Solutions to the Minimization Problem Arising in a Dark Monopole Model in Gauge Field Theory

We prove the existence of dark monopole solutions in a recently formulated Yang--Mills--Higgs theory model with technical features similar to the classical monopole problems. The solutions are obtained as energy-minimizing static spherically symmetric field configurations of unit topological charge. We overcome the difficulty of recovering the full set of boundary conditions by a regularization method which may be applied to other more complicated problems concerning monopoles and dyons in non-Abelian gauge field theories. Furthermore we show in a critical coupling situation that an explicit BPS solution may be used to provide energy estimates for non-BPS monopole solutions. Besides, in the limit of infinite Higgs coupling parameter, although no explicit construction is available, we establish an existence and uniqueness result for a monopole solution and obtain its energy bounds.


Introduction
It is well known that the existence of a magnetic monopole was first theoretically conceived by P. Curie [8] based on the electromagnetic duality observed in the Maxwell equations. Later, Dirac [10] explored the quantum mechanical implication of the presence of a magnetic monopole and demonstrated that the existence of such in nature would explain why electric charges are quantized as multiples of a small common unit. Despite the elegance of Dirac's formalism, it is not surprising that, like a Coulomb electric point charge, a magnetic point charge, or simply monopole, in the Maxwell theory would render a point singularity in the field and thus carry infinite amount of energy. A landmark development came when 't Hooft [37] and Polyakov [26] found in non-Abelian gauge field theory, known as the Yang-Mills-Higgs theory, that a singularity-free topological soliton arises as a consequence of the spontaneously broken symmetry in the vacuum manifold which behaves asymptotically like a Dirac monopole, away from local regions, has concentrated energy near the origin, where the monopole resides, is smoothly distributed in space, and carries a finite energy. Such a particle-like soliton is commonly referred to as the 't Hooft-Polyakov monopole and has since then been extensively studied. See [14, 16, 28-30, 42, 45, 46] for some expository presentations on the subject. Although monopoles remain a hypothetical construct, the concept they offer leads to many fruitful investigations of various theoretical issues, for example, the quark confinement problem [15,32,33] in light of a linear confinement mechanism for a monopole and antimonopole pair immersed in a type-II superconductor [22-24, 38, 39]. In Deglmann and Kneipp [9], a new family of monopole solitons are constructed with a characteristic feature that they do not belong to the sector of the usual asymptotically electromagnetic Dirac monopole type, thus called 'dark monopoles', which may have some relevance in the study of dark matter/energy. As in the formulation of the 't Hooft-Polyakov monopole problem [26,37], mathematically, the existence of such monopoles amounts to solving a two-point boundary value problem with solutions minimizing a correspondingly reduced radially symmetric energy functional as in the classical work of Tyupkin, Fateev, and Shvarts [41] using functional analysis. However, the method in [41] is only sufficient to allow the acquisition of a weak solution [5] since the functional is not coercive enough for us to recover the full set of boundary properties required for regularity which is not directly implied by finite-energy condition. Technically, one needs to impose boundary conditions, imposed both at the point where the monopole resides and at spatial infinity, to carry out a minimization process. Without preservation of such a full set of boundary conditions for field configurations over the associated admissible class, there is no ensurance for the attainability of the energy minimum, thus even the existence of a weak solution as a critical point of the energy functional may become problematic. Therefore, it is imperative to tackle the issue of recovering the full set of boundary conditions in the minimization treatment. Our strategy here is to use the regularization method developed in [44] for a simpler but similar Yang-Mills-Higgs monopole problem [20] with vanishing Higgs potential as in the classical Bogomol'nyi [3] and Prasad-Sommerfield [27] (BPS) limit [16,29,36]. Minimization problems of similar technical subtleties arise in other treatments of non-Abelian monopole problems [1,14,17,28,42]. Such general applicability and appeal motivate our present mathematical investigation.
Analytically, our regularization method consists of the following steps: We first solve the problem away from the singular point which permits an approximation of the concerned boundary condition at the singularity. This approximation enables us to realize a monotonicity property of a minimizing sequence such that the monotonicity property is preserved in the limit as well. Thus we are able to see that the boundary limit at the singularity would exist as a consequence. We then argue that the limiting value must be the desired one otherwise it would falsify the finiteenergy condition. Furthermore, we will use a BPS solution of the same form as in the classical studies [3,27] to estimate the monopole energy (mass) away from the BPS phase. Besides, we will obtain monopole solutions when the coupling parameters satisfy some specific conditions and demonstrate their applications.
