Four-quark structure of the Z
cð3900Þ, Zð4430Þ, and X
bð5568Þ states
Fabian Goerke,1 Thomas Gutsche,1 Mikhail A. Ivanov,2 Jürgen G. Körner,3 Valery E. Lyubovitskij,1,4,5and Pietro Santorelli6,7
1Institut für Theoretische Physik, Universität Tübingen, Kepler Center for Astro and Particle Physics, Auf der Morgenstelle 14, D-72076 Tübingen, Germany
2Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980 Dubna, Russia
3PRISMA Cluster of Excellence, Institut für Physik, Thomas GutscheJohannes Gutenberg-Universität, D-55099 Mainz, Germany
4Department of Physics, Tomsk State University, 634050 Tomsk, Russia 5
Laboratory of Particle Physics, Mathematical Physics Department, Tomsk Polytechnic University, Lenin Avenue 30, 634050 Tomsk, Russia
6
Dipartimento di Fisica, Università di Napoli Federico II, Complesso Universitario di Monte Sant’Angelo, Via Cintia, Edificio 6, 80126 Napoli, Italy
7
Istituto Nazionale di Fisica Nucleare, Sezione di Napoli, 80126 Napoli, Italy (Received 17 August 2016; published 15 November 2016)
We examine the four-quark structure of the recently discovered charged Zcð3900Þ, Zð4430Þ, and Xbð5568Þ states. We calculate the widths of the strong decays Zþc → J=ψπþ (ηcρþ, ¯D0Dþ, ¯D0Dþ), Zð4430Þþ→ J=ψπþ(ψð2sÞπþ), and Xþb → Bsπþwithin a covariant quark model previously developed by us. We find that the tetraquark-type current widely used in the literature for the Zcð3900Þ leads to a significant suppression of the ¯DDand ¯DD modes. Contrary to this a molecular-type current provides an enhancement by a factor of 6–7 for the ¯DD modes compared with the Zþc → J=ψπþ,ηcρþ modes in agreement with recent experimental data from the BESIII Collaboration. In the case of the Zð4430Þ state we test a sensitivity of the ratio RZof the Zð4430Þþ→ ψð2sÞπþand Zð4430Þþ→ J=ψπþdecay rates to a choice of the size parameterΛZð4430Þof the Zð4430Þ. Using the upper constraint for the sum of these two modes deduced from the LHCb Collaboration data we find that RZvaries from 4.64 to 4.08 whenΛZð4430Þ changes from 2.2 to 3.2 GeV. Also we make the prediction for the Zð4430Þþ→ Dþ¯D0 decay rate.
DOI:10.1103/PhysRevD.94.094017
I. INTRODUCTION
In the course of experimentally establishing the heavy meson spectrum unusual states were observed that cannot be simply interpreted in the context of a minimal constituent quark-antiquark model. Among these new states are the Zcð3900Þ, Zð4430Þ, and the Xbð5568Þ, where especially the
last resonance still needs solid experimental confirmation. The flavor structure of these states is unusual as evident from their strong decay modes; a simple quark-antiquark inter-pretation is not feasible. In the following we focus on these special states. We first collect the experimental findings.
The process eþe−→ πþπ−J=ψ has been studied by the BESIII Collaboration [1]. A structure was observed at around 3.9 GeV in the πJ=ψ mass spectrum that was christened the Zcð3900Þ state. If interpreted as a new
particle, it is unusual in that it carries an electric charge and couples to charmonium. A fit to the πJ=ψ invariant mass spectrum results in a mass of MZc¼ 3899.0 3.6ðstatÞ 4.9ðsystÞ MeV and a width of ΓZc ¼ 46 10ðstatÞ 20ðsystÞ MeV.
The cross section for eþe−→ πþπ−J=ψ between 3.8 and 5.5 GeV was measured by the Belle Collaboration[2].
This measurement lead to the observation of the state Yð4260Þ, and its resonance parameters were determined. In addition, an excess of πþπ−J=ψ production around 4 GeV was observed. This feature can be described by a Breit-Wigner parametrization with properties that are consistent with the Yð4008Þ state that was previously reported by Belle. In a study of the Yð4260Þ → πþπ−J=ψ decays, a structure was observed in the
MðπJ=ψÞ mass spectrum with 5.2σ significance, with
mass M¼ 3894.5 6.6ðstatÞ 4.5ðsystÞ MeV and width Γ ¼ 63 24ðstatÞ 26ðsystÞ MeV, where the errors are statistical and systematic, respectively. This structure can be interpreted as a new charged charmoniumlike state.
Using 586 pb of eþe− annihilation data the CLEO-c detector made an analysis atpffiffiffis¼ 4170 MeV at the peak of the charmonium resonance ψð4160Þ. The subsequent decay ψð4160Þ → πþπ−J=ψ was analyzed [3], and the charged state Zcð3900Þ was observed, which decays into πJ=ψ at a significance level of > 5σ. The value of the
mass MZc¼ 3886 4ðstatÞ 2ðsystÞ MeV and the width ΓZc¼ 37 4ðstatÞ 8ðsystÞ MeV were found to be in good agreement with the results for this resonance reported by the BES III and Belle collaborations in the decay of the
resonance Yð4260Þ. In addition CLEO-c presented the first evidence for the production of the neutral member of this isospin triplet, Z0cð3900Þ decaying into π0J=ψ at a 3.5σ
significance level.
A study of the process eþe−→ πðD ¯DÞ∓was reported by the BESIII Collaboration[4]atpffiffiffis¼ 4.26 GeV using a 525 pb−1 data sample collected with the BESIII detector
at the BEPCII storage ring. A distinct charged structure was observed in the ðD ¯DÞ∓ invariant mass distribution. When fitted to a mass-dependent-width Breit-Wigner line shape, the pole mass and width were determined to be Mpole¼ 3883.9 1.5ðstatÞ 4.2ðsystÞ MeV and Γpole¼ 24.8 3.3ðstatÞ 11.0ðsystÞ MeV. The mass and width of the structure referred to as Zcð3885Þ are 2σ and 1σ, respectively, below those of the Zcð3900Þ → πJ=ψ peak
observed by BESIII and Belle in πþπ−J=ψ final states produced at the same center-of-mass energy. The angular distribution of the πZcð3885Þ system favors a JP¼ 1þ
quantum number assignment for the structure and disfavors the assignment1− or0−. The Born cross section times the DDbranching fraction of the Zcð3885Þ is measured to be
σðeþe− → πZ∓
cð3885ÞÞ × BðZ∓cð3885Þ → ðD ¯DÞ∓Þ
¼ 83.5 6.6ðstatÞ 22.0ðsystÞ pb: ð1Þ Assuming that the Zcð3885Þ → D ¯D signal reported in [4] and the Zcð3900Þ → πJ=ψ signal are from the same
source, the ratio of partial widths is determined as ΓðZcð3885Þ → D ¯DÞ
ΓðZcð3885Þ → πJ=ψÞ
¼ 6.2 1.1ðstatÞ 2.7ðsystÞ: ð2Þ That means that the Zcð3900Þ state has a much stronger coupling to DDthan toπJ=ψ[5]. An unbinned maximum likelihood fit gives a mass of M¼ 3889.1 1.8 MeV and a width ofΓ¼28.14.1MeV (M ¼ 3891.8 1.8 MeV and Γ ¼ 27.8 3.9 MeV) for the two data sets, respectively. The pole position of this peak is calculated to be Mpole¼ 3883.91.54.2MeV and Γpole¼24.83.311.0MeV.
