The Effective Field Theory of Large-Scale Structure in the presence of
  Massive Neutrinos
  We develop a formalism to analytically describe the clustering of matter in
the mildly non-linear regime in the presence of massive neutrinos. Neutrinos,
whose free streaming wavenumber ($k_{\rm fs}$) is typically longer than the
non-linear scale ($k_{\rm NL}$) are described by a Boltzmann equation coupled
to the effective fluid-like equations that describe dark matter. We solve the
equations expanding in the neutrino density fraction $(f_\nu)$ and in $k/
k_{\rm NL}$, and add suitable counterterms to renormalize the theory. This
allows us to describe the contribution of short distances to long-distance
observables. Equivalently, we construct an effective Boltzmann equation where
we add additional terms whose coefficients renormalize the contribution from
short-distance physics. We argue that neutrinos with $k_{\rm fs}\gtrsim k_{\rm
NL}$ require an additional counterterm similar to the speed of sound ($c_s$)
for dark matter. We compute the one-loop total-matter power spectrum, and find
that it is roughly equal to $16f_\nu$ times the dark matter one for $k$'s
larger that the typical $k_{\rm fs}$. It is about half of that for smaller
$k$'s. The leading contribution results from the back-reaction of the neutrinos
on the dynamics of the dark matter. The counterterms contribute in a
hierarchical way: the leading ones can either be computed in terms of $c_s$, or
can be accounted for by shifting $c_s$ by an amount proportional to $f_\nu$.
