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Is the Y (2175) a Strangeonium Hybrid ?

Jason Ho Dordt University, [email protected]

R. Berg University of Saskatchewan

T. G. Steele University of Saskatchewan

W. Chen Sun Yat-sen University

D. Harnett University of the Fraser Valley

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Recommended Citation Ho, J., Berg, R., Steele, T. G., Chen, W., & Harnett, D. (2019). Is the Y (2175) a Strangeonium Hybrid Meson?. Physical Review D, 100 (3), 034012. https://doi.org/10.1103/PhysRevD.100.034012

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Abstract QCD Gaussian sum rules are used to explore the vector (JPC=1−−) strangeonium hybrid interpretation of the Y(2175). Using a two-resonance model consisting of the Y(2175) and an additional resonance, we find that the relative resonance strength of the Y(2175) in the Gaussian sum rules is less than 5% that of a heavier 2.9 GeV state. This small relative strength presents a challenge to a dominantly hybrid interpretation of the Y(2175).

Keywords quantum chromodynamics, exotic , hybrid mesons, mass

Disciplines Physics

Comments Access on publisher's site: https://journals.aps.org/prd/pdf/10.1103/PhysRevD.100.034012

This article is available at Digital Collections @ Dordt: https://digitalcollections.dordt.edu/faculty_work/1239 PHYSICAL REVIEW D 100, 034012 (2019)

Is the Yð2175Þ a strangeonium hybrid meson?

J. Ho, R. Berg, and T. G. Steele Department of Physics and Engineering Physics University of Saskatchewan Saskatoon, Saskatchewan, S7N 5E2, Canada

W. Chen School of Physics Sun Yat-Sen University Guangzhou 510275, China

D. Harnett Department of Physics University of the Fraser Valley Abbotsford, British Columbia, V2S 7M8, Canada

(Received 18 June 2019; published 12 August 2019)

QCD Gaussian sum rules are used to explore the vector (JPC ¼ 1−−) strangeonium hybrid interpretation of the Yð2175Þ. Using a two-resonance model consisting of the Yð2175Þ and an additional resonance, we find that the relative resonance strength of the Yð2175Þ in the Gaussian sum rules is less than 5% that of a heavier 2.9 GeV state. This small relative strength presents a challenge to a dominantly hybrid interpretation of the Yð2175Þ.

DOI: 10.1103/PhysRevD.100.034012

3 I. INTRODUCTION was predicted to be 167 MeV in the P0 model and The initial state radiation (ISR) process in eþe− anni- 212 MeV in the flux tube breaking model [12]. Another Y 2175 hilation is a very useful technique to search for vector states possible interpretation of the ð Þ is that of a strang- −− eonium hybrid meson (i.e., sgs¯ ). Masses of vector strang- (i.e., JPC ¼ 1 ) in B-factories. In 2006, the BABAR eonium hybrid mesons have been computed using several Collaboration studied the cross sections for the ISR methodologies including the flux tube model [13–16], processes eþe− → KþK−πþπ− and eþe− → KþK−π0π0 lattice QCD [17], and QCD Laplace sum rules (LSRs) up to 4.5 GeV, aiming to confirm the existence of the [18]. The flux tube model calculation of [16] found a vector Yð4260Þ in the ϕππ channels. Instead of observing the strangeonium hybrid mass of 2.1–2.2 GeV. The lattice Yð4260Þ, however, they found a new resonance structure in QCD analysis of [17] found a vector strangeonium the ϕð1020Þf0ð980Þ channel, which was named the hybrid mass between 2.4 GeV and 2.5 GeV while the Yð2175Þ [1]. (It is also known as the ϕð2170Þ [2]). This LSRs calculation of [18] found a heavier mass of resonance was later confirmed by BABAR [3–5],BES[6], ð2.9 0.3Þ GeV. Yet another possible interpretation of and Belle [7] and recently by BESIII [8,9]. Its mass and the Yð2175Þ is that of a sss¯s¯ . In [19], the masses decay width are M ¼ð2188 10Þ MeV and Γ ¼ð83 of vector sss¯s¯ were investigated. Two states 12 MeV and its quantum numbers are IGJPC 0−1−− [2]. Þ ¼ were predicted with respective masses ð2.34 0.17Þ GeV To date, the nature of the Y 2175 is still unknown. ð Þ and ð2.41 0.25Þ GeV. Other LSRs analyses of sss¯s¯ Based on strange mass predictions using a tetraquarks can be found in [20,21]. Furthermore, the 33 23 ¯ relativized potential model, only the S1 and D1 ss states Yð2175Þ has also been proposed as a molecular state of are expected to have masses close to that of the Y 2175 ð Þ ΛΛ¯ [22,23].In[22], a chromomagnetic interaction [10]. However, both interpretations are disfavored as the Hamiltonian was used to predict a hexaquark of mass corresponding resonance width predictions are signifi- 2.184 GeV that is strongly coupled to the ΛΛ¯ channel. In cantly larger than the width of the Yð2175Þ. The width 3 [23], a one--exchange potential model was used to of the 3 S1 ss¯ state was predicted to be 378 MeV using the ¯ 3 3 predict a ΛΛ mass between 2.149 GeV and 2.181 GeV. P0 decay model [11] whereas the width of the 2 D1 ss¯ state Also, the Yð2175Þ has been interpreted as a dynamically generated resonance of ϕf0ð980Þ [24–27]. Decay modes and rates will be crucial to determining the Published by the American Physical Society under the terms of nature of the Yð2175Þ.In[11], it was predicted using the the Creative Commons Attribution 4.0 International license. 3 33 Further distribution of this work must maintain attribution to P0 model that the most important decay modes of the S1 ¯ the author(s) and the published article’s title, journal citation, ss meson would be K K , KK ð1410Þ, and KK1ð1270Þ and DOI. Funded by SCOAP3. whereas the KK mode would be very weak. In [12],itwas

