Day-night effect in solar neutrino oscillations with three flavors

We investigate the effects of a nonzero leptonic mixing angle θ₁₃ on the solar neutrino day-night asymmetry. Using a constant matter density profile for the Earth and well-motivated approximations, we derive analytical expressions for the νₑ survival probabilities for solar neutrinos arriving directly at the detector and for solar neutrinos which have passed through the Earth. Furthermore, we numerically study the effects of a nonzero θ₁₃ on the day-night asymmetry at detectors and find that they are small. Finally, we show that if the uncertainties in the parameters θ₁₂ and Δm² as well as the uncertainty in the day-night asymmetry itself were much smaller than they are today, this effect could, in principle, be used to determine θ₁₃.