Investigating the role of tetraquark operators in lattice QCD studies of the $a_0(980)$ and $κ$ resonances

Abstract

The role of tetraquark operators in studying the isodoublet strange $κ$ and isovector nonstrange $a_0(980)$ scalar mesons in lattice QCD is examined using an ensemble with $m_π\approx230$ MeV and spatial extent $L$ such that $m_πL\approx4.4$. Hermitian correlation matrices using both single-meson, meson-meson, and tetraquark interpolating operators are used to extract the spectrum of finite-volume stationary states in the appropriate symmetry channels. Hundreds of local and extended tetraquark operators are explored. Determinations of the spectrum in each channel are found to be unreliable without the inclusion of at least one tetraquark operator. For example, the inclusion of tetraquark operators with isospin 1/2 and strangeness 1 quantum numbers reveals the existence of an additional energy level in the $Kη$ sub-system below the $Kη$ threshold. The implications of this on parametrizing the scattering $K$-matrix through a well-known quantization condition to extract properties of the $κ$ and $a_0(980)$ scalar meson resonances are discussed.

Figures (13)

• Figure 1: Our tetraquark operators consist of two gauge-covariantly displaced quarks (open circles) and two displaced antiquarks (solid circles) arranged in the spatial configurations shown above. The boxes indicate locations where color and spin indices are connected. The different spatial configurations we use are single-site operators (SS), doubly-displaced operators in an I configuration (DDIa and DDIb), and quadruply-displaced operators in a cross configuration (QDXa and QDXb). The letters "a" and "b" in the labels denote the different orderings of displacing the quarks and antiquarks. For each label, there are two color structures, given by Eqs. (\ref{['eq:TSdef']}) and (\ref{['eq:TAdef']}).
• Figure 2: Effective energies for the at-rest pion (left), kaon (center), and $\eta$ meson (right). The symbols with error bars indicate the Monte Carlo estimates of the effective energy functions, and fitting information is provided by the dashed and solid blue curves, which are explained in the text. Fit results for the dimensionless products of the temporal lattice spacing $a_t$ times the meson masses are shown, along with fit quality $\chi^2/N_{\rm dof}$. Fits to the correlators are done using a sum of two exponentials as the fit model. The vertical displacements of the effective energy points for the $\eta$ meson at time separations 14.5 $a_t$ and 17.5 $a_t$ is caused by using an interlace-16 time dilution scheme for the disconnected contributions. Energies were also obtained for these mesons at a variety of different momenta. The operators used to obtain the single-meson masses are discussed in the text in Sec. \ref{['sec:singlemesons']}. The reference energy $E_{\rm ref}$ is taken to be the mass of the kaon.
• Figure 3: Effective energies for the lowest six levels in the isodoublet, strangeness 1, zero-momentum, $A_{1g}$ channel. (Top two rows) Results obtained using only the operators in Tables \ref{['tab:single-meson-ops']} and \ref{['tab:two-meson-ops']}, excluding the tetraquark operator. (Bottom two rows) Results obtained using the operators in Tables \ref{['tab:single-meson-ops']} and \ref{['tab:two-meson-ops']} and also including the tetraquark operator in Table \ref{['tab:tetraq']}. Effective energy curves calculated from correlator fits are overlaid, and fit results are shown. Note that level 3 in the bottom row corresponds to the additional level found using the tetraquark operator.
• Figure 4: Determinations of the spectrum in the isodoublet strange zero-momentum $A_{1g}$ channel obtained using three different diagonalization times $\tau_D=8$ (left), $\tau_D=12$ (center), and $\tau_D=15$ (right) for the operator basis given in Tables \ref{['tab:single-meson-ops']}, \ref{['tab:two-meson-ops']}, and \ref{['tab:tetraq']}, which includes the tetraquark operator. Good agreement is found, but a significant increase in statistical uncertainties is evident for $\tau_D=15$.
• Figure 5: Overlap factors $\vert Z^{(n)}_j\vert^2$ for the twelve operators used in the correlator matrix corresponding to the isodoublet strange zero-momentum $A_{1g}$ channel. Each plot shows the factors for a single operator, with horizontal axis labelled by $n$, the index identifying the stationary states. The factors for the tetraquark operator are shown in the left-most plot in the second row from the top. Levels 3 and 2 have significant tetraquark, $K\tilde{\eta}$, and $K\tilde{\phi}$ content, where $\tilde{\eta}$ means $\overline{u}u+\overline{d}d$ and $\tilde{\phi}$ means $\overline{s}s$.