Deciphering the charged heavy quarkoniumlike states in chiral effective field theory

We generalize the framework of chiral effective field theory to study the interactions of the isovector $D^\ast\bar{D}^{(\ast)}$ and $B^\ast\bar{B}^{(\ast)}$ systems up to the next-to-leading order, in which the long-, mid-, and short-range force contributions as well as the $S$-$D$ wave mixing are incorporated. Based on the Lippmann-Schwinger equation, we fit the invariant mass distributions of the elastic channels measured by the BESIII and Belle Collaborations. Our results indicate that the four charged charmoniumlike and bottomoniumlike states $Z_c(3900)$, $Z_c(4020)$ and $Z_b(10610)$, $Z_b(10650)$ can be well identified as the $D\bar{D}^{\ast},D^\ast\bar{D}^{\ast}$ and $B\bar{B}^{\ast},B^\ast\bar{B}^{\ast}$ molecular resonances. The bound state explanations are vetoed in our framework. Our study favors the $Z_c$ and $Z_b$ states are the twin partners under the heavy quark symmetry.

Hadrons are usually classified as the conventional quark model states (qq mesons and qqq baryons) and exotic states (glueball, hybrid and multiquark states etc.). Hadron spectrum serves as a golden platform in investigating the low energy strong interactions. Since the discovery of X(3872) in 2003 by the Belle Collaboration [1], many new states in the charmonium and bottomonium energy regions have been observed [2]. Most of these so-called XY Z states cannot be easily accommodated in the mass spectra of the quark models, which stimulated the theorists to propose various possible interpretations of these unconventional ones [3][4][5][6][7][8].
In the charmonium energy region, two charged charmoniumlike structures Z c (3900) and Z c (4020) were observed by the BESIII Collaboration in the J/ψπ ± [9] and h c π ± [10] channels, respectively. The Z c (3900) was subsequently confirmed by the Belle [11] and Xiao et al [12]. Latter, the BESIII studied the (DD * ) ± and (D * D * ) ±,0 distributions and found the signals of Z c (3900) and Z c (4020) in the open charmed channels [13][14][15][16], respectively. The former was named as the Z c (3885) because the mass measured in the (DD * ) ± channel is about 15 MeV smaller than that of the J/ψπ ± channel. Enlightened by the Ockham's razor: "Entities should not be multiplied unnecessarily", we treat the Z c (3900) and Z c (3885) as the same state that was visualized in different 'microscope'. After all, the mass resolution in different measurements is inequable. In the bottomonium energy region, the Belle Collaboration discovered two charged bottomoniumlike states Z b (10610) and Z b (10650) in the Υ(nS)π ± (n = 1, 2, 3) and h b (mP )π ± (m = 1, 2) invariant mass spectra [17]. Four years later, the Belle Collaboration also observed these two structures in the BB * and B * B * channels, respectively [18].
Isospin and parity analyses indicate these Z ( ) Q (Q = c, b) states are the isovector particles with positive G-parity and negative C-parity (C-parity for the neutral members). We will denote the Z c (3900), Z c (4020) and Z b (10610), Z b (10650) as Z c , Z c and Z b , Z b respectively in the following context for simplicity. Analyses of the angular distributions favor the J P = 1 + assignment for the Z c [13,14] and Z ( ) b [19]. The J P quantum numbers of the Z c are undetermined yet, but the J P = 1 + is presumed in most works [3][4][5][6][7][8]. The min-imal quark component in these Z ( ) Q states should be QQqq (q = u, d) rather than the pure QQ since they are the charged particles. Such a quark configuration is obviously beyond the conventional mesons and baryons, so they are dubbed the exotic hadrons. Many theoretical explanations have been proposed to understand these exotica, such as the loosely bound molecular states, compact tetraquarks, kinematical effects and so on (one can consult some comprehensive reviews [3][4][5][6][7][8] for deepgoing excavations). Besides the similarities of the decay modes, the mass differences of (Z c , Z c ) and (Z b , Z b ) almost equal to the mass splittings of (D, D * ) and (B, B * ), respectively. The large comparability between the Z ( ) c and Z ( ) b suggests that they are the partners under the heavy quark flavor symmetry. The most salient feature of the Z c , Z c and Z b , Z b is their proximities to the DD * , D * D * and BB * , B * B * thresholds, respectively. Therefore, the properties of the Z are molecular resonances generated from the D * D( * ) and B * B( * ) interactions, respectively. In this work we exploit the chiral effective field theory (χEFT) to study the D * D( * ) and B * B( * ) interactions up to the