Interpretation of Y b ( 10750 ) as a tetraquark and its production mechanism

the analysis of the cross sections for the processes + ϒ ( 1 , , 3) in the center-of-mass energy range from to A new structure, called here Y b ( 10750 ) with the mass M ( Y b ) = ( 10752 . 7 ± 5 . 9 + 0 . 7 − 1 1 ) MeV and the Breit-Wigner width (cid:4) ( Y b ) = ( 35 . 5 + 17 . 6 + 3 . 9 − 11 . 3 − 3 . 3 ) MeV was observed We interpret Y b ( 10750 ) as a compact PC = 1 −− state with a dominant tetraquark component. The mass eigenstate Y b ( 10750 ) is treated as a linear combination of the diquark-antidiquark and b ¯ b components due to the mixing via gluonic exchanges shown recently to arise in the limit of large number of quark colors. The mixing angle between Y b and ϒ ( 5 S ) can be estimated from the electronic width, recently determined to be (cid:4) ee ( Y b ) = ( 13 . 7 ± 1 . 8 ) eV. The mixing provides a plausible mechanism for Y b ( 10750 ) production in high energy collisions from its b ¯ b component and we work out the Drell-Yan and prompt production cross sections for pp → Y b ( 10750 ) → ϒ ( nS ) π + π − at the LHC. The resonant part of the dipion invariant mass spectrum in Y b ( 10750 ) → ϒ ( 1 S ) π + π − and the corresponding angular distribution of π + -meson in the dipion rest frame are presented as an example. © 2020 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP 3 .


Introduction
Recently, Belle has reported an updated measurement of the cross sections for e + e − → ϒ(nS) π + π − ; nS = 1S, 2S, 3S in the e + e − center-of-mass energy range from 10.52 to 11.02 GeV. They observe a new structure, Y b (10750), in addition to the ϒ(10860)and ϒ(11020)-resonances, having the masses and Breit-Wigner decay widths shown in Table 1 [1]. The measured ranges of the product ee × B (in eV) for the three final states are also presented in Table 1. The global significance of the new structure is 5.2σ . We recall that in high statistics energy scans for the ratios R ϒ π + π − ≡ σ (e + e − → (ϒ(1S), ϒ(2S), ϒ(3S)) π + π − )/σ (e + e − → μ + μ − ) and R bb ≡ σ (e + e − → bb)/σ (e + e − → μ + μ − ), Belle had found no new structures in their 2016 analysis [2]. In the same analysis, a 90% C.L. upper limit of 9 eV was set on ee in search of a structure around 10.9 GeV in R bb [2]. We also recall that the * Corresponding author.  [3].
The combined BaBar and Belle data on R bb have been recently reanalyzed taking into account the coherent sum of the three resonances ϒ(10860), ϒ(11020), and Y b (10750) [4], and a continuum amplitude, proportional to 1/ √ s, where √ s is the center-of-mass e + e − energy. The fit parameters of the R bb lineshape are the masses, Breit-Wigner decay widths, leptonic partial decay widths, and the relative phases. The resulting resonance masses and decay widths are found to be in agreement with the ones obtained from the R ϒ π + π − scan, and one gets a number of solutions for the partial electronic widths (mathematically 8 solutions are expected), which differ in other parameters, such as the total decay widths and partial leptonic widths. Most of these solutions are likely unphysical except the first solution, in which the electronic width of Y b is given as [4]:

