\(\pi_{e 3}\) form factor \(f_-\) near the mass shell (Q889070)

Summary: 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 \(\pi_{e3}\) 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.