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PHYSICAL REVIEW D 99, 036014 (2019)

Study of the strong decays of ϕð2170Þ and the future charm- factory

† Hong-Wei Ke1,* and Xue-Qian Li2, 1School of Science, Tianjin University, Tianjin 300072, China 2School of Physics, Nankai University, Tianjin 300071, China

(Received 30 October 2018; published 19 February 2019)

The present data imply that ϕð2170Þ may not be an excited state of ϕ but is a four- state with sss¯ s¯ constituents. Furthermore, there are no two of ss¯ available to form a that fits the mass spectrum of ϕð2170Þ; thus, we suggest it should be an sss¯ s¯ state. In this scenario, we estimate its decay rates through the fall-apart mechanism. Our theoretical estimates indicate that its main decay modes 0 should be ϕð2170Þ into ϕf0ð980Þ, h1η, h1η , K1ð1270ÞK, and K1ð1400ÞK. Under this hypothesis, the ϕð2170Þ → ð890Þ0 ¯ ð890Þ0 þ − 0 0 modes K K , K K , and KLKS should be relatively suppressed. Since the width of h1 is rather large, at present, it is hard to gain precise data on BRðϕð2170Þ → h1ηÞ and 0 BRðϕð2170Þ → h1η Þ, the measurements of which may be crucial for drawing a definite conclusion about the inner assignment of ϕð2170Þ. We place our expectations on the proposed charm-tau factory, which will have much larger luminosity and better capacities.

DOI: 10.1103/PhysRevD.99.036014

I. INTRODUCTION cannot find two available mesons [1] with ss¯ constituents to form a molecular of which the mass fits the mass Very recently, a ϕð2170Þ came into the view of spectrum of ϕð2170Þ. Thus, we turn to suggest that it is an researchers because it may be a special exotic state. It was sss¯ s¯ tetraquark state. This conjecture was also considered observed via its decay into ϕ þ f0ð980Þ [1]; meanwhile, by the authors of Ref. [10]. some possible final states K0Kπ and K0K¯ 0 have not At the end of last century, a stimulating question was been seen. If it were a normal meson i.e., an excited state of ϕ þ − 0 0 raised: did multiquark states indeed exist in nature, because the decay portals into K K and KLKS would be dominant in their primary paper Gell and Mann predicted them along ϕ 0 ¯ 0 as the ground does. Moreover, even the channel K K with the simplest assignments of qq¯ for mesons and qqq for should also be seen since a sufficient phase space is available. [11]. The first proposed of qqqqs¯ with Furthermore, in Ref. [2], the theoretical evaluation on the unusual B ¼ 1 and S ¼ 1 quantum numbers would defi- ϕð2170Þ total width obviously conflicts with data if is a nitely be a multiquark state. In that assignment of the normal meson. A reasonable interpretation is needed. It is pentaquark qqqqs¯ except s¯, all other are light ones suggested that the observed ϕð2170Þ could be a molecular u d ¯ ( or types). The passion of detecting such state of ΛΛ[3] or a tetraquark state [4]. In Ref. [5], the author was very high; however, after a hard and desperate search, thinks that ϕð2170Þ is an excited qqs¯ s¯ tetraquark (q ¼ u, d). such pentaquarks were never observed experimentally. The But this assignment is questionable because no ground qqs¯ s¯ despair discouraged researchers, who decided to give up. tetraquark has ever been observed. But following conduction of more accurate experiments Being hinted at by the decay mode ϕð2170Þ → and innovated skills of analysis, many exotic mesons ϕf0ð980Þ, a natural conjecture is that ϕð2170Þ may be a have been measured. These exotic states are proposed to four-quark state with sss¯ s¯ constituents. There are two be four-quark states (molecular states or states) choices: a molecular state or a tetraquark. However, [12–21], later two pentaquarks were observed by the LHCb between two mesons, there should exist a binding energy Collaboration [22], which validates the suggestion about of about a few tens of mega--volts [6–9]; thus, we the existence of multiquark states. It validates the sugges- tion about the existence of multiquark states. However, *[email protected] we have observed that all the discovered multiquark states † [email protected] contain at least one heavy quark (c or b). This may hint that the existence of heavy quarks in the multiquark states is Published by the American Physical Society under the terms of ϕð2170Þ the Creative Commons Attribution 4.0 International license. fatal [23]. Is that the conclusion of the story? , Further distribution of this work must maintain attribution to which has come to our attention recently, could be the author(s) and the published article’s title, journal citation, identified as a four quark state with sss¯ s¯ constituents. and DOI. Funded by SCOAP3. Even though it is true, the early allegation might not be

