Strong decays and dipion transitions of Upsilon(5S)

Dipion transitions of $\Upsilon (nS)$ with $n=5, n'=1,2,3$ are studied using the Field Correlator Method, applied previously to dipion transitions with $n=2,3,4$ The only two parameters of effective Lagrangian were fixed in that earlier study, and total widths $\Gamma_{\pi\pi} (5, n')$ as well as pionless decay widths $\Gamma_{BB} (5S), \Gamma_{BB^*} (5S), \Gamma_{B^*B^*}(5S)$ and $\Gamma_{KK} (5, n')$ were calculated and are in a reasonable agreement with experiment. The experimental $\pi\pi$ spectra for $(5,1)$ and (5,2) transitions are well reproduced taking into account FSI in the $\pi\pi$.


Introduction
In a recent series of papers [1]- [3], [4] we have studied the (n, n ′ ) bottomonium dipion transitions Υ(n) → Υ(n ′ )ππ and decays Υ(n) → BB, BBπ using effective Lagrangian derived in the framework of the Field Correlator Method (FCM) [5]. This Lagrangian, as was understood in [3], contains two effective masses, playing the role of decay vertices, M ω for pionless qq pair creation, and M br for qq accompanied by one or two pions (kaons). It was found that M ω is responsible for pionless decays of the type Υ(n) → BB, BB * , B * B * , while M br enters into pionic decay transitions Υ(n) → BBπ. These are the only free parameters of the method. It was shown in [4], that both pionless dikaon widths are discussed. The dipion spectra with and without ππ FSI factors are given in section 5. Main results are discussed in the concluding section together with a short summary and perspective. where J (1) (p, k), J (2) (p, k 1 , k 2 ) are the overlap matrix elements between wave functions Ψ(q) of Υ(5S) and ϕ(q 1 )ϕ(q 2 ) of B(B * ) mesons.
It is convenient to approximate Ψ(q), ϕ(q) by a series of oscillator wave functions; indeed in Fig. 2 we show the quality of fitting of Ψ(r) by series of 5 and 15 terms. In this case the dependence on k 1 , k 2 as shown below simplifies. For the pionless overlap matrix element one can write are χ 2 fitting coefficients and ϕ k -oscillator functions for Ψ(q) and ϕ 1 -for B, B * mesons, and β 1 , β 2 are oscillator parameters for Υ(5S) and B, B * found from fitting. The factorȳ 123 defined in [3] takes into account the Dirac trace structure of the overlap vertex.
In a similar way one can define J (1) n , J (2) n for one -and two-pion emission integrals (3) Hereȳ (π) 123 ,ȳ (ππ) 123 are defined by the Dirac traces of the amplitudes and are given in [3]. As a result, the total amplitude is written as Here M 1 ∼ a, M 2 ∼ b, explicit expressions for M 1 , M 2 in terms of the integrals of overlap matrix elements J (1) , J (2) , J (0) , as in (1), are given in [3], and here we only quote results of numerical computations of M 1 , M 2 for (5,1), (5,2) and (5,3) transitions. As will be seen, both M 1 and M 2 do not depend strongly on cos θ and x, so that the main dependence of M(x, cos θ) on arguments comes from two exponential factors in (5) (some exclusion is imaginary part of M 1 , which is peaked near | cos θ| = 1).
The differential probability of dipion transition is given by where we introduced variables q ≡ M ππ , and numerical factor C 0 = 3 The B-meson decays of Υ(5S) In this section we study the pionless decays of Υ(5S), namely into BB, BB * + c.c., B * B * , B sBs , B sB * s + c.c., B * sB * s to which we ascribe numbers k = 1, 2, ..6. The corresponding formula for the width was derived in [3], namely M k is twice the reduced mass in channel k. The corresponding coefficients Z k account for spin and isospin multiplicities and (cf. similar coefficients in [11]) are as follows: Here J BB n (p k ) are overlap matrix elements wherec = ω 2(ω+Ω) , and ω, Ω are average energies of light and heavy quarks in B meson, computed in [12], ω ≈ 0.587 GeV, Ω = 4.827 GeV, ω s = 0.639 GeV, Ω s = 4.83 GeV, see Table 4 in [1].
Expanding Ψ n , ϕ B in series of oscillator functions as in [3], one obtains 11 (p) is a polynomial in p 2 , ∆ = 2β 2 1 + β 2 2 and β 1 , β 2 are oscillator parameters for Υ(nS) and B meson respectively, found from the χ 2 fitting procedure to the realistic wave function calculated in [9], and for 5S state and B meson one finds respectively where J 5 (p) = (1) I 5,11 (p)e − p 2 ∆ , and (1) I 5,11 is given in Eq. (2). Below in Table 1 the computed values of Γ k for k = 1, ...6 and with k max = 5, i.e.five oscillator terms approximating wave function of Υ(5S) are given. Computing (1) I 5,11 (p) for different number of oscillator terms k max , one can see, that values of I 5,11 (t), t = p 2 β 2 0 , β 0 ≈ 0.886 GeV, in the interval 0.2 ≤ t ≤ 2 are sensitive to k max and vary around the value |I 5,11 | ≈ 1 GeV 3/2 . We choose this value to estimate the variation of Γ k and find that for the dominant channel 3 the width changes by 6%, while Γ 4 can change by a factor of 10.
Comparing partial widths from the Table 1 with experimental limits (12)- (14), one can see, that all inequalities except the last right ones in (12) and (14) are satisfied by our theoretical values, however more work on theoretical side (explicit form of 5S wave function) and in experiment is needed.

