Exotic $\Omega\Omega$ dibaryon states in a molecular picture

TL;DR

This paper predicts the existence of exotic $ ext{Omega} ext{Omega}$ dibaryon states with specific quantum numbers using QCD sum rules, suggesting they are loosely bound molecular states with potential experimental signatures.

Contribution

It introduces a molecular picture approach with interpolating currents to calculate the masses of $ ext{Omega} ext{Omega}$ dibaryons, providing the first mass estimates for these exotic states.

Findings

Scalar $ ext{Omega} ext{Omega}$ state mass is about 3.33 GeV, 15 MeV below threshold.

Tensor $ ext{Omega} ext{Omega}$ state mass is about 3.24 GeV, possibly indicating a deeper bound state.

Predicted states may be observed in heavy-ion collision experiments.

Abstract

We investigate the exotic $\Omega\Omega$ dibaryon states with $J^P=0^+$ and $2^+$ in a molecular picture. We construct the scalar and tensor $\Omega$$\Omega$ molecular interpolating currents and calculate their masses within the method of QCD sum rules. Our results indicate that the mass of the scalar dibaryon state is $m_{\Omega\Omega, \, 0^+}=(3.33\pm0.22) \,\unit$, which is about $15 \,\mathrm{MeV}$ below the $2m_\Omega$ threshold. This result suggests the existence of a loosely bound molecular state of the $J^P=0^+$ scalar $\Omega\Omega$ dibaryon with a small binding energy around 15 MeV. The mass of the tensor dibaryon is predicted to be $m_{\Omega\Omega,\, 2^+}=(3.24\pm0.23)\, \mbox{GeV}$, which may imply a deeper molecular state of the tensor $\Omega\Omega$ dibaryon than the scalar channel. These exotic strangeness $S=-6$ and doubly-charged $\Omega\Omega$ dibaryon states may be identified in the heavy-ion collision processes.

Equations

$$J_{\mu }^{\Omega }(x)={ϵ}^{abc}\left[s_a^T(x)C{\gamma }_{\mu }{s}_b(x)\right]s_c(x),$$

$$J^{\Omega \Omega }(x)$$

$$J_{\mu \nu }^{\Omega \Omega }(x)$$

$$\Pi (q^2)$$

$${\Pi }_{\mu \nu ,\rho \sigma }(q^2)$$

$$⟨0|J^{\Omega \Omega }|X⟩$$

$$⟨0|J_{\mu \nu }^{\Omega \Omega }|X⟩$$

$$\begin{array}{rl}{P}_{0T}& =\frac{1}{16}{g}_{\mu \nu }{g}_{\rho \sigma },\text{mbox}for{J}^P=0^{+},\text{mbox}T\\ P_{0S}& =T_{\mu \nu }{T}_{\rho \sigma },\text{mbox}for{J}^P=0^{+},\text{mbox}S\\ P_{0TS}& =\frac{1}{4}(T_{\mu \nu }{g}_{\rho \sigma }+T_{\rho \sigma }{g}_{\mu \nu }),\text{mbox}for{J}^P=0^{+},\text{mbox}TS\\ P_{2S}^P& =\frac{1}{2}\left({\eta }_{\mu \rho }{\eta }_{\nu \sigma }+{\eta }_{\mu \sigma }{\eta }_{\nu \rho }-\frac{2}{3}{\eta }_{\mu \nu }{\eta }_{\rho \sigma }\right),\text{mbox}for{J}^P=2^{+},\text{mbox}S\end{array}$$

$$\begin{array}{rl}{\eta }_{\mu \nu }& =\frac{q_{\mu }{q}_{\nu }}{q^2}-g_{\mu \nu },T_{\mu \nu }=\frac{q_{\mu }{q}_{\nu }}{q^2}-\frac{1}{4}{g}_{\mu \nu },\\ T_{\mu \nu ,\rho \sigma }^{\pm }& =\left[\frac{q_{\mu }{q}_{\rho }}{q^2}{\eta }_{\nu \sigma }\pm (\mu \leftrightarrow \nu )\right]\pm (\rho \leftrightarrow \sigma ).\end{array}$$

$$\Pi \left(q^2\right)={\left(q^2\right)}^N{\int }_0^{\infty }\mathrm{d}s\frac{\rho \left(s\right)}{s^N\left(s-q^2-\mathrm{i}ϵ\right)}+\sum _{k=0}^{N-1}{b}_n{\left(q^2\right)}^k,$$

