Matter localization on brane-worlds generated by deformed defects

Localization and mass spectrum of bosonic and fermionic matter fields of some novel families of asymmetric thick brane configurations generated by deformed defects are investigated. The localization profiles of spin 0, spin 1/2 and spin 1 bulk fields are identified for novel matter field potentials supported by thick branes with internal structures. The condition for localization is constrained by the brane thickness of each model such that thickest branes strongly induces matter localization. The bulk mass terms for both fermion and boson fields are included in the global action as to produce some imprints on mass-independent potentials of the Kaluza-Klein modes associated to the corresponding Schr\"odinger equations. In particular, for spin 1/2 fermions, a complete analytical profile of localization is obtained for the four classes of superpotentials here discussed. Regarding the localization of fermion fields, our overall conclusion indicates that thick branes produce a left-right asymmetric chiral localization of spin 1/2 particles.

viable in high energy physics. Several alternative scenarios, including Gauss-Bonnet terms, f (R) gravity, tachyonic potentials, cyclic defects, and Bloch branes, have been further studied [13][14][15][16], and analogous scenarios in an expanding Universe have been approached [17,18]. The curvature nature of the brane-world, namely, to be a de Sitter, Minkowski or anti-de Sitter one, is in general obtained a posteriori, by solving the 5D Einstein field equations. In fact, the bulk and the brane cosmological constants depend upon the brane and the bulk gravitational field content, governed by curvature, and must obey the intrinsic fine-tuning, in the Randall-Sundrum-like models limit.
The analytical study of stability can be uncontrollably intricate, due to the involved structure of the scalar field coupled to gravity. To circumvent the complicated and not analytical approaches, linearized formulations have been commonly worked out. In this context, supported by the stability of deformed defect generated brane-world models, scalar, vector, and tensor perturbations are investigated throughout this work.
Localization aspects of various matter fields with spin 0, 1/2, and 1 on analytical thick braneworld models are indeed a main concern in deriving brane-world models, since they must describe our physical 4D world. The localization of the spin 1/2 fermions deserves a special attention, since there is no scalar field to couple with in this model, in contrast to thick branes generated by deforming defect mechanisms [19]. Otherwise, Kalb-Ramond fields, although already investigated [20], shall not be the main aim here. The spin 1/2 issue has been previously studied in some other contexts [21], including further coupling of more scalar fields in the action [22] and asymmetric brane-worlds generated by a plenty of scalar field potentials [8,[23][24][25][26]. In particular, asymmetric Bloch branes in the context of the hierarchy problem have been addressed in Ref. [14].
Our aim is to investigate the localization of bulk matter and gauge fields on the brane, in the context where the mass-independent potentials of the corresponding Schrödinger-like equations, regarding the 1D quantum mechanical analogue problem, can be suitably acquired from a warped metric. In particular, for a bulk mass proportional to the fermion mass term enclosed by the global action, the possibility of trapping spin 1/2 fermions on asymmetric branes is discussed and quantified.
To accomplish this aim, this paper is organized as follows. In Sect. II, a brief review of braneworld scenarios supported by an effective action driven by a (dark sector) scalar field is presented.
Warp factors and the corresponding internal brane structure are described for four different analytical models. In Sect. III, the left-right chiral asymmetric aspects of matter localization for spin 1/2 fermion fields on thick branes are investigated. Extensions to scalar boson and vector boson fields are obtained in Sect. IV and V, respectively. Final conclusions are drawn in Sect. VI.
Let one starts considering a 5D space-time warped into 4D. The most general 5D metric compatible with a brane-world spatially flat cosmological background has the form given by where e 2A(y) denotes the warp factor, and the signature (− + + + +) is employed, with M, N = 0, 1, 2, 3, 5. The g µν stands for the components of the 4D metric tensor (µ, ν = 0, 1, 2, 3). One can identify y ≡ x 4 as the infinite extra-dimension coordinate (which runs from −∞ to ∞), and notice that the normal to surfaces of constant y is orthogonal to the brane, into the bulk 1 .
The brane-world scenario examined here is setup by an effective action, driven by a (dark sector) scalar field, ζ, coupled to 5D gravity, given by where R is the 5D scalar curvature, R N Q = g BM R BN QM is the Ricci tensor, and κ 5 = (8πG 5 ) 1/2 denotes the 5D gravitational coupling constant, hereon set to equal to unity, where G 5 is the 5D Newton constant. The Einstein equations read where T ζ M N denotes the energy-momentum tensor corresponding to the matter Lagrangian, regarding the matter field ζ. After solving the 5D Einstein field equations, the bulk cosmological constant turns out, in general, to be positive or negative, thus realising a de Sitter or anti-de Sitter brane-world, respectively, generated by curvature. It realises and emulates the interplay involving the 4D and 5D cosmological constants. Some further possibilities are devised, e. g., in [9,15], however it is worth to mention that an additional scalar field can be still added in the action, whose isotropisation shall precisely define the nature of the brane-world. This the latter case is however beyond the scope of our analysis. Obviously, whatever the possibility to be considered, the thin brane limit must obey the fine-tuning relation [10] Λ 4 = κ 2 5 2 1 6 κ 2 5 σ 2 + Λ 5 , among the effective 4D and 5D cosmological constants and the brane tension σ as well.
