A possible partner state of the $Y(2175)$

We study the $Y(2175)$ using the method of QCD sum rules. There are two independent $ss\bar s\bar s$ interpolating currents with $J^{PC} = 1^{--}$, and we calculate both their diagonal and off-diagonal correlation functions. We obtain two new currents which do not strongly correlate to each other, so they may couple to two different physical states: one of them couples to the $Y(2175)$, while the other may couple to another state whose mass is about $71 ^{+172}_{-~48}$ MeV larger. Evidences of the latter state can be found in the BaBar, BESII, Belle, and BESIII experiments.


I. INTRODUCTION
In recent years there have been lots of exotic hadrons observed in hadron experiments [1], which can not be explained in the traditional quark model and are of particular importance to understand the low energy behaviours of Quantum Chromodynamics (QCD) [2][3][4][5][6][7][8]. Most of them contain heavy quarks, such as the charmonium-like XY Z states, while there are not so many exotic hadrons in the light sector only containing light u/d/s quarks. The Y (2175) is one of them, which is often taken as the strange analogue of the Y (4220) [9,10].
Besides the Y (2175), there might be another structure in the φf 0 (980) invariant mass spectrum at around 2.4 GeV, whose evidences can be found in the BaBar [12] [12] determined its mass and width to be 2.47 ± 0.07 GeV and 77 ± 65 MeV, respectively. Shen and Yuan [19] also used the BaBar [11,12] and Belle [16] data to fit its mass and width to be 2436 ± 34 MeV and 99±105 MeV, respectively. However, its statistical significance is smaller than 3.0σ. In this paper we shall study this structure as well as the Y (2175) simultaneously using the method of QCD sum rules.
Since its discovery, the Y (2175) has attracted much attention from the hadron physics community, and many theoretical methods and models were applied to study it. By using both the chiral unitary model [20,21] and the Faddeev equations [22], the authors interpreted the Y (2175) as a dynamically generated state in the φKK and φππ systems, and more states were predicted in the φπ 0 η [23] and N KK [24,25] systems. By using similar approaches, the Y (2175) was interpreted as a dynamically generated resonance by the self-interactions between the φ and f 0 (980) resonances [26], while the resonance spectrum expansion formalism by including the f 0 (980) as a resonance in the coupled ππ-KK system is also able to generate the Y (2175) in the φf 0 (980) channel [27].
Besides the dynamically generated resonance, there are many other interpretations to explain this structure. In Ref. [28] the authors interpreted the Y (2175) as a 2 3 D 1 ss meson, and calculated its decay modes using both the 3 P 0 model and the flux-tube model. In Ref. [29] the authors used a constituent quark model to interpret the Y (2175) as a hidden-strangeness baryon-antibaryon state (qqsqqs) strongly coupling to the ΛΛ channel. Later in Ref. [30] the authors applied the one-boson-exchange model to interpret the Y (2175) and η(2225) as the bound states of The e + e − → φf 0 (980) cross section. Taken from BaBar [11].
ΛΛ( 3 S 1 ) and ΛΛ( 1 S 0 ), respectively. In Ref. [31] the authors interpreted the Y (2175) as a strangeonium hybrid state and used the flux-tube model to study its decay properties. However, this interpretation is not supported by the non-perturbative lattice QCD calculations [32]. Productions of the Y (2175) were studied in Refs. [33,34] by using the Nambu-Jona-Lasinio model and the Drell-Yan mechanism, while its decay properties were studied in Refs. [35,36] via the initial single pion emission mechanism.
The method of QCD sum rules was also applied to study the Y (2175) [37,38], which method has been widely and successfully used to study hadron properties [39,40]. When using this method to investigate a physical state, one needs to construct the relevant interpolating current, but we still do not fully understand their relations: a) the interpolating current sees only the quantum numbers of the physical state, so it can also couple to some other physical states as well as the relevant threshold; b) one can sometimes construct more than one interpolating currents, all of which can couple to the same physical state. Some previous studies tell us that: 1. In Ref. [41] we systematically studied the P -wave singly heavy baryons. Theoretically, we find that they have rich internal structures, and there can be as many as three excited Ω c states of J P = 1/2 − , three of J P = 3/2 − , and one of J P = 5/2 − . For each state we can construct one interpolating current having the same internal structure. The numbers of excited Λ c /Ξ c /Σ c /Ξ ′ c states/currents are the same.
