Decay properties of the $Z_c(3900)$ through the Fierz rearrangement

We systematically construct all the tetraquark currents/operators of $J^{PC} = 1^{+-}$ with the quark configurations $[cq][\bar c \bar q]$, $[\bar c q][\bar q c]$, and $[\bar c c][\bar q q]$ ($q=u/d$), and derive their relations through the Fierz rearrangement of the Dirac and color indices. Using the transformations of $[qc][\bar q \bar c] \to [\bar c c][\bar q q]$ and $[\bar c q][\bar q c]$, we study decay properties of the $Z_c(3900)$ as a compact tetraquark state; while using the transformation of $[\bar c q][\bar q c] \to [\bar c c][\bar q q]$, we study its decay properties as a hadronic molecular state.


I. INTRODUCTION
In the past twenty years many charmonium-like XY Z states were discovered in particle experiments [1]. All of them are good multiquark candidates, and their relevant experimental and theoretical studies have significantly improved our understanding of the strong interaction at the low energy region. Especially, in 2013 BE-SIII reported the Z c (3900) + in the Y (4260) → J/ψπ + π − process [2], which was later confirmed by Belle [3] and CLEO [4]. Since it couples strongly to the charmonium and yet it is charged, the Z c (3900) + is not a conventional charmonium state and contains at least four quarks. It is quite interesting to understand how it is composed of these four quarks, and there have been various models developed to explain this, such as a compact tetraquark state composed of a diquark and an antidiquark [5,6], a loosely-bound hadronic molecular state composed of two charmed mesons [7][8][9][10], a hadro-quarkonium [11], or due to the kinematical threshold effect [12,13], etc. We refer to reviews [14][15][16][17][18] for detailed discussions.
The charged charmonium-like state Z c (3900) of J P C = 1 +− [19] has been observed in the J/ψπ and DD * channels [2,3,20,21], and there was some events in the h c π channel [22]. In a recent BESIII experiment [23], evidence for the Z c (3900) → η c ρ decay was reported with a statistical significance of 3.9σ at √ s = 4.226 GeV, and the relative branching ratio was evaluated to be 2.2 ± 0.9 at the same center-ofmass energy. This ratio has been studied by many theoretical methods/models [24][25][26][27][28][29][30][31][32], and was suggested in Ref. [33] to be useful to discriminate between the compact tetraquark and hadronic molecule scenarios. As summarized in Table I, this ratio was calculated in many molecular models, but the extracted values are highly model dependent. Hence, it would be useful to derive a model independent result, and it would be even better if one * Electronic address: hxchen@buaa.edu.cn could do this within the same framework for both the tetraquark and molecule scenarios.
Again, the Fierz rearrangement can be applied to relate them. Based on these relations, we shall extract some decay properties of the Z c (3900) in this paper.
There are eight independent [cq][cq] currents of J P C = 1 +− , which have been systematically constructed in Ref. [48]. Here we choose one of them, where C is the charge-conjugation matrix, the subscripts a · · · e are color indices, and the sum over repeated indices is taken. This current would strongly couple to the Z c (3900), if it has the same internal structure (internal symmetry) as that state. The above current is useful from the viewpoints of both effective field theory and QCD sum rules. Note that there are various quark-based effective field theories, which have been successfully applied to describe the meson and baryon systems, such as the Non-Relativistic QCD for the heavy quarkonium system [49,50]: Type-II diquark-antidiquark model [33] tetraquark 0.95 QCD sum rules [24] 0.57 QCD sum rules [25] 1.1 QCD sum rules [26] 1.28 covariant quark model [27] 4.6 +2.5 −1.7 × 10 −2 Non-Relativistic effective field theory [33] hadronic 0.12 light front model [28] molecule 0.68 × 10 −2 effective field theory [29] 1.78 covariant quark model [27] We refer to Ref. [51] for detailed review of this method. The above Lagrangian contains four four-fermion operators, which can be used to study the annihilation width of a heavy quarkonium into light particles. In this method the Fierz rearrangement is applied to decouple the Dirac and color indices that connect the short-distance part to the long-distance part [50]. Compared with this, the quark-based effective field theory for the multiquark system is much more difficult [18]. Let us attempt to do this for the Z c (3900). Based on Eq. (2) we can write down an eight-quark operator (the same argument applies for other Lagrangians containing η Z µ ): where c 0 is a constant. Then we can use the Fierz rearrangement to transform it to be λ n ab λ n cdc a γ 5 c bqc γ µ q d + 1 4 λ n ab λ n cdc a γ µ c bqc γ 5 q d − i 4 λ n ab λ n cdc a γ ν γ 5 c bqc σ µν q d Detailed discussions on this transformation will be given below.
