SO(10) Yukawa Unification with mu<0

We consider the low energy implications including particle spectroscopy of SO(10) inspired t-b-tau Yukawa coupling unification with mu<0, where mu is the coefficient of the bilinear Higgs mixing term of the minimal supersymmetric standard model (MSSM). We imploy non-universal MSSM gaugino masses induced by SO(10) invariant dimension five operators, such that the total number of fundamental parameters is precisely the same as in Yukawa unified supersymmetric SO(10) models with universal gaugino masses and mu>0. We find that t-b-tau Yukawa unification with mu<0 is compatible with the current experimental bounds, including the WMAP bound on neutralino dark matter and the measured value of the muon anomalous magnetic moment. We present a variety of benchmark points which include relatively light squarks (~ TeV) of the first two families and an example in which the bottom and top squarks are lighter than the gluino. This is quite distinct from Yukawa unification with mu>0 and universal gaugino masses in which the gluino is the lightest colored sparticle and the sqaurks of the first two families have masses in the multi-TeV range.


Introduction
Supersymmetric (SUSY) SO(10) grand unified theory (GUT), in contrast to its non-SUSY version, yields third family (t-b-τ ) Yukawa unification via the unique renormalizable Yukawa coupling 16 · 16 · 10, if the Higgs 10-plet is assumed to contain the two Higgs doublets H u and H d of the minimal supersymmetric standard model (MSSM) [1]. The matter 16-plet contains the 15 chiral superfields of MSSM as well as the right handed neutrino superfield. The implications of this Yukawa unification condition at M G ∼ 2 × 10 16 GeV have been extensively explored over the years [1,2]. In SO(10) Yukawa unification with µ > 0 and universal gaugino masses, the gluino is the lightest colored sparticle [3,4], which will be tested [5] at the Large Hadron Collider (LHC). The squarks and sleptons, especially those from the first two families, turn out to have masses in the multi-TeV range. Moreover, it is argued in [3,4] that the lightest neutralino is not a viable cold dark matter candidate in SO(10) Yukawa unification with µ > 0 and universal gaugino masses at M G Spurred by these developments we have investigated t-b-τ Yukawa unification [4,6,7] in the framework of SUSY SU (4) c × SU (2) L × SU (2) R [8] (4-2-2, for short). The 4-2-2 structure allows us to consider non-universal gaugino masses while preserving Yukawa unification. An important conclusion reached in [4,6] is that with same sign non-universal gaugino soft terms, Yukawa unification in 4-2-2 is compatible with neutralino dark matter, with gluino co-annihilation [4,5,6,9] being a unique dark matter scenario for µ > 0.
By considering opposite sign gauginos with µ < 0, M 2 < 0, M 3 > 0 (where µ is the coefficient of the bilinear Higgs mixing term, M 2 and M 3 are the soft supersymmetry breaking (SSB) gaugino mass terms corresponding respectively to SU (2) L and SU (3) c ). It is shown in in [7] that Yukawa coupling unification consistent with the experimental constraints can be implemented in 4-2-2. With µ < 0 and opposite sign gauginos, Yukawa coupling unification is achieved for m 0 300 GeV, as opposed to m 0 8 TeV for the case of same sign gauginos. The finite corrections to the b-quark mass play an important role here [7]. By considering gauginos with M 2 < 0, M 3 > 0 and µ < 0, we can obtain the correct sign for the desired contribution to (g − 2) µ [10]. This enables us to simultaneously satisfy the requirements of t-b-τ Yukawa unification in 4-2-2, neutralino dark matter and (g − 2) µ , as well as a variety of other bounds.
