Peccei-Quinn Symmetry and Nucleon Decay in Renormalizable SUSY SO(10)

We suggest simple ways of implementing Peccei-Quinn (PQ) symmetry to solve the strong CP problem in renormalizable SUSY $SO(10)$ models with a minimal Yukawa sector. Realistic fermion mass generation requires that a second pair of Higgs doublets survive down to the PQ scale. We show how unification of gauge couplings can be achieved in this context. Higgsino mediated proton decay rate is strongly suppressed by a factor of $(M_{\rm PQ}/M_{\rm GUT})^2$, which enables all SUSY particles to have masses of order TeV. With TeV scale SUSY spectrum, $p \rightarrow \overline{\nu} K^+$ decay rate is predicted to be in the observable range. Lepton flavor violating processes $\mu \rightarrow e\gamma$ decay and $\mu-e$ conversion in nuclei, induced by the Dirac neutrino Yukawa couplings, are found to be within reach of forthcoming experiments.


Introduction
Grand unified theories (GUTs) [1][2][3] are some of the best motivated extensions of the Standard Model, and have been extensively studied in the literature. Not only do they unify the various forces of nature, they also unify quarks with leptons and particles with antiparticles. Unification of gauge couplings has been known to work very well in the context of TeV scale supersymmetry (SUSY), motivated independently from the Higgs mass hierarchy perspective. SUSY GUTs based on SO(10) gauge symmetry [4,5] would unify all members of a family of quarks and leptons, including the right-handed neutrino, into a single 16-dimensional multiplet. Owing to such a grouping, SUSY SO(10) models are capable of explaining various features of the observed fermion mass spectrum. In particular, renormalizable SUSY SO(10) models which utilize a single 10 and a single 126 of Higgs fields to generate fermion masses provide an excellent fit to all of quark and lepton masses and mixings, including neutrino oscillation data, with a relatively small number of parameters [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21]. The value of the reactor neutrino mixing angle was predicted in these models before it was measured by the DayaBay collaboration [22], which turned out to be consistent with the prediction. These models may be referred to as SUSY SO(10) models with a minimal Yukawa sector.
GUTs do not shed much insight to the strong CP problem -why the QCD parameter θ takes a value less than 10 −10 . Perhaps the most compelling explanation of the strong CP problem is in terms of the Peccei-Quinn (PQ) symmetry, which is a global U (1) symmetry broken spontaneously by a Higgs field, and also explicitly by the QCD anomaly [23]. The breaking of the U (1) PQ leads to a near massless scalar, the axion [24][25][26][27][28][29], which may constitute a fraction or the entire dark matter in the universe. The spontaneous PQ symmetry breaking scale should be of order 10 11 − 10 12 GeV, to be consistent with direct experimental limits as well as indirect limits from astrophysics and cosmology [30]. It would be of great interest to combine PQ symmetry with SUSY SO(10) models with the minimal Yukawa sector, which has not been done to date. 1 We undertake this task in this paper.
SUSY SU (5) GUT has been extended to include the U (1) PQ symmetry [33]. To achieve this goal in SUSY SO(10) with a minimal Yukawa sector, we identify two key ingredients: (i) an additional 10 of Higgs field, and (ii) a singlet sector that breaks the PQ symmetry in the SUSY limit. Without the additional 10 Higgs field the color triplet partners of the Higgs bosons would survive down to the PQ scale, mediating relatively rapid proton decay (assuming TeV scale masses for SUSY particles). While the SO(10) multiplets 126 and 126 can be utilized to break the PQ symmetry, these fields with their PQ charges being opposite (so that they can have a mass term) would leave a linear combination of U (1) X and 1 For recent works on non-SUSY SO(10) × U (1) PQ models, see Ref. [31,32].
U (1) PQ unbroken, where U (1) X is part of SO(10) gauge symmetry. This surviving global U (1) will only be broken spontaneously at the electroweak scale, leading to a weak scale axion model [24,25], which is excluded by direct experiments such as K L → πa searches [30].