The mass and width of the peak observed in the DDfinal state agree with that of the Zcð3900Þ. Thus, they are quite probably the same state.
The charmoniumlike structure Zþcð3900Þ was identified
in Ref.[6]as the charged partner of the Xð3872Þ state. The Xð3872Þ meson is considered to be a four-quark state with quantum numbers IGðJPCÞ ¼ 0þð1þþÞ. The Zcð3900Þ
meson is interpreted as the isospin 1 partner of the Xð3872Þ. As in Ref.[7]it was assumed that the quantum numbers for the neutral state in the isospin multiplet were IGðJPCÞ ¼ 1þð1þ−Þ. Using standard QCD sum rules
tech-niques, the coupling constants of the ZþcJ=ψπþ, Zþcηcρþ, Zþc ¯D0Dþ, and Zþc ¯D0Dþ vertices and the corresponding decay widths were calculated with the following results:
ΓðZþ c → J=ψ þ πþÞ ¼ ð29.1 8.2Þ MeV; ΓðZþ c → ηcþ ρþÞ ¼ ð27.5 8.5Þ MeV; ΓðZþ c → ¯D0þ DþÞ ¼ ð3.2 0.7Þ MeV; ΓðZþ c → ¯D0þ DþÞ ¼ ð3.2 0.7Þ MeV: ð3Þ
The observation of a narrow structure, Xð5568Þ, in the decay sequence Xð5568Þ → B0sπ, B0s → J=ψϕ,
J=ψ → μþμ−, ϕ → KþK− was reported in[8] by the D0 Collaboration. This would be the first observation of a hadronic state with valence quarks of four different flavors. The mass and width of the new state are measured to be M¼5567.82.9ðstatÞþ0.9−1.9ðsystÞMeV=c2, and Γ ¼ 21.9 6.4ðstatÞþ5.0
−2.5ðsystÞ MeV=c2. However, in recent analysis
performed by the LHCb Collaboration an existence of the claimed Xð5568Þ state has been not confirmed [9].
The observed strong decay mode of the Xð5586Þ implies flavor structures of the type Xþðs ¯bu ¯dÞ. Since the B0sπþpair is produced in an S wave, its quantum numbers
would be JP¼ 0þ. As already pointed out in Ref.[8]the
significant difference between the mass of the Xð5568Þ and MBþ MK threshold does not favor a hadronic molecular
interpretation of the Xð5568Þ. First qualitative consider-ations point to the Xð5568Þ state being a tetraquark state. Structure issues of the Xð5568Þ have already been discussed in a number of theoretical papers [10–27] suggesting various tests both for the tetraquark and had-ronic molecular structure of the Xð5586Þ state. In Ref.[23] a few options for the interpretation of the Xð5586Þ state have been checked. It was concluded that threshold, cusp, molecular, and tetraquark approaches for the explanation of the Xð5586Þ state are all disfavored. One of the important conclusions was that the mass of theðbsqqÞ tetraquark state must be heavier than theΞbð5800Þ baryon. Also the authors of Ref. [23] deduced a lower limit for the masses of a possible ðbsudÞ tetraquark state: 6019 (6107) MeV. Complementary to Ref.[23,24]presented an analysis based on general properties of QCD to analyze the Xð5568Þ states. In particular, it was shown that the mass of the ðbsudÞ tetraquark state must be bigger than the sum of the masses of the Bs meson and the light quark-antiquark resonance
leading to an estimate of the lower limit of Mbsud∼
5.9 GeV. Reference[26]used a Bsπ-B ¯K coupled channel
analysis with an interaction derived from heavy hadron chiral perturbation theory to implement the unitarity feature of the spectrum reported by the D0 Collaboration. The analysis lead to a T-matrix momentum cutoff of Λ ¼ 2.80 0.04 GeV, which is much larger than a typical scaleΛ ≃ 1 GeV.
Reference[28]estimated the mass of the lightestðbsudÞ, 0þ tetraquark in the framework of a tightly bound diquark
model. Their semiquantitative analysis leads to a mass of about 5770 MeV that lies approximately 200 MeV above the reported Xð5568Þ state, and 7 MeV below the B ¯K threshold.
The Zð4430Þ state with mass M ¼ 4433 4 2 MeV and widthΓ ¼ 45þ18−13ðstatÞþ30−13ðsystÞ MeV has been discov-ered by the BABAR Collaboration in theπψð2sÞ invariant mass distribution in B→ Kπψð2sÞ decay [29], where ψð2sÞ is the first radial excitation of the J=ψ. Later, in Ref. [30] the Belle Collaboration updated their predictions for the mass and width of the Zð4430Þ resonance: M ¼ 4443þ15−12ðstatÞþ19−13ðsystÞ MeV and Γ ¼ 107þ86
−43ðstatÞþ74−56ðsystÞ MeV.
The BABAR Collaboration studied the decays ¯B−;0→ K0;þπ−ψð2sÞ and ¯B−;0→K0;þπ−J=ψ, but they did not see a Zð4430Þ−signal[31]. They derived upper limits for branch-ing fractions that are yet compatible with the mentioned results of the Belle Collaboration. In Ref. [32] the Belle Collaboration reported on the spin and parity of the Zð4430Þ− state constrained from a full amplitude analysis of the B0→ ψð2sÞKþπ−decay withψð2sÞ → μþμ−or eþe−. They found that the Zð4430Þ− being a JP¼ 1þ-state was
favored over the next likely state (0−) with a significance of3.4σ.
Furthermore, the Belle Collaboration did estimate for the product of branching fractions: Bð ¯B0→K−Zð4430ÞþÞ× BðZð4430Þþ→πþψð2sÞÞ¼ð3.2þ1.8
−0.9ðstatÞþ5.3−1.6ðsystÞÞ×10−5.
The BABAR Collaboration studied the decays ¯B−;0→ K0;þπ−ψð2sÞ and ¯B−;0→ K0;þπ−J=ψ, but they did not see a Zð4430Þ− signal[31]. They derived upper limits for branching fractions that are yet compatible with the mentioned results of the Belle Collaboration.
In Ref.[32]the Belle Collaboration reported on the spin and parity of the Zð4430Þ− state constrained from a full amplitude analysis of the B0→ ψð2sÞKþπ− decay with ψð2sÞ → μþμ− or eþe−. They found that the Zð4430Þ−
being a JP¼ 1þ-state was favored over the next likely state (0−) with a significance of 3.4σ.
In Ref. [33] the LHCb Collaboration confirmed the Zð4430Þ− signal in the ψð2sÞπ−-spectrum of the decay B0→ ψ0Kþπ−, and determined unambiguously the spin parity JP ¼ 1þof the Zð4430Þ state[33]. They determined
the following values for the mass and width of the Zð4430Þ− state: M¼ 4475 7ðstatÞþ15
−25ðsystÞ MeV and
Γ ¼ 172 13ðstatÞþ37
−34ðsystÞ MeV [33]. In Ref. [34] the
LCHb Collaboration concluded that the only possible explanation for internal structure of the Zð4430Þ state is a four-quark ccud bound state.