An outline of the content of the rest of the paper is as follows. In Section 2, we review the dark monopole model [9] briefly, introduce the associated minimization problem, and state our results. In Section 3, we prove the existence of an energy-minimizing solution in the general setting and establish some qualitative properties of the solution. We then comment on a BPS critical phase. In Section 4, we illustrate how to use the BPS solution to estimate the energy of a non-BPS monopole in the zero Higgs potential situation. In Section 5, we study the limiting situation when the Higgs coupling constant is set to be infinite, which also appears in the classical 't Hooft-Polyakov model and is of independent interest. This problem becomes simpler since it is a single-equation problem, although no solution is explicitly known. Due to the non-convexity of the energy functional, it is not immediate to see that the solution is unique. Nevertheless, we are able to establish a uniqueness result and derive some qualitative properties of the solution.

Dark monopole model and existence results
Following [9], use φ to denote a scalar field in the adjoint representation of a non-Abelian gauge group G such as SU (n) (n ≥ 5) and W µ (µ = 0, 1, 2, 3) a gauge field taking values in the Lie algebra of G. In spherically symmetric static limit, the scalar field and spatial components of the gauge field are represented in terms of spherical coordinates (r, θ, ϕ) by the expressions where e, v > 0 are parameters, S, Q m , M k appropriate generators of G, u(r) and f (r) real-valued profile functions, Y * lm suitable spherical harmonics, and α 0 > 0 depends on v and l in a specific way. In terms of such representation, the gauge-covariant derivatives of φ and the magnetic field induced from W i are With these, the total Yang-Mills-Higgs energy for the dark monopole model reads [9]: where λ > 0 is the Higgs coupling parameter, and (2.6) with λ p a fundamental weight of G and the updated rescaled radial variable evr → r, which is denoted by ξ in [9], so that the associated Euler-Lagrange equations of (2.6) are (2.8) subject to the boundary conditions It is clear that all the conditions except the first one, f (0) = 0, in (2.9), are consequences of finite energy. From (2.1), we see that the first one in (2.9), i.e., f (0) = 0, is required to ensure regularity of the Higgs scalar field which is not directly imposed by the finiteness of (2.6). It is this feature that needs to be dealt with care as described earlier.
In order to simplify our notation, it will be convenient to use the substitution . (2.10) Thus our existence results regarding dark monopole solitons governed by the boundary-value problem consisting of (2.7)-(2.9) may be stated as follows. (i) For any coupling and group parameters, there exists a finite-energy solution minimizing the rescaled energy (2.6) which enjoys the properties 0 < u(r), f (r) < 1 for r > 0 and that u(r) and 1 − f (r) vanish at infinity exponentially fast, and 1 − u(r) and f (r) vanish at r = 0 like power functions, following some sharp asymptotic estimates in both cases.
(ii) When γ = 0, β = 2, and α > 0, the equations are equivalent to a BPS set of first-order equations for solutions with a finite energy. This system of the BPS equations has a unique solution which depends explicitly on the free parameter α which coincides with the classical BPS monopole solution. In other words, this is a BPS situation.
(iii) In the non-BPS situation when γ = 0 and α, β > 0 are arbitrary, the equations have an energyminimizing solution such that both u(r) and f (r) are monotone functions. Furthermore, the energy of the BPS solution obtained in (ii) may be used to get some energy estimates which become exact at the critical point β = 2.
(iv) When γ = ∞, the reduced governing equation has an energy minimizing solution for any α, β > 0. Although the solution is not known explicitly, it is unique and fulfills specific pointwise bounds and its energy estimates can be obtained through some concrete computations.
In the subsequent sections, we prove various parts of the theorem. 1 In doing so, we develop our methods for minimization of energy, construction of energy-minimizing solutions, realization of asymptotic behavior, and energy estimation. Moreover, we will present and comment on the mathematical details of the results stated in the theorem.

