Final state interaction effects on the eta(b) => J/psi J/psi decay
aa r X i v : . [ h e p - ph ] O c t th International Conference on High Energy Physics, Philadelphia, 2008
Final state interaction effects on the η b → J/ψJ/ψ decay ∗ Pietro Santorelli
Universit `a di Napoli “Federico II” & INFN, Napoli, Italy
We study the effects of final state interactions on the η b → J/ψJ/ψ decay. In particular, we discuss the effects ofthe annihilation of η b into two charmed meson and their rescattering into J/ψJ/ψ . We find that the inclusion of thiscontribution may enhance the short-distance branching ratio up to about 2 orders of magnitude.
Large efforts have been invested during the past thirty years to look for η b but the evidence of its existenceemerged very recently thanks to the Babar collaboration [1]. In [1] is reported the first unambiguous evidence of η b , with a 10 σ significance, through the hindered magnetic dipole transition process Υ(3 S ) → η b γ . The mass of η b is also measured to be m η b = 9388 . +3 . − . (stat) ± . S ) → η b γ , almost nothing is known regarding the decay pattern of η b [2]. However, rough estimate of thebranching ratios of some exclusive two and three-bodies hadronic decays can be found in [3].Some golden modes have been proposed to observe η b , such as η b → J/ψJ/ψ [4] and η b → J/ψγ [5, 6]. Despitevery clean signature due to the
J/ψ in final state, these decay modes are estimated to have rather suppressedbranching ratios. Regarding the η b → J/ψJ/ψ decay mode, the original estimate [4], which was compatible with thediscovery of η b in Tevatron Run I, has been reconsidered [3, 7]. In particular, an explicit NRQCD calculation gives B r [ η b → J/ψJ/ψ ] = (0 . ÷ . × − [3] too small to be observed also in Tevatron Run II.An interesting decay channel to observe η b , η b → D ( ∗ ) D ∗ , has been proposed in [7] where the range 10 − < B r [ η b → DD ∗ ] < − and B r [ η b → D ∗ D ∗ ] ≈ B r [ η b → DD ∗ ] ∼ − and B r [ η b → D ∗ D ∗ ] ∼ − which are at oddswith the ones obtained in [7].In [9] we assumed that the long distance contribution to the final state made of two J/ψ is dominated by the DD ∗ state and the subsequent rescattering of it into two J/ψ with a charmed meson in the t − channel as is shownin figure 1. The branching ratio of η b → DD ∗ is poorly known at present. However, as we already said there aretwo theoretical determinations we will use in considering the contribution to the η b → J/ψ J/ψ . Moreover, we willneglect the contribution coming from the annihilation of the η b to D ∗ D ∗ , in agreement with the results in [3, 7].The dominance of DD ∗ intermediate state is a consequence of the large coupling of D ( ∗ ) D ( ∗ ) to J/ψ as a result ofquark models and QCD Sum Rules calculations. η b ( p ) D ( p ) D ∗ ( p , ε ) D ( k ) , D ∗ ( k, ε ) J/ψ ( p , ε ) J/ψ ( p , ε ) Figure 1: Long-distance t − channel rescattering contributions to η b → J/ψ J/ψ . ∗ To the memory of Giuseppe (Beppe) Nardulli See also very recent calculation in NRQCD at NLO in α s B r [ η b → J/ψJ/ψ ] = (2 . ÷ . × − [8]. th International Conference on High Energy Physics, Philadelphia, 2008
Figure 2: The contributions coming from the loop graphs (for definitions see text). The contributions are plotted for g η b DD ∗ /g η b JJ ≈ g η b DD ∗ /g η b JJ ≈ { , } (solid lines). The dashed lines correspond to g η b DD ∗ /g η b JJ ≈ The full amplitude which takes into account the short distance part and the contribution coming from the evaluationof the graphs in figure 1 can be written as A f ( η b ( p ) → J/ψ ( p , ε ) J/ψ ( p , ε )) = ı g η b JJ m η b ε αβγδ p α p β ǫ ∗ γ ǫ ∗ δ (cid:20) g η b DD ∗ g η b JJ (cid:16) ı A LD + D LD (cid:17)(cid:21) , (1)where A LD and D LD represent the absorbitive and the dispersive part of the graphs in figure 1, respectively. Fordetails about the calculation of the previous quantities we refer to [9, 10]. The coupling g η b JJ is obtained by usingthe results in [3] while g η b DD ∗ from the estimate of the B r [ η b → DD ∗ ] and so g η b DD ∗ g η b JJ = 1 for B r [ η b → D ¯ D ∗ ] ≈ − [3] ∈ [11 ,