next-to-leading order (NLO), and then fit the experimental data to extract the resonance parameters. As the modern theory of nuclear forces [20,21], χEFT has been extensively used to study the nucleon systems with high precision [22][23][24][25][26][27]. Within χEFT, the effective potentials of the VP and VV systems [V and P denote the (anti-)charmed/bottom vector and pseudoscalar mesons, respectively] with the definite isospin can be respectively parameterized as where p and p denote the initial and final state momenta in the center of mass system (c.m.s), respectively. ε ( ) and ε ( ) † represent the polarization vectors of the initial and final vector with q = p − p the transferred momentum and k = (p + p)/2 the average momentum. For the VV system, the number of the possible operators increases drastically due to the involvement of two new polarization vectors ε and ε † , e.g., where the ellipsis denotes the other possible combinations among q, k, ε ( ) and ε ( ) † at the NLO. Like the nuclear forces [23,24], the interactions between a pair of charmed (bottom) mesons can also be divided into the short-, mid-and long-range contributions. The χEFT does not depend on the details of the short-range dynamics (r 1/m π ), which is usually mimicked by the contact interaction. Following the spirit of Eq. (1), the contact potential of the VP system is parameterized as follows, where C i (i = 0, . . . , 7) are the unknown low energy constants (LECs). The C 0 and C 1,...,7 terms designate the leading order (LO) and the next-to-leading order (NLO) contributions, respectively. With Eq. (2), one can construct the similar form as in Eq. (5) for the contact potential of the VV system. The χEFT is very good at dealing with the long-and midrange interactions, which could be calculated to any high orders theoretically. For the VP and VV systems, the long-range interaction is provided by the one-pion-exchange (OPE), which is firmly rooted in the chiral symmetry and its spontaneous breaking of quantum chromodynamics (QCD). The mid-range force arises from the two-pion-exchange (TPE). The corresponding loop diagrams are illustrated in Fig. 1. The long-and mid-range effective potentials can be obtained from the LO chiral Lagrangians, where · · · denotes the trace in spinor space. The covariant derivative D µ = ∂ µ + Γ µ and v = (1, 0) represents the fourvelocity of heavy mesons. The H andH denote the superfield of the charmed (bottom) mesons and anti-charmed (bottom) mesons, respectively. Their expressions can be found in Refs. [28][29][30][31]. The axial coupling g 0.57 for the charmed mesons is extracted from the partial decay width of D * + → D 0 π + [2], while for the bottom ones average value g 0.52 is taken from the lattice QCD calculations [32,33]. The chiral connection Γ µ and axial-vector current u µ are formulated as: , with ϕ the matrix form of the pion triplet [30], and f π = 92.4 MeV the pion decay constant. Establishing the flavor wave functions of the I G (J P C ) = 1 [34] and unfolding Eq. (6) one can get the OPE potentials for the Z Q and Z Q states, respectively, with m π the pion mass, and q 2 = p 2 + p 2 − 2pp cos ϑ (where p = |p|, p = |p |, and ϑ is the scattering angle in the c.m.s of VP and VV). In the Breit approximation [35], the effective potential V from the scattering amplitude M reads Similarly, the mid-range potential provided by the loop diagrams in Fig. 1 can be calculated with the one-pion and twopion coupling vertices in Eq. (6) (for the calculation details one can consult Refs. [36,37]). In heavy quark limit, the two-particle-irreducible TPE potential can be formulated via a concise form, with where = q 2 + 4m 2 π , and y = 2pp cos ϑ − p 2 − p 2 . The Z Q and Z Q are observed in the e + e − → πVP and e + e − → πVV processes, respectively. So we simulate the two transitions and fit the invariant mass spectra of the VP and VV pair. The reaction is illustrated in Fig. 2, where graphs 2(a) and 2(b) describe the continuum and resonance contributions, respectively. In Fig. 2(b) we need to cope with the VP(V) rescatterings, since they account for the dynamical generation of the Z ( ) Q . Additionally, we also need to mimic the γ * → πVP(V) coupling, which can be depicted by the following effective Lagrangians where g ( ) γ designate the effective coupling constants, and F µν is the field strength tensor of the virtual photon. P ( ) µν are the antisymmetric tensors that constructed as: P µν = (P † µ u ν P † − VP: VV: where (P µ /P ) P µ /P denote the (anti)-charmed (bottom) vector/pseudoscalar meson fields (e.g., see Refs. [30,31]), and u µ is the axial-vector field.
