New Localization Method of Abelian Gauge Fields on Bloch Branes

In this paper, we study the localization of the five-dimensional <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>U</mi><mo>(</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula> gauge field coupled with a background scalar potential on symmetric and asymmetric degenerate Bloch branes. By decomposing the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>U</mi><mo>(</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula> gauge field <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>A</mi><mi>M</mi></msub></semantics></math></inline-formula> into its vector part (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mover accent="true"><mi>A</mi><mo>^</mo></mover><mi>M</mi></msub></semantics></math></inline-formula>) and scalar components, we found that the Lagrangian of the five-dimensional <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>U</mi><mo>(</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula> gauge field can be rewritten as two independent parts: one for the vector field and the other for two scalar fields. Regarding the vector part, the effective potential exhibits a volcano-like shape with finite depth. We obtain a massless vector field on both types of Bloch branes and a set of massive KK resonances. For the scalar part, their massless modes are coupled with each other, while two sets of massive scalar KK modes are independent. Similar to the vector effective potential, the scalar potentials create infinite wells for both types of degenerate Bloch brane solutions. Therefore, there is only one independent massless scalar mode and two sets of massive scalar Kaluza–Klein resonances. Furthermore, we also observed that, for the two types of Bloch brane solutions, the asymmetric parameter <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>c</mi><mn>0</mn></msub></semantics></math></inline-formula> has different effects on the localization of scalar modes.