We do not know which of them exist in nature, but we do know that, experimentally, the P -wave charmed baryons also have rich structures [42], for example, the LHCb experiment [43] observed as many as five excited Ω c states at the same time, all of which can be P -wave charmed baryons.
3. In Ref. [49] we systematically studied hidden-charm pentaquark states having spin J = 1 2 / 3 2 / 5 2 . We constructed hundreds of hidden-charm pentaquark interpolating currents, suggesting that their internal structures are rather complicated. However, the only observed hidden-charm pentaquark states so far are the P c (4380) and P c (4450) [50], and it is unbelievable that there exist hundreds of hiddencharm pentaquark states in nature.
Hence, the internal structures of (exotic) hadrons are complicated. For each internal structure we can construct the relevant interpolating current, and their relations are also complicated. Especially, there can be many interpolating currents when studying exotic hadrons, which makes them not easy to handle.
To clarify this problem, a good subject is to study the Y (2175) of J P C = 1 −− . The relevant ssss interpolating currents have been systematically constructed in Ref. [38], and there are only two independent ones. We have separately used them to perform QCD sum rule analyses, both of which can be used to explain the Y (2175). However, in Ref. [38] we only calculated the diagonal terms of these two currents, and in this work we shall further calculate their off-diagonal term to study their correlation. This can significantly improve our understanding on the relations between interpolating currents and physical states.
Another advantage to study the Y (2175) is that, experimentally, there might be another structure in the φf 0 (980) invariant mass spectrum at around 2.4 GeV, as we have discussed before. It is quite interesting to study the relations between the two independent ssss currents with J P C = 1 −− and the two possible structures in the φf 0 (980) invariant mass spectrum, both theoretically and experimentally, and both coherently and incoherently. Note that there are many charmonium-like Y states of J P C = 1 −− , so it is natural to think that there can be more than one Y states in the light sector.
This paper is organized as follows. In Sec. II, we list the two independent ssss interpolating currents with J P C = 1 −− , and discuss how to diagonalize them. In Sec. III, we use two diquark-antidiquark (ss)(ss) interpolating currents to perform QCD sum rule analyses, and obtain two new currents which do not strongly correlate to each other. In Sec. IV, we use these two new currents to calculate mass spectra, and Sec. V is a summary.

II. INTERPOLATING CURRENTS AND THEIR RELATIONS TO POSSIBLE PHYSICAL STATES
The interpolating currents having the quark content ssss and with the quantum number J P C = 1 −− have been systematically constructed in Ref. [38]. We briefly summarize the results here and discuss their relations to possible physical states.
1. There are two non-vanishing diquark-antidiquark (ss)(ss) interpolating currents with J P C = 1 −− : where a and b are color indices, C = iγ 2 γ 0 is the charge conjugation operator, and the superscript T represents the transpose of Dirac indices. These two currents are independent of each other.
2. There are four non-vanishing meson-meson (ss)(ss) interpolating currents with J P C = 1 −− : However, only two of them are independent.
3. When using local currents, we can verify the following relations between the above (ss)(ss) and (ss)(ss) currents through the Fierz transformation: In Ref. [38] we have separately used η 1µ and η 2µ to perform QCD sum rule analyses, i.e., we have calculated the diagonal terms: However, although η 1µ and η 2µ are independent of each other, they can be correlated to each other, i.e., the offdiagonal term can be non-zero: suggesting that η 1µ and η 2µ may couple to the same physical state. In this paper we shall evaluate this off-diagonal term in order to find two non-correlated currents: Then we shall use J 1µ and J 2µ to perform QCD sum rule analyses. Due to the above Eq. (11), J 1µ and J 2µ should not strongly couple to the same physical state, so we assume where Y 1 and Y 2 are two different states with J P C = 1 −− , f 1 and f 2 are decay constants, and ǫ µ is the polarization vector. Especially, we shall evaluate the mass splitting between these two states/currents.

III. QCD SUM RULE ANALYSIS
The method of QCD sum rules is a powerful and successful non-perturbative method [39,40]. In this method, we calculate the two-point correlation function at both the hadron and quark-gluon levels.
