CERN–TH/2001-211 hep-ph/0108037

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

Athanasios Dedes^{1}, Herbert K. Dreiner^{1} and Ulrich Nierste^{2}

1 Physikalisches Institut, 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 100 than in the Standard Model and well within reach of Run-II of the Tevatron. Thus an experimental search for B_{s} → μ^{+}μ^{−} is a stringent test of the mSUGRA GUT scale boundary conditions. If the decay B_{s} → μ^{+}μ^{−} is observed at Run-II of the Tevatron, then we predict the mass of lightest supersymmetric Higgs boson to be less than 118 GeV. The decay B_{s} → μ^{+}μ^{−} can also significantly probe the favoured parameter range in SO(10) SUSY GUT models, possibly excluding them.

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. The first three quantities are defined at a high, grand unified energy scale and the others at the electroweak scale. They are the boundary conditions for the renormalization group equations, which determine the physical parameters at our low scale. 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 highly predictive so it can be significantly tested by low energy precision measurements. In this letter we show that the decay B_{s} → μ^{+}μ^{−} is a stringent test of the mSUGRA scenario, in particular when correlated with (g − 2)_{μ}.

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]. It further invites a large tan β > ∼ 10 [4]. 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}. It is our purpose here to show the strong correlation between B(B_{s} → μ^{+}μ^{−}) and δa_{μ} in mSUGRA.

SUSY modifies B meson observables if tan β is large, because the b Yukawa coupling becomes sizable. Especially sensitive are quantities with a b quark chirality flip like the branching ratios 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 [5]. 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 it requires an experimental cut on the photon energy, which introduces some hadronic uncertainty.

In [5] 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. This did not include the decay B_{s} → μ^{+}μ^{−}. In contrast to the observables in [5], the branching ratio B(B_{s} → μ^{+}μ^{−}) grows like tan^{6} β [6–8], with a possible several orders of magnitude enhancement. We here go beyond this work to study B_{s} → μ^{+}μ^{−} in the mSUGRA model. Since B(B → ℓ^{+}ℓ^{−}) ∝ m_{ℓ}^{2}, the branching ratio is largest for ℓ = τ. Yet τ’s are hard to detect at hadron colliders, so that the prime experimental focus 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} < ∼ 0.06. Since in B factories the boost of the B_{d} meson is known and the considered leptonic decay rates can be 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 [6–8] the SUSY corrections to B(B_{s} → μ^{+}μ^{−}) were calculated at the one-loop level. For large tan β, higher order corrections can be large, eventually of order 1. In [9] tan β-enhanced supersymmetric QCD corrections have been summed to all orders in perturbation theory. We have incorporated these dominant higher order corrections 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 [9]. 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 CKM matrix element and f_{B_{s}} = (230 ± 30) MeV [10] is the B_{s} decay constant. 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 shortdistance physics, are normalised as in [11]. The coefficients c_{S}, c_{P} and c_{10} defined in [8] 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 NLO result for C_{A} can be well approximated by C_{A} = 2.01(\overline{m}_{t}/167 GeV)^{1.55} [12]. Here \overline{m}_{t} ≡ \overline{m}_{t}(m_{t}) is the top quark mass in the \overline{MS} scheme. \overline{m}_{t} = 167 GeV corresponds to a pole mass of m_{t} = 175 GeV. The SM prediction is given by B(B_{s} → μ^{+}μ^{−}) = (3.7 ± 1.2) · 10^{−9}, with the uncertainty (±25%) dominated by f_{B_{s}}. This is also the main hadronic uncertainty in the SUSY calculation.

During Run-I of the Tevatron CDF determined [13]

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} [14]. 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 implies 250-500 events in this example.

