#### Abstract

We apply the BPS Lagrangian method to derive BPS equations of monopole and dyon in the Yang-Mills-Higgs model, Nakamula-Shiraishi models, and their generalized versions. We argue that, by identifying the effective fields of scalar field, , and of time-component gauge field, , explicitly by with being a real constant, the usual BPS equations for dyon can be obtained naturally. We validate this identification by showing that both Euler-Lagrange equations for and are identical in the BPS limit. The value of is bounded to due to reality condition on the resulting BPS equations. In the Born-Infeld type of actions, namely, Nakamula-Shiraishi models and their generalized versions, we find a new feature that, by adding infinitesimally the energy density up to a constant , with being the Born-Infeld parameter, it might turn monopole (dyon) to antimonopole (antidyon) and vice versa. In all generalized versions there are additional constraint equations that relate the scalar-dependent couplings of scalar and of gauge kinetic terms or and , respectively. For monopole the constraint equation is , while for dyon it is which further gives lower bound to as such . We also write down the complete square-forms of all effective Lagrangians.

#### 1. Introduction

Monopole has been known to exist in nonabelian gauge theory. One of the main developments was given by ’t Hooft in [1] and in parallel with a work by Polyakov in [2], in which he showed that monopole could arise as soliton in a Yang-Mills-Higgs theory, without introducing Dirac’s string [3], by spontaneously breaking the symmetry of gauge group into gauge group. Later on, Julia and Zee showed that a more general configuration of soliton called dyon may exist as well within the same model [4]. Furthermore, the exact solutions were given by Prasad and Sommerfiled in [5] by taking some limit where . These solutions were proved by Bogomolnyi in [6] to be solutions of the first-order differential equations which turn out to be closely related to the study of supersymmetric system [7] (in this article, we shall call the limit as BPS limit and the first-order differential equations as BPS equations).

At high energy the Yang-Mills theory may receive contributions from higher derivative terms. This can be realized in string theory in which the effective action of open string theory may be described by the Born-Infeld type of actions [8]. However, there are several ways in writing the Born-Infeld action for nonabelian gauge theory because of the ordering of matrix-valued field strength [8–13]. Further complications appear when we add Higgs field into the action. One of examples has been given by Nakamula and Shiraishi in which the action exhibits the usual BPS monopole and dyon [14]. Unfortunately, the resulting BPS equations obviously do not capture essential feature of the Born-Infeld action; namely, there is no dependency over the Born-Infeld parameter. In other examples such as in [15], the monopole’s profile depends on the Born-Infeld parameter, but the BPS equations are not known so far.

In this article, we would like to derive the well-known BPS equations of monopole and dyon in the Yang-Mills-Higgs model and their Born-Infeld type extensions, which we shall call them Nakamula-Shiraishi models, using a procedure called BPS Lagrangian method developed in [16]. We then extend those models to their generalized versions by adding scalar-dependent couplings to each of the kinetic terms and derive the BPS equations for monopole and dyon. In Section 2, we will first discuss in detail the BPS Lagrangian method. In Section 3, we describe how to get the BPS equations for monopole (dyon) from energy density of the Yang-Mills-Higgs model using Bogomolny’s trick. We write explicitly its effective action and effective actions of the Nakamula-Shiraishi models by taking the ’t Hooft-Polyakov (Julia-Zee) ansatz for monopole(dyon). In Section 4, we use the BPS Lagrangian approach to reproduce the BPS equations for monopole and dyon in the Yang-Mills-Higgs model and Nakamula-Shiraishi models. Later, in Section 5, we generalize the Yang-Mills-Higgs model by adding scalar-dependent couplings to scalar and gauge kinetic terms and derive the corresponding BPS equations. We also generalize Nakamula-Shiraishi models in Section 6 and derive their corresponding BPS equations. We end with discussion in Section 7.

