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Decay properties of the Zc(3900) through the Fierz rearrangement

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1. Jiang, Y.-W., Tan, W.-H., Chen, H.-X. et al. Strong Decays of the ϕ(2170) as a Fully Strange Tetraquark State[J]. Symmetry, 2024, 16(8): 1021. doi: 10.3390/sym16081021
2. Yang, G., Ping, J., Segovia, J. Charmoniumlike tetraquarks in a chiral quark model[J]. European Physical Journal C, 2023, 83(8): 772. doi: 10.1140/epjc/s10052-023-11945-6
3. Li, N., He, H.-Z., Liang, W. et al. Light meson emissions of selected charmonium-like states within compact tetraquark configurations*[J]. Chinese Physics C, 2023, 47(6): 063102. doi: 10.1088/1674-1137/acc648
4. Chen, H.-X., Chen, W., Liu, X. et al. An updated review of the new hadron states[J]. Reports on Progress in Physics, 2023, 86(2): 026201. doi: 10.1088/1361-6633/aca3b6
5. Chen, H.-X., Yan, Y.-X., Chen, W. Decay behaviors of the fully bottom and fully charm tetraquark states[J]. Physical Review D, 2022, 106(9): 094019. doi: 10.1103/PhysRevD.106.094019
6. Chen, H.-X.. Hidden-charm pentaquark states through current algebra: from their production to decay[J]. Chinese Physics C, 2022, 46(9): 093105. doi: 10.1088/1674-1137/ac6ed2
7. Chen, H.-X.. Hadronic molecules in B decays[J]. Physical Review D, 2022, 105(9): 094003. doi: 10.1103/PhysRevD.105.094003
8. Chen, H.-X.. Decay properties of the X(3872) through the Fierz rearrangement[J]. Communications in Theoretical Physics, 2022, 74(2): 025201. doi: 10.1088/1572-9494/ac449e
9. Chen, H.-X., Chen, W., Liu, X. et al. Strong decays of fully-charm tetraquarks into di-charmonia[J]. Science Bulletin, 2020, 65(23): 1994-2000. doi: 10.1016/j.scib.2020.08.038

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Hua-Xing Chen. Decay properties of the Zc(3900) through the Fierz rearrangement[J]. Chinese Physics C. doi: 10.1088/1674-1137/abae4b
Hua-Xing Chen. Decay properties of the Zc(3900) through the Fierz rearrangement[J]. Chinese Physics C.  doi: 10.1088/1674-1137/abae4b shu
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Decay properties of the Zc(3900) through the Fierz rearrangement

    Corresponding author: Hua-Xing Chen, hxchen@buaa.edu.cn
  • 1. School of Physics, Beihang University, Beijing 100191, China
  • 2. School of Physics, Southeast University, Nanjing 210094, China

Abstract: We systematically construct all the tetraquark currents/operators of JPC=1+ with the quark configurations [cq][ˉcˉq], [ˉcq][ˉqc], and [ˉcc][ˉqq] (q=u/d), and derive their relations through the Fierz rearrangement of the Dirac and color indices. Using the transformations of [qc][ˉqˉc][ˉcc][ˉqq] and [ˉcq][ˉqc], we study decay properties of the Zc(3900) as a compact tetraquark state; while using the transformation of [ˉcq][ˉqc][ˉcc][ˉqq], we study its decay properties as a hadronic molecular state.

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    1.   Introduction
    • In the past twenty years, many charmonium-like XYZ states have been discovered in particle experiments [1]. All of these 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. In particular, in 2013, BESIII reported the Zc(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 Zc(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 [7-13], a hadro-quarkonium [8, 14, 15], or due to the kinematical threshold effect [16-19], etc. We refer to reviews [20-24] for detailed discussions.

      The charged charmonium-like state Zc(3900) of JPC=1+ [25] has been observed in the J/ψπ and DˉD channels [2, 3, 26, 27], and there were some events in the hcπ channel [28]. In a recent BESIII experiment [29], evidence for the Zc(3900)ηcρ decay was reported with a statistical significance of 3.9σ at s=4.226 GeV, and the relative branching ratio

      RZcB(Zc(3900)ηcρ)B(Zc(3900)J/ψπ),

      (1)

      was evaluated to be 2.2±0.9 at the same center-of-mass energy. This ratio has been studied by many theoretical methods/models [30-39], and was suggested in Ref. [40] to be useful to discriminate between the compact tetraquark and hadronic molecule scenarios. As summarized in Table 1, 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 to do so within the same framework for both the tetraquark and molecule scenarios.

      interpretationsRZcmethods/models
      compact tetraquark(2.3+3.31.4)×102Type-I diquark-antidiquark model [40]
      0.27+0.400.17Type-II diquark-antidiquark model [40]
      0.95QCD sum rules [30]
      0.57QCD sum rules [31]
      1.1QCD sum rules [32]
      1.28covariant quark model [33]
      hadronic molecule(4.6+2.51.7)×102Non-Relativistic effective field theory [40]
      0.12light front model [34]
      0.68×102effective field theory [35]
      1.78covariant quark model [33]

      Table 1.  The relative branching ratio RZcB(Zc(3900)ηcρ)/B(Zc(3900)J/ψπ), calculated by various theoretical methods/models.

      In this paper we study the decay properties of the Zc(3900) under both the compact tetraquark and hadronic molecule interpretations. This study is based on our previous finding that the diquark-antidiquark currents ([qq][ˉqˉq]) and the meson-meson currents ([ˉqq][ˉqq]) are related to each other through the Fierz rearrangement of the Dirac and color indices [41-51]. More studies on light baryon operators can be found in Refs. [52-54]. In the present case the Zc(3900) contains four quarks: the c, ˉc, q, ˉq quarks (q=u/d). Thus, there are three configurations:

      [cq][ˉcˉq],[ˉcq][ˉqc],and[ˉcc][ˉqq].

      Again, the Fierz rearrangement can be applied to relate them. Based on these relations, we shall extract some decay properties of the Zc(3900) in this paper.

      There are eight independent [cq][ˉcˉq] currents of JPC=1+, which have been systematically constructed in Ref. [55]. Here, we choose one of them,

      ηZμ=ϵabeϵcdeqTaCγμcbˉqcγ5CˉcTd{γμγ5},

      (2)

      where C is the charge-conjugation matrix, the subscripts ae are the color indices, and the sum over repeated indices is taken. This current would strongly couple to the Zc(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 [56, 57]:

      LNRQCD=ψ{iD0+}ψ+χ{iD0+}χ+f1(1S0)m1m2ψχχψ+f1(3S0)m1m2ψσχχσψ+f8(1S0)m1m2ψTaχχTaψ+f8(3S0)m1m2ψTaσχχTaσψ+.

      (3)

      We refer to Ref. [58] for a 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 [57].

      Compared with this, the quark-based effective field theory for the multiquark system is much more difficult [24]. Let us attempt to do this for the Zc(3900). Based on Eq. (2), we can add an eight-quark operator (the same argument applies for other Lagrangians containing ηZμ):

      L=c0×ηZμ×(ηZ,μ)=c0×(ϵabeϵcdeqTaCγμcbˉqcγ5CˉcTd{γμγ5})×(ϵabeϵcdeˉcbγμCˉqTacTdCγ5qc{γμγ5}),

      (4)

      where c0 is a constant. Next, we can use the Fierz rearrangement to transform it to

      L=c0×(+13ˉcaγ5caˉqbγμqb13ˉcaγμcaˉqbγ5qb+i3ˉcaγνγ5caˉqbσμνqbi3ˉcaσμνcaˉqbγνγ5qb14λnabλncdˉcaγ5cbˉqcγμqd+14λnabλncdˉcaγμcbˉqcγ5qdi4λnabλncdˉcaγνγ5cbˉqcσμνqd+i4λnabλncdˉcaσμνcbˉqcγνγ5qd)×(ϵabeϵcdeˉcbγμCˉqTacTdCγ5qc{γμγ5}).

      (5)

      Detailed discussions on this transformation will be given below.

      Considering that the meson operators, ˉqγ5q, ˉqγμq, ˉcγ5c, and ˉcγμc couple to the π, ρ, ηc, and J/ψ mesons (see Table 2 at Sec. 3), the above eight-quark operator can describe the fall-apart decays of the Zc(3900) into the ηcρ 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 examine further. However, their relative branching ratios can be extracted much more easily, which are useful and important for understanding the nature of the Zc(3900) [59].

      operatorsJPCmesonsJPCcouplingsdecay constants
      JS=ˉdu0++0++
      JP=ˉdiγ5u0+π+0+0|JP|π+=λπλπ=fπm2πmu+md
      JVμ=ˉdγμu1ρ+10|JVμ|ρ+=mρfρ+ϵμfρ+=208 MeV [82]
      JAμ=ˉdγμγ5u1++π+0+0|JAμ|π+=ipμfπ+fπ+=130.2 MeV [1]
      a1(1260)1++0|JAμ|a1=ma1fa1ϵμfa1=254 MeV [87]
      JTμν=ˉdσμνu1±ρ+10|JTμν|ρ+=ifTρ(pμϵνpνϵμ)fTρ=159 MeV [82]
      b1(1235)1+0|JTμν|b1=ifTb1ϵμναβϵαpβfTb1=180 MeV [95]
      IS=ˉcc0++χc0(1P)0++0|IS|χc0=mχc0fχc0fχc0=343 MeV [76]
      IP=ˉciγ5c0+ηc0+0|IP|ηc=ληcληc=fηcm2ηc2mc
      IVμ=ˉcγμc1J/ψ10|IVμ|J/ψ=mJ/ψfJ/ψϵμfJ/ψ=418 MeV [83]
      IAμ=ˉcγμγ5c1++ηc0+0|IAμ|ηc=ipμfηcfηc=387 MeV [83]
      χc1(1P)1++0|IAμ|χc1=mχc1fχc1ϵμfχc1=335 MeV [77]
      ITμν=ˉcσμνc1±J/ψ10|ITμν|J/ψ=ifTJ/ψ(pμϵνpνϵμ)fTJ/ψ=410 MeV [83]
      hc(1P)1+0|ITμν|hc=ifThcϵμναβϵαpβfThc=235 MeV [83]
      OS=ˉdc0+D+00+0|OS|D+0=mD0fD0fD0=410 MeV [108]
      OP=ˉdiγ5c0D+00|OP|D+=λDλD=fDm2Dmc+md
      OVμ=ˉcγμu1ˉD010|OVμ|ˉD0=mDfDϵμfD=253 MeV [105]
      OAμ=ˉcγμγ5u1+ˉD000|OAμ|ˉD0=ipμfDfD=211.9 MeV [1]
      D11+0|OAμ|D1=mD1fD1ϵμfD1=356 MeV [108]
      OTμν=ˉdσμνc1±ˉD+10|OTμν|D+=ifTD(pμϵνpνϵμ)fTD220 MeV
      1+

      Table 2.  Couplings of meson operators to meson states. Color indices are omitted for simplicity.

      The current ηZμ can also be investigated from the viewpoint of QCD sum rules [60, 61]. We assume it couples to the Zc(3900) through

      0|ηZμ|Zc=fZcϵμ.

      (6)

      After the Fierz rearrangement, ηZμ transforms to the long expression inside Eq. (5). Through the first and second terms, it couples to the ηcρ and J/ψπ channels simultaneously:

      0|ηZμ|ηcρ=130|ˉcaγ5ca|ηc0|ˉqbγμqb|ρ+,0|ηZμ|J/ψπ=130|ˉcaγμca|J/ψ0|ˉqbγ5qb|π+.

      (7)

      Again, these two equations can be easily used to calculate the relative branching ratio RZc. Detailed discussions on this will be given below.

      In the above equations, we have worked within the naive factorization scheme, so our uncertainty is significantly larger than the well-developed QCD factorization method [62-64], which has been widely and successfully applied to study weak and radiative decay properties of conventional (heavy) hadrons, e.g., see Refs. [65, 66]. However, given that we still do not fully understand the internal structure of the Zc(3900) (as well as all the other exotic hadrons), the naive factorization scheme at this moment can be useful. Besides, the tetraquark decay constant fZc is removed when calculating relative branching ratios, which significantly reduces our uncertainty.

      In this study, we shall examine the strong decay properties of the Zc(3900) under the naive factorization scheme. To do this we just need to replace the weak-interaction Lagrangian by some interpolating current, and apply the similar technics here together with the Fierz arrangement. Note that a similar arrangement of the spin and color indices in the nonrelativistic case was used to study strong decay properties of the Zc(3900) in Refs. [8, 67, 68].

