Two loop radion correction to $K_L$ - $K_S$ mass difference in the stabilised Randall-Sundrum brane world scenario

In the stabilised Randall-Sundrum brane world scenario, the radion can have phenomenologically testable effects, which can be measured against precisely measured electroweak physics data. We investigate the effect of two loop radion corrections to $K_L$ - $K_S$ mass difference to set a bound on the radion mass and vacuum expectation value. It is found that the leading two loop corrections are of the order $[Log(\frac{\Lambda^2}{m_\phi^2}) ]^2$ where $\Lambda$ is the cut-off scale ${\cal O}(\sim$TeV) and $m_\phi$ is the radion mass.