Probing the X(3700) through two-photon $D \bar{D}$ production in ultraperipheral collisions and at the EIC

We investigate whether photon-induced production in ultraperipheral heavy-ion collisions and at the EIC can provide a new environment to search for the scalar $D \bar{D}$ molecular candidate usually denoted as $X(3700)$. We calculate the tree-level amplitudes for $\gamma\gamma\to D^+ D^-$ and $\gamma\gamma\to D^0 \bar{D^0}$, dress the outgoing pair with a coupled-channel Bethe-Salpeter interaction in the $(D^+ D^- , D^0 \bar{D^0})$ basis, and convolute the resulting subprocess cross sections with equivalent-photon spectra. In the adopted two-channel scenario the analytic continuation of the $T$-matrix gives a bound state at $M_X=3.7162$~GeV, which is below the $D^0 \bar{D^0}$ threshold. As a consequence of this, the observed signal in the continuum is not simply an isolated narrow peak, and a characteristic near-threshold line-shape distortion is produced: around $3.70 \leq W \leq 3.85$~GeV the integrated dressed-to-bare ratios are $(13.71,0.35)$ in Pb-Pb UPCs and $(13.74,0.35)$ in $e$-Au collisions for $(D^+D^-,D^0 \bar{D^0})$. Using the BABAR-fitted central normalization $K_{\rm dressed}=0.382$, the corresponding dressed cross sections are $(24.6,14.6)~\mu{\rm b}$ in Pb-Pb UPCs and $(2.41,1.43)~\rm nb$ at the EIC. The charged-to-neutral ratios are directly accessible observables, whereas the dressed-to-bare ratios provide theoretical diagnostics of the final-state interaction. The near-threshold features are controlled by the common $\gamma\gamma\to D\bar{D}$ dynamics and are rather independent of the external photon source.