2470-0010=2019=99(3)=036014(6) 036014-1 Published by the American Physical Society HONG-WEI KE and XUE-QIAN LI PHYS. REV. D 99, 036014 (2019) completely subverted because the mass of the s quark situation was discussed by Jaffe in Ref. [26], so we also resides between that of a very light quark and the supposed omit the color-sextet state of a . The orbital angular “heavy” and the remaining constituents in the momentum between the diquark and antidiquark is 1, i.e., exotic state are all s flavor (s¯) with “middle” mass. in the p wave for guaranteeing the parity is negative. A naive analysis may provide us support for this ϕð2171Þ is a spin-1 state jJ; J12;J34i¼j1; 1; 1i, which has Ω ϕ conjecture. The masses of and , which consist of three three spin projections: jJ; Jzi¼j1; 1; i; j1; 0; i; j1; −1; i. s quarks and an ss¯, respectively, are 1672 and 1020 MeV. Since the C parity of ϕð2170Þ is odd the spin configuration This implies that the s-quark mass is around 500–600 MeV; of the tetraquark is fully determined, for example, ¯ ¯ thus, a simple estimate on the mass of the sss s tetraquark 1 state should fall in a region close to the mass of ϕð2170Þ.If j1 1i¼pffiffiffi ðj1 1i j1 0i − j1 0i j1 1i Þ ð Þ ; ; ss ; s¯ s¯ ; ss ; s¯ s¯ : 2 the assignment is true, ϕð2170Þ is indeed a tetraquark with 2 a single flavor of strangeness. The color configuration is j1; 3¯; 3¯i, which can be written Without doubt, it is absolutely important to get a better as [25] understanding of the inner structure of ϕð2170Þ. Since it only possesses s flavor, its decays would be dominated by 1 pffiffiffiffiffi ε εaefðsbsdÞðs¯ s¯ Þ: ð3Þ the modes in which the final states mainly contain s flavors. 48 abd e f Let us turn to investigate the mechanism that governs the strong decay of ϕð2170Þ. It is the so-called fall-apart Note the spin configuration of the tetraquark sss¯ s¯ is in the mechanism [24,25]. diquark and antidiquark spin bases. When it decays via the In Refs. [24,25], the authors suggested that the fall-apart fall-apart mechanism, one needs to switch a pair quark- mechanism induces the main decay modes of the tetraquark antiquark around and rearrange their spins and colors to make state. By this mechanism, the constituents in a tetraquark proper combinations for the two mesons in the final state. are rearranged into two color singular pairs by exchanging Now, let us study the decay of ϕð2170Þ via the fall-apart soft and then simply fall apart into two mesons. In mechanism. Apparently, the two-body final states with the this work, we will employ this mechanism to study the s wave are preferred if it is allowed. Since the JPC of −− decays of ϕð2170Þ. ϕð2170Þ is a 1 tetraquark, sss¯ s¯ can fall apart into two −− This paper is organized as follows. After the Introduction, mesons with the assignments as 1 and þþ þ− −þ in Sec. II, we will explore the decays of ϕð2170Þ.SectionIII 0 or 1 and 0 , is devoted to our conclusion and discussions. 1 j1 i¼ ðj1 i j0 0i þj0 0i j1 i ;Jz 2 ;m 13 ; 24 ; 13 ;m 24 II. FALL-APART DECAYS OF ϕð2170Þ þj1;mi14j0; 0i23 þj0; 0i14j1;mi23Þ; ð4Þ Since ϕð2170Þ of JP ¼ 1− is supposed to be an sss¯ s¯ ¼ tetraquark, which is in a diquark-antiquark configuration, with Jz m. ϕð2170Þ its spin state is can also fall apart into two mesons with the quantum numbers 1þþ and 1−−, jJ; J12;J34i¼j1; 1; 1i; ð1Þ  1 X j1 i¼pffiffiffi j1 ij1 i ;Jz Cm13m24 ;m13 ;m24 where J is the spin of the tetraquark sss¯ s¯, J12 is the spin of 2 ¯ ¯ m13m24 ss, and J34 is the spin of s s. The wave function (color- X  flavor-spin orbit) of the two strange quarks in the diquark þ j1 ij1 i ð Þ Cm14m23 ;m14 ;m23 ; 5 must be totally antisymmetric; for the color-spin there are m14m23 two possibilities: one is that the two s quarks reside in a color antitriplet and their total spin is 1 (constituting a with J13 ¼ J1 þ J3, J24 ¼ J2 þ J4, m13 and m24 being ¼ þ vector), whereas they can also be in a color sextet with the their projections along the Z axis, and Jz m13 m24. total spin zero. In a regular , since the third quark is Cm13m24 and Cm14m23 are corresponding Clebsch-Gordan in the color triplet, only one choice remains; i.e., ss should coefficients. − be in an antitriplet to guarantee the baryon is in the color One also notices that the IG of ϕð2170Þ is 0 ,soforsuch singlet, so the total spin of ss must be 1. However, in a strong Okubo-Zweig-Iizuka allowed decays the two final tetraquark the light diquark with an ½ss content may be in a mesons should more favorably be in IG ¼ 0− and 0þ color-sextet while the other diquark with s¯ s¯ content in an respectively. Of course, the combination of 1−; 1þ could anti-sextet, in that case, the spin of the ss system can be also work, but naively may be suppressed (further discussion zero i.e., it is a scalar or pseudoscalar. But since in the will be presented in the last section). This analysis advocates color-sextet state the QCD interaction between two con- the final states ϕð1020Þf0ð980Þ, ϕð1020Þf0ð500Þ, stituents is repulsive, even though such a sextet configu- ϕð1680Þf0ð500Þ, ωð782Þf0ð980Þ, ωð782Þf0ð500Þ, ration is not strictly forbidden, it is not favorable. The ωð1420Þf0ð500Þ, ωð1650Þf0ð500Þ, h1ð1170Þη,

