We discuss Big Bang Nucleosynthesis constraints on maximal \nu _ { \mu } \leftrightarrow \nu _ { s } mixing .
Vacuum \nu _ { \mu } \leftrightarrow \nu _ { s } oscillation has been proposed as one possible explanation of the Super Kamiokande atmospheric neutrino data .
Based on the most recent primordial abundance measurements , we find that the effective number of neutrino species for Big Bang Nucleosynthesis ( BBN ) is { { { { N _ { \nu } \mathrel { \mathchoice { \lower 3.6 pt \vbox { \halign { \cr } $ \displaystyle \hfil < % $ \cr$ \displaystyle \hfil \sim$ } } } { \lower 3.6 pt \vbox { \halign { \cr } $ \textstyle \hfil% < $ \cr$ \textstyle \hfil \sim$ } } } { \lower 3.6 pt \vbox { \halign { \cr } $ \scriptstyle \hfil% < $ \cr$ \scriptstyle \hfil \sim$ } } } { \lower 3.6 pt \vbox { \halign { \cr } $% \scriptscriptstyle \hfil < $ \cr$ \scriptscriptstyle \hfil \sim$ } } } } 3.3 .
Assuming that all three active neutrinos are light ( with masses \ll 1 MeV ) , we examine BBN constraints on \nu _ { \mu } \leftrightarrow \nu _ { s } mixing in two scenarios : ( 1 ) a negligible lepton asymmetry ( the standard picture ) ; ( 2 ) the presence of a large lepton asymmetry which has resulted from an amplification by \nu _ { \tau } \leftrightarrow \nu _ { s ^ { \prime } } mixing ( \nu _ { s ^ { \prime } } being \nu _ { s } or another sterile neutrino species ) .
The latter scenario has been proposed recently to reconcile the BBN constraints and large-angle \nu _ { \mu } \leftrightarrow \nu _ { s } mixing .
We find that the large-angle \nu _ { \mu } \leftrightarrow \nu _ { s } mixing in the first scenario , which would yield N _ { \nu } \approx 4 , is ruled out as an explanation of the Super Kamiokande data .
It is conceivably possible for the \nu _ { \mu } \leftrightarrow \nu _ { s } solution to evade BBN bounds in the second scenario , but only if 200 eV { { { { { { { { { } ^ { 2 } \mathrel { \mathchoice { \lower 3.6 pt \vbox { \halign { \cr } $ \displaystyle \hfil < $% \cr$ \displaystyle \hfil \sim$ } } } { \lower 3.6 pt \vbox { \halign { \cr } $ \textstyle \hfil < % $ \cr$ \textstyle \hfil \sim$ } } } { \lower 3.6 pt \vbox { \halign { \cr } $ \scriptstyle \hfil < % $ \cr$ \scriptstyle \hfil \sim$ } } } { \lower 3.6 pt \vbox { \halign { \cr } $% \scriptscriptstyle \hfil < $ \cr$ \scriptscriptstyle \hfil \sim$ } } } } m ^ { 2 } _ { \nu _ { \tau } % } - m ^ { 2 } _ { \nu _ { s ^ { \prime } } } \mathrel { \mathchoice { \lower 3.6 pt \vbox { \halign { \cr } $% \displaystyle \hfil < $ \cr$ \displaystyle \hfil \sim$ } } } { \lower 3.6 pt \vbox { \halign { % \cr } $ \textstyle \hfil < $ \cr$ \textstyle \hfil \sim$ } } } { \lower 3.6 pt \vbox { \halign { % \cr } $ \scriptstyle \hfil < $ \cr$ \scriptstyle \hfil \sim$ } } } { \lower 3.6 pt \vbox { % \halign { \cr } $ \scriptscriptstyle \hfil < $ \cr$ \scriptscriptstyle \hfil \sim$ } } } } 10 ^ { 4 } eV ^ { 2 } is satisfied , and if \nu _ { \tau } decays non-radiatively with a lifetime { { { { \mathrel { \mathchoice { \lower 3.6 pt \vbox { \halign { \cr } $ \displaystyle \hfil < $ \cr$% \displaystyle \hfil \sim$ } } } { \lower 3.6 pt \vbox { \halign { \cr } $ \textstyle \hfil < $ \cr% $ \textstyle \hfil \sim$ } } } { \lower 3.6 pt \vbox { \halign { \cr } $ \scriptstyle \hfil < $ \cr% $ \scriptstyle \hfil \sim$ } } } { \lower 3.6 pt \vbox { \halign { \cr } $ \scriptscriptstyle% \hfil < $ \cr$ \scriptscriptstyle \hfil \sim$ } } } } 10 ^ { 3 } years .
This mass-squared difference implies 15 eV { { { { { { { { \mathrel { \mathchoice { \lower 3.6 pt \vbox { \halign { \cr } $ \displaystyle \hfil < $ \cr$% \displaystyle \hfil \sim$ } } } { \lower 3.6 pt \vbox { \halign { \cr } $ \textstyle \hfil < $ \cr% $ \textstyle \hfil \sim$ } } } { \lower 3.6 pt \vbox { \halign { \cr } $ \scriptstyle \hfil < $ \cr% $ \scriptstyle \hfil \sim$ } } } { \lower 3.6 pt \vbox { \halign { \cr } $ \scriptscriptstyle% \hfil < $ \cr$ \scriptscriptstyle \hfil \sim$ } } } } m _ { \nu _ { \tau } } \mathrel { \mathchoice { % \lower 3.6 pt \vbox { \halign { \cr } $ \displaystyle \hfil < $ \cr$ \displaystyle \hfil \sim$% } } } { \lower 3.6 pt \vbox { \halign { \cr } $ \textstyle \hfil < $ \cr$ \textstyle \hfil \sim$ } } % } { \lower 3.6 pt \vbox { \halign { \cr } $ \scriptstyle \hfil < $ \cr$ \scriptstyle \hfil \sim$% } } } { \lower 3.6 pt \vbox { \halign { \cr } $ \scriptscriptstyle \hfil < $ \cr$% \scriptscriptstyle \hfil \sim$ } } } } 100 eV if \nu _ { s ^ { \prime } } is much lighter than \nu _ { \tau } .
We conclude that maximal ( or near maximal ) \nu _ { \mu } \leftrightarrow \nu _ { \tau } mixing is a more likely explanation of the Super Kamiokande data .