In Ref. [35] the Belle Collaboration reported that they also found a Zþð4430Þ signal in the J=ψπþspectrum of the decay ¯B0→J=ψK−πþ. They report product branching frac-tion Bð ¯B0→ K−Zð4430ÞþÞ × BðZð4430Þþ → πþJ=ψÞ ¼ ð5.4þ4.0
−1.0ðstatÞþ1.1−0.9ðsystÞÞ × 10−5. If one compares this value
with the corresponding product branching fraction for the ψð2sÞ particle (see above) and assumes that the decay rates are invariant under charge conjugation, one can derive an estimation for the branching ratio of the two decay channels
of the Zð4430Þ. We do not know how the errors of the values correlate, so we only do a rough estimation of the errors by dividing the upper limit of the one product branching fraction by the lower limit of the other one and vice versa, taking into account statistical and systematical error both in one step. We get
RZ¼ Γ
ðZð4430Þ → πψð2sÞÞ
ΓðZð4430Þ→ πJ=ψÞ ≃ 11.1þ18−8.6: ð4Þ
In the present paper we critically check the tetraquark picture for both the Zcð3900Þ and Xð5568Þ states by analyzing their strong decays. In our consideration we use the covariant quark model proposed in[36]and used in Refs.[37,38]to describe the properties of the Xð3872Þ state as a tetraquark state. First, we employ an interpretation of the Zcð3900Þ state as the isospin 1 partner of the Xð3872Þ as was suggested in Refs. [6] and [7]. We calculate the partial widths of the decays Zþcð3900Þ → J=ψπþ,ηcρþ, and
¯D0Dþ, ¯D0Dþ. We find that for a relatively small model
size parameterΛZc∼ 1.4 GeV one can reproduce the central values for the partial widths of the decays Zþc → J=ψπþ, ηcρþ as they were also obtained in Refs.[6,7]. It turns out
that, in our model, the leading Lorentz metric structure in the matrix elements describing the decays Zcð3900Þ → ¯DD
vanishes analytically. This results in a significant suppres-sion of these decay widths by the smallness of the relevant phase space factorjqj5. Since the experimental data[4]show that the Zcð3900Þ has a much more stronger coupling to
DDthan to J=ψπ, one has to conclude that the tetraquark-type current for the Zcð3900Þ is in discord with experiment.
As an alternative we employ a molecular-type four-quark current to describe the decays of the Zcð3900Þ state. In
this case we find that for a relatively large size parameter ΛZc∼ 3.3 GeV one can obtain the partial widths of the decays Zþcð3900Þ → ¯DD at the order∼15 MeV for each mode. At the same time the partial widths for decays Zþcð3900Þ → J=ψπþ, ηcρþ are suppressed by a factor of
6–7 in accordance with experimental data[4].
Let us stress, that in our manuscript we consider exotic mesons in the four-quark picture with the use of two possible configuration of quarks in these states. Note that molecular configuration does not mean that a specific size of the state with such structure is more compact than the tetraquark configuration. In this sense it is differed from hadronic molecules—extended object with clear separation of two hadrons—the constituents building the exotic state. Such hadronic molecules (extended objects) with smaller size parameter (of order of 1–2 GeV) have been considered some of us in the phenomenological Lagrangian approach based on the composite structure of exotic states as bound states of separate hadrons[39]. In the present manuscript, for the first time in order to distinguish both configurations we vary the size parameter in the same region 3.2–3.4 GeV, which is guided by experimental data. Also we would like
to mention that the size parameter is not directly related to the size of a hadron like e.g., in potential approaches. Indeed, our size parameter is related to the physical quantities like electromagnetic radii, slope of the form factors, etc.
Then, we test the tetraquark picture for the Xð5568Þ structure by analyzing its strong one-pion decay. We found that for a mass of 5568 MeV one can fit the experimental decay width by using the value of size parameter ΛXb∼ 1.4 GeV. In the case of a larger mass of 5771 MeV [28] one findsΛXb∼ 1.7 GeV. Finally, we consider the decays of the Zð4430Þ state Zð4430Þþ→J=ψ þπþ, Zð4430Þþ→ ψð2sÞ þ πþ, and Zð4430Þþ → Dþþ ¯D0in the tetraquark
picture.
The paper is organized as follows. In Sec.II, we consider the Zcð3900Þ state, a four-quark state, as a compact
tetraquark bounded by color diquark and antidiquark. In Sec.IIIwe test a molecular-type four-quark structure of the Zcð3900Þ state. In Sec.IVwe present study of the Xð5568Þ exotic state as the tetraquark four-quark state. In Sec.Vwe apply the tetraquark model for the Zð4430Þ state. Finally, in Sec. VI we summarize and conclude our results.
II. THE Zcð3900Þ AS A FOUR-QUARK STATE
WITH A TETRAQUARK-TYPE CURRENT Let us first interpret Zcð3900Þ as the isospin 1 partner of the Xð3872Þ as was suggested in Refs. [6] and [7]. Then the quantum numbers for the neutral state are IGðJPCÞ ¼ 1þð1þ−Þ. Accordingly the interpolating current for the Zþcð3900Þ state is given by
Jμ¼ iffiffiffi 2
p εabcεdec½ðuTaCγ5cbÞð ¯ddγμC¯cTeÞ
− ðuT
aCγμcbÞð ¯ddγ5C¯cTeÞ: ð5Þ
We employ a charge conjugation matrix in the form of C¼ γ0γ2, i.e., without a factor“i” as is usually employed. This allows one to simplify the calculations because of C¼ C†¼ C−1 ¼ −CT, CΓTC−1¼ Γ (“þ”for Γ ¼ S, P,
A, and “−” for Γ ¼ V, T). In what follows we drop the superscript “T” (transpose) from the spinors to avoid a complication of notation.
