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 analyse 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) [1] relates all supersymmetric parameters to just 5 real 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. 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 renormalisation group equations. Precision observables, which are affected by SUSY corrections through loop effects, play an important role in constraining the supersymmetric parameter space. The small number of parameters makes mSUGRA sufficiently predictive to be significantly tested by low energy precision measurements.

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 [2]. 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) [3–15]. It further invites a large tan β > ∼ 10. 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 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) light-by-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 [2]. Hence in order to resolve the possible ambiguity between 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 meson 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 the branching ratio 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 [16]. 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 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 [16] the possible impact of flavour-blind SUSY on other B physics observables, in particular those which enter the fit of the unitarity triangle, were studied and only small effects were found. By contrast, the branching ratio B(B → ℓ^{+}ℓ^{−}) grows like tan^{6} β [17–19], with a possible several orders of magnitude enhancement. 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. Leptonic branching ratios of 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 in SUSY, we encourage our colleagues at BaBar and BELLE to look for B_{d} → τ^{+}τ^{−} decays as well. From now on we restrict ourselves to the decay mode B_{s} → μ^{+}μ^{−}.

In [17–19] the SUSY corrections to B(B_{s} → μ^{+}μ^{−}) were calculated at the one-loop level. For large tan β, however, higher order corrections can be large, eventually of order 1. In [20] 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 [20]. 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 generalisation to B_{d} → τ^{+}τ^{−} is straightforward. The Wilson coefficients C_{S}, C_{P} and C_{A}, which contain the short-distance physics, are normalised as in [22]. The coefficients c_{S}, c_{P} and c_{10} defined in [19] 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} [23]. 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. The SM prediction is then given by B(B_{s} → μ^{+}μ^{−}) = (3.7 ± 1.2) · 10^{−9}, with the uncertainty dominated f_{B_{s}}.

Within mSUGRA the dominant coefficients are C_{S,P} since they are proportional to tan^{3} β. An approximate formula, which is good to within 50% for tan β > 45 and M_{1/2} < 600 GeV is given by |C_{S,P} | ≈ (m_{μ}F tan^{3} β)/(4M_{W} M_{1/2}). F is a loop factor which can be approximated by 1. Here we have made use of the fact that within mSUGRA the low-energy values of both μ and the squark masses are dominated by M_{1/2} through the renormalization group equations. For large values of tan β the supersymmetric contribution will dominate in B(B_{s} → μ^{+}μ^{−}), and we expect it to scale as |C_{S,P} |^{2} ∝ tan^{6} β/M_{1/2}^{2}. A similar estimate of the supersymmetric contribution to a_{μ} yields δa_{μ} ∝ tan β/M_{1/2}^{2}. Thus within mSUGRA we expect a strong correlation between B(B_{s} → μ^{+}μ^{−}) and δa_{μ}. Furthermore, if an experimental discrepancy in a_{μ} requires a supersymmetric explanantion with large tan β then we would expect B(B_{s} → μ^{+}μ^{−}) to be strongly enhanced. If however the supersymmetric explanation of δa_{μ} requires small M_{1/2} and moderate to small values of tan β then we would expect the relative enhancement of B(B_{s} → μ^{+}μ^{−}) to be much smaller.

During Run-I of the Tevatron CDF found the upper bound [24]

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} [25]. Thus 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.

FIG. 1. The SUSY 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]. Shown also, the SM prediction, the present bound by CDF [24], on B(B_{s} → μ^{+}μ^{−}) aswell as the present 1σ and 2σ bound on δa_{μ} from BNL [2]. f_{B_{s}} has been chosen to be 230 MeV.

FIG. 2. Same as in Fig.1 but with M_{1/2} = 250, M_{0} = 200 GeV.

