Left Chiral Fermion Research Articles

Gravity and matter on a pure geometric thick polynomial f(R) brane

In this paper, the solutions of the pure geometric thick $f(R)$ brane are investigated. The formulation of $f(R)$ is chosen as $f(R)=\sum_{i=1}^{n}a_{i}R^{i}+\Lambda$. For the certain value of $n$, the brane solution can be calculated, and when $n=3$, $n=4$, and $n=10$, the thick brane solutions are presented. The solutions is stable under linear tensor fluctuation. The zero mode of gravity and scalar field can be localized on thick $f(R)$ branes naturally. The zero mode of vector field and left-chiral fermion can be localized on thick $f(R)$ branes by introducing the coupling with scalar curvature $R$ of spacetime, and the massive resonant modes can be quasilocalized on the brane with the large coupling coefficients.

${\boldsymbol~K}$-fields domain wall

In this paper, we construct the single kink and double kink domain wall solutions by considering the non-canonical $K$-field in a flat five-dimensional spcaetime. The localization of chiral fermion is also investigated with the Yukawa coupling mechanism, and the corresponding massless left-chiral fermion can be localized on the $K$-field two types domain walls. The effective potentials of the chiral fermion on the two types domain walls will have different behaviors and the corresponding effective potential of the chiral fermion on the double-kink domain wall will have double well, and which means that the double kink domain wall will have mroe richer internal structures.

Coulomb's law corrections and fermion field localization in a tachyonic de Sitter thick braneworld

Following recent studies which show that it is possible to localize gravity as well as scalar and gauge vector fields in a tachyonic de Sitter thick braneworld, we investigate the solution of the gauge hierarchy problem, the localization of fermion fields in this model, the recovering of the Coulomb law on the non-relativistic limit of the Yukawa interaction between bulk fermions and gauge bosons localized in the brane, and confront the predicted 5D corrections to the photon mass with its upper experimental/observational bounds, finding the model physically viable since it passes these tests. In order to achieve the latter aims we first consider the Yukawa interaction term between the fermionic and the tachyonic scalar fields MF(T)ΨΨ̅ in the action and analyze four distinct tachyonic functions F(T) that lead to four different structures of the respective fermionic mass spectra with different physics. In particular, localization of the massless left-chiral fermion zero mode is possible for three of these cases. We further analyze the phenomenology of these Yukawa interactions among fermion fields and gauge bosons localized on the brane and obtain the crucial and necessary information to compute the corrections to Coulomb's law coming from massive KK vector modes in the non-relativistic limit. These corrections are exponentially suppressed due to the presence of the mass gap in the mass spectrum of the bulk gauge vector field. From our results we conclude that corrections to Coulomb's law in the thin brane limit have the same form (up to a numerical factor) as far as the left-chiral massless fermion field is localized on the brane. Finally we compute the corrections to the Coulomb's law for an arbitrarily thick brane scenario which can be interpreted as 5D corrections to the photon mass. By performing consistent estimations with brane phenomenology, we found that the predicted corrections to the photon mass, which are well bounded by the experimentally observed or astrophysically inferred photon mass, are far beyond its upper bound, positively testing the viability of our tachyonic braneworld. Moreover, the 5D parameters that define these corrections possess the same order, providing naturalness to our model, however, a fine-tuning between them is needed in order to fit the corresponding upper bound on the photon mass.

Resonance spectrum of a bulk fermion on branes

It is known that there are two mechanisms for localizing a bulk fermion on a brane, one is the well-known Yukawa coupling and the other is the new coupling proposed in [Phys. Rev. D 89, 086001 (2014)]. In this paper, we investigate localization and resonance spectrum of a bulk fermion on the same branes with the two localization mechanisms. It is found that both the two mechanisms can result in a volcano-like effective potential of the fermion Kaluza-Klein modes. The left-chiral fermion zero mode can be localized on the brane and there exist some discrete massive fermion Kaluza-Klein modes that quasilocalized on the brane (also called fermion resonances). The number of the fermion resonances increases linearly with the coupling parameter.

