Trapping Spin-$0$ particles on $p-$balls in $(D,1)$ dimensions

p-Balls are topological defects in $(D,1)$ dimensions constructed with $\mathcal{M}\ge 1$ scalar fields which depend radially on only $2 \le p\le D-2$ spatial dimensions. Such defects are characterized by an action that breaks translational invariance and are inspired on the physics of a brane with $D-p$ extra dimensions. Here we consider the issue of localization of bosonic states described by a scalar field $\Phi$ sufficiently weak to not disturb sensibly the defect configuration. After describing the general formalism, we consider some specify examples with $\mathcal{M}=1,2$ and $3$, looking for some region of parameters where bound and resonant bosonic states can be found. We investigate the way the influence of the defect structure, number of radial dimensions and coupling between the fields are related to the occurrence of bound and resonant states.


I. INTRODUCTION
The study of topological defects [1,2] is important due to their mathematical properties and connection to several areas of physics such as quark confinement [3], cosmology [4] and condensed matter [5]. From the point of view of multidimensional spacetimes, one can cite for instance the vortex [6] for superconductor physics in (2,1) dimensions, the magnetic monopole [7] connected to cosmology in (3,1) dimensions and brane models [8] in general (p, 1) dimensions. The simplest example of a topological defect is the kink [9], where the solution interpolates between two different vacua. The kink is extended by the concept of branes with one extra dimension [10][11][12][13][14][15][16][17][18][19][20], where the brane structure is a result of an action with a dynamical scalar field. The tentative of solving the hierarchy and the constant cosmological problems with one extra dimension always faced a fine-tuning problem [8].
Some tentative to evade this problem included to consider more than one extra dimension.
The literature has several interesting examples of topological defects with larger codimension number. In 6 dimensions with codimension 2 one can cite gravity localization on strings [21] and baby-Skyrmion branes [22]. Higher codimension topological defects are studied in [23].
In general codimension-1 brane models can be easily treated using the first-order formalism [10]. In such cases the scalar field potential and metric can be described by first-order differential equations in terms of a function W called fake superpotential. This name refers to the flat spacetime analog where a true superpotential is introduced to find BPS states [24]. The BPS formalism for some defects with radial symmetry constructed with a scalar field was introduced in Refs. [25,26]. Recently, inspired in brane models with codimension-2, such study was extended for the case of two coupled scalar fields in (3, 1)-dimensions with the aim to investigate in absence of gravity resonances and localization of particles with spin-0 [27] and spin-1/2 [28] in axial symmetric topological defects. Such analysis has shown similar results with the previous works dealing with gravity and particle localization and resonances [29] on branes with codimension 1.
In presence of gravity, the lack of a first-order formalism for brane-models with codimension higher than 1 make the analysis of field localization and resonance much more involved, needing in general numerical analysis for finding the way the scalar fields depend on the extra dimensions. However, by neglecting gravity effects, in this work we show that is possible to implement a first-order formalism to describe topological defects generated by scalar fields with pure radial dependence. Consequently, we analyze the effects in the trapping of spin-0 fields due to the higher codimension of the topological defect. This paper is presented in the following way: In Section II we consider the general fistorder formalism for a (D, 1)−dimensional flat spacetime where M ≥ 1 scalar fields depend radially on p spatial dimensions, with 2 ≤ p ≤ D − 2. Next we apply the formalism for a certain number of specific cases. Then, in Sects III, IV and V we consider defects formed respectively by one, two and three field models, looking for some aspects of localization of a spin-0 field in each system, and numerically investigating the occurrence of bound states and resonance effects. Our main conclusions concerning to a comparative analysis of the influence of p, D and M on the number and intensity of resonances are presented in Sect.

II. GENERAL FORMALISM
We start with the action with α = 0, ..., D. The (D, 1)-dimensional cartesian coordinates will be separated in D − pdimensions (x 1 , x 2 , ..., x D−p ) where the fields can be located and the remaining p transverse dimensions (x D−p+1 , ..., x D ), with 2 ≤ p ≤ D − 2, where the defect will be formed.
The potential is chosen to be where we have used a simplified notation: and so on. The explicit dependence of the potential on r, follows closely and generalizes for M scalar fields the construction of Ref. [25,26], initially motivated for avoiding the Derrick-Hobart's theorem [30]- [33]. We also suppose that the scalar fields φ i depend only on r.