Solutions to equations of motion by regularized minimization
In terms of the suppressed parameters given in (2.10), the energy functional (2.6) becomes The equations (2.7)-(2.8), or the Euler-Lagrange equations associated with (3.1), are subject to the boundary conditions stated in (2.9). For our purpose, we shall obtain solutions to (3.2)-(3.3) subject to (2.9) as an energy-minimizing configuration of the functional (3.1). To this end, set where the admissible set X is defined to be First, we note that the structure of the functional (3.1) indicates that we may always modify (u, f ) in X if necessary, to lower the energy, to achieve the property This property will be observed in our minimization study to follow. Next, let {(u n , f n )} be a minimizing sequence of (3.4). Then, for any pair of numbers 0 < a < b < ∞, {(u n , f n )} is a bounded sequence in the Sobolev space W 1,2 (a, b). By a diagonal subsequence argument, we obtain the existence of a pair u, f ∈ W 1,2 loc (0, ∞), so that I(u, f ) < ∞ and, by choosing a suitable subsequence if necessary, we may assume without loss of generality u n → u, f n → f (n → ∞) weakly in W 1,2 (a, b) and strongly in C[a, b] for any 0 < a < b < ∞. In fact, to achieve this, we may proceed as follows: For each m = 1, 2, . . . , consider the sequence{(u n , f n )} on (a, b) with a = a m = 1 m and b = b m = 1+m in the way that we extract a subsequence, for any m ≥ 1 which establishes that {(u n,n , f n,n )} is weakly convergent in W 1,2 (a, b) for any 0 < a < b < ∞ as asserted. We are yet to show that (u, f ) lies in X. In other words, we need to verify the boundary condition (2.9). To do so, we note that the tricky, and perhaps the most unnatural or indirect, part of (2.9) is f (0) = 0, which will be detailed later, and other parts are rather straightforward to see [41]. For example, assuming I(u n , f n ) ≤ η 0 + 1 (∀n), we have the uniform estimate showing that f n (r) → 1 as r → ∞ uniformly. As a consequence, {u n } is a bounded sequence in W 1,2 (0, ∞). These properties readily establish f (∞) = 1, u(0) = 1, u(∞) = 0. So it now remains to establish f (0) = 0. To this end, we proceed as follows by a regularization approach [44].
Lemma 3.1. Let {(u n , f n )} be a minimizing sequence of (3.4). Then we can modify the sequence {f n } so that it solves the boundary-value problem Proof. Let {(u n , f n )} be a minimizing sequence of (3.4) satisfying I(u n , f n ) ≤ η 0 + 1 (∀n), say. Introduce the cut-off function Then, using 0 ≤ f n ≤ 1 and the Schwarz inequality, we have Other terms are easily controlled. Hence we obtain In other words, {(u n , ξ n f n )} is also a minimizing sequence. That is, we are allowed to assume that f n satisfies the truncated condition f n (r) = 0 for r ≤ 1 n , which is seen to be regularized since the singularity of (3.1) is at r = 0.
Let {f m } be a minimizing sequence of (3.10). As before, we can assume that 0 ≤ f m ≤ 1. Thus for any 1 n < c < ∞, the sequence {f m } is bounded in W 1,2 1 n , c . A diagonal subsequence argument shows that there is a subsequence, which we still denote by {f m }, and there is an element It is clear that f n solves (3.10). So f n solves (3.7) as well. Proof. Assume that f n satisfies (3.7). Taking n → ∞ in any interval [a, b] with 0 < a < b < ∞ and iterating convergence from lower-to higher-order derivatives of the sequence, we see that the limit f solves the equation In fact, if (3.13) is false, there are constants δ > 0 and ε 0 > 0 such that Therefore, in view of (3.14), for any 0 < r 0 < δ, we have δ r 0 which diverges as r 0 → 0, contradicting the convergence of the integral ∞ 0 r 2 (f (r)) 2 dr. Using (3.13), we obtain after integrating (3.12) that (3.16) Using 0 ≤ f ≤ 1 and u(0) = 1 in (3.16), we see that that f (r) ≥ 0 when r > 0 is small. In particular, the monotonicity of f (r) holds for r > 0 small. Consequently, there is a number f 0 ≥ 0 such that lim where we have inserted (3.12) to get the right-hand-side quantity of the above. Hence, if f 0 > 0 in (3.17), then there are constants δ > 0 and ε 0 > 0 such that Integrating (3.20), we obtain which contradicts the existence of limit stated in (3.17). Therefore the lemma follows.