Dipion and dikaon transitions of Υ(5S)
In this section we discuss dipion spectra and angular distributions for the transitions (5,1), (5,2) and (5,3), as well as total dipion and dikaon widths, given by Eq. (11). The differential probability dwππ dqd cos θ is given in (6), and integrating over dx or over d cos θ we obtain one-dimensional spectrum and angular distribution The values of M, Eq.(5), were calculated using Mω M br = 6 and for M 1 , M 2 the same equations (23-25) from [3] were used as for Υ(nS) transitions with n ≤ 4.
At this point we impose on the amplitude M the soft pion property, and use the AZR to rewrite Eq.(5) in the form where exp 1 and exp 2 refer to the exponential factors in (5) and the factor f (q), later used for the FSI effects, obeys the condition f (q 2 = m 2 π ) = 1. NormalizingM to M 2 , so thatM = M br Mω f 2 π M 2 , one can insert (17) in (15) to obtain Γ ππ . The corresponding values without FSI, i.e. for f (q) ≡ 1 are given in Table 2, upper line, and called the model 1.
For the dikaon (5,1) transition one can in first approximation neglect the change of m π to m K in matrix element (5), and take it into account in phase space, also remembering that M is O 1 f 2 π , which should be replaced by In the total width Γ KK (5,1) one can write similarly to (7) Γ KK (5, 1) = C 0 µ 3 Here µ 2 K = (∆E) 2 − 4m 2 K = 0.985 GeV 2 , µ K = 0.992 GeV. As a result, approximating the ratio of integrals over dx as 1/2, one obtains where we have used f π = 93 MeV, f K = 112 MeV [13]. Correspondingly one obtains the last column in Table 2 from the second one, using (19).

Final state interaction in (5, n ′ ) transitions
One of the important new features of (5S) (and higher states like (6S) ) transitions is that a large phase space is available where both σ and f 0 resonances can be seen. In (4,1) transitions f 0 is at the edge of phase space while σ in most transitions lies near the region x = η, where amplitudes vanish and therefore no strong FSI effects are visible in (n, n ′ ) for n ≤ 4.
In (5,1), (5,2) transitions the situation is different and e.g. in the (5,1) transition the f 0 resonance is well inside the available q region.
At this point it is necessary to stress that the FSI acts differently on onepion ("a" or M 1 ) amplitude and two-pion ("b" or M 2 ) amplitude. Namely, for the case of M 1 , where two pions are emitted from two points separated by distance L ∼ 1/Γ, Γ < ∼ 0(10 MeV), the ππ interaction of range r 0 < ∼ 0.6÷0.8 fm is damped by a factor of the order of r 0 /L ∼ O(1/10). E.g. in the FSI description in [14]- [16], the relative weight of ππ amplitudes with and without FSI was estimated as ∼ (1/7).
Completely different situation occurs in b, (M 2 ), where a pair of s-wave pions with I = 0 is emitted from a point (or, rather, a region of the order of λ ∼ 0.1 fm , λ -gluonic correlation length of QCD vacuum ). Here FSI is obligatory and is given by the Omnès-Muskhelishvili solution f (q) = P (q 2 ) D(q 2 ) ; with P (q 2 ) -a polynomial normalizing f (q 2 ) at some point: we shall use normalization f (q 2 = (2m π ) 2 ) = 1; a very close result is obtained for the Adler zero normalization f (q = m π ) = 1. Hence one can write f (q 2 ) as follows (cf the corresponding factors in [14,15]).
, i = σ, f 0 (21) and δ i (q 2 ) is the ππ phase due to the i-th resonance.
In the simplest approximation one can write (22) The factors, corresponding to the resonances yield peaks, in (22) the σ peak is a wide structure, while f 0 produces a sharp peak near 1 GeV. Another feature of f f 0 (q), Eq. (22), is that it changes sign just above position of f 0 due to the jump of δ(q 2 ) nearly equal to π, near q = 1 GeV, [14,15].
The resulting curves (solid lines) are given in Figs.3 and 4 for (5,1) and in Figs. 5 and 6 for the (5,2) cases, together with the curves for the model 1 (f ≡ 1, no FSI), shown by broken lines. Note, that in Figs. 3-6 theoretical curves were fitted to the experimental width Γ exp ππ , which means that M br /f π were varied in the interval 1 ÷ 0.75.

