A gobal fit to the anomalous magnetic moment, b->s gamma and Higgs limits in the constrained MSSM

New data on the anomalous magnetic moment a_mu of the muon together with the b->s gamma decay rate are considered within the supergravity inspired constrained minimal supersymmetric model. We perform a global statistical chi^2 analysis of these data and show that the allowed region of parameter space is bounded from below by the Higgs limit, which depends on the trilinear coupling and from above by the anomalous magnetic moment a_mu. The newest b->s gamma data deviate 1.7 sigma from recent SM calculations and prefer a similar parameter region as the 2.6 sigma deviation from a_mu.


Introduction
Recently a new measurement of the anomalous magnetic moment of the muon became available, which suggests a possible 2.6 standard deviation from the Standard Model (SM) expectation [1]: ∆a µ = a exp µ − a th µ = (43 ± 16) · 10 −10 . The theoretical prediction depends on the uncertainties in the vacuum polarization and the light-by-light scattering, see e.g. the discussion in [2]. However, even with a conservative estimate of the theoretical errors, one has a positive difference ∆a µ of the order of the weak contribution to the anomalous magnetic moment, which opens a window for "new physics". The most popular explanation is given in the framework of SUSY theories [3]- [12], since the contribution of superpartners to the anomalous magnetic moment of the muon is of the order of the weak contribution and allows to explain the desired difference ∆a µ . It requires the Higgs mixing parameter to be positive [4] and the sparticles contributing to the chargino-sneutrino (χ ± − ν µ ) and neutralino-smuon (χ 0 −μ) loop diagrams to be relatively light [3].
The positive sign of µ 0 is also preferred by the branching ratio of the b-quark decaying radiatively into an s-quark -b → X s γ - [13]. Last year the observed value of b → X s γ was close to the SM expectation, so in this case the sparticles contributing to the charginosquark (χ ± −q) and charged Higgs-squark (H ± −q) loops have to be rather heavy in order not to contribute to b → X s γ .  Figure 1: The dependence of the third generation Yukawa couplings at the GUT scale as function of tan β for µ 0 > 0 and µ 0 < 0, obtained by fitting them to the low energy masses of the top, bottom and tau mass. The results are for a common mass m 0 = m 1/2 = 500 GeV, but for different masses the curves look very similar, except that the 'triple' unification point for µ 0 < 0 shifts between 42 and 48, if the common mass is shifted from 200 to 1000 GeV.
As will be shown, the small deviations from the SM for both a µ and b → X s γ require now very similar mass spectra for the sparticles.
In the Constrained Minimal Supersymmetric Model (CMSSM) with supergravity me-diated breaking terms all sparticles masses are related by the usually assumed GUT scale boundary conditions of a common mass m 0 for the squarks and sleptons and a common mass m 1/2 for the gauginos. The region of overlap in the GUT scale parameter space, where both a µ and b → X s γ are within errors consistent with the data, is most easily determined by a global statistical analysis, in which the GUT scale parameters are constrained to the low energy data by a χ 2 minimization. In this paper we present such an analysis within the CMSSM assuming common scalar and gaugino masses and radiatively induced electroweak symmetry breaking. We use the full NLO renormalization group equations to calculate the low energy values of the gauge and Yukawa couplings and the one-loop RGE equations for the sparticle masses with decoupling of the contribution to the running of the coupling constants at threshold. For the Higgs potential we use the full 1-loop contribution of all particles and sparticles. For details we refer to previous publications [18,19].
In principle, one can also require b−τ Yukawa coupling unification, which has a solution at low and high values of the ratio of vacuum expectation values of the neutral components of the two Higgs doublets, denoted tan β = H 0 2 / H 0 1 [18,19]. From Fig. 1 one observes that if the third generation Yukawa couplings at the GUT scale are constrained by the low energy top, bottom and tau masses, they become equal for µ < 0 at tan β ≈ 40, while for µ > 0 they never become equal, although the difference between the Yukawa couplings is less than a factor three. Since µ > 0 is required by ∆a µ > 0 (see below), we do not insist on Yukawa coupling unification and consider tan β to be a free parameter, except for the fact that the present Higgs limit of 113.5 GeV from LEP [20] requires tan β > 4.3 in the CMSSM [13].
We found that the allowed area of overlap between b → X s γ and a µ can be increased considerably for positive values of the common trilinear coupling A 0 at the GUT scale. For A 0 > 0 the present Higgs limit becomes more stringent than for the no-scale models with A 0 = 0, as will be shown.

