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We develop a formalism to analytically describe the clustering of matter inthe mildly non-linear regime in the presence of massive neutrinos. Neutrinos,whose free streaming wavenumber ($k_{\rm fs}$) is typically longer than thenon-linear scale ($k_{\rm NL}$) are described by a Boltzmann equation coupledto the effective fluid-like equations that describe dark matter. We solve theequations expanding in the neutrino density fraction $(f_\nu)$ and in $k/k_{\rm NL}$, and add suitable counterterms to renormalize the theory. Thisallows us to describe the contribution of short distances to long-distanceobservables. Equivalently, we construct an effective Boltzmann equation wherewe add additional terms whose coefficients renormalize the contribution fromshort-distance physics. We argue that neutrinos with $k_{\rm fs}\gtrsim k_{\rmNL}$ 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 findthat it is roughly equal to $16f_\nu$ times the dark matter one for $k$'slarger 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 neutrinoson the dynamics of the dark matter. The counterterms contribute in ahierarchical way: the leading ones can either be computed in terms of $c_s$, orcan be accounted for by shifting $c_s$ by an amount proportional to $f_\nu$.

The Effective Field Theory of Large-Scale Structure in the presence of Massive Neutrinos