On the possibility to observe higher n 3 D 1 bottomonium states in the e + e − processes
aa r X i v : . [ h e p - ph ] S e p On the possibility to observe higher n D bottomoniumstates in the e + e − processes A.M. Badalian ∗ Institute of Theoretical and Experimental Physics, Moscow, Russia
B.L.G. Bakker † Department of Physics and Astronomy, Vrije Universiteit, Amsterdam, The Netherlands
I.V. Danilkin ‡ Moscow Engineering Physics Institute, Moscow, Russia andInstitute of Theoretical and Experimental Physics, Moscow, Russia
The possibility to observe new bottomonium states with J PC = 1 −− in the region 10 . − . n + 1) S and n D states( n ≥
3) may be mixed with a rather large mixing angle, θ ≈ ◦ and this effect provides the correctvalues of Γ ee (Υ(10580)) and Γ ee (Υ(11020)). On the other hand, the S − D mixing gives rise toan increase by two orders of magnitude of the di-electron widths of the mixed ˜Υ( n D ) resonances( n = 3 , , D − wave states. The value Γ ee ( ˜Υ(3 D )) = 0 . +0 . − . keV isobtained, being only ∼ ee ( ˜Υ(5 D )) ∼
135 eV appears to be close to Γ ee (Υ(11020)) and therefore this resonance may become manifest inthe e + e − experiments. The mass differences between M ( nD ) and M (( n + 1) S ) ( n = 4 ,
5) are shownto be rather small, 50 ±
10 MeV.
I. INTRODUCTION
Recently the Belle Collaboration has observed anenhancement in the production process, e + e − → Υ( nS ) π + π − ( n = 1 , ,
3) [1]. Their fit using a sin-gle Breit-Wigner resonance yields a resonance mass10889 . .
3) MeV, slightly larger than that of Υ(10860),and a width 54 . +11 . − . MeV, which is two times smallerthan the width of Υ(10860), known from the earlier ex-periments [2], [3]. The BaBar Collaboration has also ob-served two resonance structures in the e + e − → b ¯ b crosssections between 10.54 and 11.20 GeV with the fittingparameters: M = 10876(2) MeV, Γ = 43(4) MeV and M = 10996(2) MeV, Γ = 37(3) MeV [4], which alsodiffer from the parameters of the conventional Υ(10865)and Υ(11020) resonances.Meanwhile, precise knowledge of the masses and thedi-electron widths of higher bottomonium vector statesis very important for the theory: They may provide newinformation on the details of the QCD quark-antiquarkinteraction at large distances, possible hadronic shifts ofhigher states, like Υ(10860) and Υ(11020), and S − D mixing. At present it remains unclear whether it is pos-sible to observe the higher n D ( n = 3 , ,
5) states,which have masses in the mass region considered [5]-[7].It is known that pure D − wave bottomonium stateshave very small di-electron widths [7], [8], in particu-lar, in Ref. [7] the values Γ ee ( n, D ) ∼ − ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] obtained. Therefore an observation of the D − wave res-onances in the e + e − processes seems to be not possiblenow. However, one cannot exclude that the bottomo-nium D -wave states with J P C = 1 −− , which lie abovethe open beauty threshold(s), may be mixed with thenearby S − wave states, as it takes place in the charmo-nium family, where due to S − D mixing the di-electronwidths of physical resonances, e.g. ψ (4040) and ψ (4160),have almost equal di-electron widths [9].An important feature of the bottomonium spectrum isthat the mass difference between the ( n + 1) S and nD states is small and decreases for increasing n . In [7] thevalue ∆ M ( n ) ∼ n ≥ B ¯ B and B s ¯ B s channels maybe strong [10]. Owing to such a coupling a mass shift ofthe higher resonances may occur. In particular, the massshift down of Υ(4 S ) is estimated to be ∼
50 MeV.Up to now only the 1 D − meson with J P C = 2 −− and M (1 D ) = 10161(2) MeV has been measured by theCLEO Collaboration in the cascade radiative processes[11], which lies far below the B ¯ B threshold. Here wewill discuss mostly those bottomonium states which areabove the B ¯ B theshold, and concentrate on those reso-nances which originate from pure D − wave states ( n ≥ D − wave” resonances in the e + e − processes may be possible, if owing to S − D mixing theirdi-electron widths are not small.At present the resonances Υ(10580), Υ(10860), andΥ(11020) are usually considered as pure n S ( n =4 , ,
6) states. However, in theoretical studies with dif-ferent Q ¯ Q potentials [6], [7] their di-electron widths turnout to be significantly larger than those found in exper-iment. We do not support the point of view of the au-thors of Ref. [5] who, in order to suppress the calculateddi-electron widths, took a small QCD radiative correc-tion factor β V = 0 .