One feature that results in the PQ extension of SUSY SO (10) is that an extra pair of Higgs doublets (H u , H d ) survives down to the PQ scale. If their masses were at the GUT scale, the masses of either the up-type quarks or the down-type quarks ( [33]. One important consequence of the PQ embedding of SUSY SO(10) models is that Higgsino-mediated d = 5 proton decay operators [34,35] become suppressed compared to the corresponding non-PQ models. This is a great bonus, as it has been shown that these d = 5 proton decay operators lead to rather fast proton decay in SUSY SO(10) with the minimal Yukawa sector, assuming that all the SUSY particles have masses of order TeV [20].
This problem prompted the suggestion of a mini-split SUSY spectrum in Ref. [20] with the gauginos having masses of order TeV and scalars having masses of order 100 TeV. The decay rate of the proton would be suppressed by a factor of (M PQ /M GUT ) 2 compared to the results of Ref. [20] in the SUSY SO(10) with PQ symmetry that we present here. This suppression would enable all SUSY particles to have masses of order TeV. This statement would be quantified later on in this paper. Within this setup we find that the decay rate for p → νK + is within reach of ongoing and proposed experiments. With TeV scalars, we also find that lepton flavor violating (LFV) decays µ → eγ and µ − e conversion in nuclei lie in the range that may be observed in forthcoming experiments. Such flavor violations have their origin in the neutrino Dirac Yukawa couplings which are active between M GUT and the B − L symmetry breaking scale v R ∼ 10 12 GeV. Renormalization group flow of SUSY parameters in the momentum range v R ≤ µ ≤ M GUT where ν R 's are active would transfer LFV information to the sleptons, which have masses of order TeV. This in turn would lead to LFV processes such as µ → eγ. The Dirac neutrino Yukawa couplings and the B − L breaking scale are fixed in these models owing to the minimality of the Yukawa sector, leading to crisp predictions for LFV, which depend only on the SUSY particle masses.

SUSY SO(10) with U (1) PQ
In this section we present a viable model that combines a global U (1) PQ symmetry with SO(10) gauge symmetry in the supersymmetric context. The PQ symmetry solves the strong CP problem; it also enables us to realize TeV scale super-particles consistent with proton decay limits. The model we present is an extension of the renormalizable SUSY SO (10) which preserves the minimality of the Yukawa sector. Fermion families, which belong to the 16-dimensional representations, have Yukawa couplings in these models with a single 10 and a single 126 of Higgs superfiels. The Yukawa superpotential of these models is given by:  [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21].
The model has 12 real parameters and 7 phases to fit 18 measured quantities including the neutrino oscillation parameters, leading to certain predictions. In particular, the model prediction for the reactor neutrino angle was borne out by experiments [22].
To complete the symmetry breaking, a 210 and a 126 Higgs fields need to be employed [36][37][38]. Such a model, with {210 + 126 + 126 + 10} of Higgs fields, can separately be consistent with SO(10) symmetry breaking and fermion mass generation. However, when these two requirements are combined, the model does not fare well [13,[39][40][41] for the following reason. A fit to the neutrino oscillation parameters sets the overall right-handed neutrino mass scale, and the (B − L) symmetry breaking scale v R to be (10 12 − 10 13 ) GeV.
If v R is chosen to be in the phenomenologically viable range of (10 12 − 10 13 ) GeV, certain colored Higgs multiplets would acquire masses of order v 2 R /M GUT ∼ (10 8 − 10 9 ) GeV. This would spoil perturbative unification of gauge couplings, making the model inconsistent.
This problem can be resolved, while maintaining the minimality of the Yukawa sector of Eq.   Without the PQ symmetry, SUSY SO(10) models with the minimal Yukawa sector would require a mini-split SUSY spectrum [20] to suppress the decay rate for p → νK + arising via dimension-5 operators mediated by the Higssinos. In the PQ symmetric model, these baryon number violating operators are induced only after the PQ symmetry is spontaneously broken, leading to a suppression factor of (M PQ /M GUT ) 2 in the decay rate. Thus, neither the mini-split SUSY spectrum of Ref. [20] with the minimal Yukawa sector, nor the cancellation mechanism adopted in the color-triplet Higgs Yukawa couplings with an extended Yukawa sector that also includes couplings to a 120 of Higgs boson [42,43], is necessary.