The equations of motion for the scalar fields read and the ones describing static solutions are where ∇ 2 is the p−dimensional Laplacian, defined by The energy density is given by and the total energy of the defect in the transverse volume is In order to describe the system via first-order differential equations, we implement the BPS formalism such that the total energy can be written as By setting as null the squared term we get the set of first-order differential equations whose solutions we label as φ i ± (r). Whenever these first-order equations are satisfied and using (3), the total energy reads At this point we observe the integrand can be transformed in a total derivative whether N = 2p − 2. In this way the BPS energy reads and from Eq. (11) the corresponding BPS equations read A similar result about parameter N can be obtained by considering scaling properties of the scalar fields in the energy density (9). Firstly, we define the vectors φ = (φ 1 , ..., φ M ) and r = (x D−p+1 , ..., x D ). We make the scaling transformation r → λ r and φ( r) → φ(λ r), corresponding to a change in the energy given by E → E λ , and impose (∂E λ )/(∂λ)| λ=1 = 0.
This leads to the following restrictions on N and p: i) for p = 1, N = 0; ii) for p = 2, N = 2.
Additionally by imposing the equality of the gradient and potential parts of E, we get iii) A similar result was previously found in [25], for the case where Hence, we have shown the system (6) admits topological solutions obtained from the set of self-dual equations (14) which minimize the system energy (9).
The topological character of the solutions φ i ± can be demonstrated following closely Ref. [25], with the difference that there one has p = D. For (1, 1)-dimensions we have M conserved currents j iµ = ǫ µν ∂ ν φ i , with µ = 0, 1 and i = 1, ...M. This results in M conserved quantities σ i = dφ i /dx, such that ρ = i (σ i ) 2 is the energy of the field configuration [34] and the topological charge Q T = ∞ −∞ dxρ is also the total energy of the solution. However, for the class of defects described here one must be in (D, 1)-dimensions with D ≥ 4, with the scalar fields depending on p ≥ 2 spatial dimensions. For the minimum case, with D = 4 and p = 2 we have M current tensors j iµ 1 µ 2 = ǫ µ 1 µ 2 µ 3 ∂ µ 3 φ i , where each µ 1 , µ 2 , µ 3 can assume the values 0, 1, 2. We have ∂ µ 1 j iµ 1 µ 2 = 0. For each scalar field this gives the set of two conserved densities σ ik 1 = j i0k 1 , where k 1 = 1, 2. The scalar quantity 2 can be used to define the topological charge as Q T = d rρ = −Ω∆W , which coincides with the energy of the defect in the transverse volume. Finally for general (D, 1)-dimensions with p ≥ 2 transverse This gives, for each scalar field, the set of p conserved densities σ ik 1 k 2 ...k p−1 = j i0k 1 k 2 ...k p−1 . The scalar quantity ρ = σ ik 1 k 2 ...k p−1 σ ik 1 k 2 ...k p−1 = (−1) p (p−1)! i (dφ i /dr) leads to the topological charge Q T = d rρ = (−1) p (p − 1)!Ω p ∆W , which coincides with the energy density of the defect in the transverse volume.
