Is the $Y(2175)$ a Strangeonium Hybrid Meson?

TL;DR

This study uses QCD sum-rules to investigate whether the Y(2175) meson can be a strangeonium hybrid, finding that its weak signal challenges this interpretation.

Contribution

The paper applies Gaussian sum-rules with a two-resonance model to assess the hybrid meson nature of Y(2175), providing new insights into its structure.

Findings

Y(2175) has less than 5% resonance strength compared to a 2.9 GeV state.

Results challenge the interpretation of Y(2175) as a dominant hybrid meson.

Method demonstrates the use of sum-rules for meson structure analysis.

Abstract

QCD Gaussian sum-rules are used to explore the vector ($J^{PC}=1^{--}$) 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 $Y(2175)$.

Figures

Equations

$$\Pi (q^2)=\frac{i}{D-1}\left(\frac{q_{\mu }{q}_{\nu }}{q^2}-g_{\mu \nu }\right)\int d^{D}x{e}^{iq\cdot x}⟨\Omega |\tau j_{\mu }(x)j_{\nu }(0)|\Omega ⟩$$

$$j_{\mu }=\frac{g_s}{2}\overline{s}{\gamma }^{\rho }{\gamma }_5{\lambda }^a{\tilde{G}}_{\mu \rho }^{a}s.$$

$${\tilde{G}}_{\mu \rho }^a=\frac{1}{2}{ϵ}_{\mu \rho \omega \zeta }{G}_{\omega \zeta }^a,$$

$${\Pi }^{\text{QCD}}(q^2)=\left(\frac{{\alpha }_s}{\pi }\left(-\frac{q^6}{240{\pi }^2}+\frac{5m_s^2{q}^4}{48{\pi }^2}-\frac{4q^2}{9}⟨m_s\overline{s}s⟩\right)+\frac{q^2}{36\pi }⟨\alpha G^2⟩+\frac{19{\alpha }_s{m}_s}{72\pi }⟨g\overline{s}\sigma Gs⟩\right)\log \left(\frac{-q^2}{{\mu }^2}\right)$$

$$⟨m_s\overline{s}s⟩=⟨m_s{\overline{s}}_i^{\alpha }{s}_i^{\alpha }⟩$$

$$⟨\alpha G^2⟩=⟨{\alpha }_s{G}_{\mu \nu }^a{G}_{\mu \nu }^a⟩$$

$$⟨g\overline{s}\sigma Gs⟩=⟨g_s{\overline{s}}_i^{\alpha }{\sigma }_{ij}^{\mu \nu }{\lambda }_{\alpha \beta }^a{G}_{\mu \nu }^a{s}_j^{\beta }⟩$$

$$\Pi (Q^2)=\frac{Q^8}{\pi }{\int }_{t_0}^{\infty }\frac{\mathrm{Im}\Pi (t)}{t^4(t+Q^2)}\mathrm{d}t+\cdots$$

$$\frac{1}{\pi }\mathrm{Im}\Pi (t)={\rho }^{\text{had}}(t)+\theta (t-s_0)\frac{1}{\pi }\mathrm{Im}{\Pi }^{\text{QCD}}(t),$$

$$G(\hat{s},\,\tau)=\sqrt{\frac{\tau}{\pi}}\lim_{\stackrel{{\scriptstyle N,\Delta^{2}\rightarrow\infty}}{{\tau=\Delta^{2}/(4N)}}}\frac{\big{(}-\Delta^{2}\big{)}^{N}}{\Gamma(N)}\bigg{(}\frac{d}{d\Delta^{2}}\bigg{)}^{N}\left\{\frac{\Pi(-\hat{s}-i\Delta)-\Pi(-\hat{s}+i\Delta)}{i\Delta}\right\}.$$

$$G^{\text{QCD}}(\hat{s},\tau )$$

$$⟹G^{\text{QCD}}(\hat{s},\tau )$$

$$\frac{1}{\sqrt{4\pi \tau }}{\int }_{s_0}^{\infty }{e}^{-\frac{(\hat{s}-t{)}^2}{4\tau }}\frac{1}{\pi }\mathrm{Im}{\Pi }^{\text{QCD}}(t)\mathrm{d}t,$$