      This paper is organized as follows. In Sec. 2 we systematically construct all the tetraquark currents of JPC=1+ with the quark content cˉcqˉq. There are three configurations, [cq][ˉcˉq], [ˉcq][ˉqc], and [ˉcc][ˉqq], and their relations are also derived in this section by using the Fierz rearrangement of the Dirac and color indices. In Sec. 3 we discuss the couplings of meson operators to meson states and list those which are needed in the present study. In Sec. 4 and Sec. 5 we extract some decay properties of the Zc(3900), separately for the compact tetraquark interpretation and the hadronic molecule interpretation. The obtained results are discussed and summarized in Sec. 6.

    2.   Tetraquark currents of JPC=1+ and their relations
    • By using the c, ˉc, q, ˉq quarks (q=u/d), one can construct three types of tetraquark currents, as illustrated in Fig. 1:

      Figure 1.  (color online) Three types of tetraquark currents. Quarks are shown in red/green/blue color, and antiquarks are shown in cyan/magenta/yellow color.

      η(x,y)=[qTa(x)CΓ1cb(x)]×[ˉqc(y)Γ2CˉcTd(y)],ξ(x,y)=[ˉca(x)Γ3qb(x)]×[ˉqc(y)Γ4cd(y)],θ(x,y)=[ˉca(x)Γ5cb(x)]×[ˉqc(y)Γ6qd(y)],

      (8)

      where Γi are the 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 typically calls η(x,y) the diquark-antidiquark current, and ξ(x,y) and θ(x,y) the mesonic-mesonic currents. We separately construct them as follows:

    • 2.1.   [qc][ˉqˉc] currents ηiμ(x,y)

    • There are altogether eight independent [qc][ˉqˉc] currents of JPC=1+ [55]:

      η1μ=qTaCγμcbˉqaγ5CˉcTbqTaCγ5cbˉqaγμCˉcTb,η2μ=qTaCγμcbˉqbγ5CˉcTaqTaCγ5cbˉqbγμCˉcTa,η3μ=qTaCγνcbˉqaσμνγ5CˉcTbqTaCσμνγ5cbˉqaγνCˉcTb,η4μ=qTaCγνcbˉqbσμνγ5CˉcTaqTaCσμνγ5cbˉqbγνCˉcTa,η5μ=qTaCγμγ5cbˉqaCˉcTbqTaCcbˉqaγμγ5CˉcTb,η6μ=qTaCγμγ5cbˉqbCˉcTaqTaCcbˉqbγμγ5CˉcTa,η7μ=qTaCγνγ5cbˉqaσμνCˉcTbqTaCσμνcbˉqaγνγ5CˉcTb,η8μ=qTaCγνγ5cbˉqbσμνCˉcTaqTaCσμνcbˉqbγνγ5CˉcTa.

      (9)

      Here, we have omitted the coordinates x and y for simplicity. Their combinations, η1μη2μ, η3μη4μ, η5μη6μ, and η7μη8μ have the antisymmetric color structure [qc]ˉ3c[ˉqˉc]3c[cˉcqˉq]1c, and η1μ+η2μ, η3μ+η4μ, η5μ+η6μ, and η7μ+η8μ have the symmetric color structure [qc]6c[ˉqˉc]ˉ6c[cˉcqˉq]1c.

      In the "type-II" diquark-antidiquark model proposed in Ref. [6], the ground-state tetraquarks can be written in the spin basis as |sqc,sˉqˉcJ, where sqc and sˉqˉc are the charmed diquark and antidiquark spins, respectively. There are two ground-state diquarks: the "good" one of JP=0+ and the "bad" one of JP=1+ [69]. By combining them, the Zc(3900) was interpreted as a diquark-antidiquark state of JPC=1+ in Ref. [6]:

      |0qc1ˉqˉc;1+=12(|0qc,1ˉqˉcJ=1|1qc,0ˉqˉcJ=1).

      (10)

      The interpolating current having the identical internal structure is simply the current ηZμ given in Eq. (2), which has been studied in Refs. [3032, 70] using QCD sum rules:

      ηZμ(x,y)=η1μ([uc][ˉdˉc])η2μ([uc][ˉdˉc])=uTa(x)Cγμcb(x)×(ˉda(y)γ5CˉcTb(y){ab}){γμγ5}.

      (11)

      Here, we have explicitly chosen the quark content [uc][ˉdˉc] for the positive-charged Zc(3900)+.

    • 2.2.   [ˉcq][ˉqc] currents ξiμ(x,y)

    • There are altogether eight independent [ˉcq][ˉqc] currents of JPC=1+:

      ξ1μ=ˉcaγμqaˉqbγ5cb+ˉcaγ5qaˉqbγμcb,ξ2μ=ˉcaγνqaˉqbσμνγ5cbˉcaσμνγ5qaˉqbγνcb,ξ3μ=ˉcaγμγ5qaˉqbcbˉcaqaˉqbγμγ5cb,ξ4μ=ˉcaγνγ5qaˉqbσμνcb+ˉcaσμνqaˉqbγνγ5cb,ξ5μ=λnabλncd(ˉcaγμqbˉqcγ5cd+ˉcaγ5qbˉqcγμcd),

      ξ6μ=λnabλncd(ˉcaγνqbˉqcσμνγ5cdˉcaσμνγ5qbˉqcγνcd),ξ7μ=λnabλncd(ˉcaγμγ5qbˉqccdˉcaqbˉqcγμγ5cd),ξ8μ=λnabλncd(ˉcaγνγ5qbˉqcσμνcd+ˉcaσμνqbˉqcγνγ5cd).

      (12)

      Among them, ξ1,2,3,4μ have the color structure [ˉcq]1c[ˉqc]1c[cˉcqˉq]1c, and ξ5,6,7,8μ have the color structure [ˉcq]8c[ˉqc]8c[cˉcqˉq]1c. In the molecular picture, the Zc(3900) can be interpreted as the DˉD hadronic molecular state of JPC=1+ [710]:

      |DˉD;1+=12(|DˉDJ=1|ˉDDJ=1),

      (13)

      and the relevant interpolating current is [7173]:

      ξZμ(x,y)=ξ1μ([ˉcu][ˉdc])=ˉca(x)γμua(x)ˉdb(y)γ5cb(y)+{γμγ5}.

      (14)

      Again, we have chosen the quark content [ˉcu][ˉdc].

    • 2.3.   [ˉcc][ˉqq] currents θiμ(x,y)

    • There are altogether eight independent [ˉcc][ˉqq] currents of JPC=1+:

      θ1μ(x,y)=ˉca(x)γ5ca(x)ˉqb(y)γμqb(y),θ2μ(x,y)=ˉca(x)γμca(x)ˉqb(y)γ5qb(y),θ3μ(x,y)=ˉca(x)γνγ5ca(x)ˉqb(y)σμνqb(y),θ4μ(x,y)=ˉca(x)σμνca(x)ˉqb(y)γνγ5qb(y),θ5μ(x,y)=λnabλncdˉca(x)γ5cb(x)ˉqc(y)γμqd(y),θ6μ(x,y)=λnabλncdˉca(x)γμcb(x)ˉqc(y)γ5qd(y),θ7μ(x,y)=λnabλncdˉca(x)γνγ5cb(x)ˉqc(y)σμνqd(y),θ8μ(x,y)=λnabλncdˉca(x)σμνcb(x)ˉqc(y)γνγ5qd(y).

      (15)

      Among them, θ1,2,3,4μ have the color structure [ˉcc]1c[ˉqq]1c[cˉcqˉq]1c, and θ5,6,7,8μ have the color structure [ˉcc]8c[ˉqq]8c[cˉcqˉq]1c. We will discuss their corresponding hadron states in Sec. 3.

    • 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. [41-54]. It can also be used to relate the above three types of tetraquark currents. To do this, we must use a) the Fierz transformation [74] 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. [75].

      In Eq. (5) the Fierz rearrangement is applied to local operators/currents. However, the Fierz rearrangement is actually a matrix identity, which is valid if the same quark field in the initial and final operators is at the same location. As an example, we can apply the Fierz rearrangement to transform the non-local current with the quark fields η(x,x;y,y)=[q(x)c(x)][ˉq(y)ˉc(y)] into a combination of several non-local currents with the quark fields at the same locations ξ(y,x;y,x)=[ˉc(y)q(x)][ˉq(y)c(x)].

      Altogether, we obtain the following relation between the currents ηiμ(x,x;y,y) and θiμ(y,x;y,x):

      (η1μη2μη3μη4μη5μη6μη7μη8μ)=(1/21/2i/2i/200001/61/6i/6i/61/41/4i/4i/43i/23i/21/21/20000i/2i/21/61/63i/43i/41/41/41/21/2i/2i/200001/61/6i/6i/61/41/4i/4i/43i/23i/21/21/20000i/2i/21/61/63i/43i/41/41/4)×(θ1μθ2μθ3μθ4μθ5μθ6μθ7μθ8μ),

      (16)

      the following relation between ηiμ(x,x;y,y) and ξiμ(y,x;y,x):

      (η1μη2μη3μη4μη5μη6μη7μη8μ)=(0i/61/600i/41/400i/21/200000i/2001/63i/4001/43i/2001/200001/600i/61/400i/41/200i/2000001/6i/2001/43i/4001/23i/200000)×(ξ1μξ2μξ3μξ4μξ5μξ6μξ7μξ8μ),

      (17)

      the following relation among ηiμ(x,x;y,y), ξ1,2,3,4μ(y,x;y,x), and θ1,2,3,4μ(y,x;y,x):

      (η1μη2μη3μη4μη5μη6μη7μη8μ)=(00001/21/2i/2i/20i/21/20000000003i/23i/21/21/23i/2001/2000000001/21/2i/2i/21/200i/2000000003i/23i/21/21/201/23i/200000)×(ξ1μξ2μξ3μξ4μθ1μθ2μθ3μθ4μ),

      (18)

      and the following relation between ξiμ(y,x;y,x) and θiμ(y,x;y,x):

      (ξ1μξ2μξ3μξ4μξ5μξ6μξ7μξ8μ)=(1/61/6i/6i/61/41/4i/4i/4i/2i/21/61/63i/43i/41/41/41/61/6i/6i/61/41/4i/4i/4i/2i/21/61/63i/43i/41/41/48/98/98i/98i/91/61/6i/6i/68i/38i/38/98/9i/2i/21/61/68/98/98i/98i/91/61/6i/6i/68i/38i/38/98/9i/2i/21/61/6)×(θ1μθ2μθ3μθ4μθ5μθ6μθ7μθ8μ).

      (19)
    3.   Meson operators
    • There are altogether six types of meson operators: ˉqaqa, ˉqaγ5qa, ˉqaγμqa, ˉqaγμγ5qa, ˉqaσμνqa, and \bar q_a \sigma_{\mu\nu} \gamma_5 q_a . The last two can be related to each other through

      \sigma_{\mu\nu} \gamma_5 = {{\rm i}\over2} \epsilon_{\mu\nu\rho\sigma} \sigma^{\rho\sigma} \, .

      (20)

      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 this study, we require the following couplings, as summarized in Table 2:

      1. The scalar operators J^{S} = \bar q_a q_a and I^{S} = \bar c_a c_a of J^{PC} = 0^{++} couple to scalar mesons. In Ref. [76] the authors used the method of QCD sum rules and extracted the coupling of I^{S} to \chi_{c0}(1P) to be

      \langle 0 | \bar c_a c_a | \chi_{c0}(p) \rangle = m_{\chi_{c0}} f_{\chi_{c0}} \, ,

      (21)

      where

      f_{\chi_{c0}} = 343\; {\rm{MeV}} \, .

      (22)

      See also discussions in Refs. [77-79]. The light scalar mesons have a complicated nature [80], so we shall not investigate their relevant decay channels in this study.

      2. The pseudoscalar operators J^{P} = \bar q_a {\rm i}\gamma_5 q_a and I^{P} = \bar c_a {\rm i}\gamma_5 c_a of J^{PC} = 0^{-+} couple to the pseudoscalar mesons \pi and \eta_c , respectively. We can evaluate them through [81]:

      \begin{split} & \langle 0 | \bar d_a {\rm i}\gamma_5 u_a | \pi^+(p) \rangle = \lambda_{\pi} = {f_{\pi^+} m_{\pi^+}^2 \over m_u + m_d} \, , \\ & \;\;\;\;\langle 0 | \bar c_a {\rm i}\gamma_5 c_a | \eta_c(p) \rangle = \lambda_{\eta_c} = {f_{\eta_c} m_{\eta_c}^2 \over 2 m_c} \, . \end{split}

      (23)

      3. The vector operators J^{V}_{\mu} = \bar q_a \gamma_{\mu} q_a and I^{V}_{\mu} = \bar c_a \gamma_{\mu} c_a of J^{PC} = 1^{–} couple to the vector mesons \rho and J/\psi , respectively. In Refs. [82, 83] the authors used the method of Lattice QCD to obtain

      \begin{split} & \langle 0 | \bar d_a \gamma_{\mu} u_a | \rho^+(p, \epsilon) \rangle = m_{\rho} f_{\rho^+} \epsilon_{\mu} \, , \\ & \langle 0 | \bar c_a \gamma_{\mu} c_a | J/\psi(p, \epsilon) \rangle = m_{J/\psi} f_{J/\psi} \epsilon_{\mu} \, , \end{split}

      (24)

      where

      \begin{array}{l} \;\;\;f_{\rho^+} = 208\; {\rm{MeV}} \, , \;\;\; f_{J/\psi} = 418\; {\rm{MeV}} \, . \end{array}

      (25)

      See also discussions in Refs. [84-86].

