A Detailed Study of Bogomol'nyi Equations in Two-Dimensional Generalized Maxwell-Higgs Model Using \textit{On-Shell} Method

We use a recent {\it on-shell} Bogomol'nyi method, developed in~\cite{Atmaja:2014fha}, to construct Bogomol'nyi equations of the two-dimensional generalized Maxwell-Higgs model~\cite{Bazeia:2012uc}. The formalism can generate a large class of Bogomol'nyi equations parametrized by a constant $C_0$. The resulting equations are classified into two types, determined by $C_0=0$ and $C_0\neq0$. We identify that the ones obtained by Bazeia {\it et al}~\cite{Bazeia:2012uc} are of the type $C_0=0$. We also reveal, as in the case of ordinary vortex, that this theory does not admit Bogomol'nyi equations in the Bogomol'nyi-Prasad-Sommerfield limit in its spectrum. However, when the vacuum energy is lifted up by adding some constant to the energy density then the existence of such equation is possible. Another possibility whose energy is equal to the vacuum is also discussed in brief. As a future of the \textit{on-shell} method, we find another new Bogomol'nyi equations, for $C_0\neq0$, which are related to a non-trivial function defined as a difference between energy density of potential term of the scalar field and kinetic term of the gauge field.


Introduction
Bogomolnyi method is a smart trick to reduce the second-order Euler-Lagrange equations into the first-order, whose solitonic solutions possess minimum energies [3]. For topologically nontrivial field's vacuum manifold the solutions are stable since at the boundary they map each point in coordinate space with different global minimum of the potential.
So far the Bogomolnyi equations were derived by saturating the lower bound of the corresponding static energy (the so-called off-shell approach). This method may not always give the Bogomolnyi equations easily, especially when the Lagrangian contains noncanonical terms, as in the case of k-defects [4][5][6][7][8][9][10]. Recently, two of us [1] proposed an alternative in obtaining the first-order equations by directly evaluating the Euler-Lagrange equations, later dubbed the on-shell approach. This formalism reproduces the known Bogomolnyi equations for kinks, vortices, and monopoles, as well as Dirac-Born-Infeld (DBI) kins and vortices. This is a novel result though still preliminary, since it might enable us in constructing BPS (Bogomonlyi-Prasad-Sommerfield) states for general defects. Not only it is interesting in its own right, but also these least-energy solitonic solutions might have different properties from their canonical BPS counterparts. In the context of cosmology this might shed a new light on the dynamics of defects.
Not long time ago one of us [2] studied topological vortices in the generalized Maxwell-Higgs theory, whose dynamics are controlled by two positive functions in the Lagrangian, G (|φ|) and w (|φ|). It was shown that, for several choices of G− and w−functions, there exist BPS solutions with various topology and energies (that can be greater than the canonical BPS tensions). Soon it was followed by the discovery of prescription for obtaining their analytical BPS vortex solutions [11]. The similar study was also done on generalized BPS monopoles [12][13][14] 1 .
Here in this paper we look for something more modest by following a different route. Our aim is twofold. First, we wish to improve the on-shell method so that it includes noncanonical Lagrangian. Second, by applying it to the generalized Maxwell-Higgs theory we try to construct generators that generates the corresponding Bogomol'nyi equations. It is expected that for arbitrarily positive functions G (|φ|) and w (|φ|) a large class of first-order Bogomolnyi equations (and their solutions) can be obtained.

