Tetraquarks with hidden charm and strangeness as $\varphi \text{−}\psi \left(2S\right)$ hadrocharmonium

In the hadrocharmonium picture a $\overline{c}c$ state and a light hadron form a bound state. The effective interaction is described in terms of the chromoelectric polarizability of the $\overline{c}c$ state and energy-momentum-tensor densities of the light hadron. This picture is justified in the heavy quark limit, and may successfully account for a hidden-charm pentaquark state recently observed by the LHCb Collaboration. In this work we extend the formalism to the description of hidden-charm tetraquarks, and address the question of whether the resonant states observed by LHCb in the $J/\psi \text{−}\varphi$ spectrum can be described as hadrocharmonia. This is a nontrivial question because nothing is known about the $\varphi$ meson energy-momentum-tensor densities. With rather general assumptions about energy-momentum-tensor densities in the $\varphi$ meson we show that a $\psi \left(2S\right)\text{−}\varphi$ bound state can exist, and we obtain a characteristic relation between its mass and width. We show that the tetraquark $X\left(4274\right)$ observed by LHCb in the $J/\psi \text{−}\varphi$ spectrum is a good candidate for a hadrocharmonium. We make predictions which will allow testing this picture. Our method can be generalized to identify other potential hadrocharmonia.

Figure 1

Examples of the effective potentials obtained from different values of the parameters in the intervals (5) in our Ansätze for $\varphi$-meson EMT densities.

Figure 2

The scatter plot of the decay width $\mathrm{\Gamma }(M_1,M_2,D)$ vs mass $M(M_1,M_2,D)$ of tetraquarks obtained from varying the parameters $M_1,M_2,D$, which describe the unknown $\varphi$-meson EMT form factors (4), within a wide range of the values (5). In this plot 310 different points are shown. Remarkably, even though we randomly scan a large parameter space, the $\mathrm{\Gamma }\text{−}M$ values lie approximately on a characteristic curve; see text. The crosses on the $M$ axis indicate the bounds $m_J+m_{\varphi }<M<m_{\psi }+m_{\varphi }$. For comparison we show the four tetraquarks in the $J/\psi \text{−}\varphi$ resonance region with their statistical (thin lines) and systematic (shaded areas) uncertainties and spin-parity assignments [4]. The state $X\left(4274\right)$ emerges as a candidate for the description as a hadrocharmonium. This method can be used to identify other possible hadroquarkonia.