$$\rho \left(s\right)$$

$$=f_X^2\delta \left(s-m_X^2\right)+\text{continuum},$$

$$\Pi (s_0,M_B^2)=f_X^2{m}_X^2{e}^{-m_X^2/M_B^2}={\int }_{<}^{s_0}ds{e}^{-s/M_B^2}\rho (s),$$

$$m_X^2\left(s_0,M_B^2\right)=\frac{{\int }_{<}^{s_0}dss\rho \left(s\right)e^{-s/M_B^2}}{{\int }_{<}^{s_0}ds\rho \left(s\right)e^{-s/M_B^2}},$$

$$\begin{array}[]{cc}\hline\cr\hline\cr\left<\bar{s}s\right>&-(0.8\pm 0.1)\times(0.24\pm 0.03)^{3}\,\mathrm{GeV}^{3}\\ \left<g_{s}^{2}GG\right>&(0.48\pm 0.14)\,\mathrm{GeV}^{4}\\ \left<g_{s}\bar{s}\sigma Gs\right>&-M_{0}^{2}\left<\bar{s}s\right>\\ M_{0}^{2}&(0.8\pm 0.2)\,\mathrm{GeV}^{2}\\ m_{s}&95^{+9}_{-3}\,\mathrm{MeV}\\ \hline\cr\hline\cr\end{array}$$

$$\Pi (\infty ,M_B^2)=6.98\times 10^{-12}{M}_B^{16}+3.22\times 10^{-11}{M}_B^{12}$$

$$+1.83\times 10^{-10}{M}_B^{10}-6.04\times 10^{-10}{M}_B^8+4.21\times 10^{-10}{M}_B^6,$$

$$m_{\Omega \Omega }=(3.33\pm 0.14)\text{mbox}GeV,$$

$$f_{\Omega \Omega }=\left(4.6\pm 0.7\right)\times 10^{-4}\text{mbox}{GeV}^8.$$

$$m_{\Omega \Omega }=(3.33\pm 0.42)\text{mbox}GeV,$$

$$f_{\Omega \Omega }=\left(2.1\pm 0.16\right)\times 10^{-3}{\mathrm{GeV}}^8.$$

$$m_{\Omega \Omega ,0^{+}}=(3.33\pm 0.22)\text{mbox}GeV.$$

$$m_{\Omega \Omega ,2^{+}}=(3.24\pm 0.23)\text{mbox}GeV,$$

$$f_{\Omega \Omega ,2^{+}}=\left(1.8\pm 0.7\right)\times 10^{-3}{\mathrm{GeV}}^8.$$

$$\begin{array}{rl}\rho (s)& =\frac{27s^7}{7!7!2^9{\pi }^{10}}-\frac{21m_s⟨\bar{s}s⟩s^5}{5!5!2^5{\pi }^8}-\frac{9⟨g_s^{2}GG⟩s^5}{5!5!2^{11}{\pi }^{10}}-\frac{3m_s⟨g_s\bar{s}\sigma Gs⟩s^4}{5!2^5{\pi }^8}+\frac{17{⟨\bar{s}s⟩}^2{s}^4}{5!2^6{\pi }^6}+\frac{11m_s⟨g_s^{2}GG⟩⟨\bar{s}s⟩s^3}{4!4!2^4{\pi }^8}+\frac{3⟨g_s\bar{s}\sigma Gs⟩⟨\bar{s}s⟩s^3}{2^7{\pi }^6}\\ & +\frac{5m_s⟨g_s^{2}GG⟩⟨g_s\bar{s}\sigma Gs⟩s^2}{2^{10}{\pi }^8}-\frac{17m_s{⟨\bar{s}s⟩}^3{s}^2}{12{\pi }^4}+\frac{13{⟨g_s\bar{s}\sigma Gs⟩}^2{s}^2}{2^8{\pi }^6}-\frac{221⟨g_s^{2}GG⟩{⟨\bar{s}s⟩}^2{s}^2}{4!5!2^2{\pi }^6}.\end{array}$$