Considering that the meson operators,qγ 5 q,qγ µ q, cγ 5 c, andcγ µ c couple to the π, ρ, η c , and J/ψ mesons respectively (see Table II below), the above eight-quark operator can describe the fall-apart decays of the Z c (3900) into the η c ρ and J/ψπ final states simultaneously, together with some other possible decay channels. In order to extract the widths of these decays, one still needs to do further calculations, which we shall not study any more. However, their relative branching ratios can be extracted much more easily, which are also useful and important to understand the nature of the Z c (3900) [52].
The current η Z µ can also be investigated from the viewpoint of QCD sum rules [53,54]. We assume it couples to the Z c (3900) through After the Fierz rearrangement, η Z µ transforms to the long expression inside Eq. (5). Through the first and second terms, it couples to the η c ρ and J/ψπ channels simultaneously: Again, these two equations can be easily used to calculate the relative branching ratio R Zc . Detailed discussions on this will be given below.
In the above equations we have followed the idea of the QCD factorization method [55][56][57], which has been widely and successfully applied to study weak decay properties of (heavy) hadrons. In the present study we just need to replace the weak-interaction Lagrangian by some interpolating current, and the similar technics can apply here, together with the Fierz arrangement, to study strong decay properties of the Z c (3900). Note that a similar arrangement of the spin and color indices in the nonrelativistic case was used to study strong decay properties of the Z c (3900) in Refs. [8,58,59].
This paper is organized as follows. In Sec. II we systematically construct all the tetraquark currents of , and their relations are also derived in this section by using the Fierz rearrangement of the Dirac and color indices. In Sec. III we discuss the couplings of meson operators to meson states, and list those which are needed in the present study. In Sec. IV and Sec. V we extract some decay properties of the Z c (3900), separately for the compact tetraquark interpretation and the hadronic molecule interpretation. The obtained results are discussed and summarized in Sec. VI.
II. TETRAQUARK CURRENTS OF J P C = 1 +− AND THEIR RELATIONS By using the c,c, q,q quarks (q = u/d), one can construct three types of tetraquark currents, as illustrated in Fig. 1: where Γ i are Dirac matrices, C is the charge-conjugation matrix, the subscripts a, b, c, d are color indices, and the sum over repeated indices is taken. One usually call η µ (x, y) the diquark-antidiquark current, and ξ µ (x, y) and θ µ (x, y) the mesonic-mesonic currents. We separately construct them as follows.
There are altogether eight independent [qc][qc] currents of J P C = 1 +− [48]: Here we have omitted the coordinates x and y for simplicity. Their combinations, In the "type-II" diquark-antidiquark model proposed in Ref. [6], the ground-state tetraquarks can be written in the spin basis as |s qc , sqc J , where s qc and sqc are the charmed diquark and antidiquark spins, respectively. There are two ground-state diquarks: the "good" one of J P = 0 + and the "bad" one of J P = 1 + [60]. By combining them, the Z c (3900) was interpreted as a diquark-antidiquark state of J P C = 1 +− in Ref. [6]: The interpolating current having the identical internal structure is just the current η Z µ given in Eq. (2), which has been well studied in Ref. [61]: Here we have explicitly chosen the quark content [uc][dc] for the positive-charged one Z c (3900) + .

