Yukawa unification: MSSM at large tan β

Talk given at PASCOS 2012, Merida, Mexico describing work in progress in collaboration with Archana Anandakrishnan, Christopher Plumberg and Akin Wingerter.


Introduction
In this talk we discuss SO(10) Yukawa unification and its ramifications for phenomenology. The initial constraints come from fitting the top, bottom and tau masses, requiring large tan β ∼ 50 and particular values for soft SUSY breaking parameters. We will show that this leads to a spectrum of squark and slepton masses with the first two families heavy while the third family is significantly lighter, i.e. an inverted scalar mass hierarchy. In addition, gauginos should be observable at the LHC. We perform a global χ 2 analysis, fitting the recently observed 'Higgs' with mass of order 125 GeV in addition to fermion masses and mixing angles and several flavor violating observables.
Fermion masses and mixing angles are manifestly hierarchical. The simplest way to describe this hierarchy is with Yukawa matrices which are also hierarchical. Moreover the most natural way to obtain the hierarchy is in terms of effective higher dimension operators of the form W ⊃ 16 3 10 16 3 + 16 3 10 45 This version of SO(10) models has the nice features that it only requires small representations of SO (10), has many predictions and can, in principle, find an UV completion in string theory.
at M GU T . Note, one CANNOT predict the top mass due to large SUSY threshold corrections to the bottom and tau masses, as shown in [1]. These corrections are of the form Figure 2. The forward-backward asymmetry for the process B → K * µ + µ − measured by LHCb.

Light Higgs mass
An approximate formula for the light Higgs mass is given by [7] where X t = A t − µ/ tan β. The light Higgs mass is maximized as a function of X t for X t /M SU SY = √ 6, referred to as maximal mixing. Hence we see that for large values of A t and M SU SY it is quite easy to obtain a light Higgs mass of order 125 GeV. The arrow corresponds to maximal mixing.
4. B s → µ + µ − In this section we argue that the light Higgs boson must be Standard Model-like. To do this we show that the CP odd Higgs boson, A, must have mass greater than ∼ 1 TeV and as a consequence this is also true for the CP even Higgs boson, H, and the charged Higgs bosons, H ± , as well. This is the well-known decoupling limit in which the light Higgs boson couples to matter just like the Standard Model Higgs.
Consider the branching ratio BR(B s → µ + µ − ) which in the Standard Model is ∼ 3 × 10 −9 . In the MSSM this receives a contribution proportional to ∼ tan β 6 These two results seem to be inconsistent, so we will choose to try and satisfy the LHCb data which is also consistent with results from CMS. Since we have tan β ∼ 50, our only choice is to take the CP odd Higgs mass to be large with m A ≥ 1 TeV. Hence the light Higgs is SM-like.