      4. The axialvector operators J^{A}_{\mu} = \bar q_a \gamma_{\mu} \gamma_5 q_a and I^{A}_{\mu} = \bar c_a \gamma_{\mu} \gamma_5 c_a of J^{PC} = 1^{++} couple to both the pseudoscalar mesons ( \pi and \eta_c of J^{PC} = 0^{-+} ) and axialvector mesons ( a_1(1260) and \chi_{c1}(1P) of J^{PC} = 1^{++} ). The coupling of J^{A}_{\mu} to \pi has been well measured in particle experiments [1]:

      \langle 0 | \bar d_a \gamma_{\mu} \gamma_5 u_a | \pi^+(p) \rangle = {\rm i} p_{\mu} f_{\pi^+} \, ,

      (26)

      while its coupling to a_1(1260) was evaluated by using Lattice QCD [87]:

      \langle 0 | \bar d_a \gamma_{\mu} \gamma_5 u_a | a_1(p, \epsilon) \rangle = m_{a_1} f_{a_1} \epsilon_{\mu} \, ,

      (27)

      where

      \begin{array}{l} f_{\pi^+} = 130.2\; {\rm{MeV}} \, , \;\;\; f_{a_1} = 254\; {\rm{MeV}} \, . \end{array}

      (28)

      The coupling of I^{A}_{\mu} to \eta_c and \chi_{c1}(1P) was evaluated by using Lattice QCD [83] and QCD sum rules [77]:

      \begin{split} & \langle 0 | \bar c_a \gamma_{\mu} \gamma_5 c_a | \eta_c(p) \rangle = {\rm i} p_{\mu} f_{\eta_c} \, , \\ & \langle 0 | \bar c_a \gamma_{\mu} \gamma_5 c_a | \chi_{c1}(p, \epsilon) \rangle = m_{\chi_{c1}} f_{\chi_{c1}} \epsilon_{\mu} \, , \end{split}

      (29)

      where

      \begin{array}{l} f_{\eta_c} = 387\; {\rm{MeV}} \, , \;\;\; f_{\chi_{c1}} = 335\; {\rm{MeV}} \, . \end{array}

      (30)

      See also discussions in Refs. [77, 85, 88-94].

      5. The tensor operators J^{T}_{\mu\nu} = \bar q_a \sigma_{\mu\nu} q_a and I^{T}_{\mu\nu} = \bar c_a \sigma_{\mu\nu} c_a of J^{PC} = 1^{\pm-} couple to both the vector mesons ( \rho and J/\psi of J^{PC} = 1^{–} ) and the axialvector mesons ( b_1(1235) and h_{c}(1P) of J^{PC} = 1^{+-} ). The coupling of J^{T}_{\mu\nu} to \rho and b_1(1235) was calculated through Lattice QCD [82] and QCD sum rules [95]:

      \begin{split} & \langle 0 | \bar d_a \sigma_{\mu\nu} u_a | \rho^+(p, \epsilon) \rangle = {\rm i} f^T_{\rho} (p_{\mu}\epsilon_\nu - p_\nu\epsilon_{\mu}) \, , \\& \langle 0 | \bar d_a \sigma_{\mu\nu} u_a | b_1(p, \epsilon) \rangle = {\rm i} f^T_{b_1} \epsilon_{\mu\nu\alpha\beta} \epsilon^\alpha p^\beta \, , \end{split}

      (31)

      where

      \begin{array}{l} f_{\rho}^T = 159\; {\rm{MeV}} \, , \;\;\; f_{b_1}^T = 180\; {\rm{MeV}} \, . \end{array}

      (32)

      The coupling of I^{T}_{\mu\nu} to J/\psi and h_{c}(1P) was calculated through Lattice QCD [83]:

      \begin{split}& \langle 0 | \bar c_a \sigma_{\mu\nu} c_a | J/\psi(p, \epsilon) \rangle = {\rm i} f^T_{J/\psi} (p_{\mu}\epsilon_\nu - p_\nu\epsilon_{\mu}) \, , \\ & \langle 0 | \bar c_a \sigma_{\mu\nu} c_a | h_c(p, \epsilon) \rangle = {\rm i} f^T_{h_c} \epsilon_{\mu\nu\alpha\beta} \epsilon^\alpha p^\beta \, , \end{split}

      (33)

      where

      \begin{array}{l} f_{J/\psi}^T = 410\; {\rm{MeV}} \, , \;\;\; f_{h_c}^T = 235\; {\rm{MeV}} \, . \end{array}

      (34)

      See also discussions in Refs. [96-104].

      6. The Z_c(3900) is above the D \bar D^* threshold; thus, we need the couplings of O^{P} = \bar q_a {\rm i}\gamma_5 c_a and O^{A}_{\mu} = \bar c_a \gamma_{\mu} \gamma_5 q_a to the D meson [1]:

      \begin{split} & \langle 0 | \bar d_a {\rm i}\gamma_5 c_a | D^+(p) \rangle = \lambda_D \, , \\ & \langle 0 | \bar c_a \gamma_{\mu} \gamma_5 u_a | \bar D^0(p) \rangle = {\rm i} p_{\mu} f_{D} \, , \end{split}

      (35)

      and the couplings of O^{V}_{\mu} = \bar c_a \gamma_{\mu} q_a and O^{T}_{\mu\nu} = \bar q_a \sigma_{\mu\nu} c_a to the D^* meson [105]:

      \begin{split} & \langle 0 | \bar c_a \gamma_{\mu} u_a | \bar D^{*0}(p, \epsilon) \rangle = m_{D^*} f_{D^*} \epsilon_{\mu} \, , \\ & \langle 0 | \bar d_a \sigma_{\mu\nu} c_a | D^{*+}(p, \epsilon) \rangle = {\rm i} f^T_{D^*} (p_{\mu}\epsilon_\nu - p_\nu\epsilon_{\mu}) \, , \end{split}

      (36)

      where

      \begin{array}{l} \lambda_D = \dfrac{{f_{D} m_{D^+}^2}}{m_c + m_d} \, , \;\; f_{D} = 211.9\; {\rm{MeV}} \, , \;\; f_{D^*} = 253\; {\rm{MeV}} \, . \end{array}

      (37)

      We found no theoretical study on the transverse decay constant f^T_{D^*} ; thus, we simply fit it among the decay constants, f_{\pi^+} - f_{\rho^+} - f_{\rho}^T , f_{\eta_c} - f_{J/\psi} - f_{J/\psi}^T , and f_{D} - f_{D^*} - f_{D^*}^T , to obtain

      f_{D^*}^T \approx 220\; {\rm{MeV}} \, .

      (38)

      See also discussions in Refs. [106, 107].

      7. The Z_c(3900) \rightarrow D \bar D^*_0 \rightarrow D \bar D \pi decay is kinematically allowed, so we need the coupling of O^{S} = \bar q_a c_a to the D_0^* meson [108]:

      \langle 0 | \bar d_a c_a | D_0^{*+}(p) \rangle = m_{D_0^{*}} f_{D_0^{*}} \, ,

      (39)

      where

      f_{D_0^{*}} = 410\; {\rm{MeV}} \, .

      (40)

      See also discussions in Refs. [109, 110].

    4.   Decay properties of the Z_c(3900) as a compact tetraquark state
    • In this section and the next, we use Eqs. (16-19) derived in Sec. 2 to extract some decay properties of the Z_c(3900) . The two possible interpretations of the Z_c(3900) are: a) the compact tetraquark state of J^{PC} = 1^{+-} composed of a J^P = 0^+ diquark/antidiquark and a J^P = 1^+ antidiquark/diquark [5, 6], i.e., |0_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle defined in Eq. (10); and b) the D \bar D^* hadronic molecular state of J^{PC} = 1^{+-} [710], i.e., | D \bar D^*; 1^{+-} \rangle defined in Eq. (13). Moreover, we shall study their mixing with the |1_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle and | D^* \bar D^*; 1^{+-} \rangle states, whose definitions will be given below.

      In this section, we investigate the former compact tetraquark interpretation, whose relevant current \eta^{{\cal{Z}}}_{\mu}(x,y) has been given in Eq. (11). This current can be transformed to \theta_{\mu}^i(x,y) and \xi_{\mu}^i(x,y) according to Eqs. (16-18), through which we shall extract some decay properties of the Z_c(3900) as a compact tetraquark state in the following subsections.

    • 4.1.   \eta^{{\cal{Z}}}_{\mu}\big([uc][\bar d \bar c]\big) \rightarrow \theta_{\mu}^i \big([\bar c c] + [\bar d u]\big)

    • As depicted in Fig. 2, when the c and \bar c quarks meet each other and the u and \bar d quarks meet each other at the same time, a compact tetraquark state can decay into one charmonium meson and one light meson:

      Figure 2.  (color online) The decay of a compact tetraquark (diquark-antidiquark) state into one charmonium meson and one light meson. This decay can happen through either (b) a direct fall-apart process, or (c) a process with gluon(s) exchanged, that is the {\cal{O}}(\alpha_s) corrections.

      \begin{split} [u(x) c(x)]\; [\bar d(y) \bar c(y)] \Longrightarrow & [u(x \to y^{\prime})\; c(x \to x^{\prime})]\\&\times [\bar d(y \to y^{\prime})\; \bar c(y \to x^{\prime})] \\ \Longrightarrow & [\bar c(x^{\prime}) c(x^{\prime})] + [\bar d(y^{\prime}) u(y^{\prime})] \, .\end{split}

      (41)

      The first process is a dynamical process, during which we assume that all the flavor, color, spin and orbital structures remain unchanged, so the relevant current also remains the same. The second process for |0_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle can be described by transformation (16):

      \begin{split} \eta^{{\cal{Z}}}_{\mu}(x,y) & \Longrightarrow + {1\over3}\; \theta_{\mu}^1(x^{\prime},y^{\prime}) - {1\over3}\; \theta_{\mu}^2(x^{\prime},y^{\prime}) \\ & \;\;\;\;\;\;\;\; + {{\rm i}\over3}\; \theta_{\mu}^3(x^{\prime},y^{\prime}) - {{\rm i}\over3}\; \theta_{\mu}^4(x^{\prime},y^{\prime}) + \cdots \\ & \;\;\;\;= - {{\rm i}\over3}\; I^{P}(x^{\prime}) \; J^{V}_{\mu}(y^{\prime}) + {{\rm i}\over3}\; I^{V}_{\mu}(x^{\prime}) \; J^{P}(y^{\prime}) \\ & \;\;\;\;\;\;\;\; + {{\rm i}\over3}\; I^{A,\nu}(x^{\prime}) \; J^{T}_{\mu\nu}(y^{\prime}) - {{\rm i}\over3}\; I^{T}_{\mu\nu}(x^{\prime}) \; J^{A,\nu}(y^{\prime}) + \cdots \, , \end{split}

      (42)

      where we have only kept the direct fall-apart process described by \theta_{\mu}^{1,2,3,4} , but neglected the {\cal{O}}(\alpha_s) corrections described by \theta_{\mu}^{5,6,7,8} .

