  At present it is generally believed that ``new physics'' effects contribute
to leptonic anomalous magnetic moment, a_l, via quantum loops only and they are
proportional to the squared lepton mass, (m_l)^2. An alternative mechanism for
a contribution by new physics is proposed. It occurs at the tree level and
exhibits a linear rather than quadratic dependence on m_l. This leads to a much
larger sensitivity of a_e to the new physics than was expected so far.
