  Inspired by the work of Laumon on $\varepsilon$-factors and by Deligne's
$1974$ letter to Serre, we give an explicit cohomological definition of
$\varepsilon$-factors for $\ell$-adic Galois representations over henselian
discrete valuation fields of positive equicharacteristic $p \neq \ell$, with
(not necessarily finite) perfect residue fields. These geometric local
$\varepsilon$-factors are completely characterized by an explicit list of
purely local properties, such as an induction formula and the compatibility
with geometric class field theory in rank $1$, and satisfy a product formula
for $\ell$-adic sheaves on a curve over a perfect field of characteristic $p$.
