Trapping Dirac fermions in tubes generated by two scalar fields
In this work we consider dimensional resonant Dirac fermionic states on tube-like topological defects. The defects are formed by rings in dimensions, constructed with two scalar field and , and embedded in the dimensional Minkowski spacetime. The tube-like defects are attained from a lagrangian density explicitly dependent with the radial distance relative to the ring axis and the radius and thickness of the its cross-section are related to the energy density. For our purposes we analyze a general Yukawa-like coupling between the topological defect and the fermionic field . With a convenient decomposition of the fermionic fields in left- and right- chiralities, we establish a coupled set of first order differential equations for the amplitudes of the left- and right- components of the Dirac field. After decoupling and decomposing the amplitudes in polar coordinates, the radial modes satisfy Schrödinger-like equations whose eigenvalues are the masses of the fermionic resonances. With the Schrödinger-like equations are numerically solved with appropriated boundary conditions. Several resonance peaks for both chiralities are obtained, and the results are confronted with the qualitative analysis of the Schrödinger-like potentials.
Braneworld scenarios have their origins in attempts of solution of important problems of theoretical physics such as the cosmological constant and the gauge hierarchy br1 (); br2 (); br3 (); br4 (); br5 (). In the original formulation of thin branes, the matter fields are by construction localized on a brane with energy density described by a delta function thin (), while gravity propagates in all dimensions. Usual Newton’s law can then be reproduced on the brane depending on the metric warp factor, attained after solving Einstein’s equations. Several extensions soon appeared, with smooth thick branes constructed by scalar fields thick1 (); thick2 (); thick3 (); thick4 (); thick5 (); thick6 (); thick7 (); thick8 (); thick9 (); thick10 (). A comprehensive review on this subject can be found in thick_rev (). This opened up the idea of matter fields to visit extra dimension space, with possible signal of deviations of standard model due to extra dimensions.
In general, thick branes are possibly able to trap gravitons and scalar fields. For fermions, however, the introduction of the fermion-scalar coupling is a necessary condition to ensure the normalizable zero modes. This is a known property already demonstrated by Jackiw and Rebbi jackiw () for domain walls. For some models, the massive fermionic states leak from the branes but stay for a sufficiently longer time to be characterized as resonances brss1 (); brss2 (); brss3 (); brss4 (); brss5 (); brss6 (); brss7 (); brss8 (); brss9 (). In particular, the Ref. liu () analyzes the localization of matter fields in branes constructed from a scalar field coupled to a dilaton. In liu1 (), the localization and mass spectra of various matter fields in thick brane were investigated. For fermionic Kaluza-Klein modes, bound states for both chiralities were found. In brss5 (), some of the authors of the present work have investigated the presence of massive modes for right-hand and left-hand fermions with branes with internal structure constructed by two scalar fields coupled to gravity by introducing a simple Yukawa coupling.
In this work we are interested in topological defects embedded in a flat spacetime that can be constructed following similar procedure used for modeling branes, namely, the embedding of a topological defect in one or more extra dimensions. Thus, inspired by the physics of extra dimensions, in Section II we consider -dimensional ring-like topological defects bmm1 (); bmm2 () which are embedded in a -dimensional flat spacetime forming a tube-like topological defect. In Section III, we study some aspects of localization of fermionic fields in this system. We have particular interest for resonance effects, which are studied in Section IV. Our conclusions are presented in Section V.
Ii A tube in -dimensions
A tube in -dimensions can be described by the action
We use capital letters , for all dimensions. The explicit dependence of follows closely and generalizes for two fields the construction of bmm1 (); bmm2 () for evading Derrick-Hobarts’ theorem raj1 (); raj2 (); raj3 ().