$$G^{\text{QCD}}(\hat{s},\tau ,s_0)$$

$$⟹G^{\text{QCD}}(\hat{s},\tau ,s_0)$$

$$G^{\text{QCD}}(\hat{s},\tau ,s_0)\equiv \frac{1}{\sqrt{4\pi \tau }}{\int }_0^{s_0}{e}^{-\frac{(\hat{s}-t{)}^2}{4\tau }}\left(\frac{{\alpha }_s}{\pi }\left(-\frac{t^3}{240{\pi }^2}+\frac{5m_s^2{t}^2}{48{\pi }^2}-\frac{4t}{9}⟨m_s\overline{s}s⟩\right)+\frac{t}{36\pi }⟨\alpha G^2⟩+\frac{19{\alpha }_s{m}_s}{72\pi }⟨g\overline{s}\sigma Gs⟩\right)\mathrm{d}t.$$

$${\alpha }_s(\mu )=\frac{{\alpha }_s\left(M_{\tau }\right)}{1+\frac{25}{12\pi }{\alpha }_s(M_{\tau })\log \left(\frac{{\mu }^2}{M_{\tau }^2}\right)}$$

$$M_{\tau }=1.77686\pm 0.00012\text{GeV}$$

$${\alpha }_s\left(M_{\tau }\right)=0.325\pm \mathrm{0.015.}$$

$$m_s(\mu )=m_s(2\text{GeV}){\left(\frac{{\alpha }_s(\mu )}{{\alpha }_s(2\text{GeV})}\right)}^{\frac{12}{25}}$$

$$m_s(2\text{GeV})=96_{-4}^{+8}\text{MeV}.$$

$$⟨m_s\overline{s}s⟩=-\frac{1}{2}{f}_K^2{m}_K^2,$$

$$m_K=(493.677\pm 0.016)\text{MeV}$$

$$f_K=(110.0\pm 4.2)\text{MeV}$$

$$⟨\alpha G^2⟩=(0.075\pm 0.020)\text{GeV}.$$

$$\frac{m_s⟨g\overline{s}\sigma Gs⟩}{⟨m_s\overline{s}s⟩}\equiv M_0^2=(0.8\pm 0.1){\text{GeV}}^2.$$

$${\int }_{-\infty }^{\infty }{G}^{\text{QCD}}(\hat{s},\tau ,s_0)\mathrm{d}\hat{s}={\int }_{t_0}^{\infty }{\rho }^{\text{had}}(t)\mathrm{d}t.$$

$$N^{\text{QCD}}(\hat{s},\tau ,s_0)\equiv \frac{G^{\text{QCD}}(\hat{s},\tau ,s_0)}{{\int }_{-\infty }^{\infty }{G}^{\text{QCD}}(\hat{s},\tau ,s_0)\mathrm{d}\hat{s}}$$

$$N^{\text{QCD}}(\hat{s},\tau ,s_0)=\frac{\frac{1}{\sqrt{4\pi \tau }}{\int }_{t_0}^{\infty }{e}^{-\frac{(\hat{s}-t{)}^2}{4\tau }}{\rho }^{\text{had}}(t)\mathrm{d}t}{{\int }_{t_0}^{\infty }{\rho }^{\text{had}}(t)\mathrm{d}t}.$$

$${\rho }^{\text{had}}(t)=f_1^2\delta \left(t-m_1^2\right)+f_2^2\delta \left(t-m_2^2\right).$$

$$N^{\text{had}}\left(\hat{s},\,\tau\right)=\frac{1}{\sqrt{4\pi\tau}}\Bigg{(}re^{-\frac{(\hat{s}-m_{1}^{2})^{2}}{4\tau}}+(1-r)e^{-\frac{(\hat{s}-m_{2}^{2})^{2}}{4\tau}}\Bigg{)},$$

