Vector hidden-bottom tetraquark candidate: Y(10750)

In this article, we take the scalar diquark and antidiquark operators as the basic constituents, and construct the type tetraquark current to study Y(10750) with the QCD sum rules. The predicted mass and width support the assignment of Y(10750) as the diquark-antidiquark type vector hidden-bottom tetraquark state, with a relative P-wave between the diquark and antidiquark constituents.


Introduction
Y(10750) 6.7σ e + e − → Υ(nS )π + π − n = 1, 2, 3 10.52 11.02 GeV Recently, the Belle collaboration observed a resonance structure with the global significance of in the ( ) cross-section at energies from to using the data collected with the Belle detector at the KEKB asymmetric energy collider [1]. The Breit-Wigner mass and width are and MeV, respectively.
is observed in the processes ( ), and its quantum numbers may be . In the famous Godfrey-Isgur model, the nearby bottomonium states are , and with masses , and , respectively [2], while in the QCD motivated relativistic quark model based on the quasipotential approach (the screened potential model), the corresponding masses are , and ( , and [3]), respectively [4]. Without introducing the mixing effects, the experimental mass cannot be reproduced if we assign as a conventional bottomonium state [5]. scalar (or axial-vector) antidiquark operators explicitly to construct the vector tetraquark current operators, and calculated the masses and pole residues of the vector hiddencharm tetraquark states using the QCD sum rules in a systematic way. We obtained the lowest masses of the vector hidden-charm tetraquark states up to now. Our predictions support the assignment of the exotic states , , and as the vector tetraquarks with quantum numbers , which originate from the relative P-wave between the diquark and antidiquark constituents. On the other hand, if we take the scalar ( -type), pseudoscalar (C-type), vector ( -type) and axial-vector ( -type) diquark operators as the basic constituents, and construct the vector tetraquark current operators with the quantum numbers without introducing the relative P-wave between the diquark and antidiquark constituents, we can obtain the masses of the lowest vector tetraquark states, which are about or [8]. These values are larger or much larger than the measured mass of by the BESIII collaboration [9,10], because the pseudoscalar and vector diquarks are not the favored quark configurations [8]. In Ref. [11], we took the scalar and axial-vector diquark (and antidiquark) operators as the basic constituents to construct the current operators, calculated the masses and pole residues of the hidden-bottom tetraquark states with the quantum numbers , , and systematically using the QCD sum rules, and found that the masses of the hidden- 10 In the present work, we tentatively assign as a diquark-antidiquark vector hidden-bottom tetraquark state with the quantum numbers , and construct the type tetraquark current operator to calculate its mass and pole residue using the QCD sum rules. In the calculations, we take into account the vacuum condensates up to dimension 10 in the operator product expansion, as in our previous works. Furthermore, we study the two-body strong decays of the vector hidden-bottom tetraquark candidate with the three-point correlation functions by carrying out the operator product expansion up to the vacuum condensates of dimension . In the calculations, we take into account both the connected and disconnected Feynman diagrams.
The paper is organized as follows. In Section 2, we obtain the QCD sum rules for the mass and pole residue of . In Section 3, we obtain the QCD sum rules for the hadronic coupling constants in the strong decays of , and then obtain the partial decay widths. Section 4 gives a short conclusion. rent operator couples to the diquark-antidiquark type vector hidden-bottom tetraquark states which have degenerate masses. In the present work, we choose for simplicity.
The scattering amplitude for one-gluon exchange is proportional to and is the Gell-Mann matrix. The negative (positive) sign in front of the antisymmetric antitriplet (symmetric sextet ) indicates that the interaction is attractive (repulsive), which favors (disfavors) formation of diquarks in the color antitriplet (color sextet ). We prefer diquark operators in the color antitriplet to diquark operators in the color sextet to construct the tetraquark current operators and interpolate the lowest tetraquark states.
On the phenomenological side, we take into account the non-vanishing current-hadron couplings with the same quantum numbers, and separate the contribution of the ground state vector hidden-bottom tetraquark state in the correlation function [12][13][14], which is supposed to be , where the pole residue is defined by , and is the polarization vector.
On the QCD side, we carry out the operator product expansion up to the vacuum condensates of dimension 10 in a consistent way, and take into account the vacuum condensates , , , , , , and . We then obtain the QCD spectral density using the dispersion relation, take the quark-hadron duality below the continuum threshold , and perform the Borel transform to obtain the QCD sum rules: .
