$\rho$-meson longitudinal leading-twist distribution amplitude revisited and the $D\to \rho$ semileptonic decay

TL;DR

This paper revisits the $ ho$-meson longitudinal leading-twist distribution amplitude using a Dyson-Schwinger equations model and QCD sum rules, then applies it to calculate form factors and decay widths for $D o ho$ semileptonic decays, providing improved theoretical predictions.

Contribution

It introduces a DSE-based model for the $ ho$-meson DA, refines the moments calculation with improved sum rules, and applies these to predict semileptonic decay form factors and branching fractions.

Findings

Calculated $ ho$-meson DA moments at various scales.

Determined $D o ho$ transition form factors at large recoil.

Predicted branching fractions for $D o ho u o ho o ext{leptons}$ decays.

Abstract

Motivated by our previous work [Phys. Rev. D \textbf{104}, no.1, 016021 (2021)] on pionic leading-twist distribution amplitude (DA), we revisit $\rho$-meson leading-twist longitudinal DA $\phi_{2;\rho}^\|(x,\mu)$ in this paper. A model proposed by Chang based on the Dyson-Schwinger equations (DSEs) is adopted to describe the behavior of $\phi_{2;\rho}^\|(x,\mu)$. On the other hand, the $\xi$-moments of $\phi_{2;\rho}^\|(x,\mu)$ are calculated with the QCD sum rules in the framework of the background field theory. The sum rule formula for those moments are improved. More accurate values for the first five nonzero $\xi$-moments at typical scale $\mu =1, 1.4, 2, 3~{\rm GeV}$ are given, e.g., at $\mu = 1~{\rm GeV}$, \modi{$\langle\xi^2\rangle_{2;\rho}^\| = 0.220(6) $, $\langle\xi^4\rangle_{2;\rho}^\| = 0.103(4)$, $\langle\xi^6\rangle_{2;\rho}^\| = 0.066(5)$, $\langle\xi^8\rangle_{2;\rho}^\| = 0.046(4)$ and $\langle\xi^{10}\rangle_{2;\rho}^\| = 0.035(3)$}. By fitting those values with the least squares method, the DSE model for $\phi_{2;\rho}^\|(x,\mu)$ is determined. By taking the left-handed current light-cone sum rule approach, we get the transition form factor at large recoil region, {\it i.e.} $A_1(0) = 0.498^{+0.014}_{-0.012}$, $A_2(0)=0.460^{+0.055}_{-0.047}$, $V(0) = 0.800^{+0.015}_{-0.014}$, and the ratio $r_2 = 0.923^{+0.133}_{-0.119}$, $r_V = 1.607^{+0.071}_{-0.071}$. After making the extrapolation with a rapidly converging series based on $z(t)$-expansion, we present the decay width for the semileptonic decays $D\to\rho\ell^+\nu_\ell$. Finally, the branching fractions are $\mathcal{B}(D^0\to \rho^- e^+ \nu_e) = 1.889^{+0.176}_{-0.170}\pm 0.005$, $\mathcal{B}(D^+ \to \rho^0 e^+ \nu_e) = 2.380^{+0.221}_{-0.214}\pm 0.012$, $\mathcal{B}(D^0\to \rho^- \mu^+ \nu_\mu) = 1.881^{+0.174}_{-0.168}\pm 0.005$, $\mathcal{B}(D^+ \to \rho^0 \mu^+ \nu_\mu) =2.369^{+0.219}_{-0.211}\pm 0.011$.