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A sterile neutrino in the $3+1$ scheme, where the sterile accounts forneutrino anomalies not explained solely by the weak active neutrinos, arises asa natural source for the breaking of the $\mu-\tau$ symmetry suggested byoscillation neutrino data. We explore the predictions for the Dirac CP phasesin this scenario, with and without sterile neutrino decay, and show thatcurrent limits on $\delta_{CP}$ suggest a normal hierarchy and a lightestneutrino scale below 0.1~eV as the most plausible explanation for that, whenMajorana phases are null. Other Dirac phases turn out to be non zero as well.

Dirac CP phases in a 3+1 neutrino scenario with $\mu-\tau$ symmetry