Resurrecting Minimal Yukawa Sector of SUSY SO(10)

Supersymmetric $SO(10)$ models with Yukawa coupling matrices involving only a $10_H$ and a $\overline{126}_H$ of Higgs fields can lead to a predictive and consistent scenario for fermion masses and mixings, including the neutrino sector. However, when coupled minimally to a symmetry breaking sector that includes a $210_H$ and a $126_H$, these models lead either to an unacceptably small neutrino mass scale, or to non-perturbative values of the gauge couplings. Here we show that with the addition of a $54_H$ to the symmetry breaking sector, the successful predictions of these models for fermion masses and mixings can be maintained. The $54_H$ enables a reduction of the $B-L$ symmetry breaking scale to an intermediate value of order $10^{12}$ GeV, consistent with the observed neutrino mass spectrum, while preserving perturbative gauge coupling unification. We obtain an excellent fit to all fermion masses and mixings in this framework. We analyze carefully the prediction of the model for CP violation in neutrino oscillations. Consistency with proton lifetime, however, requires a mini-split SUSY spectrum with the squarks and sleptons having masses of order 100 TeV, accompanied by TeV scale gauginos and Higgsinos. Such a spectrum may arise from pure gravity mediation, which would predict the partial lifetime for the decay $p \rightarrow \overline{\nu} K^+$ to be an order of magnitude above the current experimental limit.


Introduction
Perhaps the most attractive feature of unified models based on SO(10) gauge symmetry is that all fermions of a given family, including the right-handed neutrino, are assembled into a single representation, the 16-dimensional spinor of SO (10). The Higgs fields that can generate masses for the fermions can be inferred from the fermion bilinears, can be made diagonal and real, whence the other would have 6 complex parameters. In addition to the Yukawa couplings, the fermion mass matrices will involve 3 ratios of vacuum expectation values (VEVs), with one ratio remaining complex. This leads to a total of 12 real parameters and 7 phases.) It is remarkable that this minimal setup is able to reproduce the full fermion mass spectrum, including large leptonic mixing angles along with small quark mixing angles [2][3][4][5][6][7][8][9][10][11][12], a nontrivial feat in any quark-lepton unified framework. To its credit, this setup predicted [4,5,[7][8][9]12] a large value for the reactor neutrino mixing angle θ 13 very close to the experimental value measured subsequently by the Daya Bay collaboration [13]. Choosing any other combinations involving two of these Higgs fields will not lead to viable phenomenology. For example, using two copies of 10 H will lead to nonzero quark mixings, but the unviable mass relations m 0 µ = m 0 s and m 0 e = m 0 d will prevail in this case at the GUT scale. If two copies of 126 H is used instead, unacceptable relations m 0 τ = −3m 0 b and m 0 e = −3m 0 d will emerge. Similarly, combining a 120 H with either 10 H or 126 H will also lead to inconsistent phenomenology.
GeV, while its SM doublet components also acquire weak scale VEVs [1] (10) symmetry down to the SM, while the SU (2) L doublets from these fields along with the doublets from the 10 H break the electroweak symmetry at a lower scale. This model has been extensively studied in the literature, and has been referred to as the minimal SUSY SO(10) model.
It has long been recognized [1,14,15] that this minimal SUSY SO(10) model contains all the essential ingredients to be realistic. Furthermore, it was shown in Ref. [16] that this model has the minimal number of parameters among all supersymmetric grand unified theories. 1 The supersymmetric sector of this model is described by 26 real parameters: 3 + 12 = 15 real Yukawa couplings and 14 − 3 = 11 real superpotential parameters from the symmetry breaking sector W SSB . There are 7 complex parameters in W SSB , of which 3 can be made real. Significant effort has been put into the study of the parameter space of this theory for very good reasons.
2 Problems with the minimal model While the minimal model was found to be very successful in fitting fermion data, it was realized soon thereafter that it faced some hurdles once the symmetry breaking constraints are included [8,[17][18][19]. The overall scale of the right-handed neutrino masses comes out to be V R ∼ (10 12 −10 13 ) GeV from fits to light neutrino masses, while consistency of symmetry breaking of the model requires V R ∼ (10 15 − 10 16 ) GeV. If V R takes values in the range of (10 12 −10 13 ) GeV, then certain colored multiplets from various Higgs fields will have masses of order V 2 R /M GUT ∼ 10 10 GeV, spoiling perturbative gauge coupling unification. There is an independent issue related to proton decay mediated by the color-triplet Higgsinos of the model. The partial lifetime for the decay p → νK + would be shorter than the experimental lower limit of 5.9 × 10 33 years [20] if all the SUSY particles have masses of order TeV. While this is not a problem in itself, as the SUSY particle masses are currently unknown, a realistic version of SUSY SO(10) should also be compatible with this constraint. Assuming TeV-scale superparticle masses, it has been proposed that this issue can be overcome by a cancelation mechanism with the inclusion of a 120 H [21,22] into the minimal model. Another possibility that has been suggested to suppress proton decay rate is via non-perturbative (but uncontrollable) suppressions [23] with an asymptotically safe theory in mind [24]. Here we show explicitly the severity of this constraint in the minimal model (without a 120 H ), if all the SUSY particles have masses around a TeV.
Both of these problems, too small a scale for neutrino masses and too short a lifetime for the proton with TeV superparticles, could in principle be solved in a split-SUSY scenario [25][26][27] with the gaugino-Higgsino masses around 100 TeV and the squark-slepton masses around 10 13 GeV [12]. This would allow for an increase in the light neutrino masses, because the gauge couplings can be kept smaller for higher energies before the bosonic threshold is reached. Simultaneously, d = 5 proton decay rate would be highly suppressed, owing to the large sfermion masses. Unfortunately, this solutions turns out to have various shortcomings, both theoretical and experimental. First, the Higgs boson mass of 125 GeV discovered after this proposal was made does not allow such a large SUSY breaking scale with small Aterms [28,29]. Second, the reactor neutrino mixing angle, although large, was not large enough in [12]. 1 Third, there may be also issues with stability of the spectrum with such large values of the bosonic masses [30], although this may be an issue of naturalness only. Higgs fields needs to be extended, in view the problems it faces. One avenue is to add a 120 H to the theory in which case the fermion mass fits of the model will be significantly affected.

