Timelike scalar and vector form factors are recalculated using chiral perturbation theory and dispersion theory. It is shown that chiral perturbation theory at the one-loop level violates the final-state theorem (i.e., unitarity). In order to satisfy this theorem, chiral perturbation theory should be applied to the inverse of the form factor whose result is shown to be equivalent to the Pade-approximant method.

Using the forward dispersion relations for nn scattering, we show that the coefficients of the derivative quartic terms in the chiral Lagrangian are positive. The Skyrme term is mainly given by the p-meson contribution to the I = 1 total cross section and gives a proton mass of 1.27 GeV. The non-Skyrme term is found of comparable strength from the I = 0 S-wave n w data.The standard minimal SU (3) x SU (3) nonlinear effective Lagrangian for the pseudoscalar-meson octet is known to reproduce all the current-algebra results for processes involving soft mesons in an elegant manner.'s2 For hardmeson processes, one expects on general grounds deviations from the current-algebra prediction as found in K14 form factor^,^ q + 37r decay rates,4 and the S-wave I = 0 7r7r scattering ~e n g t h .~?~ Thus correction terms must be present to account for the experimental data.There are two interrelated approaches to account for the correction terms. The first one consists in using currentalgebra results as low-energy theorems which are supplemented by the use of the unitarity and analyticity to calculate the correction term^.^,^ This is a nonperturbative approach, and good agreement with experimental data is obt a i~~e d .~.~The second method involves the introduction of terms involving higher powers of field derivatives to the minimal Lagrangian; the strength of these terms are new parameters which must be determined either from independent experiments or from the matrix elements given by the first m e t h~d .~ Because the effective-Lagrangian method is a power-series expansion of the matrix element in terms of the invariant variables, it can only account for the singularities induced by the unitarity condition in a perturbative way.6In the region where the power-series expansion is valid3 (e.g., below the unitarity cut), the effective-Lagrangian method should give good results, as it was recently shown that the inclusion of the symmetry-breaking terms with derivative coupling is needed7 to account for fK / f , and the KI3 Callan-Treiman relation. (It has also recently been shown that the derivative quartic terms are needed to account for the possible discrepancy with data for K -3.rr rate.8 Unfortunately, this is not the whole story because the calculated K -27r and K -37r are purely real, while the physical K--2a amplitude has a phase of 40-45" which cannot be neglected. This illustrates some difficulty of using the effective Lagrangian above the unitarity cut.') From this standpoint we see that similar terms in higher powers of derivatives (quartic terms, etc.) must also be present for pion-pion scattering. A possible source for these terms may be found in the coupling of the pseudoscalar meson with heavy particles (vector, scalar mesons p , u . . .), which in the limit of low-external momenta can be eliminated from the theory, leaving a meson Lagrangian with higher powers of field derivatives. For example, in the u model,9 by expanding the u propagator in powers of k2/rnV2 ( k2 being a typical external momentum), one finds t...