#### 2. BPS Lagrangian Method

In deriving BPS equations of a model, we normally use the so-called Bogomolny’s trick by writing the energy density into a complete square form [6]. However, there are several rigorous methods which have been developed in doing so. The first one is based on Bogomolny’s trick by assuming the existence of a homotopy invariant term in the energy density that does not contribute to Euler-Lagrange equations [17]. The second method called first-order formalism which works by solving a first integral of the model, together with stressless condition [18–20]. The third method called On-Shell method which works by adding and solving auxiliary fields into the Euler-Lagrange equations and assuming the existence of BPS equations within the Euler-Lagrange equations [16, 21]. The forth method called First-Order Euler-Lagrange (FOEL) formalism, which is generalization of Bogomolnyi decomposition using a concept of strong necessary condition developed in [22], which works by adding and solving a total derivative term into the Lagrangian [23] (in our opinion the procedure looks similar to the On-Shell method by means that adding total derivative terms into the Lagrangian is equivalent to introducing auxiliary fields in the Euler-Lagrange equations. However, we admit that the procedure is written in a more covariant way). The last method, which we shall call BPS Lagrangian method, works by identifying the (effective) Lagrangian with a BPS Lagrangian such that its solutions of the first-derivative fields give out the desired BPS equations [16]. This method was developed based on the On-Shell method by one of the authors of this article and it is much easier to execute compared to the On-Shell method. We chose to use the BPS Lagrangian method to find BPS equations of all models considered in this article. The method is explained in the following paragraphs.

In general the total static energy of N-fields system, , with Lagrangian density , is defined by . Bogomolny’s trick explains that the static energy can be rewritten aswhere is a set of positive-semidefinite functions and is the boundary contributions defined by . Neglecting the contribution from boundary terms in , as they do not affect the Euler-Lagrange equations, configurations that minimize the static energy are also solutions of the Euler-Lagrange equations and they are given by known as BPS equations. Rewriting the static energy to be in the form of (1) is not always an easy task. However it was argued in [16] that one does not need to know the explicit form of (1) in order to obtain the BPS equations. By realizing that, in the BPS limit, where the BPS equations are assumed to exist, remaining terms in the total static energy are in the form of boundary terms, . Therefore we may conclude that BPS equations are solutions of .

Now let us see in detail what is inside . Suppose that in spherical coordinates the system effectively depends on only radial coordinate . As shown by the On-Shell method on models of vortices [21], the total static energy in the BPS limit can be defined aswhere is called BPS energy function. The BPS energy function does not depend on the coordinate explicitly; however in general it can also depend on explicitly in accordance with the chosen ansatz. In most of the cases if we choose the ansatz that does not depend explicitly on coordinate , then we would have . Hence, with a suitable ansatz, we could write in which is the effective field of as a function of coordinate only. Assume that can be treated with separation of variablesthis give us a pretty simple expression of , i.e.,and we could obtain in terms of the effective fields and their first-derivative.

Now we proceed to find the s from . As we mentioned must be positive-semidefinite function and we restrict that it has to be a function of and for each . The BPS equation gives solutions to as follows:with (). Positive-semidefinite condition fixes to be an even number and further there must be even number of equal solutions in . As an example if for all , then is a quadratic equation in and so we will have . The restriction on forces us to rewrite the function into partitions explicitly. This is difficult to apply on more general forms of Lagrangian, since there exists a possibility that there are terms with where . Another problem is ambiguity in choosing which terms contain nonderivative of fields that should belong to which partitions .

For more general situations, the BPS equations can be obtained by procedures explained in [16] which we describe below. On a closer look, we can consider as a polynomial equation of first-derivative fields. Seeing it as the polynomial equation of , whose maximal power is , its roots arewith and . Then we haveAs we mentioned before here must be an even number and to ensure positive-definiteness at least two or more even number of roots must be equal. This will result in some constraint equations that are polynomial equations of the remaining first-derivative fields . Repeate the previous procedures for until whose iswith being also an even number. Now all are only functions of and equating some of the roots will become constraint equations that we can solve order by order for each power series of . As an example let us take and . Then the constraint can be seen as a quadratic equation of . This give us the last constraint . Since the model is valid for all , we could write the constraint as , where all s are independent of and . Then all s need to be zero and from them we can find each . Then the BPS equations for can be found.

We can see that this more general method is straightforward for any Lagrangian. This will be used throughout this paper, since we will later use some DBI-type Lagrangian that contains terms inside square root which is not easy to write the partitions explicitly. In [16], with particular ansatz for the fields, writing is shown to be adequate for some models of vortices. Here, we show that the method is also able to do the job, at least for some known models of magnetic monopoles and dyons, using the well-known ’t Hooft-Polyakov ansatz.