To see the viability of the model and to understand the importance of introducing 10 Hmultiplet, we first write down the superpotential involving the Higgs fields consistent with the PQ charge assignment of In the next subsection we will discuss the breaking of PQ symmetry in the SUSY limit 2 An alternative approach is to add a 120 of Higgs field, which could be used to pair up some of the would-be light states. In this case the Yukawa superpotential of Eq. (2.1) would have an additional coupling matrix [42,43].
that involves SO(10) singlet fields S 1,2,3 . There we will see that all these singlet fields acquire VEVs which will be taken to be of order (10 11 − 10 12 ) GeV. Among these fields, owing to our charge assignment, only S 3 has coupling with the SO(10) non-singlet fields.
With the superpotantial of Eq. (2.2) the mass spectrum of the SM non-singlet fields can be found readily from the results given in Ref. [44]. Certain couplings are absent in our case however, due to the PQ symmetry. One must set m 3 = λ 4 = λ 11 = λ 12 = λ 13 = 0 in the results of Ref. [44]. Due to the presence of an additional 10 H -multiplet, which contains SU (2) L doublet and SU (3) c triplet fields, the weak-doublet and color-triplet mass matrices get altered compared to the results of Ref. [44]. We present these matrices here, which play important roles in the fermion mass fit and dimension-5 baryon number violation. The explicit form of the doublet and triplet mass matrices would also help understand the need for the 10 H -multiplet in the theory. From the superpotential Eq. (2.2) we find the SU (2) L doublet mass matrix to be Here the Φ 1 = (1, 1, 1) , Φ 2 = (1, 1, 15) and Φ 3 = (1, 3, 15) are the VEVs of Φ(210 H ) decomposition. And the color-triplet mass matrix is found to be It is evident from these matrices that if the 10 H field is not present, then the last row and the last column in both the mass matrices would be absent, which would result in one massless state in each sector. Electroweak symmetry breaking would generate mass to these states, which is however phenomenologically unacceptable since the light color-triplet would mediate rapid proton decay. One could avoid this by assigning S 3 a PQ charge such that the superpotential coupling S 3 10 2 H is allowed. In this case, the (1,1) entry will be nonzero in the doublet and triplet mass matrices. The color triplet would then acquire a mass of order the PQ breaking scale, of order 10 11 GeV, still leading to rapid proton decay. To give large mass to the color triplet fields and avoid rapid proton decay, the simplest choice is to extend the Higgs sector by the addition of a 10 H with a PQ charge shown in Table 1. Note that 10 H has no Yukawa coupling with the fermions owing to the PQ charge, so the Yukawa superpotential remains minimal as in Eq. (2.1).
Another important distinction between this PQ symmetric version and the conventional SUSY SO(10) GUTs without PQ symmetry is that a second pair of Higgs doublets would have mass at the PQ scale. Note that in the PQ symmetric limit S 3 → 0, the doublet mass matrix given in Eq. (2.4) would take a bolck-diagonal form: 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. (2.4). We compute these numerically in our estimate of proton decay rate. Two important parameters, denoted as r and s, which appear in the fermion mass fits are determined in terms of these superpotential parameters: The best fit values of these parameters from fermion mass spectrum are given in Eq. (3.8).
In our set-up the symmetry breaking proceeds in three steps: The  Table 2. Since the original PQ symmetry and the U (1) X symmetry commute with SU (5), the P Q charges are the same for all members of a given SU (5) multiplet. The existence of the unbroken global P Q symmetry shows that SO(10) singlet fields are necessary for consistent symmetry breaking.