Encouraged by the abundance of solutions and coannihilation channels available in the case of Yukawa unified 4-2-2 with M 2 < 0 and µ < 0, it seems natural to explore Yukawa unification in SO(10) GUT (with M 2 < 0 and µ < 0). It has been pointed out [11] that non-universal MSSM gaugino masses at M G can arise from non-singlet F-terms, compatible with the underlying GUT symmetry such as SU (5) and SO (10). The SSB gaugino masses in supergravity [12] can arise, say, from the following dimension five operator: Here λ a is the two-component gaugino field, F ab denotes the F-component of the field which breaks SUSY, the indices a, b run over the adjoint representation of the gauge group, and M P l = 2.4 × 10 18 GeV is the reduced Planck mass. The resulting gaugino mass matrix is F ab /M Pl where the supersymmetry breaking parameter F ab transforms as a singlet under the MSSM gauge group SU ( The F ab fields belong to an irreducible representation in the symmetric part of the direct product of the adjoint representation of the unified group. In SO(10), for example, If F transforms as a 54 or 210 dimensional representation of SO(10) [11], one obtains the following relation among the MSSM gaugino masses at M G : where M 1 , M 2 , M 3 denote the gaugino masses of U (1), SU (2) L and SU (3) c respectively. The low energy implications of this relation have recently been investigated in [13] without imposing Yukawa unification. The outline for the rest of the paper is as follows. In Section 2 we summarize the scanning procedure and the experimental constraints that we have employed. In Section 3 we present the results from our scan and highlight some of the predictions of an SO(10) model with µ < 0 and the non-universal MSSM gaugino masses at M G related by Eq.(3). We display some benchmark points which can be tested at the LHC. Our conclusions are summarized in Section 4.

Phenomenological Constraints and Scanning Procedure
We employ the ISAJET 7.80 package [14] to perform random scans over the fundamental parameter space. In this package, the weak scale values of gauge and third generation Yukawa couplings are evolved to M G via the MSSM renormalization group equations (RGEs) in the DR regularization scheme. We do not strictly enforce the unification condition g 3 = g 1 = g 2 at M G , since a few percent deviation from unification can be assigned to unknown GUT-scale threshold corrections [15]. The deviation between g 1 = g 2 and g 3 at M G is no worse than 3 − 4%. For simplicity we do not include the Dirac neutrino Yukawa coupling in the RGEs, whose contribution is expected to be small. The various boundary conditions are imposed at M G and all the SSB parameters, along with the gauge and Yukawa couplings, are evolved back to the weak scale M Z . In the evaluation of Yukawa couplings the SUSY threshold corrections [16] are taken into account at the common scale M SUSY = √ mt L mt R . The entire parameter set is iteratively run between M Z and M G using the full 2-loop RGEs until a stable solution is obtained. To better account for leading-log corrections, one-loop step-beta functions are adopted for gauge and Yukawa couplings, and the SSB parameters m i are extracted from RGEs at multiple scales m i = m i (m i ). The RGE-improved 1-loop effective potential is minimized at M SUSY , which effectively accounts for the leading 2-loop corrections. Full 1-loop radiative corrections are incorporated for all sparticle masses. The requirement of radiative electroweak symmetry breaking (REWSB) imposes an important theoretical constraint on the parameter space. In order to reconcile REWSB with Yukawa unification, the MSSM Higgs soft supersymmetry breaking (SSB) masses should be split in such way that m 2 H d /m 2 Hu > 1.2 at M G [17]. As mentioned above, the MSSM doublets reside in the 10 dimensional representation of SO (10) GUT for Yukawa unification condition to hold. In the gravity mediated supersymmetry breaking scenario [12] the required splitting in the Higgs sector can be generated by involving additional Higgs fields [18], or via D-term contributions [19]. Another important constraint comes from limits on the cosmological abundance of stable charged particles [20]. This excludes regions in the parameter space where charged SUSY particles, such asτ 1 ort 1 , become the lightest supersymmetric particle (LSP). We accept only those solutions for which one of the neutralinos is the LSP and saturates the WMAP bound on relic dark matter abundance.
We have performed random scans for the following parameter range: with µ < 0 and m t = 173. 1 GeV [21]. Note that our results are not too sensitive to one or two sigma variation in the value of m t [18]. We use m b (m Z ) = 2.83 GeV which is hard-coded into ISAJET. The set of parameters presented above is usually referred to as NUHM2 [22]. This choice of parameter space was informed by our previous works on t-b-τ Yukawa Unification [6,18].