      Together with Table 2, we extract the following decay channels from the above transformation:

      1. The decay of |0_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle into \eta_c\rho is contributed by both I^{P} \times J^{V}_{\mu} and I^{A,\nu} \times J^{T}_{\mu\nu} :

      \begin{split} & \langle Z_c^+(p,\epsilon) | \eta_c(p_1)\; \rho^+(p_2,\epsilon_2) \rangle \\ \approx & - {{\rm i} c_1\over3}\; \lambda_{\eta_c} m_\rho f_{\rho^+}\; \epsilon \cdot \epsilon_2 \\ & - {{\rm i} c_1\over3}\; f_{\eta_c} f^T_{\rho}\; (\epsilon \cdot p_2 \; \epsilon_2 \cdot p_1 - p_1 \cdot p_2\; \epsilon \cdot \epsilon_2) \\ \equiv & g^S_{\eta_c \rho}\; \epsilon \cdot \epsilon_2 + g^D_{\eta_c \rho}\; (\epsilon \cdot p_2 \; \epsilon_2 \cdot p_1 - p_1 \cdot p_2\; \epsilon \cdot \epsilon_2) \, , \end{split}

      (43)

      where c_1 is an overall factor, related to the coupling of \eta^{{\cal{Z}}}_{\mu}(x,y) to the Z_c(3900)^+ as well as the dynamical process (x,y) \Longrightarrow (x^{\prime}, y^{\prime}) shown in Fig. 2. The two coupling constants g^S_{\eta_c \rho} and g^D_{\eta_c \rho} are defined for the S- and D-wave Z_c(3900) \to \eta_c \rho decays:

      {\cal{L}}^S_{\eta_c \rho} = g^S_{\eta_c \rho}\; Z_{c}^{+,\mu}\; \eta_c\; \rho^{-}_{\mu} + \cdots \, ,

      (44)

      \begin{split}\quad\quad {\cal{L}}^D_{\eta_c \rho} =& g^D_{\eta_c \rho} \times \left( g^{\mu\sigma}g^{\nu\rho} - g^{\mu\nu}g^{\rho\sigma} \right) \\ &\times Z_{c,\mu}^{+}\; \partial_\rho \eta_c\; \partial_\sigma \rho^{-}_{\nu} + \cdots \, . \end{split}

      (45)

      2. The decay of |0_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle into J/\psi \pi is contributed by both I^{V}_{\mu} \times J^{P} and I^{T}_{\mu\nu} \times J^{A,\nu} :

      \begin{split} & \langle Z_c^+(p,\epsilon) | J/\psi(p_1,\epsilon_1)\; \pi^+(p_2) \rangle \\ \approx & {{\rm i} c_1 \over3}\; \lambda_{\pi} m_{J/\psi} f_{J/\psi}\; \epsilon \cdot \epsilon_1 \\ &+ {{\rm i} c_1 \over3}\; f_{\pi^+} f^T_{J/\psi}\; (\epsilon \cdot p_1 \; \epsilon_1 \cdot p_2 - p_1 \cdot p_2\; \epsilon \cdot \epsilon_1) \\ \equiv & g^S_{\psi \pi}\; \epsilon \cdot \epsilon_1 + g^D_{\psi \pi}\; (\epsilon \cdot p_1 \; \epsilon_1 \cdot p_2 - p_1 \cdot p_2\; \epsilon \cdot \epsilon_1) \, . \end{split}

      (46)

      The two coupling constants g^S_{\psi \pi} and g^D_{\psi \pi} are defined for the S- and D-wave Z_c(3900) \rightarrow J/\psi \pi decays respectively:

      {\cal{L}}^S_{\psi \pi} = g^S_{\psi \pi}\; Z_{c}^{+,\mu}\; \psi_{\mu}\; \pi^- + \cdots \, ,

      (47)

      \begin{split} {\cal{L}}^D_{\psi \pi} = g^D_{\psi \pi} \times \left( g^{\mu\rho}g^{\nu\sigma} - g^{\mu\nu}g^{\rho\sigma} \right) \times Z_{c,\mu}^{+}\; \partial_\rho \psi_\nu\; \partial_\sigma \pi^- + \cdots \, .\quad\quad \end{split}

      (48)

      3. The decay of |0_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle into \eta_c b_1 is contributed by I^{A,\nu} \times J^{T}_{\mu\nu} :

      \begin{split} \langle Z_c^+(p,\epsilon) | \eta_c(p_1)\; b_1^+(p_2,\epsilon_2) \rangle &\approx - {{\rm i} c_1 \over3}\; f_{\eta_c} f^T_{b_1}\; \epsilon_{\mu\nu\alpha\beta} \epsilon^\mu p_1^\nu \epsilon_2^\alpha p_2^\beta \\ &\equiv g_{\eta_c b_1}\; \epsilon_{\mu\nu\alpha\beta} \epsilon^\mu p_1^\nu \epsilon_2^\alpha p_2^\beta \, . \end{split}

      (49)

      This process is kinematically forbidden, but the |0_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow \eta_c b_1 \rightarrow \eta_c \omega \pi \rightarrow \eta_c + 4 \pi decay is kinematically allowed.

      4. The decay of |0_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle into \chi_{c1} \rho is contributed by I^{A,\nu} \times J^{T}_{\mu\nu} :

      \begin{split} &\;\;\;\;\;\; \langle Z_c^+(p,\epsilon) | \chi_{c1}(p_1,\epsilon_1)\; \rho^+(p_2,\epsilon_2) \rangle \\ &\approx - {c_1\over3}\; m_{\chi_{c1}} f_{\chi_{c1}} f^T_{\rho}\; (\epsilon_1 \cdot \epsilon_2\; \epsilon \cdot p_2 - \epsilon_1 \cdot p_2\; \epsilon \cdot \epsilon_2) \\ &\equiv g_{\chi_{c1} \rho}\; (\epsilon_1 \cdot \epsilon_2\; \epsilon \cdot p_2 - \epsilon_1 \cdot p_2\; \epsilon \cdot \epsilon_2) \, . \end{split}

      (50)

      This process is kinematically forbidden, but the |0_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow \chi_{c1} \rho \rightarrow \chi_{c1} \pi \pi decay is kinematically allowed.

      5. The decay of |0_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle into \chi_{c1} b_1 is contributed by I^{A,\nu} \times J^{T}_{\mu\nu} :

      \begin{split} &\;\;\;\;\;\; \langle Z_c^+(p,\epsilon) | \chi_{c1}(p_1,\epsilon_1)\; b_1^+(p_2,\epsilon_2) \rangle \\ &\approx - {c_1\over3}\; m_{\chi_{c1}} f_{\chi_{c1}} f^T_{b_1}\; \epsilon_{\mu\nu\alpha\beta} \epsilon^\mu \epsilon_1^\nu \epsilon_2^\alpha p_2^\beta \\ &\equiv g_{\chi_{c1} b_1}\; \epsilon_{\mu\nu\alpha\beta} \epsilon^\mu \epsilon_1^\nu \epsilon_2^\alpha p_2^\beta \, . \end{split}

      (51)

      This process is kinematically forbidden.

      6. The decay of |0_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle into h_c \pi is contributed by I^{T}_{\mu\nu} \times J^{A,\nu} :

      \begin{split} &\;\;\;\;\;\; \langle Z_c^+(p,\epsilon) | h_c(p_1,\epsilon_1)\; \pi^+(p_2) \rangle \\ &\approx {{\rm i} c_1\over3}\; f_{\pi^+} f^T_{h_c}\; \epsilon_{\mu\nu\alpha\beta} \epsilon^\mu p_2^\nu \epsilon_1^\alpha p_1^\beta \\ &\equiv g_{h_c \pi}\; \epsilon_{\mu\nu\alpha\beta} \epsilon^\mu p_2^\nu \epsilon_1^\alpha p_1^\beta \, . \end{split}

      (52)

      This process is kinematically allowed.

      7. The decay of |0_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle into J/\psi a_1 is contributed by I^{T}_{\mu\nu} \times J^{A,\nu} :

      \begin{split} &\;\;\;\;\; \langle Z_c^+(p,\epsilon) | J/\psi(p_1,\epsilon_1)\; a_1^+(p_2,\epsilon_2) \rangle \\ &\approx {c_1\over3}\; f^T_{J/\psi} m_{a_1} f_{a_1}\; (\epsilon_1 \cdot \epsilon_2\; \epsilon \cdot p_1 - \epsilon_2 \cdot p_1\; \epsilon \cdot \epsilon_1) \\ &\equiv g_{\psi a_1}\; (\epsilon_1 \cdot \epsilon_2\; \epsilon \cdot p_1 - \epsilon_2 \cdot p_1\; \epsilon \cdot \epsilon_1) \, . \end{split}

      (53)

      This process is kinematically forbidden, but the |0_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow J/\psi a_1 \rightarrow J/\psi \rho \pi \rightarrow J/\psi + 3 \pi decay is kinematically allowed.

      8. The decay of |0_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle into h_c a_1 is contributed by I^{T}_{\mu\nu} \times J^{A,\nu} :

      \begin{split} &\;\;\;\;\;\langle Z_c^+(p,\epsilon) | h_c(p_1,\epsilon_1)\; a_1^+(p_2,\epsilon_2) \rangle \\ &\approx {c_1\over3}\; f^T_{h_c} m_{a_1} f_{a_1}\; \epsilon_{\mu\nu\alpha\beta} \epsilon^\mu \epsilon_2^\nu \epsilon_1^\alpha p_1^\beta \\ &\equiv g_{h_c a_1}\; \epsilon_{\mu\nu\alpha\beta} \epsilon^\mu \epsilon_2^\nu \epsilon_1^\alpha p_1^\beta \, . \end{split}

      (54)

      This process is kinematically forbidden.

      Summarizing the above results, we numerically obtain

      \begin{split}& g^S_{\eta_c \rho} = - {\rm i} c_1\; 7.29 \times 10^{10}\; {\rm{MeV}}^4 \, , \\ & g^D_{\eta_c \rho} = - {\rm i} c_1\; 2.05 \times 10^{4}\; {\rm{MeV}}^2 \, , \\ & g^S_{\psi \pi} = {\rm i} c_1\; 11.87 \times 10^{10}\; {\rm{MeV}}^4 \, , \\ & g^D_{\psi \pi} = {\rm i} c_1\; 1.78 \times 10^{4}\; {\rm{MeV}}^2 \, , \\ & g_{\eta_{c} b_1} = - {\rm i} c_1\; 2.32 \times 10^{4}\; {\rm{MeV}}^2 \, , \\ & g_{\chi_{c1} \rho} = - c_1\; 6.23 \times 10^{7}\; {\rm{MeV}}^3 \, , \\ & g_{\chi_{c1} b_1} = - c_1\; 7.06 \times 10^{7}\; {\rm{MeV}}^3 \, , \\ & g_{h_c \pi} = {\rm i} c_1\; 1.02 \times 10^{4}\; {\rm{MeV}}^2 \, , \\ & g_{\psi a_1} = c_1\; 4.27 \times 10^{7}\; {\rm{MeV}}^3 \, , \\ & g_{h_c a_1} = c_1\; 2.45 \times 10^{7}\; {\rm{MeV}}^3 \, . \end{split}

      (55)

      From these coupling constants, we further obtain the following relative branching ratios, which are kinematically allowed:

      \begin{split} &{{\cal{B}}(|0_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow \eta_c\rho) \over {\cal{B}}(|0_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow J/\psi\pi)} = 0.059 \, , \\ & {{\cal{B}}(|0_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow h_c\pi) \over {\cal{B}}(|0_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow J/\psi\pi)} = 0.0088 \, , \\& {{\cal{B}}(|0_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow \chi_{c1}\rho \rightarrow \chi_{c1}\pi \pi) \over {\cal{B}}(|0_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow J/\psi\pi)} = 1.4 \times 10^{-6} \, . \end{split}

      (56)

      In addition, the following decay chains are also possible but have quite small partial decay widths:

      \begin{split}& |0_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow \eta_c b_1 \rightarrow \eta_c \omega \pi \rightarrow \eta_c + 4 \pi \, , \\ & |0_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow J/\psi a_1 \rightarrow J/\psi \rho \pi \rightarrow J/\psi + 3 \pi \, . \end{split}

      (57)
    • 4.2.   \eta^{{\cal{Z}}}_{\mu}\big([uc][\bar d \bar c]\big) \rightarrow \xi_{\mu}^i \big([\bar c u] + [\bar d c]\big)

    • As depicted in Fig. 3, when the c and \bar d quarks meet each other and the u and \bar c quarks meet each other at the same time, a compact tetraquark state can decay into two charmed mesons. This process for |0_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle can be described by transformation (17):

      Figure 3.  (color online) The decay of a compact tetraquark (diquark-antidiquark) state into two charmed mesons. This decay can happen through either (b) a direct fall-apart process, or (c) a process with gluon(s) exchanged, that is the {\cal{O}}(\alpha_s) corrections.

      \eta^{{\cal{Z}}}_{\mu}(x,y) \Longrightarrow - {{\rm i}\over3}\; \xi_{\mu}^2(x^{\prime},y^{\prime}) + {1\over3}\; \xi_{\mu}^3(x^{\prime},y^{\prime}) + \cdots \, .

      (58)

      Again, we have only kept the direct fall-apart process described by \xi_{\mu}^{2,3} , but neglected the {\cal{O}}(\alpha_s) corrections described by \xi_{\mu}^{6,7} .