2470-0010=2019=100(3)=034012(7) 034012-1 Published by the American Physical Society HO, BERG, STEELE, CHEN, and HARNETT PHYS. REV. D 100, 034012 (2019)

3 gs ρ a ˜ a predicted using the P0 model that the most important jμ ¼ s¯γ γ5λ Gμρs: ð2Þ 3 2 decay modes of the 2 D1 ss¯ meson would be KKð1460Þ, 1410 1270 KK ð Þ, KK1ð Þ, and K K . While not dominant, ˜ a In (2), s is a strange field and Gμρ is the dual the KK decay mode was predicted to have a partial width of about 0.06. In [28], it was predicted using flux tube and field strength tensor, constituent gluon models that the most important decay 1 ˜ a a modes of a vector strangeonium hybrid would be Gμρ ¼ ϵμρωζGωζ; ð3Þ 2 KK1ð1400Þ, KK1ð1270Þ, KK ð1410Þ, and KK2ð1430Þ, each containing a S-wave meson plus a P-wave meson, defined in terms of the Levi-Civita symbol, ϵμρωζ. – due to the S þ P selection rule [28 30]. Also, it was noted Between [18] and [41], the quantity Πðq2Þ from (1) has that the ϕf0ð980Þ mode could be significant. Of particular 2 been computed to leading-order (LO) in α gs within the interest are the KK, KK and KKð1460Þ modes which are s ¼ 4π predicted to be zero for a strangeonium hybrid interpreta- operator product expansion. In [18], the perturbative and tion (the usual rule that suppresses or even forbids hybrid dimension-four (i.e., 4d) quark and gluon condensate decays to pairs of S-wave mesons [29,31,32]). For the sss¯s¯ contributions were calculated. In [41], the 5d mixed, 6d tetraquark interpretation, it has been suggested that the ηϕ quark, and 6d gluon condensate contributions were calcu- O 2 channel should be one of the dominant decay modes due to lated as well as ðms Þ corrections to perturbation theory the large phase space in the fall-apart mechanism [28]. where ms is the mass. Denoting the result as ΠQCD 2 However, in [33], it was argued that the ηϕ decay mode ðq Þ to emphasize that it is a QCD calculation, we ϕ 980 η have would be greatly suppressed and that the f0ð Þ, h1 , 0 ¯ 6 2 4 2 and h1η modes would be most important. For the ΛΛ α q 5m q 4q ΠQCD 2 s − s − ¯ interpretation of the Yð2175Þ, the KK decay mode was ðq Þ¼ π 240π2 þ 48π2 9 hmsssi predicted to dominate [34]. At present, the data concerning 2 19α − 2 decay modes and rates of the Yð2175Þ is incomplete, q α 2 sms ¯σ q þ h G iþ hgs Gsi log 2 making it difficult to draw any definitive conclusions [2]. 36π 72π μ 4260 As they are both observed in ISR processes, the Yð Þ ð4Þ and Yð2175Þ states have the same quantum numbers, and are often considered as analogous states in the hidden- where charm and hidden-strange sectors, respectively [1,35,36]. ¯ ¯α α Perhaps determining the nature of one will shed light on the hmsssi¼hmssi si ið5Þ other. Since the Yð4260Þ has been identified as a good α 2 α a a candidate for charmonium hybrid cgc¯ [37–39] or hidden- h G i¼h sGμνGμνið6Þ charm tetraquark state qcq¯c¯ [40], the Yð2175Þ meson may ¯σ ¯ασμνλa a β also be interpreted as a hybrid or tetraquark candidate. hgs Gsi¼hgssi ij αβGμνsj ið7Þ In this work, we