Position-space sampling for local multiquark operators in lattice QCD using distillation and the importance of tetraquark operators for $T_{cc}(3875)^+$

TL;DR

The paper introduces position-space sampling within the distillation framework to make local multiquark operators computationally tractable in lattice QCD. The method provides an unbiased estimator by projecting momentum on sparse spatial grids and demonstrates favorable scaling, allowing efficient inclusion of local tetraquark operators alongside bilocal scattering operators. Applied to the $T_{cc}(3875)^+$, the study shows that local tetraquark operators can shift several finite-volume energy levels, underscoring their importance for a reliable spectrum, and it performs a single-channel $s$-wave Lüscher analysis to connect finite-volume energies to scattering phase shifts, revealing a virtual bound-state tendency with caveats from left-hand cuts. Overall, the approach enables robust spectroscopy of exotic hadrons in large volumes and improves the reliability of extracted scattering information, with future work extending to moving frames and multiple lattice spacings.

Abstract

Obtaining hadronic two-point functions is a central step in spectroscopy calculations in lattice QCD. This requires solving the Dirac equation repeatedly, which is computationally demanding. The distillation method addresses this difficulty by using the lowest eigenvectors of the spatial Laplacian to construct a subspace in which the Dirac operator can be fully inverted. This approach is efficient for nonlocal operators such as meson-meson and baryon-baryon operators. However, local multiquark operators with four or more (anti)quarks are computationally expensive in this framework: the cost of contractions scales with a high power of the number of Laplacian eigenvectors. We present a position-space sampling method within distillation that reduces this cost scaling by performing the momentum projection only over sparse grids rather than the full spatial lattice. We demonstrate the efficiency of this unbiased estimator for single-meson, single-baryon and local tetraquark operators. Using Wilson-clover fermions at the $SU(3)$-flavour-symmetric point, we apply this method to study the importance of local tetraquark operators for the finite-volume $T_{cc}(3875)^+$ spectrum. To this end, we extend a large basis of bilocal $DD^*$ and $D^*D^*$ scattering operators by including local tetraquark operators. The inclusion of local operators leads to significant shifts in several energy levels. Finally, we show the effect of these shifts on the $DD^*$ scattering phase shift from a single-channel $s$-wave Lüscher analysis.

Figures (7)

• Figure 1: Tensor network diagrams of two-point functions in distillation of operators relevant for the $T_{cc}$ tetraquark; a bilocal $DD^*$ (top) and a local tetraquark operator (bottom). For the bilocal $DD^*$ operator, the two terms from the Wick contraction are topologically distinct, whereas for the local tetraquark operator, they have the same topology. The spin structure is suppressed.
• Figure 2: Tensor network diagrams of the same tetraquark two-point function as in the bottom panel of Figure \ref{['fig:corr_in_distill']}. Here, all tensors involved in distillation are displayed to visualize the order in which they are contracted. In the conventional distillation method (top), the perambulators $\tau_f$ (with flavour $f$) are constructed by contracting the propagators $S_f$ with the neighboring $V$ matrices. The remaining $V$'s are contracted with $e^{-i\vb*{p}\vb*{x}'}$ which gives the rank-4 tensor $\Xi$. In the position-space sampling method (bottom), the propagators are contracted with all four $V$'s connected to it, giving the smeared propagators $S_{f,\mathrm{sm}}$. The contraction with the exponentials is then performed only over subspaces of the spatial lattice (indicated by the dashed lines). The spin and color structure is suppressed in these diagrams.
• Figure 3: Effective energies $E_\mathrm{eff}^h$ for the $D$ meson ($h = D$, top left) and the nucleon ($h = N$, bottom left) at zero momentum and for different point separations $N_\mathrm{sep}$. The red bands are results from plateau fits for $N_\mathrm{sep}=4$. The right panel displays the energies $E_{N_\mathrm{sep}}^h$ obtained from plateau fits to $E_\mathrm{eff}^h$ for the $D$ meson (top right) and the nucleon (bottom right) as a function of $N_\mathrm{sep}$. The energies $E_{N_\mathrm{sep}}^h$ were computed using momenta with $\vb*{p}^2 = 0$ and $\vb*{p}^2 = (2\pi/L)^2$, and are normalized with the $N_\mathrm{sep}=1$ value $E_1^h$.
• Figure 4: Plateau values $E_{N_\mathrm{sep}}^T$ from plateau fits to the effective energy of the local $DD^*$ and diquark-antidiquark two-point functions vs. the point separation $N_\mathrm{sep}$. The plateau values are normalized with the $N_\mathrm{sep}=4$ value $E_4^T$.
• Figure 5: Finite-volume ground state (top left) and first excited state (top right) energy from using different operator bases with and without local tetraquark 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 energies are normalized with the threshold energy using correlated ratios. We removed the $T\;\{3\}$ energies from the lower plots due to their large values compared to the others. All energies were computed on the N202 gauge ensemble.