036014-2 STUDY OF THE STRONG DECAYS OF ϕð2170Þ AND … PHYS. REV. D 99, 036014 (2019)

TABLE I. Some factors for the decay ϕð2170Þ → two mesons.

0 0 Decay mode ϕf0ð980Þ ϕf0ð500Þ h1η h1η ϕη ϕη ωf0ð980Þ K1ð1270ÞKK1ð1400ÞK Color factor p2ffiffi p2ffiffi p2ffiffi p2ffiffi p2ffiffi p2ffiffi p2ffiffi p4 ffiffi p4 ffiffi p3ffiffiffi p3ffiffiffi p3ffiffiffi p3ffiffiffi p3ffiffiffi p3ffiffiffi p3ffiffiffi 3 3 3 3 Spin factor 2 2 2 2 2 2 2 2 2 2 2 2 2 2 44 0 0 00 00 00 00 0 Flavor factor c1 c2 c1 c2 c1 c2 c1 11 Phase space factora 0.0036 0.0056 0.0054 0.0018 0.0062 0.0038 0.0053 0.0050 0.0041 a Γ ¼ 1 jpj jMj2 Ω M The partial decay width is d 32π2 m2 d , where is the hadronic transition amplitude. Supposing it is irrelevant to the solid 1 jpj angle, one can easily integrate the width over the phase space factor 8π m2.

