = 0), the yz-plane (ϕ = π/2), on the z-axis (θ = 0) and in the xy-plane (θ = π/2) follow from the reflection symmetries of the fields. In particular, we suppose that the rational map R 4 determines the reflection symmetries of the fields. Thus we impose, that for the fundamental platonic sphalerons the Higgs field Φ and the gauge field V i possess the same reflection symmetries as τ R4 and i g [∂ i τ R4 , τ R4 ], respectively 2 . For the first excited platonic sphalerons, however, the boundary conditions of the Higgs field in the planes must also be compatible with the boundary conditions at the origin and at infinity, i.e., Φ a τ a ∼ τ z . The consistent set of Higgs boundary conditions then follows from the symmetry requirements of the energy density, while the boundary conditions for the gauge field remain unchanged. For the Higgs and spherical gauge field components the proper set of boundary conditions is given in Table 1. For the dilaton field the normal derivative must vanish in the xy-, xz-and yz-plane, and on the z-axis ∂ θ φ = 0.
For the numerical construction of platonic sphalerons an adequate initial guess is mandatory. For the fundamental cubic sphalerons the initial guess is obtained by parametrizing the Higgs and gauge fields according to and the dilaton field by φ(r). This simple ansatz with the three radial functions h(r), f (r), and φ(r) is inserted into the action, the angular dependence is integrated, and a set of variational equations is obtained, similar to the set of spherically symmetric (n = 1) sphaleron equations. These ordinary differential equations are solved numerically, subject to the set of boundary conditions of the spherical (n = 1) sphalerons (in particular f (0) = 1, f (∞) = −1). The initial guess is then given by (34) with these numerically determined functions. For the first excited cubic sphalerons the initial guess is obtained analogously, only the procedure now involves the parametrization k xz-plane (ϕ = 0) yz-plane (ϕ = π/2) 1, 2

Numerical results
For convenience we rescale the coordinates r → r/gv, the gauge potential V i → vV i , and the dilaton field φ → φ/κ, and introduce the dimensionless coupling constant We further rescale the dilaton charge D → D/κgv and the energy E → E4πv/g, to obtain the scaled dilaton charge-energy relation (7) The numerical solutions are constructed with help of the software package FIDISOL [22] based on the Newton-Raphson algorithm. The solutions are obtained in spherical coordinates r, θ, ϕ. To map the infinite range of the radial variable r to the finite interval [0, 1] we introduce the compactified variabler = r/(1+r). Typical grids contain 100×30 points for the axially symmetric solutions and 80 × 25 × 25 points for the platonic solutions. The estimated relative errors are on the order of 0.01% for the axially symmetric sphalerons, and 1% for the platonic sphalerons, or better.
In the following we discuss successively spherically symmetric (N = 1), axially symmetric (N = 2, 4) and platonic (N = 4) sphalerons. We construct these excited sphaleron solutions for the Higgs self-coupling constant λ = 0.125. and study their dependence on the coupling constant α.

Spherically symmetric sphalerons
Fundamental spherically symmetric sphalerons have been studied before [19,23]. As a function of the coupling constant α two branches of fundamental sphaleron solutions exist. The lower branch of dilatonic sphalerons emerges from the spherically symmetric sphaleron of Weinberg-Salam theory, and extends up to a maximal value of the coupling constant, α max . There it merges with the upper branch of fundamental sphaleron solutions, which extends back to α = 0. On the lower branch the dimensionless energy E decreases with increasing α, whereas on the second branch it increases with decreasing α and diverges as α tends to zero. At the same time the value of the dilaton function at the origin decreases continuously along both branches.
Considering the limit of vanishing α on the upper branch, we note that the scaled energy αE and the value of the dilaton function at the origin φ(0) approach finite values, equal to the energy E, respectively φ(0), of the fundamental spherically symmetric YMD solution [15]. Indeed, introducing the scaled variabler = r/α, the Higgs fieldΦ = Φα and the gauge fieldV i = V i α, one arrives at an equivalent system of equations, in which the limit of vanishing Higgs fieldΦ corresponds to the limit of vanishing α (v → 0) on the upper branch of the original system. We exhibit the scaled energy αE and the value of the dilaton function at the origin φ(0) for the fundamental sphalerons as a function of α in Fig. 1  The figure also exhibits the first three radial excitations of the spherically symmetric sphalerons, possessing k = 2, and 3 nodes, respectively. For each radial excitation two branches of solutions appear, which merge and end at a maximal value of the coupling constant α max (k), which decreases with increasing k. At α = 0 the scaled energy of the upper branch solutions with k nodes and the scaled energy of the lower branch solutions with k + 1 nodes coincide, yielding for the scaled energy of the solutions a 'Christmas tree' like structure. For the YMHD solutions themselves the transition at α = 0 is not continuous, however. The scaled YMHD solution with k nodes approaches on its upper branch smoothly the YMD solution with k nodes [15]. But the scaled YMHD solution with k + 1 nodes emerges on its lower branch from the YMD solution with k nodes with a discontinuity at infinity, since the additional node of the gauge field function has been pushed out to infinity. Comparing these YMHD solutions with the corresponding solutions of EYMH theory [12], we observe, that their structure and their coupling constant dependence are completely analogous, in accordance with our expectation.