In the following we show that the solutions φ i ± (r) are stable under radial and time-dependent fluctuations. For such a purpose, we follow the procedure realized in Ref. [35] for domain walls with two scalar fields. Thus, we construct the function By substituting it in Eq. (5) and keeping only linear terms in the fluctuations η k n (r), we get where we have defined the matrix M and the eigenvector η n by We have verified that 1 in this section, upper and lower signals are for, respectively, BP S solutions φ i ± of Eq. (14) and we have defined the matrix G Now this in Eq. (16) leads to the useful factorization where the operators are defined as and Note that, for one scalar field (i.e., M = 1), this factorization differs from the presented in Ref. [25]. The advantage is that now one can verify explicitly that these operators are such that A † ± = B ± , that is, in p spatial dimensions, provided we impose the boundary condition a condition valid if ψ are square-integrable bound states (not scattering states). From this analysis we can rewrite Eq. (20) as which means that the ω 2 n are eigenvalues of a non-negative operator B † ± B ± . This proves that negative eigenvalues are absent and that the p−balls which satisfy the set of first-order equations given by Eq. (14) are stable. The lowest bound state is given by the zero-mode, identified as B ± η n = 0, which gives η i n = cW φ i for the M components of η n , where c is the normalization constant, such that Further, note that the presence of an explicit dependence with r in A ± and its absence in B ± (compare Eqs. (21) and (22)) introduces an asymmetry necessary for the condition A † ± = B ± to be valid. In this way there is an extension of the usual symmetric form of factorization for problems with p = 1, the (1, 1)-dimensional kink being an archetype (see Eq. (3.5) from Ref. [35]). Specific factorizations of the Hamiltonian where also attained in other contexts, for instance for quantum systems with position-dependent masses [36]. Now let us turn to the search of explicit solutions for the dependence in the radial dimension of the scalar fields. A convenient way to solve Eq. (14) is to make a change of variables dξ = 1/r p−1 dr, or equivalently ξ(r) = ln(r/r 0 ), p = 2 (27) or This coordinate transformation turns Eq. (14) in After solving this equation for φ(ξ), and back to r variable, explicit expressions for the scalar field and energy density can be easily attained. Now to form a topological defect one must chose a function W (φ 1 , φ 2 , ...φ p ) with W (r → ∞) = W (r = 0). A convenient choice is a field φ i with a kink-like pattern in r around a finite value r 0 and the remaining other fields φ j , j = i with kink or bell-shape pattern around the same value of r. In a terminology from the literature we could say the field φ i forms the defect whereas the other fields are responsible for its internal structure.
As we saw, from the D +1 spacetime dimensions, the topological defect lives in p of them.
Now we want to consider how a spin-0 particle living in the full D + 1-dimensional spacetime can be effectively trapped by the topological defect in a form of bound or resonant states.
Then we consider a scalar field Φ in a region where it is formed a radial defect constructed with the M scalar fields φ i . In the present analysis we neglect the backreaction on the topological defect by considering that the interaction between the scalar fields is sufficiently weak in comparison to the self-interaction that generates the defect. In the following, we designate Φ as the weak field and φ i , i = 1, 2, ..., M the strong ones. We write the following action describing the system as with β = 0, 1, ..., D. Here F (φ 1 , ..., φ p ) is the coupling between the weak field Φ and the topological defect. The equation of motion of the scalar field Φ is where in the former expression we decomposed the D+1-dimensional d'Alembertian between the D − p transverse dimensions and the p transverse dimensions. That is ∂ µ ∂ µ = , with µ = 0, ..., D − p and ∇ 2 T is a p-dimensional Laplacian. Now considering that the strong fields φ 1 , ..., φ p depend only in the radial direction, we have a coupling F (φ 1 , ..., φ p ) = F (r). We restrict our discussion to functions F (r) finite for all values of r, with η lim r→0 F (r) = 0.
We decompose the scalar field Φ as where  is related to the angular momentum eigenvalue. Here we have changed the transverse coordinates (x D−p+1 , ..., x D ) from the cartesian to the generalized spherical coordinates (r, ϕ, θ 1 , ..., θ p−2 ), with r defined previously and ...
The p-dimensional Laplacian is given by where L p is the p-dimensional angular momentum operator, given by (we set θ 1 = θ, θ 2 = w to ease notation) In general, for p ≥ 3 we have Now the field ξ n (t, and the amplitude ς n (r) satisfies the radial Schrödinger-like equation with the Schrödinger potential given by By requiring that Eq. (43) defines a self-adjoint differential operator in r ∈ [0, +∞), the Sturm-Liouville theory establishes the orthonormality condition for the components ς n, (r) The spherical harmonics of degree  satisfy (see Ref. [37] for a general treatment of spherical harmonics with general number of dimensions) and are polynomials of degree  with variables restricted to the unit (p − 1)-sphere, which satisfy the orthonormality condition Here Ψ m (ϕ) = e imϕ and Θ ℓm (θ) satisfies the following differential This means that Y ℓm is associated with the eigenvalue ℓ(ℓ + 1) and carries angular momentum ℓ(ℓ + 1). The index ℓ labels the irreducible representations of SO (3) whereas m labels the corresponding representations of the subgroup SO(2). For each ℓ there are 2ℓ + 1 linearly independent spherical harmonics corresponding to the various values of m. Therefore the irreducible representations of SO(3) based on Y ℓm are (2ℓ + 1) dimensional [38].
