Abstract
In this paper, we obtain a gauge invariant effective action for a bulk massless U(1) gauge vector field on a brane with codimension two by using a general Kaluza-Klein (KK) decomposition for the field. It suggests that there exist two types of scalar KK modes to keep the gauge invariance of the action for the massive vector KK modes. Both the vector and scalar KK modes can be massive. The masses of the vector KK modes m(n) contain two parts, m1( n) and m2( n), due to the existence of the two extra dimensions. The masses of the two types of scalar KK modes mϕ( n) and mφ( n) are related to the vector ones, i.e., mϕ( n) = m1( n) and mφ( n) = m2( n). Moreover, we derive two Schrödinger-like equations for the vector KK modes, for which the effective potentials are just the functions of the warp factor.
Highlights
JHEP01(2019)[021] gauge invariant effective action on the brane
By the new localization mechanism for a 1-form bulk field, we can get a gauge invariant effective action for the massive vector KK modes, which are coupled with some massless scalar KK modes
We did not choose any gauge for the bulk field, and did a general KK decomposition
Summary
We consider the model of brane world with codimension two. The line-element is given by (1.1). The action for a bulk massless U(1) gauge vector field is. With a KK decomposition of the vector field and a dimension reduction, we will obtain the effective actions for the vector KK modes. It is easy to show that the effective action of the vector zero mode is gauge invariant, while the ones for the massive KK modes are not without some mechanisms [42–45]. We have found one way to get the gauge invariant actions for the massive vector KK modes on a brane with codimension one, where our aim was to solve the Hodge duality on the brane for the q-form field [26]. With a given KK decomposition of a bulk field, the effective actions on the brane for the KK modes of the field can be derived. Where m(vn) ≡ m(1n)2 + m(2n)[2] are the masses of the vector KK modes, and m(φn) and m(φn) are the masses of the two scalar modes φ(n) and φ(n), respectively
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