Spin-polarized tunneling into helical edge states: asymmetry and conductances

We consider tunneling from the spin-polarized tip into the Luttinger liquid edge state of quantum spin Hall system. This problem arose in context of the spin and charge fractionalization of an injected electron. Renormalization of the dc conductances of the system is calculated in the fermionic approach and scattering states formalism. In lowest order of tunneling amplitude we confirm previous results for the scaling dependence of conductances. Going beyond the lowest order we show that interaction affects not only the total tunneling rate, but also the asymmetry of the injected current. The helical edge state forbids the backscattering, which leads to the possibility of two stable fixed points in renormalization group sense, in contrast to the Y-junction between the usual quantum wires.