The Localization Of Fields In Braneworlds

As an important development in extra-dimensional theories,the braneworld theory proposes that the 4-dimensional spacetime we live in is a brane in higher-dimensional spacetime.The braneworld has been widely studied to explain various open problems,such as the hierarchy problem,the mass hierarchy among three generations of fermions,the cosmological constant,and dark energy and dark matter.To recover 4-dimensional effective throry,the brane world model assumes that the fields is localized on the brane.Localization of matter has been extensively studied in the 5-dimensional braneworld models.Many 5-dimensional thick brane models have been established in the asymptotically anti-de Sitter(AdS)spacetime.However,in these models,there is a contradiction between the localization conditions of gravity and the U(1)gauge field and the free U(1)gauge field cannot be localized by the minimal coupling with gravity.In addition,in the 5-dimensional RS-like models,fermions cannot be localized by minimal coupling with gravity.The minimal coupling of gravity does not distinguish between the left and right chiralities of fermions and therefore cannot lead to a 4-dimensional chiral theory from the reduction of a high-dimensional free Dirac field.The localization of fields is commonly achieved by adding additional interactions.As gravity is described by the curvature of spacetime,this suggests that we can achieve the localization of various matter fields by the minimal coupling with gravity,and avoid introducing new assumptions.In a spacetime background containing compact dimensions,the free U(1)gauge field can be localized by the minimal coupling with gravity.Bypassing the no-go theorem of the braneworld,we constructed a smooth 6-dimensional thick brane model using a scalar field as the dynamical field,in which both gravity and the free U(1)gauge field can be localized by the minimal coupling with gravity,and Newtonian gravity and Maxwell electromagnetic force are both restored,thus avoiding the contradiction between the localization conditions of U(1)gauge field and gravity in the 5-dimensional model.However,reducing spinor fields in higher-dimensional spacetime is challenging.The mass terms and Yukawa interactions prevent the decoupling of the left and right chiralities of higher-dimensional spinors.This makes it difficult to obtain 4-dimensional free spinor fields.To solve the above difficulty,we did a systematic study of the localization problem of spinor fields from 2n+2-dimensional spacetime to 2n-dimensional spacetime.We investigated the Cl(2n+1,1)algebra and the representation of the Γmatrix in higher-dimensional spacetime.Combining with the generalized Dirac equation,we derived the localization condition of fermions and some general mathematical results.We found that the topology of spacetime affects the chiral symmetry in effective theories.Under the common topologies M4 ×R1×S1 and M4 ×R2,the spinor field cannot be localized by minimal coupling with gravity.Under the M4 × R1×S1 topology,gravity does not distinguish between the left and right chiralities,while under the M4 × R2 topology,gravity does distinguish between the left and right chiralities.In order to realize the localization of spinor fields,we introduce the general tensor coupling mechanism ΨΓMΓNΓP…TMNP...Ψ,which ensures SO(n,1)symmetry.For an odd order tensor TMNP...,the left and right chiralities of high-dimensional spinors are decoupled.We find that the special form of tensor coupling ΨΓM?MF(φ,R,RμνRμν,…)Ψallows for localization of the spinor field when the coupling function takes the form F(φ)=φn.