Monopoles on string-like models and the Coulomb's law

The t'Hooft-Polyakov monopole mass can be substantially altered, in the thick GS and HC brane-world setup, and can be employed to constrain the brane thickness limit. In this work, we comprise a brief review regarding gauge fields localization in the string-like six dimensional brane-world models setup. The correction to the Coulomb's law in two models is studied. Besides, the monopole features are investigated from the point of view of the gauge fields localization in the string-like brane-worlds and its thickness.

Moreover, the thickness of the brane plays an important role. Smooth brane-world have been investigated, presenting a thickness ∆ that drive deviations from the 4D Newton's law on scales of such a magnitude [33][34][35]. Current torsion-balance experiments yield the upper constraint ∆ 44 µm [36] for the brane thickness, that also have to be as big as the 5D Planck scale, ∆ 2 × 10 −21 cm. The 4D Planck mass m p 2.2 × 10 −8 kg and the length prescribing the 4D Newton's constant G N = p /m p are the main ingredients in our analysis, and throughout this paper, natural units shall be employed. Although widely investigated in 5D models, 6D models lack still such kind of phenomenological approach. Notwithstanding, in what follows, the 6th dimension does not play any additional role on fixing the 5D thickness in 6D models, being our approach here equivalent to the 5D ones in the literature.
't Hooft-Polyakov monopoles are physical solutions corresponding to localized topological solitons with finite energy, firstly suggested in the context of Georgi-Glashow models [37]. SU (2) monopoles are allowed to place on a warped thick brane-world. The usual monopole radius was shown to be affected by the warped geometry with sine-Gordon potentials ruling the scalar field that generates the thick brane, in Ref. [38]. As observed in that reference, the more distant from the thick brane core the monopole, the bigger its radius. Hence, one can assert that the monopole radius decreases as a function of the thick brane evanescence out of the 4th dimension into 5D, exactly attaining a maximum value at the thick brane core. In fact, the naked monopole radius, i. e., without the warp multiplicative factor, is dressed by the spontaneous symmetry breaking parameter in a Higgs-like potential, in a thick brane setup. This radius is inversely proportional to the monopole mass [39]. As the dressed -visible -monopole radius does depend on the monopole position, an upper limit on the thick brane width can be hence established.
Effective monopoles were considered in Ref. [38], without effects of the gravitational backreaction, those which have survived out the period of inflation. Therefore, the brane-world cosmology is here assumed not to be altered with respect to the usual cosmological scenario. More precisely, the current existence of SU (2) monopoles is also proposed in Ref. [40], where a monopole can be produced as a D 3 brane, whose tension was shown to coincide with the mass of a monopole in the effective action. The brane inflation may promote the formation of monopoles after the brane inflation, which are not negligible in models of brane cosmology [40].
In the present paper, we analyse the effect of the monopole mass on the thickness of two stringlike models. The Gherghetta-Shaposhnikov (GS) [17] model and the Hamilton Cigar (HC) [18] model. Additionally, the study of thickness parameter by the configurational entropy for both GS and HC model was already performed in the Ref. [4]. This paper is structured as follows: in Sect.
II, we present a brief review of string-like models in 6D, their prominent features concerning the gauge vector fields localization and the correction to the Coulomb's law as well. In Sect. III, we use the fact that the monopoles have not been observed, in order to delimit the brane thickness.
The main concepts and results present in this paper are summarized in the concluding section IV. The localization of vector gauge fields in six-dimensional (6D) scenarios has been already performed in several works [19][20][21][22][23]. On the String-like models, the confinement of vector gauge fields is performed via gravitational coupling, without any additional scalar coupling [19]. Thus, the vector field can be localized on the string-like defect just with the exponentially decreasing warp factor, in opposition to what occurs in 5D models [24,25]. In this section, we briefly review the localization of the U (1) gauge fields in these string-like models.
In this work we analyse two AdS 6 models. The first one is the thin string-like model called Gergheta-Shaposhnikov model (GS) [17]. The second one is the regular thick models called Hamilton Cigar (HC) [18]. The advantage of the GS model relies on the fact that mostly of the results can be analytically obtained. However, the regularity conditions do not hold in the GS model [18,41]. We represent the warp factor of these models in the Table below, I.
Let us now particularize the result to the GS model, where the results are analytically obtained.
In order to obtain a system where the derivative boundary conditions (7) hold for the massive modes, we need to impose a cut off point r max [17,20,21]. Hence, for the small mass regime m n < c, the following discrete spectrum m n = cπn 2 e − c 2 rmax can be obtained, where n > 1 is an integer [20,21].
For the HC model, the results are only numerically obtained [20]. However, due to the fact that asymptotically when r → ∞ the HC model converts to the GS model, the eigenfunction differences are modified only at the origin [17,[19][20][21]. These differences are fundamental for the regularity conditions to hold, modifying the corrections to the Coulomb's law.
In order to illustrate these result, we plot in Fig. 2 the massless modes Φ 0 (z) of Eq. (15) (thick lines) and the analogue quantum potential U (z) of Eq. (13) (thin lines) for the GS (dashed lines) and the HC models (continuous lines). It is worth to emphasize that both the zero modes are confined, being the GS centered at the origin, while the HC is displaced from it. The potential of the GS model is a barrel, whereas the potential of the HC model represents an infinite well.
An example of massive mode is presented in Fig 3, where one notes that both oscillating function cannot obey the normalizing condition (14). Close to the origin, the massive modes of HC model displays a non-periodical behavior, due to the brane displacement out from the origin [20][21][22][23].

