CERN-TH-2001-xxx July 2001

Stricking correlation between

(g − 2)_{μ} and the B̄_{s} → μ^{+}μ^{−} in the Minimal Supergravity Scenario

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

Abstract

After more than 20 years of both theoretical and experimental effort, Supersymmetry (SUSY) still remains a “theoretical” symmetry. Anxioucly theorists are waiting for the next round of experiments at Tevatron and LHC to see if finnaly this symmetry has something to do with reality. Supersymmetry, although broken at low energies, will reveal her structure indirectly, through deviations from the -succesive after all- Standard Model (SM) predictions and/or directly from decay signals of superparticles. Of course every deviation from the SM explained by SUSY will likely be accompanied with deviations and/or new predictions on other observables. An example of such an observable’s interplay is

i) the recent “hint” from BNL [1] for an approximate 2.6σ diviation on the muon anomalous magnetic moment,

ii) a hint from LEP/SLAC on electroweak precision measurements that the higgs boson should be light compared to TeV scale together with the fact that LEP in the end [2] “prefers” a 115.6 GeV higgs boson mass, and finally

iii) the SUSY prediction for the decay B̄_{s} → μ^{+}μ^{−} could be even three(!) orders of magnitude different than its SM prediction.

How and why these three observables are related in the Minimal Supergravity (mSUGRA) scenario [3] is the question we want to address in this letter. Since the first and the second “hint” have been extensively discussed within the context of the mSUGRA scenario in [4]-[16] and [17] respectively, we will discuss in (some) detail here only the point (iii).

The decays B̄_{s,d} → l^{+}l^{−} are probably the theoretically cleanest modes among the rare B-decays [18]. In the SM they are dominated from Z-boson penguins and box diagrams involving top quark exchanges. Within the SM the Branching fraction, B(B̄_{s} → μ^{+}μ^{−}), is helicity suppressed i.e, proportional to the final state lepton mass, and thus very small. In particular, the SM prediction gives in LO approximation,

2.4 × 10^{−9} < B(B̄_{s} → μ^{+}μ^{−}) < 4.9 × 10^{−9} (SM) ,  (1)

where the error is dominated from the variation of f_{B_{s}} = 230 ± 30 MeV [19] and the variation on |V_{ts}| = 0.04 ± 0.002. In (1) we assume fixed^{1} mass M̄S value for the top quark, m_{t}(m_{t}) = 166.6 GeV which corresponds to a pole mass m_{t} = 175 GeV [there is an additional -rather small- sencitivity of ±0.3 when scanning the top mass in its experimental value]. The above central values together with m_{b}(m_{b}) = 4.25 GeV are used throught this letter. From the experimental point of view we have only an upper bound by CDF [20] which at 95% CL is

B(B̄_{s} → μ^{+}μ^{−}) < 2.6 × 10^{−6} (expt) ,  (2)

which is three orders of magnitude above the SM prediction in (1). CDF Run II will be able to probe this value down to ∼ 10^{−8} and only LHCb [21] or BTeV will reach the SM expectation of (1). Thus any observation of the rare decay mode B̄_{s} → μ^{+}μ^{−} at CDF Run II will definitely mean Physics Beyond the SM. Apart from the new physics motivation (see below), the decay mode B̄_{s} → μ^{+}μ^{−} is an experimental challenge due to its small value predicted from the SM and also experimental favourite thanks to its unique signature.

1In the mSUGRA analysis below, m_{t}(m_{t}) is not fixed due the variation of the strong coupling α_{s} with the SUSY spectrum under the assumption of the unification of the gauge couplings. However, the effect of this variation is small compared to the hadronic uncertainty.

The SM Z-boson penguins and W-box diagrams were calculated first in [22]. SM Higgs penguins found [23] to be suppressed by m_{b}^{2}/M_{W}^{2} . Next to Leading Order (NLO) corrections to B(B̄_{s} → μ^{+}μ^{−}) can be found in [24]. New physics enters the Branching ratio B̄_{s} → μ^{+}μ^{−} through the Higgs-boson exchange diagrams [25]-[30]. In the two Higgs doublet model although there is an enhancement with tan^{2} β from boxes, penguins and self energies involving charged Higgs boson masses in the loop, there is no significant effect on B(B̄_{s} → μ^{+}μ^{−}) for 40 < tanβ < 60 [26, 27]. The importance of the Higgs boson mediated Flavour Changing Neutral Currents in even “flavour blind” SUSY models for large tan β has been noted for the first time in [32]. In the case of B̄_{s} → μ^{+}μ^{−} here, SUSY contributions with chargino and squark loop from self energy diagrams would give a tan^{3} β enhansement [29] which for large tan β leads the B(B̄_{s} → μ^{+}μ^{−}). Chargino and squark boxes and penguins behave like tan^{2} β and give a subleading effect.