Improved Version of On-shell Method
The effective, one-dimensional, Euler-Lagrange equations (6) in the on-shell method of [1] are difficult to get since the right hand side of the equations is only allowed to depend on the parameter r and the fields φ a . It was very fortunate that examples given in [1] for the non-standard theory, which were the DBI defects, have not suffered from this difficulty. However, it should not happen in general for any theory with non-standard kinetic terms, such as the Generalized Maxwell-Higgs theory discussed in this article. Here, we need to improve the on-shell method such that the right hand side of the effective Euler-Lagrange equations are allowed to depend on first derivative of the fields φ a . As a simple case, let us consider a theory with the effective degree of freedom is given by φ, in which the effective one dimensional Lagrangian L = L(r, φ, φ ′ ) and the Euler-Lagrange equation are given by The Euler-Lagrange equation can be arranged into We then need to determine what would be the expected function of f (r, φ, φ ′ ) provided that the left hand side of (2.3) can be rewritten as where h ≡ h(r, φ). Now, since h ′ = ∂h ∂r + ∂h ∂φ φ ′ , it yields that the function f must be of the form We keep the linear terms of f , in φ ′ , in the left hand side of (2.3) and move the nonlinear terms to the right hand side of (2.3). The Bogomol'nyi equation is then given by h(r, φ)φ ′ = X(φ), while the constraint equation is now where g contains all remaining non-linear terms coming from f . Notice that upon substituting the Bogomol'nyi equation into (2.3), we can get back the form of effective Euler-Lagrange equation as in the equation (6) of [1]. For multiple fields theory 2 , generalization of the above procedures are more involved. As such, for each field φ a , the effective one dimensional Euler-Lagrange equations are One should notice that the Euler-Lagrange equations are linear in φ ′′ . Taking the same procedures as in the case of a single field theory, we may write the Euler-Lagrange equation, for each φ a , as where f is linear function in φ ′ . To have the Bogomol'nyi equations, the function f b must be of the form where h b = h b (r, φ). The Bogomol'nyi equations then are given by and the constraint equations are As in [1], the topological charge can directly be obtained by inserting the Bogomol'nyi equations into the energy functional. We shall obtain, in general, where F [X a (φ)] is a general functional of X a (φ) whose form depends on the actual kinetic form of the Lagrangian. In particular, for canonical case F [X a (φ)] = X a (φ). Its integral becomes (2.14)

Generalized Maxwell-Higgs Model
As an example of application of the prescription above, let us now consider a generalized Maxwell-Higgs theory described by the following (1 + 2)-dimensional Lagrangian density [2] where F µν = ∂ µ A ν − ∂ ν A µ , D µ φ = ∂ µ + ieA µ φ, and the Minkowskian metric is η µν ≡ diag(+, −, −). Here, we take the gauge coupling e and the vacuum expectation value v of the scalar field to be real and positive. The functions G (|φ|) and w (|φ|) are constrained to be positive and depend explicitly only on the Higgs field amplitude, |φ|, but not on its derivative 3 . In this article, we will consider a static solitonic object, in particular topological vortices, in which all the fields are static. Furthermore, we will consider the spatial part of the action and write it in terms of the spherical coordinates. We chose a temporal gauge A 0 = 0 and the static fields ansatz where (r, θ) is the polar coordinates and n = ±1, ±2, . . . is an integer winding number. Notice that the Lagrangian is invariant under two-dimensional rotation and an abelian gauge transformation, SO(2)×U (1). The ansatz for the Higgs field is chosen to be invariant under subgroup of this symmetry which is the SO(2) rotational transformation with a particular choice of U (1) gauge transformation, that cancels the two-dimensional rotation. It is guaranteed that the solutions of the effective equations of motion, derived by using this ansatz, are also the solutions of the full equation of motions [17].
Using these ansatz, the static energy, proportional to the static action, can be simply written as The Euler-Lagrange equations, or equations of motion, derived from the above static energy are and The vacuum solution of the above theory (3.1) is related to the solution in which A µ = 0 and φ = v. For the case of topological vortex, we consider the case in which v = 0. For topological vortex solutions, we require the fields a and g to behave asymptotically, near the origin and the boundary, as follows How fast the functions a and g approaching their asymptotic values, namely the next leading order terms, is determined by the Bogomol'nyi equations and the explicit form of G, w, and V ; as such, the static energy (3.3) is finite.