Localization and quasilocalization of a spin-1/2fermion field on a two-field thick braneworld

Localization of a spin-$1/2$ fermion on the braneworld is an important and interesting problem. It is well known that a five-dimensional free massless fermion $\Psi$ minimally coupled to gravity cannot be localized on the Randall-Sundrum braneworld. In order to trap such a fermion, the coupling between the fermion and bulk scalar fields should be introduced. In this paper, localization and quasilocalization of a bulk fermion on the thick braneworld generated by two scalar fields (a kink scalar $\phi$ and a dilaton scalar $\pi$) are investigated. Two types of couplings between the fermion and two scalars are considered. One coupling is the usual Yukawa coupling $-\eta\bar{\Psi}\phi\Psi$ between the fermion and kink scalar, another one is $\lambda\bar{\Psi}\Gamma^{M}\partial_{M}\pi\gamma^{5}\Psi$ between the fermion and dilaton scalar. The left-chiral fermion zero mode can be localized on the brane, and both the left- and right-chiral fermion massive Kaluza-Klein modes may be localized or quasilocalized. Hence the four-dimensional massless left-chiral fermion and massive Dirac fermions, whose lifetime is infinite or finite, can be obtained on the brane.

Interacting topological insulator and emergent grand unified theory

Motivated by the Pati-Salam Grand Unified Theory, we study $(4+1)d$ topological insulators with $SU(4) \times SU(2)_1 \times SU(2)_2$ symmetry, whose $(3+1)d$ boundary has 16 flavors of left-chiral fermions, which form representations $(\mathbf{4}, \mathbf{2}, \mathbf{1})$ and $(\bar{\mathbf{4}}, \mathbf{1}, \mathbf{2})$. The key result we obtain is that, without any interaction, this topological insulator has a $\mathbb{Z}$ classification, namely any quadratic fermion mass operator at the $(3+1)d $ boundary is prohibited by the symmetries listed above; while under interaction this system becomes trivial, namely its $(3+1)d$ boundary can be gapped out by a properly designed short range interaction without generating nonzero vacuum expectation value of any fermion bilinear mass, or in other words, its $(3+1)d$ boundary can be driven into a "strongly coupled symmetric gapped (SCSG) phase". Based on this observation, we propose that after coupling the system to a dynamical $SU(4) \times SU(2)_1 \times SU(2)_2$ lattice gauge field, the Pati-Salam GUT can be fully regularized as the boundary states of a $(4+1)d$ topological insulator with a {\it thin} fourth spatial dimension, the thin fourth dimension makes the entire system generically a $(3+1)d$ system. The mirror sector on the opposite boundary will {\it not} interfere with the desired GUT, because the mirror sector is driven to the SCSG phase by a carefully designed interaction and is hence decoupled from the GUT.

Production and decay rates of excited leptons in a left-right symmetric scenario

We merge two leading beyond standard model scenarios, namely compositeness and left-right symmetry, and probe the resulting collider signatures in the leptonic case. The constraints on composite models for fermions leave open the possibility of vectorlike excitations of standard model fermions. Here we consider the possibility of low scale left-right gauge symmetry $SU(2{)}_{R}\ifmmode\times\else\texttimes\fi{}SU(2{)}_{L}\ifmmode\times\else\texttimes\fi{}U(1{)}_{B\ensuremath{-}L}$, with the simplifying assumption that the rightlike excited sector of fermions is significantly heavier than the excitations of the left chiral fermions. It is found that the right-hand currents still contribute to observable processes, and alter the existing bounds on the scale of compositeness. The cross section times branching ratio of the photon decay channel is strongly depressed, bringing down the exclusion limit of the mass of excited electrons from about 2 TeV to below 1 TeV. On the other hand, cross section times branching ratio of the Z decay channel is significantly enhanced and remains greater than that of the photon channel. We thus propose analyzing the Z decay channel in existing collider data in order to search for signature of left-right symmetry as well as excited leptons with masses above 1 TeV.