We notice that this construction breaks translational invariance, which is also present in QCD scenarios. For instance, in investigations which deal with color superconductivity, pairing with quarks with different chemical potentials results in crystalline quark matter condensates which spontaneously break translational and rotational invariance, and include spin-zero Cooper pairs color1a (); color1b (). In color2a (); color2b () the effective Lagrangian density describing the color-flavor locked (CFL) symmetry phase of QCD at high density has fields depending on the velocity of the massless Dirac fermions. With glueball effective lagrangian model the breaking of Lorentz invariance induced by the quark chemical potential affect the critical temperature for the onset of the superconductive state deconf (). The breaking of translational invariance also occurs in problems dealing with brane intersections br-inta (); br-intb (), noncommutative field theory with non-constant noncommutativity n-coma (); n-comb () and condensed matter physics dobr (); bor ().
The equations of motion for static solutions are
In this work we restrict us to configurations with radial symmetry, i.e., the fields and depends only on . One can show that the solutions of the first-order equations bmm1 ()
In this work, for generating the tube solution we consider bsr ()
This choice of with the potential was studied in Ref. bsr (). The potential has minima at and with and the equations of motion has static solutions connecting the minima as defects with internal structure known as Bloch walls. The limit turns the two-field problem into a one-field model with solution known as Ising wall. See also Refs. shif (); izq () for other solutions. The extension of this construction to dimensions leading to Bloch brane was presented in thick9 (). The richer structure of degenerate and critical Bloch branes where proposed in dah (). In thick9 () it was shown that the presence of the field is crucial to giving internal structure to the brane. In the present work the presence of the field also contributes to generate an internal structure to the tube formed. We will see that this is crucial for localizing fermions with a simple Yukawa coupling.
where Changing back to the variable we get
which has a ring profile and can be identified with the ring radius of the tube’s cross section.
The energy density of the -dimensional defect is
Here we consider finite in , which restricts the parameters to satisfy when . The Figs. 1a-b depict the energy density for fixed and several values of and coupling constant . We note that for fixed and , the behavior of the energy density changes from a lump centered in to a peak centered around . For fixed , the maximum amplitude of increases with , so large values of produce more interesting results. For large values of , there exists a value so that for , the effects of the field are strong and the defect appears as a thick tube structure whose center is localized between the origin and . On the other hand, for it is clear the predominance of the field and the defect looks like as a thin tube centered around .
This means that for larger values of we can characterize the defect as a ring in the -dimensional -plane, or as a cylindrical tube in the -dimensional space, oriented along the symmetry axis. The influence of larger values of shows that the field is responsible for the process of generating a thicker tube. The total energy in the -plane is given by , which can be identified with the mass of the ring, .
Iii fermion localization
We are interested in the localization of (1,1)-dimensional fermions in an infinite tube whose transversal section is the 2-dimensional ring. Here, the tube in consideration is the one analyzed in the previous section. In considering fermionic states on a tube-like defect, we must remark that the analysis about the existence or not of fermionic zero modes was also considered in the other contexts. One can cite fermions in the field of an abelian and nonabelian string1 (); string2 (); string3 () vortex solutions and more specifically neutrino zero modes on electroweak strings Zstring1 (); Zstring2 (); Zstring3 ().
In the following we consider the fermionic field coordinates as and the ring coordinates as . Then, after neglecting the backreaction on the tube, we consider the following fermionic action
where the matrices are defined as
and are conveniently chosen to provide Schrödinger’s equations in the plane whose potentials are supersymmetric partners. Here is a function of the scalar fields and giving the ring solution in Eqs. (9) and is the coupling constant.
After transforming the plane to polar coordinates, the equation of motion for is found as
where Greek letters are for the coordinates and we have chosen
We decouple the coordinates from by making the decomposition
and by imposing that and are the chiral components of a fermion satisfying the (1+1)-dimensional Dirac’s equation
we can rewrite Eq. (13) in the following set of equations for the chiral amplitudes and :
Now we make the useful decomposition
where the potentials are given by
Then, we have transformed the equation for fermions in a set of independent Schrödinger-like equations for the amplitudes and , allowing to get our goal of finding massive modes and analyzing their localization properties. The equations (21) and (22) allow us to adopt a probabilistic interpretation for finding massive modes of both chiralities in the tube. Here we are mainly interested in resonant states.