a µ and b → X s γ in the CMSSM
The contribution to the anomalous magnetic moment of the muon from SUSY particles are similar to that of the weak interactions after replacing the vector bosons by charginos and neutralinos. The total contribution to a µ can be approximated by [3] where m µ is the muon mass, m SU SY is an average mass of supersymmetric particles in the loop (essentially the chargino mass). In our calculations we use the complete oneloop SUSY contributions from [4] with zero phase factors and the additional logarithmic suppression factor as in eq.(1). The calculated value of a µ is shown in Fig. 2 as function of tan β . Clearly, it is approximately proportional to tan β and its sign depends on the sign of µ 0 1 . Only positive values of µ 0 are allowed for the positive deviation from the SM 1 Our sign conventions are as in Ref. [21]. and in addition the sparticles have to be rather light. However, light sparticles contribute also substantially to the b → X s γ decay rate. In the past this posed a conflict. However, if one uses in the b → X s γ calculations the running mass for the charm quark, as suggested recently by Gambino and Misiak, the SM prediction is increased by 11%. In this case the newest world average on b → X s γ is 1.7σ below the SM, as mentioned in the introduction. Such a deviation is most easily obtained for large tan β and not too heavy sparticles, as shown in Fig. 3 [15]- [17] and ∆a µ = (43 ± 16) · 10 −10 [1], which shows once more that b → X s γ and a SU SY µ prefer a relatively light supersymmetric spectrum.
To find out the allowed regions in the parameter space of the CMSSM, we fitted both the b → X s γ and a µ data simultaneously. The fit includes the following constraints: i) the unification of the gauge couplings, ii) radiative elctroweak symmetry breaking, iii) the masses of the third generation particles, iv) b → X s γ and ∆a µ , v) experimental limits on the SUSY masses, vi) the lightest superparticle (LSP) has to be neutral to be a viable candidate for dark matter. We do not impose b − τ unification, since it prefers µ 0 < 0, as shown in Fig. 1, while ∆a µ requires µ 0 > 0, as shown in Fig. 2. Yukawa unification for µ 0 > 0 can only be obtained by relaxed unification of the gauge couplings and non-universality of the soft terms in the Higgs sector [23].
The χ 2 contributions of b → X s γ and the anomalous magnetic moment a µ in the global fit are shown in Fig. 5 for A 0 = 0 and tan β = 35. As expected, the χ 2 contribution from b → X s γ is smallest for heavy sparticles, if b → X s γ is calculated with m c /m b = 0.29, while the minimum χ 2 is obtained for intermediate sparticles, if m c /m b = 0.22 is used. With the newly calculated b → X s γ values, one can see, that b → X s γ and a µ prefer a similar region of the m 0 , m 1/2 plane. Fig. 6 shows the combined χ 2 contributions from b → X s γ and a SU SY µ in the m 0 , m 1/2 plane, both in 3D and 2D, for A 0 = 0 (top) and A 0 free (bottom). In the latter case the lower 2σ contour from b → X s γ moves to the lower left corner, but for the preferred value A 0 ≈ 3m 0 , which is the maximum allowed value in the fit in order to avoid negative stop-or Higgs masses and colour breaking minima, the Higgs bound moves up considerably. The total allowed region is similar in both cases, as shown by the light shaded areas in the contour plots. The 2σ contours from the individual contributions are in good agreement with previous calculations [6,9], but in these paper a simple scan over the parameter space was performed without calculating the combined probability. In addition, A 0 = 0 was assumed.
We repeated the fits for tan β = 20 and 50, as shown in Fig. 7. For smaller values of tan β the allowed region decreases, since a µ becomes too small. At larger tan β values the region allowed by a µ and b → X s γ increases towards heavier sparticles, as expected from Eq. 1, but it is cut by the region where the charged stau lepton becomes the Lightest Supersymmetric Particle (LSP), which is assumed to be stable and should be neutral. A charged stable LSP would have been observed by its electromagnetic interactions after being produced in the beginning of the universe. Furthermore, it would not be a candidate for dark matter. The increase of the LSP-excluded area is due to the larger mixing term between the left-and right handed staus at larger tan β .
We conclude that the a µ measurement strongly restricts the allowed region of the parameter space in the CMSSM, since it excludes the µ 0 < solution, which was the preferred one from b − τ Yukawa unification. In addition, it prefers large tan β with relatively light sparticles, if the present deviation from the SM of 2.6σ persists.
At large tan β a global fit including both b → X s γ and a µ as well as the present Higgs limit of 113.5 GeV leaves a quite large region in the CMSSM parameter space. Here we left the trilinear coupling to be a free parameter, which affects both the Higgs limit constraint and the b → X s γ constraint, but in opposite ways, so that the preferred region is similar for the no-scale models with A 0 = 0 and models which leave A 0 free. The 95% lower limit on m 1/2 is 300 GeV (see Figs. 6+7), which implies that the lightest chargino (neutralino) is above 240(120) GeV. The 95% upper limit on m 1/2 is determined by the lower limit on a SU SY µ and therefor depends on tan β (see Fig. 2). For tan β =35(50) one finds m 1/2 ≤ 610(720) GeV, which implies that the lightest chargino is below 500(590) GeV and the lightest neutralino is below 260(310) GeV.    where the combined χ 2 is below 4. The regions outside this shaded region are excluded at 95% C.L.. The white lines correspond to the "two-sigma" contours, i.e. χ 2 = 4 for that particular contribution. The lower row shows the same for the fit, where A 0 was left free, in which case A 0 ≈ 3m 0 (its maximum allowed value in our fit) is preferred in the region where the stop mixing is important, i.e. regions where the left and right handed stops are not very heavy compared with the top mass. One observes that with A 0 as a free parameter the Higgs limit becomes the most important lower bound on the SUSY sparticles, while for the no-scale models with A 0 = 0 (top) the b → X s γ rate determines mainly the lower bound.