On the second Gegenbauer moment of $\rho$-meson distribution amplitude

Abstract

Using the soft pion theorem, crossing, and the dispersion relations for the two pion distribution amplitude ($2\pi$DA) we argue that the second Gegenbauer moment the $\rho$-meson DA ($a_2^{(\rho)}$) most probably is negative. This result is at variance with the majority of the model calculations for $a_2^{(\rho)}$. Using the instanton theory of the QCD vacuum, we computed $a_2^{(\rho)}$ at a low normalisation point and obtain for the ratio $ a_2^{(\rho)}/M_3^{(\pi)}$ {\it definitely negative value} in the range of $a_2^{(\rho)}/M_3^{(\pi)}\in [-2, -1]$. The range of values corresponds to a generous variation of the parameters of the instanton vacuum. The value of the second Gegenbauer moment of pion DA is positive in the whole range and is compatible with its the most advanced lattice measurement. It seems that the topologically non-trivial field configurations in the QCD vacuum (instantons) lead to qualitatively different shapes of the pion and the $\rho$-meson DAs.