Residue theorem and summing over Kaluza-Klein excitations

TL;DR

This paper develops a method using the residue theorem to sum over Kaluza-Klein excitations in warped extra dimensions, providing conditions for convergence and connecting five-dimensional propagators with four-dimensional effective theory.

Contribution

It introduces a novel approach to sum Kaluza-Klein modes via residue theorem and establishes links between 5D propagators and 4D effective theory in warped extra dimensions.

Findings

Derived sufficient conditions for series convergence of Kaluza-Klein modes.

Presented a method to sum over infinite Kaluza-Klein series using residue theorem.

Analyzed the suppression of Higgs contributions to B-meson decay operators.

Abstract

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. As an example, we demonstrate that the corrections from neutral Higgs to the Wilson coefficients of relevant operators for $B\rightarrow X_s\gamma$ contain the suppression factor $m_b^3m_s/m_{_{\rm w}}^4$ comparing with that from other sectors, thus can be neglected safely.