D. Fierz rearrangement
We have applied the Fierz rearrangement of the Dirac and color indices to systematically study light baryon and tetraquark operators/currents in Refs. [34][35][36][37][38][39][40][41][42][43][44][45][46][47]. It can also be used to relate the above three types of tetraquark currents. To do this, we need to use a) the Fierz transformation [64] in the Lorentz space to rearrange the Dirac indices, and b) the color rearrangement in the color space to rearrange the color indices. All the necessary equations can be found in Sec. 3.3.2 of Ref. [65].

III. MESON OPERATORS
There are altogether six types of meson operators: q a q a ,q a γ 5 q a ,q a γ µ q a ,q a γ µ γ 5 q a ,q a σ µν q a , andq a σ µν γ 5 q a . The last two can be related to each other through The couplings of these operators to meson states are already well understood, i.e., some of them have been measured in particle experiments, and some of them have been studied and calculated by various theoretical methods, such as Lattice QCD and QCD sum rules, etc.
In the present study we need the following couplings, as summarized in Table II: 1. The scalar operators J S =q a q a and I S =c a c a of J P C = 0 ++ couple to scalar mesons. In Ref. [66] the authors used the method of QCD sum rules and extracted the coupling of I S to χ c0 (1P ) to be where See also discussions in Refs. [67][68][69]. The light scalar mesons have a complicated nature [70], so we shall not investigate their relevant decay channels in the present study.
6. The Z c (3900) is above the DD * threshold, so we need the couplings of O P =q a iγ 5 c a and O A µ = c a γ µ γ 5 q a to the D meson [1]: 0|c a γ µ γ 5 u a |D 0 (p) = ip µ f D , and the couplings of O V µ =c a γ µ q a and O T µν = q a σ µν c a to the D * meson [95]: We do not find any theoretical study on the transverse decay constant f T D * , so we simply fit among the decay constants, See also discussions in Refs. [96,97].
7. The Z c (3900) → DD * 0 → DDπ decay is kinematically allowed, so we need the coupling of O S =q a c a to the D * 0 meson [98]: 0|d a c a |D * + where See also discussions in Refs. [99, 100].
In this section we shall investigate the former compact tetraquark interpretation, whose relevant current η Z µ (x, y) has been given in Eq. (11). This current can be transformed to θ i µ (x, y) and ξ i µ (x, y) according to Eqs. (16)(17)(18), through which we shall extract some decay properties of the Z c (3900) as a compact tetraquark state in the following subsections.
As depicted in Fig. 2, when the c andc quarks meet each other and the u andd quarks meet each other at the same time, a compact tetraquark state can decay into one charmonium meson and one light meson. This process for |0 qc 1qc; 1 +− can be described by the transformation (16): where we have only kept the direct fall-apart process described by θ 1,2,3,4 µ , but neglected the O(α s ) corrections described by θ 5,6,7,8 µ .
Together with Table II, we extract the following decay channels from the above transformation: 1. The decay of |0 qc 1qc; 1 +− into η c ρ is contributed by both I P × J V µ and I A,ν × J T µν : , where c 1 is an overall factor, related to the coupling of η Z µ (x, y) to the Z c (3900) + as well as the dynamical process (x, y) =⇒ (x ′ , y ′ ). The two coupling constants g S ηcρ and g D ηcρ are defined for the S-and D-wave |0 qc 1qc; 1 +− → η c ρ decays, respectively.
2. The decay of |0 qc 1qc; 1 +− into J/ψπ is contributed by both I V µ × J P and I T µν × J A,ν : . The two coupling constants g S ψπ and g D ψπ are defined for the S-and D-wave |0 qc 1qc; 1 +− → J/ψπ decays, respectively. Both of them contain the same overall factor c 1 .
6. The decay of |0 qc 1qc; 1 +− into h c π is contributed by I T µν × J A,ν : This process is kinematically allowed.
8. The decay of |0 qc 1qc; 1 +− into h c a 1 is contributed by I T µν × J A,ν :

This process is kinematically forbidden.
