Top Quark and Higgs Boson Masses in Supersymmetric Models

We study the implications for bounds on the top quark pole mass m_t in models with low scale supersymmetry following the discovery of the Standard Model-like Higgs boson. In the minimal supersymmetric standard model, we find that m_t>= 164 GeV, if the light CP even Higgs boson mass m_h = 125 +-2 GeV. We also explore the top quark and Higgs boson masses in two classes of supersymmetric SO(10) models with t-b-tau Yukawa coupling unification at M_GUT. In particular, assuming SO(10) compatible non-universal gaugino masses, setting m_h = 125 GeV and requiring 5% or better Yukawa unification, we obtain the result 172 GeV<= m_t<= 175 GeV. Conversely, demanding 5% or better t-b-tau Yukawa unification and setting m_t=173.2 GeV, the Higgs boson mass is predicted to lie in the range 122 GeV<= m_h<= 126 GeV.


Introduction
With the discovery by the ATLAS [1] and CMS [2] collaborations of a Standard Model (SM)-like Higgs boson with mass m h 125 − 126 GeV, the particle content of the SM is fully verified, and the top quark with mass m t = 173.18 ± 0.94 GeV [3] remains the heaviest particle. The flavor structure of the SM, and, in particular, the relatively heavy top quark mass compared with other fermion masses is an open question in the SM. Likewise, as far as the SM is concerned, there is no deep reason for the Higgs boson to have mass m h 125 − 126 GeV.
At the same time, one of the successes of the SM is a relatively accurate prediction of the top quark pole mass m t from fits to the electroweak data. The top quark 'prediction' deduced from such fits does not change much as a result of the Higgs boson discovery due to a relatively mild dependence on the latter [4,5]. This mild dependence of the electroweak fits on the Higgs boson mass can be understood from the one loop radiative corrections to the W -boson mass. While these corrections depend quadratically on the top quark mass, there is only a mild logarithmic dependence on the SM Higgs boson mass. Employing all experimental data (including m h = 125.7 GeV), fits to the electroweak data at the 95% confidence level yield 168 GeV m t 178 GeV, and 80.3 GeV M W 80.4 GeV [4] for the W -boson mass, in good agreement with the direct measurement of these quantities. Note that while it is possible to constrain the top quark in the SM from rare decays, such constraints are not compatible with the bounds obtained from the determination of the W boson mass [4,5].
Supersymmetry remains a compelling extensions of the SM and the top quark plays a key role in several features of the minimal supersymmetric standard model (MSSM). For instance, the top quark Yukawa coupling is the dominant contributor to the mechanism of radiative electroweak symmetry breaking (REWSB). We discuss in this paper how the REWSB condition restricts the top quark mass as a function of tan β, the ratio of the vacuum expectation values of the two MSSM Higgs doublets. The top quark also plays a crucial role in the calculation of radiative corrections to the lightest CP even Higgs boson mass in the the MSSM. Among other things these corrections, to leading order, are proportional to the fourth power of the top quark mass. In addition, there is a logarithmic dependence on the geometric mean M S of the stop quark masses, as well as a contribution proportional to (A t /M S ) 4 , where A t is a tri-linear soft supersymmetry breaking (SSB) term [6]. We show in this paper that a 125 GeV SM-like Higgs boson provides strong constraints on the allowed mass range for the top quark mass. This mass interval turns out to be compatible with the range obtained from fits to the electroweak data.
In a supersymmetric SO(10) grand unified theory (GUT) with t-b-τ Yukawa unification [7], it was shown in [8] that with suitable non-universal SSB gaugino masses at M GUT the lightest CP even Higgs boson mass is predicted to lie close to 125 GeV.
Such gaugino non-universality at M GUT can arise, for instance, from a non-singlet Fcomponent of the field that breaks supersymmetry. Motivated by this result we seek to provide an answer in this paper to the following question: Can models compatible with t-b-τ Yukawa unification also yield a stringent constraint for the top quark mass in good agreement with the observations?