Before we describe the corresponding formulae, we should remark that although the above contributions give sizable contributions to B̄ → Kl^{+}l^{−} mode, this is already ruled out from the B̄_{s} → μ^{+}μ^{−} mode [27]. In addition, one of course expects similar enhancement to B(B̄_{d} → μ^{+}μ^{−}) modes but this decay mode is ussually one order of magnitude smaller than the B(B̄_{s} → μ^{+}μ^{−}) which is experimentaly first evident. For B̄_{s} → τ^{+}τ^{−} and B̄_{s} → e^{+}e^{−} one expects branching ratios of order O(10^{−6}) and O(10^{−13}) respectively in the SM [18]. Compination of all these modes together with the observed b → sγ within the mSUGRA scenario is of course a very interesting subject and we postpone it for the future. For recent works on B-meson observables and the muon the anomalous magnetic moment we refer to [33, 34].

After integrating out the electroweak and SUSY scale particles the process B̄_{s} → μ^{+}μ^{−} can be discribed at low energies of order O(5 GeV) with the effective Hamiltonian^{2} for the process b → sμ^{+}μ^{−},

H_{eff} = −\frac{G_{F}}{\sqrt{2}} \frac{α(M_{Z})}{π}V_{tb}V_{ts}^{∗}\{c_{S}(μ)O_{S}(μ) + c_{P} (μ)O_{P} (μ) + c_{A}(μ)O_{A}(μ)\} ,  (3)

with the Scalar, Pseudoscalar and Axial local operators

O_{S} = m_{b}(s̄_{α}P_{R}b_{α})(μ̄μ) , O_{P} = m_{b}(s̄_{α}P_{R}b_{α})(μ̄γ_{5}μ) , O_{S} = (s̄_{α}γ^{μ}P_{L}b_{α})(μ̄γ_{μ}γ_{5}μ) , (4)

and c_{S,P,A} being the corresponding Wilson coefficients. We assume CP-violation only through the CKM phase and we do not take into account additional SUSY CP-violating phases.

analytic formula of the leading Wilson here : Sakis?

2We neglect the operators O′_{S} = m_{s}(s̄_{α}P_{L}b_{α})(μ̄μ) and O′_{P} = m_{s}(s̄_{α}P_{L}b_{α})(μ̄γ_{5}μ) since they are 50 times smaller then the leading m_{b} ones. For large values of tan β the m_{s} tan^{3} β Wilson coefficients, c′_{S,P} could be only comparable with the m_{b} tan^{2} β coefficients c_{S,P} but not competitive to the leading m_{b} tan^{3} β self energy c_{P,S}’s.

Following [27] the branching ratio for the process B̄_{s} → μ^{+}μ^{−} is given by

B(B̄_{s} → μ^{+}μ^{−}) = \frac{G_{F}^{2} α^{2}M_{B_{s}}τ_{B_{s}}}{16π^{3}} |V_{tb}V_{ts}^{∗}|^{2}\sqrt{1 − \frac{4m_{μ}^{2}}{M_{B_{s}}^{2}}} \{(1 − \frac{4m_{μ}^{2}}{M_{B_{s}}^{2}} )|F_{S}|^{2} + |F_{P} + 2m_{μ}F_{A}|^{2}\} ,

(5)

with form factors

F_{S} = −\frac{i}{2}M_{B_{s}}^{2}f_{B_{s}}(\frac{c_{S} m_{b}}{m_{b} + m_{s}} ) , F_{P} = −\frac{i}{2}M_{B_{s}}^{2}f_{B_{s}}(\frac{c_{P} m_{b}}{m_{b} + m_{s}} ) , F_{A} = −\frac{i}{2}f_{B_{s}}c_{10} . (6)

some words on the tanb ressumation here: Uli??