0 0 þ þ h1ð1170Þη ,andωð782Þf1ð1285Þ, which satisfy all the Both ΓðDs → η π Þ and ΓðDs → ηπ Þ have been mea- constraints from matching concerned quantum numbers. sured, and one can obtain the ratio of the rates of the two Γð → η0πþÞ Γð → ηπþÞ¼2 32 In the simple , ðss¯Þ1−− can be decomposed channels as Ds = Ds . . Taking into c1ϕð1020Þþc2ωð782Þ, where the values of c1 ≃ 1 into account the phase space difference 0.82, we obtain the ratio c00=c00 ¼ 1.68. Then, we can use the ratio and the and c2 ≃ 0 are estimated by fitting the decay rates of 2 1 j 00j2 þj 00j2 ¼ 1 ϕð1020Þ → KþK− and ϕð1020Þ → πþπ−[1]. In this pic- required normalization condition c1 c2 to deter- mine the modules of coefficients c00 and c00. The advantage ture, ω only contains a very tiny fraction of strange flavor; 1 2 of using the ratio instead of the widths enables us to avoid thus, those modes involving ωð782Þ in the aforementioned 00 00 some experimental errors. Finally, c2 ¼ 0.89 and c1 ¼ channels would have a very small probability of occurring 0.51 are achieved. However, in this scenario, we cannot 00 00 via the fall-apart mechanism directly, but the channel determine the relative phase between c1 and c2. Using ϕð2170Þ → f0ð980Þω still has a chance to be measured, these values, an estimate on the ratios is made as which we will discuss latter. Γðϕð2170Þ→ϕf0ð980ÞÞ∶Γðϕð2170Þ→h1ηÞ∶Γðϕð2170Þ→ 0 If f0ð980Þ and f0ð500Þ are two normal mesons h1η Þ≃1∶0.4∶0.4. We suggest experimentally searching the 0 ϕð2170Þ → η ϕð2170Þ → η [27–31], the decomposition follows ðss¯Þ0þþ ¼ c1f0ð980Þþ two channels h1 and h1 because 0 they do have substantial branching ratios and should be c f0ð500Þ. Moreover, another relation is ðss¯Þ −þ ¼ 2 0 “ ” 00η þ 00η0 1þ− seen according to our prediction. c1 c2 . For the quantum system, the only candi- ω ð980Þ ð ¯Þ ¼ ð1170Þ Even though f0 cannot be produced via the fall- date is ss 1þ− h1 . With those decompositions, we ¯ ϕð2170Þ apart mechanism, the s from the diquark and s from the may estimate the corresponding decay rates of antidiquark of the tetraquark sss¯ s¯ can annihilate into uu¯ or into the final products involving those mesons via the fall- ¯ dd, while the remaining ss¯ pair forms f0ð980Þ (see Fig. 1); apart mechanism. by this picture, ϕð2170Þ → ωf0ð980Þ still can be seen in It is widely accepted that if the fall-apart mechanism the experiment, but comparing with ϕð2170Þ → ϕf0ð980Þ, exists the dominant decay processes should be determined 2 the mode should be suppressed by α and an additional via this mechanism. Thus, we can estimate the decay rates s color matching factor. of ϕð2170Þ roughly by inputting the coefficients of relevant þ It is noted that transition D → ωπ occurs via weak decompositions, and the relations are listed in the following s interaction, namely, the charm quark turns into s þ ud, Table I. and the spectator s¯ joins the produced s quark; thus, the ss¯ Relevant factors for the decays of ϕð2170Þ are listed in pair annihilates into uu¯ or dd¯. Because of the similarity, Table I. There exists an unknown factor gFA that is the þ phenomenologically, we can use the data of Ds → ϕπ parameter corresponding to the fall-apart mechanism, and it þ and Ds → ωπ to predict the width of ϕð2170Þ → should be universal for all the processes. þ þ 0 0 00 ωf0ð980Þ. Using the ratio ΓðDs → ωπ Þ=ΓðDs → ϕπ Þ¼ At present, accurate values of the coefficients c1, c2, c1, 00 0.053 and taking the corresponding phase factors into and c2 cannot be obtained from data because, so far, there are no measurements with sufficient precision on the relevant processes available. However, we can make rough estimates using the information we have so far. That is what we are doing below. ϕ → f0ð980Þγ exists, but ϕ → f0ð500Þγ does not [1]; 0 this fact implies c1 ≃ 1. Of course, a possibility is that f0ð500Þ is a rather wide resonance, and such a radiative decay would be hard to observe. Anyhow, one can roughly assert that the probability of finding the channel ϕð2170Þ → ϕ ð500Þ f0 may be small, and then we set FIG. 1. The Feynman diagram for sss¯ s¯ → ϕf0ð980Þ transi- 0 c1 ¼ 1. tions via the fall-apart mechanism.