B. The corrections to the Coulomb's law on string-like models
The computation of the corrections for Coulomb's law in warped extra dimensional models was proposed in Ref. [42][43][44]. In these references, the interactions of the massive gauge modes with the massless left-handed fermion mode yield a correction of the Coulomb's potential V c (x) in the form [42][43][44]: where x and e 0 are usual the norm of the position vector and the charge of the fermion trapped on the 4D brane, respectively. The corrections to the Coulomb's Law are given by the ∆V c (x) [42][43][44]: where m 1 is the first excited gauge massive mode (given by the value of squared root of quantum analogue potential maximum in Eq. (12)). These corrections are suppressed by the distance x and the mass m. The c 0 is the normalization constant of gauge zero mode, obtained from Eq. (15).
Moreover, theα L 0 (z) is the fermionic normalized left zero mode in the string-like brane-worlds, which takes the form [23]:α where the = C L 0 is the normalization constant for fermions and the parameter ζ > 0. The fermionic zero mode of Eq. (21) is plotted in Fig. 4, where we note that the parameter ζ narrows the width of the fermions zero modes. In a qualitative way, Fig. 5 shows a perspective for these corrections, that can be adjusted to the experimental data. In addition, the corrections to Newton's law on regular string-like models were already studied in Ref. [35], where similar profiles were obtained.

III. MONOPOLES IN 6D WARPED THICK BRANE-WORLDS
In this section we shall study how monopoles can physically constrain parameters that rule the previously presented models. We start with the Georgi-Glashow model, consisting of an SU (2) gauge field A µ and a Higgs field φ in the adjoint representation, with the Lagrangian where the covariant derivative reads D µ = ∂ µ + igA µ and igF µν = [D µ , D ν ], for A µ = A a µ σ a and φ = φ a σ a , for σ a being generators of the SU(2) algebra.
The effective Lagrangian reads after a renormalization φ a → σ −1/2 φ a , where theṽ stands for the spontaneous symmetry breaking dressed parameter, related to the naked parameter byṽ = ve A . On the classical level, the model has two dimensionless parameters, given by the coupling constant, g, and λ. The scale is determined by m 2 , whose negative values induces the SU(2) symmetry to be spontaneously broken into the Abelian U(1), by a non-zero vacuum expectation value of the Higgs field, given by Tr φ 2 = v 2 2 = m 2 2λ . In the broken phase, the particle spectrum consists of a massless photon, electrically charged W ± bosons, and a neutral Higgs scalar, with respective masses and massive magnetic monopoles.
The scalar and vector field for the monopole read φ a = r a gr 2 H(gvr) (26) where H and K are that describe a magnetic charge with localized energy. The monopole mass reads M m = 4π where h(0) = 1. Eq. (24) yields the warp factor to be realized by the monopole mass on the thick brane. The solution in (26,27) is physically a magnetic charge with localized energy, describing a particle with finite mass and a long-range magnetic Coulomb force between monopoles.
Determining the monopole radius R m [45] is a task that makes the typical magnitude of R m to be chosen in order to balance the energy stored in the monopole magnetic field, g 2 /R m , and the energy provided by the monopole scalar field gradient (M 2 W R m ), yielding R m ≈ M −1 W . On the other hand, Eq. (24) together with the functionsṽ yield It is worth to realize that the dressed monopole mass reads yielding the productR mMm = R m M m an invariant.
Although the 't Hooft-Polyakov monopole solution is well known obtained with a SU(2) initial gauge group [46,47], whose gauge symmetry is spontaneously broken at an extremely large mass scale, of order 10 14 GeV [48]. Therefore, the naked mass and the naked monopole radius read, respectively, 10 16 GeV and 10 −30 m, onto the brane core. Since the monopole radius depends on the warp factor, the monopole radius increases out of the brane core. In order to analyse the 6D models of the previous section in this perspective, consider a cross section of the two extra dimensions with θ equals a constant, effectively providing the 5th dimension brane thickness. This can be justified by the expressions for the dressed parametersR m andM m , since the warp factor in Eq. (1) is what is taken into account for dressing such parameters. The previous analysis of magnetic monopoles yield therefore a constraint for the brane thickness.
Therefore, for the HC model, in order to avoid (unobserved) monopoles with mass scale of order TeV, taking into account that R m ∼ 1/M m , the condition R m exp(−cr * + tanh(cr * )) 10 −15 cm must be imposed, where r * denotes the brane 'surface' along the extra dimension. This condition reads exp(−c∆ HC /2 + tanh(c∆ HC /2)) 10 13 , where ∆ HC = 2r * defines the brane thickness. Eq. (30) implies c∆ HC 6.11588 and The parameter c was identified in Ref. [29] as an effective mass of dark matter particles ∼ 10 −2 GeV, implying the brane thickness The respective crosshatch region plot can be seen in Fig. 6, below. To determine the lower bound for the brane thickness, the parameter c was identified in Ref. [29] as an effective mass of dark matter particles ∼ 10 −2 GeV, implying the brane thickness ∆ HC 6.4472 × 10 −1 GeV −1 = 1.276 × 10 −16 m.
Analogously, we can also derive similar results for the GS model, wherein the condition R m exp(−cr * + tanh(cr * )) 10 −15 cm reads yielding Again, as the parameter c can be identified to the effective mass of dark matter particles ∼ 10 −2 GeV [29], it yields the brane width ∆ 29.9336 × 10 −2 GeV −1 = 5.8914 × 10 −17 m. (36) Similarly to the case of the HC model, the GS model has the region plot depicted in Fig. 7. m, which have been replaced by tighter phenomenological bounds. In this work, the corrections to the Coulomb's Law was only qualitatively studied, and its profile shares some similarities with corrections to the Newton's Law on string-like models. As a perspective, we want to adjust the scales for the c parameter, in order to get more restrictive bounds based simultaneously in the corrections to the Coulomb's Law, the informational entropy, the monopole radius and mass, and many other phenomenological data.