= Isotropic zero-sound model
{title2=$1+F_0^s[1-\frac{s}{2}\log\frac{s+1}{s-1}]=0$}
Consider a neutral, isotropic, unpolarized three-dimensional spin-one-half <Fermi liquid>, with only $F_0^s>0$ nonzero. A longitudinal density wave with <angular frequency> $\omega$ and <wavevector> magnitude $q$ has $s=\omega/(qv_F)=c_0/v_F$. The collisionless angular equation for the energy displacement $\nu(u)$ of the <Fermi surface> is
$$
(s-u)\nu(u)=uF_0^s\langle\nu\rangle,\qquad \langle\nu\rangle=\frac12\int_{-1}^1\nu(u)\,du,\qquad u=\cos\theta.
$$
Averaging gives
$$
1=\frac{F_0^s}{2}\int_{-1}^{1}\frac{u}{s-u}\,du=F_0^s\left[\frac{s}{2}\log\frac{s+1}{s-1}-1\right].
$$
The bracket decreases from infinity to zero on $1<s<\infty$: each $F_0^s>0$ gives one undamped root above the small-$q$ <particle-hole continuum>.<Oscillations in a Fermi Liquid by Lev Davidovich Landau>[Section 1, equations (11)–(15), pp.103–104]{ref}
For an original example, $F_0^s=2$ gives $s=1.16474145$. Since $F_1^s=0$ here, the <Principle of Galilean relativity> gives $m^*=m$ and taking the hydrodynamic limit before $T\to0$ gives $c_1/v_F=\sqrt{(1+F_0^s)/3}=1$. Stipulating $v_F=50\ \mathrm{m\,s^{-1}}$ gives $c_0\simeq58.24\ \mathrm{m\,s^{-1}}$ and $c_1=50\ \mathrm{m\,s^{-1}}$. These are model choices, not <helium-3> measurements. At large $F_0^s$, expanding the logarithm gives $s\sim\sqrt{F_0^s/3}$; for small positive $F_0^s$, the root approaches $1$ exponentially closely, so numerical resolution matters. A charged <Electron> liquid needs its long-range interaction included; this neutral model is not its long-wavelength density mode.
\Image[/fermi-liquid-zero-sound-root-and-collision-regimes.png]
{title=Collisionless zero-sound root and the first-sound collision condition}
{description=Original three-dimensional $F_0^s$-only model. <Zero sound> and <first sound> speeds refer to their respective limits, rather than two simultaneous solutions at a fixed collision rate. The lower panel stipulates $\omega\tau=(T/T_\times)^{-2}$ at fixed <angular frequency>; its crossover is not measured <helium-3> data.}
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