We turn now the discussion into the correlation of the three observables. The muon anomalous magnetic moment in the MSSM is set by the scale of the slepton masses. As we have seen before the B(B̄_{s} → μ^{+}μ^{−}) value is set by the CP-Odd Higgs mass which also sets the light Higgs mass in the MSSM. All these three observables have one common : they are affected substantially by large^{3} 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^{4}. Furthermore, by demanding succesful Radiative Electroweak symmetry breaking (REWSB) [35] in the mSUGRA scenario the CP-Odd Higgs mass gets reduced [17] with large values of tan β. The latter enhances the ratio tan^{3} β/M_{A}^{2} and eventually the braching fraction B(B̄_{s} → μ^{+}μ^{−}).

In Fig.1 we plot the SUSY contribution to the anomalous magnetic moment of the muon against B(B̄_{s} → μ^{+}μ^{−}) for 10 < tan β < 58 ^{5} shown on the top of the figure. We have chosen one specific point[M_{1/2} = 450, M_{0} = 350, A_{0} = 0, μ > 0 and m_{t} = 175 GeV]^{6} of the mSUGRA parameter space. The SM prediction and the CDF [20] for B(B̄_{s} → μ^{+}μ^{−}) (see eq.(1) and eq.(2)) are also shown in the figure. The dashed curve represents the B(B̄_{s} → μ^{+}μ^{−}) result before the ressumation and the solid one after. We obtain a suppresion on the B(B̄_{s} → μ^{+}μ^{−}) around 20% for tan β values around 50. The effect diminishes for small tan β values around 10. The conclusion is straightforward : 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(B̄_{s} → μ^{+}μ^{−}) > ∼ 10^{−7} will be oberved at CDF Run II with 2 fb^{−}1. With 10 fb^{−}1 we expect the sencitivity to reach a branching fraction of 1.3 × 10^{−8} which in turn means that δa_{μ} > ∼ 22 × 10^{−10} i.e, BNL will persist deviating from the SM, CDF will probe tan β values above 40. Another stricking feature from Fig.1 is the destructive interference in B(B̄_{s} → μ^{+}μ^{−}) for moderate values of 20 < ∼ tan β < ∼ 30. This is a result of a conspirancy between the pure higgs boson contributions (2HDM contributions) plus the SUSY box sneutrino+squark diagram [which are acually more or less equal in this case] against the SM contribution to cbs. This effect has been described in detail within the context of the 2HDM in [26] and is analogous in here. The B(B̄_{s} → μ^{+}μ^{−}) becomes just a half of the SM expectation and only LHCb can probe this effect. Finnaly we note that for tan β < ∼ 10 the branching fraction B(B̄_{s} → μ^{+}μ^{−}) reaches its SM expectation. We should mention also that the effect on B(B̄_{s} → μ^{+}μ^{−}) in varying f_{B_{s}} in 200-260 MeV is about ∓ 20% with respect to the values shown in Fig.(1). As a general fact, for large values of tan β > ∼ 35 the self energy Wilson coefficients are 25 to 50 times bigger than the SUSY penguins and boxes as well as the corresponding pure Higgs contributions.

3In 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 B̄_{s} → μ^{+}μ^{−} , tan β > ∼ 40.

4The 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 B̄_{s} → μ^{+}μ^{−} within the f_{B_{s}} error. We leave it out here in order to make the analysis simpler.

5The upper limit comes from the REWSB requirement.

6Actually this point coincides with the point L quoted in [36].

After making the tan β analysis in Fig.(1) for a specific mSUGRA point we now turn to discuss the branching fraction B(B̄_{s} → μ^{+}μ^{−}) and its correlation with the muon g − 2 and the higgs mass on a plane M_{1/2} − M_{0} for fixed tan β = 50, A_{0} = 0 GeV and μ > 0 ^{7} in Fig.(2). The solid contours represent the ressumed in tan β result for the branching fraction B(B̄_{s} → μ^{+}μ^{−}). First of all we obtain that the area bounded by M_{1/2} < ∼ 200 GeV and M_{0} < ∼ 350 GeV is excluded by the CDF limit in eq.(2). Secondly, CDF RUN II with 2fb^{−}1 will probe M_{1/2} and M_{0} up to 500 GeV and 1.2 TeV respectively. Thirdly, if BNL persists deviating from the SM within 2σ (11 < δa_{μ} < 75) the CDF will certainly confirm the evidence for the decay B̄_{s} → μ^{+}μ^{−} . This is the main result of this letter. We must notice here that although 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] will be able prepare the ground for the LHC after the indirect observation of the B̄_{s} → μ^{+}μ^{−} mode. Even better after 10fb^{−}1 the CDF Run II will probe M_{1/2} = 750 GeV and 1.8 TeV which in turn means masses for the heaviest superpartners of 2.4 TeV. As it happens with every observable, the decay B̄_{s} → μ^{+}μ^{−} will be like the SM one if the superpartner masses are heavy. The SM limit of eq.(1) is reached for M_{1/2} > ∼ 1.3 TeV. This region will only be explored at the LHCb. We conclude the discussion of Fig.(2) with the light Higgs boson prediction (dotdashed line) for tan β = 50 in the mSUGRA scenario. We obtain that a possible Higgs mass of less than 119 GeV is also prefered by a two orders of magnitude deviation on the B(B̄_{s} → μ^{+}μ^{−}). The recent “hint” by LEP [2] of a mass qround 115.6 GeV results in B(B̄_{s} → μ^{+}μ^{−}) > ∼ 5×10^{−7} which might be discovered at CDF with before the Higgs boson.