$$r=\frac{f_1^2}{f_1^2+f_2^2},1-r=\frac{f_2^2}{f_1^2+f_2^2},0\le r\le 1.$$

$$m_1=m_{Y(2175)}=2.188\text{GeV},$$

Full text

I Introduction

The initial state radiation (ISR) process in $e^{+}e^{-}$ annihilation is a very useful technique to search for vector states (i.e., $J^{PC}=1^{--}$) in B-factories. In 2006, the BaBar Collaboration studied the cross sections for the ISR processes $e^{+}e^{-}\to K^{+}K^{-}\pi^{+}\pi^{-}$ and $e^{+}e^{-}\to K^{+}K^{-}\pi^{0}\pi^{0}$ up to $4.5\,\text{GeV}$, aiming to confirm the existence of the $Y(4260)$ in the $\phi\pi\pi$ channels. Instead of observing the $Y(4260)$, however, they found a new resonance structure in the $\phi(1020)f_{0}(980)$ channel, which was named the $Y(2175)$ Aubert et al. (2006). (It is also known as the $\phi(2170)$ Tanabashi et al. (2018)). This resonance was later confirmed by BaBar Aubert et al. (2007, 2008); Lees et al. (2012), BES Ablikim et al. (2008), and Belle Shen et al. (2009) and recently by BESIII Ablikim et al. (2015, 2017a). Its mass and decay width are $M=(2188\pm 10)\,\text{MeV}$ and $\Gamma=(83\pm 12)\,\text{MeV}$ and its quantum numbers are $I^{G}J^{PC}=0^{-}1^{--}$ Tanabashi et al. (2018).

To date, the nature of the $Y(2175)$ is still unknown. Based on strange quarkonium mass predictions using a relativized potential model, only the $3\,{}^{3}S_{1}$ and $2\,{}^{3}D_{1}$ $s\bar{s}$ states are expected to have masses close to that of the $Y(2175)$ Godfrey and Isgur (1985). However, both interpretations are disfavoured as the corresponding resonance width predictions are significantly larger than the width of the $Y(2175)$. The width of the $3\,{}^{3}S_{1}$ $s\bar{s}$ state was predicted to be $378\,\text{MeV}$ using the ${}^{3}P_{0}$ decay model Barnes et al. (2003) whereas the width of the $2\,{}^{3}D_{1}$ $s\bar{s}$ state was predicted to be $167\,\text{MeV}$ in the ${}^{3}P_{0}$ model and $212\,\text{MeV}$ in the flux tube breaking model Ding and Yan (2007a). Another possible interpretation of the $Y(2175)$ is that of a strangeonium hybrid meson (i.e., $\bar{s}gs$). Masses of vector strangeonium hybrid mesons have been computed using several methodologies including the flux tube model Merlin and Paton (1985); Isgur and Paton (1985); Merlin and Paton (1987); Barnes et al. (1995), lattice QCD Dudek (2011), and QCD Laplace sum-rules (LSRs) Govaerts et al. (1985). The flux tube model calculation of Barnes et al. (1995) found a vector strangeonium hybrid mass of 2.1–2.2 GeV. The lattice QCD analysis of Dudek (2011) found a vector strangeonium hybrid mass between $2.4\,\text{GeV}$ and $2.5\,\text{GeV}$ while the LSRs calculation of Govaerts et al. (1985) found a heavier mass of $(2.9\pm 0.3)\,\text{GeV}$. Yet another possible interpretation of the $Y(2175)$ is that of a $ss\bar{s}\bar{s}$ tetraquark. In Chen et al. (2018a), the masses of vector $ss\bar{s}\bar{s}$ tetraquarks were investigated. Two states were predicted with respective masses $(2.34\pm 0.17)\,\text{GeV}$ and $(2.41\pm 0.25)\,\text{GeV}$. Other LSRs analyses of $ss\bar{s}\bar{s}$ tetraquarks can be found in Wang (2007); Chen et al. (2008). Furthermore, the $Y(2175)$ has also been proposed as a molecular state of $\Lambda\bar{\Lambda}$ Abud et al. (2010); Zhao et al. (2013). In Abud et al. (2010), a chromomagnetic interaction Hamiltonian was used to predict a hexaquark of mass $2.184\,\text{GeV}$ that is strongly coupled to the $\Lambda\bar{\Lambda}$ channel. In Zhao et al. (2013), a one-boson-exchange potential model was used to predict a $\Lambda\bar{\Lambda}$ mass between $2.149\,\text{GeV}$ and $2.181\,\text{GeV}$. Also, the $Y(2175)$ has been interpreted as a dynamically generated resonance of $\phi f_{0}(980)$ Napsuciale et al. (2007); Martinez Torres et al. (2008); Alvarez-Ruso et al. (2009); Gomez-Avila et al. (2009).

