Bounds on topological Abelian string-vortex and string-cigar from information-entropic measure

In this work we obtain bounds on the topological Abelian string-vortex and on the string-cigar, by using a new measure of configurational complexity, known as configurational entropy. In this way, the information-theoretical measure of six-dimensional braneworlds scenarios are capable to probe situations where the parameters responsible for the brane thickness are arbitrary. The so-called configurational entropy (CE) selects the best value of the parameter in the model. This is accomplished by minimizing the CE, namely, by selecting the most appropriate parameters in the model that correspond to the most organized system, based upon the Shannon information theory. This information-theoretical measure of complexity provides a complementary perspective to situations where strictly energy-based arguments are inconclusive. We show that the higher the energy the higher the CE, what shows an important correlation between the energy of the a localized field configuration and its associated entropic measure.


I. INTRODUCTION
In 1948, in a seminal work, Shannon [1] introduced the information theory, whose main goal was to introduce the concepts of entropy and mutual information, using the communication theory.Therein, the entropy was defined to be a measure of "randomness" of a random phenomenon.Thus, if a little deal of information concerning a random variable is received, the uncertainty decreases, which makes it possible to measure the decrement in the uncertainty, related to the quantity of transmitted information.Inspired by Shannon, Gleiser and Stamatopoulos (GS) latterly introduced a measure of complexity of a localized mathematical function [2].GS proposed that the Fourier modes of square-integrable, bounded, mathematical functions can be used to construct a measure, the so-called configurational entropy (CE).A single mode system has zero CE, whereas that one where all modes contribute with equal weight has maximal CE.In order to apply such ideas to physical models, GS used the energy density of a given spatiallylocalized field configuration, as a solution of the related partial differential equation (PDE).Hence the CE can be used to choose the best fitting trial function with energy degeneracy.
The CE has been already employed to acquire the stability bound for compact objects [3], to investigate the non-equilibrium dynamics of spontaneous symmetry breaking [4], to study the emergence of localized objects during inflationary preheating [5] and to discern configurations with degenerate-energy spatial profiles [6].Moreover, solitons were studied in a Lorentz symmetry violating (LV) framework with the aid of CE [7][8][9][10].In this context, the CE associated to travelling solitons in LV frameworks plays a prominent role in probing systems wherein the parameters are somehow arbitrary.Furthermore, the CE identifies critical points in continuous phase transitions [11].Moreover, the CE can be used to measure the informational organization in the structure of the system configuration for five-dimensional (5D) thick scenarios.In particular, the CE plays an important role to decide the most appropriate intrinsic parameters of sine-Gordon braneworld models [12], being further studied both in f (R) [13] and f (R, T ) [14] theories of gravity.In what follows, we present a brief discussion of 5D braneworld models to treat the CE in six-dimensional (6D) scenarios.
Randall-Sundrum (RS) models [15,16] proposed a warped braneworld scenario, wherein the gauge hierarchy problem is explained and the gravity zero mode is localized, reproducing four-dimensional (4D) gravity on the brane.The 5D bulk gravitons provide a small correction in the Newton law [16].However, this thin model presents singularities and drawbacks concerning the nonlocalization of spin gauge and fermion fields [17].To solve these problems, some thick models were proposed [18].