Localization of bulk matter fields, the hierarchy problem and corrections to Coulomb’s law on a pure de Sitter thick braneworld

In this paper we investigate the localization and mass spectra of various matter fields with spin 0, 1 and $1/2$ on a geometric thick brane generated by pure four-dimensional (4D) and five-dimensional positive cosmological constants without bulk scalar fields. This model possesses a 4D cosmological constant that can be made as small as one desires without fine-tuning it with the bulk cosmological constant. The Randall-Sundrum model was obtained as an analytic continuation of the flat brane limit of this braneworld configuration when the Hubble parameter disappears. We also show that within the framework of this inflating braneworld model, it is possible to formulate a mechanism for obtaining TeV mass scales from Planck ones by adding a positive thin brane, where the Standard Model fields are trapped, located at a distance ${y}_{2}$ from the origin, where the Planck thick brane resides. The brane separation must be of the same order as the inverse thickness parameter of the model in order for the mechanism to generate the desired hierarchy. This result is obtained by imposing the recovery of both the correct 4D gravitational couplings and the actually observed accelerated expansion of the Universe in our de Sitter braneworld. Regarding the localization of matter in the purely geometric thick braneworld, for spin-0 massless and massive scalar fields as well as for spin-1 vector fields, the potentials of the Kaluza-Klein (KK) modes in the corresponding Schr\"odinger equations are modified P\"oschl-Teller potentials, which lead to the localization of the scalar and vector zero modes on the brane as well as to mass gaps in the mass spectra. We also compute the corrections to Coulomb's law coming from massive KK vector modes. For spin-$1/2$ fermions, we introduce the bulk mass term $MF(z)\overline{\ensuremath{\Psi}}\ensuremath{\Psi}$ in the action, and four different cases are investigated. Localization of the massless left-chiral fermion zero mode is feasible for just two cases of $F(z)$. In the first one, we obtain Schr\"odinger equations with modified P\"oschl-Teller potentials with the corresponding mass gaps for both left- and right-chiral fermions; the number of massive KK bound states is finite (determined by the ratio $M/b$) and is the same for left- and right-chiral modes. In the second case, we get a family of solutions with an infinite number of bound states where the mass spectra for left- and right-chiral KK fermion modes are discrete. A special case resembles the one-dimensional quantum harmonic oscillator problem. For both of these cases, the mass spectrum of right-chiral fermions is shifted with respect to the mass spectrum of the left-chiral ones.

Thick branes with a nonminimally coupled bulk-scalar field

In this paper, we investigate thick branes with a nonminimally coupled background scalar field, whose solution is a single-kink or a double-kink. The effects of the nonminimal coupling constant $\xi$ on the structure of the thick branes and the localization of gravity, fermions, scalars and vectors are discussed. It is shown that each brane will split into two sub-branes as increasing the nonminimal coupling constant $\xi$. By investigating the tensor perturbation equations of gravity and the general covariant Dirac equation of fermions, we find that both the gravity zero mode and left-chiral fermion zero mode are localized at the center of the single-kink branes and localized between the two sub-branes generated by the double-kink, which indicates that the constant $\xi$ does not effect the localization of these zero modes. However, the zero mode of scalars is localized on each sub-brane (for both single-kink and double-kink branes) when $\xi$ is larger than its critical value $\xi_0$. The effects of the nonminimal coupling constant $\xi$ on the resonances of gravity and fermions with finite lifetime on the branes are also discussed.

