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Bottomonium spectroscopy with mixing of eta_b states and a light CP-odd Higgs

The mass of the eta_b(1S), measured recently by BABAR, is significantly lower than expected from QCD predictions for the Upsilon(1S) - eta_b(1S) hyperfine splitting. We suggest that the observed eta_b(1S) mass is shifted downwards due to a mixing with a CP-odd Higgs scalar A with a mass m_A in the range 9.4 - 10.5 GeV compatible with LEP, CLEO and BABAR constraints. We determine the resulting predictions for the spectrum of the eta_b(nS) - A system and the branching ratios into tau^+ tau^- as functions of m_A.

The mass of the η b (1S), measured recently by BABAR, is significantly lower than expected from QCD predictions for the Υ(1S) -η b (1S) hyperfine splitting. We suggest that the observed η b (1S) mass is shifted downwards due to a mixing with a CP-odd Higgs scalar A with a mass mA in the range 9.4 -10.5 GeV compatible with LEP, CLEO and BABAR constraints. We determine the resulting predictions for the spectrum of the η b (nS) − A system and the branching ratios into τ + τ − as functions of mA. The BABAR collaboration has recently determined the η b (1S) mass m η b (1S) with an error of only a few MeV in radiative decays Υ → γ η b of excited Υ states and the observation of peaks in the photon energy spectrum. The result from Υ(3S) decays is m η b (1S) = 9388.9 +3.1 −2.3 (stat)± 2.7 (syst) MeV [1], and the result from Υ(2S) decays is m η b (1S) = 9392.9 +4. 6 −4.8 (stat) ± 1.9 (syst) MeV [2]. The average gives [2] m η b (1S) = 9390.9 ± 3.1 MeV , This result can be compared to predictions from QCD. Recent results based on perturbative QCD are in good agreement with each other and give E hf s (1S) = 44 ± 11 MeV [3] and E hf s (1S) = 39 ± 14 MeV [4] (whereas E hf s (1S) varies over a wider range in phenomenological models [5]); in the following we will use tentatively an average value which is about two standard deviations away from the experimental result (2). The most recent result from (unquenched) lattice QCD is [6] E latQCD hf s which is within 1 σ of (2). However, the hyperfine splitting is quite sensitive to short distances or hard quark momenta. It has been argued in [7] that the perturbative results in [4] can be used for short distance corrections of the Wilson coefficient of the corresponding spin-flip operator measured on the lattice. The additional contribution δ hard E hf s (1S) to E latQCD hf s (1S) has been estimated as δ hard E hf s (1S) ∼ −20 MeV [7], which brings the lattice result (4) in good agreement with the perturbative result (3). Although this conclusion needs to be checked within perturbation theory with lattice regularization, we consider it as a support for the perturbative QCD result (3).
Whereas an explanation of the discrepancy between (2) and (3) within QCD is not excluded at present, we will elaborate below the consequences of an explanation of this discrepancy due to new physics in the form of a mixing of the η b states with a CP-odd Higgs scalar A with a mass m A in the range 9.4−10.5 GeV: as a result of such a mixing, the masses of the η b -like eigenstates of the full mass matrix can differ considerably from their values in pure QCD without the presence of the CP-odd Higgs A [8,9,10], and the mass of the state interpreted as the η b (1S) can be smaller than expected if m A is somewhat above 9.4 GeV.
In such a scenario, the masses of the states interpreted as η b (2S) and η b (3S) can also be affected, and all states can have non-negligible branching ratios into τ + τ − due to their mixing with A. According to recent results of BABAR [11], the corresponding branching ratio of the observed state is below 8% at 90% confidence level. The branching ratio into µ + µ − would be smaller by a factor m 2 µ /m 2 τ , and well below the present upper limit [12]. The investigation of these phenomena is the purpose of the present paper.