Summarizing the above results, we obtain numerically From these coupling constants, we further obtain the following relative branching ratios, which are kinematically allowed: Besides them, the following decay chains are also possible but with quite small partial decay widths: As depicted in Fig. 3, when the c andd quarks meet each other and the u andc quarks meet each other at the same time, a compact tetraquark state can decay into two charmed mesons. This process for |0 qc 1qc; 1 +− can be described by the transformation (17): Again, we have only kept the direct fall-apart process described by ξ 2,3 µ , but neglected the O(α s ) corrections described by ξ 6,7 µ . The term ξ 2 µ couples to the D * D * and D * D 1 final states, and the term ξ 3 µ couples to the DD * 0 and D 1D * 0 final states. Among them, only the |0 qc 1qc; 1 +− → DD * 0 → DDπ decay is kinematically allowed, contributed by ξ 3 where c 2 is an overall factor. Numerically, we obtain Comparing the |0 qc 1qc; 1 +− → DD * 0 → DDπ decay studied in the present subsection with the |0 qc 1qc; 1 +− → J/ψπ and |0 qc 1qc; 1 +− → η c ρ decays studied in the previous subsection, we obtain B(|0 qc 1qc; 1 +− → DD * 0 +DD * 0 → DDπ) B(|0 qc 1qc; 1 +− → J/ψπ + η c ρ) The current η Z µ (x, y) does not correlate with the two terms ξ 1 µ = −iO V µ × O P and ξ 4 µ = O A,ν × O T µν , both of which can couple to the DD * final state. This suggests that |0 qc 1qc; 1 +− does not decay to the DD * final state with a large branching ratio, so that Eqs. (57) and (59)  If the above two processes investigated in Sec. IV A and Sec. IV B happen at the same time, we can use the transformation (18), i.e., |0 qc 1qc; 1 +− can decay into one charmonium meson and one light meson as well as two charmed mesons at the same time, which process is described by the color-singlet-color-singlet currents θ 1,2,3,4 µ and ξ 1,2,3,4 µ together: Here we have kept all the terms, and there is no · · · in this equation.
Comparing the above equation with Eqs. (41) and (53), we arrive at the same conclusions as Sec. IV A and Sec. IV B, just with the overall factors c 1 and c 2 replaced by others.
D. Mixing with |1qc1qc; 1 +− The relative branching ratio R Zc calculated in Sec. IV A is just 0.059, significantly smaller than the BESIII measurement R Zc = 2.2 ± 0.9 at √ s = 4.226 GeV [23]. In this subsection we slightly modify the internal structure of the Z c (3900) to reevaluate this ratio.
We try to add this |1 qc 1qc; 1 +− component in this subsection. The interpolating current having the identical internal structure is so that |x qc 1qc; 1 +− can be described by which transforms according to Eq. (16) as: Note that the two mixing angles θ 1 and θ ′ 1 are not necessarily the same (probably not the same), but they can be related to each other, i.e., To solve this relation, we need to know the couplings of η Z µ and η Z ′ µ to |0 qc 1qc; 1 +− and |1 qc 1qc; 1 +− , which we shall not investigate in the present study. Anyway, we can plot the three ratios: as functions of the mixing angle θ ′ 1 , which are shown in Fig. 4. We find that R ψπ decreases and R ηcρ increases, so that the ratio R increases rapidly, as the mixing angle B(|x qc 1qc; 1 +− → χ c1 ρ → χ c1 ππ) B(|x qc 1qc; 1 +− → J/ψπ) = 1.5 × 10 −5 .
The first ratio R is 2.2, which is the same as the BESIII measurement R Zc = 2.2 ± 0.9 [23]. The decay of |x qc 1qc; 1 +− into two charmed mesons can be described by the current η mix µ (x, y) together with the transformation (17): Hence, |x qc 1qc; 1 +− can decay into the DD * final state, which is consistent with the BESIII observations [20,21]. Moreover, it was proposed in Ref. [58] that: to enable the decay of the Z c (3900), a constituent of a diquark must tunnel through the barrier of the diquark-antidiquark potential, but this tunnelling for heavy quarks is exponentially suppressed compared to that for light quarks, so the compact tetraquark couplings are expected to favour the open charm modes with respect to charmonium ones. According to this, c 2 may be significantly larger than c 1 , so that |x qc 1qc; 1 +− may mainly decay into two charmed mesons.