Bulk matter fields on two-field thick branes

In this paper, we obtain a new solution of a brane made up of a scalar field coupled to a dilaton. There is a unique parameter $b$ in the solution, which decides the distribution of the energy density and will affect the localization of bulk matter fields. For free vector fields, we find that the zero mode can be localized on the brane. And for vector fields coupled with the dilaton via ${\mathrm{e}}^{\ensuremath{\tau}\ensuremath{\pi}}{F}_{MN}{F}^{MN}$, the condition for localizing the zero mode is $\ensuremath{\tau}\ensuremath{\ge}\ensuremath{-}\sqrt{b/3}$ with $0<b\ensuremath{\le}1$, or $\ensuremath{\tau}>\ensuremath{-}1/\sqrt{3b}$ with $b>1$, which includes the case $\ensuremath{\tau}=0$. While the zero mode for free Kalb-Ramond fields cannot be localized on the brane, if only we introduce a coupling between the Kalb-Ramond fields and the dilaton via ${\mathrm{e}}^{\ensuremath{\zeta}\ensuremath{\pi}}{H}_{MNL}{H}^{MNL}$. When the coupling constant satisfies $\ensuremath{\zeta}>1/\sqrt{3b}$ with $b\ensuremath{\ge}1$ or $\ensuremath{\zeta}>\frac{2\ensuremath{-}b}{\sqrt{3b}}$ with $0<b<1$, the zero mode for the KR fields can be localized on the brane. For spin half fermion fields, we consider the coupling $\ensuremath{\eta}\overline{\ensuremath{\Psi}}{\mathrm{e}}^{\ensuremath{\lambda}\ensuremath{\pi}}\ensuremath{\phi}\ensuremath{\Psi}$ between the fermions and the background scalars with positive Yukawa coupling $\ensuremath{\eta}$. The effective potentials for both chiral fermions have three types of shapes decided by the relation between the dilaton-fermion coupling constant $\ensuremath{\lambda}$ and the parameter $b$. For $\ensuremath{\lambda}\ensuremath{\le}\ensuremath{-}1/\sqrt{3b}$, the zero mode of left-chiral fermion can be localized on the brane. While for $\ensuremath{\lambda}>\ensuremath{-}1/\sqrt{3b}$ with $b>1$ or $\ensuremath{-}1/\sqrt{3b}<\ensuremath{\lambda}<\ensuremath{-}\sqrt{b/3}$ with $0<b\ensuremath{\le}1$, the zero mode for left-chiral fermion also can be localized.

Localization of gravity and bulk matters on a thick anti–de Sitter brane

In this paper, we investigate the localization and the mass spectra of gravity and various bulk matter fields on a thick anti--de Sitter (AdS) brane, by presenting the mass-independent potentials of the Kaluza-Klein (KK) modes in the corresponding Schr\"odinger equations. For gravity, the potential of the KK modes tends to infinity at the boundaries of the extra dimension, which leads to an infinite number of the bound KK modes. Although the gravity zero mode cannot be localized on the AdS brane, the massive modes are trapped on the brane. The scalar perturbations of the thick AdS brane have been analyzed, and the brane is stable under the scalar perturbations. For spin-0 scalar fields and spin-1 vector fields, the potentials of the KK modes also tend to infinity at the boundaries of the extra dimension, and the characteristic of the localization is the same as the case of gravity. For spin-1/2 fermions, by introducing the usual Yukawa coupling $\ensuremath{\eta}\overline{\ensuremath{\Psi}}\ensuremath{\phi}\ensuremath{\Psi}$ with the positive coupling constant $\ensuremath{\eta}$, the four-dimensional massless left-chiral fermion and massive Dirac fermions are obtained on the AdS thick brane.

Localization of matters on Anti-de Sitter thick branes

By presenting the mass-independent potentials of the Kaluza-Klein (KK) modes in the corresponding Schr\"{o}dinger equations, we investigate the localization and mass spectra of various bulk matter fields on an AdS thick brane. For a spin 0 scalar $\Phi$ coupled with itself and the domain-wall-forming field $\phi$ via a coupling potential $V=(\lambda \phi^{2}-u^{2})\Phi^{2}+\tau\Phi^{4}$, the localization and spectrum are decided by a critical coupling constant $\lambda_{0}$. When $\lambda>\lambda_0$, the potential of the scalar KK modes in the corresponding Schr\"{o}dinger equation tends to infinite when far away from the brane, which results in that there exist infinite discrete scalar bound KK states, and the massless modes could be trapped on the AdS brane by fine-tuning of parameters. When $\lambda<\lambda_0$, the potential of the scalar KK modes tends to negative infinite when far away from the brane, hence there does not exist any scalar bound KK state. For a spin 1 vector, the situation is same like the scalar with a coupling constant $\lambda>\lambda_0$, but the zero mode can not be localized on the brane. For a spin 1/2 fermion, we introduce the usual Yukawa coupling $\eta\bar{\Psi}\phi\Psi$, and find that the localization of the fermion is decided by a critical coupling constant $\eta_0$. For $\eta > \eta_0$, the four-dimensional massless left chiral fermion and massive Dirac fermions consisted of the pairs of coupled left-hand and right-hand KK modes could be localized on the AdS brane, and the massive Dirac fermions have a set of discrete mass spectrum. While for the case $0<\eta < \eta_{0}$, no four-dimensional Dirac fermion can be localized on the AdS brane.