We now present some properties of the energy-minimizing solution obtained above.
For the asymptotic estimates (3.22), we note that the one for u(r) is easy because of the form of the equation of u in (3.25) in view of (3.26). To get the estimate for f (r) in (3.22), we use the comparison function F (r) = Ce −σr , r > 0, C ≥ 1. (3.43) Then we have which may be used in the equation of f in (3.25) to give us for γ > 0. Then, using (3.26) and the estimate for u(r) stated in (3.22), we wee that there is some large r ε > 1 such that b(r) > 1 (say), Choose C in (3.43) large enough such that F (r ε ) + f (r ε ) − 1 ≥ 0. Using this and that F + f − 1 vanishes at infinity in the differential inequality we obtain F (r) + f (r) − 1 ≥ 0 for all r ≥ r ε by the maximum principle [13]. This establishes which gives rise to the asymptotic estimate for f (r) stated in (3.22) when γ > 0.
The estimate for f (r) when γ = 0 stated in (3.24) will be established in the next section.
Thus the proof of part (i) of Theorem 2.1 is carried out. We now turn to part (ii) of Theorem 2.1. Thus, in (3.1), consider the special situation, β = 2, γ = 0, with the radial energy functional which becomes the classical Bogomol'nyi [3] and Prasad-Sommerfield [27] limit of the SU (2) Yang-Mills-Higgs monopole model with a vanishing Higgs coupling constant or zero Higgs potential density function, known as the BPS self-dual limit, with setting f → f √ α . Thus, below, we only recall some facts which are useful our study, although, for the purpose of our presentation, we keep the parameter α here in order to relate the problem to the issues of our interest.
First, note that the occurrence of spontaneously broken symmetry dictates the asymptotic condition f (∞) = f ∞ > 0, (3.51) where f ∞ is otherwise prescribed which is sometimes referred to as the monopole mass [16]. Due to the structure of (3.50), it is seen that the energy is symmetric under the change of variables and parameter: (3.52) Therefore we may assume f ∞ = 1 in (3.51) without loss of generality. We will observe this 'normalized' asymptotic condition in the sequel. That is, we again follow the boundary condition (2.9) for our problem. The Euler-Lagrange equations associated with (3.50) are which may also be obtained by setting β = 2 and γ = 0 in (3.2)-(3.3) and whose least-energy solution may be obtained by minimizing (3.50) subject to (2.9) as before.
Next, as in [3,27], and using the boundary conditions (2.9), we have Hence, we have the energy lower bound, I(u, f ) ≥ 2 √ α, which is attained when the pair (u, f ) satisfies the following BPS-type equations which may be solved [27] to yield the unique solution given explicitly by the formulas It is interesting to note that, among finite-energy solutions satisfying the boundary condition (2.9), the Euler-Lagrange equations (3.53)-(3.54) and the BPS equations (3.56)-(3.57) are equivalent, as established by Maison [21]. In contrast, for the SU (3) situation, Burzlaff [5,6] showed the existence of a non-BPS solution even within radially symmetric configurations; in the SU (2) setting, without radial symmetry assumption, Taubes [40] established the existence of an infinite family of non-BPS solutions in the BPS coupling, whose result was later extended in several important contexts [4,25,31,34,35]. Thus, in general non-Abelian gauge field theories, the equivalence statement may not be valid.