46 (our notation), which correspondsto very large value of α s ( ∼ m b ) = 0 .
317 and thereforedecreases the di-electron widths by a factor of two. More-over, in [5] and [6] the S − D mixing is not taken intoaccount.A detailed study of the di-electron widths for all nS and nD ( n = 1 , ...
6) vector states in [7] shows that thecalculated widths are ∼
25% larger for 4 S and two timeslarger for Υ(11020), while for all other states the di-electron widths agree with experiment with high accu-racy, better 3% [7]. These facts can be considered as anindirect indication of a possible S − D mixing betweenhigher vector states in bottomonium and our letter is justdevoted to this topic. II. COMPARISON OF CALCULATED RESULTSTO DATA
The study of the bottomonium spectrum done here andin [7], uses the single-channel relativistic string Hamilto-nian (RSH) with a universal potential [12]. This Hamil-tonian has been derived from the gauge-invariant mesonGreen’s function in QCD and in bottomonium it has anespecially simple form: H = ω + p + m b ω + V B ( r ) . (1)In general, the quantity ω appearing in this expression isa n operator, which has to defined by an extremum con-dition, exiting in two forms: If the extremum conditionis put on H , then one obtains the well-known spinlessSalpeter equation (SSE), thus establishing a direct con-nection between the SSE and the QCD meson Green’sfunction. In the second case the extremum condition isput on the eigenvalue, or the meson mass, which give riseto the Einbein approximation (EA) [9]. We use here theEA because it has an important advantage as comparedto the SSE: Its S-wave functions are finite at the origin,while they diverge near the origin in the SSE and need tobe regularized, adding a number of additional unknownparameters.The potential V B ( r ) in (1) is the sum of a pure scalarconfining term and a gluon-exchange part, V B ( r ) = σ r − α B ( r ) r , (2)where the vector coupling α B ( r ) is taken in two-loop ap-proximation and possesses two important features: theasymptotic freedom behavior at small distances, definedby the QCD constant Λ B ( n f ) [which is considered to beknown, because Λ B is directly expressed via the QCDconstant Λ MS ( n f ) in the M S renormalization scheme];it freezes at large distances. Details about the effectivefine-structure constant can be found in Ref. [9].
TABLE I: Spin-averaged masses in MeV/ c of the higher nD and ( n + 1) S states in the region 10 . − . n M ( nD ) 10 140 10 440 10 700 10 920 11 115 M (( n + 1) S ) 10 015 10 360 10 640 10 870 11 075 The RSH has been successfully applied to light mesons[13], heavy-light mesons [14], and heavy quarkonia [15].Within this approach relativistic corrections are takeninto account and a higher state can be considered on thesame grounds as a lower one; still at present the cou-pling to open channel(s) is neglected. Nevertheless, forhigher states the calculated masses appear to be ratherclose to the experimental ones and we can estimate pos-sible mass shifts due to a coupling to open channel(s): Acomparison does not give large shifts, ∼ ±