Results and discussion
We start with the BB widths of Υ(5S) given in Table 1. It is clear that the values Γ k give only a rough estimate and actual values Γ k depends strongly on the behaviour of the Υ(5S) wave function. This is certainly true for the Eq.(8), derived for the wave function in the one-channel approximation. In the next orders, given by the equation n )δ nm − w nm (E) = 0, this sensitivity should be weaker, since the wave function becomes complex and does not have zeros. Hence one might hope that the values Γ k yield the correct order of magnitude for all channels k = 1, ..6, with the value Mω 2ω 2 ≈ 1/2 as deduced from Γ tot (Υ(4S)). ComparingΓ k with the widths Γ k obtained for the 5S wave function approximated by 5 oscillator functions, one finds a reasonable agreement in magnitude , except for Γ 4 which is small due to nearby zero of J 5 (p).
Coming now to the total dipion widths in Table 2, one can notice, that our general expression (5), without FSI, yields reasonable order of magnitude for Γ ππ and Γ KK if M br fπ ≈ 1. Here again strong dependence on the Υ(5S) wave function persists and results for k max = 5 and k max = 15 differ several times. In view of this it is not surprising that in Table 2 theoretical widths for (5,1) and (5,2) dipion transitions have a hierarchy different from that of experimental widths; however the smallness of Γ th (5, 3) is well explained by a small phase space factor µ 3 : µ 3 (5, 3)/µ 3 (5, 1) ≈ 2.8 · 10 −2 and it is not clear, why Γ exp (5, 3) ≈ Γ exp (5, 1).
Similar results for Γ ππ , Γ KK are obtained when both FSI and AZR are taken into account.
Turning to the ππ spectra, one observes that the spectra without FSI (model 1) in Figs.3,5 have less structure in contrast to the experimental data [10], where peaks in spectra at q = 0.6 GeV for (5,2) and at q ∼ = 1.2 GeV for (5,1) are clearly seen and strong cos θ dependence is observed for the (5,2) transition, The situation is much better for the FSI-AZR approximation (model 2) in Figs. 3,5 where the σ and f 0 peaks are seen in (5,2) and (5,1) cases, and also the experimental U-form of the cos θ distribution is produced in the (5,2) transition. However the much weaker experimental cos θ dependence, Fig. 4 for the (5,1) case is better reproduced in the model 1.
As a whole, it seems, that the spectrum, especially its lower enhancement at q ≈ 0.4 GeV in both (5,1) and (5,2) transitions, can be well described by the AZR+ FSI form, where the lower peak at q ≈ 0.4 GeV is due to cancellation of two terms in (17), i.e. mainly due to AZR.
Summarizing, we have used the theory developed in previous papers [1]- [3] and applied in [3] to the subthreshold transitions (n, n ′ ), n ≤ 4. This theory does not contain free parameters, the only ones M ω and M br are defined previously in [3].
2. the sequence of inequalities between Γ BB , Γ BB * , Γ B * B * and corresponding widths for B s B s , occur naturally.
4. Dipion spectra of (5,1), (5,2) transitions require inclusion of FSI with σ and f 0 peaks and the appearance of the peak at M ππ ≈ 0.4 GeV is possible due to a nearby zero of amplitude. We stress, that our method allows to reproduce the sophisticated (5,1) spectrum in Fig.3 with good accuracy, using the same FSI parameters as for the (5,2) spectrum in Fig. 5.
5. In addition the unusual (U-type) cos θ dependence is quantitatively explained for the (5,2) transition as consequence of FSI.
We have observed strong dependence of all results on the properties of the Υ(5S) wave function, in particular on the position of its zeros, which in turn may serve to derive it from the total set of experimental data.
As a whole, our method allows to understand the basic features of all Υ(nS) transitions and decays, however more work is needed to explain all data in detail.
The authors are grateful to M.V.Danilov and S.I.Eidelman for constant support and suggestions, to P.N.Pakhlov and all members of ITEP experimental group for stimulating discussions. The financial support of grants RFFI 06-02-17012, 06-02-17120 and NSh-4961.2008.2 is gratefully acknowledged.