Localization and mass spectra of fermions on symmetric and asymmetric thick branes

A three-parameter (positive odd integer $s$, thickness factor $\ensuremath{\lambda}$, and asymmetry factor $a$) family of asymmetric thick brane solutions in five dimensions were constructed from a two-parameter ($s$ and $\ensuremath{\lambda}$) family of symmetric ones in by R. Guerrero, R. O. Rodriguez, and R. Torrealba in [Phys. Rev. D 72, 124012 (2005).]. The values $s=1$ and $s\ensuremath{\ge}3$ correspond to single branes and double branes, respectively. These branes have very rich inner structure. In this paper, by presenting the mass-independent potentials of Kaluza-Klein (KK) modes in the corresponding Schr\odinger equations, we investigate the localization and mass spectra of fermions on the symmetric and asymmetric thick branes in an anti-de Sitter background. In order to analyze the effect of gravity-fermion interaction (i.e., the effect of the inner structure of the branes) and scalar-fermion interaction to the spectrum of fermion KK modes, we consider three kinds of typical kink-fermion couplings. The spectra of left chiral fermions for these couplings consist of a bound zero mode and a series of gapless continuous massive KK modes, some discrete bound KK modes including zero mode (exist mass gaps), and a series of continuous massive KK modes, infinite discrete bound KK modes, respectively. The structures of the spectra are investigated in detail.

Bulk matters on symmetric and asymmetric de Sitter thick branes

An asymmetric thick domain wall solution with de Sitter(dS) expansion in five dimensions can be constructed from asymmetric one by using a same scalar (kink) with differentpotentials. In this paper, by presenting the mass-independentpotentials of Kaluza-Klein (KK) modes in the correspondingSchrödinger equations, we investigate the localization andmass spectra of various bulk matter fields on the symmetric andasymmetric dS thick branes. For spin 0 scalars and spin 1vectors, the potentials of KK modes in the correspondingSchrödinger equations are the modified Pöschl-Tellerpotentials, and there exist a mass gap and a series of continuousspectrum. It is shown that the spectrum of scalar KK modes on thesymmetric dS brane contains only one bound mode (the masslessmode). However, for the asymmetric dS brane with a largeasymmetric factor, there are two bound scalar KK modes: a zeromode and a massive mode. For spin 1 vectors, the spectra of KKmodes on both dS branes consist of a bound massless mode and aset of continuous ones, i.e., the asymmetric factor does notchange the number of the bound vector KK modes. For spin 1/2fermions, two types of kink-fermion couplings are investigated indetail. For the usual Yukawa coupling ηϕΨ,there exists no mass gap but a continuous gapless spectrum of KKstates. For the scalar-fermion coupling ηsin(ϕ/ϕ0)cos−δ(ϕ/ϕ0)Ψ witha positive coupling constant η, there exist some discretebound KK modes and a series of continuous ones. The total numberof bound states increases with the coupling constant η. Forthe case of the symmetric dS brane and positive η, thereare NL(NL ⩾ 1) left chiral fermion bound states (includingzero mode and massive KK modes) and NL−1 right chiral fermionbound states (including only massive KK modes). For the asymmetricdS brane scenario, the asymmetric factor a reduces the numberof the bound fermion KK modes. For large enough a, there wouldnot be any right chiral fermion bound mode, but at least one leftchiral fermion zero mode.