In SUSY the dominant coefficients are C_{S,P} since they are proportional to tan^{3} β. We would like to understand the effect of the restricted mSUGRA parameters on C_{S,P} and thus on B(B_{s} → μ^{+}μ^{−}). In mSUGRA the low-energy values of both μ and the squark masses are dominated by the (GUT scale) value of M_{1/2} through the renormalization group equations. We can thus derive an approximate formula which is good to within 50% for tan β > 45 and M_{1/2} < 600 GeV: |C_{S,P} | ≈ (m_{μ} tan^{3} β)/(4M_{W} M_{1/2}). We therefore expect B(B_{s} → μ^{+}μ^{−}) ∝ |C_{S,P} |^{2} to scale as tan^{6} β/M_{1/2}^{2}. A similar estimate of the supersymmetric contribution to a_{μ} yields (δa_{μ})_{SUSY} ∝ tan βf(M_{0})/M_{1/2}^{2}. (δa_{μ})_{SUSY} depends on slepton masses, which are less sensitive to M_{1/2} than squark masses; they are dominated by M_{0}. We have encoded the M_{0} dependence in the slowly varying function f(M_{0}). From this we conclude that B(B_{s} → μ^{+}μ^{−}) ∝ tan^{5} β (δa_{μ})_{SUSY} /f(M_{0}). It is essential that we have made the assumption of the mSUGRA GUT scale boundary conditions. Thus within mSUGRA we expect a strong correlation between B(B_{s} → μ^{+}μ^{−}) and (δa_{μ})_{SUSY}. Furthermore, if δa_{μ} ≠ 0 requires a SUSY explanation 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 a moderate value of tan β then we would expect only a moderate enhancement of B(B_{s} → μ^{+}μ^{−}). Based on the above approximate analysis we also expect the dependence of B(B_{s} → μ^{+}μ^{−}) on M_{0} to be weak.

FIG. 1. (δa_{μ})_{SUSY}, versus B(B_{s} → μ^{+}μ^{−}) for the input values of tan β (top) and 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 [13], on B(B_{s} → μ^{+}μ^{−}) as well as the present 1σ and 2σ bound on δa_{μ} from BNL [2]. We used f_{B_{s}} = 230 MeV.

We now study these effects quantitatively. For this we use the full computation of Eq.(1) including the resummed SUSY QCD corrections, and restricting ourselves to the mSUGRA parameters. In Fig.1, we show the direct correlation between B(B_{s} → μ^{+}μ^{−}) and (δa_{μ})_{SUSY} for the fixed parameters: M_{1/2} = 450 GeV, M_{0} = 350 GeV, A_{0} = 0, μ > 0 and m_{t} = 175 GeV. (This point coincides with the point L quoted in [15] for tan β = 50.) On the upper edge we show the tan β dependence. We restrict ourselves to tan β < 58 in order to guarantee radiative electroweak symmetry breaking (REWSB). We have included the SM prediction and the CDF bound from Eq.(2). 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 75% for tan β > ∼ 50. In order for mSUGRA to account for δa_{μ} within 1σ of the current BNL measurement at this parameter point, we see that we need a large value of tan β > ∼ 50. Due to the strong correlation within mSUGRA we then predict B(B_{s} → μ^{+}μ^{−}) > ∼ 10^{−7}, which is observable by CDF at Run IIa.

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 SM contribution with the combined pure Higgs boson contributions and the sneutrino-squark box diagrams. This effect has been described in detail within the context of the 2HDM in [11].

FIG. 2. Contours of B(B_{s} → μ^{+}μ^{−}) (solid) and (δa_{μ})_{SUSY} (in units 10^{−10}) (dashed) in the M_{1/2}-tan β plane. The lightest neutral CP-even Higgs mass is shown as well (dot-dashed). The shaded regions are excluded, as described in the text. The mSUGRA parameters are given at the top.

As we discussed above, we expect B(B_{s} → μ^{+}μ^{−}) to dominantly depend on the mSUGRA parameters M_{1/2} and tan β. In Fig.2 we show the B(B_{s} → μ^{+}μ^{−}) (solid) and the (δa_{μ})_{SUSY} (dashed) contours in this plane. We have fixed: M_{0} = 300 GeV, A_{0} = 0, μ > 0 and m_{t} = 175 GeV. The 2σ contours for δa_{μ} (11,75) are explicitly given. The left vertical shaded region is theoretically excluded since it does not allow for REWSB or violates the LEP chargino bound. The upper right triangular shaded region is excluded, since the LSP is not neutral. The upper left-hand corner is excluded by the CDF limit of Eq.(2). If as expected, CDF can probe down to B(B_{s} → μ^{+}μ^{−}) > ∼ 2 · 10^{−7} at RUN IIa, this corresponds to a sensitivity of (M_{1/2}, tan β) ranging from (160 GeV, 36) to (450 GeV, 52). The qualitative discussion of before is now nicely reproduced. B(B_{s} → μ^{+}μ^{−}) decreases with M_{1/2} and rapidly increases with tan β.