Employing the boundary condition from Eq. (3) one can define the MSSM gaugino masses at M G in terms of the mass parameter M 1/2 : In scanning the parameter space, we employ the Metropolis-Hastings algorithm as described in [23]. The data points collected all satisfy the requirement of REWSB, with the neutralino in each case being the LSP. After collecting the data, we impose the mass bounds on all the particles [20] and use the IsaTools package [24] to implement the various phenomenological constraints. We successively apply the following experimental constraints on the data that we acquire from ISAJET:

Yukawa Unification and Particle Spectroscopy
We next present the results of the scan over the parameter space listed in Eq.(4).
In Fig. 1 we show the results in the R − m 0 , R − tan β, R − M 1/2 and M 1/2 − m 0 planes. Gray points are consistent with REWSB and neutralino LSP. Green points satisfy particle mass bounds and constraints from BR(B s → µ + µ − ), BR(b → sγ) and BR(B u → τ ν τ ). In addition, we require that green points do no worse than the SM in terms of (g − 2) µ . Orange points belong to a subset of green points and satisfy the WMAP bounds onχ 0 1 dark matter abundance. In the M 1/2 − m 0 plane, points in brown represent a subset of yellow points that are consistent with Yukawa coupling unification to within 10%.
In the R -m 0 plane of Fig. 1 we see that with both µ < 0 and M 2 < 0, we can realize Yukawa unification consistent with all constraints mentioned in Section 2 including the one from (g − 2) µ . This is possible because for µ < 0, we can implement Yukawa unification for relatively small m 0 (∼ 500 GeV), and, in turn, (g − 2) µ obtains the desired SUSY contribution proportional to µM 2 . This is more than an order of magnitude reduction on the m 0 values required for Yukawa unification with µ > 0 and universal gaugino masses. In the present with 10% or better t-b-τ Yukawa unification Figure 1: Plots in R − m 0 , R − tan β, R − M 1/2 and M 1/2 − m 0 planes. Gray points are consistent with REWSB and neutralino LSP. Green points satisfy particle mass bounds and constraints from BR(B s → µ + µ − ), BR(b → sγ) and BR(B u → τ ν τ ). In addition, we require that green points do no worse than the SM in terms of (g − 2) µ . Orange points belong to a subset of green points and satisfy the WMAP bounds oñ χ 0 1 dark matter abundance. In the M 1/2 − m 0 plane, points in brown represent a subset of yellow points and satisfy Yukawa coupling unification to within 10%.
we obtain a relaxation also of A 0 values similar to the SU (4) c × SU (2) L × SU (2) R model in [7], with −2.5 < A 0 /m 0 < 2. Our observation about relaxing the possible range of tan β that accommodates Yukawa unified models is explicitly shown in the R -tan β plane. In the R -m 1/2 plane of Fig. 1 we see that employing the boundary conditions for gauginos presented in Eq. (5), the lightest neutralino mass can be as low as 15 GeV consistent with all constraints mentioned in Section 2 including the one from (g −2) µ . Note that it is impossible to realize a neutralino mass as this in the universal gaugino case due to the chargino mass constraint. A narrow orange strip Figure 2: Plots in the mτ -mχ0 1 , m A -mχ0 1 , mχ± 1 -mχ0 1 , and mq -mg χ planes. The gray points satisfy the requirements of REWSB andχ 0 1 LSP. Dark green points satisfy particle mass bounds and constraints from BR(B s → µ + µ − ), BR(b → sγ) and BR(B u → τ ν τ ). In addition, we require that green points do no worse than the SM in terms of (g − 2) µ . Light green points are a subset of these points which also satisfy Yukawa unification. We show in the mτ -mχ0 1 and mχ± 1 -mχ0 1 planes the unit slope lines representing the respective coannihilation channels. In the m A -mχ0 1 plane we show the line m A = 2mχ0 1 that signifies the A resonance channel.
for low M 1/2 values indicates the existence of Z and light Higgs resonance solutions for neutralino dark matter. Actually, there are two very narrow strips, one around 45 GeV and a second around 60 GeV, even though they appear as one strip in the figure. In order to better visualize the magnitude of the sparticle masses consistent with t-b-τ Yukawa unification, we present our results in the M 1/2 − m 0 plane, where the brown points correspond to Yukawa unification better than 10%.
