On the Laplace transform of the Weinberg type sum rules

We consider the Laplace transform of various sum rules of the Weinberg type including the leading non-perturbative effects. We show from the third type Weinberg sum rules that 7.5 to 8.9 <or approx. msub(s)-circumflex/msub(d)-circumflex <or approx. 22.5 to 26.6 for the sum rule scale value M approx.= Msub(rho) of the order of Msub(phi). We also deduce an upper bound to the by-product msub(u)-circumflex msub(d)-circumflex of light quark invariant masses. The first Weinberg sum rule allows us to get a lower bound on fsub(π) and on the A1 coupling to the W boson, while the second sum rule gives an upper bound on the A1 mass (Msub(A1) <or approx. 1.25 GeV). (author)