A plausible explanation of Υ(10860)
AA plausible explanation of
Υ(10860)
R. Bruschini ∗ and P. González † Departamento de Física Teórica-IFICUniversidad de Valencia-CSICE-46100 Burjassot(Valencia), Spain
Abstract
We show that a good description of the
Υ(10860) properties, in par-ticular the mass, the e + e − leptonic widths and the π + π − Υ( ns ) ( n =1 , , production rates, can be obtained under the assumption that Υ(10860) is a mixing of the conventional
Υ(5 s ) quark model state withthe lowest P − wave hybrid state. Keywords: quark; meson; potential.
The explanation of the large e + e − → π + π − Υ( ns ) ( n = 1 , , widths at √ s = 10 . ± . GeV near the
Υ(10860) peak [1, 2, 3], about two ordersof magnitude larger than those for Υ( n (cid:48) s ) → π + π − Υ(1 s ) ( n (cid:48) = 2 , , , isnowadays a theoretical challenge. This so called “anomalous” dipion produc-tion suggests that either Υ(10860) is not the standard
Υ(5 s ) meson, or thereis some overlap of Υ(5 s ) with a non standard resonance close by, or there aresome dynamical effects with much bigger influence for Υ(5 s ) than for Υ( n (cid:48) s )( n (cid:48) = 2 , , [4]. Regarding the first option a tetraquark interpretation of Υ(10860) was used in reference [5]. By assuming a nonresonant part of theamplitude with the experimentally required order of magnitude the authorsshowed that the consideration of resonant terms from intermediate f Υ( ns ) states with f standing for f (500) , f (980) and f (1270) allowed for a fit of ∗ [email protected] † [email protected] a r X i v : . [ h e p - ph ] M a r he decay distributions of Υ(10860) → π + π − Υ( ns ) ( n = 1 , , . However notheoretical justification for the order of magnitude of the nonresonant partwas given. Concerning the second and third options we shall try to shownext that they may be related in such a way to provide a plausible expla-nation of
Υ(10860) . For this purpose we develop in Section 2 a standarddescription of − (1 −− ) bottomonium states from a conventional quark po-tential model. In Section 3 dipion transitions from Υ(5 s ) are studied withinthe QCD Multipole Expansion framework through the implementation of in-termediate hybrid states. The calculated widths for these processes suggestthat a detailed explanation of data is feasible. Finally in Section 4 possibleinterpretations of Υ(10860) deriving from this explanation are discussed.
Our starting point will be the simplest yet realistic non relativistic quarkmodel description of bottomonium ( bb ) provided by a Cornell like potential[6] V C ( r ) = σr − ζr (1)where r is the b − b distance and the parameters σ and ζ stand for the stringtension and the chromoelectric coulomb strength respectively. This form forthe static potential has been justified from quenched lattice QCD calcula-tions, see for instance [7]. It should be kept in mind that in the spirit of thenonrelativistic quark model calculations σ and ζ have to be considered aseffective parameters through which some non considered corrections to thepotential may be implicitly taken into account. We shall fix the Coulombstrength to ζ = 100 MeV fm corresponding to a strong quark-gluon coupling α s = ζ (cid:126) (cid:39) . in agreement with the value derived from QCD from the hy-perfine splitting of p states in bottomonium [8]. As for σ we shall choose itsvalue altogether with the quark mass value m b to get a good fit to the massesof − (1 −− ) spin triplet bottomonium states . Thus, for σ = 873 