Strong decays of fully-charm tetraquarks into di-charmonia

We study strong decays of the possible fully-charm tetraquarks recently observed by LHCb, and calculate their relative branching ratios through the Fierz rearrangement. Together with our previous QCD sum rule study [Phys. Lett. B 773, 247 (2017)], our results suggest that the broad structure around $6.2$-$6.8$~GeV can be interpreted as an $S$-wave $cc\bar c \bar c$ tetraquark state with $J^{PC} = 0^{++}$ or $2^{++}$, and the narrow structure around 6.9~GeV can be interpreted as a $P$-wave one with $J^{PC} = 0^{-+}$ or $1^{-+}$. These structures were observed in the di-$J/\psi$ invariant mass spectrum, and we propose to confirm them in the di-$\eta_c$, $J/\psi h_c$, $\eta_c \chi_{c0}$, and $\eta_c \chi_{c1}$ channels. We also propose to search for their partner states having the negative charge-conjugation parity in the $J/\psi \eta_c$, $J/\psi \chi_{c0}$, $J/\psi \chi_{c1}$, and $\eta_c h_c$ channels.


I. INTRODUCTION
Very recently, the LHCb Collaboration reported their preliminary results on possible fully-charm tetraquarks [1]. They investigated the di-J/ψ invariant mass spectrum, where they observed a broad structure ranging from 6.2 to 6.8 GeV and a narrow structure at around 6.9 GeV with a global significance of more than 5σ. Especially, they assumed the latter narrow structure to be a resonance with the Breit-Wigner lineshape, and measured its mass and width to be either based on no-interference fit, or based on the simple model with interference. The above results are in a remarkable coincidence with our previous QCD sum rule predictions [2], as partly shown in Table I. There we used eighteen diquark-antidiquark [QQ][QQ] currents to perform QCD sum rule analyses, and calculated mass spectra of cccc and bbbb tetraquark states. Although the currents used there do not explicitly contain derivatives, they can still have both positive-and negativeparities, and their quantum numbers can be J P C = 0 ++ /0 −+ /0 −− /1 ++ /1 +− /1 −+ /1 −− /2 ++ .  [2] using the QCD sum rule method. Definitions of currents will be given in Sec. II.
There are altogether twelve currents of the positive parity, four of which well correspond to the S-wave cccc tetraquark states within the quark-model picture. We used them to perform QCD sum rule analyses, and the masses were predicted to be about 6.5 GeV [2]. Accordingly, the broad structure observed by LHCb at around 6.2-6.8 GeV [1] can be interpreted as an S-wave cccc tetraquark state. There are altogether seven currents of the negative parity. We also used them to perform QCD sum rule analyses, and the masses were predicted to be about 6.9 GeV [2]. Accordingly, the narrow structure observed by LHCb at around 6.9 GeV [1] can be interpreted as a P -wave cccc tetraquark state.
In this paper we shall study decay properties of fullycharm tetraquark states, based on our previous QCD sum rule study [2]. We shall investigate both S-and P -wave cccc tetraquark states. Assuming them to be compact diquark-antidiquark [cc][cc] states, we shall apply the Fierz rearrangement of the Dirac and color indices to calculate their relative branching ratios. This method has been used in Refs. [35][36][37] to study the Z c (3900), X(3872), and P c states. This paper is organized as follows. In Sec. II we systematically construct diquark-antidiquark [QQ][QQ] currents of both positive and negative parities, and use the Fierz rearrangement to transform them into mesonmeson [QQ][QQ] currents. Based on the obtained Fierz identities, in Sec. III we study their possible fall-apart decays, and calculate their relative branching ratios within the naive factorization scheme. The results are summarized in Sec. IV.