The nonlocal version of the four-quark interpolating current reads JμZ cðxÞ ¼ Z dx1… Z dx4δ x−X 4 i¼1 wixi ×ΦZc X i<j ðxi− xjÞ2 Jμ4qðx1;…; x4Þ; Jμ4q¼ iffiffiffi 2
p εabcεdecf½uaðx4ÞCγ5cbðx1Þ½ ¯ddðx3ÞγμC¯ceðx2Þ
− ½uaðx4ÞCγμcbðx1Þ½¯qdðx3Þγ5C¯ceðx2Þg; ð6Þ
where wi¼ mi=
P4
j¼1mj. The numbering of the
coordi-nates xiis chosen such that one has a convenient
arrange-ment of vertices and propagators in the Feynman diagrams to be calculated. The effective interaction Lagrangian describing the coupling of the meson Zc to its constituent
quarks is written in the form Lint ¼ gZcZc;μðxÞ · J
μ
ZcðxÞ þ H:c: ð7Þ The Fourier transform of the vertex function ΦZcð
P
i<jðxi− xjÞ2Þ can be calculated by using
appropri-ately chosen Jacobi coordinates
xi¼ x þ X3 j¼1 wijρj; ð8Þ where w11¼þ2w2þw3þw4 2pffiffiffi2 w12¼− w3−w4 2pffiffiffi2 w13¼þ w3þw4 2 w21¼−2w1þw3þw4 2pffiffiffi2 w22¼− w3−w4 2pffiffiffi2 w23¼þ w3þw4 2 w31¼−w1−w2 2pffiffiffi2 w32¼þ w1þw2þ2w4 2pffiffiffi2 w33¼− w1þw2 2 w41¼−w1−w2 2pffiffiffi2 w42¼− w1þw2þ2w3 2pffiffiffi2 w43¼− w1þw2 2 :
It is straightforward to check that x¼P4i¼1xiwi, and P
1≤i<j≤4ðxi− xjÞ2¼
P3
i¼1ρ2i. The vertex function is then
written as ΦZc X i<j ðxi− xjÞ2 ¼ Z d ~ω ð2πÞ12e−i~ρ ~ω~ΦZcð−~ω 2Þ; ð9Þ
where the vertex function in momentum space is chosen to have a Gaussian form
~ΦZcð−~ω
2Þ ¼ expð~ω2=Λ2
ZcÞ ð10Þ with theΛ2Z
c being an adjustable size parameter.
The coupling constant gZcin Eq.(7)is determined by the normalization condition called the compositeness condition (see Refs.[40]and[36] for details),
ZZc¼ 1 − g2Z c~Π 0 Zcðm 2 ZcÞ ¼ 0; ð11Þ whereΠZcðp
2Þ is the scalar part of the vector-meson mass
~Πμν ZcðpÞ ¼ g μν~Π Zcðp 2Þ þ pμpν~Πð1Þ Zcðp 2Þ; ~ΠZcðp 2Þ ¼ 1 3 gμν−pμpν p2 ΠμνZcðpÞ: ð12Þ
The Fourier transform of the Zc-tetraquark mass operator reads
ΠμνZcðpÞ ¼ 6 Y3 i¼1 Z d4ki ð2πÞ4i~Φ2Zcð−~ω 2Þ ×ftr½S4ðˆk4Þγ5S1ðˆk1Þγ5tr½S3ðˆk3ÞγμS2ðˆk2Þγν þ tr½S4ðˆk4ÞγνS2ðˆk2Þγμtr½S3ðˆk3Þγ5Sð1ˆk1Þγ5g; ð13Þ where ˆk1¼ k1− w1p, ˆk2¼ k2− w2p, ˆk3¼ k3þ w3p, ˆk4¼k1þk2−k3þw4p, and ω~2¼ 1=2ðk21þ k22þ k23þ k1k2− k1k3− k2k3Þ. Details of the calculation can be found in our previous papers, e.g.,[37,38].
The matrix elements of the decays Zþc → J=ψ þ πþ and Zþc → ηcþ ρþ are given by MμνðZcðp; ϵμpÞ → J=ψðq1;ϵνq1Þ þ π þðq 2ÞÞ ¼ 6ffiffiffi 2 p gZcgJ=ψgπ Z d4k1 ð2πÞ4i Z d4k2 ð2πÞ4i ~ΦZcð−~η 2Þ ~Φ J=ψð−ðk1þ v2q1Þ2Þ ~Φπð−ðk2þ u4q2Þ2Þ ×ftr½γ5S4ðk2Þγ5S3ðk2þ q2ÞγμS2ðk1ÞγνS1ðk1þ q1Þ þ tr½γμS4ðk2Þγ5S3ðk2þ q2Þγ5S2ðk1ÞγνS1ðk1þ q1Þg ¼ AJ=ψπgμνþ BJ=ψπqμ1qν2; ð14Þ MμαðZcðp; ϵμpÞ → ηcðq1Þ þ ρðq2;ϵαq2ÞÞ ¼ 6ffiffiffi 2 p gZcgηcgρ Z d4k1 ð2πÞ4i Z d4k2 ð2πÞ4i~ΦZcð−~η 2Þ ~Φ ηcð−ðk1þ v2q1Þ 2Þ ~Φ ρð−ðk2þ u4q2Þ2Þ ×ftr½γ5S4ðk2ÞγαS3ðk2þ q2ÞγμS2ðk1Þγ5S1ðk1þ q1Þ þ tr½γμS4ðk2ÞγαS3ðk2þ q2Þγ5S2ðk1Þγ5S1ðk1þ q1Þg ¼ Aηcρgμα− Bηcρqμ2qα1: ð15Þ
The argument of the Zc-vertex function is given by ~ η2¼ η2 1þ η22þ η23; η1¼ þ 1 2p ð2kffiffiffi2 1þ ð1 − w1þ w2Þq1− ðw1− w2Þq2Þ; η2¼ þ 1 2p ð2kffiffiffi2 2− ðw3− w4Þq1þ ð1 − w3þ w4Þq2Þ; η3¼ þ 12ððw3þ w4Þq1− ðw1þ w2Þq2Þ: ð16Þ
The quark masses are specified as m1¼ m2¼ mc, m3¼ m4¼ md¼ mu, and the two-body reduced masses as
v1¼ m1=ðm1þ m2Þ, v2¼ m2=ðm1þ m2Þ, u3¼ m3=ðm3þ m4Þ, and u4¼ m4=ðm3þ m4Þ. The matrix elements of the decays Zþc → ¯D0þ Dþ and Zþc → ¯D0þ Dþ read
MμνðZcðp; ϵμpÞ → ¯D0ðq1Þ þ Dþðq2;ϵνq2ÞÞ ¼ 6ffiffiffi 2 p gZcgDgD Z d4k1 ð2πÞ4i Z d4k2 ð2πÞ4i ~ΦZcð−~δ 2Þ ~Φ Dð−ðk2þ v2q2Þ2Þ ~ΦDð−ðk1þ u1q2Þ2Þ ×ftr½γ5S4ðk2þ q1Þγ5S1ðk1ÞγνS3ðk1þ q2ÞγμS2ðk2Þ − tr½γ5S4ðk2þ q1ÞγμS1ðk1ÞγνS3ðk1þ q2Þγ5S2ðk2Þg ¼ A¯DDgμν− B¯DDqμ2qν1; ð17Þ
MμαðZcðp; ϵμpÞ → ¯D0ðq1;ϵαq1Þ þ D þðq 2;ÞÞ ¼ 6ffiffiffi 2 p gZcgDgD Z d4k1 ð2πÞ4i Z d4k2 ð2πÞ4i ~ΦZcð−~δ 2Þ ~Φ Dð−ðk1þˆv1q1Þ2Þ ~ΦDð−ðk2þ ˆu4q2Þ2Þ ×ftr½S4ðk2þ q1Þγ5S1ðk1Þγ5S3ðk1þ q2ÞγμS2ðk2Þγα − tr½S4ðk2þ q1ÞγμS1ðk1Þγ5S3ðk1þ q2Þγ5S2ðk2Þγαg ¼ ADDgμαþ BDDqμ1qα2: ð18Þ
The argument of the Zc-vertex function is given by ~ δ2¼ δ2 1þ δ22þ δ23; δ1¼ − 1 2p ðkffiffiffi2 1− k2þ ðw1− w2Þðq1þ q2ÞÞ; δ2¼ þ 1 2p ðkffiffiffi2 1− k2− ð1 þ w3− w4Þq1þ ð1 − w3þ w4Þq2Þ; δ3¼ −12ðk1þ k2þ ðw1þ w2Þðq1þ q2ÞÞ: ð19Þ
The quark masses are specified as m1¼ m2¼ mc,
m3¼ m4¼ md¼ mu, and the two-body reduced masses as
ˆv2¼m2=ðm2þm4Þ,ˆv4¼m4=ðm2þm4Þ,ˆu1¼m1=ðm1þm3Þ,
and ˆu3¼ m3=ðm1þ m3Þ.