      The term \xi_{\mu}^{2} couples to the D^*\bar D^* and D^* \bar D_1 final states, and the term \xi_{\mu}^{3} couples to the D \bar D_0^* and D_1 \bar D_0^* final states. Among them, only the |0_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow D \bar D^*_0 \rightarrow D \bar D \pi decay is kinematically allowed, contributed by \xi_{\mu}^{3} = O^{A}_{\mu} \times O^{S} to be:

      \begin{split} \langle Z_c^+(p,\epsilon) | \bar D^0(p_1)D_0^{*+}(p_2) \rangle \approx & \dfrac{{\rm i} c_2}{3}\; f_D m_{D_0^*} f_{D_0^*}\; \epsilon \cdot p_1 \\ \equiv & g_{D \bar D_0^*}\; \epsilon \cdot p_1 \, , \end{split}

      (59)

      \begin{split} \langle Z_c^+(p,\epsilon) | D^+(p_1)\bar D_0^{*0}(p_2) \rangle &\approx - \dfrac{{\rm i} c_2}{3}\; f_D m_{D_0^*} f_{D_0^*}\; \epsilon \cdot p_1\\ &\equiv - g_{D \bar D_0^*}\; \epsilon \cdot p_1 \, , \end{split}

      (60)

      where c_2 is an overall factor.

      Thus, we numerically obtain

      g_{D \bar D_0^*} = {\rm i}c_2\; 6.80 \times 10^{7}\; {\rm{MeV}}^3 \, .

      (61)

      Comparing the |0_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow D \bar D^*_0 \rightarrow D \bar D \pi decay studied in the present subsection with the |0_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow J/\psi \pi and |0_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow \eta_c\rho decays studied in the previous subsection, we obtain

      {{\cal{B}}(|0_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow D \bar D_0^{*} + \bar D D_0^{*} \rightarrow D \bar D \pi) \over {\cal{B}}(|0_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow J/\psi\pi+ \eta_c\rho)} = 9.3 \times 10^{-8} \times {c_2^2 \over c_1^2} \, .

      (62)

      The current \eta^{{\cal{Z}}}_{\mu}(x,y) does not correlate with the two terms \xi_{\mu}^{1} = -{\rm i} O^{V}_{\mu} \times O^{P} and \xi_{\mu}^{4} = O^{A,\nu} \times O^{T}_{\mu\nu} , both of which can couple to the D \bar D^* final state. This suggests that |0_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle does not decay to the D \bar D^* final state with a large branching ratio,

      g_{D \bar D^{*}} \approx 0 \, ,

      (63)

      so that

      {{\cal{B}}(|0_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow D \bar D^{*} + \bar D D^{*}) \over {\cal{B}}(|0_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow J/\psi\pi + \eta_c\rho)} \approx 0 \, .

      (64)

      Eqs. (62) and (64) together suggest that |0_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle mainly decays into one charmonium meson and one light meson, other than two charmed mesons.

    • 4.3.   \eta^{{\cal{Z}}}_{\mu}\big([uc][\bar d \bar c]\big) \rightarrow \theta_{\mu}^{1,2,3,4}\big([\bar c c] + [\bar d u]\big) + \xi_{\mu}^{1,2,3,4}\big([\bar c u] + [\bar d c]\big)

    • If the above two processes investigated in Sec. 4.1 and Sec. 4.2 happen at the same time, we can use the transformation (18); i.e., |0_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle can decay into one charmonium meson and one light meson as well as two charmed mesons at the same time, the process of which is described by the color-singlet-color-singlet currents \theta_{\mu}^{1,2,3,4} and \xi_{\mu}^{1,2,3,4} together:

      \begin{split} \eta^{{\cal{Z}}}_{\mu}(x,y) \Longrightarrow & + {1\over2}\; \theta_{\mu}^1(x^{\prime},y^{\prime}) - {1\over2}\; \theta_{\mu}^2(x^{\prime},y^{\prime}) + {{\rm i}\over2}\; \theta_{\mu}^3(x^{\prime},y^{\prime}) \\ & - {{\rm i}\over2}\; \theta_{\mu}^4(x^{\prime},y^{\prime}) - {{\rm i}\over2}\; \xi_{\mu}^2(x^{\prime\prime},y^{\prime\prime}) + {1\over2}\; \xi_{\mu}^3(x^{\prime\prime},y^{\prime\prime}) \, . \end{split}

      (65)

      Here, we have kept all the terms, and there is no \cdots in this equation.

      Comparing the above equation with Eqs. (42) and (58), we obtain the same relative branching ratios as Sec. 4.1 and Sec. 4.2, with just the overall factors c_1 and c_2 replaced by others.

    • 4.4.   Mixing with |1_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle

    • The relative branching ratio {\cal{R}}_{Z_c} calculated in Sec. 4.1 is only 0.059 , which is significantly smaller than the BESIII measurement {\cal{R}}_{Z_c} = 2.2 \pm 0.9 at \sqrt{s} = 4.226 GeV [29]. In this subsection we slightly modify the internal structure of the Z_c(3900) to reevaluate this ratio.

      Actually, in the Type-II diquark-antidiquark model [6], the Z_c(3900) was interpreted as

      |0_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle = {1\over\sqrt2} \left(| 0_{qc}, 1_{\bar q \bar c} \rangle_{J = 1} - |1_{qc}, 0_{\bar q \bar c} \rangle_{J = 1} \right) \, ,

      and the ratio {\cal{R}}_{Z_c} was predicted to be 0.27^{+0.40}_{-0.17} [40]; while in the Type-I diquark-antidiquark model [5], the Z_c(3900) was interpreted as the mixing state

      |x_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle = \cos \theta_1 \; |0_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle + \sin \theta_1 \; |1_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \, ,

      (66)

      where

      |1_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle = | 1_{qc}, 1_{\bar q \bar c} \rangle_{J = 1} \, ,

      (67)

      and a small |1_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle component is able to increase this ratio to \left( 2.3^{+3.3}_{-1.4} \right) \times 10^2 [40], which is almost one thousand times larger.

      Thus, we attempt to add this |1_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle component in this subsection. The interpolating current having the identical internal structure is

      \eta^{{\cal{Z}}^{\prime}}_{\mu}(x,y) = \eta^3_{\mu}([uc][\bar d \bar c]) - \eta^4_{\mu}([uc][\bar d \bar c]) \, ,

      (68)

      so that |x_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle can be described by

      \eta^{\rm{mix}}_{\mu}(x,y) = \cos \theta^{\prime}_1\; \eta^{{\cal{Z}}}_{\mu}(x,y) + {\rm i} \sin \theta^{\prime}_1\; \eta^{{\cal{Z}}^{\prime}}_{\mu}(x,y)\, ,

      (69)

      which transforms according to Eq. (16) as:

      \begin{split} \eta^{\rm{mix}}_{\mu}(x,y) \Longrightarrow & + \left( -{{\rm i}\over3} \cos \theta^{\prime}_1 + {\rm i} \sin \theta^{\prime}_1 \right) \; I^{P}(x^{\prime}) \; J^{V}_{\mu}(y^{\prime}) \\ & + \left( + {{\rm i}\over3} \cos \theta^{\prime}_1 + {\rm i} \sin \theta^{\prime}_1 \right) \; I^{V}_{\mu}(x^{\prime}) \; J^{P}(y^{\prime}) \\ & + \left( + {{\rm i}\over3} \cos \theta^{\prime}_1 - {{\rm i}\over3} \sin \theta^{\prime}_1 \right) \; I^{A,\nu}(x^{\prime}) \; J^{T}_{\mu\nu}(y^{\prime}) \\ & + \left( - {{\rm i}\over3} \cos \theta^{\prime}_1 - {{\rm i}\over3} \sin \theta^{\prime}_1 \right) \; I^{T}_{\mu\nu}(x^{\prime}) \; J^{A,\nu}(y^{\prime}) + \cdots \, . \end{split}

      (70)

      Note that the two mixing angles \theta_1 and \theta^{\prime}_1 are not necessarily the same (and probably not the same), but they can be related to each other, i.e.,

      \theta_1 = f(\theta^{\prime}_1) \, .

      (71)

      To solve this relation, we must determine the couplings of \eta^{{\cal{Z}}}_{\mu} and \eta^{{\cal{Z}}^{\prime}}_{\mu} to |0_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle and |1_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle , which we shall not investigate in this study. Nonetheless, we can plot the three ratios

      \begin{split} & {\cal{R}}_{\psi \pi} \equiv {\Gamma(|x_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow J/\psi\pi) \over \Gamma(|0_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow J/\psi\pi)} \, , \\ & {\cal{R}}_{\eta_c \rho} \equiv {\Gamma(|x_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow \eta_c \rho) \over \Gamma(|0_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow \eta_c \rho)} \, , \\ & {\cal{R}} \equiv {{\cal{B}}(|x_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow \eta_c\rho) \over {\cal{B}}(|x_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow J/\psi\pi)} \, , \end{split}

      (72)

      as functions of the mixing angle \theta^{\prime}_1 , as shown in Fig. 4. We find that {\cal{R}}_{\psi \pi} decreases and {\cal{R}}_{\eta_c \rho} increases, so that the ratio {\cal{R}} increases rapidly as the mixing angle \theta^{\prime}_1 decreases from 0 to -10^{\rm{o}} .

      Figure 4.  The ratios (a) {\cal{R}}_{\psi\pi} \equiv \dfrac{\Gamma(|x_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow J/\psi\pi) }{ \Gamma(|0_{qc}1_{\bar q \bar c}; 1^{+-} \rangle \rightarrow J/\psi\pi)} , (b) {\cal{R}}_{\eta_c\rho} \equiv \dfrac{\Gamma(|x_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow \eta_c\rho) }{ \Gamma(|0_{qc}1_{\bar q \bar c}; 1^{+-} \rangle \rightarrow \eta_c\rho)} , and (c) {\cal{R}} \equiv \dfrac{{\cal{B}}(|x_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow \eta_c\rho) }{{\cal{B}}(|x_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow J/\psi\pi)} as functions of the mixing angle \theta^{\prime}_1 .

      Especially, after fine-tuning \theta^{\prime}_1 = -8.8^{\rm{o}} , we obtain

      \begin{split} {\cal{R}} \equiv & \dfrac{{\cal{B}}(|x_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow \eta_c\rho) }{ {\cal{B}}(|x_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow J/\psi\pi)} = 2.2 \, , \\ & \dfrac{{\cal{B}}(|x_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow h_c\pi) }{ {\cal{B}}(|x_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow J/\psi\pi)} = 0.052 \, , \\ & \dfrac{{\cal{B}}(|x_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow \chi_{c1}\rho \rightarrow \chi_{c1} \pi \pi)}{{\cal{B}}(|x_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow J/\psi\pi)} = 1.5 \times 10^{-5} \, . \end{split}

      (73)

      The first ratio {\cal{R}} is 2.2, which is the same as the BESIII measurement {\cal{R}}_{Z_c} = 2.2 \pm 0.9 [29].

      The decay of |x_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle into two charmed mesons can be described by the current \eta^{\rm{mix}}_{\mu}(x,y) together with the transformation (17):

      \begin{split} \eta^{\rm{mix}}_{\mu}(x,y) \Longrightarrow & - {{\rm i}\over3}\; \cos \theta^{\prime}_1\; \xi_{\mu}^2(x^{\prime},y^{\prime}) + {1\over3}\; \cos \theta^{\prime}_1\; \xi_{\mu}^3(x^{\prime},y^{\prime}) \\ & -\; \sin \theta^{\prime}_1\; \xi_{\mu}^1(x^{\prime},y^{\prime}) - {{\rm i}\over3}\; \sin \theta^{\prime}_1\; \xi_{\mu}^4(x^{\prime},y^{\prime}) + \cdots \, , \end{split}

      (74)

      so that

      \begin{split} & \dfrac{{\cal{B}}(|x_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow D \bar D^{*} + \bar D D^{*}) }{ {\cal{B}}(|x_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow J/\psi\pi + \eta_c\rho)} = 0.26 \times \dfrac{c_2^2}{ c_1^2} \, , \\ &\dfrac{{\cal{B}}(|x_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow D \bar D_0^{*} + \bar D D_0^{*} \rightarrow D \bar D \pi) }{ {\cal{B}}(|x_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow J/\psi\pi + \eta_c\rho)} = 2.5 \times 10^{-7} \times \dfrac{c_2^2 }{ c_1^2} \, . \end{split}

      (75)

      Hence, |x_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle can decay into the D \bar D^* final state, which is consistent with the BESIII observations [26, 27]. Moreover, it was proposed in Ref. [67] 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. However, 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. Thus, c_2 may be significantly larger than c_1 , so that |x_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle may mainly decay into two charmed mesons.

    5.   Decay properties of the Z_c(3900) as a hadronic molecular state
    • Another possible interpretation of the Z_c(3900) is the D \bar D^* hadronic molecular state of J^{PC} = 1^{+-} [710], i.e., | D \bar D^*; 1^{+-} \rangle defined in Eq. (13). Its relevant current \xi^{{\cal{Z}}}_{\mu}(x,y) has been given in Eq. (14). We can transform this current to \theta_{\mu}^i(x,y) according to transformation (19), through which we shall extract some decay properties of the Z_c(3900) as a hadronic molecular state in the following subsections.