Fermions on thick branes in the background of sine-Gordon kinks

A class of thick branes in the background of sine-Gordon kinks with a scalar potential $V(\ensuremath{\phi})=p(1+\mathrm{cos}\frac{2\ensuremath{\phi}}{q})$ was constructed by R. Koley and S. Kar [Classical Quantum Gravity 22, 753 (2005)]. In this paper, in the background of the warped geometry, we investigate the issue of localization of spin-$1/2$ fermions on these branes in the presence of two types of scalar-fermion couplings: $\ensuremath{\eta}\overline{\ensuremath{\Psi}}\ensuremath{\phi}\ensuremath{\Psi}$ and $\ensuremath{\eta}\overline{\ensuremath{\Psi}}\mathrm{sin}\ensuremath{\phi}\ensuremath{\Psi}$. By presenting the mass-independent potentials in the corresponding Schr\"odinger equations, we obtain the lowest Kaluza-Klein modes and a continuous gapless spectrum of Kaluza-Klein states with ${m}^{2}&gt;0$ for both types of couplings. For the Yukawa coupling $\ensuremath{\eta}\overline{\ensuremath{\Psi}}\ensuremath{\phi}\ensuremath{\Psi}$, the effective potential of the right chiral fermions for positive $q$ and $\ensuremath{\eta}$ is always positive; hence only the effective potential of the left chiral fermions could trap the corresponding zero mode. This is a well-known conclusion which is discussed extensively in the literature. However, for the coupling $\ensuremath{\eta}\overline{\ensuremath{\Psi}}\mathrm{sin}\ensuremath{\phi}\ensuremath{\Psi}$, the effective potential of the right chiral fermions for positive $q$ and $\ensuremath{\eta}$ is no longer always positive. Although the value of the potential at the location of the brane is still positive, it has a series of wells and barriers on each side, which ensures that the right chiral fermion zero mode could be trapped. Thus we may draw the following remarkable conclusion: for positive $\ensuremath{\eta}$ and $q$, the potentials of both the left and right chiral fermions could trap the corresponding zero modes under certain restrictions.

Localization and mass spectrum of matters on Weyl thick branes

In this paper, we study localization and mass spectrum of various matter fields on a family of thick brane configurations in a pure geometric Weyl integrable 5-dimensional space time, a non-Riemannian modification of 5-dimensional Kaluza--Klein (KK) theory. We present the shape of the mass-independent potential of the corresponding Schr\{o}dinger problem and obtain the KK modes and mass spectrum, where a special coupling of spinors and scalars is considered for fermions. It is shown that, for a class of brane configurations, there exists a continuum gapless spectrum of KK modes with any $m^2>0$ for scalars, vectors and ones of left chiral and right chiral fermions. All of the corresponding massless modes are found to be normalizable on the branes. However, for a special of brane configuration, the corresponding effective Schr\{o}dinger equations have modified P\{o}schl-Teller potentials. These potentials suggest that there exist mass gap and a series of continuous spectrum starting at positive $m^2$. There are one bound state for spin one vectors, which is just the normalizable vector zero mode, and two bound KK modes for scalars. The total number of bound states for spin half fermions is determined by the coupling constant $\eta$. In the case of no coupling ($\eta=0$), there are no any localized fermion KK modes including zero modes for both left and right chiral fermions. For positive (negative) coupling constant, the number of bound states of right chiral fermions is one less (more) than that of left chiral fermions. In both cases ($\eta>0$ and $\eta<0$), only one of the zero modes for left chiral fermions and right chiral fermions is bound and normalizable.

Chiral fermions in a spacetime with multiple warping

In a six-dimensional brane world model with multiple ${S}_{1}/{Z}_{2}$ warping, two of the 4-branes at the boundaries have coordinate-dependent brane tensions in order to implement the orbifolded boundary conditions consistently. Such brane tension is shown to be equivalent to a scalar field distribution on the brane [D. Choudhury and S. SenGupta, Phys. Rev. D 76, 064030 (2007).]. We show that in such a scenario a massless left chiral fermion on the 4-brane localizes naturally on the standard model 3-brane located at one edge of the compact manifold while the massless right chiral fermion wave function, as well as the wave functions for the massive fermion modes, peak away from this brane. This offers a mechanism of obtaining a massless chiral fermion with only one of its chiral components present in our brane.