For the general case, the irreducible representations of SO(p) based on hyperspherical harmonics have dimension given by [39] and we have an orthonormal set of hyperspherical harmonics which have extra indices that are labels of the irreducible representations of the following chain of subgroups of SO(p): Let us illustrate how this works with one more example. The generalization for even larger values of p demands additional work but is straightforward.
• For p = 4 an orthonormal set of hyperspherical harmonics have extra indices that are labels of the irreducible representations of the following chain of subgroups of SO (4): , with the following specific constructions: One can make the decomposition Y ℓm = W ℓ (w)Y ℓm (θ, ϕ), where Y ℓm (θ, ϕ) are the usual spherical harmonic described in the p = 3 case, and W ℓ (w) satisfies the following differential equation where prime means derivative with respect to the argument.
which shows that Φ n is a massive (D − p + 1)-dimensional Klein-Gordon field with mass M n .
In order to investigate numerically the massive states, firstly we consider the region near the origin (r ≪ r 0 ). Since we are considering only functions ηF (r) finite, the Schrödinger-like where V (0) = lim r→0 ηF (r), whose nonsingular solution at r = 0 is On the other hand, for  ≥ 1 the Schrödinger-like potential is dominated by the contribution of the angular momentum proportional to 1/r 2 , and the nonsingular solutions in r = 0, given by Both functions (55) and (57) are used as an input for the numerical method. From this approximation we can calculate ς(r min ) and dς/dr(r min ), to be used for the Runge-Kutta method to determine ς(r) from the Schrödinger-like equation. We define the probability for finding scalar modes with mass M n and angular momentum inside the p−ball of radius r 0 as is [32] here r min ≪ r 0 is used as the initial condition and r max is the characteristic box length used for the normalization procedure, being a value where the Schrödinger potentials are close to zero and where the massive modes ς(r) oscillate as planes waves. Resonances are characterized by peaks in the plots of P as a function of M n . The thinner is a peak, the longer is the lifetime of the resonance. This finishes the part of the general formalism. In the remaining of this work we will solve some specific examples with one, two and three scalar fields.
The number of parameters involved in the models considered here led us to make some restrictions in order to better identify the effect of the number of transverse dimensions for the occurrence of bound and/or resonant states. For instance, we have identified that an increasing in  reduces the possibility of the occurrence of bound states. Case  = 0 is special, since in this case we have lim r→0 V sch (r) is finite or even zero, in opposition to Then without loosing generality we have chosen to study states with  = 0 and  = 2.

III. A ONE-FIELD MODEL
In this section we will consider the model [25] W with q = 1, 3, 5.... The first-order equation, described by Eq. (29), has solution given by The case q = 1 corresponds to the usual kink solution of the φ 4 model in the variable ξ.
Back to r variable, explicit expressions for the scalar field and energy density can be easily attained: with ξ(r) given by Eq. (27), for p = 2 or (28), for p = 3, 4, .... Fig. 1 shows plots of φ(r) for fixed λ, r 0 and several values of q and p. The scalar field φ(r) interpolates between zero and φ c , with i) φ c = 1 for either q = 1 or p = 2 and ii) φ c = tanh q [λ/(q(p − 2)r p−2 0 )] for q > 1 and p = 2. For q = 1 the larger is p, the lower is φ c . We note from the figure that the φ(r) configurations is now of two kinks connected at r = r 0 with a flat region around r 0 that grows with q. The internal kink runs from r = 0 to r = r 0 and has a compacton character whereas the external kink goes from r = r 0 to r → ∞ and is a semi-compacton.
As p increases we see that the internal kink (for r < r 0 ) has its thickness reduced whereas We have a) p = 2, b) p = 3 and c) p = 4. Curves are for q = 3 (black), 5 (red), 7 (green) and 9 (blue).
the external kink (for r > r 0 ) has its thickness increased. Fig. 2 shows plots of the radial energy density ρ(r) for fixed λ, r 0 and several values of q ≥ 2 and p. From the figures we see that the energy density is characterized by two peaks: a higher and thinner one, centered at r < r 0 and a lower and thicker one, centered at r > r 0 . The distance between the peaks grows with q, enlarging the region around r = r 0 where ρ ∼ 0, at the price of reducing the height of the peaks. For fixed q, the effect of the increasing of p is an increasing of height and thinness of the peak at r < r 0 and a corresponding decreasing of the peak at r > r 0 .
