Abstract
We classify possible boundary conditions of a 6d Dirac fermion Ψ on a rectangle under the requirement that the 4d Lorentz structure is maintained, and derive the profiles and spectrum of the zero modes and nonzero KK modes under the two specific boundary conditions, (i) 4d-chirality positive components being zero at the boundaries and (ii) internal chirality positive components being zero at the boundaries. In the case of (i), twofold degenerated chiral zero modes appear which are localized towards specific directions of the rectangle pointed by an angle parameter θ. This leads to an implication for a new direction of pursuing the origin of three generations in the matter fields of the standard model, even though triple-degenerated zero modes are not realized in the six dimensions. When such 6d fermions couple with a 6d scalar with a vacuum expectation value, θ contributes to a mass matrix of zero-mode fermions consisting of Yukawa interactions. The emergence of the angle parameter θ originates from a rotational symmetry in the degenerated chiral zero modes on the rectangle extra dimensions since they do not feel the boundaries. In the case of (ii), this rotational symmetry is promoted to the two-dimensional conformal symmetry though no chiral massless zero mode appears. We also discuss the correspondence between our model on a rectangle and orbifold models in some details.
Highlights
Considering extra dimensions has been a fascinating direction for deriving better understandings on various aspects of the standard mode (SM) especially during the past two decades, e.g. on the hierarchy problem between the electroweak scale and an ultraviolet scale
The boundary conditions (BCs) of them at the circumference of extra dimensions play a significant role in the determination of mode functions, especially in the zero modes which correspond to the lowest modes of the effective mass appearing after the Kaluza–Klein (KK) decomposition among the extra spacial directions
A powerful and widely used precept for determining a class of BCs is the variational principle. An advantage of this method originates from the characteristic that we derive the equation of motion (EOM) of higher dimensional fields and necessary conditions for BCs simultaneously
Summary
Considering extra dimensions has been a fascinating direction for deriving better understandings on various aspects of the standard mode (SM) especially during the past two decades, e.g. on the hierarchy problem between the electroweak scale and an ultraviolet scale [24,25,26,27,28,29,30,31] with orbifolding), under which zero mode profiles become chiral, degenerated and quasi-localized Considering this direction can lead to a simultaneous explanation of the three features of the SM, 4d chirality, three generations and mass hierarchies of the quarks and leptons. We have discussions on a rectangle, which is a simple 6d generalization of an interval in 5d, without introducing nontrivial backgrounds (e.g. solitonic configurations among the extra dimensions) or boundary-localized terms at the leading order.. Like the 5d case on an interval, the following 6d Dirac mass term M can be written down, where the mass parameter M is expected to describe directions and magnitudes of localized profiles at the zero modes. Mode profiles of degenerate massless 4d chiral fermions can generate large mass splitting when (6d) fermions couple to a (6d) scalar with a vacuum expectation value (VEV)
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