Relation between proton decay and PMNS phase in the minimal SUSY $SO(10)$ GUT

Proton decay is one of the most important predictions of the grand unified theory (GUT). In the supersymmetric (SUSY) GUT, proton decays via the dimension-five operators need to be suppressed. In the $SO(10)$ model where ${\bf 10}+\overline{\bf 126}$ Higgs fields couple to fermions, neutrino oscillation parameters including the CP-violating Pontecorvo-Maki-Nakagawa-Sakata (PMNS) phase can be related to the Yukawa couplings to generate the dimension-five operators in the unified framework. We show how the suppressed proton decay depends on the PMNS phase, and stress the importance of the precise measurements of the PMNS phase as well as the neutrino 23-mixing angle. These become especially important if the SUSY particles are found around less than a few TeV at LHC and proton decays are observed at Hyper-Kamiokande and DUNE experiments in the near future.


Introduction
The measurement of the parameters in the neutrino oscillations is one of the recent major progress in particle physics. Neutrinos must be very light compared to the other fermions in the standard model (SM). The lightness of neutrinos can be explained by the seesaw mechanism [1,2], which can open a window to the ultra high scale physics. In that sense, the precise measurements of the neutrino mixing angles and the PMNS phase [3,4,5,6] are important not only for the scientific interests to archive the accurate values, but also to probe the physics beyond the SM. In fact, their precise measurements are very important to determine the detail structure of the neutrino mass matrix.
The quantum numbers of quarks and leptons are disparate each other, and it has been discussed whether quarks and leptons can be described in a unified framework. In the SO(10) grand unified theory (GUT), the entire fermion species (including a right-handed neutrino) can be unified in one spinor multiplet in each generation, and the description of fermion masses and mixings can be developed in a unified framework. There are several ways to construct a model in the SO (10) GUT. Among them, the model with 126 Higgs representation has an attractive feature that the fermion coupling to 126 can generate both left-and righthanded Majorana neutrino masses as well as a part of the Dirac masses of the fermions. In that model, the smallness of quark mixing angles and two large neutrino mixings can be naturally explained if the 10-dimensional Higgs coupling generates the third generation Yukawa couplings dominantly [7].
Proton decay is one of the most important predictions of GUTs. The supersymmetry (SUSY) can solve the hierarchy problem between the weak scale and the GUT scale. The SUSY GUTs, however, suffer from constraints of dimension-five proton decay operators [8], which can be generated by the Yukawa interaction via color triplet Higgs exchange: More than 10 years ago, one of the authors and his collaborators discussed how the proton decay amplitudes can be suppressed by the specific Yukawa structure, which can be connected to the parameters in the neutrino oscillations, and claimed that the neutrino 13-mixing angle θ 13 cannot be zero and θ 13 ∼ 0.1 under the condition [9]. In these arguments, it was assumed that light neutrino mass matrix is dominated by the type II contribution from the left-handed Majorana neutrino mass. In Ref. [10], we describe the sum rule of the fermion mass matrices in the form that the coupling matrices are given by the parameters in the lepton sector. Using the algebraic expression, we argue that the similar Yukawa structure can be applied even if the type I seesaw contribution is incorporated to the neutrino mass matrix in the case where the vacuum expectation value (VEV) of 126, v R , is large.
In this paper, we describe the Yukawa structure which is suitable to suppress the proton decay amplitude in the SO(10) model, and study how the Yukawa structure depends on the neutrino oscillation parameters. In order to explain the proton decay suppression, we adopt the minimal SO(10) model, in which the 10 + 126 Higgs fields couple to fermions [11,12,13,14,15,16,17,18]. We use the χ 2 fits in the minimal SO(10) model, obtained in Ref. [10] to show the consequence numerically. In the case where v R > ∼ 10 16 GeV, the suppressed proton decay width depends on the parameters, and we show the numerical calculation in terms of the neutrino 23-mixing angle and the PMNS phase. The accurate measurements of the oscillation parameters are important if the dimension-five proton decays are suppressed but it will be observed in the near future.
2 The importance of the PMNS phase to the structure of the neutrino mass matrix CP violation of the neutrino oscillations is important to learn the nature of the fundamental symmetry. Not only because of the fundamental curiosity, but also to understand the structure of the neutrino mass matrix the precise measurement of the PMNS phase is necessary.