Other contraints here : dark matter b to s gamma e.t.c I think the paper by Roszckowski et.al [15] is enough here Some words also on the recent paper [37]. Herbi could you please write this here??

The real motivation for large tan β values comes from the SO(10) Yukawa unification. It has recently been found [38, 39] that a narrow region consistent with the current g − 2 constraint and b → sγ survives. This region must have a light μ and M_{1/2} parameters. This implies also light CP-Odd Higgs boson mass probably less than 300 GeV. But this is what exactly the B̄_{s} → μ^{+}μ^{−} decay mode needs to be significant. As an application, we find the B(B̄_{s} → μ^{+}μ^{−}) for the best fitted points found in [38]: 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(B̄_{s} → μ^{+}μ^{−}) > ∼ 10^{−5} which already is excluded by CDF [20]. The only way out from the B̄_{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 B̄_{s} → μ^{+}μ^{−} is just around the corner.

7Needed not to comply with g − 2 and b → sγ constraints.

Conclusions here: Herbi, Uli??

References

[1] H. N. Brown et al. [Muon g-2 Collaboration], Phys. Rev. Lett. 86, 2227 (2001) [hepex/0102017]. For a theory review see, A. Czarnecki and W. J. Marciano, Phys. Rev. D 64, 013014 (2001) [hep-ph/0102122].

[2] The ALEPH, DELPHI, L3 and OPAL Collaborations, hep-ex/0107030.

[3] H. P. Nilles, Phys. Lett. B 115 (1982) 193; Nucl. Phys. B 217 (1983) 366; A. H. Chamseddine, R. Arnowitt and P. Nath, Phys. Rev. Lett. 49 (1982) 970; R. Barbieri, S. Ferrara and C. A. Savoy, Phys. Lett. B 119 (1982) 343; L. Hall, J. Lykken and S. Weinberg, Phys. Rev. D 27 (1983) 2359; S. K. Soni and H. A. Weldon, Phys. Lett. B 126 (1983) 215.

[4] J. L. Feng and K. T. Matchev, Phys. Rev. Lett. 86, 3480 (2001) [hep-ph/0102146].

[5] U. Chattopadhyay and P. Nath, Phys. Rev. Lett. 86, 5854 (2001) [hep-ph/0102157].

[6] J. R. Ellis, D. V. Nanopoulos and K. A. Olive, Phys. Lett. B 508, 65 (2001) [hepph/0102331].

[7] R. Arnowitt, B. Dutta, B. Hu and Y. Santoso, Phys. Lett. B 505, 177 (2001) [hepph/0102344].

[8] K. Choi, K. Hwang, S. K. Kang, K. Y. Lee and W. Y. Song, Phys. Rev. D 64, 055001 (2001) [hep-ph/0103048].

[9] S. P. Martin and J. D. Wells, Phys. Rev. D 64, 035003 (2001) [hep-ph/0103067].

[10] D. F. Carvalho, J. R. Ellis, M. E. Gomez and S. Lola, hep-ph/0103256.

[11] Y. G. Kim and M. M. Nojiri, hep-ph/0104258.

[12] J. R. Ellis, S. Heinemeyer, K. A. Olive and G. Weiglein, hep-ph/0105061.

[13] V. Barger and C. Kao, hep-ph/0106189.

[14] G. Belanger, F. Boudjema, A. Cottrant, R. M. Godbole and A. Semenov, hepph/0106275.

[15] L. Roszkowski, R. Ruiz de Austri and T. Nihei, hep-ph/0106334.

[16] A. B. Lahanas and V. C. Spanos, hep-ph/0106345; A. B. Lahanas, D. V. Nanopoulos and V. C. Spanos, hep-ph/0107151.

