Missing Mass Matters: Neutrinophilic Scalar in $D$-Meson Decays

A light neutrinophilic scalar $\phi$, which couples predominantly to neutrinos, can be probed through leptonic meson decays for scalar masses $10\mathrm{M}\mathrm{e}\mathrm{V}\lesssim m_{\phi }\lesssim \mathcal{O}(100)\mathrm{M}\mathrm{e}\mathrm{V}$. We study neutrinophilic-scalar emission in $D$-meson decays at BESIII and the future Super Tau-Charm Facility (STCF), focusing on $D_{(s)}^{+}\to e^{+}{\bar{\nu }}_{\alpha }\phi$. Accounting for the $m_{\phi }$-dependent missing-mass squared ($M_{\mathrm{m}\mathrm{i}\mathrm{s}\mathrm{s}}^2$) distribution within the experimentally selected region, we find the resulting $D^{+}$ constraint to be substantially weaker than previous estimates based on the integrated three-body decay rate. Extending the $M_{\mathrm{m}\mathrm{i}\mathrm{s}\mathrm{s}}^2$ coverage provides sensitivity to heavier scalars. With the full kinematic range of $M_{\mathrm{m}\mathrm{i}\mathrm{s}\mathrm{s}}^2$ and an integrated luminosity of $1{\mathrm{a}\mathrm{b}}^{-1}$, the projected STCF sensitivities surpass the existing kaon bound for $m_{\phi }\gtrsim 380\mathrm{M}\mathrm{e}\mathrm{V}$ in the $D^{+}$ decay and $m_{\phi }\gtrsim 310\mathrm{M}\mathrm{e}\mathrm{V}$ in the $D_s^{+}$ decay. We further discuss direct searches for $D_{(s)}^{+}\to {\mu }^{+}{\bar{\nu }}_{\alpha }\phi$ and their potential for probing the neutrinophilic scalar. A complete analytic expression for the three-body decay width is also derived with full dependence on the scalar and charged-lepton masses, including its behavior in the $m_{\phi }\to 0$ and $m_{\mathrm{\ell }}\to 0$ limits.