use QCD Gaussian sum rules (GSRs) methods to study the strangeonium hybrid possibility for are respectively the 4d strange quark condensate, the 4d Yð2175Þ. In contrast to previous analyses of strangeonium gluon condensate, and the 5d mixed strange quark con- hybrids using LSRs [18], the use of GSRs enables an densate. In (5)–(7), subscripts on strange are Dirac μν i μ ν exploration of the possibility of multiple states with hybrid indices, superscripts are color indices, and σ ¼ 4 ½γ ; γ . components, allowing us to examine the scenario of a In computing (4), divergent integrals were handled through hybrid component of the Yð2175Þ. We find little evidence dimensional regularization in D ¼ 4 þ 2ϵ dimensions at in support of the Yð2175Þ having a significant strangeo- MS-renormalization scale μ. A dimensionally regularized μ 2 nium hybrid component. γ5 satisfying fγ5; γ g¼0 and γ5 ¼ 1 was used following the prescription of [42]. Also, TARCER [43],aMathematica II. THE CORRELATOR AND implementation of the recurrence relations of [44,45],was GAUSSIAN SUM RULES employed to reduce the set of needed integral results to a small, well-known collection. An irrelevant polynomial in We investigate vector strangeonium hybrids through the 2 q has been omitted from (4) as it ultimately does not correlator Z contribute to the GSRs used in this article (see below). 2 i qμqν D iq·x Included in this omitted polynomial are the 6d quark and Πðq Þ¼ −gμν d xe hΩjτjμðxÞjνð0ÞjΩi D−1 q2 gluon condensate contributions, both of which are constant for this channel as discussed in [41]. ð1Þ 2 The quantity Πðq Þ in (1) is related to its imaginary part, where D is spacetime dimension and where the current jμ is i.e., the hadronic spectral function, through a dispersion given by relation

034012-2 IS THE Yð2175Þ A STRANGEONIUM HYBRID MESON? PHYS. REV. D 100, 034012 (2019) Z Q8 ∞ ImΠðtÞ where ρhadðtÞ represents the resonance contribution to ΠðQ2Þ¼ dt þ ð8Þ π 4 2 ImΠðtÞ and θðt − s0Þ is a Heaviside step function shifted t0 t ðt þ Q Þ to the continuum threshold parameter s0. 2 2 at Euclidean momentum Q ≡ −q > 0.In(8), t0 is a In (8), to eliminate subtraction constants as well as the production threshold and represents subtraction aforementioned polynomials omitted from (4) and to constants, collectively a third degree polynomial in Q2.On enhance the resonance contribution relative to the con- the left-hand side of (8), we identify Π with ΠQCD of (4).On tinuum contribution to the integral on the right-hand side, the right-hand side, we partition the hadronic spectral some transform is typically applied leading to some function using a resonance-plus-continuum decomposition, corresponding variant of QCD sum rules. Laplace sum rules, for example, are a common choice (e.g., see [46– 1 1 49]). Here, we instead choose to work with (lowest-weight) Π ρhad θ − ΠQCD π Im ðtÞ¼ ðtÞþ ðt s0Þ π Im ðtÞ; ð9Þ GSRs defined as [50]

rffiffiffi   τ ð−Δ2ÞN d N Πð−sˆ − iΔÞ − Πð−sˆ þ iΔÞ Gðs;ˆ τÞ¼ lim : ð10Þ 2 2 π N;Δ →∞ ΓðNÞ dΔ iΔ τ¼Δ2=ð4NÞ

