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

The generalized Ward-Takahashi identity (gWTI) in the pion sector for broken isotopic symmetry is derived and used for the model-independent calculation of the longitudinal form factor $f_{-}$ of the $π_{e3}$ vector vertex. The on-shell $f_{-}$ is found to be proportional to the mass difference of the pions 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 value two times higher than the previous estimate from the quark model. Off-shell form factors are known to be ambiguous because of the gauge dependence and the freedom in the parameterization of the fields. The near-mass-shell $f_{-}$ appears to be an exception, allowing for experimental verification of the consequences of the gWTI. We calculate the near-mass-shell $f_{-}$ using the gWTI and dispersion techniques. The results are discussed in the context of the conservation of vector current (CVC) hypothesis.