Bogomol'nyi Equations
In order to obtain the Bogomol'nyi equations, following the prescription in section 2, we rewrite the Euler-Lagrange equations into and The first term on the right hand side of equation (4.2) contains first derivative of field a, a ′ (r). It can be turned into a non-derivative fields dependence by using the Bogomol'nyi equations as we will show later in detail. Now, let us introduce some auxiliary fields into the Euler-Lagrange equations as follows and (4.4) where X and Y are the auxiliary functions that depend only on the fields a and g, but not their derivatives, and do not depend explicitly on r. From these equations, we can extract the Bogomol'nyi equations which are and The Bogomol'nyi equations are supplemented by the constraint equations and Notice that we have substituted the first term on the right hand side of the constraint equation (4.8) by using the Bogomol'nyi equation (4.5). Substituting further the Bogomol'nyi equations into the constraint equations yields and ∂Y ∂g Next, we solve those constraint equations by dividing each of them into terms that depend on the explicit power of r. Solving those terms independently, this process yields several equations: The problem is now reduced to finding the auxiliary functions, X and Y , which solve the above (constraint) equations. The fist equation in (4.11) implies that X is independent of a. The general solution for Y can be obtained by solving the first equation in (4.12) which is given by Y 2 (g, a) = a 2 g 2 w + C 0 (a), where C 0 is an arbitrary function of a. However, for nontrivial solutions, the second equation in (4.11) restricts the function C 0 ∝ a 2 . In general, we may write the solution for Y to be Y 2 (g, a) = a 2 g 2 w + C 0 , where now C 0 is just a constant. Since the first equation in (4.11) gives X ≡ X(g), all the auxiliary functions are essentially separable functions. Writing all the auxiliary functions to be separable, without loss of generality we can take X a = 1, Y a = a, and Y 2 g = g 2 w + C 0 . Using Y g = ± g 2 w + C 0 , we obtain from the second equation in (4.11) where C 1 is an integration constant. Therefore we obtain that the Bogomol'nyi equation (4.5) depends on functions w and G, while the Bogomol'nyi equation (4.6) depends only on function w. It will be useful later to define functions Using the previously obtained functions: X g and Y g , we are left with only one constraint equation, the second equation in (4.6), which in terms of functions S and R is simply written as From now on, we will use ′ ≡ ∂ ∂g if it not defined explicitly. The Bogomol'nyi equations can simply be rewritten as follows So, we can say that the equations in (4.15) generate the Bogomol'nyi equations in (4.17) for the generalized Maxwell-Higgs model (3.1) once we fix the functions: w and G, and the constants: C 0 and C 1 , while the constraint equation (4.16) determines the form of potential V once we know all these functions and constants. At first sight, the constraint equation (4.16) is different from the standard one obtained in [2] which, in our conventions, can be written as However, we will show later in the next section that the constraint equation (4.18) of [2] is a particular case of our constraint equation (4.16).

Bogomol'nyi equations for
In this subsection, we consider a particular simple class of solutions. This class of solutions is provided by taking C 0 = 0, for which we obtain S = ±q and It is tempted to expect from the above integral that w ≡ w(g 2 ) which happens to be the case in all Bogomol'nyi equations of [2]. One can also check that all functions of w, G, and V in each Bogomol'nyi equations of [2] are solutions to the constraint equation (4.16). In this case, the Bogomol'nyi equations can be simply written as Using the fact that X ′ g = (RG) ′ = ±2e 2 v 2 gw, the constraint equation can be rewritten as The solution to this differential equation is Here, this constant can actually be set to zero by shifting the potential V in the action. Furthermore, we will see later that by imposing a condition that the energy of the vortex to be finite, this constant is forced to be zero. In this case, it turns out that the potential (4.24) also solves the constraint equation (4.18), and thus it is the same as the constraint equation in [2]. The potentials obtained in [2] can be derived simply by using the constraint (4.24) with a particular choice of the functions and parameters: (a). Standard Maxwell-Higgs model G = 1; w = 1;

