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The generalized Ward-Takahashi identity (gWTI) in the pion sector for brokenisotopic symmetry is derived and used for the model-independent calculation ofthe longitudinal form factor $f_{-}$ of the $\pi_{e3}$ vector vertex. Theon-shell $f_{-}$ is found to be proportional to the mass difference of thepions and the difference between the vector isospin $T = 1$ and scalar isospin$ T = 2 $ pion radii. A numerical estimate of the form factor yields a valuetwo times higher than the previous estimate from the quark model. Off-shellform factors are known to be ambiguous because of the gauge dependence and thefreedom in the parameterization of the fields. The near-mass-shell $f_{-}$appears to be an exception, allowing for experimental verification of theconsequences of the gWTI. We calculate the near-mass-shell $f_{-}$ using thegWTI and dispersion techniques. The results are discussed in the context of theconservation of vector current (CVC) hypothesis.

$\pi_{e3}$ form factor $f_{-}$ near the mass shell