036014-3 HONG-WEI KE and XUE-QIAN LI PHYS. REV. D 99, 036014 (2019)

FIG. 3. The Feynman diagram for sss¯ s¯ → ϕf0ð980Þ transition ¯ ð980Þ FIG. 2. The Feynman diagram for sss¯ s¯ → K1K transitions. as f0 is regarded as a multiquark state.

Γðϕð2170Þ → ω ð980ÞÞ∶Γðϕð2170Þ → account, we have f0 different, and the relevant Feynman diagram is shown in ϕf0ð980ÞÞ ≈ 0.068∶1. Fig. 3. In this case, ϕð2170Þ → ϕf0ð980Þ is suppressed ¯ Along the same line, since there are no valence uu by α2 compared to the aforementioned case in or dd¯ components in the tetraquark sss¯ s¯, ϕð2170Þ cannot s which f0ð980Þ is supposed to be a normal meson. Now, fall apart into K1ð1270ÞK or K1ð1400ÞK.Toproduce the ratio Γðϕð2170Þ→ϕf0ð980ÞÞ∶Γðϕð2170Þ→h1ηÞ∶ K1ð1270ÞK K1ð1400ÞK ss¯ 0 or ,an pair in the tetraquark Γðϕð2170Þ→h1η Þ is close to 1∶1.6∶1.6.However,asis ¯ ¯ annihilates into uu or dd. The leading Feynman diagram is well understood, f0ð980Þ may be a mixture of the ss¯ state Fig. 2. The color and spin factors are presented in Table I, and a multiquark state; thus, according to our estimate, one and the production process is somehow similar to can roughly evaluate the fraction of each constituent, and ϕð2170Þ → ϕ ð980Þ α2 ¯ f0 but is suppressed by s. For the cc this would answer a long-standing question about the α ¯ α system, s is about 0.39 [32], whereas for the ss case, s identity of f0ð980Þ. Obviously, precise measurement on may be slightly larger, but the suppression exists. Moreover, ϕð2170Þ → ϕf0ð980Þ would be very helpful. there are double quark-rearrangements (at initial and final At last, we can estimate the results if ϕð2170Þ is an excited sides); thus, an extra factor gFA is introduced. qqs¯ s¯ tetraquark (q ¼ u, d) as suggested. Naturally, the α ∼ 0 5 ∼ 1 Γðϕð2170Þ→ If we set s . and gFA , we expect following decay portals would dominate the total width of K1ð1270ÞKÞ∶Γðϕð2170Þ→ϕf0ð980ÞÞ∼0.31∶1.Interms ϕð2170Þ; they are qq¯ þ ss¯ and qs¯ þ sq¯, which can be of ΓðΥð4SÞ → Υð1SÞππÞ ≈ ΓðΥð4SÞ → Υð2SÞππÞ,we realized via the fall-apart mechanism. In this case, we expect Γðϕð2170Þ → ð1270Þ Þ estimate K1 K to be close to ϕð2170Þ → ϕf0ð500Þ to be the main decay channel rather Γðϕð2170Þ → K1ð1400ÞKÞ. We should ask whether ¯ than ϕð2170Þ → ϕf0ð980Þ. Γðϕð2170Þ → K1ð1270ÞKÞ and Kð890Þ0K¯ ð890Þ0, KþK−,andK0 K0 can be experimen- ¯ L S Γðϕð2170Þ→K1ð1400ÞKÞ should be close to Γðϕð2170Þ → tally measured. Since the relative orbital