#### 3. The ’t Hooft-Polyakov Monopole and Julia-Zee Dyon

The model is described in a flat -dimensional space-time whose Minkowskian metric is . The standard Lagrangian for BPS monopole, or the Yang-Mills-Higgs model, has the following form [1, 2]:with gauge group symmetry and , , being a triplet real scalar field in adjoint representation of . The potential is a function of which is invariant under gauge transformations. Here we use Einstein summation convention for repeated index. The definitions of covariant derivative and field strength tensor of the Yang-Mills gauge field are as follows:with being the gauge coupling and being the Levi-Civita symbol. The Latin indices denote the “vector components” in the vector space of algebra with generators , where is Pauli’s matrix. The generators satisfy commutation relation and their trace is . With these generators, the scalar field, gauge field, adjoint covariant derivative, and field strength tensor can then be rewritten in a compact form, respectively, as , ,These lead to the Lagrangian

Varying (9) with respect to the scalar field and the gauge field yieldswith additional Bianchi identitywhere . Throughout this paper, we will consider only static configurations. The difference between monopole and dyon is whether is zero or nonzero, respectively. For monopole, the Bianchi identity becomesHere and are the spatial indices. For dyon, , there are additional equations of motion for “electric” part since the Gauss law is nontrivial,where .

We could write the energy-momentum tensor by varying the action with respect to the space-time metric. The energy density is then given by component,In [5], it is possible to obtain the exact solutions of the Euler-Lagrange equations in the BPS limit, i.e., , but still maintaining the asymptotic boundary conditions of , and we define a new parameter such thatThe last two terms can be converted to total derivativeafter employing the Gauss law (16) and Bianchi identity (15). They are related to the “Abelian” electric and magnetic fields identified in [1], respectively. Since the total energy is , the total derivative terms can be identified as the electric and magnetic charges accordinglywith denoting integration over the surface of a 2-sphere at . Therefore the total energy is since the other terms are positive semidefinite. The total energy is saturated if the BPS equations are satisfied as follows [24]:Solutions to these equations are called BPS dyons; they are particularly called BPS monopoles for . The energy of this BPS configuration is simply given byAdding the constant contained in and is somehow a bit tricky. We will show later using BPS Lagrangian method that this constant comes naturally as a consequence of identifying two of the effective fields.

Employing the ’t Hooft-Polyakov, together with Julia-Zee, ansatz [1, 2, 4]where and as well, denotes the Cartesian coordinate. Notice that the Levi-Civita symbol in ((22a), (22b), and (22c)) mixes the space-index and the group-index. Substituting the ansatz ((22a), (22b), and (22c)) into Lagrangian (9) we can arrive at the following effective Lagrangian:where ; otherwise it means taking derivative over the argument. As shown in the effective Lagrangian above there is no dependency over angles coordinates and despite the fact that the ansatz ((22a), (22b), and (22c)) depends on and . Thus we may derive the Euler-Lagrange equations from the effective Lagrangian (23) which are given byLater we will also consider the case for generalize Lagrangian of (9) by adding scalar-dependent couplings to the kinetic terms as follows [25]:The equations of motions are now given byIn [25, 26], they found BPS monopole equations and a constraint equation . Using our method in the following sections, we obtain the similar BPS monopole equations and constraint equation. Furthermore, we generalize it to BPS dyon equations with a more general constraint equation.

There are other forms of Lagrangian for BPS monopole and dyon which were presented in the Born-Infeld type of action by Nakamula and Shiraishi in [14]. The Lagrangian for BPS monopole is different from the BPS dyon. The Lagrangians are defined such that the BPS equations ((21a), (21b), (21c), and (21d)) satisfy the Euler-Lagrange equations in the usual BPS limit. The Lagrangian for monopole and dyon is given, respectively, by [14]with being the Born-Infeld parameter and the potential is taken to be the same as in (9). It is apparent that, even though , . Using the ansatz ((22a), (22b), and (22c)), both Lagrangians can be effectively written asWe can see immediately that . However, by assuming the BPS equations is valid beforehand we would get . Hence from both Lagrangians, we could obtain the same BPS equations when we turn off the “electric” part for monopole.

#### 4. BPS Equations in Yang-Mills-Higgs and Nakamula-Shiraishi Models

Here we will show that the BPS Lagrangian method [16] can also be used to obtain the known BPS equations for monopole and dyon in the Yang-Mills model (9) and the Nakamula-Shiraishi models, (27) and (28). To simplify our calculations, from here on we will set the gauge coupling to unity, .

##### 4.1. BPS Monopole and Dyon in Yang-Mills-Higgs Model

Writing the ansatz ((22a), (22b), and (22c)) in spherical coordinates,we find that there is no explicit dependent in all fields above. Therefore we propose that the BPS energy function for the case of monopole, where , should take the following form:Since , we have the BPS LagrangianBefore showing our results, for convenience we define through all calculations in this article , and .