The explicit expression of the QCD spectral density and the technical details for calculating the Feynman diagrams can be found in Refs. [6,15].

Y(10750)
We obtain the QCD sum rules for the mass of the vector hidden-bottom tetraquark candidate as the ratio, ⟨qq⟩ = −(0.24± 0.01 GeV) 3 ⟨qg s σGq⟩ = m 2 0 ⟨qq⟩ at the energy scale [12][13][14]16], take the mass listed in "The Review of Particle Physics" [17], and set the u and d quark masses to zero. Furthermore, we take into account the energy-scale dependence of the parameters on the QCD side from the renormalization group equation [18,19], , and , and for the flavors 5, 4 and 3, respectively [17]. As we study the vector hidden-bottom tetraquark state, it is better to choose the flavor and then evolve all input parameters to the ideal energy scale . The Borel parameter is a free parameter. The continuum threshold parameter is also a free parameter, but we can borrow some ideas from the mass spectrum of the conventional mesons and the established exotic mesons to put additional constraints on so as to avoid contamination from the excited and continuum states. In the conventional QCD sum rules, there are two basic criteria (i.e. "pole dominance on the hadron side" and "convergence of the operator product expansion") that need to be obeyed. In the QCD sum rules for the multiquark states, we add two additional criteria, (i.e. "appearance of the flat Borel platforms" and "satisfying the modified energy scale formula"), since in the QCD sum rules for the conventional mesons and baryons we cannot obtain very flat Borel platforms due to the lack of higher dimensional vacuum condensates to stabilize the QCD sum rules. We search for the optimal values of the two parameters that satisfy the four criteria by trial and error.
In Refs. [15,20,21], we studied the hidden-charm and hidden-bottom tetraquark states (which consist of a diquark-antidiquark pair in a relative S-wave) with the QCD sum rules, and explored for the first time the energy scale dependence of the extracted masses and pole residues.
In the heavy quark limit , the heavy quark Q serves as a static well potential and attracts the light quark q to form a diquark in the color antitriplet , while Qq 3 c the heavy antiquark serves as another static well potential and attracts the light antiquark to form an antidiquark in the color triplet . The diquark and antidiquark then attract each other to form a compact tetraquark state.
The favored heavy diquark configurations are the scalar and axial-vector diquark operators and in the color antitriplet [22,23]. If there exists an additional P-wave between the light quark and heavy quark, we get the pseudoscalar and vector diquark operators and in the color antitriplet without introducing the additional Pwave explicitly, as multiplying can change the parity, and the P-wave effect is included in the underlined . On the other hand, we can introduce the P-wave explicitly, and obtain the vector and tensor diquark operators and in the color antitriplet.
We take the C, , , , and type diquark and antidiquark operators (also the and type diquark operators, which have both components and ) as the basic constituents to construct the tetraquark current operators with , , , and , and to interpolate the hidden-charm or hidden-bottom tetraquark states. The P-wave, if any, is between the light quark and heavy quark (or between the light antiquark and heavy antiquark), in other words the P-wave lies inside the diquark or antidiquark, while the diquark and antidiquark are in the relative S-wave [8,11,15,20,21]. In this case, we introduce the effective heavy quark mass and virtuality to characterize the tetraquark states, and suggest the energy scale formula for choosing the optimal energy scales of the QCD spectral densities [15,20,21].
On the other hand, if there exists a relative P-wave between the diquark and antidiquark constituents, we have to consider its effect and modify the energy scale formula, denotes the energy cost of the relative P-wave [6,7]. lies near and , and the energy gap between the masses of and ( and ) is about ( ) in the potential models [2,3]. As we study the vector hidden-bottom tetraquark state, there exists a relative P-wave between the bottom diquark and bottom antidiquark constituents, and the relative P-wave is estimated to cost about . We can then modify the energy scale formula to, Chinese Physics C Vol. 43, No. 12 (2019) 123102 where we choose the updated value [24]. The value is reasonable, as the QCD sum rules indicate that the ground state hidden-bottom tetraquark mass is about [11]. The vector hidden-bottom tetraquark mass is estimated to be , which is in excellent agreement with (or at least compatible with) the experimental value of obtained by the Belle collaboration [1].