We consider two classes of SO(10) models with a minimal set of SSB parameters at M GUT in which t-b-τ Yukawa unification is realized. The most well studied one has universal SSB gaugino mass terms but non-universal Higgs SSB terms, m 2 Here m 2 Hu,H d stand for the up/down type Higgs SSB masses 2 . The second class of models, mentioned earlier, assumes universal Higgs SSB mass 2 terms, but the gaugino SSB masses at M GUT are non-universal. Allowing the top quark mass to vary in the interval 0 < m t < 220 GeV, we scan the characteristic SSB parameter space for both classes of models and show that the t-b-τ Yukawa unification condition yields to a relatively narrow interval for the masses of the top quark and the light CP even Higgs boson. An upper bound for the top quark mass is obtained by imposing perturbativity on the top Yukawa coupling up to M GUT . It is possible [9,10] to relax the universality of SSB Higgs mass 2 terms along with the non-universality of SSB gaugino mass terms, but we restrict ourselves here to the above two classes.
The outline for the rest of the paper is as follows. In section 2 we summarize the scanning procedure and the experimental constraints applied in our analysis. In Section 3 we present the results for the SO(10) model in which t-b-τ Yukawa unification is achieved via non-universal SSB gaugino masses and universal up and down Higgs SSB mass 2 terms (m 2 Hu = m 2 H d = m 2 10 ). Using the fact that the light CP even Higgs boson mass is in the interval 123 GeV < m h < 127 GeV, we show that the top quark mass lies in the range 164 GeV < m t < 205 GeV. Imposing 5% or better t-b-τ Yukawa unification, the allowed top quark mass range shrinks to 168 GeV < m t < 180 GeV. In Section 4 we present results for the SO(10) model with universal SSB gaugino mass terms but which also requires that m 2 Hu < m 2 H d [11]. In this case, we find the top quark mass range 165 GeV < m t < 200 GeV. Imposing 5% or better t-b-τ Yukawa unification condition, the interval for the top quark mass is reduced to 170 GeV < m t < 178 GeV. We also consider b-τ Yukawa unification in this case and, as expected, the constraint on the top quark mass is relaxed to 168 GeV < m t < 200 GeV. Our conclusions are presented in Section 5.

Phenomenological constraints and scanning procedure
We employ the ISAJET 7.84 package [12] to perform random scans over the parameter space. In this package, the weak scale values of gauge and third generation Yukawa couplings are evolved to M GUT 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 GUT , since a few percent deviation from unification can be assigned to unknown GUT-scale threshold corrections [13]. With the boundary conditions given at M GUT , all the SSB parameters, along with the gauge and third family Yukawa couplings, are evolved back to the weak scale M Z . In evaluating the Yukawa couplings the SUSY threshold corrections [14] are taken into account at a common scale M S = √ mt L mt R . The entire parameter set is iteratively run between M Z and M GUT using the full 2-loop RGEs until a stable solution is obtained. To better account for the leading-log corrections, one-loop step-beta functions are adopted for the gauge and Yukawa couplings, and the SSB scalar mass parameters m i are extracted from RGEs at appropriate scales m i = m i (m i ).The RGEimproved 1-loop effective potential is minimized at an optimized scale M S , which effectively accounts for the leading 2-loop corrections. Full 1-loop radiative corrections are incorporated for all sparticle masses.
