BONN-???,  CERN–TH/2001-???,  hep-ph/yymmnnn

Correlation of B_{s} → μ^{+}μ^{−} and (g − 2)_{μ} in Minimal Supergravity

A. Dedes^{1}, H. K. Dreiner^{1} and U. Nierste^{2}

1Physikalisches Institut der Universität Bonn, Nußallee 12, D-53115 Bonn, Germany

2 CERN, TH Division, CH-1211 Geneva 23, Switzerland

We analyze the rare decay mode B_{s} → μ^{+}μ^{−} in the minimal supergravity scenario (mSUGRA). We find a strong correlation with the muon anomalous magnetic moment (g−2)_{μ}. An interpretation of the recently measured excess in (g − 2)_{μ} in terms of mSUGRA corrections implies a substantial supersymmetric enhancement of the branching ratio B(B_{s} → μ^{+}μ^{−}): if (g − 2)_{μ} exceeds the Standard Model prediction by 3 · 10^{−9}, B(B_{s} → μ^{+}μ^{−}) is larger by a factor of XXX — YYY than in the Standard Model and well within reach of Run-II of the Tevatron. a remark on Higgs masses or SO(10)?

Supersymmetry (SUSY) is an attractive and widely studied extension of the Standard Model (SM). The minimal supergravity model (mSUGRA) [3] relates all supersymmetric parameters to just 5 quantities: the universal scalar and gaugino masses M_{0} and M_{1/2}, the trilinear term A_{0} the ratio tan β of the two Higgs vacuum expectation values and sgn μ, where μ is the Higgsino mass parameter.^{1} [4] These quantities are defined at a high, grand unified energy scale and determine the physical parameters, which are then evolved to low energies by the renormalization group equations. Precision measurements, which are affected by SUSY corrections through loop effects, play an important role to constrain the supersymmetric parameter space. The small number of parameters makes mSUGRA sufficiently predictive to be significantly tested by low energy precision observables.

Recently the Brookhaven National Laboratory (BNL) reported an excess of the muon anomalous magnetic moment a_{μ} = (g − 2)_{μ}/2 over its SM value [1]. The difference δa_{μ} = a_{μ}^{exp} − a_{μ}^{SM} = (43 ± 16) · 10^{−10} corresponds to a 2.6σ deviation from the SM. An mSUGRA interpretation of this anomaly implies μ > 0 (in the sign convention with M_{1/2} > 0 and equal signs of the diagonal elements of the chargino mass matrix) [5–14,?,16,17]. It further invites a large tan β. The discrepancy in the case of a_{μ} is by itself not significant enough to justify the claim of new physics. Especially since the calculation

1We do not consider new CP-violating phases which could occur in μ and A_{0}.

of a_{μ}^{SM} involves two hadronic quantities: the hadronic contributions to the photon self-energy, which must be obtained from other experiments, and the (smaller) lightby-light scattering contribution, which can only be estimated with hadronic models. A more conservative estimate of the latter would reduce the BNL anomaly to a 2σ effect. Hence in order to resolve the possible ambiguity between the mSUGRA and alternative explanations of δa_{μ} one ideally wishes to study other observables whose sensitivity to supersymmetric loop corrections is correlated with δa_{μ}. SUSY modifies B physics observables if tan β is large, because the b Yukawa coupling becomes sizable. Especially sensitive are quantities which involve a chirality flip in the b quark line like B(B → X_{s}γ) and B(B → ℓ^{+}ℓ^{−}). In mSUGRA the low energy value for the trilinear term A_{t} is dominated by M_{1/2} with A_{t} < 0 for M_{1/2} > 0 [35]. Then μ > 0 implies that the charged-Higgs-top loop and the chargino-stop loop tend to cancel in B(B → X_{s}γ), so that the sensitivity of B(B → X_{s}γ) to mSUGRA corrections is weakened. A further disadvantage of this decay mode is that experimentally it requires a cut on the photon energy, which introduces some hadronic uncertainty. In [35] the possible impact of flavor-blind SUSY on other B physics observables, in particular those which enter the fit of the unitarity triangle, has been studied and only small effects have been found. By contrast, in SUSY B(B → ℓ^{+}ℓ^{−}) grows like tan^{6} β [32,33,30], with possible order-of-magnitude enhancements. Since B(B → ℓ^{+}ℓ^{−}) ∝ m_{ℓ}^{2}, the branching ratio is largest for ℓ = τ. Yet τ’s are hard to detect, so that the prime experimental focus at hadron colliders is on the search for B_{s} → μ^{+}μ^{−}. B factories running on the Υ(4S) resonance produce no B_{s} mesons. Branching ratios of leptonic B_{d} mesons are smaller by a factor of |V_{td}/V_{ts}|^{2}. Since in B factories the boost of the B_{d} meson is known and the considered leptonic decay rates are substantially enhanced over their SM values, we encourage our colleagues at BaBar and BELLE to look for B_{d} → τ^{+}τ^{−} decays as well. From now on we specify to the decay mode B_{s} → μ^{+}μ^{−}for simplicity. In [32,33,30] SUSY corrections to B(B_{s} → μ^{+}μ^{−}) has been calculated at the one-loop level. For large tan β, however, higher order corrections can be large, eventually of order 1. In [18] tan β-enhanced supersymmetric QCD corrections have been summed to all orders in perturbation theory. These dominant higher-order terms can be easily incorporated by replacing the b Yukawa coupling h_{b} ∝ m_{b} tan β with h_{b}^{eff} = h_{b}/(1 + Δm_{b}), where Δm_{b} ∝ μ tan β depends on the gluino and sbottom masses and can be found in [18]. Δm_{b} is positive for μ > 0. The dominant contribution to B(B_{s} → μ^{+}μ^{−}) is proportional to h_{b}^{eff 4}, so that the inclusion of Δm_{b} tempers the large-tan β behaviour.

