ar X iv : h ep - p h / 05 11 12 6 v 1 1 0 N ov 2 00 5 Scalar Mesons in B - decays

We summarize some persistent problems in scalar spectroscopy and discuss what could be learned here from charmless B-decays. Recent experimental results are discussed in comparison with theoretical expectations: a simple model based on penguin dominance leads to various symmetry relations in good agreement with recent data; a factorisation approach yields absolute predictions of rates. For more details, see [1]. WHY STUDYING SCALARS IN B-DECAYS There are various reasons for studying scalar particles in B-decays: 1. Dominance of S-wave resonances with little background from crossed channels In (B → 1,2,3)-decays the masses of (1,2) resonances can extend to M < ∼ 5 GeV. Then there is little overlap with resonances in crossed channels (2,3) or (1,3). This is very different from D decays where resonance masses extend only up to ∼ 1.5 GeV and in general there is a large overlap. Furthermore, in the final 2-body systems S-wave interactions are dominant. 2. New source of glueballs The elementary subprocess b → sg with an isolated gluon is rather well understood theoretically and is described by a penguin diagram. The decay rate has been calculated in next-to-leading order of perturbative QCD as [2] B(b → sg) = (5±1)×10−3. (1) The gluon may give rise to production of a glueball which could show up as a resonance in the system X of 2-body decays B → K(∗) + X . This process adds to the other well known gluon rich processes like: central production in pp collisions, J/ψ → γX and pp̄ annihilation near threshold. 3. Non-charm final states with strangeness The decays b → sqq̄ are dominated again by the gluonic penguin process whereas the electroweak tree diagrams occur at the level of 20% only. In the leading penguin approximation the decays b → suū, sdd̄, sss̄ occur with the same fraction and have been calculated to amount to ∼ 2× 10−3 each. In the corresponding hadronic 2-body final states B → xy, if x and y are members of SU(3) multiplets X ,Y each, one obtains various symmetry relations [3]. Hopefully, this will ultimately identify the members of the lightest scalar nonet and the mixing properties. PROBLEMS OF LIGHT SCALAR MESON SPECTROSCOPY The interest in light scalar mesons originates from the following expectations: 1. The existence of glueballs This is a requirement from the first days of QCD and may be the most urgent open problem of the theory at the fundamental level. In lattice QCD, quenched approximation, the lightest glueball appears in the 0++ channel with a mass of 1400-1800 MeV [4]. The effect of unquenching is under study but realistic estimates are still difficult, especially because of the large quark masses. An alternative QCD approach is based on QCD sum rules [5] where the lightest glueball is centered around 1000-1400 MeV. 2. Multiplets of qq̄ and exotic bound states There is no general consensus on the members of the lightest qq̄ nonet, i.e. the parity partner of π,K,η,η ′. In addition, there is the possibility of tetraquarks [6], bound states of di-quarks. The list of scalar particles provided by the PDG [7] with mass M < ∼ 1.8 GeV includes I = 0: f0(600) (or σ ), f0(980), f0(1370), f0(1500), f0(1710); I = 1 2 : κ(900) (?), K ∗ 0 (1430); I = 1: a0(980), a0(1450). There are two typical scenarios for the classification of these states: I. One nonet below and one above 1 GeV The nonet of lower mass includes σ , κ, f0(980), a0(980), either qq̄ (see, for example, Ref. [8] and Van Beveren [9]) or qqq̄q̄ [6] bound states. The higher mass states could then make a qq̄ nonet with members K∗ 0 (1430) and a0(1450); in the isoscalar sector the three states f0(1370), f0(1500) and f0(1710) could be, as originally proposed in [10], a superposition of the glueball and the two members of the isoscalar nonet. II. One nonet above 1 GeV In this scheme the σ and κ with the parameters given are not considered as physical states to be classified along the lines we discuss here. The qq̄ nonet is rather formed by a0(980) (or also a0(1450)), f0(980), K∗ 0 (1430) and f0(1500) [11, 12] whereas two higher mass nonets including f0(1370) have been proposed in [13]. The ππ S-wave is interpreted as being dominated by a very broad object, centered around 1 GeV, the lower part could be responsible for the σ(500) effect. This broad state (Γ > 500 MeV) has been proposed as representing the isoscalar glueball by various arguments [13, 12]. There are states whose identity is in doubt as can be seen by the large uncertainty in mass and width estimated by the PDG: σ(500), f0(1370) with no single branching ratio or ratio of such numbers accepted by PDG and finally κ or K∗(800) not carried in the main listing of PDG. We will add a few remarks