3 Family Model
The previous results depended solely on SO(10) Yukawa unification for the third family. We now consider a complete three family SO(10) model for fermion masses and mixing, including neutrinos [9,5]. The model also includes a D 3 × [U (1) × Z 2 × Z 3 ] family symmetry which is necessary to obtain a predictive theory of fermion masses by reducing the number of arbitrary parameters in the Yukawa matrices. In the rest of this talk we will consider the new results due to the three family analysis. We shall consider the superpotential generating the effective fermion Yukawa couplings. We then perform a global χ 2 analysis, including precision electroweak data which now includes both neutral and charged fermion masses and mixing angles. The superspace potential for the charged fermion sector of this model is given by: W ch.f ermions = 16 3 10 16 3 + 16 a 10 χ a (10) where 45 is an SO(10) adjoint field which is assumed to obtain a VEV in the B -L direction; and M is a linear combination of an SO(10) singlet and adjoint. Its VEV M 0 (1 + αX + βY ) gives mass to Froggatt-Nielsen states. Here X and Y are elements of the Lie algebra of SO (10) with X in the direction of the U (1) which commutes with SU (5) and Y the standard weak hypercharge; and α , β are arbitrary constants which are fit to the data.
are SO(10) singlet 'flavon' fields, andχ are a pair of Froggatt-Nielsen states transforming as a 16 and 16 under SO(10). The 'flavon' fields are assumed to obtain VEVs of the form After integrating out the Froggatt-Nielsen states one obtains the effective fermion mass operators in Fig. 4. We then obtain the Yukawa matrices for up and down quarks, charged leptons and Note, the superpotential (Eqn. 10) has many arbitrary parameters. However, at the end of the day the effective Yukawa matrices have many fewer parameters. This is good, because we then obtain a very predictive theory. Also, the quark mass matrices accommodate the Georgi-Jarlskog mechanism, such that m µ /m e ≈ 9m s /m d . We then add 3 real Majorana mass parameters for the neutrino see-saw mechanism. The anti-neutrinos get GUT scale masses by mixing with three SO(10) singlets {N a , a = 1, 2; N 3 } transforming as a D 3 doublet and singlet respectively. The full superpotential is given by We assume 16 obtains a VEV, v 16 , in the right-handed neutrino direction, and S a = M a for a = 1, 2 and S 3 = M 3 . The effective neutrino mass terms are given by all assumed to be real. Finally, upon integrating out the heavy Majorana neutrinos we obtain the 3 × 3 Majorana mass matrix for the light neutrinos in the lepton flavor basis given by where the effective right-handed neutrino Majorana mass matrix is given by: with 6. Global χ 2 analysis Just in the fermion mass sector we can see that the theory is very predictive. We have 15 charged fermion and 5 neutrino low energy observables given in terms of 11 sector of the theory. However in order to include the complete MSSM sector we perform the global χ 2 analysis with 24 arbitrary parameters at the GUT scale given in Table 1. Note, this is to be compared to the 27 arbitrary parameters in the Standard Model or the 32 parameters in the CMSSM. In 2007, Albrecht et al. [5] performed a global χ 2 analysis of this theory. Two of the tables from their paper are exhibited in Fig. 6. This analysis included 27 low energy observables and a reasonable fit to the data was only found for m 16 = 10 TeV. Note, the Higgs mass was predicted to be 129 GeV. There is now a discovery by CMS and ATLAS of a boson with mass of order 125 GeV. This may well be the Higgs boson with Standard Model couplings, but further data is needed before such a conclusion can be confirmed. In the mean time we shall assume it is the lightest Higgs boson of the MSSM. In this case, we must conclude (based on our analysis) that it must look very much like the Standard Model Higgs boson, since as can be seen from Fig. 6, the CP odd Higgs mass, m A = 842 GeV, so the theory is in the Higgs decoupling limit. In this work we have decided to extend the analysis of Albrecht et al. to values of m 16 ≥ 10 TeV, including more low energy observables such as the light Higgs mass, the neutrino mixing angle θ 13 and lower bounds on the gluino and squark masses coming from recent data. This is work in progress in collaboration with Archana Anandakrishnan, Christopher Plumberg and Akin Wingerter, along with significant help from Radovan Dermisek and Aditi Raval. We perform a three family global χ 2 analysis. We are using the code developed by Radovan Dermisek to renormalize the parameters in the theory from the GUT scale to the weak scale, perform electroweak symmetry breaking and calculate squark, slepton, gaugino masses, as well as quark and lepton masses and mixing angles. We also use the Higgs code of Pietro Slavich (suitably revised for our particular scalar spectrum) to calculate the light Higgs mass and SUSY Flavor v2.0 [10] to evaluate flavor violating B decays.
There are 24 arbitrary parameters defined mostly at the GUT scale and run down to the weak scale where the χ 2 function is evaluated. However the value of m 16 has been kept fixed in our analysis, so that we can see the dependence of χ 2 on this input parameter. Thus with 23 arbitrary parameters we fit 37 observables, giving 14 degrees of freedom. The χ 2 function has been minimized using the CERN package, MINUIT. This work is on going so we just present the complete results for one value of m 16 = 20 TeV (see Table 2). The fit is quite good with χ 2 /d.o.f. = 1.8. However, note that we have not taken into account correlations in the data, so we will just use χ 2 as a indicator of the rough goodness of the fit. In Table 3 we see that the value of χ 2 decreases as m 16 increases, but our analysis shows that the m 16 ∼ 20 TeV minimizes χ 2 , i.e. we have found that χ 2 slowly increases for m 16 > 20 TeV.

Conclusion and plans for the future
We have presented an analysis of a theory satisfying Yukawa unification and large tan β. The results are encouraging but as usual there are some good and some bad features.
Some of the good features are - • gauge coupling unification is satisfied • the Higgs mass is of order 125 GeV and Standard Model like • there is an inverted scalar mass hierarchy, with the following consequence at the LHC -• gluinos predominantly decay viag → ttχ 0 , tbχ − , bbχ 0 , etc.

And bad features -
• fine-tuned by at least 0.1 % • and gluinos are light.

Work in progress -
• We would like to understand why χ 2 increases for values of m 16 > 20 TeV or is there an upper bound on m 16 ? • The gaugino masses are small. We want to evaluate the χ 2 dependence as a function of the gluino mass or, in other words, ascertain the sensitivity of χ 2 on varying the gaugino masses? • Finally, we would like to consider alternative sets of boundary conditions at the GUT scale which may be consistent with Yukawa unification. In particular, we will consider the so-called "DR3" scheme and non-universal gaugino masses as may be obtained with (a) non-trivial gauge kinetic functions or (b) "mirage mediation." In the "DR3" scheme [11] we have SO(10) boundary conditions at M GU T , however the third family scalar masses are taken to be smaller than the first two, i.e. m (3) 16 ≤ m (1,2) 16 . In addition, Higgs doublet mass splitting is accomplished with a U(1) X D-term, D X , where U(1) X is the   is found in SO(10)/[SU(5) × U(1) X ]. Hence, m 2 i = Q i X D X + (m 0 i ) 2 and the charges, Q i X , are given in Table 4. One possible scheme for obtaining non-universal gaugino masses was contemplated in [12] using a non-trivial gauge kinetic function with a chiral 54 dimensional representation of SO (10) obtaining an F-term VEV, but NO scalar VEV. This gives gaugino masses in the ratio M 3 : M 2 : M 1 = 1 : −3/2 : −1/2. In addition they take µ < 0 and U(1) X D-term splitting of squark, slepton and Higgs masses. In the end they obtain a totally different spectrum than with either "Just So" Higgs splitting or "DR3" splitting.
Finally, it has been shown that non-universal gaugino masses can be obtained with "mirage mediation" boundary conditions at M GU T [13]. We would have gaugino masses given by where α is an arbitrary parameter fixing the ratio of the contributions of gravity vs anomaly mediation. In addition we would take µ < 0 and U(1) X D-term splitting for scalars.