The considered branching ratio can be expressed as

B(B_{s} → μ^{+}μ^{−}) =

6.0 · 10^{−7} (\frac{|V_{ts}|}{0.040})^{2} (\frac{f_{B_{s}}}{230 MeV})^{2} \frac{m_{μ}^{2}}{m_{B_{s}}^{2}} \sqrt{1 − \frac{4m_{μ}^{2}}{m_{B_{s}}^{2}}}

\{(1 − \frac{4m_{μ}^{2}}{M_{B_{s}}^{2}} ) |\frac{m_{B_{s}}^{2}C_{S}}{m_{μ}} |^{2} + |\frac{m_{B_{s}}^{2}C_{P}}{m_{μ}} − 2C_{A}|^{2}\}. (1)

Here |V_{ts}| = 0.040 ± 0.002 is the relevant element of the Cabibbo-Kobayashi-Maskawa matrix and f_{B_{s}} = (230 ± 30) MeV [21] is the decay constant of the B_{s} meson. In (1) we have kept the dependence on the lepton mass m_{μ}, so that the generalization to B_{d} → τ^{+}τ^{−} is straightforward. The Wilson coefficients C_{S}, C_{P} and C_{A}, which contain the short-distance physics, are normalized as in [29]. The coefficients c_{S}, c_{P} and c_{10} defined in [30] are related to ours by C_{S} = −2c_{S} sin^{2} θ_{W}, C_{A} = −2c_{10} sin^{2} θ_{W} and C_{P} = 2c_{P} sin^{2} θ_{W}. Within the SM C_{S} and C_{P} are negligibly small and the next-to-leading order result for C_{A} can be well approximated by C_{A} = 1.99(\overline{m}_{t}/166.6 GeV)^{1}.55 [27]. Here \overline{m}_{t} ≡ \overline{m}_{t}(m_{t}) is the top quark mass in the \overline{MS} scheme. \overline{m}_{t} = 166.6 GeV corresponds to a pole mass of m_{t} = 175 GeV. f_{B_{s}} dominates the uncertainty of the SM prediction, which reads B(B_{s} → μ^{+}μ^{−}) = (3.7 ± 1.1) · 10^{−9}. During Run-I of the Tevatron CDF found the upper bound [22]

B(B_{s} → μ^{+}μ^{−}) < 2.6 × 10^{−6}  at 95% C.L. (2)

The single event sensitivity of CDF at Run-IIa is estimated to be 1.0 · 10^{−8} for an integrated luminosity of 2 fb^{−1} [23]. That is, if mSUGRA corrections enhance B(B_{s} → μ^{+}μ^{−}) to e.g. 5 · 10^{−7}, one will see 50 events in Run-IIa. Run-IIb may collect 10-20 fb^{−1} of integrated luminosity, which will imply 250-500 events in this example.

The Wilson coefficients for a type-II two-Higgs doublet model can be found in [29]: contribution from charged Higgs loops to C_{S} and C_{P} grow at most as tan^{2} β and, in most of the parameter space, tend to diminish B. SUSY corrections to these coefficients, however, grow like tan^{3} β and can enhance B by several orders of magnitude [32,33,30]. Hence the signatures of the extended Higgs sector and of the superparticles are very different. The resummation of the tan β-enhanced SUSY-QCD corrections comprised in h_{b}^{eff} can be easily implemented into the formulae of [32,33,30] by replacing the leading tan^{3} β term with tan β^{3}/(1+Δm_{b})^{2}. The dominant dependence of δa_{μ} and B(B_{s} → μ^{+}μ^{−}) on the SUSY parameters is given by