10 MeV forΥ(10580) and Υ(11020). Still it remains unclear why forΥ(10860) the calculated and experimental masses coin-cide. It seems possible that no hadronic shift occurs inthis case.For our analysis it is of great importance that anothereffect, namely, the production of virtual light quark pairs,is taken into account. This effect gives rise to a flatteningof the confining potential [16] and due to this flatteningphenomenon correlated downward shifts of the massesof the higher states occur, in particular, the shift of the6 S -state is ∼
40 MeV.The spectrum and di-electron widths of higher bot-tomonium states have several characteristic features.1. In the numbers given in Table I the theoretical error ±
15 MeV is not included; it mostly comes from anuncertainty in our knowledge of the pole (current) b -quark mass, taken here equal to m b (pole) = 4 . nD states( n = 3 , ,
5) occur just in the mass region 10 . − . nD states may slightly differ, as is thecase for Υ(10580) and Υ(11020).2. The mass difference between the n D and ( n +1) S states∆ n = M ( nD ) − M (( n + 1) S ) , (3)decreases for growing n : from ∼
140 MeV for n = 1(from experiment), ∼
60 MeV for n = 3 up to thesmall value ∼
40 MeV for n = 5. Due to such asmall difference the probability of the S − D mix-ing between higher bottomonium vector states in-creases.3. While the n D state (for a given n ≥
3) is mixedwith the ( n + 1) S state, such a mixed “ D − wave”state, denoted below as ˜Υ( nD ), will have a signifi-cantly larger di-electron width than a pure D − wavestate, even if the mixing angle is not large.In the case of charmonium, the almost equal di-electron widths of ψ (4160) and ψ (4040), also foundin experiment, have been obtained only for a largemixing angle, namely, θ ∼ = 35 ◦ [9]. For ψ (3686)and ψ (3770) the mixing angle, θ ∼ = 10 ◦ , is signif- icantly smaller [17], [18]; nevertheless, the experi-mental value Γ ee (3770) = 0 .
247 keV appears to be ∼
10 times larger than that of a pure 1 D state.4. The di-electron widths of pure n D bottomoniumstates are very small, ∼ (1 −
2) eV. They are de-noted below as Γ ee ( nD ), and given in Table II TABLE II: The di-electron widths (in keV) of pure ( n + 1) S and n D states in bottomonium from [7] and experimentalnumbers from [3]. n ee ( nD ) 0 . × − . × − . · − . × − . × − Γ ee (( n + 1) S ) 0.614 0.448 0.37 0.316 0.274Γ exp (Υ(( n + 1) S )) 0.612(11) 0.443(8) 0.272(29) 0.31(7) 0.13(3) Γ ee ( nD )Γ ee (( n +1) S ) × For the ground state Υ(9460) we have obtainedΓ ee (Υ(9460)) = 1 .
317 keV, in great agreement with theexperimental number, equal to 1 . ± .
02 keV. Also, asseen from Table II, the values Γ ee ( nS ) ( n = 2 ,
3) coin-cide with precise accuracy with the experimental widthsof Υ(10023) and Υ(10355). For the low-lying statesthe ratios r ( m/n ) = Γ ee ( mS ) / Γ ee ( nS ) of the calculatedwidths (Γ ee (1 S ) = 1 .
317 keV, Γ ee (2 S ) = 0 .
614 keV, andΓ ee (3 S ) = 0 .
448 keV) are found to be r (2 /
1) = 0 . r (3 /
1) = 0 . r (3 /
2) = 0 . r exp (2 /
1) = 0 . r exp (3 /
1) = 0 . r exp (3 /
2) = 0 . e + e − dynamics it isimportant that in our analysis the same QCD radiativecorrection factor, β V = 1 − π α s (2 m b ) is taken. Thisfactor is cancelled in the ratios of the di-electronic widthsand this result indicates that the calculated values of thewave function (w. f.) at the origin are defined witha good accuracy. Then β V can be extracted from theabsolute values of Γ ee ( nS ) ( n ≤ β V =0 .
