Single eta production in heavy quarkonia: breakdown of multipole expansion

The $\eta$ production in the $(n,n')$ bottomonium transitions $\Upsilon (n) \to \Upsilon (n') \eta, $ is studied in the method used before for dipion heavy quarkonia transitions. The widths $\Gamma_\eta(n,n')$ are calculated without fitting parameters for $n=2,3,4,5, n'=1$. Resulting $\Gamma_\eta(4,1)$ is found to be large in agreement with recent data. Multipole expansion method is shown to be inadequate for large size systems considered.

On theoretical side the dominant approach for both dipion and single η and π production is the Multipole Expansion Method (MEM) (see [6,7] and refs. therein), where it is asumed that heavy quarks emit two gluons and the latter are converted into meson(s) by a not clarified mechanism. An essential requirement for this mechanism in QED is the smallness of the source size r 0 as compared to the wavelength, so that r 0 k ≪ 1.
In reality both for charmonium and bottomonium transitions r 0 k > ∼ 1, but it is not this parameter which invalidates MEM for heavy quarkonia. It appears, that in QCD there is another important length parameter, the QCD vacuum correlation length λ, which makes it impossible to emit freely gluons at points separated by distance r, r > λ.
The value of λ was found on the lattice and analytically, λ < ∼ 0.2 fm [8]. Since r.m.s. radii of all excited cc, bb states are larger than 0.5 fm 1 all vacuum gluons there are correlated forming the QCD string and emission of additional gluons (if any) implies formation of heavy hybrids. All this is considerd in detail in Field Correlator Method (FCM) [9].
One can make an independent check of MEM in application to the bottomonium level calculation. Here MEM yields nonperturbative correction to the levels expressed via gluonic condensate [10]. Comparison to the experimental data shows (see [11] and Table 1 below) that for all level splittings except (2S − 1S) in bottomonium, MEM prediction is more than 50% off, while for charmonium MEM does not work at all. Thus one concludes that only at distances below or equal 0.2 fm, MEM can give reasonable results, while for all states of charmonium and all excited states of bottomonium (where sizes are much larger than vacuum correlation length λ) the application of MEM is unjustifiable. A similar failure of MEM is found in applications to dipion bottomonium transitions, where using MEM one can fit dipion spectra in Υ(2, 1)ππ, but not in Υ(3, 1)ππ and Υ(4, 2)ππ [6,7]. In contrast to that, FCM as will be discussed below explains both spectra and cos θ dependence for all dipion transitions in universal approach with two fixed parameters.
In FCM large distances are under control and not single gluons but combined effect of all gluons in the string defines the dynamics.
In particular, single eta emission in heavy quarkonia proceeds via string breaking due to qq pair creation with simultaneous emission of π or η. The flavor SU(3) violation in η production then resides in difference of threshold positions and wave functions for BB and B sBs (DD and D sDs ) intermediate states.
As it is clear, the dynamics of FCM for η emission does not depend strongly on heavy quark mass, and only sizes of initial and final heavy quarkonia states and intermediate heavy-light mesons enter in the form of overlap matrix elements.
In contrast to that, MEM predicts a strong dependence on the heavy quark mass, Γ η (n, ). In addition in [7] a strong suppression of the ratio Γ η /Γ ππ with the growth of the energy release ∆M = M(n) − M(n ′ ), Γ η /Γ ππ ∼ p 3 η (∆M ) 7 is predicted for higher excited states of quarkonia and bottomonia, which does not agree with experiment (see below).
This latter result is very large, indeed the corresponding ratio Γ(Υ(4,1)η) is ≈ 0.4 and theoretical estimates (1) from [6] for a similar ratio yields 3.3 · 10 −3 .Thus, the experimental ratio is very large as compared to MEM predictions [6,7]. All this suggests that another mechanism can be at work in single η production and below we exploit the approach based on the Field Correlator Method (FCM) recently applied to Υ(n, n ′ )ππ transitions with n ≤ 3 in [14,15], n ≤ 4 in [16] and n = 5 in [17,18].
In this paper we confront MEM and FCM and show that recent experimental data on single η production in Υ(4S) − Υ(1S) transition give a strong support to the FCM result and cannot be explained in the framework of MEM.
The method essentially expoits the mechanism of Internal Loop Radiation (ILR) with light quark loop inside heavy quarkonium and has two fundamental parameters -mass vertices in chiral light quark pair qq creation M br ≈ f π and pair creation vertex without pseudoscalars, M ω ≈ 2ω, where ω(ω s ) is the average energy of the light (strange) quark in the B(B s ) meson. Those are calculated with relativistic Hamiltonian [12] and considered as fixed for all types of transitions ω = 0.587 GeV, ω s = 0.639 GeV (see Appendix 1 of [14] for details).
Any process of heavy quarkonium transition with emission of any number of Nambu-Goldstone (NG) mesons is considered in ILR as proceeding via intermediate states of BB, BB * + c.c., B sBs etc. (or equivalently DD etc.) with NG mesons emitted at vertices.
For one η or π 0 emission one has diagrams shown in Fig.1, where dashed line is for the NG meson. As shown in [14,15,16], based on the chiral Lagrangian derived in [19], the meson emission vertex has the structure The paper is devoted to the explicit calculation of single η emission widths in bottomonium Υ(n, 1)η transitions with n = 2, 3, 4, 5. Since theory has no fitting parameters (the only ones, M ω and M br are fixed by dipion transitions) our predictions depend only on the overlap matrix elements, containing wave functions of Υ(nS), B, B s , B * , B * s . The latter have been computed previously in relativistic Hamiltonian technic in [12] and used extensively in dipion transitions in [16,17,18].
The paper is organized as follows. In section 2 general expressions for process amplitudes are given; in section 3 results of calculations are presented and discussed and a short summary and prospectives are given.