The energy density for q = 1 is characterized for a peak centered around r = r 0 , and does not depend sensibly on p. Here we will consider the coupling F (φ) = φ 2 , corresponding to the Schrödinger-like potential     Table I.  and c) q = 7.    Table II, which shows that, for q = 5 the occurrence of bound states is more frequent for p = 4 and p = 5 whereas for q = 7 this occurs for p = 3 and p = 4. Curves are for p = 2 (black), 3 (red), 4 (green) and 5 (blue).

IV. A TWO-FIELD MODEL
In this section we will consider the model With this choice of W , the potentialṼ (φ, χ) = (1/2)(W 2 φ + W 2 χ ) was introduced in Ref. [40] to construct Bloch walls. The limit s → 0.5 turns the two-field problem into a one-field one, recovering the model of an Ising wall. This can be better seen in the explicit solutions φ(ξ) and χ(ξ) bellow. The equation of motion for the scalar fields, Eq. (14) is rewritten, after a change of variables dξ = 1/r p−1 dr, as with solution Back to r variable, explicit expressions for the scalar field profiles and consequently for the energy density can be easily attained. We have, for p ≥ 2, and ξ(r) given by Eq. (27), for p = 2 and by Eq. (28), for p=3,4,....  Indeed, for small s the lump centered at r = 0 occurs only for p = 2. For p ≥ 3 there appears a broad peak for centered at a 0 < r < r 0 . The larger is p, the higher and thinner is this peak. For p ≥ 6 the peak for small s turns to be the higher in comparison to those occurring for larger values of s. Fig. 7d shows this effect for p = 8. The influence of λ on the energy density can be seen in Fig. 8 for p = 2. Note that the increase of λ turns the energy density more centered around r = r 0 . At the same time this reduces the relative maximum of the energy density for lower values of s in comparison to the higher ones. We noted a similar behavior with the variation of λ for p = 3.
First of all we consider the case  = 2. For the potentials shown in Fig. 9, and also for higher values of p, the existence of bound states was investigated and some results could be found according to Table III bound states. As explained above, s → 0.5 recover a one-field φ model. Then, the presence of a second scalar field χ contributes to trapping spin-0 particles. In other words, p−balls with more internal structure are more able to trap scalar particles. Also, for fixed s and larger number p of transverse dimensions, bound states occur with larger masses for potential V 1 than for potential V 2 . This shows that, for a trapping mechanism, the quadratic coupling χ 2 is better that the quartic one φ 2 χ 2 . Moreover, a multidimensional p−ball with larger p is a better trapping mechanism, as V 1 (r → ∞) and V 2 (r → ∞) grows with p. This is confirmed in Table III. Indeed, the larger is p, the greater is the number of bound states. Also, the asymptotic values V 1 (r → ∞) and V 2 (r → ∞) grow with λ and η. This signals that in order to grow the probability for the occurrence of bound states one must have η, λ ≫ 1 and decrease the ratio λ/η. This can be seen in Fig. 10 where λ/η = 3/10 (compare with 9 where λ/η = 1). Corresponding eigenvalues are described in Table IV. From the results for λ = 30 we see that for λ/η = 3/10 and s = 0.06 there occur bound states for p ≥ 4 (compare with the results of Table III for λ/η = 1 where bound states appear for p ≥ 4). Now note that the potentials for coupling F 1 are characterized by a local maximum at r = r 0 whereas for F 2 there is a local minimum at r = r 0 surrounded by two local maxima.
The higher local peak for V 1 in comparison to the two local ones for V 2 suggests that, with the same set of parameters, resonant states are most probable with quadratic coupling F 1 than with quartic coupling F 2 . This is in accord to the behavior of couplings concerning to the occurrence of bound states. We also see that the height of the local maxima grows with the decreasing of s, favoring the appearance of resonances. Then we expect the presence of a second scalar field χ to be important for the increasing in the number and lifetime of resonances. We also found that for all other parameters fixed, the best choice for reducing the asymptotical value of V 1 and simultaneously increasing the difference between the local maxima and minima is to keep η, λ ≫ 1 and increase the ratio λ/η. This can be seen in Fig. 11, for λ/η = 10/3 (compare with Fig. 9, where λ/η = 1 and with Fig. 10, where λ/η = 3/10). However, this occurs at the price of making the barrier thinner. Then we expect that an increasing of λ/η increases the chances for getting a larger number of resonances, but with lower lifetimes.