MeV/fm and m b = 4793 MeV a nice description of the spectral masses is obtained, asshown in Table 1.Some comments are in order. First, the significant discrepancy betweenthe calculated mass of the s state, MeV, and the experimental mea-sured mass at . MeV may be indicating mixing of the s and d states.So, the measured resonance would have a dominant s component, whereasa not yet discovered resonance at about MeV would have a dominant d component. Second, the discrepancy between the calculated mass of the s state, MeV, and the experimental measured mass at . MeV2 l States M nl (MeV) M P DG (MeV) s . ± . s . ± . d . ± . s . ± . d s . ± . d s . +3 . − . d s . +10 . − . Table 1:
Calculated − (1 −− ) bottomonium masses, M nl , from V C ( r ) with σ = 873 MeV/fm, ζ = 100 MeV fm and m b = 4793 MeV. The spectral notation nl where n ( l ) indicates the principal (orbital angular momentum) number has been used forthe states. For the ns and the d states the masses of the closest experimental Υ resonances from [3], M P DG , are quoted for comparison. indicates the need for including the effect of the first S − wave open bottommeson-meson channel BB in the potential when crossing the BB threshold,see [9]. Third, the natural assignment of Υ(10860) is to the
Υ(5 s ) state sincethe corresponding peak observed in the e + e − → bb cross section is about the s calculated energy [1]. It should be kept in mind though that some mixingwith the d state can also be expected.As for the error in the calculated masses the effectiveness of the parame-ters makes difficult to quantify it. We should expect for instance relativisticeffects to be more important for the low lying states. Then, having chosenthe values of the parameters as to fit these states may produce a non phys-ical mass shift for the high lying ones. In this sense the 25 MeV differencebetween the calculated mass of Υ(5 s ) and the quoted value for the mass of Υ(10860) might be taken as a very rough estimate of the error.It is easy to check that the calculated S states provide a very gooddescription of the measured ratios Γ(Υ( n s ) → e + e − )Γ(Υ( n s ) → e + e − ) and the correct order ofmagnitude for radiative transitions to P states, for example Γ(Υ(3 s ) → χ b (1 p ) γ )Γ(Υ(3 s ) → χ b (1 p ) γ ) Exp = 420 ± and Γ(Υ(3 s ) → χ b (2 p ) γ )Γ(Υ(3 s ) → χ b (2 p ) γ ) Exp = 2 . . ± . . Dipion transitions Υ( n i s ) → π + π − Υ( n f s ) Let us now center on the dipion transitions between − (1 −− ) s states: Υ( n i s ) → π + π − Υ( n f s ) . In QCD these processes involve the emission of two gluons andthe conversion of gluons into pions. As far as the heavy quark system movesslowly and its size is small compared to the pion system a non relativistictreatment based on the QCD Multipole Expansion (QCDME) makes sense,see [10] and references therein. Then the transition rate, dominated by dou-ble electric dipole transitions, can be expressed as [11] Γ(Υ( n i s ) → π + π − Υ( n f s )) = CG | F n i n f | (2)where C is a constant whose value can be fixed from a fit to data (see belowin this section), G is the phase space factor G = 34 M n f s M n i s π (cid:126) (cid:90) d M ππ K (cid:115) − m π M ππ ( M ππ − m π ) (3)with M ππ the dipion invariant mass, and K = (cid:113) ( M n i s + M n f s ) − M ππ (cid:113) ( M n i s − M n f s ) − M ππ M n i s (4)is the recoil momentum of Υ( n f s ) in the rest frame of Υ( n i s ) . The transitionmatrix element F n i n f is given by F n i n f = (cid:88) n