80 for all low-lying states. This value of β V shows thatin bottomonium the one-loop QCD corrections decreasethe di-electron widths by only 20% (while in [5] β V ≃ . β V ≃ . B ¯ B threshold weobtain widths which are two times larger for the 6 S state and ∼
25% larger for the 4 S vector state. Thereasons behind such a suppression of the di-electronwidths for higher states has been discussed in [6], where,however, the S − D mixing is not taken into account.In particular, there it has been demonstrated that thedi-electron widths, calculated in the framework of the Cornell coupled-channel model [20], are not suppressed.Moreover, we expect that an open channel cannot essen-tially modify the w.f. at the origin, because, as shown in[21], the w.f. at the origin of a four-quark system (like Q ¯ Qq ¯ q ) is much smaller than that of a meson ( Q ¯ Q ). Itmeans that a continuum channel, considered as a par-ticular case of a four-quark system, cannot significantlyaffect the meson w.f. at the origin. Therefore we assumehere that in bottomonium, as well as in the charmoniumfamily, the w.f. at the origin, and as a consequence thedi-electron widths, decrease mostly due to the S − D mixing.To get into agreement with the experimental valueΓ ee (Υ(10580)) = 0 . S − D mixing with the fitting angle, θ = (27 ± ◦ ,which appears to be not small (see Table III).Surprisingly, for the 5 S state the calculatedwidth coincides with the experimental central value,Γ ee (Υ(10860)) = 0 . ≤ S − D mixing takes place or not.To answer this question, more precise measurements ofΓ ee (10860) are needed. For an illustration we give inTable III the width for the mixing angle θ = 27 ◦ . Itsvalue Γ ee (Υ(10860)) = 0 .
23 keV coincides with the lowerbound of the experimental number.For Υ(11020) its di-electron width, Γ ee (11020) =(0 . ±
3) keV is two times smaller than the calculatednumber for θ = 0 and by 26% smaller than for θ = 27 ◦ .To obtain such a small width we have taken a larger mix-ing angle for Υ(11020), considereing this resonance notas a pure 6 S state. Good agreement with experimentis obtained for the mixing angle (40 ± ◦ , for which al-most the same number occurs for ˜Υ(5 D ), the mixed 5 D TABLE III: The di-electron widths of the ( n + 1) S and n D states (in keV) without mixing ( θ = 0) and with S − D mixing ( θ = 27 ◦ ). The experimental numbers are taken from[3]. Theory Experiment θ = 0 θ = 27 ◦ Γ ee (4 S ) 0.37 0.275 0.272 ± ee (3 D ) 1 . × − ee (5 S ) 0.316 0.232 0.31 ± ee (4 D ) 1 . × − ee (6 S ) 0.274 0.199 0.13 ± ee (5 D ) 1 . × − state, namely ( Γ ee (Υ(11020)) = 0 . ee ( ˜Υ(5 D )) = 0 . θ ∼ = 35 ◦ has been extracted in [13] to obtain thedi-electron widths of ψ (4040), ψ (4160), and ψ (4415) inagreement with experiment. III. SUMMARY AND CONCLUSION
Our study of higher D − wave states shows that theirmasses are close to those of the ( n + 1) S resonances andtheir di-electron widths are not small, ≥
70 eV, if the S − D mixing is taken into account. There are threearguments in favor of such a mixing:1. Suppression of the di-electron widths of Υ(10580)and Υ(11020).2. Strong coupling to the B ¯ B ( B s ¯ B s ) channel, whichhas become manifest in the recent observationsof the resonances in the processes like e + e − → Υ( nS ) π + π − ( n = 1 , ,
3) [1] and supported by thetheoretical analysis in [10].3. Similarity with the S − D mixing in the charmoniumfamily.The important question arises whether it is possible toobserve mixed D − wave states in e + e − experiments. Ourcalculations give M (3 D ) ∼ ee ( ˜Υ(3 D )) ∼
95 eV, whichis three times smaller than Γ ee (Υ(10580)). For such awidth an enhancement from this resonance in the e + e − processes will be suppressed, as compared to the peak ofthe Υ(10580) resonance.The di-electron width of Υ(10860) contains a ratherlarge experimental error and therefore one cannot drawa definite conclusion concerning the possibility of 5 S − D mixing, while for the 4 D state the mass 10920 ± D )(with the mass 11115 ± e + e − processes depend also on otherunknown parameters, like the total width and branchingratio to hadronic channels, the possibility to observe amixed 5 D -wave state, even for equal di-electron widths,might be smaller than for Υ(11020). In [4] only theΥ(11020) resonance has been observed in the mass re-gion around 11 GeV. Still one cannot exclude that dueto an overlap with an unobserved ˜Υ(5 D ) resonance, theshape and other resonance parameters of the conven-tional Υ(11020) resonance can be distorted. Acknowledgments
This work is supported by the Grant NSh-4961.2008.2.One of the authors (I.V.D.) is also supported by the grantof the
Dynasty Foundation and the
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