$$\begin{array}{rl}\rho (s)& =\frac{27s^7}{7!7!2^{13}{\pi }^{10}}-\frac{7m_s⟨\bar{s}s⟩s^5}{5\times 5!\times 2^{12}{\pi }^8}-\frac{31⟨g_s^{2}GG⟩s^5}{5^2{2}^{27}{\pi }^{10}}-\frac{3877m_s⟨g_s\bar{s}\sigma Gs⟩s^4}{5!2^{20}{\pi }^8}+\frac{{⟨\bar{s}s⟩}^2{s}^4}{3\times 2^{12}{\pi }^6}+\frac{491m_s⟨g_s^{2}GG⟩⟨\bar{s}s⟩s^3}{3^3{2}^{21}{\pi }^8}\\ & +\frac{4645⟨g_s\bar{s}\sigma Gs⟩⟨\bar{s}s⟩s^3}{3^2{2}^{19}{\pi }^6}+\frac{865m_s⟨g_s^{2}GG⟩⟨g_s\bar{s}\sigma Gs⟩s^2}{9\times 2^{20}{\pi }^8}-\frac{5m_s{⟨\bar{s}s⟩}^3{s}^2}{3\times 2^5{\pi }^4}+\frac{2725{⟨g_s\bar{s}\sigma Gs⟩}^2{s}^2}{3\times 2^{19}{\pi }^6}-\frac{23⟨g_s^{2}GG⟩{⟨\bar{s}s⟩}^2{s}^2}{3^3{2}^{14}{\pi }^6}.\end{array}$$

$$\begin{array}{rl}\rho (s)& =\frac{3s^7}{7!7!2^{12}{\pi }^{10}}-\frac{57m_s⟨\bar{s}s⟩s^5}{5\times 7!2^{11}{\pi }^8}+\frac{2693⟨g_s^{2}GG⟩s^5}{15\times 7!2^{24}{\pi }^{10}}-\frac{46073m_s⟨g_s\bar{s}\sigma Gs⟩s^4}{3\times 7!2^{19}{\pi }^8}+\frac{3{⟨\bar{s}s⟩}^2{s}^4}{7!2^5{\pi }^6}+\frac{2653m_s⟨g_s^{2}GG⟩⟨\bar{s}s⟩s^3}{5\times 3^4{2}^{18}{\pi }^8}\\ & +\frac{3011⟨g_s\bar{s}\sigma Gs⟩⟨\bar{s}s⟩s^3}{5\times 3^3{2}^{19}{\pi }^6}+\frac{2083m_s⟨g_s^{2}GG⟩⟨g_s\bar{s}\sigma Gs⟩s^2}{3^3{2}^{20}{\pi }^8}-\frac{5m_s{⟨\bar{s}s⟩}^3{s}^2}{3^2{2}^3{\pi }^4}-\frac{269{⟨g_s\bar{s}\sigma Gs⟩}^2{s}^2}{3\times 2^{18}{\pi }^6}-\frac{317⟨g_s^{2}GG⟩{⟨\bar{s}s⟩}^2{s}^2}{3^3{2}^{16}{\pi }^6}.\end{array}$$

$$\begin{array}{rl}\rho (s)& =\frac{m_s⟨\bar{s}s⟩s^5}{5\times 7!\times 2^{12}{\pi }^8}+\frac{367⟨g_s^{2}GG⟩s^5}{7!2^{23}{\pi }^{10}}+\frac{18389m_s⟨g_s\bar{s}\sigma Gs⟩s^4}{5\times 3^3{2}^{21}{\pi }^8}-\frac{{⟨\bar{s}s⟩}^2{s}^4}{5!2^6{\pi }^6}-\frac{1183m_s⟨g_s^{2}GG⟩⟨\bar{s}s⟩s^3}{45\times 2^{21}{\pi }^8}-\frac{18389⟨g_s\bar{s}\sigma Gs⟩⟨\bar{s}s⟩s^3}{45\times 2^{18}{\pi }^6}\\ & -\frac{5⟨g_s^{2}GG⟩{⟨\bar{s}s⟩}^2{s}^2}{3^3{2}^{17}{\pi }^6}-\frac{1927m_s⟨g_s^{2}GG⟩⟨g_s\bar{s}\sigma Gs⟩s^2}{3^2{2}^{21}{\pi }^8}+\frac{m_s{⟨\bar{s}s⟩}^3{s}^2}{16{\pi }^4}-\frac{11477{⟨g_s\bar{s}\sigma Gs⟩}^2{s}^2}{3^2{2}^{19}{\pi }^6}.\end{array}$$