In the quark sector, the mixing angles are all small, and the hierarchical structure of the quark mass matrices can be simply obtained. The possible exception is the (1,1) element of the mass matrix. When the mass matrix is reconstructed from various observations, there may be a cancellation in the (1,1) element. It is well-known that there is an empirical relation between the quark mass ratio and the Cabibbo angle which may be related to the smallness of the (1,1) element (compared to down quark mass) of the down-type quark mass matrix in the basis where up-type quark mass matrix is diagonal. In the lepton sector, on the other hands, the neutrino mixing angles are large and the hierarchical structure is not obvious yet. As in the quark sector, there may be a cancellation when the neutrino mass matrix is reconstructed from observations. The neutrino mass matrix (in the basis where the charged lepton mass matrix is diagonal) is expressed as where U is the unitary mixing matrix determined by three neutrino mixing angles θ 12 , θ 23 , θ 13 and a phase δ PMNS . Using a convention by Particle Data Group [19], we obtain the (1,1) and (1,2) elements as (M ν ) 11 = (m 1 cos 2 θ 12 + m 2 sin 2 θ 12 ) cos 2 θ 13 + e −2iδ PMNS m 3 sin 2 θ 13 , (M ν ) 12 = cos θ 13 [(m 2 − m 1 ) cos θ 12 sin θ 12 cos θ 23 +e −iδ PMNS sin θ 13 sin θ 23 (m 3 − e 2iδ PMNS (m 1 cos 2 θ 12 + m 2 sin 2 θ 12 ) .
In this expression, mass eigenvalues m 1 , m 2 , m 3 are complex, and one of three can be made to be real. Because sin θ 13 and ∆m 2 sol /∆m 2 atm are the same order, it is possible that both (1,1) and (1,2) elements are cancelled 1 in the normal hierarchy (NH) case, ∆m 2 sol /∆m 2 atm ∼ m 2 /m 3 . We remark that the PMNS phase has to be chosen if those elements vanishes contrary to the quark quark sector, and thus the accurate measurement of the PMNS phase is quite important to determine the structure of the neutrino mass matrix.
Let us suppose that both (1,1) elements and (1,2) elements are completely zero [20,21] (M ν ) 11 = (M ν ) 12 = 0, to see what condition is needed to suppress those elements. There are four equations for real degree of freedom. We obtain a relation among the phase δ PMNS and mixing angles θ 12 , θ 23 , θ 13 and the ratio of the mass squared differences ∆m 2 12 /∆m 2 23 , as well as |m 1 | and two Majorana phases of the mass eigenvalues. For the point of view that the constraint can be verified from the neutrino oscillations, the relation among the phase, mixing angles and masses is the most important. We obtain [22,23] cos δ PMNS = The atmospheric neutrino oscillations constrains sin 2θ 23 (up to matter effects), and thus tan θ 23 still has a large ambiguity. We plot the relation between δ PMNS and θ 23 in Fig.1 using the experimental measurements for the other parameters. The experimental global fit results are overlapped in the figure. One can see that the current experimental results are consistent with the conditions of the suppressed (1,1) and (1,2) elements. About a decade ago, one of the authors (Y.M.) and his collaborators argued that the (1,1) and (1,2) elements should be small and the 13-mixing angle is predicted to be a non-zero value and about 0.1 if the proton decay amplitudes are suppressed in a renormalizable SO(10) model [9]. Now, the 13-mixing angle is accurately measured, and the key ingredient of the suppression shifts to the accurate values of the PMNS phase and 23-mixing angle. While the PMNS phase lie around δ PMNS /π ∼ −0.5 in the global analysis, the 23-mixing angle still have ambiguity depending on which experiments have a major weight. In fact, the global analysis in Ref. [5,6] shows that the best fit lies at θ 23 < π/4, however, it seems that the current best fit without the short-baseline reactor experiments lies at θ 23 > π/4. If the 13-mixing angle Figure 1: The plot from the relation in Eq. (6). Along the green line, the (1,1) and (1,2) elements can become zero in the neutrino mass matrix in the basis where the charged-lepton mass matrix is diagonal. The dotted lines are drawn using the 3σ range of the mixing parameters. We also overlap the current 1σ (red), 2σ (blue), 3σ (orange) region of the global analysis in Ref. [6]. The star symbol shows the current best fit of the global analysis. is 1σ smaller than the current center value, the best fit would shift to θ 23 > π/4 even in the global analysis. More experimental data are needed.