hyb (cid:82) d r r R n i s ( r ) rR n hyb p ( r ) (cid:82) d r (cid:48) r (cid:48) R n hyb p ( r (cid:48) ) r (cid:48) R n f s ( r (cid:48) ) M n i s − M n hyb p (5)where R stands for the radial wave function and the sum runs over a com-plete set of color singlet intermediate states of angular momentum , eachof them containing a bb color octet. We identify these intermediate states ashybrids (( bb ) + gluon ) denoted by their principal ( n hyb ) and orbital angularmomentum ( l = 1 ) quantum numbers.Potentials for hybrid states have been derived in quenched lattice QCD[12] and parametrized in reference [13]. We shall assume that the dominantcontribution to the sum in (5) comes from the hybrid states with orbitalangular momentum corresponding to the deepest hybrid potential called V Π u . The lowest energy hybrid is indeed the p state of V Π u . At short andintermediate distances this potential has been parametrized as V Π u ( r ) = (cid:18) . r r + 0 . r (cid:19) (cid:126) + E Π u . . . . . . . . r [fm]1 . . . . . V ( r ) [ G e V ] Figure 1:
Hybrid potential V Π u ( r ) with r = 0 . fm, E = 31 . MeV and E Π u =992 . MeV (dashed line) versus vibrational potential V vib ( r ) (solid line). where r (cid:39) . fm and E Π u is an additive constant, while at large r it reads V Π u ( r ) → σr (cid:114) π (cid:126) σr + E where E is another additive constant so that E Π u − E (cid:39) . MeV fm r , as toensure that the two parametrizations connect smoothly. It is important torealize that E Π u corresponds to the energy of the ground state + − gluelump(formed by a gluon bound to a bb color octet located at the origin). Thisenergy has been estimated to be between MeV and
MeV [14].This parametrization of V Π u resembles the form of the deepest vibrationalstring potential derived in reference [15] V vib ( r ) = σr (cid:114) π (cid:126) σr (6)except for its short range behavior since V Π u becomes a repulsive Coulombpotential (with a reduced strength as compared to ζ ) instead of the constantpotential resulting from V vib ( r → . This is illustrated in Figure 1 where V vib ( r ) has been drawn versus V Π u ( r ) with E Π u (cid:39) MeV and E (cid:39) MeV.As a matter of fact the reduced short range repulsion has little effect onthe masses of the intermediate states we are interested in, and on the calcu-lation of F n i n f , so that one can safely use the simpler compact expression of5rocess Γ (keV) B QCDME B P DG
Υ(3 s ) → π + π − Υ(1 s ) 0 .
936 (4 . ± . × − (4 . ± . × − Υ(3 s ) → π + π − Υ(2 s ) 0 .
575 (3 . ± . × − (2 . ± . × − Υ(4 s ) → π + π − Υ(1 s ) 6 .
932 (3 . ± . × − (8 . ± . × − Υ(4 s ) → π + π − Υ(2 s ) 3 .
995 (1 . ± . × − (8 . ± . × − Υ(5 s ) → π + π − Υ(1 s ) 655 . . ± . × − (5 . ± . × − Υ(5 s ) → π + π − Υ(2 s ) 115 . . ± . × − (7 . ± . × − Υ(5 s ) → π + π − Υ(3 s ) 20 . . +0 . − . ) × − (4 . +1 . − . ) × − Table 2:
Calculated widths and branching fractions B QCDME for dipion transi-tions between Υ( ns ) states within the QCDME framework. The errors in B QCDME come from the errors in the experimental values of the total widths. Experimentalbranching fractions from [3], B P DG , are quoted for comparison. V vib ( r ) instead of V Π u ( r ) . For the sake of simplicity and for an easy compari-son to other vibrational potentials used in the literature within the QCDMEframework we shall use V vib ( r ) henceforth.It turns out that the mass of the lowest hybrid state, M p = 10888 MeV,is pretty close to the calculated mass of the s state, M s = 10865 MeV (themasses for the higher hybrid states are M p = 11082 MeV, M p = 11267 MeV...) This gives rise to an enhancement of the amplitudes (5) for
