  One hope to solve the cosmological constant problem is to identify a symmetry
principle, based on which the cosmological constant can be reduced either to
zero, or to a tiny value. Here, we note that requiring that the vacuum state is
Lorentz invariant significantly reduces the theoretical value of the vacuum
energy density. Hence, this also reduces the discrepancy between the observed
value of the cosmological constant and its theoretical expectation, down from
123 orders of magnitude to 56 orders of magnitude. We find that, at one loop
level, massless particles do not yield any contribution to the cosmological
constant. Another important consequence of Lorentz symmetry is stabilization of
the gravitational hierarchy: the cosmological constant (divided by Newton's
constant) does not run as the quartic power of the renormalization group scale,
but instead only logarithmically.