Equipped with the above effective potentials, the VP and VV production amplitudes U(E, p) can be obtained by solving the following Lippmann-Schwinger equation (LSE), where M(E, p) denotes the production vertex from Eq. (11) and E is the invariant mass of the paired VP(V). The Green's function G(E, q) is given as with µ and m th the reduced mass and threshold of the VP(V) systems, respectively. The potentials in Eqs. (5) and (7)- (10) are given in the plane wave helicity state basis in the c.m.s of the VP(V) systems, whereas the physical observables are usually defined in terms of partial waves, i.e., the | sj basis (where , s and j represent the orbital angular momentum, total spin and total angular momentum of the VP(V) systems, respectively). So it is desirable to obtain the above effective potentials in the partial wave decomposition. This can be eas-ily done via [38] V , =ˆdp with Y m the spherical harmonics. The remaining matrix element s, m j −m |V|s, m j −m in spin space can be directly calculated with the coupled spin multiplets |1, m s , which are the products of one-body spin states. As demonstrated in the nucleon systems, the Sand Dwave mixing effect plays an important role [22][23][24][25]. This effect can be easily taken into account in the LSE framework, in which the effective potential becomes a 2 × 2 matrix. After performing the partial wave decomposition via Eq. (14), the contact potential that incorporates the S-D mixing reads, whereC s , C s , and C sd are the so-called partial wave LECs. Their values will be fixed by fitting the experimental data. Iteration of the potential V , in the LSE requires suppressing the high momenta contribution to avoid divergence, since the χEFT is only valid in low momenta region q Λ χ ≈ 1 GeV. The Gaussian regulator is commonly used [24,27,39], i.e., V , → V , exp(−p 2 /Λ 2 − p 2 /Λ 2 ), where Λ is the cutoff parameter. For the nucleon-nucleon scattering when the high order corrections are included [24,27], the cutoff parameter Λ is normally chosen to be around 0.5 GeV. We leave it as a free parameter and determine its value by fitting the experimental lineshapes.
In terms of the production amplitude in Eq. (12), the differential decay width for γ * → πVP(V) reads where √ s is the center-of-mass energy of the e + e − collision. k 1 and k * 2 are the three-momentum of the spectator π in the c.m.s. of e + e − and the three-momentum of P(V) in the c.m.s of VP(V), respectively. We essentially have four free parameters [three partial wave LECs in Eq. (15) and a cutoff Λ] to fit the experimental lineshapes. For the Z ( ) c and Z ( ) b states, we try to fit the D * D( * ) and B * B( * ) invariant mass distributions measured by the BE-SIII [14,16] and Belle [18] Collaborations, respectively. The fitted lineshapes and parameters are given in Fig. 3 and Table I, respectively. We find the experimental data can be fitted quantitatively well with the potentials up to the NLO in our approach. Four sharp peaks appear around 3.88, 4.02, 10.61 and 10.65 GeV for each distribution, which correspond to the Z c (3900), Z c (4020), Z b (10610) and Z b (10650) signals in experiments, respectively. With the fitted parameters in Table I as inputs, we search for the poles of the T -matrix in the second (unphysical) Riemann sheet, which can be achieved through analytical continuation of the Green's function G(p + i ) in Eq. (13), where G a and G b denote the Green's function defined in the first (physical) and second Riemann sheet, respectively. We find a pole for each system in the second Riemann sheet with the pole positions given in Table I b . This can be qualitatively understood. When the γ * 'emits' a pion, the residual phase spaces for the VP(V) systems are small. Thus once the VP(V) are created near their thresholds, they move slowly and have enough time to interact with each other. If the interaction is attractive enough, a bound state is formed, which could not decay into its component mesons. If the interaction is not attractive enough but has a barrier to confine the two mesons for a finite time, a molecular resonance with certain lifetime is produced.