Υ(5 s ) ascompared to Υ( n i s ) ( n i < . More precisely, by making use of a sufficientnumber of hybrid states (equal or greater than ) as to assure convergence ofthe sum in (5) and fixing the constant C = 6 . × − to get the experimental Υ(2 s ) → π + π − Υ(1 s ) width we can reproduce nicely the order of magnitudefor all the Υ( n i s ) → π + π − Υ( n f s ) widths with n i ≤ , with the exception of Υ(5 s ) → π + π − Υ(3 s ) as can be checked in Table 2.A look in detail at this table shows that the calculated widths from Υ(3 s ) are in perfect agreement with data; from Υ(4 s ) the calculated widths arebigger (by at most a factor as should be expected if the experimentalresonance has some d mixing (for the suppression of dipion decays from d states see [16]). Regarding Υ(5 s ) the calculated dipion widths to Υ(1 s ) and Υ(2 s ) have the correct order of magnitude differing from data by at most afactor whereas in the decay to Υ(3 s ) the calculated width is one order ofmagnitude lower than data.It should be kept in mind though that there are several sources of er-ror in the calculated widths. First, in the fixing of C : as we rely on thePDG average value of the Υ(2 s ) → π + π − Υ(1 s ) width to fix it we estimatea small error; this can be taken as a minimum possible error since theexperimental dispersion of data is much bigger. Second, in the truncated se-6 . . . . . . M ππ [GeV]0 . . . . . . . . . d Γ / d M ππ Figure 2:
Calculated dipion invariant mass distribution for
Υ(5 s ) → π + π − Υ(1 s ) . ries of intermediate states; by comparing the calculated widths with differentnumber of terms we estimate this error to be another . Third, in the useof the QCDME because of its expected lost of accuracy when increasing n i due to the higher size of the initial state; this can not be trustly estimated.Nonetheless, the good values obtained for the decays of Υ(3 s ) and Υ(4 s ) make us confident that the calculational error in the Υ(5 s ) case does notaffect the calculated order of magnitude.A more precise interpretation of the results in Table 2 requires an analysisof the dipion invariant mass distribution d Γ dM ππ in the way it was carried outfor instance in reference [5]. Our calculated d Γ dM ππ from (2), plotted in Fig-ure 2, should be identified as the nonresonant part of the amplitude (see forcomparison Figure 2 in [5]). Thus, our model provides a physical justificationto the educated guess done in [5] for the S − wave nonresonant amplitude (aswe do not consider any mixing of Υ(5 s ) with Υ(4 d ) we have no D − term).Regarding additional contributions to the amplitude a look at the exper-imental representations of d Γ dM ππ versus M ππ for the dipion decay Υ(5 s ) → π + π − Υ( n f s ) , see [5] and [17], shows clearly enhancements suggesting thepresence of resonant terms where the two pions are produced via a + (0 ++ , ++ ) resonance. In the QCDME framework these would correspond to contribu-tions to the amplitude where the conversion of the two gluons to two pionstakes place through a + (0 ++ , ++ ) resonance. Following reference [18] a f (500) contribution would contain, up to a dimensional constant, a factor M ππ − m π M f − M ππ substituting the factor M ππ − m π in (3). As the numerator peaks7 R ( n ) R ( n ) Exp .
19 0 . ± .
052 0 .
51 0 . ±
113 0 .