Soon after the works of RS, an axially symmetric warped 6D model was proposed by Gergheta-Shaposhnikov [19], called string-like defect (SD).This scenario further provided the resolution of the mass hierarchy and a smaller correction to the Newtonian potential [19], besides the non-requirement of fine tuning between the bulk cosmological constant and the brane tension, for the cancellation of the 4D cosmological constant [19].Besides, the localization of gauge zero modes is spontaneous even in the thin brane case [20,21].Fermions fields are trapped through a minimal coupling with an U (1) gauge background field [22,23].Later, other 6D, spherically symmetric, models were employed to explain the generations of fundamental fermions [24,25] and the resolution of the mass hierarchy of neutrinos as well [26].Nevertheless, the SD model is a thin model and it leads to some irregularities [27].Due to it, some 6D thick models were proposed to solve these remaining issues [28][29][30][31][32][33][34][35][36][37][38][39][40][41][42].In Ref. [28], a topological abelian Higgs vortex was used to construct a regular scenario in which the dominant energy conditions hold, however solely numerical solutions have been found.Similarly, Ref. [31,32], looking for an exact vortex solution, shows that the energy density and the angular pressure are similar.This condition is likewise verified for the Resolved Conifold scenario [37][38][39].Finally, for the String-Cigar [33][34][35][36], the transverse space is represented by a cigar soliton, which is a stationary solution for the Ricci flow [43][44][45].The dominant energy conditions are also satisfied in this model.
Therefore, in this paper we investigate the entropic measure both in the Torrealba topological Abelian string (TA) [31,32] and String-Cigar (HC) [33][34][35][36] in 6D scenarios due, its analytic properties.The main aims of our work is to find bounds for 6D string defects based upon the CE concept and to establish a value for the thickness of the configuration responsible for extremizing the CE.
This paper is organized as follows: in Sect.II a briefly review string-like defects is present, whereas in Sect.III the CE bounds the parameters of TA and HC scenarios.We expose the conclusions and perspectives in Sect.IV, accordingly.
The 4D Planck mass (M P ) and the bulk Planck mass (M 6 ) are related through the volume of the transverse of space as [19,33,35,36]: In addition, the energy-momentum tensor T N M = diag (t 0 , t 0 , t 0 , t 0 , t r , t θ ) components are given by [19,33] where the κ = 8π is the 6D gravitational constant, Λ is the 6D (negative) cosmological constant and the prime denotes the derivative with respect to the radial coordinate r.
For the vacuum solution, the warp factor for the Gergheta-Shaposhnikov String Like Defect (SD) model is proposed as [19][20][21][22][23]: where the parameters c is a constant, which connects the 6D Newtonian constant and the 6D cosmological constant, and R 0 is the radius of compactification of transverse space.See that, in the limit where r → 0, only the first condition of Eq. ( 4) holds.
From constraints by this ansatz and the regular conditions in the Eq. ( 4), the solutions of fields and warp factors are numerically obtained in Ref. [28].On the other hand, by imposing conditions on the function P (r) ≡ 0, Torrealba [31,32] obtained an analytical solution, named Topological Abelian Higgs string (TA): where the parameter β is similar to the parameter c in the SD model, and δ is a thickness parameter which, for small values, reproduces the thin Gergheta-Shaposhnikov model in Eq. ( 5).Moreover, Ref. [31] concludes that, for the localization of gauge fields zero mode, the thickness of the model can not exceed the value Now, in the TA (6) string, two of the conditions (4) are verified.
To observe the correspondence between the regular condition in Eq. ( 4) and the energy momentum tensor we plot the σ(r) the warp factors ( 5), ( 6) and (8) in Fig. 1 and γ(r) in Fig. 2, whereas the energy momentum tensor in Fig. 3 for TA and HC in Fig. 4. Concerning the HC scenario, wherein all metric conditions (4) hold, the dominant energy condition t 0 ≥ |t i | , (i = r, θ) [29,30,40]   In the next section, we shall analyze these string models from the CE point of view.