as being and , guaranteeing the eigenvalues to be nonnegative. In this way it is forbidden the existence of tachyonic modes. The eigenfunctions and establish a complete set of orthonormal functions satisfying
Further note that the action given by Eq. (11) can be integrated in the dimensions in order to obtain a standard action for massive Dirac fermions
Now we consider the issue of existence of the zero-mode which is obtained from
For fixed the hamiltonian is a quantum mechanical supersymmetric partner of the hamiltonian with superpotential
hence the solution for the zero mode is
In our analysis we are considering the Yukawa coupling, . Hence, because the integral is finite for all the zero mode in non-normalizable for all . Since the zero-mode of is non-normalizable, we conclude that the spectra of and are identical due to the spontaneous breaking of supersymmetry in this quantum-mechanical system gan (); ortiz ()
Iv Numerical Results
For our purposes we consider a simple Yukawa’s coupling, . Interesting considerations for this and other couplings in models of two scalar fields can be found in cdh ().
In order to investigate numerically the massive states, firstly we consider the region near the origin ( ) where
Then for , the functions , and are finite and the potentials and are dominated by the contributions of the angular momentum proportional to . In this way, for left chirality the potential is reduced to
and the solutions finite in are
For right chirality we have
whose solutions finite in are
We now define the probability for finding fermions inside the tube of radius as
Here is used as the initial condition and 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 oscillate as plane waves.
From the energy density considerations after Figs. 1a-1b, larger values of favor the existence of a Schrödinger potential with structure similar a tube barrier em . The Fig. 2 depicts the Schrödinger-like potential and for , , , and fixed and . The potentials in general diverge in , assume a form of a barrier around and asymptote to zero as , indicating the possible presence of resonances. The increasing of turns the barrier of the potential higher, whereas the increasing of turns it thinner. We noted that influences on the behavior of the potential for but has no sensible influence on the barrier and, we also observe that the increasing of turns the potential barrier wider.
Figs. 3a-c show some results of for left- and right-fermions. The plots are for , , and and for various values of . The plots shows several thin peaks of resonances. The thinner is the peak, the larger is the lifetime of the corresponding resonance. We note for there are fewer but longer-lived resonance peaks, in comparison to . Note also that in general the masses of the resonances increase gradually for lower values of , but the relation was always verified, guaranteeing a condition for no backreaction of the fermions in the ring. The larger lifetime of the modes agrees with the correspondingly larger barrier of the Schrödinger-like potential around (compare with Figs. 2 ). This property of is also responsible for the presence of resonance modes, in general, more massive for lower values of which are closely related to the larger influence of the field in the internal structure of the defect. However, in this limit the number of the resonances is reduced because the mass values can not be higher than the tube-barrier. In this way, for finding long-lived resonances, there appears to be a physical compromise between a thinner tube (larger values of which unfavor the presence of the field) and the Yukawa coupling (smaller values of , related to a greater influence of the field ).
V Remarks and conclusions
We have studied the localization of (1+1)-dimensional fermionic fields in a generalized tube-like topological defect whose cross-section is a ring constructed with two scalar fields. Firstly, we have considered a general coupling between the defect and the fermion field carefully contructed to provide a supersymmetric quantum mechanical description of the chiral amplitudes related to the left- and right-fermionic components. Consequently, the Hamiltonians describing the chiral amplitudes are supersymmetric partners forbiding the existence of tachyonic modes. For the Yukawa coupling it was shown that the zero-mode is non-normalizable and that the spectra of both chiralities are identical due to the spontaneous breaking of supersymmetry. Such result is corroborate by the numerical analysis of the Hamiltonian spectra. Also, it was found that larger couplings and are more effective for finding resonances after a fine-tuning of the contant characterizing the internal structure of the defect.
As a further comment we would like to point out that the Yukawa coupling is useful for studies of the electromagnetic charge of the ring and some effects like charge fractionalization. These studies are currently under consideration.
The authors thank CAPES, CNPq and FAPEMA for financial support. R Casana and AR Gomes thank MM Ferreira Jr. for discussions.
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