Beyond leading twist: $\rho$ meson decay constants and distribution amplitudes in a self-consistent light-front quark model

In this study, we present a comprehensive analysis of decay constants and chiral-even and chiral-odd distribution amplitudes (DAs) up to twist 4 for the $\rho$ meson in the standard light-front quark model (LFQM) based on the Bakamjian-Thomas (BT) construction. For the $\rho$ meson, which possesses both longitudinal $(h=0)$ and transverse $(h=\pm 1)$ polarizations, two types of decay constants, $f_{\rho}^{\parallel}$ and $f_{\rho}^{\perp}$, arises accordingly. We demonstrate that these decay constants can be self-consistently extracted from both local ($z^\mu=0$) and nonlocal ($z^\mu\neq 0$) matrix elements $\langle 0 | \bar{q}(z)\, \Gamma\, q(-z) | \rho(P,h) \rangle$, with $\Gamma = (\gamma^\mu, \sigma^{\mu\nu}, \gamma^\mu\gamma_5, \mathbf{1})$, in a manner independent of current components, polarizations, and reference frames. In particular, we emphasize the role of nonlocal matrix elements involving axial-vector and scalar currents, where mixing between $f_\rho^{\parallel}$ and $f_\rho^{\perp}$ occurs. We show that this mixing is consistently resolved through the BT construction, ensuring the proper extraction of these decay constants. Additionally, we investigate the structure of chiral-even DAs ($\phi_{2;\mathrm{V}}^\parallel, \phi_{3;\mathrm{V}}^\parallel, \psi_{3;\mathrm{A}}^\perp, \phi_{4;\mathrm{V}}^\parallel$) and chiral-odd DAs ($\phi_{2;\mathrm{T}}^\perp, \phi_{3;\mathrm{T}}^\perp, \psi_{3;\mathrm{S}}^\parallel, \phi_{4;\mathrm{T}}^\perp$) beyond leading twists, and present their corresponding $\xi$-moments and Gegenbauer moments. These results provide deeper insight into the nonperturbative structure of vector mesons and demonstrate the robustness and self-consistency of the LFQM based on the BT framework.