III. CONFIGURATIONAL ENTROPY IN THE VORTEX-STRING SCENARIO
The configurational entropy (CE) [2] represents an original quantity, employed to quantify the existence of nontrivial spatially localized solutions in field configuration space.The CE is useful to bound the stability of various self-gravitating astrophysical objects [47], bound states in LV scenarios [7], in compact objects like Q-balls [3], and in modified theories of gravity as well [13].The CE is linked to the energy of a localized field configuration, where low energy systems are correlated with small entropic measures [2].
The CE can be obtained [2] by the Fourier trans-form of the energy density t 0 (r) [12,13], yielding It is worth to remark that we will consider structures with spatially localized, square-integrable, bounded energy density functions t 0 (r).The modal fraction reads [2][3][4]6] Next, the normalized modal fraction is defined as the ratio of the normalized Fourier transformed function and its maximum value f max , namely, f (ω) = f (ω)/f max .A localized and continuous function f (ω) yields the following definition for the CE: Therefore, we use this concept to obtain the CE in the Abelian string-vortex and the string-cigar contexts.By substituting the warp factor (6) in the energy density given by Eq. (3a), it yields It represents a localized density of energy, as can be verified in Fig. 3. Now, the Fourier transform of (10) reads which is a localized function having the normalized modal fraction: For the numerical evaluation of Eq. ( 9), with the input of Eq. ( 12), it is necessary to explicit here the expression of the parameter β, as defined in Refs.[31,32]: Let us remember that κ is the 6D gravitational constant, being Λ < 0 the 6D cosmological constant.Besides, this imposition over the range of the parameter β is necessary to prevent values larger than Planck mass [34].
Hence the profile of CE in Eq. ( 9) for the function in Eq. ( 12) is presented in the Fig. 5, for S(δ), and in the Fig. 6, for S(β).
It is verified in Fig. 5 that the maximum of CE occurs for δ crit ≈ 0.09β.This result bounds the thickness of TA model in two regions: the first one for δ → 0, which endorses the thin Gergheta-Shaposhnikov model of Eq. (5), and the second one for δ > δ crit .However, an upper bound thickness limit is provided in Eq. (7).Thus, for δ = 0, the constraint on the TA model thickness with q = 1 in the Eq. ( 7   Furthermore, another important physical information is presented in Fig. 6, where the minimal CE occurs when the parameter β tends to zero.Perceive that the mass hierarchy of Eq. ( 2) for the TA model is exposed in Eq. ( 6) as In the case where M P M 6 , the parameter β to tends to zero, once δ is bounded by Eq. (7).Thus, the CE exhibits this stable behaviour in Fig. 6 and to small values of β there corresponds to small values of CE.
For the HC model, where we have only the c parameter, the energy density of Eq. (3a) yields Again, the energy density is localized as can be verified in the Fig. 4. The Fourier transforms of above equation reads and its normalized modal fraction yields Setting the expression and the range of the c parameter as defined in Refs.[19,34,36] c = 2κ 5 (−Λ), with 0 < c ≤ 1 (19) it is possible to plot the S(c) by integrating Eq. ( 18), using, (9).Fig. 7 represents the result.By considering the mass hierarchy of Eq. ( 2) in the model of Eq. ( 8) provided by the result of M P M 6 is verified when c tends to zero.This also agrees with the profile exhibited for CE in Fig. 7.The intrinsic braneworld models parameters have been further constrained by analyzing the experimental, phenomenological and observational aspects in, e.
g., [12,46].In particular, Ref. [12] provides a refined analysis wherein the CE further restricts the range parameters of a 5D sine-Gordon thick braneworld model, namely, the AdS bulk curvature and the braneworld thickness.Here this procedure was applied to 6D thick braneworld models and we verified for the TA model the constraints on the thickness parameter δ in the Eq. ( 14).Besides, in both TA and HC models, the minimal CE reflects the expected result obtained from the mass hierarchy in these models.

IV. DISCUSSION AND CONCLUSIONS
In this work we have investigated the CE in the context of the topological abelian string-vortex and stringcigar scenarios.We have shown that the informationtheoretical measure of 6D dimensional braneworld models opens new possibilities to physically constrain, for example, parameters that are related to the brane thickness.The CE provides the most appropriate value of this parameter that is consistent with the best organizational structure.The information measure regarding the system organization is related to modes related to the braneworld model.Hence the constraints of the parameters that we obtained, for the TA and the HC string models, provide the range of the parameters associated to the most organized braneworld models, with respect to the information content of these models.The CE demonstrates the expected limit of parameters that agrees with the mass hierarchy of these 6D models.It provides further physical aspects to models, where strictly energybased arguments do not provide further conclusions of the physical parameters.

Figure 4 :
Figure 4: tM (r) in HC model with c = 0.5 and κ = 1.The dominant energy condition is satisfied.

Figure 5 :
Figure 5: S(δ) Configurational entropy as a function of the thickness parameter δ, for different values of the parameter β.

Figure 6 :
Figure 6: S(β) Configurational entropy as a function of the parameter β, for different values of the parameter δ.

Figure 7 :
Figure 7: S(δ) Configurational entropy of the HC string model, as a function of the parameter c.