    • 5.1.   \xi^{{\cal{Z}}}_{\mu}\big([\bar c u][\bar d c]\big) \longrightarrow \theta_{\mu}^{i}\big([\bar c c] + [\bar d u]\big)

    • As depicted in Fig. 5, when the c and \bar c quarks meet each other and the u and \bar d 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 | D \bar D^*; 1^{+-} \rangle can be described by transformation (19):

      Figure 5.  (color online) The decay of a hadronic molecular state into one charmonium meson and one light meson. This decay can happen through either (b) a direct fall-apart process, or (c) a process with gluon(s) exchanged, that is the {\cal{O}}(\alpha_s) corrections.

      \begin{split} \xi^{{\cal{Z}}}_{\mu}(x,y) \Longrightarrow& -\dfrac{1}{6}\; \theta_{\mu}^1(x^{\prime},y^{\prime}) - {1\over6}\; \theta_{\mu}^2(x^{\prime},y^{\prime}) - \dfrac{{\rm i}}{6}\; \theta_{\mu}^3(x^{\prime},y^{\prime})\\ & - {{\rm i}\over6}\; \theta_{\mu}^4(x^{\prime},y^{\prime}) + \cdots = + {{\rm i}\over6}\; I^{P}(x^{\prime}) \; J^{V}_{\mu}(y^{\prime}) \\ &+ {{\rm i}\over6}\; I^{V}_{\mu}(x^{\prime}) \; J^{P}(y^{\prime}) - \dfrac{{\rm i}}{6}\; I^{A,\nu}(x^{\prime}) \; J^{T}_{\mu\nu}(y^{\prime}) \\ &- \dfrac{{\rm i}}{6}\; I^{T}_{\mu\nu}(x^{\prime}) \; J^{A,\nu}(y^{\prime}) + \cdots \, , \end{split}

      (76)

      where we have only kept the direct fall-apart process described by \theta_{\mu}^{1,2,3,4} , but neglected the {\cal{O}}(\alpha_s) corrections described by \theta_{\mu}^{5,6,7,8} .

      We repeat the same procedures as those performed in Sec. 4.1, and extract the following coupling constants from this transformation:

      \begin{split} & h^S_{\eta_c \rho} = \dfrac{{\rm i} c_4}{ 6} \lambda_{\eta_c} m_\rho f_{\rho^+} = {\rm i} c_4\; 3.65 \times 10^{10}\; {\rm{MeV}}^4 \, , \\[-1pt] & h^D_{\eta_c \rho} = \dfrac{{\rm i} c_4}{ 6} f_{\eta_c} f^T_{\rho} = {\rm i} c_4\; 1.03 \times 10^{4}\; {\rm{MeV}}^2 \, , \\[-1pt] & h^S_{\psi \pi} = \dfrac{{\rm i} c_4}{ 6} \lambda_{\pi} m_{J/\psi} f_{J/\psi} ={\rm i} c_4\; 5.93 \times 10^{10}\; {\rm{MeV}}^4 \, , \\[-1pt] & h^D_{\psi \pi} = \dfrac{{\rm i} c_4 }{ 6} f_{\pi^+} f^T_{J/\psi} = {\rm i} c_4\; 0.89 \times 10^{4}\; {\rm{MeV}}^2 \, , \\[-1pt] & h_{\eta_c b_1} = \dfrac{{\rm i} c_4}{ 6} f_{\eta_c} f^T_{b_1} = {\rm i} c_4\; 1.16 \times 10^{4}\; {\rm{MeV}}^2 \, , \\[-1pt] & h_{\chi_{c1} \rho} = \dfrac{c_4}{ 6} m_{\chi_{c1}} f_{\chi_{c1}} f^T_{\rho} = c_4\; 3.12 \times 10^{7}\; {\rm{MeV}}^3 \, , \\[-1pt] & h_{\chi_{c1} b_1} = \dfrac{c_4}{ 6} m_{\chi_{c1}} f_{\chi_{c1}} f^T_{b_1} = c_4\; 3.53 \times 10^{7}\; {\rm{MeV}}^3 \, , \\[-1pt] & h_{h_c \pi} = \dfrac{{\rm i} c_4}{ 6} f_{\pi^+} f^T_{h_c} = {\rm i} c_4\; 0.51 \times 10^{4}\; {\rm{MeV}}^2 \, , \\[-1pt] & h_{\psi a_1} = \dfrac{c_4}{ 6} f^T_{J/\psi} m_{a_1} f_{a_1} = c_4\; 2.13 \times 10^{7}\; {\rm{MeV}}^3 \, , \\[-1pt] & h_{h_c a_1} = \dfrac{c_4}{ 6} f^T_{h_c} m_{a_1} f_{a_1} = c_4\; 1.22 \times 10^{7}\; {\rm{MeV}}^3 \, . \end{split}

      (77)

      The above coupling constants are related to the S- and D-wave | D \bar D^*; 1^{+-} \rangle \to \eta_c \rho decays, the S- and D-wave | D \bar D^*; 1^{+-} \rangle \rightarrow J/\psi \pi decays, and the | D \bar D^*; 1^{+-} \rangle \rightarrow \eta_c b_1 , \chi_{c1} \rho , \chi_{c1} b_1 , h_c \pi , J/\psi a_1 , h_c a_1 decays, respectively. All of them contain an overall factor of c_4 .

      Using the above coupling constants, we further obtain

      \begin{split} & {{\cal{B}}(| D \bar D^*; 1^{+-} \rangle \rightarrow \eta_c\rho) \over {\cal{B}}(| D \bar D^*; 1^{+-} \rangle \rightarrow J/\psi\pi)} = 0.059 \, , \\ & {{\cal{B}}(| D \bar D^*; 1^{+-} \rangle \rightarrow h_c\pi) \over {\cal{B}}(| D \bar D^*; 1^{+-} \rangle \rightarrow J/\psi\pi)} = 0.0088 \, , \\ & {{\cal{B}}(| D \bar D^*; 1^{+-} \rangle \rightarrow \chi_{c1}\rho \rightarrow \chi_{c1}\pi \pi) \over {\cal{B}}(| D \bar D^*; 1^{+-} \rangle \rightarrow J/\psi\pi)} = 1.4 \times 10^{-6} \, . \end{split}

      (78)

      These values are surprisingly the same as Eqs. (56), obtained in Sec. 4.1 for the compact tetraquark state |0_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle .

    • 5.2.   \xi^{{\cal{Z}}}_{\mu}\big([\bar c u][\bar d c]\big) \longrightarrow \xi^i_{\mu}\big([\bar c u] + [\bar d c]\big)

    • Assuming the Z_c(3900) to be the D \bar D^* hadronic molecular state of J^{PC} = 1^{+-} , it can naturally decay to the D \bar D^* final state, of which the fall-apart process can be described by

      \xi^{{\cal{Z}}}_{\mu}(x,y) \Longrightarrow \xi_{\mu}^1(x^{\prime},y^{\prime}) = -{\rm i}\; O^{V}_{\mu}(x^{\prime}) \; O^{P}(y^{\prime}) + \{ \gamma_{\mu} \leftrightarrow \gamma_5 \} \, .

      (79)

      The decay of | D \bar D^*; 1^{+-} \rangle into the D \bar D^{*} final state is contributed by this term to be

      \begin{split} \langle Z_c^+(p,\epsilon) | D^+(p_1) \bar D^{*0}(p_2,\epsilon_2) \rangle \approx &-{\rm i}c_5\; \lambda_D m_{D^*} f_{D^*}\; \epsilon \cdot \epsilon_2 \\ & \equiv h_{D \bar D^*}\; \epsilon \cdot \epsilon_2 \, , \end{split}

      (80)

      \begin{split} \langle Z_c^+(p,\epsilon) | \bar D^0(p_1) D^{*+}(p_2,\epsilon_2) \rangle \approx & -{\rm i}c_5\; \lambda_D m_{D^*} f_{D^*}\; \epsilon \cdot \epsilon_2 \\ & \equiv h_{D \bar D^*}\; \epsilon \cdot \epsilon_2 \, , \end{split}

      (81)

      where c_5 is an overall factor, and is likely larger than c_4 . Numerically, we obtain

      h_{D \bar D^*} = -{\rm i}c_5\; 2.95 \times 10^{11}\; {\rm{MeV}}^4 \, .

      (82)

      Comparing the | D \bar D^*; 1^{+-} \rangle \rightarrow D \bar D^{*} decay studied in the present subsection with the | D \bar D^*; 1^{+-} \rangle \rightarrow J/\psi \pi and | D \bar D^*; 1^{+-} \rangle \rightarrow \eta_c\rho decays studied in the previous subsection, we obtain

      {{\cal{B}}(| D \bar D^*; 1^{+-} \rangle \rightarrow D \bar D^{*} + \bar D D^{*}) \over {\cal{B}}(| D \bar D^*; 1^{+-} \rangle \rightarrow J/\psi\pi + \eta_c\rho)} = 25 \times {c_5^2 \over c_4^2} \, .

      (83)

      The current \xi^{{\cal{Z}}}_{\mu}(x,y) does not correlate with the term \xi_{\mu}^{3} = O^{A}_{\mu} \times O^{S} , so that | D \bar D^*; 1^{+-} \rangle does not decay into the D \bar D_0^{*} final state

      {{\cal{B}}(| D \bar D^*; 1^{+-} \rangle \rightarrow D \bar D_0^{*} + \bar D D_0^{*} \rightarrow D \bar D \pi) \over {\cal{B}}(| D \bar D^*; 1^{+-} \rangle \rightarrow J/\psi\pi + \eta_c\rho)} \approx 0 \, .

      (84)

      Eqs. (83) and (84) suggest that | D \bar D^*; 1^{+-} \rangle mainly decays into two charmed mesons, other than one charmonium meson and one light meson. This conclusion is opposite to the one obtained in Sec. 4.2 for the compact tetraquark state |0_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle .

    • 5.3.   Mixing with the | D^* \bar D^*; 1^{+-} \rangle

    • Similarly to Sec. 4.4, we add a small | D^* \bar D^*; 1^{+-} \rangle component

      | D^* \bar D^*; 1^{+-} \rangle = | D^* \bar D^* \rangle_{J = 1} \, ,

      (85)

      to | D \bar D^*; 1^{+-} \rangle in this subsection to reevaluate the ratio {\cal{R}}_{Z_c} . The interpolating current having the same internal structure as | D^* \bar D^*; 1^{+-} \rangle is

      \xi^{{\cal{Z}}^{\prime}}_{\mu}(x,y) = \xi^2_{\mu}([\bar c u][\bar d c]) \, ,

      (86)

      so that we can use

      \xi^{\rm{mix}}_{\mu}(x,y) = \cos \theta^{\prime}_2\; \xi^{{\cal{Z}}}_{\mu}(x,y) + {\rm i} \sin \theta^{\prime}_2\; \xi^{{\cal{Z}}^{\prime}}_{\mu}(x,y) \, ,

      (87)

      to described the mixed molecular state

      |D^{(*)} \bar D^*; 1^{+-} \rangle = \cos \theta_2 \; |D \bar D^*; 1^{+-} \rangle + \sin \theta_2 \; |D^* \bar D^*; 1^{+-} \rangle \, .