Decay modes and rates will be crucial to determining the nature of the $Y(2175)$. In Barnes et al. (2003), it was predicted using the ${}^{3}P_{0}$ model that the most important decay modes of the $3\,{}^{3}S_{1}$ $s\bar{s}$ meson would be $K^{*}K^{*}$, $KK^{*}(1410)$, and $KK_{1}(1270)$ whereas the $KK$ mode would be very weak. In Ding and Yan (2007a), it was predicted using the ${}^{3}P_{0}$ model that the most important decay modes of the $2\,{}^{3}D_{1}$ $s\bar{s}$ meson would be $KK(1460)$, $KK^{*}(1410)$, $KK_{1}(1270)$, and $K^{*}K^{*}$. While not dominant, the $KK$ decay mode was predicted to have a partial width of about 0.06. In Ding and Yan (2007b), it was predicted using flux tube and constituent gluon models that the most important decay modes of a vector strangeonium hybrid would be $KK_{1}(1400)$, $KK_{1}(1270)$, $KK^{*}(1410)$, and $KK_{2}(1430)$, each containing a $S$-wave meson plus a $P$-wave meson, due to the $S+P$ selection rule Isgur et al. (1985); Page et al. (1999); Ding and Yan (2007b). Also, it was noted that the $\phi f_{0}(980)$ mode could be significant. Of particular interest are the $KK$, $K^{*}K^{*}$ and $KK(1460)$ modes which are predicted to be zero for a strangeonium hybrid interpretation (the usual rule that suppresses or even forbids hybrid decays to pairs of S-wave mesons Isgur et al. (1985); Close and Page (1995); Page (1997)). For the $ss\bar{s}\bar{s}$ tetraquark interpretation, it has been suggested that the $\eta\phi$ channel should be one of the dominant decay modes due to the large phase space in the fall-apart mechanism Ding and Yan (2007b). However, in Ke and Li (2018), it was argued that the $\eta\phi$ decay mode would be greatly suppressed and that the $\phi f_{0}(980)$, $h_{1}\eta$, and $h_{1}\eta^{\prime}$ modes would be most important. For the $\Lambda\bar{\Lambda}$ interpretation of the $Y(2175)$, the $KK$ decay mode was predicted to dominate Dong et al. (2017). At present, the data concerning decay modes and rates of the $Y(2175)$ is incomplete, making it difficult to draw any definitive conclusions Tanabashi et al. (2018).

As they are both observed in ISR processes, the $Y(4260)$ and $Y(2175)$ states have the same quantum numbers, and are often considered as analogous states in the hidden-charm and hidden-strange sectors respectively Aubert et al. (2005, 2006); Ablikim et al. (2017b). Perhaps determining the nature of one will shed light on the other. Since the $Y(4260)$ has been identified as a good candidate for charmonium hybrid $\bar{c}gc$ Zhu (2005); Kou and Pene (2005); Palameta et al. (2018) or hidden-charm tetraquark state $qc\bar{q}\bar{c}$ Chen et al. (2016), the $Y(2175)$ meson may also be interpreted as a hybrid or tetraquark candidate.

In this work, we use QCD Gaussian sum-rules (GSRs) methods to study the strangeonium hybrid possibility for $Y(2175)$. In contrast to previous analyses of strangeonium hybrids using LSRs Govaerts et al. (1985), the use of GSRs enables an exploration of the possibility of multiple states with hybrid components, allowing us to examine the scenario of a hybrid component of the $Y(2175)$. We find little evidence in support of the $Y(2175)$ having a significant strangeonium hybrid component.