$$\begin{array}{rl}\rho (s)& =\frac{3s^7}{7!7!2^7{\pi }^{10}}+\frac{47m_s⟨\bar{s}s⟩s^5}{7!2^9{\pi }^8}-\frac{188399⟨g_s^{2}GG⟩s^5}{9\times 7!2^{22}{\pi }^{10}}-\frac{25{⟨\bar{s}s⟩}^2{s}^4}{7!24{\pi }^6}+\frac{822245m_s⟨g_s\bar{s}\sigma Gs⟩s^4}{9\times 7!2^{16}{\pi }^8}+\frac{6149m_s⟨g_s^{2}GG⟩⟨\bar{s}s⟩s^3}{3^5{2}^{18}{\pi }^8}\\ & -\frac{16823⟨g_s\bar{s}\sigma Gs⟩⟨\bar{s}s⟩s^3}{3^4{2}^{15}{\pi }^6}-\frac{18401{⟨g_s\bar{s}\sigma Gs⟩}^2{s}^2}{3^2{2}^{17}{\pi }^6}-\frac{25m_s{⟨\bar{s}s⟩}^3{s}^2}{3^3{2}^2{\pi }^4}-\frac{385⟨g_s^{2}GG⟩{⟨\bar{s}s⟩}^2{s}^2}{3^4{2}^{11}{\pi }^6}+\frac{3671m_s⟨g_s^{2}GG⟩⟨g_s\bar{s}\sigma Gs⟩s^2}{3^4{2}^{17}{\pi }^8}.\end{array}$$

Full text

QCD sum rules, molecular state, dibaryon

Introduction: The history of multiquark configurations can go back to the establishment of the quark model (QM) by Gell-Mann Gell-Mann (1964) and Zweig Zweig (1964), where the tetraquark $qq\bar{q}\bar{q}$ and pentaquark $qqqq\bar{q}$ configurations were proposed outside of the conventional meson and baryon states. In the past seventeen years, there have been great progress on the explorations of tetraquark and pentaquark states, with the observations of the so called $XYZ$ and $P_{c}$ states Liu et al. (2019); Chen et al. (2016); Esposito et al. (2016); Richard (2016); Ali et al. (2017); Lebed et al. (2017); Guo et al. (2018); Olsen et al. (2018).

A dibaryon is another kind of multiquark system composed of two color-singlet baryons, such as the deuteron (a loosely $np$ bound state in ${}^{3}S_{1}$ channel Urey et al. (1932)). In 1964, the non-strange dibaryon sextet $D_{IJ}$ (with $IJ=01,10,12,21,03$ and $30$) was proposed by Dyson and Xuong in SU(6) symmetry Dyson and Xuong (1964). The $D_{01}$, $D_{10}$ and $D_{12}$ dibaryons have been identified as the deuteron ground state, the virtual ${}^{1}S_{0}$ isovector state and an isovector $J^{P}=2^{+}$ state at the $\Delta N$ threshold respectively Dyson and Xuong (1964). Recently, the $d^{*}(2380)$ state was confirmed by the WASA detector at COSY Adlarson et al. (2011, 2013, 2014, 2015), which was considered as the $\Delta\Delta$ dibaryon in the $D_{03}$ channel Yuan et al. (1999); Gal and Garcilazo (2013); Chen et al. (2015a); Huang et al. (2014, 2015). Moreover, the H-dibaryon predicted by Jaffe Jaffe (1977) is still attractive both in experimental and theoretical aspects Ohnishi et al. (2016); Beane et al. (2011, 2013). For more introduction about dibaryons, one can consult the recent review paper in Ref. Clement (2017).

Comparing to the $NN$ and H dibaryons, the investigation on the $\Omega\Omega$ system has received much less research interest. The interaction between two $\Omega$ baryons has not been adequately understood experimentally and theoretically. Nevertheless, one would expect that the $\Omega\Omega$ dibaryon will be stable against the strong interaction, since $\Omega$ is the only stable state in the decuplet $\bf{10}$ baryons Tanabashi et al. (2018a). From the properties of $\Omega$, we know the baryon number of $\Omega\Omega$ is $2$ and the stangeness $S=-6$, which is the most strange dibaryon state. Under the restriction of Pauli exclusion principle, the total wave function of the $\Omega\Omega$ system should be antisymmetric, which results in the even total spin $S=0$ or $S=2$ for the S-wave ($L=0$) coupling. The spin-parity quantum numbers is thus $J^{P}=0^{+}$ or $2^{+}$ and there is no isospin for such a system.