Without these singlets, the P Q symmetry would be broken by Higgs doublets, leading to weak scale axion excluded by experiments.
fields Table 2: U (1) P Q charges of the color-triplets with P Q = (5P Q − X)/4. Note that except ∆ c and ∆ c multiplets, there is an associated isospin-doublet partner originating from the same Higgs field and carrying identical U (1) P Q charge as that of the color-triplet.

PQ symmetry breaking via singlet fields
To break the PQ symmetry consistently, we introduce three singlet fields S 1,2,3 that carry non-trivial charges under U (1) PQ . Their PQ charges are listed in Table 1. From various experimental bounds, the PQ breaking scale f a is restricted to be within the range 10 10 GeV f a 10 12 GeV. The superpotantial consisting of the SO(10) singlet fields is given by: 3 In the SUSY preserving limit all the F -terms must vanish. They are given by This sets S 1 = 0, and one combination of S 2 and S 3 is fixed. The undetermined VEV leads to a flat direction in S 2,3 . This flat direction is lifted once soft SUSY breaking terms are included. The full potential, including soft SUSY breaking terms of the singlet sector is given by: Straightforward calculation shows that including soft SUSY breaking, all VEVs of the singlet fields are fixed. We find (treating SUSY breaking terms perturbatively) Here we have defined This shows that the singlet sector can consistently break U (1) PQ symmetry. All three singlets acquire masses of order the PQ scale. We shall use the VEV of S 3 as an independent parameter for our fermion mass fit and proton decay calculations, assuming that S 3 ∼ 10 11 GeV. baryon number violating superpotential will take the form A ν ijkρ as [20,[45][46][47][48][49] as:

Suppression of proton decay rate
where the amplitude functionsÂ ν 1bcρ and the Yukawa couplings H (i) , F (i) are defined in Ref. [20]. The parameters x, y, z are defined as [20,50]: (2.26) Here, H and F are Yukawa coupling matrices obtained within the model from a fit to fermion masses and mixings. The best fit values of these matrices from fermion data are given in Eqs. (3.10)-(3.9) in the next section. With these as input one can compute the proton decay rate following the procedure explained in Ref. [20].

Results
In this section we present our results. First we discuss the fit to the fermion masses and mixings, then we present the details of the proton decay rate calculation, then we discuss the gauge coupling unification within our framework, and finally address lepton flavor violation predicted by the model.

Fit to fermion masses and mixings
The Yukawa couplings of the model are given in Eq. (2.1). This is identical to the case of SUSY SO(10) without PQ symmetry. So we closely follow the definitions and the parametrizations of Ref. [20] in the context of non-PQ SO(10) model. Following the same notation as Ref. [20], at the GUT scale, one has the SO(10) relations: Here Y U,D,E are the MSSM Yukawa couplings of up-quarks, down-quarks and charged leptons. We have defined Here   [55]. For neutrino observables, the low energy values are taken from Ref. [56]. With these inputs, we do the RGE running of the Yukawa couplings [57,58], the CKM parameters [59] and the effective couplings of the neutrino d = 5 operator, κ [60][61][62] within the SM up to the SUSY-scale, which we choose to be 1 TeV. Above 1 TeV the full MSSM is restored, so we use the relevant MSSM RGEs [63][64][65] and evolve them up to the GUT scale. For the charged fermion observables, we fit to these evolved values at the GUT scale. Since the requirement from the neutrino data is to have right-handed neutrinos at some lower scale than the GUT scale in the type-I seesaw scenario, in the fit procedure we include the threshold corrections due to the right-handed neutrinos from the v R scale to the GUT scale [20] (see also Ref. [66] For our numerical analysis we fix tan β = 10. Our best fit observables are presented in Table 3. We see from this Table that the model gives an excellent fit to the data, with a total χ 2 = 6.3. Most of the contributions to χ 2 arises from the d-quark mass, which is 2.3 σ below the central value. All other observables are within 1 σ, providing an excellent overall fit.
