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Using 2.92 fb$^{-1}$ of electron-positron annihilation data collected at acenter-of-mass energy of $\sqrt{s}= 3.773$ GeV with the BESIII detector, wepresent an improved measurement of the branching fraction $\mathcal{B}(D^+ \to\omega e^+ \nu_{e}) = (1.63\pm0.11\pm0.08)\times 10^{-3}$. The parametersdefining the corresponding hadronic form factor ratios at zero momentumtransfer are determined for the first time, we measure them to be $r_V =1.24\pm0.09\pm0.06$ and $r_2 = 1.06\pm0.15 \pm 0.05$. The first and seconduncertainties are statistical and systematic, respectively. We also search forthe decay $D^+ \to \phi e^+ \nu_{e}$. An improved upper limit $\mathcal{B}(D^+\to \phi e^+ \nu_{e}) < 1.3 \times 10^{-5}$ is set at 90\% confidence level.

Measurement of the form factors in the decay $D^+ \to \omega e^+ \nu_{e}$ and search for the decay $D^+ \to \phi e^+ \nu_{e}$