On ’t Hooft–Polyakov Monopole, Julia–Zee Dyon, and Higgs Field, throughout the Generalized Bogomoln’...

Lukasz Andrzej Glinka

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On ’t Hooft–Polyakov Monopole, Julia–Zee Dyon, and Higgs Field, throughout the Generalized Bogomoln’yi Equations

Lukasz Andrzej Glinka

B.M. Birla Science Centre, Hyderabad, India


In this paper, making use of the’tHooft–Polyakov–Julia–Zeeansatz for the SU(2) Yang–Mills–Higgs gauge field theory, we present the straightforward generalization of the Bogomoln’yi equations and its several consequences. Particularly, this is shown that this idea is able to generate new types of non-abelian both dyons and magnetic monopoles and, moreover, that within the new model the scalar field can be described through the Coulomb potential, whereas, upto aconstant, the non-abelian gauge field becomes the Wu–Yang monopole.

Cite this article:

  • Glinka, Lukasz Andrzej. "On ’t Hooft–Polyakov Monopole, Julia–Zee Dyon, and Higgs Field, throughout the Generalized Bogomoln’yi Equations." Applied Mathematics and Physics 2.3 (2014): 119-123.
  • Glinka, L. A. (2014). On ’t Hooft–Polyakov Monopole, Julia–Zee Dyon, and Higgs Field, throughout the Generalized Bogomoln’yi Equations. Applied Mathematics and Physics, 2(3), 119-123.
  • Glinka, Lukasz Andrzej. "On ’t Hooft–Polyakov Monopole, Julia–Zee Dyon, and Higgs Field, throughout the Generalized Bogomoln’yi Equations." Applied Mathematics and Physics 2, no. 3 (2014): 119-123.

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1. Yang–Mills–Higgs Equations

The Yang–Mills–Higgs theory is the non-abelian Yang–Mills theory of the gauge field [1] coupled to the Higgs field [2]. For the SU(N) gauge group, the Lagrangian has the form


where a = 1,...,N2 -1 is a gauge group index, and


are the strength tensor and the gauge-covariant derivative, respectively, while g is a coupling constant, and are the (antisymmetric) structure constants of the Lie algebra su (N) whose elements are the infinitesimal generators Ta of SU(N)


and SU(N) is produced by formal exponentiation of su(N). Furthermore,


m is a mass parameter, and λ is a dimensionless constant. The energy density


where is the electric field and is the magnetic field induction, is minimized HYMH = 0 for the Higgs vacuum


which is degenerated, that is the set of vacuum expectation values of the Higgs field forms a sphere on which all the points are equivalent because the gauge transformation connects them. Moreover, for φ0 ≠ 0 there is the spontaneous symmetry breakdown mechanism SU(N) → U(1)N-1. The field equations for the Yang–Mills–Higgs theory (1) are


and the Bianchi identities


are satisfied for the dual field strength tensor

2. ’t Hooft–Polyakov–Julia–Zee Ansatz

Dyon, a hypothetical object carrying a non-zero both electric and magnetic charges, was for the first time considered through Julian Seymour Schwinger [3, 4], who applied this model as the phenomenological alternative to quarks to describe a particle analogous to the flavor-neutral J/ψ meson consisting of a charm quark and a charm antiquark, known as charmonium, discovered soon later [5, 6]. Particularly, the Yang–Mills–Higgs system was considered for the case of the SU(2) gauge group, that is when the structure constants are the three-dimensional Levi–Civita symbols εabc and the infinitesimal generators in the fundamental representation are given by the Pauli matrices σa as . In such gauge field theory, a smooth nonsingular solution to the Yang–Mills–Higgs equations, arising from the assumptions of spherical symmetry and time independence of both the non-abelian gauge field and the Higgs field


known as the ’t Hooft–Polyakov ansatz, was considered in the pioneering papers authored through Gerardus ’t Hooft [7] and A.M. Polyakov [8, 9], which with the boundary conditions


is known as the ’t Hooft–Polyakov monopole. In the context of gauge field theories, dyons were for the first time considered as the solutions to the Yang–Mills–Higgs equations through Bernard Julia and Anthony Zee [10], who supplemented the ’t Hooft–Polyakov ansatz by


where C is a constant. The’t Hooft–Polyakov–Julia–Zee ansatz is compatible with the Lorentz gauge which gives


that is , and, particularly, is satisfied through each spherically symmetric function K.

