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

It is well known that the potential energy of 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. We address the problem of regularization of the arising ultraviolet divergence in the general case, with “ultraviolet” referring to small values of the holographic coordinate associated with the inverse energy scale in holographic duality. We show that in the case of the widely used soft-wall holographic models many ultraviolet divergences can appear in principle, although the appearance of more than two different divergences looks somewhat exotic in practice. Some possible subtraction schemes are discussed. Different schemes lead to a different constant shift of the potential energy, which entails a certain scheme dependence of holographic predictions for the constant term in the resulting Cornell-like confinement potentials.