To date, the $\Omega\Omega$ dibaryon states have been studied in a quark potential model Wang et al. (1995), the chiral SU(3) quark model Zhang et al. (2000), and the lattice QCD simulations Buchoff et al. (2012); Gongyo et al. (2018). In a quark potential model of Ref. Wang et al. (1995), the authors calculated the effective interaction between two $\Omega$ baryons by including the quark delocalization and color screening. They found that the mass of the scalar $\Omega\Omega$ system was heavier than the $2m_{\Omega}$ threshold, resulting in a weekly repulsive interaction. This result was supported by the lattice QCD calculation at a pion mass of 390 MeV in Ref. Buchoff et al. (2012), where the weakly repulsive interactions were found for both the $S=0$ and $S=2$ $\Omega\Omega$ systems. In the chiral SU(3) quark model, the structure of the scalar $\Omega\Omega$ dibaryon was studied by solving a resonating group method equation Zhang et al. (2000). Their result suggested a deep attraction with binding energy around 100 MeV. In Ref. Yamada et al. (2015), the HAL QCD Collaboration investigated the interaction between two $\Omega$ baryons at $m_{\pi}=700$ MeV and found that the $\Omega\Omega$ potential has a repulsive core at short distance and an attractive well at intermediate distance. The phase shift obtained from the potential showed moderate attraction at low energies. Recently, the HAL QCD Collaboration has performed the $(2+1)$-flavor lattice QCD simulations on the $(\Omega\Omega)_{0^{+}}$ dibaryon at nearly physical pion mass $m_{\pi}=146$ MeV Gongyo et al. (2018). They found an overall attraction for the scalar $\Omega\Omega$ dibaryon with a small binding energy $B_{\Omega\Omega}=1.6$ MeV. These conflicting results from the phenomenological models and lattice simulations are inspiring more theoretical studies for the $\Omega\Omega$ dibaryon systems. In this work, we shall study the $\Omega\Omega$ dibaryons in both the scalar ${}^{1}S_{0}$ and tensor ${}^{5}S_{2}$ channels in the QCD sum rule method.

QCD Sum Rules for Dibaryon Systems: In the past several decades, the QCD sum rule has been used as a powerful non-perturbative approach to investigate the hadron properties, such as the hadron masses, magnetic moments, decay widths and so on Shifman et al. (1979); Reinders et al. (1985). To study the dibaryon systems in QCD sum rules, we need to construct the $\Omega\Omega$ interpolating currents by using the local Ioffe current for the $\Omega$ baryon Ioffe (1981, 1983)

[TABLE]

in which $s(x)$ represents the strange quark field, $a,b,c$ are the color indices, $\gamma_{\mu}$ is the Dirac matrix, $C=i\gamma_{2}\gamma_{0}$ is the charge conjugation matrix and $T$ the transpose operator. The $\Omega\Omega$ dibaryon interpolating currents are then composed in the molecular picture as

[TABLE]

with $J^{P}=0^{+}$, and

[TABLE]

with $J^{P}=2^{+}$. With these interpolating currents, we consider the two-point correlation functions for $\Omega\Omega$ dibaryon systems

[TABLE]

where $J^{\Omega\Omega}(x)$ and $J^{\Omega\Omega}_{\mu\nu}(x)$ can respectively couple to the scalar and tensor dibaryon states we interested in

[TABLE]

in which $f_{S}$ and $f_{T}$ are the coupling constants, $\epsilon_{\mu\nu}$ is the polarization tensor coupling to the spin-2 state. Noting that $J^{\Omega\Omega}_{\mu\nu}(x)$ is a symmetric operator, it can induce both the $J^{P}=0^{+}$ and $2^{+}$ invariant functions with the following projectors Chen et al. (2014, 2017a, 2017b)

[TABLE]

where

[TABLE]

The projectors $P_{0T}$, $P_{0S}$ and $P_{0TS}$ in Eq. (8) can be used to pick out different invariant functions induced by the the trace part (T), traceless symmetric part (S) and their cross term part (TS) from the tensor current respectively, which all couple to the $J^{P}=0^{+}$ channel with different coupling constants.

