= Landau parameter
{c}
{title2=$F_\ell^s,F_\ell^a$}
For a three-dimensional isotropic unpolarized spin-one-half <Fermi liquid>, define $f^s=(f_{\uparrow\uparrow}+f_{\uparrow\downarrow})/2$ and $f^a=(f_{\uparrow\uparrow}-f_{\uparrow\downarrow})/2$. Dimensionless <Landau parameters> use the <spin-summed density of states> per <volume>:
$$
N^*(0)=\frac{m^*p_F}{\pi^2\hbar^3},\qquad N^*(0)f^{s,a}(\cos\theta)=\sum_{\ell=0}^{\infty}F_\ell^{s,a}P_\ell(\cos\theta).
$$
$P_\ell$ is a <Legendre polynomial>; $s,a$ label spin symmetry. No extra $(2\ell+1)$ multiplies this expansion.<Renormalization Group Approach to Interacting Fermions by Ramamurti Shankar>[Section VIII, angular interaction function; normalization stated explicitly here]{ref} The superscript $s$ on $F_0^s$ is unrelated to the dimensionless wave speed $s$ below.
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