  Recent experiments reported that quantum Hall chiral edge state-mediated
Josephson junctions (chiral Josephson junctions) could exhibit Fraunhofer
oscillations with a periodicity of either $h/e$ [Vignaud \textit{et
al}.,~Nature~(2023)] or $h/2e$ [Amet \textit{et
al}.,~Science~\textbf{352}~966~(2016)]. While the $h/e$-periodic component of
the supercurrent had been anticipated theoretically before, the emergence of
the $h/2e$-periodicity is still not fully understood. In this work, we show
that the chiral edge states coupled to the superconductors become chiral
Andreev edge states. In short junctions, the coupling of the chiral Andreev
edge states can cause the $h/2e$-magnetic flux periodicity. Our theory resolves
the long-standing puzzle concerning the appearance of the $h/2e$-periodicity in
chiral Josephson junctions. Furthermore, we explain that when the chiral
Andreev edge state couple, a pair of localized Majorana modes appear at the
ends of the Josephson junction, which are robust and independent of the phase
difference between the two superconductors. As the $h/2e$-periodicity and the
Majorana zero modes have the same physical origin, the Fraunhofer oscillation
period can be used to identify the regime with Majorana zero modes.