In scanning the parameter space, we employ the Metropolis-Hastings algorithm as described in [15]. The data points collected all satisfy the requirement of radiative electroweak symmetry breaking (REWSB) [16], with the neutralino in each case being the LSP. After collecting the data, we impose the mass bounds on all the particles [17] and use the IsaTools package [18] and SuperIso v2.3 [19] to implement the following phenomenological constraints: We also implement the following mass bounds on the sparticle masses:

SO(10) GUT with non universal gauginos masses
One of the main motivations of supersymmetric SO(10) GUT, in addition to gauge coupling unification, is matter unification. The spinor representation of SO(10) unifies all matter fermions of a given family in a single multiplet (16 i ), which also contains the right handed neutrino (ν R ). Another virtue of SO(10) is that, in principle, the two MSSM Higgs doublets can be accommodated in a single ten dimensional (10 H ) representation, which then yields the following Yukawa couplings Here i, j = 1, 2, 3 stand for family indices and the SO(10) indices have been omitted for simplicity. For the third generation quarks and leptons, the interaction in Eq.(1) yields the following Yukawa coupling unification condition at In gravity mediated supersymmetry breaking scenario [26], implementing Eq.
can place significant constraints on the supersymmetric spectrum [27]. These constraints depend on the particular boundary conditions for sparticle SSB masses chosen at M GUT . If the SSB gaugino mass terms are assumed to be universal at M GUT , m 2 is required in order for Yukawa coupling unification to be consistent with radiative electroweak symmetry breaking (REWSB). This splitting may arise, for example, via a D-term contribution to all scalar masses, or it can be generated via "Just-So" splitting [11]. The results of this scenario are presented in Section 4.
Alternatively, it is possible to achieve t-b-τ Yukawa coupling unification consistent with REWSB by assuming the gaugino SSB mass terms to be non-universal at M GUT . In this case, non-universality among the Higgs SSB mass terms is not needed.
It has been pointed out [28] that non-universal MSSM gaugino masses at M GUT can arise from non-singlet F -terms, compatible with the underlying GUT symmetry. The SSB gaugino masses in supergravity [26] 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, and the indices a, b run over the adjoint representation of the gauge group. The resulting gaugino mass matrix is F ab /M P , where the supersymmetry breaking parameter F ab transforms as a singlet under the MSSM If F transforms as a 54 or 210 dimensional representation of SO(10) [28], one obtains the following relation among the MSSM gaugino masses at M GUT : where M 1 , M 2 , M 3 denote the gaugino masses corresponding to U (1), SU (2) L and SU (3) c , respectively. In order to obtain the correct sign for the desired contribution to (g − 2) µ , we choose M 1 > 0, M 2 > 0 and M 3 < 0. Notice that, in general, if F ab transforms non trivially under SO (10), the SSB terms such as the trilinear couplings and scalar mass terms are not necessarily universal at M GU T . However, we can assume, consistent with SO(10) gauge symmetry, that the coefficients associated with terms that violate the SO(10)-invariant form are suitably small, except for the gaugino term in Eq.(3). We also assume that D-term contributions to the SSB terms are much smaller compared with contributions from fields with non-zero auxiliary F -terms. Employing the boundary condition from Eq.(4), one can define the MSSM gaugino masses at M GUT in terms of the mass parameter M 1/2 : Note that M 2 and M 3 have opposite signs which is important in implementing Yukawa coupling unification as shown in [9]. In order to quantify Yukawa coupling unification, following [29], we define the quantity R tbτ as, We have performed random scans in the fundamental parameter space as follows: Here m 16 is the universal SSB mass for MSSM sfermions, m 10 is the universal SSB mass term for the up/down MSSM Higgs doublets, M 1/2 is the gaugino mass parameter, tanβ is the ratio of the vacuum expectation values (VEVs) of the two MSSM Higgs doublets, A 0 is the universal SSB trilinear scalar interaction (with corresponding Yukawa couplings factored out), m t denotes the top-quark mass and µ > 0 sets the sign for the bi-linear SSB Higgs mixing term whose absolute value is fixed by requiring REWSB.
In Figure 1, we present our results in the m t -tanβ and m t -M S planes. Gray points are consistent with REWSB and neutralino LSP. Blue points form a subset of the gray ones and satisfy sparticle mass bounds and other constraints described in Section 2. Orange points belong to a subset of blue points and satisfy the lightest CP even Higgs boson mass bound 123 GeV ≤ m h ≤ 127 GeV.