II The Correlator and Gaussian Sum-Rules

We investigate vector strangeonium hybrids through the correlator

[TABLE]

where $D$ is spacetime dimension and where the current $j_{\mu}$ is given by

[TABLE]

In (2), $s$ is a strange quark field and $\tilde{G}^{a}_{\mu\rho}$ is the dual gluon field strength tensor,

[TABLE]

defined in terms of the Levi-Civita symbol, $\epsilon_{\mu\rho\omega\zeta}$.

Between Govaerts et al. (1985) and Ho et al. (2018), the quantity $\Pi(q^{2})$ from (1) has been computed to leading-order (LO) in $\alpha_{s}=\frac{g_{s}^{2}}{4\pi}$ within the operator product expansion (OPE). In Govaerts et al. (1985), the perturbative and dimension-four (i.e. 4d) quark and gluon condensate contributions were calculated. In Ho et al. (2018), the 5d mixed, 6d quark, and 6d gluon condensate contributions were calculated as well as $\mathcal{O}(m_{s}^{2})$ corrections to perturbation theory where $m_{s}$ is the strange quark mass. Denoting the result as $\Pi^{\text{QCD}}(q^{2})$ to emphasize that it is a QCD calculation, we have

[TABLE]

where

[TABLE]

are respectively the 4d strange quark condensate, the 4d gluon condensate, and the 5d mixed strange quark condensate. In (5)–(7), subscripts on strange quarks are Dirac indices, superscripts are colour indices, and $\sigma^{\mu\nu}=\frac{i}{4}[\gamma^{\mu},\,\gamma^{\nu}]$. In computing (4), divergent integrals were handled through dimensional regularization in $D=4+2\epsilon$ dimensions at $\overline{\text{MS}}$-renormalization scale $\mu$. A dimensionally regularized $\gamma_{5}$ satisfying $\{\gamma_{5},\gamma^{\mu}\}=0$ and $\gamma_{5}^{2}=1$ was used following the prescription of Chanowitz et al. (1979). Also, TARCER Mertig and Scharf (1998), a Mathematica implementation of the recurrence relations of Tarasov (1996, 1997), was employed to reduce the set of needed integral results to a small, well-known collection. An irrelevant polynomial in $q^{2}$ has been omitted from (4) as it ultimately does not contribute to the GSRs used in this article (see below). Included in this omitted polynomial are the 6d quark and gluon condensate contributions, both of which are constant for this channel as discussed in Ho et al. (2018).

The quantity $\Pi(q^{2})$ in (1) is related to its imaginary part, i.e., the hadronic spectral function, through a dispersion relation

[TABLE]

at Euclidean momentum $Q^{2}\equiv-q^{2}>0$. In (8), $t_{0}$ is a hadron production threshold and $\cdots$ represents subtraction constants, collectively a third degree polynomial in $Q^{2}$. On the left-hand side of (8), we identify $\Pi$ with $\Pi^{\text{QCD}}$ of (4). On the right-hand side, we partition the hadronic spectral function using a resonance-plus-continuum decomposition,

[TABLE]

where $\rho^{\text{had}}(t)$ represents the resonance contribution to $\mathrm{Im}\Pi(t)$ and $\theta(t-s_{0})$ is a Heaviside step function shifted to the continuum threshold parameter $s_{0}$.

In (8), to eliminate subtraction constants as well as the aforementioned polynomials omitted from (4) and to enhance the resonance contribution relative to the continuum contribution to the integral on the right-hand side, some transform is typically applied leading to some corresponding variant of QCD sum-rules. Laplace sum-rules, for example, are a common choice (e.g., see Shifman et al. (1979a, b); Reinders et al. (1985); Narison (2007)). Here, we instead choose to work with (lowest-weight) GSRs defined as Bertlmann et al. (1985)

[TABLE]

Discussions of how to evaluate definition (10) for a correlator such as (4) can be found in Bertlmann et al. (1985); Orlandini et al. (2001); Harnett and Steele (2001). Substituting (9) into (8) and applying (10), we find