A relatively light CP-odd Higgs scalar can appear, e. g., in non-minimal supersymmetric extensions of the Standard Model (SM) as the NMSSM (Next-to-Minimal Supersymmetric Standard Model) [10,13,14,15,16,17]. Its mass has to satisfy constraints from LEP, where it could have been produced in e + e − → Z * → Z H and H → A A (where H is a CP-even Higgs scalar). For m A > 10.5 GeV -where A would decay dominantly into bb -and m H < 110 GeV, corresponding LEP constraints are quite strong [18]. For 2 m τ < m A < 10.5 GeV, A would decay dominantly into τ + τ − and values for m H down to ∼ 86 GeV are allowed [18] even if H couples to the Z boson with the strength of a SM Higgs boson.
In fact, searches for e + e − → Z * → Z H with H → bb indicate a light excess of events (of ∼ 2.3 σ significance) for m H ∼ 95 − 100 GeV [18], which could be explained by a reduced branching ratio BR(H → bb) ∼ 0.1 and a dominant branching ratio BR(H → A A) ∼ 0.9 [16,17] if A decays dominantly into τ + τ − . The possible explanation of this excess of events at LEP is an additional motivation for a CP-odd Higgs scalar with a mass below 10.5 GeV. Allowing for m H somewhat below 100 GeV, such a scenario would also alleviate the "little fine tuning problem" of supersymmetric extensions of the SM [15,16,17].
Finally CLEO [19] and BABAR [11] have searched for a light CP-odd scalar A with A → τ + τ − in radiative Υ(1S) and Υ(3S) decays respectively, which mainly constrains the range m A < ∼ 9.4 GeV. In this work we focus on the range 9.4 GeV < ∼ m A < 10.5 GeV, which would be the most relevant for strong η b − A mixing effects as advocated in [10].
Apart from m A , such mixing effects depend in a calculable way on the model-dependent coupling of A to b quarks. The corresponding coupling, normalized with respect to the coupling of the SM Higgs scalar to b quarks, will be denoted by X d . In models with two Higgs doublets H u and H d , where H u couples to up quarks and H d to down quarks and leptons, one has [10,13,14,15,16,17] where cos θ A denotes the SU(2) doublet component of the CP-odd scalar A, and tan β = H u / H d . For tan β ≫ 1, X d can equally satisfy X d ≫ 1 [10]. Below we proceed as follows: i) First we assume that, in the absence of a CP-odd Higgs scalar, ii) We diagonalise the η b (nS) − A mass matrix (n = 1, 2, 3) and require that one eigenvalue coincides with the mass measured by BABAR within errors (1); this condition gives us an allowed strip in the X d − m A plane, which depends only weakly on the assumed masses of η 0 b (2S) and η 0 b (3S). The resulting values of X d as function of m A allow to determine the remaining eigenvalues of the mass matrix, and the decompositions of the eigenvectors in terms of A and η 0 b (nS). Finally the A components of the eigenstates allow us to determine their partial widths and to estimate the branching ratios into τ + τ − . For The diagonal elements m The off-diagonal elements δm 2 n can be computed in the framework of a non-relativistic quark potential model in terms of the radial wave functions at the origin [8,9]. These can be considered as identical for vector and pseudocalar states, and be determined from the measured Υ → e + e − decay widths. Substituting recent values for these widths (see [10] for details) we find δm 2 1 ≃ (0.14 ± 10%) GeV 2 × X d , δm 2 2 ≃ (0.11 ± 10%) GeV 2 × X d , δm 2 3 ≃ (0.10 ± 10%) GeV 2 × X d .
We estimated the errors from higher order QCD corrections to the relation between the radial wave functions at the origin and Γ(Υ(nS) → e + e − ) [8,9] to be ∼ 10%.
