Ultraviolet regularization of energy between two static sources in the bottom-up holographic approach to strong interactions

It is well known that the potential energy between two heavy quarks carries an important information about the physics of confinement. Using the Wilson loop confinement criterion and the Nambu-Goto string action, this energy can be derived within the bottom-up holographic approach to strong interactions. We recapitulate the standard holographic derivation of the potential between two static sources with emphasis on the physical interpretation of the results. After that we consider the problem of regularization of arising ultraviolet divergence in a general case. Here the term "ultraviolet" means small values of the holographic coordinate which is related with the inverse energy scale in the holographic duality. We show that in the case of widely used Soft-Wall holographic models, in principle, many ultraviolet divergences may appear, although in practice the appearance of more than two different divergences looks somewhat exotic. Some possible subtraction schemes are discussed. Different schemes lead to a different constant shift of the potential energy and this entails a certain scheme-dependence of holographic predictions for constant term in resulting Cornell-like confinement potentials.