Chiral symmetry, the angular content of the vector current in QED and QCD, and the holographic description of hadrons

We perform a general chiral symmetry and unitarity based analysis of a local process of the fermion-antifermion creation from the vacuum by a high-energy photon as well as an explicit partial wave analysis of the vector current in QED and QCD. It turns out that such a local process proceeds necessarily via a certain superposition of the $S$- and $D$-wave contributions. These constraints from chiral symmetry and unitarity are confronted then with the well-known theoretical and experimental results on $e^+e^-\to\gamma\to e^+e^-$, $e^+e^-\to\gamma\to \mu^+\mu^-$, and $e^+e^-\to\gamma\to q\bar{q}$ in the ultrarelativistic limit. It is shown that these well-known results are consistent with the $S+D$-wave structure of the vertex and are inconsistent with the pure $S$-wave interpretation of the vertex. Then a free quark loop in the $1^{--}$ channel, representing the leading term in the Operator Product Expansion, contains both $S$-wave and $D$-wave contributions. This fact rules out the possibility that there is only one radial trajectory for the $\rho$-mesons with the fixed $S$-wave content. It also implies that all holographic models that assume a pure $S$-wave content of the $\rho$-meson have to fail to satisfy the matching conditions at the ultraviolet border $z=0$.