We finally calculate the two-body decay widths. The relevant spin kinematical formulas have been collected in the Appendix. Note that momentum of the daughter vector particle is chosen to be q1 in Eq. (A1). In addition the matrix element is expressed through the dimensionless invariant amplitudes A1, and A2 in Eq. (A3). In order to adjust the notation in Eqs.(14),(15),(17), and(18)to those given in the Appendix, one has to replace q1↔ q2 in Eqs. (15) and (17) and then introduce the dimensionless form factors A1¼ A=m and A2¼ mB (p2¼ m2) where the sign “þ” stands for Eqs. (14) and (18) and “−” for Eqs.(15)and(17), respectively. The expressions for helicity amplitudes via A1and A2are given in Eq.(A5). The two-body decay widths are now calculated using Eq.(A9).
As a consequence of the subtraction of the two traces in the matrix elements in Eqs. (17) and(18) we found that ADD ¼ ADD≡ 0 analytically. This results in a significant
suppression of the decay widths due to the D-wave sup-pression factor ofjq1j5. In the calculation of the quark-loop diagrams we have only one free parameter ΛZc, the size parameter of the Zcstate. The other model parameters have
been fixed in previous papers[36–38,41]from analysis of hadron processes involving light and heavy quarks,
mu=d ms mc mb λ
0.241 0.428 1.67 5.05 0.181 GeV: ð20Þ Here mqare the constituent quark masses andλ is an infrared
cutoff parameter responsible for the quark confinement. The size parameters of theπ ρ, D, D, J=ψ, and ηc have been
fixed as
Λπ Λρ ΛD ΛD ΛJ=ψ Ληc
0.871 0.624 1.600 1.529 1.738 3.777 GeV: ð21Þ For the Zcð3900Þ mass we use the actual value 3.886 GeV. We adjust the size parameter ΛZc in such a way as to be close to the central value for the decay Zþc → J=ψ þ πþ obtained in Refs.[6,7]. If the parameterΛZc is varied in the regionΛZc¼ 2.25 0.10 GeV the numerical values of the decay widths vary as
ΓðZþ c → J=ψ þ πþÞ ¼ ð27.9þ6.3−5.0Þ MeV; ΓðZþ c → ηcþ ρþÞ ¼ ð35.7þ6.3−5.2Þ MeV; ΓðZþ c → ¯D0þ DþÞ ∝ 10−8 MeV; ΓðZþ c → ¯D0þ DþÞ ∝ 10−8 MeV: ð22Þ
Here and in the following an increasing of the size parameter leads to a decreasing of the decay width. Since the experimental data[4] show that the Zcð3900Þ has a much
more stronger coupling to DD than J=ψπ, one has to conclude that the tetraquark-type current for Zcð3900Þ is in
discord with experiment.
Moreover, we expect that a realistic value of the size parameterΛZcis about 3 GeV. UsingΛZc¼ 3.3 1.1 GeV we get a significant suppression for the Zþc → J=ψ þ πþ
and Zþc → ηcþ ρþ modes, and the rates for the modes Zþc → ¯D0þ Dþand Zþc → ¯D0þ Dþbecome much more negligible, ΓðZþ c → J=ψ þ πþÞ ¼ ð4.3þ0.7−0.6Þ MeV; ΓðZþ c → ηcþ ρþÞ ¼ ð8.0þ1.2−1.0Þ MeV; ΓðZþ c → ¯D0þ DþÞ ∝ 10−9 MeV; ΓðZþ c → ¯D0þ DþÞ ∝ 10−9 MeV: ð23Þ
III. THE Zcð3900Þ AS A FOUR-QUARK STATE WITH A MOLECULAR-TYPE CURRENT
We describe the Zþcð3900Þ as the charged particle in the isotriplet with a molecular-type current given by (see Ref.[42])
Jμ¼ 1ffiffiffi 2
p ½ð ¯dγ5cÞð¯cγμuÞ þ ð ¯dγμcÞð¯cγ5uÞ: ð24Þ
Its nonlocal generalization is given by
JμZcðxÞ ¼ Z dx1… Z dx4δ x−X 4 i¼1 wixi ΦZc X i<j ðxi− xjÞ2 Jμ4qðx1;…; x4Þ; Jμ4q ¼ 1ffiffiffi 2 p fð ¯dðx3Þγ5cðx1ÞÞð¯cðx2Þγμuðx4ÞÞ þ ð ¯dðx3Þγμcðx1ÞÞð¯cðx2Þγ5uðx4ÞÞg: ð25Þ
The Fourier transform of the Zc mass operator is written as
ΠμνZcðpÞ ¼ 92 Y3 i¼1 Z d4ki ð2πÞ4i ~Φ2Zcð−~ω 2Þftr½γ 5S1ðˆk1Þγ5S3ðˆk3Þtr½γμS4ðˆk4ÞγνS2ðˆk2Þ þ tr½γμS 1ðˆk1ÞγνS3ðˆk3Þtr½γ5S4ðˆk4Þγ5S2ðˆk2Þg ð26Þ
with ˆki and ~ω2 being defined as in the previous section.
The matrix elements of the decays Zþc → J=ψ þ πþ and Zþc → ηcþ ρþ are given by
MμνðZcðp; ϵμpÞ → J=ψðq1;ϵνq1Þ þ π þðq 2ÞÞ ¼ 3ffiffiffi 2 p gZcgJ=ψgπ Z d4k1 ð2πÞ4i Z d4k2 ð2πÞ4i ~ΦZcð−~η 2Þ ~Φ J=ψð−ðk1þ v1q1Þ2Þ ~Φπð−ðk2þ u4q2Þ2Þ ×ftr½γ5S1ðk1ÞγνS2ðk1þ q1ÞγμS4ðk2Þγ5S3ðk2þ q2Þ þ tr½γμS1ðk1ÞγνS2ðk1þ q1Þγ5S4ðk2Þγ5S3ðk2þ q2Þg ¼ AJ=ψπgμνþ BJ=ψπqμ1qν2; ð27Þ MμαðZcðp; ϵμpÞ → ηcðq1Þ þ ρðq2;ϵαq2ÞÞ ¼ 3ffiffiffi 2 p gZcgηcgρ Z d4k1 ð2πÞ4i Z d4k2 ð2πÞ4i~ΦZcð−~η 2Þ ~Φ ηcð−ðk1þ v1q1Þ 2Þ ~Φ ρð−ðk2þ u4q2Þ2Þ ×ftr½γ5S1ðk1Þγ5S2ðk1þ q1ÞγμS4ðk2ÞγαS3ðk2þ q2Þ þ tr½γμS1ðk1Þγ5S2ðk1þ q1Þγ5S4ðk2ÞγαS3ðk2þ q2Þg ¼ Aηcρgμα− Bηcρqμ2qα1: ð28Þ
The argument of the Zc-vertex function ~η2 and the specification of the quark masses are identical to those given in the
previous section.