Flat potential
A slight advantage of our constraint equation (4.23) is that the potential V can be safely taken to be zero. Unlike the one in [2], or equation (4.18), setting V = 0 will not give us a solution. In the limit of the coupling at which the potential V = 0, the Bogomol'nyi equations would corresponds to the BPS equations for vortex; this is similar to the case of BPS (Bogomol'nyi-Prasad-Sommerfield) monopole [15]. This limit is known as the BPS limit which is essential if one wants to construct the supersymmetric version of the theory. In this case the solution for G is given by where C 2 is an non-zero integration constant related to the non-zero constant in (4.24). Although the constraint (4.18) is not suitable for the case of V = 0, the solution (4.25) can actually be obtained from it by setting the potential to be constant V = 1 . This is related to the fact, as we will discuss in the next section, that the finiteness energy requires a shift in the potential by a constant. Nevertheless, the Bogomol'nyi, or to be precise BPS, equations now become Here, the function G depends on the function w and the constants (C 1 and C 2 ). We present some of the examples, with C 2 = 1, as follows • w = 2(g 2 + 1); Later, we will find that all of the above examples turn out to give infinite energy. This can be seen due to the presence of singularity of the corresponding BPS equations near the boundary. As an example, consider the configuration (c) above in which w = 2(g 2 + 1) and C 1 = −3 gives G = e 4 v 4 (g 2 − 1) 2 (g 2 + 3) 2 . The BPS equations are . (4.27) The second equation blows up at infinity, since g(r → ∞) → 1. On the other hand, there should be many possibilities of G(g) such that it satisfies the boundary conditions. For example, we can take  It is amusing that the combination of G and w above, when inserted into the equations (4.26), produces precisely the equations for ordinary BPS Maxwell-Higgs vortices (up to some overall constants), Since we know that BPS vortices exist, so do these flat potential BPS generalized Maxwell-Higgs vortices. However, there is a subtle here that the functions w and G now can be singular near the boundary, or g → 1.

Bogomol'nyi equations for
For a general case, we can rewrite the constraint equation (4.16) in terms of R and S as follows Unlike the C 0 = 0 case, the right hand side of the contraint equation above is more complicated and it is very difficult to write it as a total derivative of some functions and hence difficult to find the solution. However, we may try to follow what we did as in the C 0 = 0 case and write the constraint (4.31) simply as To have a total derivative, we are tempted to identify where C 3 is just a constant which we can just add to the constraint equation above by shifting 1 The value of C 3 needs to be non-zero otherwise it would not be consistent with C 0 = 0 since S 2 = g 2 . With this identification, we obtain that This is consistent with the C 0 = 0 solutions, in which we have to take C 3 = 0 in order for G to be non-trivial. However, this is a little bit peculiar because G dependence of w is in contradiction with the C 0 = 0 solutions. We might expect that G is still independent of w, or arbitrary, for this more general case in which the constant C 0 is arbitrary. It turns out that this solution can not lead to the finite energy solution as discussed in the next section. Although the constraint equation (4.31) does not seem to have a solution, let us write explicitly the Bogomol'nyi equations: Even if we are able to find solutions for the constraint equation (4.31), it is not guaranteed that those solutions will have finite energy. We will see later that there are some possibilities in which the solutions to the constraint equation (4.31) would give a finite energy.

Static Energy
The static energy, given by formula (3.3), can be rewritten into a nicer form by using Bogomol'nyi equations (4.17), which is defined as the static energy of vortex. We could define a function Q ≡ 2πa GR Substituting this equation into the constraint (4.31) yields or we can also write Now let us see if the vortices have finite energy using the Derrick's Theorem [16,17]. We can write the scaled static energy of (3.3) to be where 0 < λ < ∞ is the scale factor. There is a stationary point if we vary the E(λ) over λ at which is finite and positive. It means there are some vortices with finite energy. Furthermore, the virial theorem requires E gauge = E pot . Consider a simple case which both energy densities are equal pointwise. After substituting the Bogomol'nyi equations (4.17), it yields that Substituting this into the equation (5.3), or (5.4), implies that S 2 = g 2 , or it means C 0 = 0. Therefore if we assume that the energy can be written as an integral over a form dQ then the C 0 = 0 vortices will have infinite energy. Even if we do not use this assumption and just use the equation (5.6), we can show that by taking a first derivative of the equation (5.6) over g and using the Bogomol'nyi equations (4.17). This is equal to the constraint (4.31) providing that S 2 = g 2 , which also concludes that C 0 = 0. Therefore we may ignore the C 0 = 0 Bogomol'nyi equations as they are not physical since their energy is infinite.