Considering the real scalar field action, Eq. (2), one can compute the stress-energy tensor which, supposing that both the scalar field and the warp factor dynamics depend only upon the extra coordinate, y, leads to an explicit dependence of the energy density in terms of the field, ζ, and of its first derivative, dζ/dy, as With the same constraints on ζ about the dependence on y, the equations of motion currently known from [3,4], which arise from the above action, are through a variational principle relative to the scalar field, ζ, and through a variational principle relative to the metric, or equivalently to A, manipulated to result into 3 dA dy after an integration over y.
For the scalar field potential written in terms of a superpotential, w, as the above equations are mapped into first-order equations [3,4] as and for which the solutions can be found straightforwardly through immediate integrations [3] (see also Ref. [12] and references therein). The energy density follows from Eq. (9) as The analysis of localization aspects of brane-world scenarios shall be constrained by some known examples, I, II, III and IV , for which the warp factor, A(y), and the energy density, T 00 (y), can be analytically computed. The model I is supported by a sine-Gordon-like superpotential given by which reproduces the results from Ref. [4]. The model II corresponds to a deformed λζ 4 theory with the superpotential given by Models III and IV are deformed topological solutions from Ref. [28] supported by superpotentials and where the parameter a fixes the thickness of the brane described by the warp factor, e 2A(y) . Besides exhibiting analytically manipulable profiles, the above superpotentials have already been discussed in the context of thick brane localization [4,5,12]. The models I and II are respectively motivated by sine-Gordon and λζ 4 theories, and models III and IV are obtained (also analytically) from deformed versions of the λζ 4 model [13]. In particular, models III and IV can also be mapped onto tachyonic Lagrangian versions of scalar field brane models [6,12,28].
From the above superpotentials, the respective solutions for ζ(y) are set as ζ IV (y) = y where one has suppressed any additional (irrelevant) constant of integration for convenience, and one has just considered the positive solutions 2 .
The obtained expressions for the warp factor as resulting from Eq. (11) are respectively given by A IV (y) = − 1 12 where integration constants are introduced as to set a normalization criterium for which A(0) = 0.
The solutions for A I and A II are depicted in Fig. 1. The corresponding localized energy densities computed through Eq. (12) are respectively given by The brane scenarios for models from I to IV are depicted in Fig. (1) for the warp factors and in Fig. (2) for the energy densities, from which one can observe that models from I to IV give rise to thick branes, most of them with no internal structures. In fact, only the potential that controls the scalar field from model II allows the emergence of thick branes that host internal structures in the form of a layer of a novel phase enclosed by two separate interfaces, inside which the energy density of the matter field gets more concentrated. It is related to the extension/localization of the warp factor, namely when the profiles depicted in Fig. (1) approach to a plateu form in the region very inside of the brane, the corresponding internal structure is observed through its energy profile.
The appearance of negative energy densities in the plots for T 00 may be related to a predominance of the scalar field potential over the kinetic-like term related to the coordinate y. Speculatively, it indicates that the vacuum minimal energy can be adjusted by the inclusion of some additional term, eventually related to the cosmological constant.
The localization of bulk matter fields on thick branes generated by each one of these models shall be identified in the following sections. Spin 0, spin 1/2, and spin 1 fields shall evolve coupled to gravity and, as usual, the bulk matter field contribution to the bulk energy shall be neglected.
It means that the obtained solutions hold in the presence of the bulk matter, without disturbing the bulk geometry. Dirac action for a spin 1/2 fermion with a mass term can be expressed as [21,26] Here ω R = 1 4 ωRS R ΓRΓS is the spin connection, where and F (z) is some general scalar function, providing a mass term with a kink-like profile, which from this point is written in terms of a conformal variable z such that dz = e −A(y) dy regards a transformation to conformal coordinates. This kind of mass term is introduced in the action, for it has played a critical role on the localization of fermionic fields on a Minkowski brane. The components of the spin connection ω M with respect to (1) are ω µ = 1 2 (∂ z A)γ µ γ 5 +ω µ , wherê ω α = 1 4ωμν α ΓμΓν is the spin connection derived from the metric g µν =ẽμ µẽν ν ημν. Thus, the equation of motion corresponding to the action (29) reads The 5D Dirac equation can be hence studied by taking spinors with respect to 4D effective fields. In this way the chiral splitting yields where L n (z) and R n (z) are the well-known KK modes, and ψ Rn (x µ ) = +γ 5 ψ Rn (x µ ) [ψ Ln (x µ ) = γ 5 ψ Ln (x µ )] is the right-chiral [left-chiral] component of a 4D Dirac field, respectively. In addition, the sum over n can be both continuous and discrete. Assuming that γ µ (∂ µ +ω µ )ψ (R,L)n = m n ψ (L,R)n , the L n (z) and R n (z) functions should then satisfy the subsequent coupled equations, The associated Schrödinger-like equations can be thus acquired for the left and right-chiral KK modes of fermions, respectively, as: where the mass-independent potentials are given by Note that the Schrödinger-like equations (33) and (34) can be transformed into U † U L n = m 2 n L n and U U † R n = m 2 n R n , where U ≡ ∂ z + e A M F (z). This observation is based upon supersymmetric quantum mechanics, implying that the mass squared is non-negative.