The matrix elements of the decays Zþc → ¯D0þ Dþ and Zþc → ¯D0þ Dþ read
MμνðZcðp; ϵμpÞ → ¯D0ðq1Þ þ Dþðq2;ϵνq2ÞÞ ¼ 9ffiffiffi 2 p gZcgDgD Z d4k1 ð2πÞ4i Z d4k2 ð2πÞ4i~ΦZcð−~δ 2Þ ~Φ Dð−ðk2þ v4q1Þ2Þ ~ΦDð−ðk1þ u1q2Þ2Þ ×ftr½γμS1ðk1ÞγνS3ðk1þ q2Þtr½γ5S4ðk2Þγ5S2ðk2þ q1Þg ¼ A¯DDgμν− B¯DDqμ2qν1; ð29Þ
MμαðZcðp; ϵμpÞ → ¯D0ðq1;ϵαq1Þ þ D þðq 2;ÞÞ ¼ 9ffiffiffi 2 p gZcgDgD Z d4k1 ð2πÞ4i Z d4k2 ð2πÞ4i~ΦZcð−~δ 2Þ ~Φ Dð−ðk1þ ˆv1q1Þ2Þ ~ΦDð−ðk2þˆu4q2Þ2Þ ×ftr½γ5S1ðk1Þγ5S3ðk1þ q2Þtr½γμS4ðk2ÞγαS2ðk2þ q1Þg ¼ ADDgμαþ BDDqμ1qα2: ð30Þ
The argument of the Zc-vertex function is given by ~ δ2¼ δ2 1þ δ22þ δ23; δ1¼ − 1 2p ðkffiffiffi2 1þ k2þ ð1 þ w1− w2Þq1þ ðw1− w2Þq2ÞÞ; δ2¼ þ 1 2p ðkffiffiffi2 1þ k2− ðw3− w4Þq1þ ð1 − w3þ w4Þq2Þ; δ3¼ þ 12ð−k1þ k2þ ð1 − w1− w2Þq1− ðw1þ w2Þq2ÞÞ: ð31Þ
The quark masses are specified as m1¼m2¼mc, m3¼m4¼ md¼mu, and the two-body reduced masses as ˆv2¼ m2= ðm2þ m4Þ, ˆv4¼m4=ðm2þm4Þ, ˆu1¼ m1=ðm1þ m3Þ, and
ˆu3¼ m3=ðm1þ m3Þ.
As a guide to adjust the parameter ΛZc we take the experimental values for decay widths given in Ref.[4]. If the parameterΛZc is varied in the limitsΛZc ¼ 3.3 0.1 GeV the numerical values of decay widths vary according to
ΓðZþ c → J=ψ þ πþÞ ¼ ð1.8 0.3Þ MeV; ΓðZþ c → ηcþ ρþÞ ¼ ð3.2þ0.5−0.4Þ MeV; ΓðZþ c → ¯D0þ DþÞ ¼ ð10:0þ1.7−1.4Þ MeV; ΓðZþ c → ¯D0þ DþÞ ¼ ð9.0þ1.6−1.3Þ MeV: ð32Þ
Thus a molecular-type current for the vertex function of the Zc is in accordance with the experimental observation
[4] that Zcð3900Þ has a much stronger coupling to DD
than to J=ψπ.
IV. XbAS A TETRAQUARK
Let us first interpret Xb as a tetraquark state with the
quantum numbers JP ¼ 0þ. Then the interpolating current
for the Xbð5568Þ is given by J¼ εabcεdecðuT
aCγ5bbÞð ¯ddγ5C¯sTeÞ: ð33Þ
The nonlocal version of the four-quark interpolating current reads JþX bðxÞ ¼ Z dx1… Z dx4δ x−X 4 i¼1 wixi ×ΦXb X i<j ðxi− xjÞ2 Jþ4qðx1;…; x4Þ; Jþ4q¼ εabcεdec½uaðx3ÞCγ5bbðx1Þ½ ¯ddðx4Þγ5C¯seðx2Þ;
ð34Þ where wi¼mi=P4j¼1mj. The effective interaction Lagrangian describing the coupling of the meson Xb to
its constituent quarks takes the form Lint ¼ gXbX
−
bðxÞ · JþXbðxÞ þ H:c: ð35Þ The Fourier transform of the Xb-tetraquark mass oper-ator is given by ΠXbðp 2Þ ¼ 6Y3 i¼1 Z d4ki ð2πÞ4i ~Φ2Xbð−~ω 2Þ × tr½γ5S1ðˆk1Þγ5S3ðˆk3Þtr½γ5S2ðˆk2Þγ5S4ðˆk4Þ; ð36Þ where ˆk1¼k1−w1p, ˆk2¼k2−w2p, ˆk3¼k3þw3p, ˆk4¼ k1þk2−k3þw4p, and ω~2¼ 1=2ðk21þ k22þ k23þ k1k2− k1k3− k2k3Þ.
The matrix element of the decay XþbðpÞ → Bsðq1Þ þ πþðq 2Þ reads MðZb → Bsþ πþÞ ¼ 6gXbgBsgπ Z d4k1 ð2πÞ4i Z d4k2 ð2πÞ4i ~ΦXbð−~η 2Þ ~Φ Bsð−ðk1þ v1q1Þ 2Þ ~Φ πð−ðk2þ u4q2Þ2Þ × tr½γ5S1ðk1Þγ5S2ðk1þ q1Þγ5S4ðk2Þγ5S3ðk2þ q2Þ ¼ GXbBsπ; ð37Þ
where the arguments of the Xb-vertex function are given by ~ η2¼ η2 1þ η22þ η23; η1¼ − 1 2p ð2kffiffiffi2 1þ ð1 þ w1− w2Þq1þ ðw1− w2Þq2Þ; η2¼ þ 1 2p ð2kffiffiffi2 2− ðw3− w4Þq1þ ð1 − w3þ w4Þq2Þ; η3¼ þ 12ððw3þ w4Þq1− ðw1þ w2Þq2Þ: ð38Þ
The quark masses are specified as m1¼mb, m2¼ ms,
m3¼ mu, m4¼ md, and the two-body reduced masses as
v1¼m1=ðm1þm2Þ, v2¼m2=ðm1þm2Þ, u3¼m3=ðm3þm4Þ, and u4¼ m4=ðm3þ m4Þ.
The two-body decay width is given by ΓðXb→ Bsþ πÞ ¼
jq1j
8πM2 Xb
G2XbBsπ; ð39Þ wherejq1j is the momentum of the daughter particles in the rest frame of the Xb.
