Axialvector tetraquark candidates for Zc(3900), Zc(4020), Zc(4430), and Zc(4600)

We construct the axialvector and tensor current operators to systematically investigate the ground and first radially excited tetraquark states with quantum numbers using the QCD sum rules. We observe one axialvector tetraquark candidate for and , two axialvector tetraquark candidates for the , and three axialvector tetraquark candidates for .


Introduction
The LHCb collaboration in 2019 performed an angular analysis of weak decays using protonproton collision data, inspected versus planes, and observed two possible resonant structures in the vicinity of the energies and , respectively [1]. There were two tentative proposals of the structure in the vicinity of , namely the type vector tetraquark state with [2] and the first radially excited type tetraquark state with [3]. In this study, the subscripts P, S, V, A, and T represent the pseudoscalar, scalar, vector, axialvector, and tensor color-antitriplet diquark states, respectively.
The BESIII collaboration in 2013 reported the charged charmonium-like resonance in the invariant mass spectrum of the process with and , respectively [4]. The Belle collaboration also observed in the same process [5]; furthermore, the CLEO collaboration confirmed the existence of the [6]. At around the same time, the BESIII collaboration observed the charmonium-like resonance in the vicinity of the threshold in the electron-positron scattering process [7]. Moreover, the BESIII collabora- tion observed the charmonium-like resonance in the invariant mass spectrum of the electron-positron collisions [8]. Presently, and are listed in The Review of Particle Physics as the same particle [9]. In 2014, the LHCb collaboration performed a four-dimensional fit of the scattering amplitude for the decay in proton-proton collisions and obtained the first independent confirmation of the charmonium-like resonance , determining its quantum numbers to be [10]. In 2017, the BESIII collaboration established the quantum numbers of the charmonium-like resonance as [11]. There are several possible explanations for the exotic states and , including the molecular states (from heavy quark symmetries [12,13], QCD sum rules [14,15], light-front quark model [16], one-pion exchange model [17], and phenomenological Lagrangian approach [18]), tetraquark states (from the diquark model with the effective Hamiltonian [19], QCD sum rules [20][21][22][23], and the potential model [24]), triangle singularities (in rescattering amplitudes) [ the superscripts i, j, k, m, and n are color indexes with values obeying the antisymmetric tensor and the charge conjugation matrix . If we preform the charge conjugation (and parity) transform (and ), the axialvector current operators and tensor current operator have the following properties: where the coordinates and .
The diquark operators in the attractive color-antitriplet channel have five spinor structures, where , C, , , and or correspond to the scalar, pseudoscalar, vector, axialvector, and tensor diquark operators, respectively. The favorable quark-quark correlations are the scalar diquark and axialvector diquark in the color-antitriplet channel from the QCD sum rules [33]. If we introduce a relative Pwave between the light quark and heavy quark, we can obtain the pseudoscalar diquark operator and vector diquark operator without explicitly introducing the additional P-wave, as multiplying a can change the parity, and the P-wave effect is embodied in the underlined . Because the pseudoscalar and vector diquark states (or the P-wave diquark states) have larger masses compared to the scalar and axialvector diquark states, we choose the scalar diquark and axialvector diquark to construct the four-quark current operators to interpolate the lower tetraquark states.
The tensor heavy diquark operators and have both axialvector and vector constituents, where the space indexes j, , 2, 3. The tensor diquark operators also play an important role in constructing the tetraquark current operators [34]. We multiply the tensor diquark (antidiquark) operators with the axialvector or vector antidiquark (diquark) operators to project the axialvector and vector constituents and construct the four-Chinese Physics C Vol. 44, No. 6 (2020) 063105 quark current operators and . Thereafter, we use and to represent the vector and axialvector constituents of the tensor diquark operators, respectively.
The four-quark current operators , , and potentially couple to the type, type, and type axialvector hidden-charm tetraquark states with the spin-parity-charge-conjugation , respectively. Meanwhile, the current operator potentially couples to both the type axialvector tetraquark state with and the vector tetraquark state with . Hereafter, we do not distinguish the negative or positive electric-charge of tetraquark states, as they have degenerate masses.
We insert a complete set of hadron states that have nonvanishing couplings with the four-quark current operators and into the Green functions and to obtain the hadronic representation [35,36]. Then, we separate the ground state axialvector and vector tetraquark state contributions from other contributions, such as the higher excited tetraquark and continuum states, to obtain the results, where Z represents the axialvector tetraquark states; Y represents the vector tetraquark states; , and are the pole residues or current-tetraquark coupling constants, where the antisymmetric tensor , and depicts polarization vectors of the axialvector and vector tetraquark states, which satisfy the summation formula, The diquark-antidiquark type four-quark current operators and potentially couple to the diquarkantidiquark type hidden-charm tetraquark states. We perform Fierz rearrangements to those currents in both the spinor space and color space to obtain a series of colorsinglet-color-singlet (or meson-meson) type current operators, for example, while the constituents, like , , etc, potentially couple to the meson-meson type scattering or tetraquark molecular states.
