Research Article BPS Equations of Monopole and Dyon in (2) Yang-Mills-Higgs Model, Nakamula-Shiraishi Models, and Their Generalized Versions from the BPS Lagrangian Method Ardian Nata Atmaja 1 and Ilham Prasetyo 1,2 1 Research Center for Physics, Indonesian Institute of Sciences (LIPI), Kompleks Puspiptek Serpong, Tangerang 15310, Indonesia 2 Departemen Fisika, FMIPA, Universitas Indonesia, Depok 16424, Indonesia Correspondence should be addressed to Ardian Nata Atmaja; ardi002@lipi.go.id and Ilham Prasetyo; ilham.prasetyo@sci.ui.ac.id Received 28 August 2018; Accepted 26 November 2018; Published 10 December 2018 Academic Editor: Diego Saez-Chillon Gomez Copyright © 2018 Ardian Nata Atmaja and Ilham Prasetyo. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Te publication of this article was funded by SCOAP 3 . We apply the BPS Lagrangian method to derive BPS equations of monopole and dyon in the (2) Yang-Mills-Higgs model, Nakamula-Shiraishi models, and their generalized versions. We argue that, by identifying the efective felds of scalar feld, , and of time-component gauge feld, , explicitly by = with  being a real constant, the usual BPS equations for dyon can be obtained naturally. We validate this identifcation by showing that both Euler-Lagrange equations for  and  are identical in the BPS limit. Te value of  is bounded to ||<1 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 fnd a new feature that, by adding infnitesimally the energy density up to a constant 4 2 , 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 = −1 , while for dyon it is (− 2 )=1− 2 which further gives lower bound to  as such ≥|2√ 1− 2 |. We also write down the complete square-forms of all efective Lagrangians. 1. Introduction Monopole has been known to exist in nonabelian gauge theory. One of the main developments was given by ’t Hoof 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 (3) gauge group into (1) gauge group. Later on, Julia and Zee showed that a more general confguration of soliton called dyon may exist as well within the same model [4]. Furthermore, the exact solutions were given by Prasad and Sommerfled in [5] by taking some limit where  → 0. Tese solutions were proved by Bogomolnyi in [6] to be solutions of the frst-order diferential equations which turn out to be closely related to the study of supersymmetric system [7] (in this article, we shall call the limit  → 0 as BPS limit and the frst-order diferential equations as BPS equations). At high energy the Yang-Mills theory may receive contri- butions from higher derivative terms. Tis can be realized in string theory in which the efective action of open string the- ory 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 feld strength [8–13]. Further complications appear when we add Higgs feld 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 profle depends Hindawi Advances in High Energy Physics Volume 2018, Article ID 7376534, 17 pages https://doi.org/10.1155/2018/7376534