We adjust the parameter ΛXb for two values of the Xb mass, (i) mXb ¼ 5567.8 MeV as reported by the D0 Collaboration [8], and (ii) mXb ¼ 5771 MeV as was obtained in[28]. The numerical values of the decay widths can be calculated to be mXb¼ 5.568 GeV; ΛXb¼ ð1.36 0.05Þ GeV; ΓðXb→ BsπÞ ¼ ð21.9 3.5Þ MeV; mXb¼ 5.771 GeV; ΛXb¼ ð1.66 0.05Þ GeV; ΓðXb→ BsπÞ ¼ ð21.7 3.5Þ MeV: ð40Þ V. Zð4430Þ AS A TETRAQUARK
The interpolating tetraquark current of the Zð4430Þ state with JP¼ 1þ fixed by the LHCb Collaboration [33]
has the same structure as the tetraquark current for the Zc
state [see Eqs.(5)and(6)]. Similarity of the Zð4430Þ and Zc states also concerns the effective interaction Lagrangian describing the coupling of Zð4430Þ to its constituent quarks,
Lint¼ gZZμðxÞ · JμZðxÞ þ H:c:; ð41Þ
where JμZðxÞ ¼ JμZ
cðxÞ with a specific value of the size parameter ΛZ.
In the case of Zð4430Þ we consider two strong decay modes Zð4430Þ → J=ψ þ π and Zð4430Þ → ψð2sÞ þ π, which are calculated by analogy with the case of Zc→
J=ψ þ π in the tetraquark picture. A new feature is that we should specify the vertex function of the ψð2sÞ state. By analogy with the oscillator potential model it should
emulate the node structure of theψð2sÞ. In our calculations we use the following form of theψð2sÞ-vertex function:
~Φψð2sÞð−k2Þ ¼ expðk2=Λψð2sÞÞ½1 − αexpðk2=Λψð2sÞÞ; ð42Þ
where α is a free parameter, encoding the node structure of the ψð2sÞ meson. It is fixed at α¼1.0172 from the description of the leptonic decay constant fψð2sÞ¼291MeV. For convenience, we use the same size parameterΛJ=ψ ¼ Λψð2sÞ ¼ 1.738 GeV for J=ψ, and its radial excitation ψð2sÞ
state. For the Zð4430Þ mass we use the actual value 4.478 GeV.
Now let us turn to the discussion of our results for the Zð4430Þ → J=ψ þ π and Zð4430Þ → ψð2sÞ þ π decay widths. We have a single free parameter: theΛZð4430Þ-size parameter of the Zð4430Þ state. We use the present upper limit for the total width of the Zð4430Þ state Γ ≤ 212 MeV deduced from the averaged value Γ ¼ 181 31 MeV in Particle Data Group[43]as the upper limit for the sum of the widths of two modes Zð4430Þ → J=ψ þ π and Zð4430Þ → ψð2sÞ þ π. It constrains the choose of the size parameter ΛZ. In particular, we found that
ΛZ≥ 2.2 GeV, which supports the compact ðc ¯cd ¯uÞ
tetra-quark interpretation of the Zð4430Þ state. In Table I we present our numerical results for the partial decay widths ΓJ=ψ ≐ ΓðZð4430Þ → J=ψ þ πÞ, and Γψð2sÞ≐ ΓðZð4430Þ → ψð2sÞ þ πÞ decay widths, their sum Γ ¼ ΓJ=ψ þ Γψð2sÞ, and their ratio RZ¼ Γψð2sÞ=ΓJ=ψ for
variation of ΛZ from 2.2 to 3.2 GeV. One can see that the decay width of the Zð4430Þ → ψð2sÞ þ π process dominates over the one of Zð4430Þ → J=ψ þ π by a factor RZ≃ ð4.36 0.28Þ.
Finally, we make the prediction for the Zð4430Þþ → Dþþ ¯D0 decay rate. This process is described by the invariant matrix element, which is expressed in terms of three relativistic amplitudes Bi,ði ¼ 1; 2; 3Þ as
MμαβðZð4430Þðp; μÞ → Dðq1;αÞ þ ¯Dðq2;βÞÞ
¼ B1qμϵq1q2αβþ B2ϵq1μαβþ B3ϵq2μαβ: ð43Þ
TABLE I. Zð4430Þ decay rates.
ΛZð4430Þ(GeV) ΓJ=ψ (MeV) Γψð2sÞ(MeV) Γ (MeV) RZ
2.2 37.4 173.7 211.1 4.64 2.3 31.7 144.7 176.4 4.56 2.4 26.9 120.6 147.5 4.48 2.5 22.9 100.8 123.7 4.40 2.6 19.4 84.4 103.8 4.35 2.7 16.5 70.9 87.4 4.30 2.8 14.1 59.7 73.8 4.23 2.9 12.0 50.4 62.4 4.20 3.0 10.3 42.7 53.0 4.15 3.1 8.8 36.3 45.1 4.13 3.2 7.6 31.0 38.6 4.08
The Zð4430Þþ→ Dþþ ¯D0 decay rate is calculated according to the formula
ΓðZð4430Þþ→ Dþþ ¯D0Þ ¼ jq1j 12πM2 Z B21M2Zjq1j4þB22 3M2 Dþþ 1þ M2Z M2D0 jq1j2 þB2 3 3M2 D0þ 1þ M2Z M2Dþ jq1j2 þB1B2jq1j2ðMZ2þM2Dþ−M2D0Þ þB1B3jq1j2ðM2ZþM2 D0−M 2 DþÞ þB2B3ð3ðM2Z−MD2þ−M2D0Þ−jq1j 2Þ : ð44Þ
Our numerical result forΛZð4430Þvaried from 2.2 to 3.2 GeV isΓðZð4430Þþ→ Dþþ ¯D0Þ ¼ 23.5 15.6 MeV.
VI. SUMMARY AND CONCLUSIONS Let us summarize the main results of our paper. Presently two possible four-quark configurations for exotic states are tested experimentally and theoretically: the tetraquark (compact) configuration corresponding to the coupling of color diquark and antidiquark and molecular (extended) configuration corresponding to the coupling of two sepa-rate mesons. We have critically checked both possible four-quark pictures (tetrafour-quark and molecular scenario) in the case of the Zcð3900Þ state. For the case of the Xð5568Þ and
Zð4430Þ states we considered only the tetraquark picture. Our study has been done by analyzing strong decays of the exotic state. The strong decays have been calculated in the framework of the covariant quark model previously devel-oped by us. First, we have interpreted the Zcð3900Þ state as
the isospin 1 partner of the Xð3872Þ. We have calculated the partial widths of the decays Zþcð3900Þ → J=ψπþ,ηcρþ, and ¯D0Dþ, ¯D0Dþ. It turned out that the leading metric Lorentz structure in the matrix elements describing the decays Zcð3900Þ → ¯DD vanishes analytically. This
results in a significant D-wave suppression of these decays through the appearance of the phase space factor propor-tional tojqj5. Since the experimental data from the BESIII Collaboration show that Zcð3900Þ has a much more
stronger coupling to DDthan to J=ψπ, we have concluded that the tetraquark-type current for the Zcð3900Þ is in disaccord with experiment. As an alternative we have employed a molecular-type four-quark current to describe the Zcð3900Þ state. In this case we found that for a
relatively large model size parameter of ΛZc∼ 3.3 GeV one can obtain partial widths for the decays Zcð3900Þ →
¯DD that are close to ∼15 MeV for each mode. At
the same time the partial widths for the decays Zcð3900Þ → J=ψπ; ηcρ are suppressed by a factor of 6–7
in accordance with experimental data.
