Unraveling $K(1690)$ as a pseudoscalar $ud\bar{d}\bar{s}$ tetraquark state

TL;DR

The paper addresses whether the strange meson $K(1690)$ can be interpreted as a compact light tetraquark with content $ud\bar{d}\bar{s}$ and $J^P=0^-$ by applying QCD sum rules. It develops a framework with ten interpolating currents for the $ud\bar{d}\bar{s}$ system, computes the two-point correlator up to dimension-8, and crucially includes the full tri-gluon condensate term $\langle g^3 f G^3 \rangle$ with infrared divergence cancellation to obtain IR-safe, reliable sum rules. The main result finds hadron masses in the range $m_X\approx 1.60$–$1.70$ GeV for currents $P_6$, $P_3$, and $V_3$, consistent with $K(1690)$, and shows that including $\langle g^3 f G^3 \rangle$ increases the predicted masses by roughly 20% for some currents. Collectively, these findings support a compact tetraquark interpretation of $K(1690)$ while acknowledging alternative explanations and signaling the need for decay-channel studies to further test the internal structure.

Abstract

The recent observed $K (1690)$ has been identified as a supernumerary pseudoscalar resonance signal in the strange-meson spectrum predicted by quark model calculations. It is the best candidate of a strange crypto-exotic state. In this work, we systematically study the hadron masses of $ud\bar{d}\bar{s}$ tetraquark states with $J^P = 0^-$ in the method of QCD sum rules (QCDSR). For ten interpolating currents, we calculate the correlation functions up to dimension-8 nonperturbative condensates. To calculate the tri-gluon condensate, we comprehensively consider the contributions from different operators with and without covariant derivatives. The infrared (IR) safety can be guaranteed for the completely calculated tri-gluon condensate by properly addressing the IR divergences in Feynman diagrams. It is demonstrated that the tri-gluon condensate provides significant contributions to the sum-rule analyses in these light tetraquark systems. Our results support the interpretation of $K (1690)$ resonance to be a pseudoscalar $ud\bar{d}\bar{s}$ tetraquark state.

Figures (7)

• Figure 1: Cancellation of $\langle (GG)G\rangle$ and $\langle G(DDG)\rangle$ condensates formed by two different quark propagators.
• Figure 2: IR safety for $\langle (DG)(DG)\rangle$ condensate: (a) cancellation of $\langle (DG)(DG) \rangle$ by any two quark propagators; (b) the propagator containing $\langle (DG)(DG) \rangle$ has no IR divergence.
• Figure 3: Cancellation of $\langle GGG \rangle$ and $\langle G(DDG)\rangle$ condensates formed by the single quark propagator.
• Figure 4: The Feynman diagrams involved in our calculations for the $ud\bar{d}\bar{s}$ tetraquark systems. A quark line with dots contains terms with the color factor $\delta^{i j}$ in $S^{i j} (x)$.
• Figure 5: The OPE convergence (a) and variation of hadron mass with $s_0$ (b) for the interpolating current $P_3$.