4pi periodic Josephson current through a Quantum Spin-Hall edge

The helical edge state of a quantum spin-Hall insulator can carry a supercurrent in equilibrium between two superconducting electrodes (separation L, coherence length ?). We calculate the maximum (critical) current Ic that can flow without dissipation along a single edge, going beyond the short-junction restriction L?? of earlier work, and find a dependence on the fermion parity of the ground state when L becomes larger than ?. Fermion-parity conservation doubles the critical current in the low-temperature, long-junction limit, while for a short junction Ic is the same with or without parity constraints. This provides a phase-insensitive, dc signature of the 4?-periodic Josephson effect.