Discussions of how to evaluate definition (10) for a correlator such as (4) can be found in [50–52]. Substituting (9) into (8) and applying (10), we find Z ∞ 2 1 −ðsˆ−tÞ 1 GQCDðs;ˆ τÞ ≡ pffiffiffiffiffiffiffiffi e 4τ ImΠQCDðtÞdt ð11Þ 4πτ 0 π Z Z ∞ 2 ∞ 2 1 −ðsˆ−tÞ 1 −ðsˆ−tÞ 1 ⇒ QCD ˆ τ ffiffiffiffiffiffiffiffi 4τ ρhad ffiffiffiffiffiffiffiffi 4τ ΠQCD G ðs; Þ¼p e ðtÞdt þ p e π Im ðtÞdt: ð12Þ 4πτ t0 4πτ s0

Subtracting the continuum contribution, Z ∞ 2 1 −ðsˆ−tÞ 1 ffiffiffiffiffiffiffiffi 4τ ΠQCD p e π Im ðtÞdt; ð13Þ 4πτ s0 from (11) and (12) leads to subtracted GSRs Z 1 s0 ˆ− 2 1 QCD −ðs tÞ QCD G ðs;ˆ τ;s0Þ ≡ pffiffiffiffiffiffiffiffi e 4τ ImΠ ðtÞdt ð14Þ 4πτ 0 π Z 1 ∞ ˆ− 2 QCD −ðs tÞ had ⇒ G ðs;ˆ τ;s0Þ¼pffiffiffiffiffiffiffiffi e 4τ ρ ðtÞdt: ð15Þ 4πτ t0

Finally, calculating ImΠQCDðtÞ from (4) and substituting the result into the right-hand side of (14), we find

Z 3 2 2 s0 2 1 −ðsˆ−tÞ αs t 5ms t 4t t 2 19αsms QCD ˆ τ ≡ ffiffiffiffiffiffiffiffi 4τ − − ¯ α ¯σ G ðs; ;s0Þ p e 2 þ 2 hmsssi þ h G iþ hgs Gsi dt: ð16Þ 4πτ 0 π 240π 48π 9 36π 72π

Note that the definite integral in (16) can be evaluated in Renormalization-group (RG) improvement of (16) amounts to replacing α and m by running quantities at terms of error functions.p Theffiffiffiffiffi kernel of the subtracted GSRs pffiffiffi s s is a Gaussian of width 2τ centred at sˆ. As discussed in the scale μ2 ¼ τ (e.g., [50,54]). The one-loop, MS [41,51–53], GSRs are particularly well suited to the study running coupling at nf ¼ 4 active quark flavors is of multiresonance hadron models as, by varying sˆ, excited α ðMτÞ and ground state resonances can be probed with similar α μ s sð Þ¼ 25 μ2 ð17Þ 1 þ α ðMτÞ logð 2Þ sensitivity. 12π s Mτ

