Solving the left-hand cut problem in lattice QCD: $T_{cc}(3875{)}^{+}$ from finite volume energy levels

We discuss a novel effective-field-theory-based approach for extracting two-body scattering information from finite volume energies, serving as an alternative to Lüscher’s method. By explicitly incorporating one-pion exchange, we overcome the challenging left-hand cut problem in Lüscher’s method and can handle finite volume energy levels both below and above the left-hand cut. Applied to the lattice data for $D{D}^{*}$ scattering at a pion mass of 280 MeV, as an illustrative example, our results reveal the significant impact of the one-pion exchange on $P$-wave and $S$-wave phase shifts. The pole position of the $T_{cc}(3875{)}^{+}$ state, extracted from the finite-volume energy levels at this pion mass while taking into account left-hand cut effects, range corrections and partial-wave mixing, is consistent with a near-threshold resonance. This study demonstrates, for the first time, that two-body scattering information can be reliably extracted from lattice spectra including the left-hand cut.

Figure 1

Schematic illustration of the approach employed in this study. $V$ denotes the effective potential in chiral EFT, involving the OPE and contact interactions; $E_{\mathrm{FV}}$ stands for the finite volume energy levels in lattice simulations, used here as input.

Figure 2

Fit results for the center-of-mass energy $E_{\mathrm{cm}}=\sqrt{E^2-{\mathbf{P}}^2}$ of the $D{D}^{*}$ system normalized by $E_{D{D}^{*}}=M_D+M_{D^{*}}$, for the heavier charm quark mass and two different volumes from Ref. [23] in various FV irreps. The lattice energy levels are shown by open circles, squares and triangles; the blue and green points in the irreps $T_1^{+}(0),A_1^{-}(0)$, and $A_2(1)$ are used as input in this analysis as well as in the scattering analysis of Ref. [23]. The orange symbols, slightly shifted to the right for transparency, represent the results of our full calculation (Fit 2), including pions. For each irrep, we indicate the lowest partial waves, which contribute to it. Our results in the irrep $A_2(4)$ are predictions. The solid and dot-dashed lines correspond to the noninteracting $D{D}^{*}$ and $D^{*}{D}^{*}$ energies, respectively.

Figure 3

Phase shifts in the ${}^3{S}_1$ (left panel) and ${}^3{P}_0$ (right panel) partial waves extracted from the FV energy levels $E_{\mathrm{FV}}$ calculated in lattice QCD. Red bands represent the results of our 3-parameter fits to $E_{\mathrm{FV}}$ without the OPE (Fit 1, upper panel) and with the OPE (Fit 2, lower panel), including the $1\sigma$ uncertainty. Green dots in the left panel are the phase shifts extracted from $E_{\mathrm{FV}}$ using the single-channel Lüscher quantization conditions in Ref. [23]. Green dots in the right panel are extracted in this study using the same method. Blue bands are the results of the 4-parameter fits of $E_{\mathrm{FV}}$ using the ERE in Ref. [23]. Orange lines in the left panel correspond to $ip=\pm |p|$ from unitarity, normalized to $E_{D{D}^{*}}$. The gray vertical dashed line denotes the position of the branch point of the left-hand cut nearest to the threshold.