Volume stabilization in a warped flux compactification model

We investigate the stability of the extra dimensions in a warped, codimension two braneworld that is based upon an Einstein-Maxwell-dilaton theory. The braneworld solution has two 3-branes, which are located at the positions of the conical singularities. For this type of brane solution the relative positions of the branes ( the shape modulus) are determined via the tension-deficit relations, if the brane tensions are fixed. However, the volume of the extra dimensions ( the volume modulus) is not fixed in the context of the classical theory. Hence, we discuss the one-loop effective potential of the volume modulus for a massless, minimally coupled scalar field. Given the scale invariance of the background solution, the form of the modulus effective potential can only be determined from the sign of the logarithmic term in the effective potential. This term can be evaluated via heat kernel analysis and we show that in most cases the volume modulus is stabilized. In the actual evaluation, due to a lack of knowledge of the UV contributions from the conical branes in a six-dimensional spacetime, we consider its four-dimensional counterpart. We then go on to discuss the mass scale of the modulus itself and find that it becomes comparable to the gravitational scale for mild degrees of warping. Then, we make some suggestions on the original six-dimensional model. Finally, we close this article, after discussing some phenomenological implications relating to the hierarchy problem of the fundamental energy scales and the smallness of the effective vacuum energy on the brane. We find that one-loop corrections in this model appear to alleviate some of these problems.