Fig.2 also nicely shows the cross-correlation between (δa_{μ})_{SUSY} and B(B_{s} → μ^{+}μ^{−}). If both B(B_{s} → μ^{+}μ^{−}) and (δa_{μ})_{SUSY} are found in disagreement with the SM and are measured with a 50% and 20% accuracy, respectively, then for given M_{0}, this fixes tan β to better than 15% and M_{1/2} to better than 30%.

FIG. 3. Contour plots of the B(B_{s} → μ^{+}μ^{−}) (solid) and on (δa_{μ})_{SUSY} (dashed) the plane M_{0}, M_{1/2} for mSUGRA parameter values shown. The shaded regions are excluded as described in the text. Contours of the light Higgs boson mass (dot-dash line) are also shown.

It is conventional to discuss physics in mSUGRA in the (M_{1/2}, M_{0})-plane. In Fig.3 we show the contours of B(B_{s} → μ^{+}μ^{−}) (solid) and (δa_{μ})_{SUSY} (dashed) in this plane, for tan β = 50, A_{0} = 0, μ > 0 and m_{t} = 175 GeV. Again we include the CDF bound Eq.(2) and the Higgs mass contours. The left shaded region is excluded through the requirement of REWSB or the chargino bound. The lower right shaded region is excluded through the requirement of a neutral LSP. A sensitivity of B(B_{s} → μ^{+}μ^{−}) > ∼ 2 · 10^{−7} at CDF now corresponds to a sensitivity of M_{1/2} < ∼ 450 GeV and M_{0} < ∼ 1.2 TeV, respectively. We also see the weak dependence of B(B_{s} → μ^{+}μ^{−}) on M_{0}.

While CDF is not able to see squark masses directly up to 1.5 TeV (corresponding to M_{1/2} = 500 GeV, 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 10 fb^{−1} CDF will probe M_{1/2} = 700 GeV and M_{0} =1.7 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 [16]. 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. 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.

In our figures we have omitted further constraints on the mSUGRA parameter space, in order to clearly show the correlation between B(B_{s} → μ^{+}μ^{−}) and (δa_{μ})_{SUSY}. The most significant further constraint comes from the measurement of B(B → X_{s}γ) [17], whose prediction is less certain in the large tan β region [18,5]. If we take the conservative approach of [19], then we can exclude values of M_{1/2} < ∼ 250 GeV in Fig.2 for tan β > ∼ 25. In Fig.3 the exclusion curve lies just to the left of the B(B_{s} → μ^{+}μ^{−}) = 5 · 10^{−7} iso curve, and values to the left of this curve are excluded. For a discussion of the constraints from supersymmetric dark matter see for example [4,19] and references therein.

The large values of tan β we have been considering are theoretically well motivated within SUSY SO(10) Yukawa unification. There a narrow parameter region can explain the observed δa_{μ} while still being consistent with the constraint from b → sγ [20,21]. 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 we expect B(B_{s} → μ^{+}μ^{−}) to be strongly enhanced. As an example we determine B(B_{s} → μ^{+}μ^{−}) for the best fitted points found in [21]: M_{A} = 110 GeV, mχ̃_{1} < ∼ 250 GeV, A_{t} < ∼ 5 TeV, mt̃ < ∼ 1 TeV and tan β ≃ 50. Within the hadronic uncertainties B(B_{s} → μ^{+}μ^{−}) > ∼ 10^{−5} which is already excluded by CDF [13]. Thus the SO(10) models should be reconsidered in the light of B(B_{s} → μ^{+}μ^{−}). The only escape we can see is to have M_{A} > ∼ 200 GeV, or smaller values of tan β < 45. One has to then 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 branching ratio B(B_{s} → μ^{+}μ^{−}) in mSUGRA scenarios. If the reported excess in a_{μ} [2] is caused by mSUGRA corrections with large tan β, one faces more than an order of magnitude enhancement of B(B_{s} → μ^{+}μ^{−}) over its SM value. This is within reach of Run-II of the Tevatron. The combined measurements significantly constrain the mSUGRA parameters, allowing a determination of tan β and M_{1/2}. 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 barely compatible with the present upper bound on B(B_{s} → μ^{+}μ^{−}).

A.D. would like to acknowledge financial support from the Network RTN European Program HPRN-CT-2000-00148. We would like to thank G. Isidori, K. Mizukoshi and X. Tata for comparing results.

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