$\infty-\infty$: vacuum energy and virtual black-holes

Abstract

We discuss other contributions to the vacuum energy of quantum field theories and quantum gravity, which have not been considered in literature. As is well known, the presence of virtual particles in vacuum provides the so famous and puzzling contributions to the vacuum energy. As is well known, these mainly come from loop integrations over the four-momenta space. However, we argue that these also imply the presence of a mass density of virtual particles in every volume cell of space-time. The most important contribution comes from quantum gravity $S^{2}\times S^{2}$ bubbles, corresponding to virtual black hole pairs. The presence of virtual masses could lead to another paradox: the space-time itself would have an intrinsic virtual mass density contribution leading to a disastrous contraction - as is known, no negative masses exist in general relativity. We dub this effect {\it the cosmological problem of second type}: if not other counter-terms existed, the vacuum energy would be inevitably destabilized by virtual-mass contributions. It would be conceivable that the cosmological problem of second type could solve the first one. Virtual masses renormalize the vacuum energy to an unpredicted parameter, as in the renormalization procedure of the Standard Model charges. In the limit of $M_{Pl}\rightarrow \infty$ (Pauli-Villars limit), virtual black holes have a mass density providing an infinite counter-term to the vacuum energy divergent contribution $M_{Pl} \rightarrow \infty$ (assuming $M_{UV}=M_{Pl}$). Therefore, in the same Schwinger-Feynman-Tomonaga attitude, the problem of a divergent vacuum energy could be analogous to the {\it put-by-hand} procedure used for Standard Model parameters.