II. CURRENTS AND FIERZ IDENTITIES
All the diquark-antidiquark [QQ][QQ] currents without derivatives have been systematically constructed in Ref. [2], with only one current of J P C = 2 ++ missing: Here Q a is the heavy quark field with the color index a, and Γ µν αβ is the projection operator, Γ αβ;µν = g αµ g βν + g αν g βµ − 1 2 g αβ g µν .
There are altogether twelve currents of the positive parity. Four of them well correspond to the S-wave [QQ][QQ] tetraquark states within the quark-model picture, and we shall only investigate these four currents in the present study. There are altogether seven currents of the negative parity, and we shall study all of them in the present study. Detailed expressions are given in the following subsections, together with the Fierz identities to transform them into meson-meson [QQ][QQ] currents.
A. Currents of the positive parity There are two S-wave diquarks, the "good" diquark of J P = 0 + and the "bad" one of J P = 1 + (other are "worse") [38]. We can combine them to construct Swave tetraquark states. To do this we follow the diquarkantidiquark model proposed in Refs. [39,40]. In this model the S-wave tetraquark states can be written in the spin basis as |s,s J , where s = s QQ ands = sQQ are the diquark and antidiquark spins, respectively.
There are altogether four S-wave QQQQ tetraquark states, denoted as |X; J P C : Similar to the quark-model picture, there are two Swave diquark fields: We can combine them to construct four tetraquark currents corresponding to Eqs. (7): The tensor diquark field Q T a Cσ µν γ 5 Q b couples to both J P = 1 + and 1 − channels. However, its positive-parity Besides it, there exists another current directly corresponding to |X 3 ; 1 +− : but this current contains both positive-and negativeparity components, so we do not investigate it in the present study.
Here η 0 ++ 1,2,3,4,5 are the meson-meson currents of J P C = 0 ++ , η 1 +− 6α,7α are of J P C = 1 +− , and η 2 ++ 8αβ,9αβ,10αβ are of J P C = 2 ++ : Note that there are the same number of color-octetcolor-octet meson-meson currents, while all of them can be related to the above color-singlet-color-singlet mesonmeson currents, e.g., B. Currents of the negative parity P -wave tetraquark states contain one unit of orbital excitation. In the diquark-antidiquark picture this or-bital excitation can be between the diquark and antidiquark, or it can also be inside the diquark/antidiquark. Hence, P -wave tetraquark states are more complicated than S-wave ones. In a recent Ref. [28] the authors used a nonrelativistic potential quark model to systematically classify all the P -wave QQQQ tetraquark states, where they found altogether twenty states. However, without using derivatives, we can not construct all their corresponding tetraquark currents. Instead, in the present study we shall use all the negative-parity tetraquark currents without derivatives to study their decay properties.
All the diquark-antidiquark [QQ][QQ] currents of the negative parity have been systematically constructed in Ref. [2]. There are two currents of J P C = 0 −+ : There is only one current of J P C = 0 −− : There are two currents of J P C = 1 −+ : There are two currents of J P C = 1 −− : − Q T a Cγ µ Q bQa σ αµ CQ T b . Similar to Eqs. (7), we can write P -wave tetraquark states in the spin-parity basis as |s P ,sP J , where P and P are the diquark and antidiquark parities, respectively. The above tetraquark currents correspond to the following P -wave QQQQ tetraquark states: The realistic physical states can be different from these states. However, if there exists some tetraquark current well (better) coupling to the physical state, the results extracted from this current should also be (more) consistent with that state. Hence, we investigate all the negative-parity tetraquark currents without derivatives in the present study, given that the internal structure of P -wave QQQQ tetraquark states are still unknown.

III. RELATIVE BRANCHING RATIOS
In this section we investigate possible decay channels of fully-charm tetraquark states, both qualitatively and quantitatively. According to the recent LHCb experiment [1], we assume the S-and P -wave cccc tetraquark states to have the masses about 6.5 GeV and 6.9 GeV, respectively.
As depicted in Fig. 1, when one heavy quark and one heavy antiquark meet each other and the rest heavy quark and antiquark also meet each other at the same time, a compact diquark-antidiquark [cc][cc] state can fall-apart decay into two charmonium mesons. This process can be described by the Fierz identities given in Eqs. (14)(15)(16)(17) and Eqs. (29)(30)(31)(32)(33)(34)(35). To study it we need the couplings of charmonium operators to charmonium states, which have been well studied in the literature [46][47][48][49] and summarized here in Table II.  can also couple to this channel, so that the state |X 1 ; 0 ++ can decay into the J/ψJ/ψ final state. Similarly, we can derive six other possible channels to be η c η c , χ c0 χ c0 , χ c1 χ c1 , h c h c , η c χ c1 , and J/ψh c . Among them, the J/ψJ/ψ, η c η c , and η c χ c1 channels are kinematically allowed.
In principle one needs the coupling of J 0 ++ 1 to |X 1 ; 0 ++ as an input to quantitatively calculate partial decay widths of these channels. We define this coupling to be while it is not necessary any more if one only calculates relative branching ratios. Moreover, because couplings of meson operators to meson states are well studied but couplings of tetraquark currents to tetraquark states are not, the decay constant f X1 is not so well determined compared to the meson decay constants listed in Table II. Accordingly, relative branching ratios can be calculated more reliably than partial decay widths. To calculate relative branching ratios, we just need to keep f X1 as an unfixed parameter, calculate partial decay widths, and finally remove f X1 . Still take Eq. (14) as an example, the couplings of |X 1 ; 0 ++ to the J/ψJ/ψ and η c η c channels can be extracted from it to be: After calculating the two partial decay widths Γ |X1;0 ++ →J/ψJ/ψ and Γ |X1;0 ++ →ηcηc , we can remove  [49] the parameter f X1 and obtain: Similarly, we can add the η c χ c1 channel and obtain: Following the same procedures, we shall separately investigate the S-and P -wave cccc tetraquark states in the following subsections.
Quantitatively, we assume masses of the S-wave cccc tetraquark states to be about 6.5 GeV, and obtain: Relative branching ratios of |X 3 ; 1 +− are not given, because we only derive one of its possible decay channels.
• Eq. (31) suggests the possible decay channels of |X 7 ; 0 −− to be J/ψη c and J/ψχ c1 . These two channels are both kinematically allowed.
Quantitatively, we assume masses of the P -wave cccc tetraquark states to be about 6.9 GeV, and obtain:

IV. SUMMARY AND DISCUSSIONS
Very recently, the LHCb Collaboration reported their preliminary results on possible fully-charm tetraquarks [1]. They investigated the di-J/ψ invariant mass spectrum, where they observed a broad structure ranging from 6.2-6.8 GeV and a narrow structure at around 6.9 GeV with a global significance of more than 5σ. Their results are in a remarkable coincidence with our previous QCD sum rule predictions [2], as discussed already in Sec. I.
In this paper we have calculated as many as possible relative branching ratios, and the results are given in Eqs. (41)(42)(43)(44)(45)(46)(47)(48)(49)(50) and summarized in Table III. Before drawing conclusions, let us generally discuss about their uncertainty. In the present study we have worked within the naive factorization scheme, so our uncertainty is larger than the well-developed QCD factorization method [50][51][52], that is at the 5% level when being applied to study weak and radiative decay properties of conventional (heavy) hadrons [53]. On the other hand, the tetraquark decay constants, such as f X1 , are removed when calculating relative branching ratios. This significantly reduces our uncertainty. Hence, we roughly estimate our uncertainty to be at the X +100% − 50% level. The LHCb experiment [1] observed a broad structure at around 6.2-6.8 GeV. It can be interpreted as an Swave cccc tetraquark state, whose mass was predicted to be about 6.5 GeV in our previous QCD sum rule study [2]. To study its fall-apart decays, we use the four cccc currents, which well correspond to the four S-wave cccc tetraquark states. The obtained results are summarized in the 3rd-6th rows of Table III. Our results suggest that the broad structure observed by LHCb at around 6.2-6.8 GeV [1] has the quantum numbers J P C = 0 ++ or 2 ++ . We propose to confirm it in the di-η c channel. This channel is also helpful to determine its quantum numbers and understand its internal structure. Besides, we propose to search for another J P C = 1 +− state in the J/ψη c channel also at around 6.5 GeV.
The LHCb experiment [1] observed a narrow structure at around 6.9 GeV. It can be interpreted as a Pwave cccc tetraquark state, whose mass was predicted to be also about 6.9 GeV in our previous QCD sum rule study [2]. To study its fall-apart decays, we investigate all the negative-parity tetraquark currents without derivatives, and the results are summarized in the 7th-13th rows of Table III. Their correspondences to physical states are not so clear. However, if some of them well (better) couples to the physical state, the results extracted from this current should also be (more) consistent with that state. Note that the internal structure of P -wave cccc tetraquark states are still unknown and difficult to be known, since there can be as many as twenty states [28].
Our results suggest that the narrow structure observed by LHCb at around 6.9 GeV [1] has the quantum numbers J P C = 0 −+ or 1 −+ . We propose to confirm it in the di-η c , J/ψh c , η c χ c0 , and η c χ c1 channels. These channels are helpful to determine its quantum numbers as well as understand its internal structure. We also propose to search for the J P C = 0 −− and 1 −− states in the J/ψη c , J/ψχ c0 , J/ψχ c1 , and η c h c channels at around 6.9 GeV. III: Relative branching ratios of the S-and P -wave cccc tetraquark states. In the second column we write these state in the spin-parity basis as |s P cc ,sP cc J ; s P cc andsP cc are the diquark and antidiquark spins-parities, respectively; the subscripts a/b denote |Xcc, Ycc a J = 1 √ 2 (|Xcc, Ycc J + |Ycc, Xcc J ) and |Xcc, Ycc b J = 1 √ 2 (|Xcc, Ycc J − |Ycc, Xcc J ). In the 3rd-7th columns we show branching ratios relative to the J/ψJ/ψ channel, such as B(X→ηcηc) B(X→J/ψJ/ψ) in the 4th column. In the 8th-11th columns we show branching ratios relative to the J/ψηc channel, such as B(X→J/ψχ c0 ) B(X→J/ψηc) in the 9th column.