in the mτ -mχ0 1 , m A -mχ0 1 , mχ± 1 -mχ0 1 , and mq -mg planes. Gray points shown in this figure satisfy the requirements of REWSB andχ 0 1 LSP. Dark green points satisfy the particle mass bounds and constraints from BR(B s → µ + µ − ), BR(b → sγ) and BR(B u → τ ν τ ). the green points do no worse than the SM in terms of (g − 2) µ . The light green points represent a subset of the dark green points, and correspond to 10% or better t-b-τ Yukawa unification. This choice of color coding is influenced from displaying the sparticle spectrum with and without t-b-τ Yukawa unification, while still focussing on all the other experimental constraints. The idea is to show the myriad of solutions that implement Yukawa unification and are consistent with all known experimental bounds except that on relic dark matter density from WMAP. The appearance of a variety of Yukawa unified solutions with a very rich sparticle spectrum is a characteristic feature of µ < 0 [7].
We can see in Fig. 2 that a variety of coannihilation and resonance scenarios are compatible with Yukawa unification and neutralino dark matter. Included in the m A -mχ0 1 plane is the line m A = 2mχ0 1 which shows that the A-funnel region is compatible with Yukawa unification. In the mτ -mχ0 1 plane in Fig. 2, we draw the unit slope line which indicates the presence of stau coannihilation scenarios. From the mχ± 1 -mχ0 1 plane, it is easy to recognize the light Higgs (h) and Z resonance channels. We expect that other coannihilation channels like the stop coannihilation scenario are also consistent with Yukawa unification, although we have not found them, perhaps due to lack of statistics.
Let us remark on the low mass neutralino solutions that we have found in our model (with µ < 0). Because of the M G scale gaugino mass relations in Eq. (5), it is possible in principle to have small M 1 values, thus giving rise to a light neutralino. The neutralino mass nonetheless is bounded from below because of the relic density bounds on dark matter. The SO(10) model with non-universal gaugino masses, as in this paper, has all the ingredients to bring down the neutralino mass to the lowest possible value consistent with the various constraints. The solution with the neutralino (mass ∼ 43 GeV) is consistent with Yukawa unification and corresponds the Z-resonance dark matter scenario.
Finally, in Table 1 we present some benchmark points for the SO(10) t-b-τ Yukawa unified model with µ < 0 and non-universal gaugino masses. All of these points contain WMAP compatible with neutralino dark matter and satisfy the constraints mentioned in Section 2. Point 1 depicts a solution with essentially perfect Yukawa unification corresponding to the A funnel region. Points 2 and 3 correspond to the light Higgs and Z-resonance solutions, while point 4 represents the stau coannihilation solution. It is interesting to note that for the light Higgs and Z-resonance solutions, there is an upper bound on the gluino mass. Employing the boundary condition in Eq.(5) this turns out to be m g ≈ 700 GeV. Hence, the light Higgs and Z-resonance solutions are not compatible with this model if the gluinos are founded to be heavier than ≈ 700 GeV.

Conclusion
We have shown that SO(10) t-b-τ Yukawa unification with µ < 0 and non-universal gaugino masses is nicely consistent with all available experimental data. We have considered a variety of WMAP compatible neutralino dark matter scenarios, including some examples in which the LSP neutralino can be rather light, about half the Zboson mass (Z-resonance solution) or the SM-like Higgs mass (light Higgs resonance solution). Neutralino dark matter solutions corresponding to the A-funnel region and stau-coannihilation are also shown to exist. With µM 2 > 0, the SUSY contributions to the muon anomalous magnetic moment can help provide better agreement than the SM with the experimental data. Finally, in comparison to SO(10) with µ > 0 and universal gaugino masses, there exist some important differences, even though the number of fundamental parameters in the two cases are the same. The lack of WMAP compatible neutralino dark matter in the µ > 0 case is one of them. Also, with µ < 0, we find examples in which the first two squark families are relatively light (∼ TeV), and the third family b and t squarks can be lighter than the gluino (which happens to be the lightest colored sparticle in SUSY SO(10) with µ > 0 ).
Note Added: As we were finishing this work, Stuart Raby pointed out that M. Badziak, M Olechowski and S. Pokorski are also investigating SO(10) Yukawa unification with µ < 0