on these problematic states which will be of relevance for our discussion of B decays. Isoscalar channel: f0(600) or σ and f0(1370) Most definitive experimental results on these states can be obtained from the 2 → 2 scattering processes ππ → ππ ,KK̄,ηη applying an energy independent partial wave analysis (EIPWA); in this case unitarity provides important constraints in the full energy range. Recently, results on D and B decays as well as pp̄ → 3 particles with higher statistics became available. There is no general constraint on the mass dependence of the amplitude which can be affected by various dynamical effects. So far, in these processes no EIPWA over the full energy range has been performed, so an optimal description of data for a particular model parametrization is selected. A promising new approach towards EIPWA in D-decays has been presented at this conference by Meadows [14]. Concerning the ππ interaction there is a general consensus that there exists indeed a broad state with the width of the order of the mass, but the parameters depend on the mass range considered, a feature which is known already since about 30 years. 1. Low mass range Mππ < ∼ 0.9 . . .1.2 GeV. In this region the complex ππ amplitude moves along the unitarity circle to its top (phase 90◦) where a rapid circular motion follows from f0(980). An early analysis has been performed by the Berkeley collaboration [15], they found a state, σ , with Mσ = 660±100 MeV, Γσ = 640±140 MeV. Recently, results from D-decays by E791 [16], FOCUS [17] and from J/ψ → ωππ by BES [18] have been interpreted in terms of a σ with similar mass, although good fits based on a K matrix parametrization have been obtained without such a state [17]. On the theoretical side, parametrizations of such data using the low mass χPT constraints lead to a low mass pole with Mσ ∼ 450 MeV and Γσ ∼ 450 . . .600 MeV (see, e.g. Refs. [19, 20] and the reports by Bugg and Pelaes [21]). 2. Extended mass range 500 ≤ M ≤ 1800 MeV In case of a broad state the parameters should be determined from the energy interval where its influence is important and this includes the inelastic region above 1 GeV. All analyses of ππ scattering in this region find again one broad state, but with a higher mass than before, in a range around 1000 MeV and with large width > 500 MeV. The first analysis along these lines goes back again 30 years [22] and in Table 1 we list the pole positions from K matrix fits of various analyses. The fits by Estabrooks [23] refer to the four solutions of an EIPWA of elastic ππ scattering [24] as well as of the ππ → KK̄ reaction. In all solutions of the EIPWA the S-wave amplitude above 1 GeV follows a circular path with some inelasticity in the Argand diagram ( Im T vs. Re T ) which can be fitted by a broad resonance. Superimposed is a smaller circle corresponding to a resonance [23] with parameters close to what is known today as f0(1500). No additional pole, such as f0(1370), is seen in this analysis. A similar picture is found [12] for the inelastic channels ππ → ηη and ππ → KK̄ comprising the broad background and f0(1500) with the interference pattern ππ → KK̄: background f0(1500) constructive interference ππ → ηη : background f0(1500) destructive interference (2) This broad state is seen in a variety of processes and has been dubbed f0(1000) in [25]. Later arguments have been presented that this broad state be a glueball [13, 12]. This state also appears in decay processes although it may happen that the higher mass tails are suppressed for dynamical reasons. As an example, we quote the study by BES [26] of the final state J/ψ → ωK+K− where the large S-wave background (“σ”) extends up to about 2 GeV. A significant flat background has also been observed recently in the gluon rich channel J/ψ → γKK̄ by BES [27]. Apparently, the state f0(1370) shows up if reactions other than (1)-(3) in Table 1 without unitarity constraints are included in the fits. Whereas f0(980), “ f0(1000)” and f0(1500) are clearly seen as circles in the Argand diagrams, no such circle has ever been shown to exist for f0(1370). Before such a behaviour is demonstrated, this state could hardly be considered as established. The strong interference between background and f0(1500), leads to very different mass spectra, depending on the relative phase, which could easily simulate a “new state” f0(1370). TABLE 1. Position of broad state in the T matrix of ππ scattering according to various K matrix fits to data from reactions (1) ππ → ππ , (2) ππ → KK̄, (3) ππ → ηη ,ηη ′, (4) pp̄ annihilation and (5) J/ψ decays Authors mass (MeV) width (MeV) channels CERN-Munich [22] 1049 500 1 Estabrooks [23] 750 800-1000 1,2 Au, Morgan & Pennington [25] 910 700 1,2,5 Anisovich and Sarantsev [13] 153