The influence of the variation of the angular momentum  can be seen in Fig. 12, where and the nonsingular solutions in r = 0 are given by Eq. (57), used for calculating the relative probability P . keeping the mass separation between the peaks almost unaltered. Figs. 13b and 13d shows the resonance peaks for coupling F 2 (φ, χ). Comparing this with Figs. 13a and 13c (related to F 1 (χ)), we see that for coupling F 2 (φ, χ) the number and masses of resonances is strongly reduced in comparison to the case of coupling F 1 (χ). Indeed, even for s = 0.06, Fig. 13b shows a pair of neighbor peaks, with only one with relative probability close to one. The sequence of almost equally spaced peaks present for coupling F 1 (χ) (corresponding to Fig.   13a) is now absent. This shows that coupling F 2 (φ, χ) is less effective for the occurrence of The plots correspond to potentials of Fig. 11. resonances. The simplest choice is to consider p−balls in (3, 1)-dimensions. In this case the only possibility is to construct a radial defect with p = 2 spatial dimensions. This is a tube-like defect and has already been studied by us in Refs. [41,42]. Requiring ρ(r) finite in r = 0 restricts the parameters to satisfy λs ≥ 1/2 when λ > 1. For large values of λ, there exists a value s 0 so that for 1 2λ < s < s 0 , the effect of the field χ is stronger and the defect appears as a thick tube structure whose center is localized between the origin and r 0 . The decomposition of the weak scalar field Φ(t, where the spherical harmonic is Y ℓ (ϕ) = e iℓϕ and ς n,ℓ (r) satisfies a (2, 1)-dimensional Klein-Gordon equation.
B. p−balls in (4, 1)-dimensions In this case we have two possibilities: i) to construct a radial defect with p = 2 spatial transverse dimensions and (2, 1) longitudinal dimensions, or ii) to construct a radial defect with p = 3 spatial transverse dimensions and (1, 1) longitudinal dimensions. In the following we will consider these two possibilities separately.

p−balls in (4, 1)-dimensions with p = 2 transverse dimensions
The defect is characterized by a potential which generates, respectively, kink-like and lump-like solutions for the scalar fields φ(r) and χ(r), as well as energy density T 00 (r) with the same profile found for the case analyzed in Sect. IV A for (3, 1)−dimensions. This is expected since we have the same number (two) of transverse dimensions. All would follow the same as in Sec. IV A: the decomposition of the spherical harmonics Y ℓ , Schrödingerlike potentials V sch and relative probabilities P . The difference is that in the present case of (4, 1)−dimensions we have (2, 1) longitudinal dimensions. This is reflected in the longitudinal part ξ nℓ of the decomposition of the weak scalar field Φ(t, x 1 , x 2 ), which now satisfy a (2, 1)-dimensional Klein-Gordon equation.

V. A THREE-FIELD MODEL
In this section we will consider some three-field models. The numerical analysis of bound and resonant states follows the same prescription done in the two previous sections and we will not pursue in this direction here, focusing mainly in the analysis of the Schrödinger-like potentials. We start with a simple extension of the previous model, given by [43] W where s is a real parameter. The equation of motion for the scalar fields, Eq. (14) is rewritten, after a change of variables dξ = 1/r p−1 dr, as One solution connecting the minima (±1, 0, 0) of the potential is [43] φ(ξ) = tanh(2λsξ), (82) with 0 ≤ s ≤ 0.5 and 0 ≤ ϑ < 2π, where now ϑ is a new parameter of the model. Back to r variable, explicit expressions for the scalar field profiles and consequently for the energy density can be easily attained. We have, for p ≥ 2, with τ p given by Eq. (72). One can interpret the φ field as forming a host hypersphere, with the fields χ and σ giving its internal structure. The balancing of the internal fields is given by the parameter s. We can also consider the real scalar fields χ and σ as the real and imaginary part of a complex scalar field ζ, with the model given by A simple coupling is F 3 = |ζ| 2 = (χ 2 + σ 2 ). The explicit solutions χ(r), σ(r) shows that this coupling recovers the results obtained for F 1 (χ) from Sect. IV. Another coupling is recover coupling F 1 . For coupling F 4 , the Schrödinger-like potential is with b = b 1 cos 2 (ϑ) + b 2 sin 2 (ϑ). Figure 15 shows potential (meaning a quadratic coupling with fields χ or σ) is crucial for the occurrence of resonances.