      (88)

      The current \xi^{\rm{mix}}_{\mu}(x,y) transforms according to Eq. (19) to be

      \begin{split} \xi^{\rm{mix}}_{\mu}(x,y) \Longrightarrow & + \left( + {{\rm i}\over6}\cos \theta^{\prime}_2 - {{\rm i}\over2}\sin \theta^{\prime}_2 \right) \; I^{P}(x^{\prime}) \; J^{V}_{\mu}(y^{\prime}) \\ & + \left( + {{\rm i}\over6}\cos \theta^{\prime}_2 + {{\rm i}\over2} \sin \theta^{\prime}_2 \right) \; I^{V}_{\mu}(x^{\prime}) \; J^{P}(y^{\prime}) \\ &+ \left( - {{\rm i}\over6}\cos \theta^{\prime}_2 + {{\rm i}\over6} \sin \theta^{\prime}_2 \right) \; I^{A,\nu}(x^{\prime}) \; J^{T}_{\mu\nu}(y^{\prime}) \\ &+ \left( - {{\rm i}\over6}\cos \theta^{\prime}_2 - {{\rm i}\over6} \sin \theta^{\prime}_2 \right) \; I^{T}_{\mu\nu}(x^{\prime}) \; J^{A,\nu}(y^{\prime}) + \cdots \, . \end{split}

      (89)

      After fine-tuning \theta^{\prime}_2 = -8.8^{\rm{o}} , we obtain

      \begin{split} {\cal{R}}^{\prime} \equiv & {{\cal{B}}(|D^{(*)} \bar D^*; 1^{+-} \rangle \rightarrow \eta_c\rho) \over {\cal{B}}(|D^{(*)} \bar D^*; 1^{+-} \rangle \rightarrow J/\psi\pi)} = 2.2 \, , \\ & {{\cal{B}}(|D^{(*)} \bar D^*; 1^{+-} \rangle \rightarrow h_c\pi) \over {\cal{B}}(|D^{(*)} \bar D^*; 1^{+-} \rangle \rightarrow J/\psi\pi)} = 0.052 \, , \\ &{{\cal{B}}(|D^{(*)} \bar D^*; 1^{+-} \rangle \rightarrow \chi_{c1}\rho \rightarrow \chi_{c1} \pi \pi) \over {\cal{B}}(|D^{(*)} \bar D^*; 1^{+-} \rangle \rightarrow J/\psi\pi)} = 1.5 \times 10^{-5} \, , \end{split}

      (90)

      the values of which are the same as those in Eqs. (78) obtained in Sec. 4.1 for the mixed compact tetraquark state |x_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle . In fact, we can also plot the following three ratios

      \begin{split} & {\cal{R}}^{\prime}_{\psi \pi} \equiv {\Gamma(|D^{(*)} \bar D^*; 1^{+-} \rangle \rightarrow J/\psi\pi) \over \Gamma(|D \bar D^*; 1^{+-} \rangle \rightarrow J/\psi\pi)} \, , \\ &{\cal{R}}^{\prime}_{\eta_c \rho} \equiv {\Gamma(|D^{(*)} \bar D^*; 1^{+-} \rangle \rightarrow \eta_c \rho) \over \Gamma(|D \bar D^*; 1^{+-} \rangle \rightarrow \eta_c \rho)} \, , \\ & {\cal{R}}^{\prime} \equiv {{\cal{B}}(|D^{(*)} \bar D^*; 1^{+-} \rangle \rightarrow \eta_c \rho) \over {\cal{B}}(|D^{(*)} \bar D^*; 1^{+-} \rangle \rightarrow J/\psi\pi)} \, , \end{split}

      (91)

      as functions of the mixing angle \theta^{\prime}_2 , and the obtained figures are identical to Fig. 4, where {\cal{R}}_{\psi \pi} , {\cal{R}}_{\eta_c \rho} , and {\cal{R}} are shown as functions of \theta^{\prime}_1 .

      We also obtain

      \begin{split} & {{\cal{B}}(|D^{(*)} \bar D^*; 1^{+-} \rangle \rightarrow D \bar D^{*} + \bar D D^{*}) \over {\cal{B}}(|D^{(*)} \bar D^*; 1^{+-} \rangle \rightarrow J/\psi\pi + \eta_c\rho)} = 67 \times {c_5^2 \over c_4^2} \, , \\ &{{\cal{B}}(|D^{(*)} \bar D^*; 1^{+-} \rangle \rightarrow D \bar D_0^{*} + \bar D D_0^{*} \rightarrow D \bar D \pi) \over {\cal{B}}(|D^{(*)} \bar D^*; 1^{+-} \rangle \rightarrow J/\psi\pi + \eta_c\rho)} \approx 0 \, , \end{split}

      (92)

      suggesting that |D^{(*)} \bar D^*; 1^{+-} \rangle mainly decays into two charmed mesons.

    6.   Summary and discussions
    • In this paper we systematically construct all the tetraquark currents/operators of J^{PC} = 1^{+-} with the quark content c \bar c q \bar q ( q = u/d ). There are three configurations: [cq][\bar c \bar q] , [\bar c q][\bar q c] , and [\bar c c][\bar q q] , and for each configuration we construct eight independent currents. We use the Fierz rearrangement of the Dirac and color indices to derive their relations, through which we study the strong decay properties of the Z_c(3900) :

      ● Using the transformation of [qc][\bar q \bar c] \to [\bar c c][\bar q q] , we study the decay properties of the Z_c(3900) as a compact diquark-antidiquark tetraquark state into one charmonium meson and one light meson.

      ● Using the transformation of [qc][\bar q \bar c] \to [\bar c q][\bar q c] , we study the decay properties of the Z_c(3900) as a compact diquark-antidiquark tetraquark state into two charmed mesons.

      ● We use the transformation of the [qc][\bar q \bar c] currents to the color-singlet-color-singlet [\bar c c][\bar q q] and [\bar c q][\bar q c] currents, and obtain the same relative branching ratios as the above results.

      ● Using the transformation of [\bar c q][\bar q c] \to [\bar c c][\bar q q] , we study the decay properties of the Z_c(3900) as a hadronic molecular state into one charmonium meson and one light meson.

      ● Through the [\bar c q][\bar q c] currents themselves, we study the decay properties of the Z_c(3900) as a hadronic molecular state into two charmed mesons.

      Our results suggest that the possible decay channels of the Z_c(3900) are: a) the two-body decays Z_c(3900) \to J/\psi\pi , Z_c(3900) \to \eta_c\rho , Z_c(3900) \to h_c\pi , and Z_c(3900) \to D \bar D^{*} , b) the three-body decays Z_c(3900) \rightarrow \chi_{c1}\rho \rightarrow \chi_{c1}\pi \pi and Z_c(3900) \rightarrow D \bar D_0^{*} + \bar D D_0^{*} \rightarrow D \bar D \pi , and c) the many-body decay chains Z_c(3900) \rightarrow J/\psi a_1 \rightarrow J/\psi \rho \pi \rightarrow J/\psi + 3 \pi and Z_c(3900) \rightarrow \eta_c b_1 \rightarrow \eta_c \omega \pi \rightarrow \eta_c + 4 \pi . Their relative branching ratios are summarized in Table 3, where we have investigated the following interpretations of the Z_c(3900) :

      channels|0_{qc}1_{\bar q \bar c}; 1^{+-} \rangle|x_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle(\theta^{\prime}_1 = -8.8^{\rm{o}}) |D \bar D^*; 1^{+-} \rangle|D^{(*)} \bar D^*; 1^{+-} \rangle(\theta^{\prime}_2 = -8.8^{\rm{o}})
      \dfrac {{\cal{B}}(Z_c \rightarrow \eta_c\rho)}{ {\cal{B}}(Z_c \rightarrow J/\psi\pi)}}0.0592.2 (input)0.0592.2 (input)
      {{\cal{B}}(Z_c \rightarrow h_c\pi)}{ {\cal{B}}(Z_c \rightarrow J/\psi\pi)}0.00880.0520.00880.052
      \dfrac {{\cal{B}}(Z_c \rightarrow \chi_{c1}\rho \rightarrow \chi_{c1} \pi \pi) }{{\cal{B}}(Z_c \rightarrow J/\psi\pi)}}1.4 \times 10^{-6}1.5 \times 10^{-5}1.4 \times 10^{-6}1.5 \times 10^{-5}
      \dfrac {{\cal{B}}(Z_c \rightarrow D \bar D^{*} + \bar D D^{*})}{{\cal{B}}(Z_c \rightarrow J/\psi\pi + \eta_c\rho)}}\approx 00.26 t_125 t_267 t_2
      \dfrac {{\cal{B}}(Z_c \rightarrow D \bar D_0^{*} + \bar D D_0^{*} \rightarrow D \bar D \pi)} {{\cal{B}}(Z_c \rightarrow J/\psi\pi + \eta_c\rho)}}9.3 t_1 \times 10^{-8}2.5 t_1 \times 10^{-7}\approx 0\approx 0

      Table 3.  Relative branching ratios of the Z_c(3900) evaluated through the Fierz rearrangement. \theta_{1,2}^{\prime} are the two mixing angles defined in Eqs. (69) and (87), which are fine-tuned to be \theta^{\prime}_1 = \theta^{\prime}_2 = -8.8^{\rm{o}}, so that \dfrac{{\cal{B}}(|x_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow \eta_c\rho) }{ {\cal{B}}(|x_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow J/\psi\pi)} = \dfrac{{\cal{B}}(|D^{(*)} \bar D^*; 1^{+-} \rangle \rightarrow \eta_c\rho)}{ {\cal{B}}(|D^{(*)} \bar D^*; 1^{+-} \rangle \rightarrow J/\psi\pi)} = 2.2 [29]. In this table, we do not take into account the phase angle \phi between S- and D-wave coupling constants.

      ● In the second and third columns of Table 3, |0_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle and |x_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle denote the compact tetraquark states of J^{PC} = 1^{+-} , as defined in Eq. (10) and Eq. (66), respectively. In particular, we have considered the mixing between the compact tetraquarks states

      | 0_{qc}1_{\bar q \bar c}; 1^{+-} \rangle \oplus | 1_{qc}1_{\bar q \bar c}; 1^{+-} \rangle \rightarrow | x_{qc}1_{\bar q \bar c}; 1^{+-} \rangle \, .

      (93)

      Using the mixing angle \theta^{\prime}_1 = -8.8^{\rm{o}} , we obtain

      \begin{split} {{\cal{B}}\left(| x_{qc}1_{\bar q \bar c}; 1^{+-} \rangle \rightarrow J/\psi\pi \!:\! \eta_c\rho \!:\! h_c\pi \!:\! \chi_{c1}\rho (\rightarrow \pi \pi) \!:\! D \bar D^{*} \!:\! D \bar D_0^{*} (\rightarrow \bar D \pi) \right) \over {\cal{B}}(| x_{qc}1_{\bar q \bar c}; 1^{+-} \rangle \rightarrow J/\psi\pi)} \approx 1 \!:\! 2.2 ({\rm{input}}) \!:\! 0.05 \!:\! 10^{-5} \!:\! 0.82 t_1 : 10^{-6} t_1 . \end{split}

      (94)

      ● In the fourth and fifth columns of Table 3, | D \bar D^*; 1^{+-} \rangle and |D^{(*)} \bar D^*; 1^{+-} \rangle denote the hadronic molecular states of J^{PC} = 1^{+-} , defined in Eq. (13) and Eq. (88), respectively. Especially, we have considered the mixing between the hadronic molecule states

      | D \bar D^*; 1^{+-} \rangle \oplus |D^{*} \bar D^*; 1^{+-} \rangle \rightarrow |D^{(*)} \bar D^*; 1^{+-} \rangle \, .

      (95)

      Using the mixing angle \theta^{\prime}_2 = -8.8^{\rm{o}} , we obtain

      \begin{split} &{{\cal{B}}\left(|D^{(*)} \bar D^*; 1^{+-} \rangle \rightarrow J/\psi\pi : \eta_c\rho : h_c\pi \, : \chi_{c1}\rho (\rightarrow \pi \pi) : D \bar D^{*} \right) \over {\cal{B}}(|D^{(*)} \bar D^*; 1^{+-} \rangle \rightarrow J/\psi\pi)} \\ \approx &\,1 \, : 2.2 ({\rm{input}}) : 0.05 : 10^{-5} \, : 210 t_2 \, . \end{split}

      (96)

      In the above expressions, we have used the recent BESIII measurement {\cal{R}}_{Z_c} \equiv \dfrac{{\cal{B}}(Z_c(3900) \rightarrow \eta_c\rho) }{ {\cal{B}}(Z_c(3900) \rightarrow J/\psi\pi)} = 2.2 \pm 0.9 [29] as an input to determine the mixing angles \theta^{\prime}_1 and \theta^{\prime}_2 . The ratio t_1 \equiv {c_2^2 / c_1^2} is the parameter measuring which process happens more easily, the process depicted in Fig. 2(b) or the process depicted in Fig. 3(b). Generally, the exchange of one light quark with another light quark seems to be easier than the exchange of one light quark with another heavy quark [67, 111]. Thus, it can be the case that t_1 \geq 1 . As discussed in Sec. 5.2, c_5 is likely larger than c_4 , so that the other ratio t_2 \equiv {c_5^2 / c_4^2} \geq 1 .

      The above relative branching ratios calculated in the present study turn out to be very different, which may be one of the reasons why many multiquark states were observed in only a few decay channels [75]. 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 but neglected the {\cal{O}}(\alpha_s) corrections described by color-octet-color-octet meson-meson currents. This means that there can be other decay channels.

