Testing the tetraquark mixing framework from QCD sum rules for $a_0(980)$

According to a recent proposal of the tetraquark mixing framework, the two light-meson nonets in the $J^P=0^{+}$ channel, namely, the light nonet composed of $a_0(980)$, $K_0^{*}(800)$, $f_0(500)$, $f_0(980)$, and the heavy nonet of $a_0(1450)$, $K_0^{*}(1430)$, $f_0(1370)$, $f_0(1500)$, can be expressed by linear combinations of the two tetraquark types, one type containing the spin-0 diquark and the other with the spin-1 diquark. Among various consequences of this mixing model, one surprising result is that the second tetraquark with the spin-1 diquark configuration is more important for the light nonet. In this work, we report that this result can be supported by the QCD sum rule calculation. In particular, we construct a QCD sum rule for the isovector resonance $a_0(980)$ using an interpolating field composed of both tetraquark types and then perform the operator product expansion up to dimension 10 operators. Our sum rule analysis shows that the spin-1 diquark configuration is crucial in generating the $a_0(980)$ mass. Also, the mixed correlation function constructed from the two tetraquark types is found to have large strength, which seems consistent with what the tetraquark mixing framework is advocating. On the other hand, the correlation function from the interpolating field with the spin-0 diquark configuration alone fails to predict the $a_0(980)$ mass mostly by the huge negative contribution from dimension 8 operators.

Figure 1

The Borel curves contributing to ${\widehat{\mathrm{\Pi }}}_{0,0}^{\mathrm{OPE}}$ [Eq. (26)], plotted separately for each OPE dimension as specified in the inset.

Figure 2

The Borel curve for ${\widehat{\mathrm{\Pi }}}_{0,0}^{\mathrm{OPE}}$, that is, the sum of all the lines in Fig. 1.

Figure 3

The Borel curve for the $a_0(980)$ mass extracted from Eq. (31) using the interpolating field $J_0$ only.

Figure 4

The Borel curves contributing to ${\widehat{\mathrm{\Pi }}}_{1,1}^{\mathrm{OPE}}$ [Eq. (28)], plotted separately for each OPE dimension as specified in the inset.

Figure 5

The Borel curve for ${\widehat{\mathrm{\Pi }}}_{1,1}^{\mathrm{OPE}}$, that is, the sum of all the lines in Fig. 4.

Figure 6

The Borel curve for the $a_0(980)$ mass extracted from Eq. (31) using the interpolating field $J_1$ only.

Figure 7

The Borel curves contributing to ${\widehat{\mathrm{\Pi }}}_{0,1}^{\mathrm{OPE}}$ [Eq. (27)], plotted separately for each OPE dimension as specified in the inset.

Figure 8

The Borel curve for the mixed correlator, ${\widehat{\mathrm{\Pi }}}_{0,1}^{\mathrm{OPE}}$ [Eq. (27)].

Figure 9

The Borel curves for the $a_0(980)$ mass extracted from Eq. (31) using the full interpolating field $J_{a_0}^L$ [Eq. (19)] are presented here with three solid lines with different $s_0$ as specified. The dashed line is the same curve as in Fig. 6 plotted here again for a clear comparison.