However, we must be careful in performing the Fierz rearrangements, where the rearrangements in the spinor space and color space are quite non-trivial, and the scenarios of the diquark-antidiquark type tetraquark states and meson-meson type molecular states are considerably different.
According to the arguments by Selem and Wilczek, a diquark-antidiquark type tetraquark may be described by two diquarks trapped in a double potential well, where the two potential wells are separated apart by a barrier [37]. At long distances, the diquark and antiquark serve as point color charges and attract each other strongly, just like in the quark and antiquark bound states. However, when the two diquarks approach each other, the attractions between the quark and antiquark in different diquarks decrease the bonding energy of the diquarks and tend to destroy the diquarks. Those effects (beyond the naive one-gluon exchange force) increase when the dis-Chinese Physics C Vol. 44, No. 6 (2020) 063105 063105-3 tance between the diquark and antidiquark decreases, and a repulsive interaction between the diquark and antidiquark emerges. If this repulsion is large enough, it leads to a barrier between the diquark and antidiquark [38]. The two potential wells that are separated by a barrier can provide good descriptions of the diquark-antidiquark type tetraquark states [38]. Meanwhile, in the dynamical picture of the tetraquark states, the large spatial separation between the diquark and antidiquark leads to a small wave-function overlap between the quark-antiquark pair [39], and the rearrangements in the spinor space and color space are highly suppressed.
It is difficult to account for the non-local effects between the diquark and antidiquark pair in the fourquark currents and directly in practical calculations. For example, the current can be modified to to account for the non-locality by adding a finite ; however, it is highly difficult to deal with the finite both at the hadron side and at the QCD side in a consistent manner. Hence, we express the current in terms of the Taylor series of , Π µν (p) ϵ then we express the correlation function likewise in terms of the Taylor series of , where the components with , 1, 2, represent the contributions of the order . In this article, we study the leading order contributions and . The effects beyond the leading order frustrate the Fierz rearrangements of the diquarkantidiquark type currents into a series of color-singletcolor-singlet (meson-meson) type currents freely.
We employ the Feynman diagram drawn in Fig. 1 to describe the lowest order contributions in the correlation functions for the diquark-antidiquark type four-quark currents and use the Feynman diagrams drawn in Fig. 2 to describe the corresponding lowest order contributions in the correlation functions for the color-singlet-color-singlet type four-quark currents. The Feynman diagram drawn in Fig. 1 cannot be freely factorized into the two Feynman diagrams drawn in Fig. 2 due to the barrier (or spatial separation) between the diquark and antidiquark [38,39]. When a quark (antiquark) in the diquark (antidiquark) penetrates the barrier, the Feynman diagram drawn in Fig. 1 is factorizable in color space. In this case, the non-factorizable diagrams start at the order [40].
In Ref. [40], Lucha, Melikhov, and Sazdjiand argued that the diquark-antidiquark type four-quark currents can be changed into color-singlet-color-singlet (mesonmeson) type currents through the Fierz transformation. The Feynman diagrams, which make contributions to the quark-gluon operators of the order and in accomplishing the operator product expansion are factorizable, and they are canceled out by the contributions of the two-meson scattering states at the phenomenological side. Furthermore, the factorizable parts (in color space) of the Feynman diagrams of the order are also canceled out by contributions from the two-meson scattering states (or more precisely, the free two-meson states). The relevant non-factorizable contributions start at the order . We do not agree with their viewpoint, as there is a repulsive barrier [37,38] or a large spatial separation [39] embodied in the non-local effects to prevent freely performing the Fierz transformation, although at the present time we cannot take into account non-local effects in the QCD sum rules and have to assume the leading order approximations and . Our viewpoint is that the relevant contributions begin at the order , and it is not necessary or it is very difficult to perform the Fierz transformation to separate factorizable and non-factorizable contributions in the color space. Hence, we should take into account both the factorizable and nonfactorizable Feynman diagrams for the diquark-antidiquark type currents.
When the quark or antiquark penetrates the barrier, we can perform the Fierz rearrangements and study the effects of the scattering states. Thus, we explore the contributions of the meson-meson type scattering states (in other words, the two-meson loops) to the Green function for the four-quark current as a representative example, where , and the and are the hadronic coupling constants. We resort to the bare quantities and so as to absorb the divergent terms that appear in the integrals in calculating the self-energies , , etc. The self-energies after renormalization result in a finite energy-dependent width to modify the dispersion relation, where the experimental value of the total decay width [4] (or [9]), and the zero width approximation of spectral densities at the phenomenological side are considered reasonable [41]. In this study, we neglect contributions of the meson-meson type scattering states or the two-meson loops, and the predictions remain robust.