Also a quadratic coupling with field φ is of secondary importance, when compared with the effect of similar couplings with the other two fields that form the defect. A quadratic coupling with only the φ field has no effect concerning to resonances. This illustrates the importance of the secondary fields χ, σ that give to the defect an internal structure. Comparing Figs.
15a and 15b we see that for p = 3 we have more possibility for resonant states in comparison to p = 4. The increasing of V 4 (r → ∞) with p shows that an intermediate value of p is better for attaining bound states, in a similar conclusion achieved in Sect. III for one-field models. For p = 2 the potential V 4 is a monotonically decreasing function, and there is neither bound nor resonant states.
Other couplings can be considered, but to further illustrate the generality of the construction of p-balls, here we choose to consider another model, restricted to a Z 2 × Z 2 symmetry in the χ and σ axis [43]: For s > 0 and −1 < g < 1 the corresponding potential V (φ, χ, σ) has six minima given by For s = 0 the only solution connecting the minima v 1,2 is the one-field limit given by χ = σ = 0 and φ(ξ) = tanh(ξ). Nontrivial solutions for the three scalar fields can be obtained connecting minima v 3,4 to v 5,6 and are given, with sg 2 = 1, by [43] φ(ξ) = g 2 (1 + tanh(ξ/g)), Back to r variable, and for p ≥ 2, we can obtain the following expressions for the scalar fields: with ξ(r) given by Eq. (27), for p = 2 or (28), for p = 2, 3, .... Figure 16 shows plots of φ(r), χ(r) and σ(r) for p = 2 and fixed λ, r 0 and several values of s. From the figure we see that all the three fields have a kink-like structure around r = r 0 .

VI. REMARKS AND CONCLUSIONS
In this work we introduced p-balls as topological defects in (D, 1) dimensions constructed with M ≥ 1 scalar fields which depend radially on only 2 ≤ p ≤ D − 2 spatial dimensions. A quadratic coupling χ 2 is better than the quartic one φ 2 χ 2 concerning to the occurrence of either bound or resonant states. We have verified the presence of the second scalar field χ contributes to the increasing of trapping spin-0 particles. In these two-field models, the larger is p, the greater is the number of bound states, but the number of resonances is roughly the same. We have also explored some three-field models, finding from the analysis of the Schrodinger-like potential that in some cases an intermediate value of p is better for the occurrence of bound states.
Concerning to the influence of , the probability of occurrence of bound states with  = 0 is higher in comparison to states with larger values of . This was verified numerically.
On the other hand, for resonances the analysis of the Schrödinger potential shows that the behavior depends on the model. In our one-field model there is no such possibility for  = 0 whereas the possibility of occurrence of such states is low but grows with the increasing of . On the other hand for our two-field model the number of resonant states seems to accompany the tendency of bound states, with a decreasing in number with .
It is worthwhile to observe that we where able to establish a connection between the number of extra dimensions and the capability of trapping massive states. Such a study was attained by analyzing qualitatively the Schrodinger-like potentials and by performing numerical analysis. For all class of models considered, we could identify that initially there is an increasing of the number of bound states with p. For the one-field model this occurs up to a certain number of transverse dimensions; in this way there is an optimal number of transverse dimensions for trapping states. On the other hand, for the two-field model considered, we have investigated up to p = 7 and the number of bound states always grow with p, whereas the number of resonances seems to be independent of p. Also we have confirmed that, for fixed p, the growing of the internal structure of the defect (connected with larger values of q for the one-field model and smaller values of s in the two-field and three-field models) lead to an increasing of the number of bound and/or resonance states.
Whether our results represent a general characteristic of the p-balls or the main conclusions are results of the particularities of the models here described deserves to be better understood. Advances in this direction will be reported elsewhere.