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QCD where [2] G s;ˆ τ;s0 QCD ˆ τ ≡ R ð Þ N ðs; ;s0Þ ∞ QCD ð28Þ −∞ G ðs;ˆ τ;s0Þdsˆ Mτ ¼ 1.77686 0.00012 GeV ð18Þ which, from (15) and (27), implies that α 0 325 0 015 R 2 sðMτÞ¼ . . : ð19Þ 1 ∞ −ðsˆ−tÞ pffiffiffiffiffi e 4τ ρhadðtÞdt QCD 4πτ t0 N ðs;ˆ τ;s0Þ¼ R : ð29Þ Since the previous analysis of strangeonium hybrid mesons ∞ ρhad t0 ðtÞdt using QCD sum rules [18], the condensate parameters and quark masses are now known more precisely. In addition to III. ANALYSIS METHODOLOGY AND RESULTS the inclusion of higher-dimensional condensates terms in (4), we update the values and uncertainties in the QCD Previous work using LSRs used a single-narrow reso- parameters used in [18]. The running strange quark mass is nance model and resulted in a strangeonium hybrid mass prediction significantly heavier than the Yð2175Þ [18]. 12 α ðμÞ 25 Compared with that analysis, we include 5d and 6d m ðμÞ¼m ð2 GeVÞ s ð20Þ s s α ð2 GeVÞ condensate terms in the operator product expansion and s use updated QCD parameter values. Also, as outlined where [2] above, Gaussian sum rules have the ability to probe multiple states in the spectral function. We can therefore 8 update and extend the analysis of Ref. [18] and test the m ð2 GeVÞ¼96þ MeV: ð21Þ s −4 hypothesis of a Yð2175Þ hybrid component by using a double-narrow resonance model for the hadronic spectral The value of the RG-invariant 4d strange quark con- function densate is known from the partially conserved axial current, had 2 2 2 2 1 ρ ðtÞ¼f1δðt − m1Þþf2δðt − m2Þ: ð30Þ ¯ − 2 2 hmsssi¼ 2 fKmK; ð22Þ This double narrow-resonance model in (15) provides the where [2,55] hadronic contribution, i.e., the right-hand side, to the NGSRs (29), 493 677 0 016 mK ¼ð . . Þ MeV ð23Þ ˆ− 2 2 ˆ− 2 2 1 −ðs m1Þ −ðs m2Þ Nhadðs;ˆ τÞ¼pffiffiffiffiffiffiffiffi re 4τ þð1 − rÞe 4τ ; ð31Þ 4πτ fK ¼ð110.0 4.2Þ MeV ð24Þ where the normalized couplings are defined as For the 4d gluon condensate, we use the value from [56], 2 2 f1 f2 2 1 − 0 ≤ ≤ 1 hαG i¼ð0.075 0.020Þ GeV: ð25Þ r ¼ 2 2 ; r ¼ 2 2 ; r : ð32Þ f1 þ f2 f1 þ f2

For the 5d mixed strange quark condensate, we use the We fix one of our modeled resonances (m1) using the estimate from [57,58], experimental value given in Refs. [2,59],

m gs¯σGs sh i 2 2 m1 ¼ mY 2175 ¼ 2.188 GeV; ð33Þ ≡ M0 ¼ð0.8 0.1Þ GeV : ð26Þ ð Þ hmsss¯ i and the additional resonance (m2) provides the necessary Integrating (15) with respect to sˆ gives degrees of freedom in the model for the possibility that the Z Z Yð2175Þ decouples (i.e., that m1 has normalized cou- ∞ ∞ QCD had pling r ≈ 0). G ðs;ˆ τ;s0Þdsˆ ¼ ρ ðtÞdt: ð27Þ −∞ t0 We choose the width of the Gaussian kernel to be τ ¼ 10 GeV4, in line with our previous GSRs analysis The quantity on the left-hand side (LHS) of (27) is the of light hybrids [41]. Since this resolution is much larger than the experimental width of the Y 2175 (i.e., lowest-weight finite energy sum rule (FESR), and, as pffiffiffi ð Þ shown in [50], the spectral function decomposition (9) τ ≫ m1Γ), the narrow width model is an excellent only reproduces the QCD prediction at high energy scales if approximation for the Yð2175Þ. For the undetermined s0 is constrained by (27). To isolate the information in the resonance m2, we assume that it is similarly narrow GSRs that is independent of the FESR constraint (27),we compared to the Gaussian kernel resolution; this define normalized GSRs (NGSRs) [51], assumption is revisited in the results of our analysis