Table 1
Measured masses and decay widths (in MeV), and ranges of ee × B (in eV) of the ϒ(10860), ϒ(11020), and the new structure Y b (10750). The first uncertainty is statistical and the second is systematic (Belle [1] (3) antitriplet-triplet representation [5,6]. However, it can have a small bb component due to the mixing via gluonic exchanges. The behavior of QCD for large-N c , where N c is the number of colors, has been worked out long ago by 't Hooft [7]. With the quarkgluon coupling as L QCD = g QCDq λ A g A μ γ μ q and λ A being the N 2 c − 1 SU (N c ) matrices, the large-N c limit is considered as g QCD → 0,  1 We argue that the production mechanism of Y b (10750) proceeds through its bb component, which arises from the mixing . A non-vanishing mixing is induced by nonplanar diagrams [12], allowing the direct production of Y b (10750) in high energy collisions. Using this, Drell-Yan [13] and prompt production cross sections [14] for Y b (10750) are presented for the LHC. We estimate the Y b − ϒ(5S) mixing angle from ee (Y b ) in Eq. (1) In contrast to the decays of ϒ(10860) and ϒ(11020), whose dipionic transitions (ϒ(1S), ϒ(2S), ϒ(3S)) π + π − are dominated by the resonant Z ± b (10650) and Z ± b (10610) states [15], the decay Y b (10750) → Z ± b (10650) π ∓ is kinematically forbidden, and Dalitz analysis in the decay Y b → ϒ(1S) π + π − will show a band structure in the m π + π − invariant mass, revealing clear evidence of two scalars, f 0 (500) and f 0 (980), and the tensor J P C = 2 ++ meson, f 2 (1270) [16]. In other two decays Y b → (ϒ(2S), ϒ(3S)) π + π − , only the broad f 0 (500)-meson is present.
With higher statistics data anticipated with the Belle-II detector, this distribution, as well as other properties of Y b (10750), will be well measured, allowing us to discriminate the tetraquark picture 1 A tetraquark interpretation [8,9] had been put forward for the Y b (10890), a resonance observed by Belle more than a decade ago [10,11], together with Y b (10860), identified with ϒ(5S). In subsequent data by Belle [2], two states Y b (10890) and ϒ(10860) were found to have the same mass within 2σ , essentially closing the window for an additional resonance. This seems to have changed with the an-
In brief, the exchange of a gluon between the two quark loops in Fig. 1(a) produces the interaction by which a genuine tetraquark pole may form in the intermediate state. Fig. 1(b) displays the non-perturbative version of Fig. 1(a). In the language introduced by 't Hooft for the large-N c expansion [7], non-planar gluon exchanges between the two fermion loops mean topologically one handle and produce a mixing coefficient f of order 2 A non-vanishing mixing with charmonia is also predicted in the alternative extension to large N c based on Witten's picture of large-N c baryons [26]. These "generalized tetraquarks" are made by N c − 1 antisymmetric quarks bound to N c − 1 antisymmetric antiquarks [27][28][29][30]. Non-vanishing mixing with quarkonia has been noted in [31]. Albeit suppressed at large N c by the exponential factor e −N c /2 , when extrapolated back to N c = 3 one finds a result not dissimilar from (2). Thus, production in the e + e − -annihilation of resonances such as Y b (10750), in addition to the bottomonia spectral lines and with a small ee , is a significant signature of tetraquarks.