3 The proton decay suppression in the minimal SO(10) model In the SO(10) models, the structure of the neutrino mass matrix can be related to the Yukawa couplings of the charged fermions. Therefore, proton decay amplitudes can be related to the neutrino mixing parameters. In this section, we explain how the smallness of the (1,1) and (1,2) elements discussed in the previous section is connected to the proton decay suppression using the minimal SO(10) model. The minimal SO(10) model [11] is defined as that in which the Yukawa interaction is minimal, namely, only H : 10 and∆ : 126 Higgs representations couple to the fermions ψ : 16 by renormalizable interaction: Due to the SO(10) algebra, the coupling matrices are symmetric, h ij = h ji and f ij = f ji . The minimality assumption is often addressed even to the Higgs sector to break the SO (10) symmetry, but we impose the minimality just on the Yukawa interactions in this paper. The Yukawa coupling (after SO (10) symmetry is broken down to SM) is given by linear combinations of the h and f matrices, and the fermion mass matrices are obtained as where r and r 2 depend on the Higgs mixing (doublet Higgs mixing in 10 and 126), and h and f are original Yukawa matrices h and f multiplied by Higgs mixings, where the unitary matrices U and V are the diagonalizing matrices of the doublet Higgs matrix M doublet : UM doublet V T is diagonal, and v u and v d are the VEVs of up-and down-type Higgs fields. The detail of the convention is found in Ref. [9]. We note that r ∼ m b /m t if f 33 is small, and tan β = v u /v d is roughly related to r 1 , The right-handed Majorana neutrino mass matrix is obtained as where v R is a VEV of 126. Practically, we denote The seesaw neutrino mass matrix can be written as [1,2] where M L is the left-handed neutrino Majorana mass which comes from SU(2) L triplet coupling [24], ℓℓ∆ L . In the SO(10) model, the 126 Higgs also includes the SU(2) L triplet ∆ L and the Yukawa coupling generates both M L and M R . Therefore, M L is also proportional to the coupling matrix f and we denote In the SO(10) model, if there is 210 or 54 Higgs representation, the VEV of ∆ L , ∆ L = v L , can be obtained as v 2 weak /M ∆ , where M ∆ is the mass of the SU(2) L triplet. The dimension-five operator is given in the form of superpotential as In the minimal SO(10) model, the dimension-five operators generated by the exchange of the colored triplets, (3, 1, −1/3) + (3, 1, 1/3), can be written as where X and Y are the diagonalization unitary matrices of the colored triplet mass matrix M T : We denote that M T 1 is the lightest triplet Higgs mass. See Ref. [9] for more details. The dimension-five proton decay amplitude depends on (1) the masses of SUSY particles, (2) the triplet Higgs mass, and (3) the structure of the Yukawa couplings, for the model parameters. Among them, we here discuss the Yukawa structure to suppress the amplitudes. People often consider the contribution only from the left-handed operator C L , since the contribution from the right-handed operator C R is smaller than C L . However, for the current experimental bounds, there must be a cancellation among the left-handed contributions, and the contribution from C R itself can exceeds the bounds. In fact, it turns out that the minimal SU(5) model conflicts with the bounds due to the contribution from C R [25,26]. In SO(10) model, on the other hands, there are additional freedoms to cancel the amplitudes in the triplet Higgs mixings, and the amplitudes can be canceled, which is performed in Ref. [27]. However, the signs of the f ij h kl coefficients are opposite between C L and C R due to the algebraic reason that the triplet components in H and∆ have opposite "D-parities". This fact makes the simultaneous cancellation of both C L and C R unnatural [9]. More concretely, for the left-handed operator, the triplet Higgs mixing in Eq. (16) can be chosen to obtain C ijkl L ∼ (Y u ) ij (Y u + bf ) kl /M T 1 very roughly so that the amplitudes from C 112l L and C 121l L are suppressed. Note that the ij part of the operator in the minimal SU(5) model (without a Higgs mixing) is given as Y d (or Y e ). Even if the i = j = 1 contribution is the size of the down quark Yukawa coupling, it can exceed the current experimental bound. Then, the right-handed operator given in Eq. (18) can be roughly written as and a ≃ 2X 12 cannot be zero as long as the charged fermion masses and mixing are fit by the h and f matrices. The right-handed operator generates the four-fermi proton decay operator by Higgsino-dressing, u R d R s L ν τ and u R s R d L ν τ , and thus, the dominant contribution can be obtained as C 1133 R y t y τ V ts +C 1233 R y t y τ V td . As a consequence, if the elements f 11 and f 12 are small (in the basis where the quark mass matrices are diagonal), both the left-and right-handed contributions can be suppressed simultaneously.