[17] S. Ambrosanio, A. Dedes, S. Heinemeyer, S. Su and G. Weiglein, hep-ph/0106255 and references therein.

[18] For an excelent review on CP-violation and Rare decays see, A. J. Buras, hepph/9806471.

[19] For the most updated lattice value on f_{B_{s}} see, C. Bernard, Nucl. Phys. Proc. Suppl. 94, 159 (2001) [hep-lat/0011064].

[20] F. Abe et al. [CDF Collaboration], Phys. Rev. D 57, 3811 (1998).

[21] For a review on the B decays at the LHC see for instance, P. Ball et al., hepph/0003238.

[22] T. Inami and C. S. Lim, Anti-K0,” Prog. Theor. Phys. 65, 297 (1981) [Erratum-ibid. 65, 1772 (1981)].

[23] B. Grzadkowski and P. Krawczyk, Z. Phys. C 18, 43 (1983); P. Krawczyk, Z. Phys. C 44, 509 (1989).

[24] G. Buchalla and A. J. Buras, Nucl. Phys. B 412, 106 (1994) [hep-ph/9308272]; M. Misiak and J. Urban, Phys. Lett. B 451, 161 (1999) [hep-ph/9901278].

[25] X. G. He, T. D. Nguyen and R. R. Volkas, Phys. Rev. D 38, 814 (1988); J. L. Hewett, S. Nandi and T. G. Rizzo, Phys. Rev. D 39, 250 (1989); M. J. Savage, Phys. Lett. B 266, 135 (1991); Y. Grossman, Nucl. Phys. B 426, 355 (1994) [hep-ph/9401311]. For B_{s} → τ^{+}τ^{−} see, W. Skiba and J. Kalinowski, Nucl. Phys. B 404, 3 (1993).

[26] H. E. Logan and U. Nierste, Nucl. Phys. B 586, 39 (2000) [hep-ph/0004139].

[27] C. Bobeth, T. Ewerth, F. Kruger and J. Urban, hep-ph/0104284. For the numerical results on B(B̄_{s} → μ^{+}μ^{−}) presented here we follow the formulae of this article.

[28] S. R. Choudhury and N. Gaur, Phys. Lett. B 451, 86 (1999) [hep-ph/9810307].

[29] K. S. Babu and C. Kolda, Phys. Rev. Lett. 84, 228 (2000) [hep-ph/9909476].

[30] P. H. Chankowski and L. Slawianowska, Phys. Rev. D 63, 054012 (2001) [hepph/0008046].

[31] C. Huang, W. Liao, Q. Yan and S. Zhu, Phys. Rev. D 63, 114021 (2001) [hepph/0006250].

[32] C. Hamzaoui, M. Pospelov and M. Toharia, Phys. Rev. D 59, 095005 (1999) [hepph/9807350].

[33] A. Bartl, T. Gajdosik, E. Lunghi, A. Masiero, W. Porod, H. Stremnitzer and O. Vives, hep-ph/0103324.

[34] A. Ali and E. Lunghi, hep-ph/0105200.

[35] L.E. Ibañez and G.G. Ross, Phys. Lett. 110 (1982) 215; K. Inoue, A. Kakuto, H. Komatsu and S. Takeshita, Progr. Theor. Phys. 68 (1982) 927; Progr. Theor. Phys. 71 (1984) 413; J. Ellis, D.V. Nanopoulos and K. Tamvakis, Phys. Lett. B 121 (1983) 123; L. Alvarez-Gaumé, J. Polchinski and M. Wise, Nucl. Phys. B 221 (1983) 495; J. Ellis, J.S. Hagelin, D.V. Nanopoulos and K. Tamvakis, Phys. Lett. B 125 (1983) 275;

[36] M. Battaglia et al., hep-ph/0106204.

[37] J. L. Feng, K. T. Matchev and Y. Shadmi, hep-ph/0107182.

[38] T. Blazek, R. Dermisek and S. Raby, hep-ph/0107097.

[39] H. Baer and J. Ferrandis, hep-ph/0106352.

Figure 1: The predicted SUSY contribution to the muon anomalous magnetic δa_{μ} (bottom), versus the predicted branching ratio B̄_{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 B̄_{s} → μ^{+}μ^{−} together with the SM prediction and the present bound by CDF [20], B̄_{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.

Figure 2: Contour plots (solid lines) of the predicted Branching Ratio B̄_{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.