We calculate all Feynman diagrams in performing the operator product expansion to obtain the QCD spectral representation of the Green functions (or correlation functions) and . In the analytical calculations, we consistently take into account vacuum condensates (by selecting the quark-gluon operators of the orders with ) up to dimension and factorize higher dimensional condensates into lower dimensional condensates by assuming vacuum saturation. After obtaining the analytical expressions of the Green functions at the quark-gluon level, we obtain the spectral representation via the dispersion relation. Then, we match the hadronic representation with the QCD representation of the Green functions (or correlation functions) below the continuum threshold parameters and carry out the Borel transformation with regard to to obtain the QCD sum rules: ρ(s) =ρ 0 (s) + ρ 3 (s) + ρ 4 (s) + ρ 5 (s) + ρ 6 (s) + ρ 7 (s) + ρ 8 (s) + ρ 10 (s), , the is the Borel parameter, the subscript i in the components of the QCD spectral densities represents the dimensions of the vacuum condensates, We neglect the cumbersome analytical expressions of spectral densities at the quark-gluon level in saving printed pages. We refer to Ref.
[20] for technical details on calculating the Feynman diagrams. In contrast, we refer to Refs. [20, 23] for explicit expressions of spectral densities at the quark-gluon level for the axialvector current and tensor current . In this study, we recalculate those QCD spectral densities, and use the formula with to deal with the Chinese Physics C Vol. 44, No. 6 (2020) 063105 higher dimensional vacuum condensates, where is the Gell-Mann matrix. This routine leads to slight but neglectful differences compared to old calculations. For the currents and , we neglect the tiny contributions of , which originate from operators such as .
We derive Eq. (16) with regard to , then reach the QCD sum rules for the tetraquark masses by eliminating the pole residues through a fraction, Thereafter, we refer the QCD sum rules in Eq. (16) and Eq. (19) as QCDSR I.
If we take into account the contributions of the first radially excited tetraquark states in the hadronic representation, we obtain the QCD sum rules, where is the continuum threshold parameter. Subsequently, we introduce the notations , , and resort to the subscripts 1 and 2 to represent the ground state tetraquark state and the first radially excited tetraquark state for simplicity. We rewrite the QCD sum rules as QCD τ where we introduce the subscript to denote the QCD representation. We derive the QCD sum rules in Eq. (21) with respect to to get From Eqs. (21)- (22), we obtain the QCD sum rules, i j τ where the indexes . We derive the QCD sum rules in Eq. (23) with respect to to get . (24) The squared masses satisfy the equation, where the indexes and . Finally, we solve above equation analytically to obtain two solutions [30],  We assume the standard values or conventional values of the vacuum condensates , , , at the typical energy scale [35,36,42], and take the modified minimal subtraction mass of the charm quark from the Particle Data Group [9]. We must evolve the quark condensate, mixed quark condensate, and modified minimal subtraction mass to a special energy scale to warrant the parameters in QCD spectral densities with the same energy scale. Then, we account for the energy-scale dependence of the input parameters at the quark-gluon level, for the quark flavors , 4, and 3, respectively [9,43]. As we explore the hiddencharm tetraquark states, we choose the flavor and search for the best energy scales .
The Okubo-Zweig-Iizuka super-allowed decays are expected to easily take place. The energy gaps maybe have the relations and . The charmonium masses are GeV, GeV, and GeV from the Particle Data Group [9], GeV, 0.94 GeV. We choose the continuum threshold parameters to be and 0.95 GeV tentatively and vary the continuum threshold and Borel parameters to satisfy the following four criteria: 1. The ground state tetraquark state or single-pole term makes the dominant contribution at the hadron side; 2. The operator product expansion is convergent below continuum thresholds, and the higher dimensional vacuum condensates make a minor contribution; 3. The Borel platforms appear in both the lineshapes of the tetraquark masses and pole residues with variations of the Borel parameters; 4. The masses of the tetraquark states satisfy the energy-scale formula. After trial and error, we reach the feasible continuum threshold parameters and Borel windows. We also acquire the optimal energy scales of the spectral densities at the quark-gluon level and the contributions of the ground state tetraquark states for the QCDSR I, as shown in Table 1. In general, for the continuum threshold parameters , we can assume that any values satisfy the relation , where the subscript denotes the ground states, as there is an energy gap between the ground state and the first radial excited state. For the conventional S-wave quark-antiquark mesons, the energy

dc]
A s 0 = 21.0 ± 1.0 GeV 2 gaps vary from to , i.e., [9]. In the QCD sum rules for conventional quark-antiquark mesons, we usually choose the values [42]. In Table 1, the continuum threshold parameters satisfy the relation or GeV. It is reasonable that as the values %, where the subscript denotes the maximum values, the contributions of the are significantly suppressed, if there are any present. In Table 1, we write the continuum threshold parameters as rather than as for the and tetraquark states to retain the same form as in our previous work [23]. In Ref. [23], we study the axialvector tetraquark state and choose continuum threshold parameters as .