71 0 . ± . Table 3:
Calculated leptonic width ratios R ( n ) from Υ(5 s ) , compared to experi-mental values R ( n ) Exp from [3]. at large M ππ (remember that M ππ ≤ M (5 s ) − M ( n f s ) ), and the denomina-tor at M ππ = M f (500) it is clear that the closer to the difference between M f (500) and ( M (5 s ) − M ( n f s )) the more important this contribution. Asthe mass of f (500) is about ( M (5 s ) − M (3 s )) we expect it to be dominantfor Υ(5 s ) → π + π − Υ(3 s ) and subdominant for Υ(5 s ) → π + π − Υ( n f s ) with ( n f = 1 , . This provides a qualitative explanation of the order of magni-tude discrepancy between the calculated nonresonant width and data in the Υ(5 s ) → π + π − Υ(3 s ) case. (As for the estimation of other resonant contri-butions like the ones coming from intermediate Z ± b π ∓ states a theoreticalcalculational scheme has not been completely developed yet). Υ(10860)
The previous results on dipion decays point out to a possible interpretation of
Υ(10860) as the standard
Υ(5 s ) state. Further support to this interpretationseems to be provided by the leptonic width ratios calculated from Υ(5 s ) as R ( n ) ≡ Γ(Υ(5 s ) → e + e − )Γ(Υ( ns ) → e + e − ) = | R Υ(5 s ) (0) | | R Υ( ns ) (0) | M ns ) M s ) (7)As can be checked from Table 3 the resuls for n = 1 , , (for n = 4 mixingwith the d state should be taken into account) are in perfect agreementwith experimental ratios Γ(Υ(10860) → e + e − ) Exp
Γ(Υ( ns ) → e + e − ) Exp . Notice that this also precludesa significant mixing of the
Υ(5 s ) with the Υ(4 d ) state.However, this interpretation can not be maintained when dipion decays Υ(5 s ) → π + π − h b ( np ) are examined. From the experimental point of view theproduction rates of Υ(10860) → π + π − h b ( np ) and Υ(10860) → π + π − Υ( n f s ) are of the same order of magnitude [19]. From the theoretical side theQCDME has no predictive power for these E − M transitions (the onlyavailable data for n i < , Γ(Υ(3 s ) → π + π − h b (1 p )) < (2 . ± . × − MeV,does not allow for the fixing of the unknown constants). Nonetheless a simpli-fied order of magnitude estimate can be obtained by approximating hadronic8ransition rates by gluon emission rates. Following reference [11] we cancalculate the ratio
Γ(Υ(5 s ) → π + π − h b (1 p ))Γ(Υ(3 s ) → π + π − h b (1 p )) ≈ Γ(Υ(5 s ) → gg h b (1 p ))Γ(Υ(3 s ) → gg h b (1 p )) = ( M s − M p ) ( M s − M p ) | g , | | g , | where (notice that the potential V Π u ( r ) does not have S − wave hybrid states[13]) g n i , ≡ (cid:88) n hyb (cid:82) d r r R n i s ( r ) rR n hyb p ( r ) (cid:82) d r (cid:48) r (cid:48) R n hyb p ( r (cid:48) ) R h b (1 p ) ( r (cid:48) ) M n i s − M n hyb p In our spin independent quark potential model V C ( r ) the spin singlet h b (1 p ) and the spin triplet χ b (1 p ) are degenerate. Then using R h b (1 p ) ( r ) = R χ b (1 p ) ( r ) we get Γ(Υ(5 s ) → π + π − h b (1 p ))Γ(Υ(3 s ) → π + π − h b (1 p )) ≈ . × This theoretical ratio is at least two order of magnitude smaller than data
Γ(Υ(10860) → π + π − h b (1 p )) Exp
Γ(Υ(3 s ) → π + π − h b (1 p )) Exp > . × making the interpretation of Υ(10860) as the standard
Υ(5 s ) state untenable.The simplest possible alternative is to interpret Υ(10860) as a result ofthe mixing of
Υ(5 s ) with the first hybrid that we shall call henceforth H b (1 p ) .This seems quite natural for the Υ(5 s ) and the H b (1 p ) masses are both closeto the measured mass of Υ(10860) (for the sake of simplicity we do not includeany possible mixing with the