In order to lead these results to the standard 4D action for a massless fermion, and a series of massive chiral fermions, the action S = n d 4 x √ −gψ n [γ µ (∂ µ +ω µ ) − m n ] ψ n is employed, for orthonormalization conditions If in the formulae (32a) and (32b), by setting m n = 0, thus it yields Hence, either the massless left-or right-chiral KK fermion modes can be localized on the brane, being the another one non-normalizable.
By taking F (z) = ζ(z), regarding Eqs. the fermion and the background scalar field, assures the localization mechanism for fermions [29].
For the majority brane-world models, the scalar field ζ is, usually, a kink, being an odd function of the extra dimension. Here we do not necessarily impose this condition, in order to not preclude asymmetric solutions, with respect to the extra dimension.
In what follows the profile of the above left-right potentials is depicted in Fig. (3) for different values of the localization parameter a. In fact, the potentials V L,R (z) have asymptotic behaviors that tend to zero from up, as y → ±∞, for all models from I to IV . In the model I, at y = 0 the potential V R (z) attains its maximum positive value, a global maximum, for a = 1. The potential V R (z) changes to a volcano-type profile along the interval of 1 < a < 2, such that for a = 2, 3, 4, . . . an asymmetric behavior emerges and produces a totally odd symmetric well-barrier profile in the limit of a → 0. Except for 0 < a 1, the zero mode of left-and right-chiral fermions can not be trapped. All potentials for the model II are asymmetric (except for a = 0, which is nonsense in the brane context), have maxima at y = 0 and tends to zero at y → ±∞, and there is no bound state for right-chiral fermions. In particular, for V R,L (y) when a = 1, the minima occur at y ∼ ±0.87.

IV. MATTER LOCALIZATION FOR SPIN 0 SCALAR FIELDS
The localization of scalar fields on thick branes generated by deformed defects can also be considered from this point. In particular, an interesting approach on domain walls can be also found in Ref. [27]. In fact, a massive scalar field coupled to gravity can be described by the following action, where m 0 denotes the effective mass of a bulk scalar field, Φ, and from where one can check whether spin 0 matter fields can be trapped on the thick brane. By employing the metric (1), the associated equation of motion from the action in Eq. (39) reads Hence, by the KK decomposition Φ(x µ , z) = n χ n (x µ )ξ n (z)e −3A/2 , where ξ n is assumed to satisfy the 4D Klein-Gordon equation ∂ µ ( √ −gg µν ∂ ν ) / √ −g − m 2 n ξ n (x µ ) = 0, being m n the 4D mass of the KK excitation of the scalar field. Then the scalar KK mode ξ n (z) is ruled by the following equation: This equation is a Schrödinger one, with effective potential given by The profile of the above scalar boson potential is depicted in Fig. (4) (solid (black) lines) for different values of the localization parameter a. For m 0 = 1, only brane scenarios with a 2 provide conditions to have a localized scalar field. Even in this case, such localized states behave much more as resonances than as bound states, given that it can be tunneled out of the potential.
Bound states appear only for non integer values of the brane width such that a < 1, which shall correspond to typical volcano-type potentials.
One now turns to spin 1 vector fields and begins with the 5D action of a vector field where F M N = ∂ [M A N ] denotes the field strength tensor. A 5D spin 1 field can be now studied via M (x ρ )τ n (z). The action of the 5D massless vector field (43) is invariant under the following gauge transformation: where F (x ρ , z) denotes any arbitrary regular scalar function, for M = µ, 5. The field component [26], by this gauge. In fact, Eq. (44) yields By choosing F (x ρ , z) = − n a (n) 5 (x ρ ) τ n (z)dz, [26] thenÃ 5 = 0, hence the action (43) is led to the spacetime action where ( ) = ∂ z . Given a set of orthonormal functions τ n (z), playing the role of spin 1 Kaluza-Klein modes, and the decomposition of the vector field A µ (x ρ , z) = n a (n) µ (x µ )τ n (z)e −A/2 , the action (46) reads ν] stands for the 4D field strength tensor. The KK modes τ n (z) satisfy the Schrödinger equation where the mass-independent potential reads [35] The profile of the above vector boson potential is depicted in Fig. (4)  have maxima at y = 0 and minima at y → ±∞, and there is no bound state for right-chiral fermions, but, again, for V R,L (y) when a = 1.
It is worth to mention that, for the localization of a fermion zero mode, the mass term M F (z)ΨΨ was considered in the 5D action. An interesting approach concerning such mass term in Eq. (29) has been studied, corresponding to the so-called singular dark spinors [30,31]. Such massive mass dimension one quantum fields are prime candidates for the dark matter problem, also presenting possible signatures at LHC [32]. It generates a slightly different action responsible for spin 1/2 matter fields localization [33,34]. Such approach can be also extended in the context of deformed defects here presented.