In Table 4 we list the predictions of the model in the fermion sector for quantities that are currently unknown. The central value of the CP violating Dirac phase relevant for neutrino oscillations is found to be δ P M N S = 327 0 . However, it should be noted that large deviations from the best fit are possible, owing to existence of nearby local minima with acceptable χ 2 . The possible variation of this phase as a function of χ 2 was presented in Fig.   1 of Ref. [20]. Similar results should also hold in our current setup, although we have not investigated this in detail.
In the neutrino sector, the ordering of masses is normal, with the effective mass for neutrinoless double beta decay found to be m ββ 5 meV. While this is well below the current limit from Kamland-Zen experiment of (61 − 165) meV [67], future improvements can potentially probe this prediction.
The Yukawa coupling parameters corresponding to the best fit are given by:   Table 4: Predictions corresponding to the best fit values presented in Table 3 for type-I dominance seesaw scenario. m i are the light neutrino masses, M i are the right handed neutrino masses, α 21,31 are the Majorana phases following the PDG parametrization, m cos = i m i , m β = i |U ei | 2 m i 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
For the proton decay rate calculation, we proceed in the following way: -We first solve the stationary conditions corresponding to the superpotential of Eq.
(2.2). These conditions, which can be found in Ref. [44], are solved for the mass parameters E, Φ 1 , m 1 , m 2 and m 5 . Note that m 5 does not play any role for proton decay calculation. We follow the the symmetry breaking pattern shown in Eq. (2.11).
-We set λ 1 = 1 and Φ 3 = M GUT . We choose M GUT = 2 × 10 16 GeV. Furthermore, we fix v R = v R = S 3 = 10 12 GeV. Then the dimensionless free parameters of the theory are: λ 2 , λ 3 , λ 4 , λ 5 , λ 10 and λ 13 which we take to be real. There is one free parameter with dimension of mass Φ 2 , which is taken to be complex. We scan over this parameter set {λ 2 , λ 3 , λ 4 , λ 5 , λ 10 , λ 13 , Φ 2 } such that the dimensionless couplings are restricted to be |λ -With each parameter set, satisfying all of the above constraints, we compute the proton decay amplitude function defined in Eq. (2.25) that contains the Yukawa couplings, for which we use the best fit values given in Eqs. (3.9)-(3.10) by going to the physical basis of fermions (for details see Ref. [20]).
-By using the amplitude function Eq. (2.25) calculated as above, we estimate the minimum value of the sfermion mass, m S that satisfies all the experimental proton decay bounds: τ p (p → K + ν) ≥ 5.9 × 10 33 years and τ p (p → π + ν) ≥ 3.9 × 10 32 years [68]. For this computation we fix the Wino mass to be m W = 1 TeV. We have taken the relevant nuclear matrix elements from Ref. [69].
In Fig. 2, we present our result on the allowed range for SUSY scalar masses arising from proton lifetime limits obtained by following the procedure outlined above. This figure shows the probability distribution as a function of the minimum required sfermion mass, m S , to satisfy the experimental lower bounds on proton lifetime when m S is varied in the range 3 TeV ≤ m S ≤ 100 TeV. We see that, owing to the suppression in the rate from PQ symmetry, SUSY scalar masses of 3 TeV is fully consistent. Even lower value, e.g., 1 TeV is found to be consistent with proton decay.
A sample point that is consistent with the proton decay bounds for m S = 3 TeV is given by the following parameter set:  We see from the analysis above that proton lifetime for decay into νK + is close to the current experimental limit. While it is difficult to make this statement more precise owing to uncertainties in the superpotential parameters, we expect proton decay to be within reach of ongoing and forthcoming deep underground experiments.