It is interesting to note that REWSB can arise even for m t values as low as 10 GeV, but tanβ values are then constrained to be in a narrow range. For a given top quark mass we have a well defined tanβ interval from the requirement of REWSB. For instance, from the m t -tanβ plane, one sees that in order to have REWSB with m t ≈ 10 GeV, the value of tanβ should be around 3 or so. Applying all the collider and B-physics constraints except the lightest Higgs boson mass bound, we obtain for the top quark mass the bound 125 GeV m t 208 GeV (blue points in Figure 1). Applying next the Higgs boson mass bound 123 GeV ≤ m h ≤ 127 GeV, the top quark is expected lie in the interval 164 GeV m t 205 GeV. Similar observations can be made from the m t -M S plane. It is evident from this plane that no matter how heavy the stop mass, the top quark cannot be lighter than 164 GeV in low scale supersymmetry when light CP-even Higgs boson is in this the interval 123 GeV ≤ m h ≤ 127 GeV. This lower bound is very close to the values obtained from fits to the electroweak data [4,5].
In Figure 2 we present the results in the R tbτ -m t and R tbτ -m h planes. The color coding is the same as in Figure 1. The green points in the R tbτ -m h plane form a subset of blue points and satisfy the bound 172.3 GeV ≤ m t ≤ 174.1 GeV.
The horizontal lines in the two planes shown in Figure 2 correspond to 5% t-b- τ Yukawa coupling unification (R tbτ = 1.05). From the R tbτ -m t plane, we see that without requiring any constraints other than REWSB, t-b-τ Yukawa unification better than 5% predicts that the top quark cannot be lighter than 140 GeV. The top quark mass is further constrained by imposing the phenomenological constraints mentioned in Section 2. Even without accounting for the bound on the light CP even Higgs mass (blue points), the top quark mass is found to lie in the range 166 GeV < m t < 192 GeV. While it is not evident from Figure 2, from of the constraints on the top quark mass implemented in blue, the most severe comes from observation of the decay B s → µ + µ − . It is well known that in low scale supersymmetry models, this flavorchanging decay receives contributions from the exchange of the pseudoscalar Higgs boson A [30], and its branching ratio is proportional to (tanβ) 6 /m 4 A . Since t-b-τ Yukawa unification happens for large tanβ(≈ 47) and it prefers relatively small values for the CP odd pseudoscalar mass (m A < 2 TeV) in this model [8], the top quark mass is severely constrained by the B s → µ + µ − decay. Applying the light CP even Higgs mass bound, the top quark is predicted to have a mass in the narrow window 170 GeV ≤ m t ≤ 178 GeV.
Next let us consider the constraints on the Higgs boson mass. From the R tbτ -m h plane, taking into account the collider and B-physics bounds (but excluding the top quark mass bound), t-b-τ YU better than 5% requires that the light CP-even Higgs boson mass m h > 119 GeV. After imposing the 1σ top quark mass bound, the model predicts that the Higgs mass lies in the range 122 GeV ≤ m h ≤ 126 GeV, in good agreement with the current experimental observations. We display the correlation between the top quark and Higgs boson masses in the  Figure 1 with the addition of red points which form a subset of blue points and satisfy R tbτ < 1.05.
presence of Yukawa unification, in the m t -m h plane in Figure 3. The right panel is a zoomed-in version of the left panel in this figure. The color coding is the same as in Figure 1, with the addition of red points which form a subset of blue points and satisfy R tbτ < 1.05. The sharp edge towards the right in the m t -m h plane shows that one requires need a heavy top quark in order to obtain a Higgs boson mass of 125 GeV. We also observe that requiring 5% or better Yukawa unification makes the Higgs and top quark masses strongly correlated. Requiring Yukawa unification along with a 125 GeV Higgs boson mass, one predicts the top quark mass to be in the interval 172 GeV ≤ m t ≤ 175 GeV. Conversely, requiring Yukawa unification along with the top quark mass to be at its experimentally observed central value, the Higgs boson mass is predicted to be in the range 124 GeV ≤ m h ≤ 126 GeV.
