Possible partner state of the $Y(2175)$

We study the $Y(2175)$ using the method of QCD sum rules. There are two independent $ss\overline{s}\overline{s}$ interpolating currents with $J^{PC}=1^{--}$, and we calculate both their diagonal and their off-diagonal correlation functions. We obtain two new currents that do not strongly correlate to each other, so they may couple to two different physical states: one of them couples to the $Y(2175)$, while the other may couple to another state whose mass is evaluated to be $2.41\pm 0.25\mathrm{GeV}$. Evidence of the latter state can be found in the BABAR [B. Aubert et al. (BABAR Collaboration), Phys. Rev. D 76, 012008 (2007)], BESII [M. Ablikim et al. (BES Collaboration), Phys. Rev. Lett. 100, 102003 (2008)], Belle [C. P. Shen et al. (Belle Collaboration), Phys. Rev. D 80, 031101 (2009)], and BESIII [M. Ablikim et al. (BESIII Collaboration), Phys. Rev. D 91, 052017 (2015)] experiments.

Figure 1

The BABAR [14, 15], Belle [19], BESII [18], and BESIII [20] experiments observing the $Y(2175)$ as well as the fit performed in Ref. [22]. (a) The $e^{+}{e}^{-}\to \varphi f_0(980)$ cross section. Taken from BABAR [14]. (b) The $K^{+}{K}^{-}{\pi }^{+}{\pi }^{-}$ invariant mass distribution in the $K^{+}{K}^{-}{f}_0(980)$ threshold region. The fits are done by including no (dashed line), one (solid line) and two (dotted line) resonances. Taken from BABAR [15]. (c) The $e^{+}{e}^{-}\to \varphi {\pi }^{+}{\pi }^{-}$ cross section with two incoherent Breit-Wigner functions, the $\varphi (1680)$ and the $Y(2175)$. Taken from Belle [19]. (d) Fits to the BABAR [14] and Belle [19] measurements of the $e^{+}{e}^{-}\to \varphi f_0(980)$ cross sections with two coherent Breit-Wigner functions, performed by Shen and Yuan and taken from Ref. [22]. (e) The $\varphi f_0(980)$ invariant mass spectrum. Taken from BESII [18]. (f) The $\varphi f_0(980)$ invariant mass spectrum. Taken from BESIII [20].

Figure 2

Feynman diagram related to the quark-gluon mixed condensate $⟨g_s\overline{s}\sigma Gs⟩$.

Figure 3

Left: $\left|\frac{{\mathrm{\Pi }}_{{\eta }_1{\eta }_2}(M_B^2)}{{\mathrm{\Pi }}_{{\eta }_1{\eta }_1}(M_B^2)}\right|$ (solid line) and $\left|\frac{{\mathrm{\Pi }}_{{\eta }_1{\eta }_2}(M_B^2)}{{\mathrm{\Pi }}_{{\eta }_2{\eta }_2}(M_B^2)}\right|$ (dotted line), as functions of the Borel mass $M_B$, when taking $s_0=6.0{\mathrm{GeV}}^2$. Right: $\left|\frac{{\mathrm{\Pi }}_{J_1{J}_2}(M_B^2)}{{\mathrm{\Pi }}_{J_1{J}_1}(M_B^2)}\right|$ (solid line) and $\left|\frac{{\mathrm{\Pi }}_{J_1{J}_2}(M_B^2)}{{\mathrm{\Pi }}_{J_2{J}_2}(M_B^2)}\right|$ (dotted line), as functions of the Borel mass $M_B$, when taking $s_0=6.0{\mathrm{GeV}}^2$.

Figure 4

The correlation function ${\mathrm{\Pi }}_{J_1{J}_1}(M_B^2)$ as a function of $s_0$ in units of ${\mathrm{GeV}}^{10}$. The curves are obtained by taking $M_B^2=2.0{\mathrm{GeV}}^2$ (short-dashed line), $3.0{\mathrm{GeV}}^2$ (solid line), and $4.0{\mathrm{GeV}}^2$ (long-dashed line).

Figure 5

Mass calculated using the current $J_{1\mu }$, as a function of the threshold value $s_0$ (left) and the Borel mass $M_B$ (right). In the left panel, the short-dashed/solid/long-dashed curves are obtained by setting $M_B^2=2.0/3.0/4.0{\mathrm{GeV}}^2$, respectively. In the right panel, the short-dashed/solid/long-dashed curves are obtained by setting $s_0=5.5/6.0/6.5{\mathrm{GeV}}^2$, respectively.