V. DECAY PROPERTIES OF THE Zc(3900) AS A HADRONIC MOLECULAR STATE
Another possible interpretation of the Z c (3900) is the DD * hadronic molecular state of J P C = 1 +− [7-10], i.e., |DD * ; 1 +− defined in Eq. (13). Its relevant current ξ Z µ (x, y) has been given in Eq. (14). We can transform this current to θ i µ (x, y) according to the transformation (19), through which we shall extract some decay properties of the Z c (3900) as a hadronic molecular state in the following subsections.
As depicted in Fig. 5, when the c andc quarks meet each other and the u andd quarks meet each other at the same time, a hadronic molecular state can decay into one charmonium meson and one light meson. This process for |DD * ; 1 +− can be described by the transformation (19): where we have only kept the direct fall-apart process described by θ 1,2,3,4 µ , but neglected the O(α s ) corrections described by θ 5,6,7,8 µ .
We repeat the same procedures as those done in Sec. IV A, and extract the following coupling constants from this transformation: The above coupling constants are related to the S-and D-wave |DD * ; 1 +− → η c ρ decays, the S-and D-wave |DD * ; 1 +− → J/ψπ decays, and the |DD * ; 1 +− → η c b 1 , χ c1 ρ, χ c1 b 1 , h c π, J/ψa 1 , h c a 1 decays, respectively. All of them contain an overall factor c 4 . Using the above coupling constants, we further obtain These values are surprisingly the same as Eqs. (51), obtained in Sec. IV A for the compact tetraquark state |0 qc 1qc; 1 +− .
Assuming the Z c (3900) to be the DD * hadronic molecular state of J P C = 1 +− , it can naturally decay to the DD * final state, which fall-apart process can be described by itself: The decay of |DD * ; 1 +− into the DD * final state is contributed by this term to be where c 5 is an overall factor, and it is probably significantly larger than c 4 . Numerically, we obtain Comparing the |DD * ; 1 +− → DD * decay studied in the present subsection with the |DD * ; 1 +− → J/ψπ and |DD * ; 1 +− → η c ρ decays studied in the previous subsection, we obtain The current ξ Z µ (x, y) does not correlate with the term ξ 3 µ = O A µ × O S , so that |DD * ; 1 +− does not decay into the DD * 0 final state: Eqs. (78) and (79) to |DD * ; 1 +− in this subsection to reevaluate the ratio R Zc . The interpolating current having the same internal structure as |D * D * ; 1 +− is so that we can use to described the mixed molecular state |D ( * )D * ; 1 +− = cos θ 2 |DD * ; 1 +− +sin θ 2 |D * D * ; 1 +− .
After fine-tuning θ ′ 2 = −8.8 o , we obtain which values are the same as Eqs. (73), obtained in Sec. IV A for the mixed compact tetraquark state |x qc 1qc; 1 +− . Actually, we can also plot the following three ratios as functions of the mixing angle θ ′ 2 , and the obtained figures are just identical to Fig. 4, where R ψπ , R ηcρ , and R are shown as functions of θ ′ 1 . We also obtain suggesting that |D ( * )D * ; 1 +− mainly decays into two charmed mesons.

VI. SUMMARY AND DISCUSSIONS
In this paper we systematically construct all the tetraquark currents/operators of J P C = 1 +− with the quark content ccqq.  Altogether we have investigated the two-body decays of the Z c (3900) into J/ψπ, η c ρ, h c π, and DD * . We have also investigated the three-body decays Z c (3900) → χ c1 ρ → χ c1 ππ and Z c (3900) → DD * 0 +DD * 0 → DDπ. Their relative branching ratios are calculated and summarized in Table III, where we have investigated the following interpretations of the Z c (3900): • In the second and third columns of Table III, |0 qc 1qc; 1 +− and |x qc 1qc; 1 +− denote the compact tetraquark states of J P C = 1 +− , defined in Eq. (10) and Eq. (61), respectively.