Proposed solution to the problems
Consistency has been shown in this case [21,22,31] (or in the non-supersymmetric case recently [32]). An alternative, which we discuss in this paper, is to modify the symmetry breaking sector without affecting the Yukawa sector. This is achieved by adding a 54 H , without the need for a 120 H . Thus the proposed model has the Higgs content of {10 H + 126 H + 126 H + 210 H + 54 H }.
One can compare the number of parameters that are introduced by the addition of a 54 H with that for a 120 H . Since 54 H has no Yukawa couplings to the fermions in 16, the only new parameters are the 6 new complex couplings that appear in W SSB . After removing one phase by field redefinition of 54 H , we arrive at 26 + 12 − 1 = 37 real parameters, keeping the model still minimal among the renormalizable versions [16]. In contrast, adding a 120 H 1 It has to be said however, that in the fit of Ref. [12] only the upper experimental limit has been put on θ 13 , so a new χ 2 fit with the measured value of θ 13 may well give a better and acceptable value.
The advantage of introducing a 54 H is that the symmetry breaking can proceed in three steps that is consistent with gauge coupling unification: The first step of symmetry breaking occurs at M GU T ≈ 10 16 GeV, which leaves the rank of The problem of rapid proton decay via d = 5 operators can be kept under control only with a mini-split SUSY scenario. We will show that a SUSY spectrum with TeV scale gauginos and Higgsinos and 100 TeV squark and sleptons would suppress the most dangerous mode sufficiently to be in accord with the experimental limit. Such a mini-split SUSY spectrum is compatible with pure gravity mediated SUSY breaking [33,34], which we shall elaborate on. Within such a framework, the partial lifetime for the decay p → νK + is found to be about an order of magnitude above the present experimental lower limit.
We perform a new fit to fermion data, taking into account the two new ingredients of this proposed scenario: a mini-split SUSY case, and the extra intermediate We have found that an excellent fit to all light fermion masses and mixings is possible in this scenario.