Employing , where is (23) and is (33), we can consider it as a quadratic equation of either or . Here we show the roots of (or ) first which areThe two roots will be equal, , if the terms inside the square root is zero, which later can be considered as a quadratic equation for (or ) with rootsAgain, we need the terms inside the square root to be zero for two roots to be equal, . The last equation can be written in power series of ,Demanding it is valid for all values of , we may take , which is just the same BPS limit in [5]. From the terms with quadratic and zero power of , we obtainwhich impliesInserting this into (34) and (35), we reproduce the known BPS equations for monopole,

Now let us take and consider the BPS limit, . In this BPS limit, we can easily see from the effective Lagrangian (23) that the Euler-Lagrange equations for both fields and are equal. Therefore it is tempted to identify . Let us write it explicitly aswhere is a real-valued constant. With this identification, we can again use (32) as the BPS energy function for dyon and hence give the same BPS Lagrangian (33). Now the only difference, from the previous monopole case, is the effective Lagrangian (23) which takes a simpler formHere we still keep the potential and we will show later that must be equal to zero in order to get the BPS equations using the BPS Lagrangian method.

Applying (41) and solving as quadratic equation for (or ) give us two rootswithNext, requiring , we obtainwhere we arrange in power series of , i.e.,Again, ; we get . Solving the last equation, which must be valid for all values of , we conclude from -terms, for nontrivial solution, and from the remaining terms we havewhich give usThe BPS equations are thenSince and are real valued, should take values . They become the BPS equations for monopole ((40a) and (40b)) when we set . We can see that this constant is analogous to the constant , or precisely , in (18); see [27] for detail. Substituting into ((49a) and (49b)), we get the same BPS equations as in [5, 24]. Here we can see the constant is naturally bounded as required by the BPS equations ((49a) and (49b)).

##### 4.2. BPS Monopole and Dyon in Nakamula-Shiraishi Model

In this subsection we will show that the Lagrangians (27) and (28) of Nakamula-Shiraishi model do indeed possess the BPS equations ((40a) and (40b) and (49a) and (49b)), respectively, after employing the BPS Lagrangian method. Substituting (29) and (32) into and following the same procedures as the previous subsection give us the roots of ,whereSolving gives uswhereThen the last equation gives us , or ,which again gives usand thus we have and , with ,the same BPS equations ((40a) and (40b)) for monopole. The other choice of potential will result in the same BPS equations with opposite sign relative to the BPS equations of ,

For dyon, using the same identification (41), we have the effective Lagrangian (30) shortened toEquating the above effective Lagrangian with , using the same BPS energy density (32), and solving this for give uswhereSolving for gives uswhereWe may set , but this will imply which is not what we want. Requiring valid for all values of , the terms with give us or . The terms with implyThis is indeed solved by the remaining terms which implyThis again gives usand hence, for ,the same BPS equations ((49a) and (49b)) for dyon. Similar to the monopole case choosing will switch the sign in the BPS equations. It is apparent that, in the limit of , the BPS equations for dyon become the ones for monopole. This indicates that, in the BPS limit and , , since in general, even though in the limit of , .

Now we know that the method works. In the next sections, we use it in some generalized Lagrangian whose BPS equations, for monopole or dyon, may or may not be known.

#### 5. BPS Equations in Generalized Yang-Mills-Higgs Model

In this section, we use the Lagrangian (25) whose effective Lagrangian is given byWe will see later that it turns out that and are related to each other by some constraint equations.

##### 5.1. BPS Monopole Case

In this case, the BPS equations are already known [25, 26]. Setting and employing we getand from we have the roots of (or )The terms inside the curly bracket in the square root must be zero in which after rearranging in power series of we obtain ,These implyand hencewhere is a positive constant. The BPS equations are given bywith a constraint equation , where is a positive constant. This constant can be fixed to one, , by recalling that in the corresponding nongeneralized version, in which , we should get back the same BPS equations of (40a) and (40b).

##### 5.2. BPS Dyon Case

As previously setting and employing we getwithand from we have the roots of whereRequiring we obtain ,Similar to the monopole case these implywhere is a positive constant and it can also be fixed to demanding that at we should get the same BPS equations ((49a) and (49b)). At , we get back the constraint equation (78) for monopole case. These give us the BPS equationsin which at we again get back the BPS equations for monopole case ((79a) and (79b)).

#### 6. BPS Equations in Generalized Nakamula-Shiraishi Model

Here we present the generalized version of the Nakamula-Shiraishi models (27) and (28) for both monopole and dyon, respectively.

##### 6.1. BPS Monopole Case

A generalized version of (27) is defined bywhere after inserting the ansatz, we write its effective Lagrangian asUsing the similar BPS Lagrangian (33), we solve as a quadratic equation of (or ) first as such the roots are given bywith