We perform a dispersive analysis of the $omegapi$ electromagnetic transition form factor, using as input the discontinuity provided by unitarity below the $omegapi$ threshold and including for the first time experimental data on the modulus measured
from $e^+e^-toomegapi^0$ at higher energies. The input leads to stringent parameterization-free constraints on the modulus of the form factor below the $omegapi$ threshold, which are in disagreement with some experimental values measured from $omegato pi^0gamma^*$ decay. We discuss the dependence on the input parameters in the unitarity relation, using for illustration an $N/D$ formalism for the P partial wave of the scattering process $omegapi to pipi$, improved by a simple prescription which simulates the rescattering in the crossed channels. Our results confirm the existence of a conflict between experimental data and theoretical calculations of the $omegapi$ form factor in the region around 0.6 GeV and bring further arguments in support of renewed experimental efforts to measure more precisely the $omegatopi^0gamma^*$ decay.
Motivated by the discrepancies noted recently between the theoretical calculations of the electromagnetic $omegapi$ form factor and certain experimental data, we investigate this form factor using analyticity and unitarity in a framework known as the
method of unitarity bounds.We use a QCD correlator computed on the spacelike axis by operator product expansion and perturbative QCD as input, and exploit unitarity and the positivity of its spectral function, including the two-pion contribution that can be reliably calculated using high-precision data on the pion form factor. From this information, we derive upper and lower bounds on the modulus of the $omegapi$ form factor in the elastic region. The results provide a significant check on those obtained with standard dispersion relations, confirming the existence of a disagreement with experimental data in the region around 0.6 GeV.
Starting from the divergent character of the perturbative expansions in QCD and using the technique of series acceleration by the conformal mappings of the Borel plane, I define a novel, non-power perturbative expansion for the Adler function, which
simultaneously implements renormalization-group summation and has a tamed large-order behaviour. The new expansion functions, which replace the standard powers of the coupling, are singular at the origin of the coupling plane and have divergent perturbative expansions, resembling the expanded function itself. Confronting the new perturbative expansions with the standard ones on specific models investigated recently in the literature, I show that they approximate in an impressive way the exact Adler function and the spectral function moments. Applied to the $tau$ hadronic width, the contour-improved and the renormalization-group summed non-power expansions in the ${overline{rm MS}}$ scheme lead to the prediction $alpha_s(M_tau^2)= 0.3192~^{+ 0.0167}_{-0.0126}$, which translates to $alpha_s(M_Z^2)= 0.1184~^{+0.0020}_{-0.0016}$.
We investigate the scalar K pi form factor at low energies by the method of unitarity bounds adapted so as to include information on the phase and modulus along the elastic region of the unitarity cut. Using at input the values of the form factor at
t=0 and the Callan-Treiman point, we obtain stringent constraints on the slope and curvature parameters of the Taylor expansion at the origin. Also, we predict a quite narrow range for the higher order ChPT corrections at the second Callan-Treiman point.
