Residue theorem and summing over Kaluza-Klein excitations

Understand

Applying the equations of motion together with corresponding boundary conditions of bulk profiles at infrared and ultraviolet branes, we verify some lemmas on the eigenvalues of Kaluze-Klein modes in framework of warped extra dimension with the custodial symmetry $SU(3)_c\times SU(2)_L\times SU(2)_R\times U(1)_X\times P_{LR}$.

• Using the lemmas and performing properly analytic extensions of bulk profiles, we present the sufficient condition for a convergent series of Kaluze-Klein excitations and sum over the series through the residue theorem.
• The method can also be applied to sum over the infinite series of Kaluze-Klein excitations in unified extra dimension.
• Additional, we analyze the possible connection between the propagators in five dimensional full theory and the product of bulk profiles with corresponding propagators of exciting Kaluze-Klein modes in four dimensional effective theory, and recover some relations presented in literature for warped and unified extra dimensions respectively.