Finite energy for
Notice that the requirement for the energy of the solution to be finite, for the point wise case, forces us to set the constant in (4.24) to be zero. In this case, the static energy (3.3) can be simplified to Recalling that (RG) ′ = 2e 2 v 2 gw, we can obtain the aforementioned function Q = 2π GR e 2 a. Now, the static energy is simply written as Using formula (4.19), it yields 5 E Sol = 2πv 2 n lim r→0 w d(g 2 ) + C 1 . (5.12) Here, we have assumed that Q(r → ∞) = 0 and hence GR(r → ∞) = O(1). In other words, we assume that GR is not singular near the boundary. This can be shown to be satisfied in general by writing GR = √ 2e 2 V G, using the equation (5.6). Recalling that near the boundary, the potential V approaches the vacuum solution, in which V = 0, then it only requires that G(r → ∞) = O(1). The finiteness of energy also requires lim r→0 w d(g 2 ), or √ V G(r → 0), to be finite. Since all these functions (w, G and V ) are functions of g, we may rewrite it as w d(g 2 ) g=0 , or √ V G(g = 0), to be finite. 5 Notice that for any polynomial w-function, w(g) ∼ g m with m ≥ 0, the limr→0 w d(g 2 ) always yields zero. In this case the topological charge is solely determined by the constant C1. On the other hand, we can also easily construct a rational w-function, say w = 1 (g 2 + 1) 2 . (5.10) This function is positive and regular at the origin, whose (indefinite) integral gives w d(g 2 ) = − 1 (g 2 + 1) . (5.11) The limit then yields -1. In this particular case, the charge would depend on (C1 − 1).
The static energy can be proportional to the topological charge Q T op = 2πv 2 |n| as such E Sol = C Q Q T op , where C Q ≥ 0. In the case of C Q = 1, we obtain that lim r→0 w d(g 2 ) + C 1 = ±1, (5.13) which means the static energy equal to the standard vortex. For example for w = 1, we have C 1 = ±1. In the list of examples in 4.1, the (a), (c), and (d) are of this type. If C Q > 1 then the static energy is higher than the standard vortex, E Sol > Q T op , and they are determined by lim r→0 w d(g 2 ) + C 1 > 1. (5.14) The example (b) in 4.1 is in this type in which the static energy E Sol = 3Q T op . There are also some interesting Bogomol'nyi equations in which C Q < 1, or E BP S < Q T op , and the condition is given by The most interesting of one is when C Q = 0, or E BP S = 0, with a condition This raises a question, do Bogomol'nyi solutions with zero energy exist? A rigorous answer needs a rigorous proof. In this paper we do not attempt to answer it. We just note that if we choose the following set of functions and parameter w = 2g 2 − 1, G = 1, C 1 = 0, (5.17) we can end up with the following Bogomolnyi equation whose potential is V = e 2 ν 4 2 g 4 1 − g 2 2 , an S 0 surrounded by an S 1 vacuum topology. The equation satisfies both regularity at the origin and finiteness of energy. Due to the vacuum manifold, this is an example of nontopological soliton discussed in [2,18] 6 For Bogomol'nyi topological solitons we need potential whose vacuum manifold is nontrivial. For example if w = 1 then it yields C1 = 0. Now we can set G = g 4 /(1 − g 2 ) 2 such that the theory still has the standard symmetry breaking Higgs potential V = 1 2 e 2 v 4 (1 − g 2 ) 2 . However these functions do not satisfy the near origin condition for the Bogomol'nyi equation; i.e., is singular at the origin.