δa_{μ} ∝ \frac{tan β}{μM_{2}} ,  B(B_{s} → μ^{+}μ^{−}) ∝ \frac{m2 ; q̃ tan^{6} β}{M_{A^{0}}^{4}} , (3)

where mq̃ represents the mass of stop and scharm, M_{2} is the wino mass parameter and M_{A^{0}} ≃ M_{H^{0}} is the mass of the pseudoscalar Higgs. Hence (3) reveals that the correlation of the two quantities in a general MSSM is weak and restricted to the fact that both grow with powers of tan β. The much stronger correlation within mSUGRA scenarios only comes in after relating the low energy parameters in (3) to the GUT parameters: Sakis and Herbi, please correct and continue, old text: All these three observables have one common : they are affected substantially by large^{2} values of tan β . In addition, in the mSUGRA scenario, these scales are related : they are equal at the GUT scale, with a common mass M_{0} and driven at the Electroweak (EW) through their RGEs which are mostly affected by the gaugino masses, assumed to be equal to M_{1/2} at the GUT scale^{3}. Furthermore, by demanding succesful Radiative Electroweak symmetry breaking (REWSB) [37] in the mSUGRA scenario the CP-Odd Higgs mass gets reduced [19] with large values of tan β. The latter enhances the ratio tan^{3} β/M_{A}^{2} and eventually the braching fraction B(B_{s} → μ^{+}μ^{−}). end of old text

In Fig.1 we plot the SUSY contribution to the anomalous magnetic moment of the muon against B(B_{s} → μ^{+}μ^{−}) for 10 < tan β < 58 ^{4} shown on the top of the figure. We have chosen one specific point[M_{1/2} = 450 GeV, M_{0} = 350 GeV, A_{0} = 0, μ > 0 and m_{t} = 175 GeV]^{5} of the mSUGRA parameter space. The SM prediction and the CDF bound from (2) for B(B_{s} → μ^{+}μ^{−}) are also shown in the figure. The solid (dashed) curve represents the B(B_{s} → μ^{+}μ^{−}) result with (without) resummation of the tan β-enhanced SUSY-QCD corrections. In this example the resummation suppresses B(B_{s} → μ^{+}μ^{−}) by 20% for tan β = 50. This looks a bit smallish to me. What is the value of μ? Hence in order for SUSY to account for a new physics contribution δa_{μ} within 1σ of the current BNL measurement [1] we need a large value of tan β > ∼ 50. At the same time we obtain B(B_{s} → μ^{+}μ^{−}) > ∼ 10^{−7}, which will be oberved by CDF

2In the case of the Higgs mass the word “large” means tan β > ∼ 10, in the case of the muon g − 2, tan β > ∼ 20 and as we will see in the case of B_{s} → μ^{+}μ^{−}, tan β > ∼ 40. 3The value for the trilinear soft SUSY breaking couplings, A_{0} is set to zero throughout this analyses. We have explicitly check that its variation within ±500 GeV modifies our result for B_{s} → μ^{+}μ^{−}within the f_{B_{s}} error. We leave it out here in order to make the analysis simpler. 4The upper limit comes from the REWSB requirement. 5Actually this point coincides with the point L quoted in [38].

at Run IIa. Another interesting feature from Fig.1 is the destructive interference in B(B_{s} → μ^{+}μ^{−}) for moderate values of 20 < ∼ tan β < ∼ 30. This is a result of a cancellation of the pure higgs boson contributions (2HDM contributions) and the SUSY box sneutrino-squark diagrams (which are similar in this case) with the SM contribution to cbs. This effect has been described in detail within the context of the 2HDM in [29]. We finally that varying f_{B_{s}} in 200-260 MeV changes B(B_{s} → μ^{+}μ^{−}) by roughly ∓ 25% with respect to the values shown in Fig.(1).