Υ(4 d ) state). We may then write | Υ(10860) (cid:105) ≈ cos θ | Υ(5 s ) (cid:105) + sin θ | H b (1 p ) (cid:105) (8)Let us first emphasize that the good description of the π + π − Υ( n f s ) decaysand the leptonic width ratios obtained from Υ(5 s ) points out to a smallmixing angle. Then, following reference [20] we write sin θ ≈ (cid:104) Υ(5 s ) | δ H| H b (1 p ) (cid:105) M s − M H b (1 p ) (9)where δ H is proportional to the E transition operator since Υ(5 s ) and H b (1 p ) have orbital angular momentum and respectively (notice thatin reference [20] the mixing of Υ(1 s ) with a different hybrid is considered).Hence we can rewrite the mixing as sin θ ≈ A (cid:82) d r r R s ( r ) rR H b (1 p ) ( r ) M s − M H b (1 p ) = A (2 × − fm / MeV ) A has units MeV/fm. By defining A ≡ aσ where σ = 873 MeV / fm stands for the confining strength for standard as wellas hybrid states we get sin θ ≈ . a being a a dimensionless constant.As this mixing allows for H b (1 p ) to decay to e + e − through its couplingto Υ(5 s ) we can estimate Γ( H b (1 p ) → e + e − ) ≈ . a Γ(Υ(5 s ) → e + e − ) Then, taking into account that the calculated leptonic width ratios from
Υ(5 s ) leave very small room for corrections we may reasonably assume a to be at most of order . This corresponds to a mixing of at most a fewpercent. Thus, the good description of the π + π − Υ( n f s ) decays previouslyobtained from Υ(5 s ) is also preserved if we reasonably assume, from HeavyQuark Spin Symmetry, that H b (1 p ) → π + π − Υ( n f s ) is somewhat suppressedagainst H b (1 p ) → π + π − h b ( np ) . In this regard let us remind that ( S b ¯ b ) H b (1 p ) =( S b ¯ b ) h b (1 p ) = 0 (cid:54) = ( S b ¯ b ) Υ( ns ) = 1 .The remaining issue has to do with the dipion decays Υ(10860) → π + π − h b ( np ) .According to our discussion above, the Υ(5 s ) → π + π − h b (1 p ) decay shouldgive a small contribution. So, we should have Γ(Υ(10860) → π + π − h b ( np )) ≈ sin θ Γ( H b (1 p ) → π + π − h b ( np )) Then, using the experimental widths
Γ(Υ(10860) → π + π − h b (1 p )) Exp = (1 . ± . × − MeV,
Γ(Υ(10860) → π + π − h b (2 p )) Exp = (2 . ± . × − MeVand sin θ ≤ . we can predict Γ( H b (1 p ) → π + π − h b (1 p )) ≥ . ± . MeV Γ( H b (1 p ) → π + π − h b (2 p )) ≥ . ± . MeVCertainly these predictions should not be taken for granted unless they wereevaluated in an independent manner. Unfortunately, the QCDME has nopredictive power for the H b (1 p ) → π + π − h b ( np ) decays since the lack of dataon hybrids makes impossible to fix confidently the unknown constants. Fur-thermore we do not know of any other effective theoretical approach be-ing (successfully) applied to the calculation of these decays. Instead wecan only add that the predicted values, although large as compared to the Υ(5 s ) → π + π − Υ( n f s ) widths, may represent a small branching fraction ifwe rely on constituent quark model estimates for the width of the −− , P − wave hybrid states [21]. In these models the hybrid H b (1 p ) would domi-nantly decay to open bottom meson-meson channels, with a width of the10rder of GeV. This large width might compensate the small sin θ factor togive a significant contribution to the open bottom meson-meson decays of Υ(10860) . The other way around, a thorough independent analysis of thesedecays, which is completely out of the scope of this letter, could constrain thevalues of the hybrid width and serve as a stringent test of this kind of models.If these were confirmed, there would be little hope of a direct clean exper-imental signal of such a broad H b (1 p ) , or more precisely of the orthogonalcombination to (8) mostly dominated by H b (1 p ) . This would make our pro-posal, if correct, the only practical available manner to infer the existence of H b (1 p ) . Meantime we may only consider the proposed mixing interpretationas a plausible explanation of Υ(10860) .This work has been supported by Ministerio de Ciencia, Innovación y Uni-versidades of Spain and EU Feder under grant FPA2016-77177-C2-1-P andby SEV-2014-0398. R. B. acknowledges the Ministerio de Ciencia, Innovacióny Universidades of Spain for a FPI fellowship.