(These errors play only a minor role for our results.) In order that one eigenvalue of M 2 coincides with the BABAR result (1) subsequently denoted as m obs , m A in (7) has to satisfy  The decomposition of the eigenstates into the states before mixing can be written as (10) It turns out that the coefficients P i,j can be expressed analytically in terms of the eigenvalues m 2 ηi of M 2 : In Fig. 3 we show our results for the A-components P i,4 for all 4 eigenstates together with the error bands from m obs , m η 0 b (1S) and δm 2 i . In the case of η 1 ≡ η obs , only the coefficients P 1,1 and P 1,4 differ significantly from 0. This allows to express the branching ratio BR(η 1 → τ + τ − ) as where Γ τ τ A is the partial width for the width of the state η 0 b (1S) without mixing with A, and Γ tot A the total width of A (without mixing). For Γ τ τ A we have where we would have to identify M with m A if A decays on its mass shell. Since the dependence on M originates from phase space integrals, M has to be identified with the mass of the decaying state (which is actually always close to m A ) in our case. Since the BR(A → τ + τ − ) is typically ∼ 90% (for large tan β), we take Γ tot A ∼ 1.1 Γ τ τ A in (12).  It is remarkable that, once we insert eqs. (11) and (13) into the expression (12) for BR(η 1 → τ + τ − ), all dependence on X d cancels: even for m A → 9.4 GeV where P 2 1,4 → 1, one has Γ tot A → 0 from X d → 0, and hence the first term in the denominator in (12) always dominates. The numerical result is Since the analytic expression for this BR is proportional to (m 2 obs − m 2 η 0 b (1S) ) 2 in our approach, its lowest possible value is quite sensitive to the smallest allowed value for this difference.
Turning to the remaining heavier eigenstates, expressions similar to (12) are always very good approximations for the branching ratios into τ + τ − , since the eigenstates consist essentially of just one η 0 b (nS) state and the CPodd Higgs A. However, the branching ratios vary now with m ηi and hence with m A ; the results are shown graphically in Fig. 4 assuming Γ η 0 b (1S) ∼ 10 MeV and For larger (smaller) total widths these branching ratios would be somewhat smaller (larger).
An important issue is the production rate of the eigenstates η i in radiative decays of excited Υ states, notably in Υ(3S) → γ η i . For a pure CP-odd scalar A, the BR (Υ(3S) → γ A) is given by the Wilczek formula [22] BR (Υ(nS) → γ A) F is an m A dependent correction factor, which includes three kinds of corrections (the relevant formulas are summarized in [23]): bound state, QCD, and relativistic corrections. Unfortunately these corrections become unreliable for m A > ∼ 8 GeV; a naïve extrapolation of the known corrections leads to a vanishing correction factor F for m A > ∼ 8.8 GeV [10] as relevant here. Thus it is difficult to predict the branching ratios BR (Υ(3S) → γ η i ): if the BR (Υ(3S) → γ A) is assumed to vanish, the production of the states η i has to rely on their η 0 b (nS) components. Otherwise, negative interference effects could appear leading to suppressed branching ratios. Hence it cannot be guaranteed that the (kinematically accessible) part of the spectrum shown in Fig. 2 is actually visible in radiative Υ(3S) decays in the form of a peak in the photon energy spectrum [11].
In view of the possibly quite low photon energies and/or large backgrounds, the photons can well escape undetected even if the process Υ(3S) → γ η i occurs with a non-negligible rate. In this case, as advocated in [24], the A components of η i can still manifest themselves through a breakdown of lepton universality in the form of an excess of τ + τ − final states in Υ(3S) → l + l − [9,10]. However, in the case of a vanishing correction factor F in (15), this phenomenon would disappear as well.
On the other hand, if the existence of a CP-odd scalar A and a CP-even scalar H with m H ∼ 95 − 100 GeV (and a dominant H → A A branching ratio of ∼ 90%) is responsible for the excess of events at LEP as noted above, it becomes important to test this scenario at the LHC: The standard search channels for a SM-like CPeven scalar H would fail, and the final states from H → A A with m A below the bb threshold would be difficult to detect. Proposals for a verification of this scenario at the LHC have been made recently in [25,26,27].
To conclude, if the mixing with a CP-odd Higgs scalar is responsible for the discrepancy between the BABAR measurement of m η b (1S) and the expectations from QCD, it can manifest itself in the form of a completely distorted spectrum of states as shown in Fig. 2. The branching ratios into τ + τ − would be non-vanishing, albeit below the present experimental upper limit (for the lowest lying state). These manifestations of a light CP-odd Higgs scalar in Υ physics at (Super) B factories [28] would be complementary to its possible discovery at the LHC.