Axially symmetric sphalerons
For the axially symmetric sphalerons we observe the same pattern for the fundamental and excited solutions as for the spherically symmetric sphalerons. For a given n, the scaled energy of the fundamental and excited sphaleron branches also exhibits a 'Christmas tree' like structure, as shown in Fig. 1 for n = 2. For α → 0, the lower fundamental YMHD sphaleron branches emerge smoothly from the corresponding sphalerons of Weinberg-Salam theory [6]. The scaled upper YMHD sphaleron branches with k nodes approach smoothly the corresponding axially symmetric YMD solutions with k nodes [18], and the lower YMHD sphaleron branches with k + 1 nodes emerge with a discontinuity at infinity.

Cubic sphalerons
Turning now to the numerical results for the platonic sphalerons with cubic symmetry, we again observe the same pattern as for the spherically and axially symmetric sphalerons. The scaled energy of the fundamental and first excited platonic sphaleron branches exhibits the typical 'Christmas tree' like structure, as seen in Fig. 2. where the value of the dilaton function at the origin is also shown.
In particular, the lower branch of the fundamental cubic YMHD sphaleron emerges smoothly from the cubic sphaleron of Weinberg-Salam theory [8], the scaled upper branch of the fundamental cubic YMHD sphaleron approaches the fundamental cubic YMD solution [24], and the scaled upper branch of the first excited cubic YMHD sphaleron approaches the first excited cubic YMD solution, giving first evidence for the existence of a whole sequence of cubic YMD solutions [24].  Comparison of the platonic sphalerons and the axially symmetric sphalerons reveals, that the platonic sphalerons exist for slightly larger values of the coupling constant α than the axially symmetric sphalerons. For all values of α, for which both types of sphalerons coexist, the energy of the platonic sphalerons is slightly smaller than the energy of the axially symmetric sphalerons.
Defining the dimensionless energy density ǫ by we present surfaces of constant total energy density ǫ tot in Fig. 3 for the first excited cubic sphaleron for α = 0.15 and λ = 0.125. Again we conclude that the shape of the energy density of platonic solutions is determined primarily by the rational map [8,19]. Interestingly, the energy density of the excited solutions exhibits a cubic shape both for large and for small constant values, revealing a small interior cube within a larger exterior cube. Surfaces of constant energy density of the gauge field ǫ gauge ( 1 2 e 2κφ Tr(F µν F µν )) exhibit an analogous pattern, as also seen in Fig. 3.

Conclusions
We have constructed numerically sphaleron solutions of YMHD theory, possessing spherical, axial, and cubic symmetry. We have shown, that to each fundamental sphaleron solution excited sphaleron solutions exist. In particular, we have obtained the first excited platonic solutions. Previously, only fundamental platonic solutions were known [7,8,19,25]. For all these types of sphaleron solutions, and in particular also for the fundamental and excited sphaleron solutions, two branches of solutions exist, which extend from α = 0 to a value α max , where they merge and end. The value of α max depends on the particular type of solution. α max decreases with increasing Chern-Simons number of the solution [19] and also with increasing excitation of the solution. Thus all solutions reveal the same general coupling constant dependence.
The excited solutions fall into two classes, distinguished by their boundary conditions and Chern-Simons numbers. We observe, that the solutions of both classes alternate with increasing excitation energy. Given a particular fundamental sphaleron solution, its scaled energy and the energy of its excitations exhibits a 'Christmas tree' like structure. While we have constructed only the first excited platonic sphalerons, we conjecture, that higher excitations of platonic solutions exist as well, analogous to the sequences of excited spherically symmetric and axially symmetric sphalerons.
In the limit α → 0, the lower fundamental YMHD sphaleron branches emerge from the corresponding sphaleron solutions of Weinberg-Salam theory. All other branches merge into solutions of YMD theory in the limit α → 0, after appropriate rescaling. While the limiting spherically and axially symmetric YMD solutions were known before [15,18], we have obtained new numerical evidence for the existence of platonic YMD solutions [24].
We consider the construction of platonic solutions in the presence of a dilaton as a step towards obtaining gravitating solutions without rotational symmetries. By analogy with the simpler spherically symmetric case, we expect that gravitating platonic sphalerons will exhibit similar properties as the dilatonic platonic sphalerons studied here.