References [1] D. Santel et al. Measurements of the Υ (10860) and Υ (11020) resonancesvia σ ( e + e − → Υ( nS ) π + π − ) . Phys. Rev. , D93(1):011101, 2016.[2] K. F. Chen et al. Observation of anomalous Upsilon(1S) pi+ pi- andUpsilon(2S) pi+ pi- production near the Upsilon(5S) resonance.
Phys.Rev. Lett. , 100:112001, 2008.[3] M. Tanabashi et al. Review of Particle Physics.
Phys. Rev. ,D98(3):030001, 2018.[4] Stephen Lars Olsen, Tomasz Skwarnicki, and Daria Zieminska. Non-standard heavy mesons and baryons: Experimental evidence.
Rev. Mod.Phys. , 90(1):015003, 2018.[5] Ahmed Ali, Christian Hambrock, and M. Jamil Aslam. A Tetraquarkinterpretation of the BELLE data on the anomalous Upsilon(1S)pi+pi- and Upsilon(2S) pi+pi- production near the Upsilon(5S) res-onance.
Phys. Rev. Lett. , 104:162001, 2010. [Erratum: Phys. Rev.Lett.107,049903(2011)].[6] E. Eichten, K. Gottfried, T. Kinoshita, K. D. Lane, and Tung-Mow Yan.Charmonium: The Model.
Phys. Rev. , D17:3090, 1978. [Erratum: Phys.Rev.D21,313(1980)]. 117] Gunnar S. Bali. QCD forces and heavy quark bound states.
Phys. Rept. ,343:1–136, 2001.[8] S. Titard and F. J. Yndurain. The l = 1 hyperfine splitting in bottomo-nium as a precise probe of the QCD vacuum.
Phys. Lett. , B351:541–545,1995.[9] P. Gonzalez. Generalized screened potential model.
J. Phys. ,G41:095001, 2014.[10] Yu-Ping Kuang, Ted Barnes, Changzheng Yuan, and Hai-Xuan Chen.Charmonium transitions.
Int. J. Mod. Phys. , A24S1:327–364, 2009.[11] Yu-Ping Kuang and Tung-Mow Yan. Predictions for Hadronic Transi-tions in the B anti-B System.
Phys. Rev. , D24:2874, 1981.[12] K. J. Juge, J. Kuti, and C. J. Morningstar. Ab initio study of hybridanti-b g b mesons.
Phys. Rev. Lett. , 82:4400–4403, 1999.[13] Eric Braaten, Christian Langmack, and D. Hudson Smith. Born-Oppenheimer Approximation for the XYZ Mesons.
Phys. Rev. ,D90(1):014044, 2014.[14] Gunnar S. Bali and Antonio Pineda. QCD phenomenology of staticsources and gluonic excitations at short distances.
Phys. Rev. ,D69:094001, 2004.[15] Roscoe Giles and S. H. H. Tye. The Application of the Quark-ConfiningString to the psi Spectroscopy.
Phys. Rev. , D16:1079, 1977.[16] Peter Moxhay. Hadronic Transitions of d Wave Quarkonium.
Phys.Rev. , D37:2557, 1988.[17] A. Garmash et al. Amplitude analysis of e + e − → Υ( nS ) π + π − at √ s =10 . GeV.
Phys. Rev. , D91(7):072003, 2015.[18] Lowell S. Brown and Robert N. Cahn. Chiral Symmetry and ψ (cid:48) → ψππ Decay.
Phys. Rev. Lett. , 35:1, 1975.[19] I. Adachi et al. First observation of the P -wave spin-singlet bottomo-nium states h b (1 P ) and h b (2 P ) . Phys. Rev. Lett. , 108:032001, 2012.[20] Tommy Burch and Doug Toussaint. Hybrid configuration content ofheavy S wave mesons.
Phys. Rev. , D68:094504, 2003.1221] F. Iddir, S. Safir, and O. Pene. Do 1- c anti-c g hybrid meson exist, dothey mix with charmonium?