At the hadronic level, the invariant structure of correlation function $\Pi(q^{2})$ can be expressed as a dispersion relation

[TABLE]

where $b_{n}$ is an unknown subtraction constant. The spectral function can be usually written as a sum over $\delta$ functions by inserting intermediate states $|n\rangle$ with the same quantum numbers as the interpolating current

[TABLE]

where we adopt the “narrow resonance” approximation to describe the spectral function, and $m_{X}$ is the mass of the lowest-lying resonance $X$.

The correlation functions can also be calculated as the functions of various QCD condensates at the quark-gluonic level, using the operator product expansion(OPE) method. These results shall be equal to the correlation function in Eq. (10) via the quark-hadron duality, after performing the Borel transform to remove the unknown subtraction constants and suppress the continuum contributions

[TABLE]

in which $s_{0}$ is the continuum threshold and $M_{B}$ is the Borel mass. Then we can calculate the hadron mass as

[TABLE]

in which the spectral density $\rho(s)$ is evaluated in the quark-gluonic level as the function of various QCD condensates up to dimension-10, including the quark condensate $\langle\bar{s}s\rangle$, quark-gluon mixed condensate $\langle g_{s}\bar{s}\sigma\cdot Gs\rangle$, gluon condensate $\langle g_{s}^{2}GG\rangle$ and so on. For the dibaryon currents $J^{\Omega\Omega}(x)$ and $J^{\Omega\Omega}_{\mu\nu}(x)$, the expressions of spectral densities are lengthy so that we will show them in the Supplemental Material.

Prediction for the scalar $\Omega\Omega$ Dibaryon with $J^{P}=0^{+}$: We use the following values for various QCD parameters in our numerical analyses Tanabashi et al. (2018b); Yang et al. (1993); Jamin (2002); Gimenez et al. (2005); Ioffe and Zyablyuk (2003); Ovchinnikov and Pivovarov (1988); Chen et al. (2008)

[TABLE]

In Eq. (13), the hadron mass is extracted as a function of two free parameters: the Borel mass $M_{B}$ and the continuum threshold $s_{0}$. For the numerical analysis, we study the OPE convergence to determine the lower bound on the Borel mass $M_{B}$, requiring the quark condensate and the four-quark condensate be less than one third of the perturbative term while the dimension-10 condensate contribute less than $1\%$. For the tensor current $J^{\Omega\Omega}_{\mu\nu}(x)$, we show the two-point correlation function numerically of the trace part (T) coupling to $J^{P}=0^{+}$

[TABLE]

in which we take $s_{0}\to\infty$. We plot the correlation function $\Pi(\infty,\,M_{B}^{2})$ term by term in Fig. 1, in which the quark condensate $m_{s}\left<\bar{s}s\right>$ and four-quark condensate $\left<\bar{s}s\right>^{2}$ are much larger than any other non-perturbative term. Accordingly, the lower bound on the Borel mass can be obtained as $M_{B}^{2}\geq 5.5$ GeV2. In principle, the upper bound on the Borel mass can be obtained by studying the pole contribution. However, the high power behavior ($s^{7}$ for the perturbative term) in the dibaryon spectral density leads to small pole contribution and thus fails to constrain the upper bound on the Borel mass Chen et al. (2015b). Instead, we shall require good stability of the Borel curves. For the continuum threshold parameter, an optimized choice of $s_{0}$ is the value minimizing the variation of the hadron mass with the Borel mass. As shown in Fig. 2, the working region of $s_{0}$ can be obtained as 11.7 GeV2 $\leq s_{0}\leq 12.7$ GeV2.