The Higgs potential and the vacuum structure
The supersymmetric Higgs sector of SO(10) involving {10 H + 126 H + 126 H + 210 H + 54 H } has been studied in Ref. [35]. We shall follow closely the notation of that paper. The most general renormalizable Higgs superpotential is: 1 By denoting the vacuum expectation values in the Pati-Salam SU(4) c ×SU(2) L ×SU(2) R notation as This symmetry breaks further down to the SM once a nonzero V R is generated. Eq. (4.7) is the condition necessary for inducing a nonzero V R .
The Higgs spectrum of this theory has been worked out in Ref. [35]. It is easy to verify for this spectrum that with V R Here the mass term for σ field and the quartic coupling are determined in terms of the fundamental parameters of Eq. (4.1), but their actual forms are not so relevant for our analysis. After minimizing the potential we get At M GU T we have the following SO(10) relations: Here v u and v d are the VEVs of the MSSM fields H u and H d , and we define as usual While the light neutrino masses receive contributions from type-I seesaw as well as type-II seesaw, the magnitude of the latter turns out to be small. This is because the weak triplet(s) in the model have masses of order the GUT scale. In our analysis we keep only the type-I contribution to neutrino masses.
where we have defined As noted before, this Yukawa sector has 12 real parameters and 7 phases to fit 18 measured quantities. Without loss of generality one can go to a basis where the symmetric matrix F is real and diagonal, whence H becomes a general complex symmetric matrix. The parameters r and c R can be made real, whilst s will remain a complex parameter.
To fit the fermion masses and mixings we perform a χ 2 -analysis, with the pull and the χ 2 -function defined as:     We shall use these best fit parameters in our evaluation of proton lifetime, discussed in the next section.

Masses (in GeV) and
Mixing parameters

Prediction of the model for neutrino CP violation
Corresponding to the best fit, the predicted quantities in our model are presented in Table III. We note that the best fit value of the CP violating parameter δ CP for neutrino oscillations is 17 0 , which is not however a firm prediction. In Fig. 1, we show the variation of this phase corresponding to the case of χ 2 < 20, which should all be acceptable. For example, if δ CP is measured to be 3π/2, the model is still acceptable, with a χ 2 per degree of freedom around 0.9.
is the effective mass parameter for beta-decay and m ββ = | i U 2 ei m i | is the effective mass parameter for neutrinoless double beta decay.

Proton decay calculation
At this point we can estimate the proton decay rate or, better, we can determine the minimal allowed value of the sfermion mass (assumed here for simplicity to be universal) from the proton decay constraint. We assume that these rates are dominated by wino exchange and take as a benchmark the value of its mass to be 1 m wino = 1 TeV. Different fits are possible and the resulting sfemions mass scale m S depends very much on that.
To compute proton decay rate, we define the relevant amplitude functions A ν ijkρ as [47]: 2) and the x, y, z parameters are defined as [48]: Here, M T is the mass matrix for the color triplets which mediate the d = 5 proton decay The rates for the dominant proton decay processes in our model are given by: (7.7) The nuclear matrix elements appearing in Eqs. (7.6)-(7.7) are given in Appendix A, Eq.
(A.14). The values of the dressing functions f (a, b) are given by: We define It is tempting to maximize the proton lifetime with respect to the parameters (x, y, z) appearing in Eq. (7.1), as was done in Ref. [48]. However, we found that maximizing proton lifetime with respect to (x, y, z) does not maximize it with the full set of model parameters. In particular, M −1 T 11 /N 2 d appears in the amplitude, which should be included in the maximization process. The constraints we obtain are much stronger than the ones quoted in Ref. [48].
To get the constraints on the SUSY scale from proton decay we followed the following procedure: • We first set λ 1 = 1 and V 3 = M GU T = 2 × 10 16 GeV. Then we randomly chose the perturbative values of the real input parameters λ 2 , λ 3 , λ 10 , and of the complex input parameters λ 4 , λ 11 , λ 12 , λ 13 . Finally we chose complex V 2 of the same order as as a function of λ 5 , which does not enter in any other quantity and is thus free. We demanded that the choice of parameters reproduce r and s (with at most 10% error) given in Eq. (6.9); • We took the Yukawa couplings from Eqs. (6.10)-(6.11) and calculated the proton decay amplitude.
• With parameters so chosen, we calculated the minimal m S (sfermion mass) which satisfies all bounds for proton decay.
For these choices we get which is within 10% of the original r, s from (6.9).
We thus conclude that with a mini-split SUSY spectrum, the model is compatible with proton lifetime constraints.  8 Embedding of the model in pure gravity mediation As shown by the proton decay analysis, the model requires a heavy SUSY particle spectrum, with lighter gauginos and Higgsinos. The latter will provide a dark matter candidate if the lightest of these fermions mass is of order TeV. While the squark and slepton masses can be varied at will without affecting gauge coupling unification (as they belong to complete multiplets of GUT), the same is not true for the gauginos and to some extend the Higgsinos.
Their masses being at the TeV scale would preserve gauge coupling unification as in the MSSM.
Such a mini-split SUSY spectrum may have a natural origin in pure gravity mediated SUSY breaking [33,34]. In this scheme, all mass parameters, including the µ term for the MSSM Higgsino mass, arise from SUSY breaking mediated by gravity. Gaugino masses are zero at tree-level, but are induced by anomaly mediated contributions, as well as by Higgsino threshold effects. Thus, these models have naturally light gauginos. The mass spectrum of the gauginos is given by [34]: Here terms proportional to L arise from Higgsino threshold. The µ term and the Bµ term in this scheme have the form where c, c are order one coefficients. Thus the Higgsino threshold corrections to the gaugino masses are of the same order as the anomaly mediated corrections. Consequently, the mass ratios M 2 /M 3 is not exactly predicted.
It should be noted that the neutral Wino is the lightest SUSY particle in this scheme, and can serve as the dark matter. For a thermal dark matter, relic abundance would require the Wino mass M 2 to be near 2.7 TeV. However, in these models, Wino dark matter can be produced non-thermally, via the decay of the gravitino, in which case the 2.7 TeV mass constraint does not hold. In fact, the preferred range for Wino dark matter is below 1 TeV [33]. The Wino cannot be below about 200 GeV, as that would modify BBN significantly.
With Wino mass in the range 200 GeV to 1 TeV, the gluino can have mass of order (3 − 20) TeV. The squark and slepton masses are of order (100 − 300) TeV. This is precisely the spectrum preferred in the model from proton decay constraints. Within this scenario, the squark and slepton masses cannot be much above 300 TeV, as opposed to general split-SUSY models. This leads to a prediction for the partial lifetime for p → νK + within the model: It cannot exceed a factor of 10 compared to the current lower limit of 5.9 × 10 33 yrs. Thus the model is testable in proton decay searches. In addition, the Wino LSP and its charged partners may be observable at colliders. Furthermore, flavor violation arising from SUSY particle exchange such as µ → eγ are highly suppressed in this scenario, owing to the large masses of the sleptons and squarks.