35] for 10 − ≤ B r [ η b → DD ∗ ] ≤ − [7]. (2)The numerical values of the on-shell strong couplings g JDD , g
JDD ∗ and g JD ∗ D ∗ are taken from QCD Sum Rules [11],from the Constituent Quark Meson model [12] and from relativistic quark model [13] findings which are compatibleeach other. We used ( g JDD , g
JDD ∗ , g JD ∗ D ∗ ) = (6 , , D ( ∗ ) mesons in figure 1 we have introduced the t − dependance of these couplings by means of the function F ( t ) = Λ − m D ( ∗ ) Λ − t . (3)No first-principles calculation of Λ exists, so, following the authors of [14], we write Λ = m R + α Λ QCD , where m R isthe mass of the exchanged particle ( D or D ∗ ), Λ QCD = 220
M eV and α ∈ [0 . , .
2] [14]; with this values, the allowedrange for Λ is given by: 2 . < Λ < . GeV .In figure 2, left panel (right panel) the ratio r A = A LD g η b DD ∗ /g η b JJ ( r D = D LD g η b DD ∗ /g η b JJ ) is plottedas a function of α for the allowed value and the range of couplings ratio. Moreover, the dashed lines are for g η b DD ∗ /g η b JJ ≈
26 which correspond to the central value in the allowed range for η b → DD ∗ estimated in Ref. [7].It is clear that for g η b DD ∗ /g η b JJ ≈ α .Very different is the case in which the annihilation of η b into DD ∗ is large [7]. The effects of final-state interactionscould be large and depend strongly on the value of α (cfr gray bands in figure 2). We use dimensionless strong coupling constants in all cases. In particular we use the ratio g JDD ∗ /m J/ψ instead of the dimensional G JDD ∗ (GeV − ) usually found in literature. th International Conference on High Energy Physics, Philadelphia, 2008Starting from the estimate of the short-distance part in [3] we are able to give the allowed range for the fullbranching ratio B r [ η b → J/ψ J/ψ ] = 0 . × − ÷ . × − , (4)where the lower bound corresponds to the corresponding one in [3], while the upper bound is obtained using theupper value in [3] and for α = 2 . g η b DD ∗ /g η b JJ = 35. The wide range for B r [ η b → J/ψ J/ψ ] in Eq. (4) depends onthe large theoretical uncertainty of the estimate of B r [ η b → DD ∗ ] and on the dependence on α parameter. It shouldbe observed that in [14] the preferred value for α is α ≈ . D and D ∗ in t -channel, whereas adirect calculation or measurement of the η b → DD ∗ process is in order.Finally we give an estimate of the discovery potential of the decay mode in the LHC experiments. Each J/ψ in thefinal state can be reconstructed by means of its muonic decay mode which represents about 6% of the total width,so we have B r [ η b → J/ψ J/ψ → µ ] ≈ × − ÷ × − . Moreover, assuming, as in [3], that i) the η b productioncross section at LHC is about 15 µ b and ii) the integrated luminosity (per year) is about 300 fb − , the theoreticallyexpected events are between 80 and 2 × . Experimentally we have to consider also the product of acceptanceand efficiency for detecting J/ψ decay to µ + µ − which is of the order of 0 . J/ψ must be tagged by µ + µ − pair and also allowits reconstruction through e + e − mode, we can have 3 ÷ η b at LHC through the 4-lepton mode exists. Acknowledgments
I thank G. De Nardo and D. Monorchio for a discussion on the experimental results from
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