We have learned that the proton decay amplitude can be suppressed if both f 11 and f 12 are small for the given SUSY spectrum and triplet Higgs mass. The next issue is whether the smallness of the elements are realized both in the f matrix and neutrino mass matrix. The simplest realization is the case where the SU(2) L triplet contribution dominates the neutrino mass matrix: M ν = M L , and the neutrino mass matrix is proportional to f . This can happen if the right-handed neutrino decouples and the type I contribution is negligible. The fermion masses and mixings can be fit in the model that the fermions couples to 10 + 126 + 120 Higgs representations and the predictions can be obtained from the proton decay suppression [9,23]. In the minimal model, on the other hand, the triplet dominant neutrino mass does not provide a good fit, and the type I contribution is necessary [17,18]. In general, if the type I term contributes, the f 11 and f 12 components are not correlated to the elements in the neutrino mass matrix. However, we can show that the coupling f can be written as a linear combination of M ν , M e and M ν M −1 e M ν algebraically, and the smallness of those elements can be correlated in the case where v R is large (c R v R > ∼ 10 16 GeV) [10]. In the limit of v R → ∞, we obtain where The square root matrix √ A is defined as ( √ A) 2 = A, and there are 2 3 = 8-fold branches in a 3 × 3 matrix. As one can see, in the case √ K 2 = K, it corresponds to the triplet dominant case. The detail algebraic description is given in Ref. [10]. On the other hand, for the fit for c R v R ∼ 10 13 GeV, which can give a best fit actually, f 11 and f 12 are not small (irrespective of the elements of the neutrino mass matrix), and the proton decay amplitudes are not suppressed.
Here, we show concrete examples of the f matrix (in GeV unit) obtained in Ref. [10] in the form of rf v u to compare with M d = Y d v d = rv u (h + f ) in the basis where the down-type quark mass matrix is real (positive) diagonal: Those two f 11 components are different by 1 digit. Roughly speaking, the lifetime are different by 2 digit, and the best fit solution does not satisfy the current experimental bound for the squarks mass to be around 2 TeV. In the solution for c R v R > ∼ 10 16 GeV, not only the elements are small, but also the f 11 and f 12 elements are correlated to the (1,1) and (1,2) elements of the neutrino mass matrix and thus they depends on θ 23 and δ PMNS as seen in the previous section. As a consequence, we find that the suppressed proton lifetime depends on the neutrino oscillation parameters.

Numerical study of the neutrino oscillation parameter dependence of the proton decay
The (1,1) and (1,2) elements of the neutrino mass matrix are suppressed along the curve shown in Fig.1. As explained in the previous section, the smallness of the f 11 and f 12 is correlated to the (1,1) and (1,2) elements of the neutrino mass matrix for the case of v R > ∼ 10 16 GeV. Therefore, we expect that the proton decay can be suppressed along the curve qualitatively. Numerically, even if (M ν ) 11 = (M ν ) 12 , the f 11 and f 12 elements are not exactly zero due to the type I contribution. Besides, the smallness of f 11 and f 12 to suppress the contribution from C R is needed in the basis where the quark mass matrices are diagonal. Thus, the suppression of the proton decay happens at a bit different from the curve shown in Fig.1, and δ PMNS -θ 23 dependence depends on the detail fits of the charged fermion masses and mixings. In this section, we show the numerical calculation of the proton lifetime in order to confirm that the qualitative expectation is realized using the explicit fits given in Ref. [10].
In order to show the numerical calculation of the partial lifetime p → Kν, we will setup in the following ways. As given in Eqs. (17), (19), the dimension-five operators C ijkl L,R are given by the linear combinations of h ij h kl , f ij h kl , h ij f kl and f ij f kl . We will choose the three coefficients in C L to cancel the left-handed contribution to the wino-dressed four-fermi nucleon decay amplitudes, A L (p → Kν τ ), A L (n → πν τ ), and A L (n → Kν τ ). The two coefficients in C R are chosen to cancel the Higgsino-dressed four-fermi decay amplitudes A R (n → πν τ ),and A R (n → Kν τ ). As we stressed, the coefficients of f ij h kl have opposite signs in C L and C R . Therefore, the coefficients are all fixed by the above conditions up to the overall factor, and the dominant contribution to p → Kν is the Higgsino-dressed four-four fermi operator (ud) R (sν τ ) L and (us) R (dν τ ) L , as explained in the previous section. We note that this method does not provide the minimization of the partial decay width. The partial amplitudes are not necessarily canceled exactly to minimize the decay width. For example, there are additional freedom to cancel the amplitudes between the wino-dressed four-fermi operator (ud) L (sν τ ) L and the Higgsino-dressed operator (ud) R (sν τ ) L , by choosing a phase in the Yukawa coupling, the wino and Higgsino masses. Our purpose to use this method is to show that the qualitative picture given in the previous section appears in the numerical calculation.