Our extracted masses are all consistent with the experimental measurements [14,16,18], but the widths in our study are smaller than those of the experimental data. We do not consider the inelastic channel J/ψπ [Υ(nS)π] and h c π [h b (mP )π] contributions (see Refs. [40][41][42] for a couplechannel approach). These inelastic channels would contribute additional partial decay widths. These inelastic processes occur at very short distance and cannot be accommodated within the χEFT framework. On the other hand, the coupling strength between Z ( ) Q and the inelastic channels is not strong, since the experimental measurements indicate that the elastic channels dominate the decay widths of Z c [13] and Z ( ) b [18]. Therefore, the corrections from the inelastic channels to the widths of Z ( ) Q shall not be significant. From Fig. 3, the signal lineshapes deviate from the moderate Breit-Wigner distribution, which are dramatically distorted by the strong coupling of VP(V). The classical Breit-Wigner function is not good enough to describe these typical very-near-threshold states.
Inspecting the fitted parameters in Table I, one notices that the rescatterings inside the VP and VV systems proceed predominantly via the S-wave interactions. They can be de-scribed almost by one set of parameters respectively, which is guaranteed by the heavy quark spin symmetry [29,43]. In addition, the LO LECC s for the charmed and bottom systems are consistent with each other within uncertainties, which is the reflection of heavy quark flavor symmetry [29,44,45]. The sensible difference of the NLO LEC C s for the D * D( * ) and B * B( * ) systems encodes the heavy quark flavor symmetry breaking effect. The value of the cutoff Λ also resides in the region (Λ Λ χ ) where the χEFT works healthily. The cutoff for the B * B( * ) systems is larger than that of the D * D( * ) , since the interaction radius (R ∼ 1/Λ) for the B * B( * ) is shorter than that of the D * D( * ) . It is well known that the bottom mesons are heavier than the charmed ones.
We also attempt to fit the data with the LO effective potentials solely (OPE plus the LO contact terms), but cannot reproduce the experimental lineshapes well (purple dot-dashed lines in Fig. 3 Although there are no guidances to judge the values ofC s , the χEFT imposes strong constrains to the Λ, which has to be smaller than the typical hard scale, i.e., the ρ meson mass m ρ 0.77 GeV. Therefore, we can conclude that either from the fitting quality or the rationality of parameters, the bound state explanations are not favored. As elucidated above, the Z ( ) Q states can be well identified as the molecular resonances. In the resonance scenario, their decay behaviors can be explained qualitatively well. In contrast to the bound state, a resonance naturally dissolves to their components after interacting within finite time, which contributes to the dominant decay mode. The decays with final states of a heavy quarkonium and a light meson, [Qq] + [Qq] → [QQ] + [qq] proceeds with less probability, which are induced by much shorter range interaction (compared to 1/Λ χ ). At the hadron level, these decays take place via exchanging a heavy meson [Qq], which is generally suppressed. This is why the partial widths from the inelastic channel contributions are much smaller than those of the elastic channels in experiments [13,18].
In summary, we systematically study the D * D( * ) and B * B( * ) effective potentials with the χEFT up to the NLO to draw a clear picture of their interactions. With these potentials, we investigate the internal structures of the experimentally observed Z   [17] generated from the analogue of nuclear forces in heavy meson sectors. The heavy quark symmetry and its breaking effect are both reflected in the parameters. The fittings with the LO potentials give rise to the bound states, which is repudiated either by the above-threshold masses or the validity of χEFT. The decay behaviors of the Z ( ) c and Z ( ) b states can also be qualitatively interpreted in the resonance picture. In our study, the Z ( ) Q signals can be fully reproduced by the VP(V) rescatterings, where the initial states πVP(V) are assumed to be produced from point-like sources. We do not need additional structures around the colliding energies.
Besides the XY Z states, more and more new states have been observed in experiments (such as the P c [46] and very recently reported X 0,1 states at LHCb [47]), thus a model independent way is urgently called for to illuminate the nature of these new hadrons. The systematical generalization of the χEFT to the heavy meson systems is very successful in this work, which helps us to pin down the inner structures of the Z ( ) c and Z ( ) b states. This framework can also be applied to investigate whether the other near-threshold states (e.g., P c and X 0,1 ) have the same origin, i.e., the dynamically generated resonances (bound states) from the analogue of nuclear forces in different sectors. This would undoubtedly deepen our understandings of the low energy behaviors of QCD. This project is supported by the National Natural Science Foundation of China under Grant 11975033.