Within these parameter values, we plot the Borel curves of the extracted hadron mass in Fig. 2. These Borel curves behave very good stability and give mass prediction of the scalar $\Omega\Omega$ dibaryon with $J^{P}=0^{+}$ (T) as

[TABLE]

in which the errors come from the uncertainties of $M_{B}$, $s_{0}$ and various QCD parameters in Eq. (14). The corresponding coupling constant can also be evaluated as

[TABLE]

As indicated in Eq. (8), the traceless symmetric part (S) and cross term (TS) in the tensor correlation function $\Pi_{\mu\nu,\,\rho\sigma}(q^{2})$ can also couple to the scalar $\Omega\Omega$ channel with $J^{P}=0^{+}$. However, our analyses indicate that the dibaryon sum rule from the traceless symmetric part (S) is unstable against the Borel mass, which cannot provide reliable mass prediction. For the case of TS cross term, the perturbative term in the OPE series is absent, and thus we will not use this invariant structure to study the scalar dibaryon.

Besides the tensor current, we shall also investigate the $\Omega\Omega$ diabryon with $J^{P}=0^{+}$ by using the scalar interpolating current $J^{\Omega\Omega}(x)$ in Eq. (2). Before performing the mass sum rule analysis, we study the spectral density and find that $\rho(s)$ is negative in a broad region 2 GeV2 $\leq s\leq 12$ GeV2, as shown in Fig. 3 ($\kappa=1.0$). Such behavior of the spectral density is distinct from the situations of the tensor current for all $J^{P}=0^{+}$ and $2^{+}$ channels, in which the negative regions of the spectral densities are rather small. In the case of the scalar current, the negative quark-gluon mixed condensate $m_{s}\left<g_{s}\bar{s}\sigma Gs\right>$ gives more contribution to the spectral density than those in the tensor current. To eliminate such negative effect, we consider the violation of factorization assumption by varying the four-quark condensate $\langle\bar{s}\bar{s}ss\rangle=\kappa\left<\bar{s}s\right>^{2}$ Reinders et al. (1985). As shown Fig. 3, the behavior of spectral density is good enough for $\kappa=1.7$ of the factorization assumption. We will use such modified spectral density in our mass sum rule analysis of the scalar current.

According to the same criteria for $s_{0}$ and $M_{B}$ as above, we choose parameters 11.5 GeV2 $\leq s_{0}\leq 12.7$ GeV2 and 5.0 GeV2 $\leq M_{B}^{2}\leq 12$ GeV2. The Borel curves in Fig. 4 show very good stability and the dibaryon mass is extracted as

[TABLE]

with the coupling constant

[TABLE]

Cosidering both Eq. (16) and Eq. (18), the mass of the scalar $\Omega\Omega$ dibaryon state with $J^{P}=0^{+}$ can be predicted as

[TABLE]

This value is about 15 MeV below the $2m_{\Omega}$ threshold Tanabashi et al. (2018b), suggesting the existence of a loosely bound molecular state of the scalar $\Omega\Omega$ dibaryon. Within errors, our prediction of the binding energy is in good agreement with the recent HAL QCD result Gongyo et al. (2018), while much smaller than the chiral SU(3) quark model calculation Zhang et al. (2000).

Prediction for the tensor $\Omega\Omega$ Dibaryon with $J^{P}=2^{+}$: To investigate the tensor $\Omega\Omega$ dibaryon state, we use the projector $P_{2S}^{P}$ in Eq. (8) to pick out the tensor invariant structure in $\Pi_{\mu\nu,\,\rho\sigma}(q^{2})$. Using this invariant function, we perform similar analysis as for the scalar channel. Unfortunately, we find that the Borel curves do not stabilize for any value of $s_{0}$ for the tensor channel, as shown in Fig. 5. The reason for that is the sign reverse of the quark condensate $m_{s}\left<\bar{s}s\right>$ contribution, which is the dominant non-perturbative term in the OPE series.

In order to make a mass prediction, we need to fix the value of $s_{0}$ under an additional criterion. The mass splitting between the tensor and scalar $\Omega\Omega$ dibaryons is mainly due to the spin-spin interaction of the two $\Omega$ baryons. This is similar with the mass difference between the non-strange dibaryons $D_{10}$ and $D_{12}$ in SU(6) symmetry, where $m_{D_{12}}-m_{D_{10}}\approx 284$ MeV Dyson and Xuong (1964); Clement (2017). For the heavier $\Omega\Omega$ systems, it is thus reasonable to fix the continuum threshold by assuming $\sqrt{s_{0,2^{+}}}-\sqrt{s_{0,0^{+}}}\approx$0.1-0.3 GeV. This criterion leads to the working region 12.9 GeV2 $\leq s_{0}\leq 14.4$ GeV2 (central value $s_{0}=13.7$ GeV2) in the tensor sum rule analysis. We plot the mass curves depending on $M_{B}^{2}$ in Fig. 5. One can trust this approximation providing a reliable mass prediction to the tensor $\Omega\Omega$ dibaryon with $J^{P}=2^{+}$