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had QCD FIG. 1. Double-narrow resonance model N ðs;ˆ τÞ (solid blue curve) and compared to QCD prediction N ðs;ˆ τ;s0Þ (orange points) for τ ¼ 10 GeV4. Central values of the QCD condensates and the corresponding fitted parameters have been used. presented below. To determine the remaining unknown significant. As a further validation of our results, we note quantities fm2;r;s0g in our model we seek the best fit of that our mass prediction for m2 is consistent with previous the sˆ dependence of the QCD prediction and hadronic LSRs analyses [18]. model by minimizing the χ2, The key aspect of our results (35)–(37) is the small relative resonance strength r ≤ 3.3% of the Yð2175Þ sˆ Xmax compared to m2, which seems to preclude a predominant χ2 had ˆ τ − QCD ˆ τ 2 ðr; m2;s0Þ¼ ½N ðs; Þ N ðs; s0Þ ; ð34Þ hybrid component of the Yð2175Þ. We can obtain a more sˆ min conservative bound on r by calculating the s0 dependence ˆ ˆ of r (i.e., choosing s0 and only fitting r and m2) and then where we use 161 equally spaced s points with smin ¼ −10 4 ˆ 30 4 considering the variation of r within the region of uncer- GeV and smax ¼ GeV . This region safely enc- tainty in s0 from (35). The result of this analysis leads to the loses the resonances resulting from our analysis as out- ≤ 5% lined below. Note that the minimization is constrained by bound r as shown in Fig. 2. A similar analysis for m2 the physical condition 0 ≤ r ≤ 1 in (32). The resulting is shown in Fig. 3. prediction of the resonance parameters and continuum onset is

opt 2 s0 ¼ 9.7 1.0 GeV ð35Þ

2 90 0 16 m2 ¼ mfit ¼ . . GeV ð36Þ

r ≤ 0.033: ð37Þ

The uncertainties in (35)–(37) are obtained by varying the values of the QCD input parameters, and calculating the deviation from the central values in quadrature. Errors are dominated by the variation in hαG2i. An upper bound on r is provided because of the r ≥ 0 constraint. Figure 1 shows that the fit between the QCD prediction and hadronic model is excellent; there is no evidence of any deviations that FIG. 2. Predicted coupling r to Yð2175Þ state as a function of would suggest a need to refine the model (e.g., inclusion of pffiffiffi the continuum onset s0. The vertical band highlights the a numerically large width τ ∼ m2Γ for m2). Figure 1 also optimized value of continuum onset sopt with corresponding 2 2 0 shows that the fitted region −10 GeV < s<ˆ 30 GeV error (35). The physical constraint r>0 has been imposed in the encloses the regions where the NGSRs are numerically analysis.

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LSRs approach is its comparable sensitivity to multiple states in a hadronic spectral function. This allowed us to explore the relative coupling to the hybrid current (2) of the Yð2175Þ and an additional unknown resonance. We found excellent agreement between the QCD prediction and hadronic model, and determined an upper bound r ≤ 5% for the relative coupling strength of the Yð2175Þ. In other words, we found no evidence for a significant strangeo- nium hybrid component of the Yð2175Þ. Recently, a structure of mass ð2239 13.3Þ MeV and width ð139.8 24.0Þ MeV (where we have combined statistical and systematic uncertainties) was observed in eþe− → KþK− with the BES III detector [60]. If the structure can be identified with the Yð2175Þ, then the 3 FIG. 3. Predicted vector strangeonium hybrid mass m2 as a observed KK decay mode would disfavor the 3 S1 strang- function of the continuum onset s0. The vertical band highlights sss¯s¯ opt eonium meson, strangeonium hybrid, and tetraquark the optimized value of continuum onset s0 with corresponding interpretations. On the other hand, if the structure cannot be error (35). identified with the Yð2175Þ, then the lack of observed KK 3 decay mode would disfavor the 2 D1 strangeonium meson IV. DISCUSSION and ΛΛ¯ interpretations. Clearly, further experimental and theoretical studies are needed. In summary, we have used QCD GSRs to study the strangeonium hybrid interpretation of the Yð2175Þ. ACKNOWLEDGMENTS Compared to a previous LSRs analysis of vector strang- eonium hybrids [18], our calculation includes 5d and 6d We are grateful for financial support from the Natural condensate contributions, strange quark mass corrections to Sciences and Engineering Research Council of Canada perturbation theory, and updated QCD parameter values. (NSERC), and the Chinese National Youth Thousand Furthermore, the advantage of the GSRs approach over the Talents Program.

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