[TABLE]

Subtracting the continuum contribution,

[TABLE]

from (11) and (12) leads to subtracted GSRs

[TABLE]

Finally, calculating $\mathrm{Im}\Pi^{\text{QCD}}(t)$ from (4) and substituting the result into the right-hand side of (14), we find

[TABLE]

Note that the definite integral in (16) can be evaluated in terms of error functions. The kernel of the subtracted GSRs is a Gaussian of width $\sqrt{2\tau}$ centred at $\hat{s}$. As discussed in Orlandini et al. (2001); Harnett and Steele (2001); Harnett et al. (2011); Ho et al. (2018), GSRs are particularly well-suited to the study of multi-resonance hadron models as, by varying $\hat{s}$, excited and ground state resonances can be probed with similar sensitivity.

Renormalization-group (RG) improvement of (16) amounts to replacing $\alpha_{s}$ and $m_{s}$ by running quantities at the scale $\mu^{2}=\sqrt{\tau}$ (e.g., Bertlmann et al. (1985); Narison and de Rafael (1981)). The one-loop, $\overline{\text{MS}}$ running coupling at $n_{f}=4$ active quark flavours is

[TABLE]

where Tanabashi et al. (2018)

[TABLE]

Since the previous analysis of strangeonium hybrid mesons using QCD sum-rules Govaerts et al. (1985), the condensate parameters and quark masses are now known more precisely. In addition to the inclusion of higher-dimensional condensates terms in (4), we update the values and uncertainties in the QCD parameters used in Govaerts et al. (1985). The running strange quark mass is

[TABLE]

where Tanabashi et al. (2018)

[TABLE]

The value of the RG-invariant 4d strange quark condensate is known from PCAC,

[TABLE]

where Tanabashi et al. (2018); Rosner et al. (2015)

[TABLE]

For the 4d gluon condensate, we use the value from Narison (2012),

[TABLE]

For the 5d mixed strange quark condensate, we use the estimate from Beneke and Dosch (1992); Belyaev and Ioffe (1982),

[TABLE]

Integrating (15) with respect to $\hat{s}$ gives

[TABLE]

The quantity on the LHS of (27) is the lowest-weight finite energy sum-rule (FESR), and, as shown in Bertlmann et al. (1985), the spectral function decomposition (9) only reproduces the QCD prediction at high energy scales if $s_{0}$ is constrained by (27). To isolate the information in the GSRs that is independent of the FESR constraint (27), we define normalized GSRs (NGSRs) Orlandini et al. (2001),

[TABLE]

which, from (15) and (27), implies that

[TABLE]

III Analysis Methodology and Results

Previous work using LSRs used a single-narrow resonance model and resulted in a strangeonium hybrid mass prediction significantly heavier than the $Y(2175)$ Govaerts et al. (1985). Compared with that analysis, we include 5d and 6d condensate terms in the OPE and use updated QCD parameter values. Also, as outlined above, Gaussian sum-rules have the ability to probe multiple states in the spectral function. We can therefore update and extend the analysis of Ref. Govaerts et al. (1985) and test the hypothesis of a $Y(2175)$ hybrid component by using a double-narrow resonance model for the hadronic spectral function

[TABLE]

This double narrow-resonance model in (15) provides the hadronic contribution, i.e., the right-hand side, to the NGSRs (29),

[TABLE]

where the normalized couplings are defined as

[TABLE]

We fix one of our modelled resonances ($m_{1}$) using the experimental value given in Refs. Tanabashi et al. (2018); Chen et al. (2018b),

[TABLE]

and the additional resonance ($m_{2}$) provides the necessary degrees of freedom in the model for the possibility that the $Y(2175)$ decouples (i.e., that $m_{1}$ has normalized coupling $r\approx 0$).