angular momentum 0 ¯ 0 ϕf0ð500ÞÞ.Inthiscase,ϕð2170Þ → K ð890Þ K ð890Þ , between the daughter mesons is l ¼ 1, then, because the ϕð2170Þ → KþK− ϕð2170Þ → K0 K0 reactions occur near the threshold, the 3-momentum is small; , and L S only receive a thus, the p-wave suppression will remarkably reduce the p-wave suppression, but not color rearrangement suppres- production rate, compared to the s-wave case. A rough sions, different from the aforementioned case. p2 estimate of the suppression factor is 2 ∼ 0.08. Moreover, to m III. CONCLUSIONS take into account additional factors that may affect evalu- ation, we adopt the suppression factor for the p wave using With the study on the multiquark structures going deeper the data Γðψð2SÞ → ηcγÞ and Γðψð2SÞ → χ0γÞ and where and deeper, many unanswered puzzles in this stimulating the 3-momentum of final mesons is close to that in field have emerged; namely, sharp contradiction between ϕð2170Þ → Kð890Þ0K¯ ð890Þ0. With the ratio Γðψð2SÞ → theoretical prediction and experimental observation reminds ηcγÞ=Γðψð2SÞ → χ0γÞ, the production rate of p wave in the us that our understanding of the exotic is far from case of ϕð2170Þ is suppressed and is about 0.034 times satisfactory. For example, many theoretical models confirm smaller than that for the s wave. Thus, Γðϕð2170Þ → the existence of Xð5568Þ; however, all experimental collab- 0 ¯ 0 K ð890Þ K ð890Þ Þ∶Γðϕð2170Þ → ϕf0ð980ÞÞ is estimated orations have offered negative reports [37–40] except the þ − as ∼0.01∶1. Meanwhile, Γðϕð2170Þ → K K Þ and D0 Collaboration [41]. To compromise the contradiction Γðϕð2170Þ → 0 0Þ Γðϕð2170Þ → KLKS should be close to between theory and experiment, we suggested that a destruc- Kð890Þ0K¯ ð890Þ0Þ. tive interference between the molecular state and tetraquark For other p-wave decays of ϕð2170Þ into ϕη and ϕη0 suppressed the concerned decay portals [42].Eventhough incorporating the phase factors, we estimate Γðϕð2170Þ → the molecular state and tetraquark are different configura- 0 ϕηÞ∶Γðϕð2170Þ → ϕη Þ∶Γðϕð2170Þ → ϕf0ð980ÞÞ ∼ 0.015∶ tions composed of four quarks, they have the same overall 0.025∶1. quantum numbers: spin, parity, etc. Actually, any of them f0ð980Þ may also be considered as a molecular state [33] can convert into the other via a color rearrangement (or, say, or a tetraquark [34–36]; if so, the picture would be slightly via a quark-antiquark swap). Therefore, generally, the two