At the hadron level we simplify its Lorentz structure to be: and express Π(q 2 ) in the form of the dispersion relation: Here ρ(s) is the spectral density, for which we adopt a parametrization of one pole dominance for the ground state Y and a continuum contribution: At the quark-gluon level, we insert J 1µ and J 2µ into Eq. (14), and calculate the correlation function using the method of operator product expansion (OPE). After performing the Borel transformation at both the hadron and quark-gluon levels, we obtain After approximating the continuum using the spectral density of OPE above a threshold value s 0 , we obtain the sum rule equation We can use this equation to calculate M Y through The sum rules for the currents η 1µ and η 2µ have been separately calculated and given in Eqs. (13) and (14) of Ref. [38]. In this paper we revise these calculations by adding the diagram shown in Fig. 2. We write them as Π η1η1 (q 2 ) and Π η2η2 (q 2 ) in the present study, which are transformed to be Π η1η1 (M 2 B ) and Π η2η2 (M 2 B ) after the Borel transformation. The results are shown in Eqs. (24) and (25), which do not change significantly compared to Ref. [38]. In Eqs. (24) and (25) we have calculated the OPE up to twelve dimension, including the strange quark mass, the perturbative term, the quark condensate ss , the gluon condensate g 2 s GG , the quark-gluon mixed condensate g ss σGs , and their combinations ss 2 , ss 3 , ss 4 , g ss σGs 2 , ss g ss σGs , ss 2 g ss σGs , g 2 s GG ss , g 2 s GG ss 2 , g 2 s GG g ss σGs , and g 2 s GG ss g ss σGs .

IV. NUMERICAL ANALYSIS
In this section we use the currents J 1µ and J 2µ to perform QCD sum rule analyses. Take J 1µ as an example. First we study the convergence of the operator product expansion, which is the cornerstone of the reliable QCD sum rule analysis. To do this we require that the D = 10 and D = 12 terms be less than 5%: After fixing s 0 = 6.0 GeV 2 , we find that this condition is satisfied when M 2 B is larger than 2.0 GeV 2 . We also show the relative contribution of each term to the correlation function Π J1J1 (M 2 B ) in Fig. 4. We find that in the region 2.0 GeV 2 < M 2 B < 4.0 GeV 2 , the perturbative term (D = 0) gives the most important contribution, and the convergence is quite good.
A common problem, when studying multiquark states using QCD sum rules, is how to differentiate the multiquark state and the relevant threshold, because the interpolating current can couple to both of them. For the case of the Y (2175), its relevant threshold is the φf 0 (980) around 2.0 GeV, which J 1µ and J 2µ can both couple to. Moreover, the Y (2175) is not the lowest state in the 1 −− channel containing ss, and J 1µ and J 2µ may also couple to the φ(1680) (for example, see the Belle experiment [16] observing the φ(1680) and Y (2175) at the same time).
If this happens, the resulting correlation function should be positive. Fortunately, we find that the correla- This fact indicates that J 1µ and J 2µ both couple weakly to the lower state φ(1680) as well as the φf 0 (980) threshold, so the states they couple to, as if they can couple to some states, should be new and possibly exotic states. However, due to the above negative contributions to the correlation functions, the pole contribution is not large enough. This small pole contribution also suggests that the continuum contribution is important, which demands a careful choice of the parameters of QCD sum rules. Accordingly, in the present study we require that the extracted mass have a dual minimum dependence on both the threshold value s 0 and the Borel mass M B .
, as functions of the Borel mass MB, when taking s0 = 6.0 GeV 2 . Right: Still using J 1µ as an example, we show the mass obtained using Eq. (20) as a function of the threshold value s 0 and the Borel mass M B in Fig. 6. We find that there is a mass minimum at around 2.4 GeV when taking s 0 to be around 6.0 GeV 2 , and at the same time the Borel mass dependence is weak at around 3.0 GeV 2 . Accordingly, we fix s 0 to be around 6.0 GeV 2 and M 2 B to be around 3.0 GeV 2 , and choose our working regions to be 5.0 GeV 2 < s 0 < 7.0 GeV 2 and 2.0 GeV 2 < M 2 B < 4.0 GeV 2 . These regions are moderately large enough for the mass prediction, where the mass is extracted to be Here the uncertainty is due to the Borel mass M B , the threshold value s 0 , and various condensates [1,[51][52][53][54][55][56][57].