Flat potential
As we mentioned previously, finiteness in the static energy requires the constant in the constraint equation (4.24) to be zero, or 2e 2 V = R 2 G. Taking V = 0 is not possible in this case and might cause the resulting energy to be infinite. Nevertheless, let us just ignore the requirement for finite energy and allow the potential V = 0. The static energy in this case can be written as Substituting the BPS equations (4.26), we obtain Indeed, we find that the static energy is infinite which comes from the last term on the right hand of equation (5.21). This infinity can be removed by adding a constant potential in the action 7 . The constant potential needed to remove this infinity is equal to the potential computed using equation (5.6) with a given solution for G is (4.25). Therefore if we take the potential to be non-zero constant in the first place, we will have no problem in taking the finite energy equation (5.6), and thus the static energy will be finite. The first term on the right hand side of equation (5.21)depends on a √ G at the boundaries. As we mentioned previously, there is a subtlety in function G if we impose regularity on the Bogomol'nyi equations (4.26). To have Bogomol'nyi equations (4.26) that respect appropriate boundary conditions, G ∼ 1 − g 2 −2m , for some positive integer m. Although G is infinite near the boundary, by taking appropriate leading order of function a as such it is going to zero faster than 1/ √ G, we could obtain the static energy which is It is interesting that the arbitrary choice of C 2 results in different value of E Sol .

Finite energy for
From the previous discussion, it is clear that the finite-energy equation (5.6) strongly restricts the constant C 0 = 0. Therefore the Bogomol'nyi equations for C 0 = 0 would not give a finite static energy of the vortex. However, we should recall that the equation (5.6) is not the only result of the Derrick's theorem followed by the virial theorem. There is another, more general, result of the virial theorem that the finiteness of energy requires ∞ 0 dr r R 2 G 2e 2 − V = 0, (5.24) in which the integrand is non-zero pointwise. Up to now, we do not know how to substitute the definite integral equation (5.24) into the constraint equation (4.16). What we can do is that we can try to rewrite the constraint equation (4.16) to be the following Using (RG) ′ = 2e 2 v 2 g 2 w/S, it can be simplified further to Indeed, there are many solutions for f , in terms of parameter r, that satisfy this condition. One of them is given by the special Laguerre functions with the following integral [19] ∞ 0 dr r e −r L n (r) = 0, L n (r) = e r d n dr n r n e −r , (5.28) where L n is the Laguerre functions for n > 1. Substituting the function f , in terms of r, into the constrain equation (5.26), and exploiting the Bogomol'nyi equations (4.17), it yields solution for a as follows where now ′ ≡ d dr and C a is an integration constant. Finding suitable function for f (r), that satisfy the boundary conditions (3.6), might give us the explicit form of functions w(g) and later also G(g).

Summary
The main purpose of this article is to show how the on-shell method, developed in [1], can be used to find the Bogomol'nyi equations of the generalized Maxwell-Higgs theory in three-dimensional spacetime [2]. In particular, we improved the on-shell method to allow the terms in the equations of motion, that would later be identified as the constraint equations, to depend on the derivative of the fields. The improvement is necessary to tackle a particular type of theory such as the one considered in this article. This might open some possibilities to improve and modify the on-shell method in obtaining the Bogomol'nyi equations of the other non-standard theories.
In the case of the generalized Maxwell-Higgs theory, we found that the Bogomol'nyi equations can be classified into two types which are parametrized by a constant C 0 . The first type is for C 0 = 0 in which we obtained the standard Bogomol'nyi equations as shown in [2,11]. An advantage of using the on-shell method is that we obtained the constraint equation (4.23) that can be applied for the case of zero potential. Although it turns out that the resulting energy is infinite, we were able to show that the static energy could be finite by adding an appropriate non-zero constant to the potential. We also discussed possibilities for the existence of vortices with the energy is equal to the vacuum. From what we know, this has not been discussed in the literature so far and it might be interesting to study the physical properties of this vortices compared to the vacuum.
The second new type Bogomol'nyi equations, that we found here, is when C 0 = 0. These equations are relatively new and we do think they could not be obtained easily using the standard off-shell, or Hamiltonian, method. It turned out that these equations are related to the difference between the energy density of potential term of the scalar field and kinetic term of the gauge field which is given by a non-trivial function f . If the function f is a constant then the requirement for finite energy vortex forces this constant to be zero and hence gives us back the the first type of Bogomol'nyi equations, C 0 = 0. The requirement for finite energy vortex on the Bogomol'nyi equations of the second type, C 0 = 0, restrict further the function f such that its integral over whole two-dimensional space is zero. Here, we do not attempt to find the explicit expressions of the Bogomol'nyi equations of the second type since they will be discussed in the future work.