The possible existence of maximal or near–maximal lepton mixing constitutes an intriguing challenge for fundamental theories of flavour .
We study the phenomenological consequences of maximal and near–maximal mixing of the electron neutrino with other ( x =tau and/or muon ) neutrinos .
We describe the deviations from maximal mixing in terms of a parameter \epsilon \equiv 1 - 2 \sin ^ { 2 } \theta _ { ex } and quantify the present experimental status for | \epsilon| < 0.3 .
We show that both probabilities and observables depend on \epsilon quadratically when effects are due to vacuum oscillations and they depend on \epsilon linearly if matter effects dominate .
The most important information on \nu _ { e } –mixing comes from solar neutrino experiments .
We find that the global analysis of solar neutrino data allows maximal mixing with confidence level better than 99 % for 10 ^ { -8 } eV ^ { 2 } \mathrel { \hbox to 0.0 pt { \lower 4.0 pt \hbox { $ \sim$ } } \raise 1.0 pt \hbox { $ < $ } % } \Delta m ^ { 2 } \mathrel { \hbox to 0.0 pt { \lower 4.0 pt \hbox { $ \sim$ } } \raise 1.0 pt% \hbox { $ < $ } } 2 \times 10 ^ { -7 } eV ^ { 2 } .
In the mass ranges \Delta m ^ { 2 } \mathrel { \hbox to 0.0 pt { \lower 4.0 pt \hbox { $ \sim$ } } \raise 1.0 pt% \hbox { $ > $ } } 1.5 \times 10 ^ { -5 } eV ^ { 2 } and 4 \times 10 ^ { -10 } eV ^ { 2 } \mathrel { \hbox to 0.0 pt { \lower 4.0 pt \hbox { $ \sim$ } } \raise 1.0 pt \hbox { $ < $ } % } \Delta m ^ { 2 } \mathrel { \hbox to 0.0 pt { \lower 4.0 pt \hbox { $ \sim$ } } \raise 1.0 pt% \hbox { $ < $ } } 2 \times 10 ^ { -7 } eV ^ { 2 } the full interval | \epsilon| < 0.3 is allowed within \sim 4 \sigma ( 99.995 % CL ) .
We suggest ways to measure \epsilon in future experiments .
The observable that is most sensitive to \epsilon is the rate [ NC ] / [ CC ] in combination with the Day–Night asymmetry in the SNO detector .
With theoretical and statistical uncertainties , the expected accuracy after 5 years is \Delta \epsilon \sim 0.07 .
We also discuss the effects of maximal and near–maximal \nu _ { e } -mixing in atmospheric neutrinos , supernova neutrinos , and neutrinoless double beta decay .