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Supersymmetric SO(10) GUT Models with Yukawa Unification and a Positive Mu Term

Supersymmetric grand unified models based on SO(10) gauge symmetry have many desireable features, including the unification of Yukawa couplings. Using D-term contributions to scalar masses arising from the breakdown of SO(10), Yukawa coupling unification only to 30% can be achieved in models with a positive superpotential Higgs mass. The superparticle mass spectrum is highly constrained, and yields relatively light top squarks and charginos. Surprisingly, the pattern of GUT scale soft SUSY breaking masses are close to those found in the context of inverted hierarchy models. Our analysis supports the idea that the low energy MSSM parameter space is an approximate SO(10) inspired fixed point.

Supersymmetric grand unified theories (SUSYGUTS) based on the gauge group SO (10) are especially attractive [1]. Not only do they unify the three forces of the Standard Model (SM), but they unify the matter content of each generation into a single 16 dimensional irreducible representation of SO(10):ψ (16). Theψ (16) includes not only the matter superfields of the minimal supersymmetric standard model (MSSM), but also a gauge singlet superfieldN c which includes a right handed neutrino. The gauge singlet superpotential mass term M N can be of order M GU T , and leads to sub-eV scale masses for left handed neutrinos (in accord with SuperK results on atmospheric neutrino oscillations), while the right handed neutrinos decouple via the well-known see-saw mechanism [2]. In the simplest models, the two Higgs doublet superfields of the MSSM reside in a single ten dimensional representation φ (10). Then SO(10) SUSYGUT models contain a superpotential interaction term where f is the Yukawa coupling which leads to masses for quarks and leptons (at this stage we neglect intergenerational mixing effects). Thus, the Yukawa couplings of each generation are assumed to be unified above the GUT scale. More sophisticated treatments of the Yukawa matrices can allow for predictions of all SM fermion masses and mixing angles in terms of just a few parameters [3]. Here we will focus only on third generation Yukawa couplings, since they will be large, and can have a substantial impact on the spectrum of superpartners.
In much the same way that the three gauge couplings of the MSSM can be extrapolated from their weak scales values to their GUT scale values using renormalization group (RG) evolution, so too can the Yukawa couplings be evolved from the weak to the GUT scale to test models with Yukawa coupling unification. In the bottom-up approach used in ISAJET v7.58 [4], we begin with the weak scale values of m b , m t and m τ in the DR regularization scheme: m τ (M Z ) = 1.7463 GeV, m b (M Z ) = 2.92 GeV, and m t (m t ) = 163.4 GeV. We calculate the corresponding Yukawa couplings, and evolve both Yukawa and gauge couplings to higher energies using two-loop RG equations (RGEs) [5]. Once M GU T is determined (by the point at which the SU(2) and U(1) gauge couplings meet), we evolve gauge couplings, Yukawa couplings and soft SUSY breaking (SSB) mass terms to the weak scale M weak , where electroweak symmetry is broken radiatively (REWSB), and where the entire SUSY particle mass spectrum can be calculated. At this stage, the Yukawa couplings can be updated to include SM and MSSM loop corrections [6,7], and the RGE process is iterated until a convergent spectrum of SUSY particle masses is obtained. In this way, the GUT scale values of the Yukawa couplings depend on the SUSY particle mass spectrum.
In previous reports [8][9][10], unification of Yukawa couplings was investigated within the context of SO(10) SUSYGUT models. It is well known that in the mSUGRA model, with universal SSB masses at the GUT scale, a high degree of Yukawa coupling unification only occurs for negative values of the superpotential Higgs mass term µ, and for values of the ratio of Higgs field vevs tan β ≡ vu v d ∼ 50 [7,9]. For such high values of tan β, the scalar potential no longer has the appropriate form at the weak scale to accommodate REWSB. Generally, the SSB down Higgs mass squared m 2 H d gets driven to negative values before the up Higgs mass squared m 2 Hu . However, even if scalar masses are universal above the GUT scale, they will receive D-term mass contributions at the GUT scale arising from the breakdown of SO(10) gauge symmetry. Thus, scalar masses are shifted by an amount where M 2 D parametrizes our ignorance of the exact breakdown mechanism for SO(10), and can have either positive or negative values of order M 2 weak . Thus, the model is characterized by the following free parameters: The value of tan β will be restricted by the requirement of Yukawa coupling unification to be close to ∼ 50. In this model, for positive values of M 2 D , the GUT scale values of m 2 Hu and m 2 H d are split, and m 2 Hu gets a head start on running towards negative values. For a sufficiently large value of M 2 D , REWSB can be recovered, and viable supersymmetric models with a high degree of third generation Yukawa coupling can be generated [9].
In previous works, model parameter space was mapped out under the restriction of GUT scale Yukawa unification to 5% [9], and implications of the model for cosmological neutralino relic density, direct detection of dark matter, radiative decays b → sγ and collider searches were determined [10]. A favorable relic density was obtained over much of model parameter space owing to s-channel neutralino pair annihilation via the very wide A and H Higgs poles at large tan β [11]. However, the decay width for b → sγ was found to be very large and generally in disagreement with experimental limits unless one entered the decoupling regime, where model parameters began becoming unnatural. The large b → sγ branching fraction at large tan β and µ < 0 is well known [12]. However, for µ > 0, the b → sγ branching ratio can be in accord with experimental limits at large tan β.
In addition, the recent measurement by the E821 experiment [13] of the muon anomalous magnetic moment a µ = (g − 2)/2 was found to deviate from SM predictions [14] by 2.6σ. If the deviation is interpreted in terms of supersymmetric models, then it disfavors models with µ < 0, assuming positive SSB gaugino masses [15]. For these reasons, it seemed prudent to re-examine Yukawa unification for positive values of µ, relaxing the ad-hoc 5% criteria used in Refs. [9,10].