Discussion and conclusion
In this paper we have resurrected the minimal Yukawa sector of SUSY SO (10) We have performed a fermion fit in this scenario including threshold effects from V R to M GU T . In addition, proton decay constraints require the SUSY spectrum to be mini-split, with the gauginos and Higgsinos having masses of order TeV and the squarks and sleptons of order 100 TeV. In our new fit to fermion data we have incorporated the mini-split SUSY spectrum as well. Such a scenario has a natural embedding in pure gravity mediation. In this rendition proton lifetime for decay into νK + cannot exceed another order of magnitude compared to the present lower limit of 5.9 × 10 33 yrs. The model can also be tested at colliders with the discovery of Wino LSP along with its charged partners.

A Details of proton decay calculation
Here we provide more details on the proton decay calculation.
Corresponding to the solution of the stationary conditions Eqs. (4.4) -(4.8), the 4 × 4 Higgs doublet mass matrix is given by The matrix in Eq.
The normalization factors are defined as The expressions for p i and q i can be found in a straightforward way from the left and the right eigenvectors corresponding to the zero eigenvalue of Eq. (A.1): With these definitions, the parameters r and s of Eq. (6.6) which are determined from the fermion mass fit can be expressed in terms of the superpotential parameters: In our numerical scan for proton decay amplitude, we impose the constraints on r and s from fermion mass fit in conjunction with Eq. (A.9).
The mass matrix for the color triplets Higgs(inos) mediating d = 5 proton decay is given by The matrix in Eq. (A.10) is written in a basis where the row vector

B RGEs from TeV scale to SUSY scale
In this Appendix we collect the relevant RGEs for split-SUSY relevant for evolution from 1 TeV to the squark and slepton mass scale (m) of order 100 TeV. The 2-loop renormalizationgroup equations for the gauge couplings are given by [25]: where t = lnμ withμ being the renormalization scale. Here the convention used is g 2 1 = (5/3)g 2 . Eq. (B.1) is scheme-independent up to the two-loop order.
In the effective theory belowm, the β-function coefficients are We have extended the one-loop RGE for the d = 5 neutrino mass operator to the case of split-SUSY: Furthermore, the right-handed neutrinos have masses of order V R , and they will contribute to the evolution of parameters above V R . Here we provide the new RGEs including these effects. The coupling of ν c with a SM singlet field σ is defined to be W ⊃ Y R 2 ν c ν c σ.