The general non-BPS situation with γ = 0
In this section, we establish part (iii) of Theorem 2.1. Thus we consider the energy (3.1) when γ = 0 such that the energy functional assumes the form As noted in the previous section, the arbitrary asymptotic limit given in (3.51) may be normalized to fit into that stated in (2.9) through the rescaling of parameters set in (3.52). The Euler-Lagrange equations associated with (4.1) are As before, it is readily shown that (4.2)-(4.3) has a solution (u, f ) that minimizes the energy (4.1), satisfying the boundary condition (2.9) and enjoying the property 0 < u(r), f (r) < 1 for r > 0. Thus, using (3.16) with γ = 0, we see that f (r) > 0 for all r > 0. It is less obvious to see that u(r) is also monotone as we now show below. Proof. We first show that u is nonincreasing. Suppose otherwise that there are 0 < a < b < ∞ so that u(a) < u(b). Let r 1 ∈ (0, b) satisfy Therefore u(r) > u(r 1 ) for all r ∈ (r 1 , b). Since u(r) → 0 as r → ∞, we have a unique r 2 > r 1 satisfying (In fact, r 2 > b.) Now modify u by setting u(r) = u(r 1 ), r ∈ (r 1 , r 2 );ũ(r) = u(r), r / ∈ (r 1 , r 2 ). (4.6) Then (ũ, f ) ∈ X but I(ũ, f ) < I(u, f ) which is false. In fact we have only to compare the energies over the interval (r 1 , r 2 ). That is, we are to show thatJ < J where We recall by the definition of r 1 that u (r 1 ) = 0 and u (r 1 ) ≥ 0 since r 1 is a local minimum point. Inserting this information into (4.2), we find since u(r 1 ) > 0. On the other hand, in view of (4.9) and the fact that f (r) increases. Consequently u(r) can only be nonincreasing. If u is not strictly decreasing, it must be a constant in an interval. So we arrive at a contradiction by using the equation (4.2) because it implies that r 2 f (r) is constant, which is false since f (r) increases. Thus the lemma follows.
We now estimate the energy carried by (u, f ). For convenience, we denote the energy (3.50) by I(u, f ; α, β). Then we have by (3.55) the lower estimate Moreover, using (3.55) again, we have Summarizing (4.11) and (4.12), we have the energy lower bound To get some upper estimates for the energy, we make the decomposition (4.14) Now use the BPS solution (3.58), denoted as (u BPS , f BPS ), as a test configuration to get Thus, inserting (u BPS , f BPS ) into the right-hand side of (4.14) and using (4.15), we have Summarizing (4.13) and (4.16), we obtain the estimates of the energy of the solution pair (u, f ) as follows It is interesting that when β = 2 we arrive at the BPS situation, I(u, f ) = 2 √ α, as anticipated, which is hardly surprising. Note also that, except for the critical situation β = 2 where (4.17) becomes equality, in any non-BPS situation β = 2, inequalities in (4.17) are strict because the pair (u BPS , f BPS ) does not satisfy the coupled equations (4.2)-(4.3).
In Figure 1 we plot the energy lower and upper bounds given in (4.17) for I(u,f ) √ α as functions of β. We now verify the asymptotic estimate for f (r) stated in (3.24). For this purpose, we integrate (4.3) and use (3.18) to get r 2 f (r) = r 0 βf (ρ)u 2 (ρ) dρ. (4.18) In view of the estimate for u(r) stated in (3.22), we see that the right-hand side is a bounded quantity for r > 0. Thus, integrating (4.18), we have which establishes (3.24). We note that the explicit BPS solution (3.58) confirms the asymptotic estimates stated in Lemma 3.3 in the case when γ = 0.

The situation when γ = ∞
We now turn to part (iv) of Theorem 2.1. In this situation, following [9] and with our notation, the energy functional now reads with the associated Euler-Lagrange equation subject to the corresponding boundary condition: This type of problems also occur in other situations in gauge field theory (e.g., a discussion in the next section). Due to such separate interest, we summarize our existence and uniqueness results regarding (5.2) as follows.
Theorem 5.1. The boundary value problem (5.2) has a solution which minimizes the energy (5.1). Furthermore, such a solution satisfies the properties that 0 < u(r) < 1 for r > 0, u(r) strictly decreases, and In fact, any finite-energy solution of (5.2) enjoys the additional properties where ε ∈ (0, 1) may be taken to be arbitrarily small. Besides, any nonnegative solution u to (5.2) satisfies the global pointwise lower bounds e − αβ 2 r < u(r) < 1, r > 0, (5.5) and is unique. In particular, subject to the boundary conditions in (5.2), the energy (5.1) has a unique minimizer.
Proof. As before, it is not hard to prove that (5.2) has a solution which minimizes the energy (5.1) and decreases monotonically. The energy estimates (5.3) will be obtained later.
