Casimir force in brane worlds: Green's function vs Zeta function

Casimir force encodes the structure of the field modes as vacuum fluctuations and so it is sensitive to theextra dimensions of brane worlds. Now, in flat spacetimes of arbitrary dimension the two standardapproaches to the Casimir force, Greens function, and zeta function yield the same result, but for braneworld models this was only assumed. In this work we show that both approaches yield the same Casimirforce in the case of universal extra dimensions and Randall-Sundrum scenarios with one and two branesadded by p compact dimensions. Essentially, the details of the mode eigenfunctions that enter the Casimirforce in the Greens function approach get removed due to their orthogonality relations with a measureinvolving the right hypervolume of the plates, and this leaves just the contribution coming from the zetafunction approach. The present analysis corrects previous results showing a difference between the twoapproaches for the single brane Randall-Sundrum; this was due to an erroneous hypervolume of the platesintroduced by the authors when using the Greens function. For all the models we discuss here, theresulting Casimir force can be neatly expressed in terms of two four-dimensional Casimir forcecontributions: one for the massless mode and the other for a tower of massive modes associated withthe extra dimensions.