036014-4 STUDY OF THE STRONG DECAYS OF ϕð2170Þ AND … PHYS. REV. D 99, 036014 (2019) configurations can mix with each other, and the resultant gain more information about the quark model and non- physical would be a mixture of them, i.e., perturbative QCD. As long as the pure qq¯, molecular state, jϕi¼c1jMiþc2jTi, where jϕi; jMi; jTi refer to the exotic and tetraquark have the same overall quantum numbers, meson, molecular state, and tetraquark, respectively. For a they may exist in a mixture. In other words, as an exotic decay, the hadronic transition matrix element can be written meson (or baryon) behaves peculiarly, i.e., cannot be h j jϕi¼ h j j iþ h j j i ¯ as f Heff c1 f Heff M c2 f Heff T , and when we explained by the simple qq structure, one is tempted to square the amplitude, an interference term emerges. That is refer to it as an exotic hadron containing four quarks or a the interference between contributions of the molecular state hybrid state. For a four-quark state, there are two possible and tetraquark. The coefficients c1 and c2 in principle can be configurations: a molecular state and tetraquark. They complex and are determined by nonperturbative QCD. At could mix to result in a physical meson. The mechanism present, we cannot obtain them from the underlying theory inducing the mixing is nonperturbative QCD effects which (the nonperturbative QCD) yet. The interference might be are still not well understood. Even though there are some constructive or destructive, and so far, we can only fix them phenomenological models to approach these effects, in fact, by fitting data, namely, determine if it is destructive via a none of them can be fully satisfactory. At present, we phenomenological study. Indeed, it is a bold conjecture and should try to collect all useful information by studying needs further verification by both theoretical calculations and relevant physical quantities, the mass spectra, production “ ” more accurate experimental measurements. and decay rates, etc., to guess a more reasonable hadronic The decay modes of ϕð2170Þ imply that the assignment of structure. The more information we collect, the closer to being an excited state of ϕ is disfavored. Some authors have reality we will be. Thus, we should try our best to “ ϕð2170Þ suggested that it should be an exotic state. More concretely, investigate more exotic" cases. Moreover, is a sss¯ s¯ special case that might be made of four strange quarks its mass and decay behaviors hint that it may be an ¯ ¯ tetraquark state. Such a structure is special because it may (ssss); it would be relevant to more interesting physics. All decay via the so-called fall-apart mechanism into hadrons of the constituents have strangeness 1 or -1, and the identity that possess dominantly strange constituents. Employing the might simplify the picture and enable us to learn more fall-apart mechanism, we estimate the decay modes of about the physics picture that does not show up for the exotic mesons containing various flavors. ϕð2170Þ, which are supposed to be its dominant portals. We suggest measuring all decay modes of ϕð2170Þ If f0ð980Þ is a simple meson with ss¯ structure, our estimate because the data will inform us of its assignment. If the shows that Γðϕð2170Þ→ϕf0ð980ÞÞ∶Γðϕð2170Þ→h1ηÞ∶Γ 0 data can decide ϕð2170Þ to be an sss¯ s¯ tetraquark, just as ðϕð2170Þ→h1η Þ≃1∶0.4∶0.4, Γðϕð2170Þ → ωf0ð980ÞÞ∶Γ we mentioned in the Introduction, the existence of multi- ðϕð2170Þ → ϕf0ð980ÞÞ ≈ 0.068∶1,andΓðϕð2170Þ → quark state with only s flavor is confirmed and our scope K1ð1270ÞKÞ∶Γðϕð2170Þ → K1ð1400ÞKÞ∶Γðϕð2170Þ → would be widened. ϕf0ð980ÞÞ ∼ 0.31∶0.31∶1.Inthiscase,Γðϕð2170Þ→ ð890Þ0 ¯ ð890Þ0Þ Γðϕð2170Þ → þ −Þ From our discussion, one can notice that to gain more K K , K K , and solid knowledge on the structure of exotic states and Γðϕð2170Þ→ 0 0Þ KLKS are suppressed by about two orders concerned dynamics is not easy because many inputs compared to Γðϕð2170Þ → ϕf0ð980ÞÞ. adopted in the computations possess large errors. This If f0ð980Þ is a four-quark state, the decay ϕð2170Þ → means that accurate data are the precondition for drawing ϕf0ð980Þ is suppressed, and the ratio Γðϕð2170Þ→ 0 definite conclusions. So far, the available facilities cannot ϕf0ð980ÞÞ∶Γðϕð2170Þ→h1ηÞ∶Γðϕð2170Þ→h1η Þ≃ would offer data with satisfactory accuracy in the energy range of be close to 1∶1.6∶1.6. charm; thus, we lay hope in the future charm-tau factory, Supposing ϕð2170Þ is an excited qqs¯ s¯ tetraquark which is planned to be built in China. Since the luminosity ¯ ðq¼u;dÞ, ϕð2170Þ→ϕf0ð500Þ, ϕð2170Þ → K1ð1270ÞK, of the new facility would be enhanced by several orders ¯ and ϕð2170Þ → K1ð1400ÞK are expected to be the more than that of BEPC II and some new detection main decay channels. Even though ϕð2170Þ → technique will be used, we can be optimistic that the Kð890Þ0K¯ ð890Þ0, ϕð2170Þ → KþK−, and ϕð2170Þ → quality of data will be much improved and the statistics can 0 0 reach a very high level. Then, we may renew our compu- KLKS are p-wave suppressed modes, since the 3-momenta for these channels are not too small, they should be tations based on the more accurate data and draw definite observed experimentally. conclusion not only about ϕð2170Þ but also about many Along with all other subjects in the hadron physics, a four-quark states and pentaquarks. better understanding of the exotic state structure and their production and decay mechanisms is badly needed. We all ACKNOWLEDGMENTS know that the fundamental theory of is QCD; however, the nonperturbative QCD that governs the One of us (Ke) would like to acknowledge a fruitful hadron physics is still not yet understood, so various discussion with Doctor Wen-Biao Yan. This study was reasonable phenomenological models are adopted by supported by the National Natural Science Foundation of researchers. The study on exotic states may help us to China under Contracts No. 11375128 and No. 11675082.

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