In Ref. [11], we studied systematically the scalar, axial-vector and tensor diquark-antidiquark type hidden-bottom tetraquark states (where the bottom diquark and bottom antidiquark are in relative S-wave) with the QCD sum rules, and chose the continuum threshold parameters as , which works well and is consistent with the assumption [17]. Here, we assume and choose the continuum threshold parameter as .
In the numerical calculations, we observe that the continuum threshold parameter , the Borel parameter and the energy scale GeV work well. The pole contribution from the ground state tetraquark candidate is about 47%-70%, and the pole dominance is satisfied. The predicted mass is about , which certainly obeys the modified energy scale formula.
We also observe that the contributions of the vacuum condensates , , and are large, and their values change quickly with the variation of the Borel parameter in the region . As the convergent behavior is bad, we have to choose . In the Borel window, , the contributions of the vacuum condensates , , and satisfy the hierarchy , where we use the symbol to denote the contribution of the vacuum condensate of dimension n. The contributions of the vacuum condensates and are very small and cannot affect the convergence behavior of the operator product expansion. The contribution of the vacuum condensates of dimension 10 is 2% -6%. We thus draw the conclusion that the operator product expansions converge well.
We can now obtain, Eq. (11), the numerical values of the mass and pole residue of the tetraquark candidate using the QCD sum rules in Eqs. (6) - (7). Taking into account the uncertainties of the input parameters, the predicted mass and pole residue as function of the Borel parameter are shown in Figs.1-2.
It is obvious from Figs.1-2 that both the mass and pole residue are on platforms in the Borel window. The four criteria of the QCD sum rules for the vector tetraquark states are all satisfied [15,20,21], and we expect to make reliable and reasonable predictions.
The numerical value from the QCD sum rules is in excellent agreement with (or at least compatible with) the experimental value of obtained by the Belle collaboration [1] (see Fig. 1), which favors the assignment of as the diquark-antidiquark type vector hidden-bottom tetraquark state with a relative P-wave between the diquark and antidiquark constituents. The relative P-wave between the constituents hampers the rearrangement of quarks and antiquarks in the color and Dirac spinor spaces in order to form the quark-antiquark type meson pairs, which can explain (or is compatible with) the   In the charm sector, the calculations based on the QCD sum rules favor the assignments of , and as the vector tetraquark states with a relative P-wave between the scalar (or axial-vector) diquark and scalar (or axial-vector) antidiquark pair [6,7]. Furthermore, the QCD sum rules favor the assignment of as the scalar-diquark-scalar-antidiquark type scalar tetraquark state, where the diquark and antidiquark constituents are in the relative S-wave [25]. Analogous arguments also hold in the bottom and charm sectors, but unambiguous assignments require more experimental data and more theoretical work. , the calculations lead to a mass of about , which is smaller than the S-wave hidden-bottom tetraquark masses [11], and hence should be rejected.

Y(10750)
We now turn to the study of the partial decay widths of as a vector hidden-bottom tetraquark candidate with the three-point QCD sum rules, and write down the three-point correlation functions, On the phenomenological side, we insert a complete set of intermediate hadronic states with the same quantum numbers as the current operators into the three-point correlation functions, and isolate the ground state contributions [12][13][14], where we use the definitions for the decay constants and hadronic coupling constants, , , and are the polarization vectors of the conventional mesons and the tetraquark candidate , respectively, and , , , , and are the hadronic coupling constants. In the calculations, we observed that the hadronic coupling constant is zero at the leading order approximation, and so we neglect the process .
The lowest scalar nonet mesons are usually assigned as the tetraquark states, and the higher scalar nonet mesons as the conventional quark-antiquark states [26-28]. Here, we assume with the symbolic quark structure Considering the components of the correlation functions in Eqs. (14)- (19), we carry out the operator product expansion up to the vacuum condensates of dimension 5. We then calculate the connected and disconnected Feynman diagrams taking into account the perturbative terms, quark condensate and mixed condensate, and neglect the tiny contributions of the gluon condensate. We obtain the QCD spectral densities from the dispersion relation, where we match the hadron side with the QCD side of the components , and perform the double Borel transform with respect to and setting in the hidden-bottom channels and in the open-bottom channels. The QCD sum rules for the hadronic coupling constants are then, Chinese Physics C Vol. 43, No. 12 (2019) 123102 123102-7