General formalism
The process of single NG boson emission in bottomonium transition is described by two diagrams depicted in Fig.1, (a) and (b) which can be written according to the general formalism of FCM [14,16,17] as (we consider η emission), see Appendix for more detail, For the diagram of Fig. 1(a) while M (2) η , corresponding to the diagram of Fig.1(b), has the same form, but without NG boson energy in the denominator of (7). The overlap integrals of Υ(nS) and BB * wave functions with emission of η with momentum k are denoted by J (i) n (p, k), the corresponding integrals without η emission are given by J n ′ (p). Finally we define all quantities in the denominator of (7); in M (1) η one omits ω η and k in (8). Finally after taking Dirac traces in amplitudes and accounting for the p-wave of emitted η one can represent the matrix element M (i) η as follows: Indices i ′ i in e i ′ il in (9) refer to the Υ(n ′ S) and Υ(nS) polarizations respectively.
One can see from the general structure of Γ η , that the main effect comes from the difference

Results and discussion
We consider here the single η emission in bottomonium transitions Υ(n, 1)η with n = 2, 3, 4, 5. The corresponding values of ∆M * , ∆M * s , ω η , k are given in the Table  2. Table 2. Mass parameters of Υ(n, n ′ )η transitions (all in GeV, k in GeV/c). The resulting values of Γ η (n, n ′ ) have been computed as in (12) with ω = 0.587 GeV and ω s = 0.639 GeV, calculated earlier in [12], see Table 4 of [14], and with wave functions fitted to the realistic wave functions in [16], (set I), while in set II parameters of the Υ(nS) wave function were changed by 10-15%.
Summarizing, we have calculated the single η production width Γ η (n, n ′ ) for Υ(n, 1)η transitions with n = 2, 3, 4, 5. We have found that Γ η (4, 1) are of the order of and larger than Γ ππ (4, 1). This fact is in agreement with the latest measurements in [13] of Γ exp η (4, 1). We have shown that Γ η (n, 1), n = 4, 5 is large (∼ O(1 kev)) for typical (realistic) parameters of Υ(nS) wave functions, but can occasionally drop near zero for slightly varied form of wave function, as it happens for n = 3. This high sensitivity is connected to the oscillating character of excited bottomonium wave functions 2 . Our calculations do not contain fitting parameters; the only two parameters M br , M ω are fixed by previous comparison with dipion data. One should stress that η production in bottomonium is not suppressed in our approach as compared to η production in charmonium transitions. This is in contrast with the results of method of [6,7]. We have given arguments why the dipion transitions in high excited states of heavy quarkonia as well as single η and π emission cannot be reliably calculated within the MEM method of [6,7], widely used now. As it is seen in Tables 3 and 4 the sequence of experimental data [4], [5], [13] contradict predictions of MEM. Recently a new calculation was done of Γ η (n, n ′ ) in [20] where also BB * etc. intermediate states were taken into account as well as in our approach. The authors however did not use wave functions of hadrons involved, but rather exploited fitted coupling constants and formfactors, and specific form of matrix elements, which makes it difficult to compare with our method directly. and Z is (we put m b ∼ = Ω) It is important, that we are looking for the P -wave of emitted η, and hence for P wave of relative BB * motion, hence the integral (A.2) should yield the term pk. This indeed happens, when one approximatesΨ n ,ψ n as series of oscillator wave functions and (A.2) has the form I n (p, k) =ȳ η n23 e − p 2 ∆n − k 2 4β 2 2 (0) I n (p). (A.8) In the process of dq integration in (A.2) one changes the integration variable q i → q ′ i − u n + O(k i ) with = β 2 , ∆ n are oscillator parameters, found by chi 2 procedure.
Thus result of d 3 q integration yields In an analogous way one obtains for J n ′ (p) in (A.1) the form J n ′ (p) =ȳ and finally one writes as in (9) M (1) n (n, n ′ ) =