Importance of local tetraquark operators for $T_{cc}(3875)^+$

Abstract

The doubly charmed tetraquark $T_{cc}(3875)^+$ observed at LHCb has attracted considerable interest in recent years. To accurately determine its finite-volume spectrum, a variational analysis using a large basis of operators, including bilocal scattering operators, but also local tetraquark operators, should be employed. Using Wilson-clover fermions at the $SU(3)$-flavour-symmetric point, we investigated the importance of local tetraquark operators for the $T_{cc}$ spectrum by adding them to a large basis of bilocal $DD^*$ and $D^*D^*$ scattering operators. We performed this calculation using the distillation framework combined with a position-space sampling method that we recently developed. This method makes local tetraquark operators affordable in distillation. Upon including local tetraquark operators, we observe significant shifts in the estimates of several energy levels. Finally, we show the effect of these shifts on the $DD^*$ scattering phase shifts obtained from a single-channel $s$-wave Lüscher analysis.

Figures (3)

• Figure 1: Tensor-network diagrams of two-point functions in distillation of operators relevant for the $T_{cc}$ tetraquark; a bilocal $D^{(*)}D^*$ (left) and a local tetraquark operator (right). For bilocal $D^{(*)}D^*$ operators, the two-point functions are computed by contracting the perambulators $\tau_f$ with the mode doublets $\Phi$. For local tetraquark operators, the smeared fermion propagators $S_{f,\mathrm{sm}}$ are constructed from the perambulators and subsequently contracted with the momentum projections $e^{\pm i\vb*{p}\vb*{x}^{(\prime)}}$ using position-space sampling (indicated by the dashed lines). The spin and color structure is suppressed in these diagrams.
• Figure 2: Finite-volume ground state (top left) and first excited state (top right) energy estimates from using three different operator bases: only local operators, only bilocal operators, and local and bilocal operators. The x-axis denotes the last operator group added to the basis for computing $E_n$; groups to the left were already present (except $T\;\{3\}$ when only bilocal operators are used). The red bands show the results from using all 12 operators. The lower panels show the same results, but the energy estimates are normalized with the threshold energy using correlated ratios. We removed the $T\;\{3\}$ energy estimates from the lower plots due to their large values compared to the others. All energies were computed on the N202 gauge ensemble.
• Figure 3: Left panel: Estimate of the finite-volume $cc\overline{u}\overline{d}$, isospin-0 spectrum in the $T_1^+$ irrep for the operator basis consisting only of bilocal scattering operators and for the full operator basis including also local tetraquark operators. The energy estimates are normalized with the threshold energy using correlated ratios. The spectrum is shown for the N202 and the B450 gauge ensembles. The horizontal lines denote the non-interacting $D^{(*)}D^*$ energies with the integer back-to-back momentum squared in the parentheses. Right panel: Estimate of the $s$-wave $DD^*$ scattering phase shifts shown as $q_\mathrm{cm}\cot{\delta_0(q_\mathrm{cm})}$ extracted from the finite-volume spectra for both operator bases. The c.m. scattering momenta $q_\mathrm{cm}$ are normalized with the pion mass $m_\pi$. The green dashed line displays the pole condition $q_\mathrm{cm}\cot{\delta_0(q_\mathrm{cm})} = \pm \sqrt{-q_\mathrm{cm}^2}$ and the vertical gray dashed line shows the start of the $u$-channel cut.