While there is a few GeV theoretical error in the calculation of the Higgs mass [31], the strong correlation between the Higgs boson mass, the top quark mass and Yukawa unification seems to be quite compelling in this class of models.

SO(10) GUT with universal gauginos masses
In this section we consider t-b-τ Yukawa unification in a supersymmetric SO(10) GUT model with universal SSB gauginos masses and "Just so" Higgs mass splitting [11]. This GUT scale boundary condition is commonly known as the non-universal Higgs mass (NUHM2) model. The SSB parameters include m 16 , M 1/2 , m H d , m Hu , A 0 , tanβ. This model predicts a heavy sfermion spectrum (m 16 20 TeV) but relatively light gaugino masses [29]. For instance, the gluinos in this scenario are not heavier than 3 TeV or so [32], which can be tested at the LHC. We next investigate the allowed range for the top quark and Higgs boson masses in this scenario.
We have performed random scans for the following ranges of parameters: In Figure 4, we present results in the m t -tanβ and m t -M S planes. The color coding is the same as in Figure 1 In contrast to the SO(10) model discussed in Section 3, one can see from the m ttanβ plane that there is no restriction in this version of SO(10) on either the top quark mass or the value of tanβ from the point of view of REWSB. After applying sparticle mass bounds and constraints from B-physics, the lower bound on the top quark mass is dramatically increased (blue points in Figure 4). The lower bound for blue points in Figure 4 appears to be higher compared to the bounds obtained from Figure 1. However, this is likely due to a lack of statistics, as indicated in the m t -M S plane, Figure 4. Indeed, we have collected far less data in this case compared to the model presented in Section 3. A more exhaustive study will likely fill the regions around the isolated blue points in m t -tanβ plane. The main reason for a lack of extensive statistics for this model is that we are mainly interested in the correlation between the top quark and Higgs boson masses, and this is amply illustrated by the data that we have collected as is obvious by focusing on the orange points in Figure 4. As a reminder to the reader, the orange points form a subset of the blue ones and require the Higgs boson to have a mass in the range 123 GeV ≤ m h ≤ 127 GeV. This constraint on m h requires, in turn, that m t is restricted to lie in the range 164 GeV < m t < 200 GeV.
In Figure 5 we show the results in the R tbτ -m t and R tbτ -m h planes. The color coding is the same as in Figure 4. The green points in the R tbτ -m t plane belong to a subset of blue points and satisfy the 1σ top quark mass bound 172.3 GeV ≤ m t ≤ 174.1 GeV.
As shown in Figure 5, requiring REWSB and t-b-τ Yukawa unification better than 5% requires that m t is heavier than 140 GeV and lighter than 188 GeV. The top quark mass gets confined to the range 166 GeV < m t < 180 GeV after applying the collider and B-physics constraints on the data from Section 2 (excluding the Higgs boson mass bound). This bound is virtually unchanged after applying the experimental bound on the Higgs mass (orange points in the figure). The most severe constraint which gives rise to the blue region comes from the decay b → sγ. This is because in low scale supersymmetry the dominant contribution to the b → sγ branching ratio is proportional to µA t tanβ, while Yukawa unification with universal gaugino masses at M GUT requires A t 15 TeV [29]. The SUSY contribution to BR(B s → µ + µ − ) is suppressed in this case because, as pointed out in [32], m A > 3 TeV in this scenario. The results in the m t -m h plane in Figure 6 show the correlation between the top quark and Higgs boson masses in the presence of Yukawa unification. This figure is similar in spirit to Figure 3 and shares its color coding.