Finally, we have tested a tetraquark picture for the Xð5568Þ and Zð4430Þ structure by analyzing their strong one-pion decay. In the analysis of the Bsπ decay mode
of the Xð5568Þ, we found that one can fit the exper-imental decay width using a mass of 5568 MeV by taking the value of the parameter to beΛXb ∼ 1.4 GeV. In the case of a larger mass 5771 MeV one finds ΛXb∼ 1.7 GeV. In the case of the Zð4430Þ state we considered the modes with J=ψ, and its first radial excitationψð2sÞ. We showed that the decay width of the Zð4430Þ → ψð2sÞ þ π process dominates over the one of Zð4430Þ → J=ψ þ π by a factor RZ ¼ ð4.36 0.28Þ and
the sum of the two decay rates of the Zð4430Þ satisfies the upper limit for the total width of the Zð4430Þ if the size parameterΛZð4430Þ ≥ 2.2 GeV. It means that the Zð4430Þ state is a good candidate for the compact tetraquark state. Our prediction for the Zð4430Þþ → Dþþ ¯D0 decay width isΓðZð4430Þþ → Dþþ ¯D0Þ ¼ 23.5 15.6 MeV.
ACKNOWLEDGMENTS
This work was supported by the German
Bundesministerium für Bildung und Forschung (BMBF) under Project No. 05P2015—ALICE at High Rate (Grant No. BMBF-FSP 202): “Jet and fragmentation processes at ALICE and the parton structure of nuclei and structure of heavy hadrons,” Tomsk State University Competitiveness Improvement Program, and the Russian Federation program “Nauka” (Contract No. 0.1526.2015, 3854). M. A. I. acknowl-edges the support from the Precision Physics, Fundamental Interactions and Structure of Matter (PRISMA) cluster of excellence (Mainz University). M. A. I. and J. G. K. thank the Heisenberg-Landau grant for partial support.
APPENDIX: SPIN KINEMATICS FOR THE DECAY 1þ → 1−þ 0−
The matrix element
M¼ h1−ðq1; ρÞ; 0−ðq2ÞjTj1þðp; μÞi ðA1Þ can be described by the three sets of amplitudes: (i) invariant amplitudes, (ii) helicity amplitudes, and (iii) (LS) ampli-tudes. In this Appendix we derive the relations between the three sets of amplitudes.
The product of the parities of the two final state mesons is (þ1), which matches the parity of the initial state. Thus the two final state mesons must have even relative orbital momenta. In the present case these are L¼ 0, 2. The spins s1 and s2of the two final state mesons couple to the total spin S¼ 1. Thus one has the two (LS) amplitudes
A01; A21: ðA2Þ
There are two covariantsKμρ1 ¼ mgμρ andK2μρ ¼m1qμ1qρ2 that describe the matrix element. These define the invariant amplitudes A1 and A2 according to
M¼ ðA1Kμρ1 þ A2Kμρ2 Þεμε1ρ: ðA3Þ There are two independent helicity amplitudes Hλλ1 ðλ ¼ λ1Þ,
Hþ1þ1; H00: ðA4Þ
From parity one has H−1−1¼ Hþ1þ1. In order to relate the helicity amplitudes to the invariant amplitudes we work in the rest system of the decay meson and define the z-direction to be along the momentum direction of meson 1. The helicity amplitudes can be related to the invariant amplitudes using the momenta and polarization vectors ϵρ1ðÞ ¼ ð0;∓ 1; −i;0Þ=pffiffiffi2;ϵρ1ð0Þ ¼ ðjq1j;0;0;E1Þ=m1;ϵμðÞ ¼ ð0; ∓ 1;−i; 0Þ=pffiffiffi2;ϵμð0Þ ¼ ð0;0;0;1Þ;qμ1¼ ðE1;0;0;jq1jÞ;qμ2¼ ðE2;0; 0;−jq1jÞ. One can then express the helicity amplitudes in
terms of the invariant amplitudes. The relations can be calculated to be H00¼ − m m1E1A1− 1 m1jq1j 2A 2; Hþ1þ1¼ H−1−1 ¼ −mA1; ðA5Þ
where the magnitude of the final state three-momentum is given byjq1j¼pffiffiffiffiffiffiffiffiffiffiffiffiffiQþQ−=2m with Q¼m2−ðm1m2Þ2¼ 2ðq1q2∓m1m2Þ.
The coefficients of the matrix relating the (LS) and helicity amplitudes can be calculated from the product of two C.G. coefficients according to [44]
hJM; LSjJM; λ1λ2i ¼ 2L þ 1 2J þ 1 1=2
hLS; 0μjJμihs1s2; λ1;−λ2jSμi; ðA6Þ
where μ ¼ λ1− λ2. One obtains A01 A21 ¼ ffiffiffi 1 3 r 2 1 ffiffiffi 2 p −pffiffiffi2 Hþ1þ1 H00 : ðA7Þ We can thus relate the (LS) amplitudes to the invariant amplitudes Ai. The relations read
A01¼ − ffiffiffi 1 3 r 1 m1ðmð2m1þ E1ÞA1þ jq1j 2A 2Þ; A21¼ ffiffiffi 2 3 r 1 m1ðmðE1− m1ÞA1þ jq1j 2A 2Þ: ðA8Þ
The (LS) amplitude A21 can be seen to have the correct D-wave threshold behavior proportional to j~q1j2 by taking the relation ðE1− m1Þ ¼ jq1j2=ðE1þ m1Þ into account.
The rate for the decay process 1þðpÞ → 1−ðq1Þ þ 0−ðq 2Þ is given by Γ ¼ 1 8π 1 2s þ 1 jq1j m2ðjHþ1þ1j 2þ jH −1−1j2þ jH00j2Þ ¼ 1 8π 1 2s þ 1 jq1j m2ðjA01j 2þ jA 21j2Þ; ðA9Þ where2s þ 1 ¼ 3.
We assume that the (1−) meson decays into two
pseudoscalar mesons as in the cascade decay Zc→ D þ Dð→ D þ πÞ. We treat the cascade decay in the narrow width approximation. The differential decay distribution for the cascade decay is given by
dΓðZc → D þ Dð→ D þ πÞÞ d cosθ ¼ BðD→ D þ πÞ 1 24π jq1j m2 × 3 8ð1 þ cos2θÞHT þ 34sin2θHL ; ðA10Þ
where HT ¼ jHþ1þ1j2þ jH−1−1j2, HL ¼ jH00j2. For the
cascade decay Zc → π þ J=ψð→ lþl−Þ we again have
dΓðZc → π þ J=ψð→ lþl−ÞÞ d cosθ ¼ BðJ=ψ → lþl−Þ 1 24π jq1j m2 × 3 8ð1 þ cos2θÞHTþ 34sin2θHL : ðA11Þ
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