Gauge coupling unification
We now comment on the gauge coupling unification in our model. Note that the automatic gauge coupling unification of the MSSM is not realized in our scenario. This is because of the appearance of an additional SU ( Multiplet, φ Running coefficient    Table 5. For this illustration, we set M SUSY = 1 TeV and used the one-loop RG evolution of gauge couplings. parameter m 5 also does not enter in proton decay calculation and is fixed by one of the stationary conditions). The corresponding unification scale is found to be ∼ 10 16 GeV and the unified gauge coupling is α −1 GUT = 24.46, with these, the expected d = 6 proton decay p → e + π 0 lifetime can be estimated [70] to be τ p ∼ 9 × 10 34 yrs, which is above the current experimental lower limit set by Super-Kamiokande, τ p > 1.6 × 10 34 yrs [71]. In Table. 5, along with the unification constraints, we also estimate the lifetime for the other two cases.
For the case of (6, 1, 4/3), the lifetime for the p → e + π 0 mode came out to be smaller than the present experimental lower limit. However, these are just estimates, since we have not included the threshold corrections from the other fields except the color-sextet. Each of these scenarios can be considered viable once all the threshold corrections are taken into account.

Lepton Flavor Violation
In this section we discuss lepton flavor violating (LFV) rare decays predicted by our model.
where, Y ν is the Dirac neutrino Yukawa coupling in a basis where the charged lepton and right-handed neutrino mass matrices are diagonal. Assuming scalar mass universality and gaugino mass unification as is usually done in constrained MSSSM, the number of parameters in the SUSY breaking sector is reduced to the set: {m 0 , m 1/2 , A 0 , sgn(µ), tan β}, where, m 0 is the common scalar mass, m 1/2 is the common gaugino mass, A 0 the universal tri-linear coupling and µ the Higgs mass term. For LFV calculation, we restrict ourselves to the cMSSM scenario that will also fix the µ parameter from electroweak symmetry breaking condition. We choose sgn(µ) > 0 (although the LFV results are essentially the same for negative µ), and set tan β = 10 corresponding to the fermion fit. The fit to fermion spectrum fully determines the Dirac coupling matrix Y ν and the right-handed neutrino masses M R k in our framework. As a result we can now compute the rates for LFV processes as functions of only the SUSY breaking parameters. With our choice, this set is reduced to {m 0 , m 1/2 , A 0 }. (3.14) Furthermore, the µ term is fixed from the symmetry breaking constraint: Now, the branching ratio of i → j γ is given by: The amplitude functions A L,R can be found in Refs. [73][74][75]. The amplitudes are evaluated in the mass insertion approximation and we include loop contributions from both the neutralinos and charginos. For our model, the relevant amplitude is From our fermion fit presented in Sec. 3.1, we compute:  Br(µ → eγ) < 4.2 × 10 −13 by the MEG experiment and the gray horizontal dashed line represents the projected sensitivity Br(µ → eγ) < 6×10 −14 by the MEG II experiment [76].
figures we see that part of the parameter space is excluded by the current experimental limit, and that future improvement can probe SUSY spectrum as large as 10 TeV.
We note that the branching ratio for the decay τ → µγ is suppressed in our framework.
When we choose parameters to satisfy the current limit on µ → eγ branching ratio, we obtain Br(τ → µγ) = 8 × 10 −11 , which is well below the experimental limit. The rare decay processes i → j γ provides the most stringent constraints on the parameter space of this class of models. Other LFV processes such as i → 3 j and µ − e transition in nuclei get the main contribution from the i → j γ penguin, with the photon attached to a lepton pair or a quark pair. Such diagrams are enhanced by a tan β factor compared to box diagrams, and will dominate. As a result, simple relations for their branching ratios can be derived [74]: Br(µ − e in Ti) ≈ 6 × 10 −3 Br(µ → eγ) and Br( i → 3 j ) ≈ 7 × 10 −3 Br( i → j γ). Future experiments can probe µ−e transition in nuclei that will be sensitive to the level of O(10 −18 ).
These models can be experimentally probed by future experiments in µ − e conversion in nuclei, µ → 3e decay as well as µ → eγ transition.

Discussion and conclusion
In this paper we have presented a U (1) PQ embedding of SUSY SO(10) models. We have adopted renormalizable SO (10)