The sharp right edge in the m t -m h plane is retained in this version of the SO(10) model, and it shows that one needs a heavy top quark in order to obtain a 125 GeV Higgs boson. We also note that 5% or better Yukawa unification (red points) makes the Higgs and top quark mass interdependence stronger as in the SO(10) model of Section 3. Requiring the Higgs boson to have a mass of 125 GeV yields a top quark mass in the interval 168 GeV ≤ m t ≤ 177 GeV. However, fixing the top quark mass in this case does not yield a sharp prediction for the Higgs boson mass. For instance, for the measured central value of m t , the Higgs boson mass lies in the range 113 GeV ≤ m h ≤ 131 GeV.
It is interesting to consider b-τ Yukawa unification which may be more natural with non-universal SSB mass terms in the Higgs sector. In Figure 7 we present results in the m t -m h plane with brown points signifying 5% or better b-τ Yukawa unification. The rest of the color coding is the same as in Figure 1. We can see two distinct brown regions corresponding to b-τ Yukawa unification in this figure. The island-like region is nothing but a somewhat larger region than the red region of Figure 6. It is larger because b-τ Yukawa unification is less restrictive than t-b-τ Yukawa unification. Unlike t-b-τ Yukawa unification, b-τ Yukawa unification is realized for both large and small tanβ values. The region formed by the isolated brown points in Figure 7 corresponds to the case of small tan β. As pointed out in [33], b-τ Yukawa unification for small tanβ values requires that the sfermion SSB mass terms are larger than 5 TeV or so. Moreover, the non-universal SSB sfermion mass term terms at M GUT are need to be based on their SU(5) representation. While we do not have non-universal SSB  mass terms characteristic of SU (5), the top quark mass is free in our analysis. As a result, b-τ Yukawa unification does occur for small tanβ values, but it occurs in an experimentally unacceptable region for the top quark mass, namely 200-210 GeV.

Conclusions
We have tried to understand the top quark mass from the requirement that t-b-τ Yukawa coupling unification occurs at M GUT . For this, we consider two SO(10) GUT models, with one model having universal SSB gaugino masses but non-universal SSB Higgs mass terms. In the second example we have universal Higgs SSB mass terms but non-universal SSB gaugino masses. We have also considered the correlation between the Higgs boson and top quark masses. The upper bound m t ∼ 210 GeV or so on the top quark mass comes from requiring perturbativity of the top quark Yukawa coupling up to scale M GUT . Radiative electroweak symmetry breaking imposes a lower bound on the top quark mass, which depends on tan β. In the model with non-universal SSB gaugino masses, applying all the collider and B-physics constraints including the bound on the Higgs boson mass yields the interval 164 GeV m t 205 GeV. Further, imposing Yukawa coupling unification narrows this mass range, namely 172 GeV m t 175 GeV.
For the model with non-universal SSB Higgs mass terms, imposing the collider bounds including the bound from the Higgs boson mass, one arrives at a similar range for the top quark mass as the previously quoted result. However, the allowed top quark mass range after imposing Yukawa coupling unification is somewhat more relaxed than in the previous case, to wit 168 GeV m t 177 GeV, which is quite consistent with the experimental data. Furthermore requiring b-τ instead of t-b-τ Yukawa unification in this model relaxes the top quark range, 167 GeV m t 182 GeV, for large tanβ values. The small tanβ scenario in this model predicts a top quark mass in the neighborhood of 200 GeV or so, and is, therefore, disfavored.
The correlation between the Higgs boson mass and the top quark mass is also very interesting, particularly in the model with non-universal SSB gaugino mass terms. By imposing t-b-τ Yukawa coupling unification and requiring the top quark mass to be close to its observed central value, the Higgs boson mass is founded to lie in the range 124 GeV m h 126 GeV. This correlation between m t and m h from Yukawa unification is not as strong in the model with non-universal SSB Higgs mass terms. In this case we obtain the result 113 GeV m h 131 GeV.