      Based on Table 3 as well as Eqs. (94) and (96), we conclude this paper:

      ● The relative branching ratios \dfrac{{\cal{B}}(|0_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow \eta_c\rho) }{ {\cal{B}}(|0_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow J/\psi\pi)} and \dfrac{{\cal{B}}(| D \bar D^*; 1^{+-} \rangle \rightarrow \eta_c\rho) }{{\cal{B}}(| D \bar D^*; 1^{+-} \rangle \rightarrow J/\psi\pi)} are both around 0.059, significantly smaller than the BESIII measurement {\cal{R}}_{Z_c} = 2.2 \pm 0.9 at \sqrt{s} = 4.226 GeV [29]. However, we can add a small |1_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle component to |0_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle to obtain \dfrac{{\cal{B}}(|x_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow \eta_c\rho) }{ {\cal{B}}(|x_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow J/\psi\pi)} = 2.2 ; we can also add a small |D^{*} \bar D^*; 1^{+-} \rangle component to | D \bar D^*; 1^{+-} \rangle to obtain \dfrac{{\cal{B}}(|D^{(*)} \bar D^*; 1^{+-} \rangle \rightarrow \eta_c\rho) }{{\cal{B}}(|D^{(*)} \bar D^*; 1^{+-} \rangle \rightarrow J/\psi\pi)} = 2.2 . Note that if the relevant mixing angles change dynamically, the ratio {\cal{R}}_{Z_c} would also change dynamically.

      ● The relative branching ratios of the | D \bar D^*; 1^{+-} \rangle decays into one charmonium meson and one light meson are the same as those of the |0_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle decays. After taking proper mixing angles, the relative branching ratios of the |D^{(*)} \bar D^*; 1^{+-} \rangle decays into one charmonium meson and one light meson are also the same as those of the |x_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle 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.

      |0_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle mainly decays into one charmonium meson and one light meson, but |x_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle might mainly decay into two charmed mesons after taking into account the barrier of the diquark-antidiquark potential (see detailed discussions in Ref. [67] proposing c_2 \gg c_1 ). Both | D \bar D^*; 1^{+-} \rangle and |D^{(*)} \bar D^*; 1^{+-} \rangle mainly decay into two charmed mesons.

      It is useful to generally discuss our uncertainty. In this study, we have worked within the naive factorization scheme; thus, our uncertainty is greater than that of well-developed QCD factorization method [6264], which is at 5% when being applied to study the weak and radiative decay properties of conventional (heavy) hadrons. On the other hand, the tetraquark decay constant f_{Z_c} is removed when calculating relative branching ratios. This significantly reduces our uncertainty because this parameter has not yet been accurately determined. Hence, we estimate our uncertainty to be approximately X^{+100\%}_{-\; 50\%} .

      Next,let us compare our results with other theoretical calculations. First, we compare them with the QCD sum rule results obtained in Refs. [30, 31], where the Z_c(3900) is assumed to be a compact diquark-antidiquark tetraquark state. In this study, we find that decays of the Z_c(3900) into J/\psi\pi and \eta_c\rho can happen through both the S-wave and D-wave, and we have calculated these two amplitudes together, as shown in Eqs. (43-48); we can also calculate them individually and obtain

      \begin{split} & {{\cal{B}}(|0_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow \eta_c\rho)_{S{-}{\rm{wave}}} \over {\cal{B}}(|0_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow J/\psi\pi)_{S{-}{\rm{wave}}}} = 0.24 \, , \\ & {{\cal{B}}(|0_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow \eta_c\rho)_{D{-}{\rm{wave}}} \over {\cal{B}}(|0_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow J/\psi\pi)_{D{-}{\rm{wave}}}} = 0.82 \, . \end{split}

      (97)

      In the former equation we have only considered the S-wave amplitudes, and in the latter, only the D-wave amplitudes. The QCD sum rule study in Ref. [30] only considers the S-wave amplitudes, where they obtained

      \begin{split} & \Gamma(Z_c(3900) \rightarrow \eta_c\rho) = 27.5 \pm 8.5\; {\rm{MeV}} \, , \\ & \Gamma(Z_c(3900) \rightarrow J/\psi\pi) = 29.1 \pm 8.2\; {\rm{MeV}} \, , \end{split}

      (98)

      so that

      {{\cal{B}}(Z_c(3900) \rightarrow \eta_c\rho)_{S{-}{\rm{wave}}} \over {\cal{B}}(Z_c(3900) \rightarrow J/\psi\pi)_{S{-}{\rm{wave}}}} = 0.95^{+0.47}_{-0.36} \, .

      (99)

      The QCD sum rule study in Ref. [31] only considers the D-wave amplitudes, where they obtained

      \begin{split} &\Gamma(Z_c(3900) \rightarrow \eta_c\rho) = 23.8 \pm 4.9\; {\rm{MeV}} \, , \\ &\Gamma(Z_c(3900) \rightarrow J/\psi\pi) = 41.9 \pm 9.4\; {\rm{MeV}} \, , \end{split}

      (100)

      so that

      {{\cal{B}}(Z_c(3900) \rightarrow \eta_c\rho)_{D{-}{\rm{wave}}} \over {\cal{B}}(Z_c(3900) \rightarrow J/\psi\pi)_{D{-}{\rm{wave}}}} = 0.57^{+0.20}_{-0.16} \, .

      (101)

      Hence, our results are more or less consistent with the QCD sum rule calculations [30, 31]. Here, we would like to note that the D-wave decay amplitudes are important and cannot be neglected:

      \begin{split} & {{\cal{B}}(|0_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow \eta_c\rho)_{D{-}{\rm{wave}}} \over {\cal{B}}(|0_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow \eta_c\rho)_{S{-}{\rm{wave}}}} = 0.51 \, , \\ & {{\cal{B}}(|0_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow J/\psi\pi)_{D{-}{\rm{wave}}} \over {\cal{B}}(|0_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow J/\psi\pi)_{S{-}{\rm{wave}}}} = 0.15 \, . \end{split}

      (102)

      In fact, there is still one parameter not considered in our calculations: the phase angle \phi between the S- and D-wave decay amplitudes. For completeness, we shall investigate its relevant uncertainty in Appendix A.

      Following this, we compare our results with Ref. [40], where the authors assumed the Z_c(3900) to be a hadronic molecular state and used the Non-Relativistic Effective Field Theory (a framework based on HQET and NRQCD) to obtain

      {{\cal{B}}(Z_c(3900) \rightarrow \eta_c\rho) \over {\cal{B}}(Z_c(3900) \rightarrow J/\psi\pi)} = 0.046^{+0.025}_{-0.017} \, .

      (103)

      This value is well consistent with our result

      {{\cal{B}}(| D \bar D^*; 1^{+-} \rangle \rightarrow \eta_c\rho) \over {\cal{B}}(| D \bar D^*; 1^{+-} \rangle \rightarrow J/\psi\pi)} = 0.059 \, .

      (104)

      Finally, we propose the BESIII, Belle, Belle-II, and LHCb Collaborations to search for those decay channels not yet observed, in order to better understand the nature of the Z_c(3900) .

      We thank Fu-Sheng Yu and Qin Chang for helpful discussions.

    Appendix A: Uncertainties due to phase angles
    • There are two different effective Lagrangians for the Z_c(3900) decay into the \eta_c \rho final state, as given in Eqs. (44) and (45):

      {\cal{L}}^S_{\eta_c \rho} = g^S_{\eta_c \rho}\; Z_{c}^{+,\mu}\; \eta_c\; \rho^{-}_{\mu} + \cdots \, ,\tag{A1}

      (A1)

      {\cal{L}}^D_{\eta_c \rho} = g^D_{\eta_c \rho} \times \left( g^{\mu\sigma}g^{\nu\rho} - g^{\mu\nu}g^{\rho\sigma} \right) Z_{c,\mu}^{+}\; \partial_\rho \eta_c\; \partial_\sigma \rho^{-}_{\nu} + \cdots \, . \tag{A2}

      (A2)

      There are also two different effective Lagrangians for the Z_c(3900) decay into the J/\psi \pi final state, as given in Eqs. (47) and (48):

      {\cal{L}}^S_{\psi \pi} = g^S_{\psi \pi}\; Z_{c}^{+,\mu}\; \psi_{\mu}\; \pi^- + \cdots \, , \tag{A3}

      (A3)

      {\cal{L}}^D_{\psi \pi} = g^D_{\psi \pi} \times \left( g^{\mu\rho}g^{\nu\sigma} - g^{\mu\nu}g^{\rho\sigma} \right) Z_{c,\mu}^{+}\; \partial_\rho \psi_\nu\; \partial_\sigma \pi^- + \cdots \, . \tag{A4}

      (A4)

      There can be a phase angle \phi between g^S_{\eta_c \rho} and g^D_{\eta_c \rho} , as well as between g^S_{\psi \pi} and g^D_{\psi \pi} . This parameter is unknown and therfore not fixed, because QCD sum rules dictate that one can only calculate the modular square of the decay constant, such as |f_{\eta_c}|^2 . This might also be the case for Lattice QCD and the light front model. For example, see the different definitions of f_{\eta_c} in Refs. [83, 91].

      We rotate this phase angle between all S- and D-wave coupling constants to be \phi = \pi , and revise the previous calculations. The results are summarized in Table A1. In particular, using the mixing angle \theta^{\prime}_1 = \theta^{\prime}_2 = -10.1^{\rm{o}} , we obtain

      channels|0_{qc}1_{\bar q \bar c}; 1^{+-} \rangle|x_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle(\theta^{\prime}_1 = -10.1^{\rm{o}}) |D \bar D^*; 1^{+-} \rangle|D^{(*)} \bar D^*; 1^{+-} \rangle(\theta^{\prime}_2 = -10.1^{\rm{o}})
      \dfrac{{\cal{B}}(Z_c \rightarrow \eta_c\rho)}{ {\cal{B}}(Z_c \rightarrow J/\psi\pi)}0.362.2 (input)0.362.2 (input)
      \dfrac{{\cal{B}}(Z_c \rightarrow h_c\pi) }{ {\cal{B}}(Z_c \rightarrow J/\psi\pi)}0.00180.00380.00180.0038
      \dfrac{{\cal{B}}(Z_c \rightarrow \chi_{c1}\rho \rightarrow \chi_{c1} \pi \pi) }{ {\cal{B}}(Z_c \rightarrow J/\psi\pi)}2.8 \times 10^{-7}1.2 \times 10^{-6}2.8 \times 10^{-7}1.2 \times 10^{-6}
      \dfrac{{\cal{B}}(Z_c \rightarrow D \bar D^{*} + \bar D D^{*}) }{ {\cal{B}}(Z_c \rightarrow J/\psi\pi + \eta_c\rho)}\approx 00.059 t_13.9 t_25.2 t_2
      \dfrac{{\cal{B}}(Z_c \rightarrow D \bar D_0^{*} + \bar D D_0^{*} \rightarrow D \bar D \pi)}{ {\cal{B}}(Z_c \rightarrow J/\psi\pi + \eta_c\rho)}1.5 t_1 \times 10^{-8}2.0 t_1 \times 10^{-8}\approx 0\approx 0

      Table A1.  Relative branching ratios of the Z_c(3900) evaluated through the Fierz rearrangement. The two mixing angles are fine-tuned to be \theta^{\prime}_1 = \theta^{\prime}_2 = -10.1^{\rm{o}}, so that \dfrac{{\cal{B}}(|x_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow \eta_c\rho) }{{\cal{B}}(|x_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow J/\psi\pi)} = \dfrac{{\cal{B}}(|D^{(*)} \bar D^*; 1^{+-} \rangle \rightarrow \eta_c\rho)}{ {\cal{B}}(|D^{(*)} \bar D^*; 1^{+-} \rangle \rightarrow J/\psi\pi)} = 2.2 [29]. In this table, we fix the phase angle \theta between all the S- and D-wave coupling constants to be \theta = \pi.

      \begin{split} {{\cal{B}}\left(| x_{qc}1_{\bar q \bar c}; 1^{+-} \rangle \rightarrow J/\psi\pi\; : \eta_c\rho : h_c\pi : \chi_{c1}\rho (\rightarrow \pi \pi) : D \bar D^{*} : D \bar D_0^{*} (\rightarrow \bar D \pi)\; \right) \over {\cal{B}}(| x_{qc}1_{\bar q \bar c}; 1^{+-} \rangle \rightarrow J/\psi\pi)} \approx \; 1 : \; 2.2\; ({\rm{input}})\; : 0.004 : 10^{-6} \; \, : 0.19\; t_1 : \; 10^{-7}\; t_1\; \, ,\end{split} \tag{{A5}}

      (A5)

      \begin{split} {{\cal{B}}\left(|D^{(*)} \bar D^*; 1^{+-} \rangle \rightarrow J/\psi\pi :\; \eta_c\rho :\; h_c\pi :\; \chi_{c1}\rho (\rightarrow \pi \pi)\; :\; D \bar D^{*}\; \right) \over {\cal{B}}(|D^{(*)} \bar D^*; 1^{+-} \rangle \rightarrow J/\psi\pi)} \approx \,1 \, : 2.2 ({\rm{input}})\; : 0.004 : 10^{-6}\; \, : 16 t_2 \, . \end{split} \tag{{A6}}

      (A6)
Reference (111)

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