Doubly charm tetraquark channel with isospin $1$ from lattice QCD

Experimentally, the doubly charm tetraquark channel $cc\bar{q}\bar{q}$ with $q\text{\hspace{0.17em}⁣}=\text{\hspace{0.17em}⁣}u,d$ features an exotic hadron, $T_{cc}$, with isospin $I\text{\hspace{0.17em}⁣}=\text{\hspace{0.17em}⁣}0$ near the $D{D}^{\ast }$ threshold, while no peak was observed for $I\text{\hspace{0.17em}⁣}=\text{\hspace{0.17em}⁣}1$. We present a lattice QCD study of this channel with $I\text{\hspace{0.17em}⁣}=\text{\hspace{0.17em}⁣}1$, $J^P\text{\hspace{0.17em}⁣}=\text{\hspace{0.17em}⁣}{1}^{+}$ and $m_{\pi }\simeq 280$MeV. Finite-volume energies calculated across five charm quark masses consistently feature a positive energy shift with respect to non-interacting energies, indicating repulsive interaction at energies near threshold. These energies are used to compute the $D{D}^{\ast }$ scattering amplitude using both the standard Lüscher method and the recently proposed effective-field-theory-based approach in the plane-wave basis, which incorporates the long-range interactions and the left-hand cut. Both analyses render a small negative scattering length and the scattering amplitude that does not feature any poles in the energy region near the $D{D}^{\ast }$ threshold, in line with LHCb results. We identify that the Wick contraction resembling $t$-channel isovector-vector meson exchanges between $D$ and $D^{\ast }$ plays a key role in distinguishing between the $I=0$ and $I=1$ channels, leading to repulsion in the $I=1$ and attraction in the $I=0$ channel.