$\rho {\gamma }^{*}\to \pi (\rho )$ transition form factors in the perturbative QCD factorization approach

In this paper, we studied the $\rho {\gamma }^{*}\to \pi$ and $\rho {\gamma }^{*}\to \rho$ transition processes and made the calculations for the $\rho \pi$ transition form factor $Q^4{F}_{\rho \pi }(Q^2)$ and the $\rho$-meson electromagnetic form factors, $F_{\mathrm{LL},\mathrm{LT},\mathrm{TT}}(Q^2)$ and $F_{1,2,3}(Q^2)$, by employing the perturbative QCD (PQCD) factorization approach. For the $\rho {\gamma }^{*}\to \pi$ transition, we found that the contribution to form factor $Q^4{F}_{\rho \pi }(Q^2)$ from the term proportional to the distribution amplitude combination ${\varphi }_{\rho }^T(x_1){\varphi }_{\pi }^P(x_2)$ is absolutely dominant, and the PQCD predictions for both the size and the $Q^2$-dependence of this form factor $Q^4{F}_{\rho \pi }(Q^2)$ agree well with those from the extended anti-de Sitter/QCD models or the light-cone QCD sum rule. For the $\rho {\gamma }^{*}\to \rho$ transition and in the region of $Q^2\ge 3{\mathrm{GeV}}^2$, furthermore, we found that the PQCD predictions for the magnitude and their $Q^2$-dependence of the $F_1(Q^2)$ and $F_2(Q^2)$ form factors agree well with those from the QCD sum rule, while the PQCD prediction for $F_3(Q^2)$ is much larger than the one from the QCD sum rule.

Figure 1

The LO diagrams for the $\rho \pi$ transition and $\rho$-meson electromagnetic form factor, with the symbol • representing the virtual photon vertex.

Figure 2

(a) the PQCD predictions for the $Q^2$-dependence of the $\rho \pi$ transition form factor $Q^4{F}_{\rho \pi }(Q^2)$ and (b) the theoretical predictions from AdS/QCD (dashed curve) or the light-cone QCD sum rules (dark region) and from the PQCD factorization approach (solid curve).

Figure 3

The PQCD predictions for $Q^2$ dependence of the $\rho$-meson form factors $F_{{\lambda }_1{\lambda }_2}(Q^2)$ with different polarizations for initial and final states. The short dashed, dots, and dot-dashed curves represent the form factors $F_{\mathrm{TT}}(Q^2)$, $F_{\mathrm{LT}}(Q^2)$, and $F_{\mathrm{LL}}(Q^2)$, respectively.

Figure 4

The PQCD predictions for $Q^2$-dependence of the Lorentz-invariant $\rho$-meson electromagnetic form factors $F_{1,2,3}(Q^2)$. The short dashed, dots, and dot-dashed curves represent the electric [$F_1(Q^2)$], the magnetic [$F_2(Q^2)$], and the quadruple [$F_3(Q^2)$] form factors, respectively.