In Fig.1 we plot the mSUGRA contribution to the anomalous magnetic moment of the muon against B(B_{s} → μ^{+}μ^{−}) for 10 < tan β < 58 ^{∗} 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]^{†} 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 50% for tan β = 50. Hence in order for mSUGRA to account for δa_{μ} within 1σ of the current BNL measurement [2] we need a large value of tan β > ∼ 50. Due to the strong correlation we have shown within mSUGRA we then predict B(B_{s} → μ^{+}μ^{−}) > ∼ 10^{−7}, which should be observed by CDF at Run IIa.

FIG. 3. Contour plots of the B(B_{s} → μ^{+}μ^{−}) (solid) on the plane M_{0}, M_{1/2} 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. Contours of the light Higgs boson mass (red dot-dash line) and of the δa_{μ} [in units 10^{−10}] (dash) are also depicted.

Another interesting feature in Fig.1 is the dip in the curve of B(B_{s} → μ^{+}μ^{−}) for moderate values of 20 < ∼ tan β < ∼ 30. This is the result of a destructive interference of the pure higgs boson contributions (2HDM contributions) and the SUSY box sneutrino-squark diagrams (which are similar in this case) on the one side with the SM contribution. This effect has been described in detail within the context of the 2HDM in [22]. The uncertainty in Fig.1 is mainly given by f_{B_{s}}. Changing

∗The upper limit comes from the REWSB requirement. †This point coincides with the point L quoted in [28].

it in the range 200-260 MeV changes B(B_{s} → μ^{+}μ^{−}) by roughly ∓ 25% with respect to the values shown in Fig.1.

In Fig.3 we show the dependence of B(B_{s} → μ^{+}μ^{−}) on the mSUGRA parameters M_{1/2} and M_{0} for fixed tan β = 50, A_{0} = 0 GeV, and μ > 0. The solid contours represent B(B_{s} → μ^{+}μ^{−}) with resummed SUSY-QCD corrections. The lower left-hand corner in the figure is excluded by the CDF limit given in Eq.(2) and indicated in the figure. If as expected, CDF can probe down to B(B_{s} → μ^{+}μ^{−}) > ∼ 2 · 10^{−7} at RUN IIa, we see from Fig.3 that this corresponds to a sensitivity of M_{1/2} < ∼ 500 GeV and M_{0} < ∼ 1.2 TeV, respectively. The shaded-dotted region represents the 2σ range (11 < δa_{μ} < 75) of the BNL experiment. It nicely reveals the correlation with B(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 directly 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} CDF will probe M_{1/2} = 750 GeV and 1.8 TeV (for tan β = 50) which in mSUGRA corresponds to masses for the heaviest superpartners of 2.4 TeV. We conclude the discussion of Fig.3 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 [29] of a mass around 115.6 GeV results in B(B_{s} → μ^{+}μ^{−}) > ∼ 5 × 10^{−7} which would be measured before the Higgs boson is discovered.

Other contraints here : dark matter b to s gamma e.t.c I think the paper by Roszckowski et.al [14] is enough here Some words also on the recent paper [30]. Herbi could you please write this here??

The large values of tan β we have been considering are theoretically very well motivated within supersymmetric SO(10) Yukawa unification. There it has recently been found [31,32] that a narrow parameter region can explain the observed δa_{μ} while still being consistent with the constraint from b → sγ. This is not within the context of mSUGRA. However, in this parameter region both μ and M_{1/2} are light, while the CP-odd Higgs boson mass is less than 300 GeV, and tan β ≈ 50. Therefore the B_{s} → μ^{+}μ^{−} decay is strongly enhanced and the bound from CDF in Eq.(2) further restricts the allowed parameter space. As an example we determine B(B_{s} → μ^{+}μ^{−}) for the best fitted points found in [32]: 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 is already excluded by CDF [24]. Thus the SO(10) models should be reconsidered in the light of the experimental bound on B(B_{s} → μ^{+}μ^{−}). The only possible escape we can see for these 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. Or, turning it around, if an SO(10) GUT model is the correct description of nature then the decay B_{s} → μ^{+}μ^{−}must be 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} [2] 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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