The axial-vector $a_1(1260)$-meson longitudinal twist-2 distribution amplitude $phi_{2;a_1}^| (x,mu )$ within the framework of QCD sum rules under the background field theory is investigated. By considering the vacuum condensates up to dimension-six and the perturbative part up to next-to-leading order QCD corrections, the moments at initial scale $mu_0=1~{rm GeV}$ are $langle xi_{2;a_1}^{|;2}rangle |_{mu_0} = 0.210 pm 0.018$, $langle xi_{2;a_1}^{|;4}rangle |_{mu_0} = 0.091 pm 0.007$, and $langle xi_{2;a_1}^{|;6}rangle |_{mu_0} = 0.052 pm 0.004$ respectively. Secondly, the transition form factors (TFFs) for $Dto a_1(1260)$ under the light-cone sum rules are given. When taking squared momentum transfer to zero, we obtain $ A(0) = 0.130_{ - 0.015}^{ + 0.013}$, $V_1(0) = 1.899_{ - 0.127}^{ + 0.119}$, $V_2(0) = 0.211_{ - 0.020}^{ + 0.018}$, and $V_0(0) = 0.235_{ - 0.025}^{ + 0.026}$. With the extrapolated TFFs for the physically allowable region, the differential decay widths and total branching ratios for the processes $D^{0(+)} to a_1^{-(0)}(1260)ell^+ u_ell$ can be obtained, i.e. ${cal B}(D^0to a_1^-(1260) e^+ u_e) = (5.421_{-0.697}^{+0.702}) times 10^{-5}$, ${cal B}(D^+to a_1^0(1260) e^+ u_e) = (6.875_{-0.884}^{+0.890}) times 10^{-5}$, ${cal B}(D^0to a_1^-(1260) mu^+ u_mu)=(4.864_{-0.641}^{+0.647}) times 10^{-5}$, ${cal B}(D^+ to a_1^0(1260) mu^+ u_mu)=(6.169_{-0.821}^{+0.813}) times 10^{-5}$.
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