Our procedure is as follows. We generate random samples of model parameters We then calculate the non-universal scalar masses according to formulae given above, and enter the parameters into the computer program ISASUGRA. ISASUGRA is a part of the ISAJET package [4] which calculates an iterative solution to the 26 coupled RGEs of the MSSM. We required unification of third generation Yukawa couplings at the GUT scale to 45%. Our requirement of 45% unification is determined by defining the variables r bτ , r tb and r tτ , where for instance r bτ = max(f b /f τ , f τ /f b ). We then require R = max(r bτ , r tb , r tτ ) < 1.45. We were able to find many solutions fulfilling the above criteria, with the best overall unification achieved for R = 1.28, or Yukawa coupling unification to 28%. The top and tau Yukawa couplings could frequently unify to very high precision, It is particularly intriguing that model solutions with the best Yukawa coupling unification cluster about model parameters with These particular boundary conditions were found by Bagger et al. from a rather different approach [18], by looking analytically for fixed point behavior in third generation SSB scalar masses which would give rise to SUSY models with a radiatively driven inverted scalar mass hierarchy (RIMH). In these models, one may begin with GUT scale scalar masses beyond the TeV scale. RG evolution drives third generation and Higgs scalar masses towards zero, while scalars of the first two generations remain beyond the TeV range. In this way, multi-TeV first and second generation scalar masses act to suppress SUSY flavor and CP violating processes, while still fulfilling conditions of naturalness, which mainly apply to the sub-TeV third generation and Higgs scalar SSB masses. The RIMH mechanism is viable for SO(10) based models with Yukawa coupling unification upon implementation of the above specific set of scalar mass boundary conditions [19,20]. Our results here are obtained using a bottomup approach, and indicate that for µ > 0, a high degree of Yukawa coupling unification can only be obtained using approximately the boundary conditions Eq  [19,20].
For Yukawa unified solutions with µ < 0, we found a weaker correlation for the soft masses around m 10 = √ 2m 16 [9] but we find no correlation for the trilinear parameter. In Fig. 2, we show values of selected weak scale sparticle masses generated from the Yukawa unified model. In frame a), we show the m A vs. m h plane of Higgs masses. The light scalar h has a mass clustering about the region m h ∼ 115 − 130 GeV, while m A ranges between 100 − 500 GeV, and is generally lower than m A values generated in models with lower values of tan β. Both the h and A (and also the heavy Higgs H) may be accessible to Higgs searches at the Fermilab Tevatron [21], and the low values of m A may give rise to measureable rates for B → µ + µ − decay [22].
In frame b), we show the mt 1 vs. mb 1 plane. We always find mb 1 > mt 1 , in contrast to Yukawa unified models with µ < 0. The value of mt 1 ranges between 100 − 600 GeV for mb 1 < 1 TeV. Frame c) shows the m χ ± 1 vs. mτ 1 plane. We note that mτ 1 ∼ 400 GeV, and is likely accessible to a linear e + e − collider operating with √ s ∼ 800 GeV [23,24].
Finally, in frame d), we show the mẽ R vs. mũ R plane. In this case, mẽ R and mũ R typically lie beyond 1 TeV. Such high mass values for first and second generation scalars can act to suppress many CP violating processes via a decoupling solution; they are generally not sufficiently heavy to suitably suppress the most dangerous flavor violating processes, such as K − K mixing [25]. A sample sparticle mass spectrum is shown in Table 1 for a Yukawa unified model with µ > 0. In this case, a µ = 16.6 × 10 −10 , within the 2σ limit from E821 [13]; for our other models, a µ typically varies around (10 ± 6) × 10 −10 .
The cosmological relic density of neutralinos has been calculated in Ref. [10] for Yukawa unified models with µ < 0. Little should change by switching to µ > 0: the pseudoscalar A and heavy scalar Higgs H will still have large widths of order 20 − 50 GeV due to the large b and τ Yukawa couplings, and will be light enough that χ 0 1 χ 0 1 annihilation can take place efficiently through s-channel annihilation. In addition, the rates for direct detection of relic neutralinos will remain large, as in the µ < 0 case [10]. The rate for b → sγ can be substantially different for µ > 0 compared to the µ < 0 result, and regions of parameter space certainly exist where this decay rate falls within experimental limits. Explicit results for the RIMH model with µ > 0 have been shown in Ref. [20]. Finally, the value of a µ has been calculated in Ref. [15] for Yukawa unified models with µ < 0 and for RIMH models with µ > 0. Regions of model space with an acceptable a µ certainly exist for the µ > 0 case. Further results along these lines will be presented in a forthcoming publication.
Finally, we note that Yukawa unified models with µ > 0 have also been recently reported by Blazek et al. [26]. These authors use a top-down approach and adopt independent values for m Hu and m H d rather than D-term splitting amongst scalar masses. We verify that in this case also Yukawa unified solutions can be obtained, although in our approach these generally occur at the 35-55% level. The solutions typically have m Hu ∼ √ 2m 16 , with m Hu < m H d , and A 0 ≃ −2m 16 .
It is well known that the three standard model gauge couplings approximately unify when extrapolated to the scale M GU T ≃ 2×10 16 GeV. This may be regarded as a coincidence, or as evidence for SUSYGUTs. Similarly, as a consequence of Yukawa unification, the clustering of the SSB parameters about an approximate SO(10) inspired fixed point, if taken seriously, can be considered as evidence for a supersymmetric SO(10) grand unified model.