3. Bogomol’nyi–Prasad–Sommerfield Limit

The only known analytical solution occurs for the Bogomol’nyi–Prasad–Sommerfield (BPS) limit [11, 12], that is λ → 0. Then, the Higgs field is massless but the full gauge invariance remains broken due to its non-zero vacuum expectation value. It is easy to see that the energy calculated by (5)


where is the scalar field potential., and


The minimum of (14) in the BPS limit occurs for the Bogomol’nyi equations


The Gauss law for electricity for gives


whereas the Gauss law for magnetism for leads to


and, in this manner,


Consequently, one receives


where is the fine-structure constant. Through Einstein’s relation , the effective mass M satisfies the Bogomol’nyi bound


And the equality holds in the BPS limit. For the’t Hooft–Polyakov monopole, in the BPS limit, one has


the total energy density of such a solution is


and the Bogomol’nyi equations (17)-(16) become


The subject of non-abelian both magnetic monopoles and dyons has already met a certain interest, Cf. the Refs. [13-44][13] and references therein.

4. Generalization of Bogomol’nyi Equations

Let us generalize the BPS solution to any λ. Considering the Yang–Mills–Higgs energy in the form , where






and are dimensionless constants, and


where the following consistency conditions hold


Applying the Gauss laws (18) and (19), one receives a


and after careful calculations, one receives


The minimum of (35) is not only the Higgs vacuum, but also and , giving the BPS limit, or equivalently


and then the effective mass is


If, moreover, this mass identically vanishes, that is




then one has the condition


which includes, as the particular situation, the BPS limit, but in general V is not necessary zero. Similarly, for (38) the particular case


is true if and only if (41), also for the BPS limit. In general, the bound


which for α1 = sinαc, α2 = cosαc is the Bogomol’nyi bound, is true for


The condition (32) gives eight in equivalent solutions


whereas the condition (33) presented in the following form


and parameterized as A +B = 1, also leads to eight solutions


where βc must not be related to αc in general. Then, one obtains


what is consistent for


The ’t Hooft–Polyakov–Julia–Zee ansatz


leads to the relations


which, along with (36) and (37), lead to the system of equations for a dyon


For the ’t Hooft –Polyakov monopole, J = 0, (57), (58), (59) and (60) are


and, therefore, one obtains the solutions


Consequently, the Higgs field and the gauge field are


whereas simultaneously the scalar field potential V satisfies the following differential equation


where we have introduced the constant


Interestingly, in the BPS limit, the ’t Hooft–Polyakov monopole is described through H = 0 and K = ±1. Then, φa = 0, for K = 1 also , while for K = 1 k


where is the Wu–Yang monopole solving the SU(2) Yang–Mills theory without the Higgs field [45]. Nevertheless, one can also consider the situation wherein G = 0, or, equivalently


Then, the gauge field is either trivial or (71), while the Higgs field is


and, moreover, the equation (69) becomes


and its straightforward integration gives two possible solutions δ


where and are the integration constants. However, for the special case and , one obtained the unique solution


which is the Coulomb potential. In such a situation, taking and , where is any initial value of , which particularly can be identified with a minimal scale, the configuration of fields is established throughout the gauge field given through the solution (71), where as the Higgs field has the following form


5. Summary

We have seen that generalization of the Bogomoln’yi equations is realizable, and leads to potentially new results for non-abelian both dyons and magnetic monopoles. It should be emphasized that the idea of this paper was based on the author’s monograph [46], and we hope for its further development.


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