arXiv:0808.0315 (cross-list from hep-th) [ps, pdf, other] Title: Can the cosmological "constant" run? - It may run Authors: Ilya L. Shapiro, Joan Sola Comments: LaTeX, 36 pages, extended discussion, typos corrected, references added. Sections 2 and 4 include our complete refutation of the criticisms from the authors of arXiv:0712.3040 [hep-th] against our previous papers on this subject Subjects: High Energy Physics - Theory (hep-th); Astrophysics (astro-ph); General Relativity and Quantum Cosmology (gr-qc); High Energy Physics - Phenomenology (hep-ph) Using standard quantum field theory, we discuss several theoretical aspects of the possible running of the cosmological constant (CC) term in Einstein's equations. The basic motivation for the present work is to emphasize that this possibility should also be taken into account when considering dynamical models for the dark energy (DE), which are nowadays mainly focused on identifying the DE with the energy density associated to one or more ad hoc scalar fields. At the same time, we address some recent criticisms that have been published (or privately communicated to us) attempting to cast doubts on the fundamental possibility of such running. In this work, we argue that while there is no comprehensive proof of the CC running, there is no rigorous proof of the non-running either. In particular, some purported "non-running theorem" recently adduced in the literature is, in our opinion, completely insubstantial and formally incorrect. The way to the CC running is, therefore, still open and we take here the opportunity to present a pedagogical review of the present state of the art in this field, including a a brief historical account.