Hadroproduction and Drell-Yan cross sections for
In Ref. [14], the hadroproduction cross sections for ϒ(5S) and ϒ(6S) in pp and pp collisions were calculated at the Tevatron and LHC, using the Non-Relativistic QCD framework [32]. The calculation has adopted a factorization ansatz to separate the short-and long-distance effects.
Secondly, to obtain the absolute cross section for Y b (10750) production, we estimate the ϒ(10860) cross section in NRQCD, following the calculation presented in [14]. One starts from the formula: Here, i and j denote a generic parton inside a proton, f i (x 1 ) and f j (x 2 ) are the parton distribution functions (PDFs) [33], the label Q denotes the quantum numbers of the bb-pair, which are labeled as 2S+1 L c J (color c, spin S, orbital angular momentum L and total angular momentum J ), O [Q ] are the corresponding long-distance matrix elements (LDMEs), andσ is a partonic cross section. The leading-order partonic processes for the S-wave configurations are: GeV 3 , respectively. Summing over the partonic processes shown above, and using the branching ratios from the PDG, yields the [14].
The corresponding cross sections for the processes pp → are obtained by using the scaling relation given in Eq. (6). For the LHC at √ s = 14 TeV, cross sections are given in Table 2 for the indicated ranges of p T (Y b ) and rapidity |y|, separately for ATLAS/CMS and for LHCb. Theoretical uncertainties in these cross sections are almost a factor 10, dominated by the uncertainties on the Color-Octet LDMEs, as well as on the ratio on the r.h.s. in Eq. (6). To estimate the expected number of events, we use 1 pb for the cross section, which lies in the middle of the indicated ranges, yielding O (10 4 ) signal events at the LHCb, and an order of magnitude larger for the other two experiments, ATLAS and CMS. The discovery channel μ + μ − π + π − , with the μ + μ − mass constrained by the ϒ(nS) (nS = 1S, 2S, 3S) masses, involves a pair of charged pions. Thus, the background is a stumbling block, but hopefully this can be overcome, with the additional constraint of the Y b (10750) mass. In addition to the mixing mechanism utilized here, there maybe direct production of the tetraquark, which would add incoherently to the previous results. Thus, the numbers presented in Table 2 give lower bounds to the expected Y b (10750) production in pp collisions.
The Drell-Yan production cross sections and differential distributions in the transverse momentum and rapidity of the J P C = 1 −− exotic hadrons φ(2170), X(4260) and Y b (10890) at the hadron colliders LHC and Tevatron have been calculated in [13].
We update these calculations for the production of Y b (10750) at the LHC for √ s = 14 TeV, and present results for pp → Y b (10750) → (ϒ(nS) → μ + μ − ) π + π − taking into account the current mass of Y b (10750) and the measured quantity ee × B, whose ranges are measured by Belle [1] and given in Table 1. In deriving the distributions and cross sections, we have included the order α s QCD corrections, resummed the large logarithms in the small transverse momentum region in the impact-parameter formalism, and have used two sets of parton distribution functions: MSTW (Martin-Stirling-Thorne-Watt) PDFs [34] and CTEQ10 [35]; the details can be seen in [13]. Numerical results for the cross section are given in Table 2, where the p T and rapidity |y| ranges for the ATLAS and CMS (called LHC 14 there), and for the LHCb, are indicated. These cross sections yield O (300) events for the current ATLAS/CMS luminosity (140 fb −1 ), and O (10) events for the LHCb (9 fb −1 ), but could be higher by a factor 2. The Drell-Yan cross Table 2 Total cross sections (in pb) for the processes pp → Y b (10750) → (ϒ(nS) → μ + μ − ) π + π − (n = 1, 2, 3) at the LHC ( √ s = 14 TeV), assuming the transverse momentum range 3 GeV < p T < 50 GeV. The rapidity range |y| < 2.5 is used for ATLAS and CMS (called LHC 14), and the rapidity range 2.0 < y < 4.5 is used for the LHCb.
The error estimates in the QCD production are from the variation of the central values of the Color-Octet LDMEs and the various decay branching ratios, as discussed in Ref. [14]. Contributions from ϒ(1S, 2S, 3S) are added together in the Drell-Yan production mechanism as in Ref. [13].

QCD (gg)
Drell-Yan sections are theoretically more accurate, but suffer from the small rates compared to the hadroproduction cross sections at the LHC.

Dipion invariant mass spectra and angular distributions in
The amplitudes of the e + e − → Y b → ϒ(nS) P P process, where P ( ) is a pseudoscalar, have been calculated in [9] as a sum of the Breit-Wigner resonances and non-resonating continuum contributions, with the latter adopted from [36]. The differential cross section is then written as [9]: where s and m P P are the squared invariant masses of the e + e − -pair and a pair of two final pseudocsalars, θ is the angle between the momenta of Y b and P in the P P rest frame, where I = 0 for π + π − , I = 0, 1 for K + K − , and I = 1 for ηπ 0 .
Details are given in [9].
We concentrate on the process Y b (10750) → ϒ(1S) π + π − , in which the σ = f 0 (500), f 0 (980), and f 2 (1270) resonances con- where S runs over possible I = 0 scalar resonances, and |k| is the magnitude of the π + -meson three momentum in the π + π − rest frame. The m π + π − and cos θ distributions for e + e − → Y b → ϒ(1S) π + π − , normalized by the measured cross section σ Belle ϒ(1S) π + π − = (1.61 ± 0.16) pb of the older Belle data [10] were fitted in [9], which determined various coupling constants. Since these distributions are not available for the new Belle data [1], we show in Fig. 2 only the resonant contributions, using the relevant input parameters from [9]. This illustrates the anticipated spectral shapes, which will be modified in detail as the non-resonant contribution is included. The fit can only be undertaken as the experimental measurements become available.

Summary
In this work, we have presented a tetraquark-based interpretation of the Belle data on the new structure Y b (10750) in e + e − annihilation, invoking the tetraquark-bb mixing anticipated in the large-N c limit. The bb-component is used to predict the hadroproduction and Drell-Yan cross sections at the LHC. A crucial test of our model is in the m π + π − and cos θ distributions, whose resonant contribution is worked out, which is not expected in other dynamical schemes, such as Y b (10750) interpreted as a D-wave bb state, with a very large S − D mixing [17]. The tetraquark-QQ mixing scheme suggested here has wider implications.