For the triplet Higgs mass, we suppose M T 1 ≃ 2 × 10 16 GeV. Because the triplet Higgs mixings are multiplied, (M −1 T ) 11 ≃ X 11 Y 11 /M T 1 depends on the detail of the triplet Higgs mass matrix. As shown in section 2, the freedom of the bottom Yukawa coupling (namely, the freedom of tan β in MSSM) depends on r 1 = U 11 /V 11 . Naively, the tan β freedom comes from the couplings H∆Φ and H∆Φ, where Φ is a 210 Higgs representation, and Y 11 can relates to the tan β freedom. Therefore, to show the numerical results, we choose (M −1 T ) 11 = m b tan β/m t /(2×10 16 ) GeV −1 . In the fit in Ref. [10], we use tan β = 10. The Higgsino-dressing amplitude of p → Kν τ depends on τ Yukawa coupling, and thus the decay amplitude is roughly proportional to tan 2 β and the partial lifetime is proportional to 1/ tan 4 β under these setups.
The squark masses and Higgsino mass µ is also important to calculate the proton decay amplitudes. We choose mq = µ = 2 TeV. We note that the amplitude is roughly proportional to µ/m 2 q . For the parameters of hadron matrix elements, we use numerical values given in Ref. [25].
In Fig.2, we show the plot of τ (p → Kν) in the δ PMNS -θ 23 plane. As expected, the partial proton lifetime is larger near the curve given in Fig.1. Near the curve, the lifetime is about 10 times bigger than the current experimental bound τ (p → Kν) > 0.59 × 10 34 years [28]. As described, the lifetime depends on the squark and Higgsino masses µ and tan β. In the calculation, we use mq = µ = 2 TeV, and tan β = 10. For tan β = 20, the lifetime along the curve can reach the current bound for the same SUSY masses. The accurate measurements of the δ PMNS phase and the 23-mixing will be very important if the SUSY particles are found at the LHC. Figure 2: We show the numerical result of the partial lifetime of the p → Kν decay. The detail setup is given in the text. As expected, the lifetime can be larger near the curve given in Fig.1. The current experimental bound is τ (p → Kν) EXP > 0.59 × 10 34 years [28].

Conclusions
The accurate measurement of the PMNS phase is important not only to accumulate the fundamental parameter in natural science, but also to determine the hierarchical structure of the neutrino mass matrix. If the SUSY particles are found at the LHC, the structure can open the window of the unified picture of the flavor physics. The accurate measurement of the neutrino 23-mixing angle is also important to obtain the structure. For the current experimental situation, the deviation of the maximal mixing θ 23 = π/4 is recently reported [29], and both θ 23 > 0 and θ 23 < 0 are possible within the statistical errors. It is expected that their accurate measurements can be done in the near future at Tokai-to-HyperKamiokande [30] and DUNE experiments [31]. The facilities can also raise the bounds of the baryon number violating nucleon decays. We learned that those two physics can be related in the SO(10) model, and the accurate measurements of the oscillation parameters are important to calculate proton decay amplitudes.
The proton decay amplitudes are suppressed at a curve given in Fig.1 if the VEV (v R ) of 126 Higgs representation, which reduces the rank of the SO(10) gauge group, is more than about 10 16 GeV. To realize this qualitative behavior, the size of the (1,1) and (1,2) elements needs to be correlated between the 126 coupling matrix f and the neutrino mass matrix. This algebraic reason is given in the minimal SO(10) model in which the fermions couple to only 10+126 Higgs fields in Ref. [10]. We expect that this qualitative behavior can be realized even in the extended models, for example, the model that the fermions can also couple with 120 Higgs field, since it is due to the algebraic feature. The concrete quantitative numbers can be surely different due to the fit of the fermion masses and mixings. The minimal SO(10) model is instructive to learn how this qualitative behavior of the suppressed proton decay is obtained. The important ingredient is v R > ∼ 10 16 GeV. As noted in Ref. [10], the influence of v R ≪ 10 16 GeV to low energy physics is the induction of flavor changing neutral currents in the right-handed charged sleptons, which can enlarge the electron electric dipole moment, and we can construct a big picture of the unified framework by the future experimental results.