We obtain the corresponding parameters for QCDSR II using trial and error, see Table 2. In this study, we employ the energy scale formula with the effective charm quark mass (or constituent charm quark mass) to restrain the tetraquark masses and energy scales of the spectral densities [31]. The energy scale formula can enhance contributions of the ground state tetraquark states remarkably at the hadron representation and improve the convergent behaviors of the operator product expansion at the QCD representation by enhancing the contributions of the lower dimensional vacuum condensates. This is feasible for the hidden-charm tetraquark and hidden-charm pentaquark states [44].
From Table 1 and Table 2, we find that the contributions of the single-pole terms (the ground state tetraquark states) are about 40% -60% for QCDSR I. The corresponding contributions of the two-pole terms (the ground state tetraquark states and the first radially excited tetraquark states) are about 70% -80% for the QCDSR II, which satisfies the pole dominance criterion well. In QCDSR II, the contributions of the ground state tetra- quark states are approximately 30%-45%, which is significantly lower than the corresponding ground state tetraquark contributions in the QCDSR I. For the ground state tetraquark masses and pole residues, we prefer the predictions from QCDSR I. In numerical calculations, we find that the contributions of vacuum condensates of dimension 10 (the largest dimension) are of percent level at the Borel widows for both the QCDSR I and QCDSR II. The minor contributions warrant good convergent behaviors of the operator product expansion.
We consider all the uncertainties of input parameters next. We obtain the numerical values of the masses and pole residues of the ground state tetraquark states and the first radially excited tetraquark states , shown in Table 3 and Table 4. The ground state tetraquark masses from the QCDSR I and the radially excited tetraquark masses from the QCDSR II satisfy the energy scale formula , where the updated value of the effective charm quark mass (or constituent charm quark mass) is adopted [23]. In Table 4, we also present the central values of the ground state tetraquark masses and pole residues extracted from the QCDSR II at the ideal energy scales shown in Table 1. We examine Table 4 and observe that the ground state tetraquark masses cannot satisfy the energy scale formula, hence we will discard those values. This is the shortcoming of the QCDSR II.
[uc]  Fig. 3, we plot the ground state tetraquark masses from QCDSR I and the first radially excited tetraquark masses from QCDSR II with respect to variations of the Borel parameters in significantly larger regions than the Borel windows, which are shown in Table 1 and Table 2. Fig. 3  In Table 3, we present the diquark spin , antidiquark spin , and total spin S of the hidden-charm tetraquark states. Table 3 shows that the and tetraquark states (that have tetraquark structures and , respectively) have slightly larger masses than the tetraquark state (which has the structure ). This is reasonable, as the most favored diquark configurations or quark-quark correlations from the attractive interaction in the color-antitriplet (color-triplet) channel induced by one-gluon exchange are the scalar diquark (antidiquark) states. In previous studies, the scalar, pseudoscalar, vector, and axialvector diquarks states were investigated with the QCD sum rules, the scalar and axialvector heavy diquark states in the color-antitriplet have almost degenerate masses, or have almost the same typical quark-quark correlation lengths, the mass gaps between the scalar and axialvector heavy diquark states are very small or tiny [33]. Furthermore, this agrees with the predictions of the simple constituent diquark-antidiquark model [27]. The vector (or P-wave) diquark states and are expected to have larger masses than the axialvector (or S-wave) diquark states and , as there is a relative P-wave between the light and heavy quark. In the case of the traditional charmed mesons, the energy exciting a P-wave costs about from the Particle Data Group [9], If the energy exciting a P-wave in the qc diquark systems also costs approximately , the tetraquark state has the largest ground state mass, which is even larger than the masses of the first radially excited states of the hidden-charm tetraquark states and with the quantum numbers , as exciting two P-waves costs about , which is larger than the energy gap between the ground state hidden-charm tetraquark state and the first radial excitation of the hidden-charm tetraquark states. We investigate the ground states and the first radially excited states of the type, type, and type tetraquark states and the ground state type tetraquark state with the quantum numbers using QCD sum rules in a systematic manner. The predicted tetraquark masses are in favor of assigning and as the ground state and the first radially excited state of the type axialvector tetraquark states, respectively; as the ground state type axialvector tetraquark state or type axialvector tetraquark state; as the first radially excited type axialvector tetraquark state or type axialvector tetraquark state, or the ground state type axialvector tetraquark state. Further experimental and theoretical studies are requiered to identify the unambiguously.