PHYSICAL REVIEW D 99, 036014 (2019) Study of the strong decays of ϕð2170Þ and the future charm-tau 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-quark state with sss¯ s¯ constituents. Furthermore, there are no two mesons of ss¯ available to form a molecule that fits the mass spectrum of ϕð2170Þ; thus, we suggest it should be an sss¯ s¯ tetraquark 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 → Ãð890Þ0 ¯ Ãð890Þ0 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 hadron of which the mass fits the mass Very recently, a meson 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 KÃ0KÆπÆ and KÃ0K¯ Ã0 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 Ã0 ¯ Ã0 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. baryons [11]. The first proposed pentaquark 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 quarks are light ones suggested that the observed ϕð2170Þ could be a molecular u d ¯ ( or types). The passion of detecting such pentaquarks 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 tetraquarks 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-electron-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 diquark. The orbital angular “heavy” charm quark 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 gluons 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 quantum number 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 baryon, 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.