In fact, the result about the limit of u (r) as r → 0 in (5.4) holds as a consequence of the estimate (3.23). Furthermore, near r = ∞, the differential equation in u in (5.2) may be approximated by the linear equation η = αβ 2 η which leads to the exponential decay estimates stated in (5.4) as well. Below we elaborate on (5.5) and the uniqueness of a nonnegative solution in detail.
Let u ≥ 0 be a nonnegative solution of (5.2). Then u(r) > 0 for all r > 0 otherwise there is some r 0 > 0 such that u(r 0 ) = 0, u (r 0 ) = 0, resulting in u(r) = 0 for all r > 0 by the uniqueness of a solution to the initial value problem of an ordinary differential equation, which is false. Moreover, using the maximum principle in (5.2), we have u < 1. Hence we have u < 1 2 αβu. Now let η denote the left-hand-side exponential function in (5.5). Then η = 1 2 αβη and (η − u) > 1 2 αβ(η − u). Thus, using the boundary condition that η − u vanishes at r = 0 and r = ∞, we get (η − u)(r) < 0 for all r > 0 in view of the maximum principle. So (5.5) follows.
Note. Since the energy functional (5.1) is not convex, the uniqueness of a critical point of it is generally not ensured. Our theorem however asserts that (5.2) has a unique minimizer as a solution to (5.2).
We now estimate the energy the unique minimizer of (5.1) carries. First, let u be a finite-energy solution of (5.2). As a critical point of (5.1), we see that the rescaled function u δ (r) = u(δr) satisfies dI(u δ ) dδ To get a lower estimate of the energy, we take v as a test function satisfying v(0) = 1, v(∞) = 0, and use the BPS method to obtain so that the lower bound is attained when v solves v + αβ which happens to be the lower bound function in (5.5). On the other hand, using (5.15) as a test function, we obtain an upper estimate for the energy Therefore, combining (5.14) and (5.16), we get the lower and upper estimates of the energy I(u) as follows: 2αβ < I(u) < 2αβ(1 + 2 ln 2), (5.17) where the right-hand side inequality in (5.17) is strict since (5.15) does not satisfy (5.2). We note that the profile (5.15) suggests that we may further improve the upper bound in (5.17) by taking a trial undetermined profile v a (r) = e −ar (a > 0) and minimizing the function Consequently we obtain an improvement upon (5.17): 2αβ < I(u) < 2αβ(1 + 4 ln 2), (5.20) where the right-hand side inequality is again strict because v a (r) does not satisfy (5.2). This result is (5.3) which presents a significant improvement over (5.17) since (1 + 2 ln 2) − (1 + 4 ln 2) > 7 16 . It will be interesting to study whether the boundary value problem (5.2) has a unique solution without assuming u ≥ 0.
Our study may find applications in other related problems. As an example, consider the SO(3) Georgi-Glashow model [17,43] described by the Lagrangian density where a = 1, 2, 3 is the group index, A µ = (A a µ ) a gauge field, φ a scalar field in the adjoint representation of the gauge group, m, λ > 0, and are the field strength tensor and gauge-covariant derivative, such that the spontaneously broken symmetry results in the vector field mass M W , Higgs boson mass M H , and the mass ratio , given by [17] respectively. The static spherically symmetric monopole soliton assumes the hedgehog form [17,26,37]: φ a = H(r) er 2 x a , A a 0 = 0, A a i = aij x j (1 − K(r)) er 2 , (5.24) whose energy is E = − R 3 L dx which in turn is given by [17]  where c 0 = e 2 4π is the fine-structure constant. Furthermore, with the rescaled radial variable M W r → r and the substitution u = K, f = H r , the energy (5.25) becomes E = M W c 0 C( ), (5.26) where [12,17] C( ) = ∞ 0 (u ) 2 + (u 2 − 1) 2 2r 2 + f 2 u 2 + r 2 2 (f ) 2 + 2 8 r 2 f 2 − 1 2 dr, (5.27) and u, f are subject to the same boundary conditions stated in (2.9). This functional is covered as a special case of (3.1). In particular, C(0) = 1 since the minimizer of the right-hand side of (5. Note that, in [12,17], by using numerical solutions, it is estimated that C(∞) = 1.787. This result is consistent with our estimates above.