  We revisit the idea of self-tuning the observed cosmological constant to a
vanishing value and promote it to a selection criterion of brane-world models,
in which our Universe is described by a 3-brane embedded in a 5d bulk. As a
concrete setup, we consider a bulk scalar field $\phi$ described by a general
Lagrangian $P(X,\phi)$ with $X=-(\partial\phi)^2/2$. By requiring that the
model enforces the 4d curvature of the maximally symmetric 3-brane world-volume
to vanish independently of the 4d effective vacuum energy, only two
possibilities remain: one with a canonical bulk kinetic term and the other with
an unconventional bulk kinetic term similar to a Cuscuton field. Further
demanding the absence of bulk singularity, the latter is selected as a unique
possibility within the class of models. At the background level, the solution
can accommodate any warp factor profile free from bulk singularity and with a
finite effective 4d Planck mass. In a cosmological context, our solution would
describe our (almost) flat Universe at late times, with a bulk warp factor
profile expected to be determined by the evolution of the Universe before
dilution of the matter fields by cosmic expansion. Eventually, a simple
analysis is performed in the bulk showing no obvious instability around the
background solution. A full stability analysis taking into account brane
bending modes is nevertheless necessary and left for future work.
