QCD Correlation Functions and the Shape of $K_{\mathrm{\ell }3}$ Form Factors

Bounds on the expansion coefficients of the strangeness changing $K\pi$ form factors were derived recently from analyticity and unitarity, using as input suitable correlation functions calculated by perturbative QCD in the Euclidian region. We investigate two types of invariant amplitudes and their corresponding dispersion relations, and show that they lead to similar results for the shape of the vector and scalar $K_{\mathrm{\ell }3}$ form factors.