• Relative branching ratios of the |DD * ; 1 +− decays into one charmonium meson and one light meson are the same as those of the |0 qc 1qc; 1 +− decays.
After taking proper mixing angles, relative branching ratios of the |D ( * )D * ; 1 +− decays into one charmonium meson and one light meson are also the same as those of the |x qc 1qc; 1 +− decays. This suggests that one may not discriminate between the compact tetraquark and hadronic molecule scenarios by only investigating relative branching ratios of the Z c (3900) decays into one charmonium meson and one light meson.
Hence, the D-wave decays are important and so can not be neglected. We also obtain:   The first one is consistent with Ref. [33], and the second one is consistent with Ref. [25].
• Actually, there is still one parameter not considered in above analyses, that is the phase angle θ between S-and D-wave coupling constants, for example, between g S ηcρ and g D ηcρ : g S ηcρ /|g S ηcρ | = e iθ × g D ηcρ /|g D ηcρ | .
This parameter is unknown and so not fixed, because in QCD sum rules one can only calculate the modular square of the decay constant, such as |f ηc | 2 . This might also be the case for Lattice QCD and light front model, for example, see the different definitions of f ηc in Refs. [73,81].
We fix the phase angle between all the S-and Dwave coupling constants to be θ = π, and redo the previous calculations. The results are summarized in Table IV.
Based on Tables III and IV, we conclude this paper. In this paper we systematically construct all the tetraquark currents/operators of J P C = 1 +− with the quark configurations [cq][cq], [cq][qc], and [cc][qq] (q = u/d), and derive their relations through the Fierz rearrangement of the Dirac and color indices. Using these relations, we study decay properties of the Z c (3900) under both the compact tetraquark and hadronic molecule interpretations, within the same framework.
In both the compact tetraquark and hadronic molecule scenarios we obtain the same relative branching ratios of the Z c (3900) decays into one charmonium meson and one light meson, such as R Zc ≡ B(Zc(3900)→ηcρ) B(Zc(3900)→J/ψπ) . Especially, the BESIII measurement R Zc = 2.2 ± 0.9 at √ s = 4.226 GeV [23] can be explained in the compact tetraquark picture after considering the mixing shown in Eq. (91), while it can also be explained in the hadronic molecule picture after considering the similar mixing shown in Eq. (92). Both the hadronic molecule states |DD * ; 1 +− and |D ( * )D * ; 1 +− mainly decay into two charmed mesons, while the mixed compact tetraquark state |x qc 1qc; 1 +− may also mainly decay into two charmed mesons as discussed in Ref. [58].
Our results suggest that the possible decay channels of the Z c (3900) are: a) the two-body decays Z c (3900) → J/ψπ, Z c (3900) → η c ρ, Z c (3900) → h c π, and Z c (3900) → DD * , b) the three-body decays Z c (3900) → χ c1 ρ → χ c1 ππ and Z c (3900) → DD * 0 +DD * 0 → DDπ, and c) the many-body decay chains Z c (3900) → J/ψa 1 → J/ψρπ → J/ψ + 3π and Z c (3900) → η c b 1 → η c ωπ → η c + 4π. Their relative branching ratios calculated in the present study, under both the compact tetraquark and hadronic molecule interpretations, are very much different: This might be one of the reasons why many multiquark states are only observed in a few decay channels [65]. We note that in order to extract the above results, we have only considered the leading-order fall-apart decays described by color-singlet-color-singlet meson-meson currents/operators, but neglected the O(α s ) corrections described by color-octet-color-octet meson-meson currents/operators. To end this paper, we propose the BE-SIII, Belle, Belle-II, and LHCb Collaborations to search for those decay channels not observed yet, in order to better understand the nature of the Z c (3900).