We choose the width of the Gaussian kernel to be $\tau=10\,\text{GeV}^{4},$ in line with our previous GSRs analysis of light hybrids Ho et al. (2018). Since this resolution is much larger than the experimental width of the $Y(2175)$ (i.e., $\sqrt{\tau}\gg m_{1}\Gamma$), the narrow width model is an excellent approximation for the $Y(2175)$. For the undetermined resonance $m_{2}$, we assume that it is similarly narrow compared to the Gaussian kernel resolution; this assumption is revisited in the results of our analysis presented below. To determine the remaining unknown quantities $\{m_{2},r,s_{0}\}$ in our model we seek the best fit of the the $\hat{s}$ dependence of the QCD prediction and hadronic model by minimizing the $\chi^{2}$,

[TABLE]

where we use 161 equally spaced $\hat{s}$ points with $\hat{s}_{\text{min}}=-10\,\text{GeV}^{4}$ and $\hat{s}_{\text{max}}=30\,\text{GeV}^{4}$. This region safely encloses the resonances resulting from our analysis as outlined below. Note that the minimization is constrained by the physical condition $0\leq r\leq 1$ in (32). The resulting prediction of the resonance parameters and continuum onset is

[TABLE]

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 $\langle\alpha G^{2}\rangle$. An upper bound on $r$ is provided because of the $r\geq 0$ constraint. Figure 1 shows that the fit between the QCD prediction and hadronic model is excellent; there is no evidence of any deviations that would suggest a need to refine the model (e.g., inclusion of a numerically-large width $\sqrt{\tau}\sim m_{2}\Gamma$ for $m_{2}$). Figure 1 also shows that the fitted region $-10\,\text{GeV}^{2}<\hat{s}<30\,\text{GeV}^{2}$ encloses the regions where the NGSRs are numerically significant. As a further validation of our results, we note that our mass prediction for $m_{2}$ is consistent with previous LSRs analyses Govaerts et al. (1985).

The key aspect of our results (35)–(37) is the small relative resonance strength $r\leq 3.3\%$ of the $Y(2175)$ compared to $m_{2}$, which seems to preclude a predominant hybrid component of the $Y(2175)$. We can obtain a more conservative bound on $r$ by calculating the $s_{0}$ dependence of $r$ (i.e., choosing $s_{0}$ and only fitting $r$ and $m_{2}$) and then considering the variation of $r$ within the region of uncertainty in $s_{0}$ from (35). The result of this analysis leads to the bound $r\leq 5\%$ as shown in Figure 2. A similar analysis for $m_{2}$ is shown in Figure 3.

IV Discussion

In summary, we have used QCD GSRs to study the strangeonium hybrid interpretation of the $Y(2175)$. Compared to a previous LSRs analysis of vector strangeonium hybrids Govaerts et al. (1985), our calculation includes 5d and 6d condensate contributions, strange quark mass corrections to perturbation theory, and updated QCD parameter values. Furthermore, the advantage of the GSRs approach over the 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 $Y(2175)$ and an additional unknown resonance. We found excellent agreement between the QCD prediction and hadronic model, and determined an upper bound $r\leq 5\%$ for the relative coupling strength of the $Y(2175)$. In other words, we found no evidence for a significant strangeonium hybrid component of the $Y(2175)$.

Recently, a structure of mass $(2239\pm 13.3)$ MeV and width $(139.8\pm 24.0)$ MeV (where we have combined statistical and systematic uncertainties) was observed in $e^{+}\,e^{-}\,\rightarrow\,K^{+}\,K^{-}$ with the BES III detector Ablikim et al. (2019). If the structure can be identified with the $Y(2175)$, then the observed $KK$ decay mode would disfavour the $3\,{}^{3}S_{1}$ strangeonium meson, strangeonium hybrid, and $ss\bar{s}\bar{s}$ tetraquark interpretations. On the other hand, if the structure can not be identified with the $Y(2175)$, then the lack of observed $KK$ decay mode would disfavour the $2\,{}^{3}D_{1}$ strangeonium meson and $\Lambda\bar{\Lambda}$ interpretations. Clearly, further experimental and theoretical studies are needed.

Acknowledgments

We are grateful for financial support from the Natural Sciences and Engineering Research Council of Canada (NSERC), and the Chinese National Youth Thousand Talents Program.