Geometric local epsilon factors

Inspired by the work of Laumon on $\epsilon$-factors and by Deligne’s $1974$ letter to Serre, we give an explicit cohomological definition of $\epsilon$-factors for $\mathrm{\ell }$-adic Galois representations over henselian discrete valuation fields of positive equicharacteristic $p\mathrm{\ne }\mathrm{\ell }$, with (not necessarily finite) perfect residue fields. These geometric local $\epsilon$-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 $\mathrm{\ell }$-adic sheaves on a curve over a perfect field of characteristic $p$.