[TABLE]

with the coupling constant

[TABLE]

The above predicted mass is about 100 MeV below the $2m_{\Omega}$ threshold, which may imply a deeper molecular state of the tensor $\Omega\Omega$ dibaryon than the scalar channle. This result is different from the weakly repulsive interaction for the tensor $\Omega\Omega$ system obtained by the lattice QCD calculation at a pion mass of 390 MeV in Ref. Buchoff et al. (2012).

Summary and Discussion: In this work, we have investigated the scalar and tensor $\Omega\Omega$ dibaryon states in ${}^{1}S_{0}$ and ${}^{5}S_{2}$ channels with $J^{P}=0^{+}$ and $2^{+}$ respectively in the framework of QCD sum rules. We construct the scalar and tensor $\Omega\Omega$ dibaryon interpolating currents in a molecular picture, by which the spectral densities and two-point correlation functions are calculated up to dimension ten condensates.

We use different projectors to pick out spin-0 and spin-2 invariant structures from the tensor correlation function, and find that all the trace part (T), the traceless symmetric part (S) and the cross term (TS) couple to the $0^{+}$ dibaryon. However, the numerical analyses show that only the trace part from the tensor current and the scalar current can provide stable mass sum rules. Accordingly, we make reliable mass prediction for the scalar $\Omega\Omega$ dibaryon with $J^{P}=0^{+}$ to be $m_{\Omega\Omega,\,0^{+}}=(3.33\pm 0.22)\,\mbox{GeV}$. This value suggests the existence of a loosely bound scalar $\Omega\Omega$ dibaryon with a small binding energy around 15 MeV. Our result supports the attractive interaction existing in the scalar $\Omega\Omega$ channel, with the small binding energy in agreement with the HAL QCD simulation Gongyo et al. (2018). For the tensor $\Omega\Omega$ system, our result provides a mass prediction around $m_{\Omega\Omega,\,2^{+}}=(3.24\pm 0.23)\,\mbox{GeV}$, which may imply a deeper molecular state of the tensor $\Omega\Omega$ dibaryon than the scalar channel.

Since their strangeness $S=-6$ and masses are below the two-$\Omega$ threshold, the $\Omega\Omega$ dibaryons can only decay under the weak interaction. In such a molecular system, an $\Omega$ component in the $\Omega\Omega$ dibaryon can decay as a free particle while another $\Omega$ acting as the spectator throughout the whole process. Therefore the dominant decay modes for the scalar $\Omega\Omega$ dibaryon are $\Omega\Omega\to\Omega^{-}+\Lambda+K^{-}$, $\Omega\Omega\to\Omega^{-}+\Xi^{0}+\pi^{-}$ and $\Omega\Omega\to\Omega^{-}+\Xi^{-}+\pi^{0}$, while only the latter two processes exist for the tensor channel. Moreover, both of the scalar and tensor $\Omega\Omega$ dibaryons may decay into the $\Xi\Xi K$ final states. Such exotic strangeness $S=-6$ and doubly-charged $\Omega\Omega$ dibaryon states may be produced and identified in the heavy-ion collision experiments in the future.

ACKNOWLEDGMENTS

We thank Prof. Shi-Lin Zhu for useful discussions. This project is supported by the National Natural Science Foundation of China under Grants No. 11722540, the Fundamental Research Funds for the Central Universities.

Appendix A Supplemental Material

We collect here the expressions of spectral densities for both the scalar and tensor interpolating currents $J^{\Omega\Omega}(x)$ and $J^{\Omega\Omega}_{\mu\nu}(x)$ described in the text.

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For the scalar current with $J^{P}=0^{+}$:

[TABLE]

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For the trace part of the tensor current ($J^{P}=0^{+}$, T)

[TABLE]

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For the traceless part of the tensor current ($J^{P}=0^{+}$, S)

[TABLE]

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For the cross term of the tensor current ($J^{P}=0^{+}$, ST)

[TABLE]

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For the tensor current ($J^{P}=2^{+}$, S)

[TABLE]