Similarly, we use J 2µ to perform QCD sum rule analyses. Choosing the same working regions 5.0 GeV 2 < s 0 < 7.0 GeV 2 and 2.0 GeV 2 < M 2 B < 4.0 GeV 2 , the mass is extracted to be The above result is shown in Fig. 7 as a function of the threshold value s 0 and the Borel mass M B .
As we have discussed in previous sections, J 1µ and J 2µ may couple to two different physical states. Using the same working region, we evaluate the mass splitting between these two states/currents to be The above result is shown in Fig. 8 as a function of the threshold value s 0 and the Borel mass M B .

V. SUMMARY AND DISCUSSIONS
In this work we apply the method of QCD sum rules to study the Y (2175) by using local ssss interpolating currents with J P C = 1 −− . The relevant diquarkantidiquark (ss)(ss) and meson-meson (ss)(ss) interpolating currents have been systematically constructed in Ref. [38], where their relations have also been derived. There we found two independent currents, so there are (at least) two different internal structures. In Ref. [38] we have calculated the two diagonal terms using the two diquark-antidiquark (ss)(ss) currents η 1µ and η 2µ , and in this work we further calculate their off-diagonal term We find two new currents J 1µ and J 2µ with the mixing angle θ = −5.0 o : J 2µ = sin θ η 1µ + cos θ i η 2µ .
These two currents do not strongly correlate to each other, suggesting that they may couple to different physical states.
We use J 1µ and J 2µ to perform QCD sum rule analyses. Especially, we find that J 1µ and J 2µ both couple weakly to the lower state φ(1680) as well as the φf 0 (980) threshold, so the states they couple to, as if they can couple to some states, should be new and possibly exotic states. Accordingly, we assume J 1µ and J 2µ separately couple to two different states with the same quantum number J P C = 1 −− , whose masses are extracted to be These results do not change significantly compared with those obtained in Ref. [38]. However, their mass splitting depend significantly on the mixing angle, and we use J 1µ and J 2µ with θ = −5.0 o to evaluate it to be The mass extracted using J 2µ is consistent with the experimental mass of the Y (2175), suggesting that J 2µ may couple to the Y (2175); while the mass extracted using J 1µ is a bit larger, suggesting that the Y (2175) may have a partner state whose mass is around 71 +172 − 48 MeV larger. Because J 1µ and J 2µ are two ssss interpolating currents with J P C = 1 −− , both the Y (2175) and its possible partner state should be vector mesons containing large strangeness components. Note that our results do not definitely suggest that they are ssss tetraquark states, because the interpolating current sees only the quantum numbers of the physical state, that is J P C = 1 −− . We can further use Eq. (7), which is derived from the Fierz transformation, to obtain that the Y (2175) and its possible partner state can both be observed in the φf 0 (980) channel, while the latter may also be observed in the φf 1 (1420) channel, as if kinematically allowed.
Experimentally, the Y (2175) has been well established by the BaBar, BESII, BESIII, and Belle experiments. Besides it, there might be another structure in the φf 0 (980) invariant mass spectrum at around 2.4 GeV. This might be the partner state of the Y (2175), which is coupled by the current J 1µ . To end this paper, we note that the two mass values we obtained, 2.34 ± 0.17 GeV  In the left panel, the short-dashed/solid/long-dashed curves are obtained by setting M 2 B = 2.0/3.0/4.0 GeV 2 , respectively. In the right panel, the short-dashed/solid/long-dashed curves are obtained by setting s0 = 5.5/6.0/6.5 GeV 2 , respectively. and 2.41±0.25 GeV, are both around 2.4 GeV, indicating that there might be even more complicated structures in this region, such as two coherent resonances. We also note that there are many charmonium-like Y states of J P C = 1 −− , so it is natural to think that there can be more than one Y states in the light sector. Accordingly, we propose to carefully study the structure in the φf 0 (980) invariant mass spectrum at around 2.4 GeV in future experiments.