While we fixed M_{1/2} and M_{0} in Fig.(1) we next display the impact of B(B_{s} → μ^{+}μ^{−}) on the M_{1/2} − M_{0} plane for fixed tan β = 50, A_{0} = 0 GeV and μ > 0 in Fig.(2). The solid contours represent B(B_{s} → μ^{+}μ^{−}) with resummed SUSY-QCD corrections. First we find that the area bounded by M_{1/2} < ∼ 200 GeV and M_{0} < ∼ 350 GeV is excluded by the CDF limit in eq.(2). Second, CDF RUN IIa will probe M_{1/2} and M_{0} up to 500 GeV and 1.2 TeV respectively. Third, the shaded-dotted region representing the 2σ range (11 < δa_{μ} < 75) of the BNL experiment nicely reveals the correlation with B_{s} → μ^{+}μ^{−}: if the BNL anomaly persists and is due to mSUGRA corrections, CDF will certainly confirm this evidence through the observation of B_{s} → μ^{+}μ^{−}. Hence while CDF is not able to see squark masses up to 1.5 TeV [corresponding to M_{1/2} = 500 GeV and M_{0} = 1.2 TeV], it will nevertheless be able to prepare the ground for LHC by observing the B_{s} → μ^{+}μ^{−}mode. Even better after 10fb^{−}1 the CDF Run II will probe M_{1/2} = 750 GeV and 1.8 TeV (for tan β = 50) which in turn means masses for the heaviest superpartners of 2.4 TeV. We conclude the discussion of Fig.(2) with the light Higgs boson prediction (dot-dashed line) for tan β = 50 in the mSUGRA scenario: we find that Higgs masses less than 119 GeV are correlated with an enhancement of B(B_{s} → μ^{+}μ^{−}) over the SM value by two orders of magnitude. The recent “hint” by LEP [2] of a mass around 115.6 GeV results in B(B_{s} → μ^{+}μ^{−}) > ∼ 5 × 10^{−7} which will be measured before Higgs boson discovery.

Other contraints here : dark matter b to s gamma e.t.c I think the paper by Roszckowski et.al [16] is enough here Some words also on the recent paper [39]. Herbi could you please write this here??

A strong theoretical motivation for a large tan β is SO(10) Yukawa unification. It has recently been found [40,41] that a narrow parameter region is consistent with the current g − 2 constraint and b → sγ. In this region μ and M_{1/2} are light. This implies also a light CP-odd Higgs boson mass probably less than 300 GeV, again enhancing the B_{s} → μ^{+}μ^{−} decay. As an application, we find the B(B_{s} → μ^{+}μ^{−}) for the best fitted points found in [40]: M_{A} = 110 GeV, mχ̃_{1} < ∼ 250 GeV, A_{t} < ∼ 5 TeV, mt̃ < ∼ 1 TeV and tan β ≃ 50. We find that within the hadronic uncertainties B(B_{s} → μ^{+}μ^{−}) > ∼ 10^{−5} which already is excluded by CDF [22]. The only way out from the B_{s} → μ^{+}μ^{−}contraint in the SO(10) Models is to have M_{A} > ∼ 200 GeV [or smaller values of tan β < 45]. That in turn means that one has to relax the Yukawa unification assumption by relying on large GUT threshold corrections. In few words: If an SO(10) GUT model supposed to describe the nature then the decay B_{s} → μ^{+}μ^{−}is just around the corner.

In conclusion we have found a striking correlation between the muon anomalous magnetic moment a_{μ} and the decay B_{s} → μ^{+}μ^{−} in mSUGRA scenarios. If the reported excess of O(4 · 10^{−9}) in the measured value of a_{μ} [1] is caused by mSUGRA corrections, one faces a huge, order-of-magnitude enhancement of B(B_{s} → μ^{+}μ^{−}) over its SM value. The enhanced branching ratio is within reach of Run-II of the Tevatron. A measurement of B(B_{s} → μ^{+}μ^{−}) will further constrain the mass of the lightest Higgs bosons. SO(10) SUSY explanations of the measured a_{μ} are hardly compatible with the present upper bound on B(B_{s} → μ^{+}μ^{−}).

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FIG. 1. The predicted SUSY contribution to the muon anomalous magnetic δa_{μ} (bottom), versus the predicted branching ratio B_{s} → μ^{+}μ^{−} (left) for the input values of tan β (top) in an indicative mSUGRA point [M_{1/2} = 450, M_{0} = 350, A_{0} = 0, μ > 0, m_{t} = 175 GeV]. We plot the both LO and NLO corrections to B_{s} → μ^{+}μ^{−} together with the SM prediction and the present bound by CDF [22], B_{s} → μ^{+}μ^{−}< 2.6 × 10^{−6} at 95% CL. The regions where the SUSY contribution to the muon anomalous magnetic moment is such that the present deviation with the SM is recovered at 1σ and 2σ is also shown. f_{B_{s}} has been chosen to be 230 MeV.

FIG. 2. Contour plots (solid lines) of the predicted Branching Ratio B_{s} → μ^{+}μ^{−}on the plane M_{0}, M_{1/2} in the mSUGRA scenario and for tan β = 50, A_{0} = 0, μ > 0, m_{t} = 175 GeV. The fully shaded region is excluded from both theoretical [REWSB(left-upper corner),not neutralino LSP(down)] and experimental [susy and higgs searches (mainly left strip till 1 TeV M_{0} approximately)] facts. The shaded-dotted region shows the prediction of this mSUGRA point for the muon anomalous magnetic moment [in units 10^{−10}] being within 2σ of the value reported in [1]. Contours with the light Higgs boson mass equal to 115,120,121 are also shown (dotted-dashed lines). Here f_{B_{s}} = 230 MeV.