Enhanced $\Gamma(p\to K^0\mu^+)/\Gamma(p\to K^+\bar{\nu}_\mu)$ as a Signature of Minimal Renormalizable SUSY $SO(10)$ GUT

The ratio of the partial widths of some dimension-5 proton decay modes can be predicted without knowledge of SUSY particle masses, and thus allows us to experimentally test various SUSY GUT models without discovering SUSY particles. In this paper, we study the ratio of the partial widths of the $p\to K^0\mu^+$ and $p\to K^+\bar{\nu}_\mu$ decays in the minimal renormalizable SUSY $SO(10)$ GUT. In the model, we expect that the Wilson coefficients of dimension-5 operators responsible for these modes are on the same order and that the ratio of $p\to K^0\mu^+$ and $p\to K^+\bar{\nu}_\mu$ partial widths is $O(0.1)$. Hence, we may be able to detect both $p\to K^0\mu^+$ and $p\to K^+\bar{\nu}_\mu$ decays at Hyper-Kamiokande, thereby gaining a hint for the minimal renormalizable SUSY $SO(10)$ GUT. Moreover, since this partial width ratio is quite suppressed in the minimal $SU(5)$ GUT, it allows us to distinguish the minimal renormalizable SUSY $SO(10)$ GUT from the minimal $SU(5)$ GUT. In the main body of the paper, we perform a fitting of the quark and lepton masses and flavor mixings with the Yukawa couplings of the minimal renormalizable $SO(10)$ GUT, and derive a concrete prediction for the partial width ratio based on the fitting results. We find that the partial width ratio generally varies in the range 0.05-0.6, confirming the above expectation.


Introduction
is O(0.1) in the minimal renormalizable SUSY SO(10) GUT. To this end, we determine the fundamental Yukawa couplings Y 10 , Y 126 through a fitting of the quark and lepton Yukawa couplings and neutrino data, as has been performed in Refs. [12]- [30], and calculate the partial width ratio based on the fitting results.
Previously, enhancement of partial width ratio Γ(p → K 0 µ + )/Γ(p → K +ν µ ) in SO(10) GUT models compared to the minimal SU(5) GUT is claimed in Refs. [31,32], but only based on a qualitative argument. Our paper is the first study where this ratio is predicted concretely and quantitatively in the minimal renormalizable SUSY SO(10) GUT, with the fundamental Yukawa couplings Y 10 , Y 126 determined through a numerical fitting.
The basic reason that Γ(p → K 0 µ + )/Γ(p → K +ν µ ) is O(0.1) in the minimal renormalizable SUSY SO(10) GUT is understood as follows. In the model, the ratio of the Wilson coefficients of dimension-5 operators responsible for the p → K 0 µ + decay and those for the p → K +ν µ decay, is proportional to (Y 10 ) u L j /(Y 10 ) d L j or (Y 126 ) u L j /(Y 126 ) d L j . Here (Y 10 ) u L j denotes (1,j)-component of Y 10 in the flavor basis where, when we write the Yukawa coupling as ψ i (Y 10 ) ij ψ j , the lefthanded up-type quark component of ψ i has the diagonalized up-type quark Yukawa coupling. (Y 10 ) d L j , (Y 126 ) u L j , (Y 126 ) d L j are defined in the same way. Y 10 , Y 126 are linear combinations of the down-type and up-type quark Yukawa matrices Y d , Y u , due to the relations Y u = Y 10 + r 2 Y 126 , Y d = r 1 (Y 10 + Y 126 ). Moreover, these linear combinations are generic, because situations where Y 10 ∝ Y u , Y 126 ∝ Y d or Y 10 ∝ Y d , Y 126 ∝ Y u would not reproduce the correct charged lepton Yukawa matrix Y e . Therefore, considering the large hierarchy y u /y t ≪ y d /y b , we expect that the components (Y 10 ) u L j , (Y 10 ) d L j , (Y 126 ) u L j , (Y 126 ) d L j are all on the order of the down quark Yukawa coupling y d times the mixing angle between the right-handed down quark and a state with flavor index j, and are not proportional to the up quark Yukawa coupling y u .
Hence, both (Y 10 ) u L j /(Y 10 ) d L j and (Y 126 ) u L j /(Y 126 ) d L j are O(1) and so is the ratio of the Wilson coefficients of dimension-5 operators for the p → K 0 µ + and the p → K +ν µ decays. The Wino-dressing diagrams give almost the same contribution for the two modes, if the 1st and 2nd generation left-handed squarks are mass-degenerate. As a result, the partial width ratio Γ(p → K 0 µ + )/Γ(p → K +ν µ ) is determined by the ratio of baryon chiral Lagrangian parameters, which lies in the range (1−D +F ) 2 /(1+D +F ) 2 = 0.085 to (1−D +F ) 2 /(1−D/3+F ) 2 = 0.30, and thus the partial width ratio is O(0.1).
This paper is organized as follows. In Section 2, we describe the minimal renormalizable SUSY SO(10) GUT and present formulas for the partial widths of the p → K +ν µ and p → K 0 µ + decays. In Section 3, we roughly estimate the partial width ratio Γ(p → K 0 µ + )/Γ(p → K +ν µ ) in the minimal renormalizable SUSY SO(10) GUT without numerically determining the fundamental Yukawa couplings Y 10 , Y 126 , and compare it to the partial width ratio in the minimal SU(5) GUT. In Section 4, we numerically determine Y 10 , Y 126 through a fitting of the quark and charged lepton Yukawa couplings and neutrino mass matrix, and calculate Γ(p → K 0 µ + )/Γ(p → K +ν µ ) based on the fitting results. Section 5 summarizes the paper.
2 Minimal Renormalizable SUSY SO(10) GUT We consider a SUSY SO(10) GUT model that contains chiral superfields H, ∆, ∆ in 10, 126, 126 representation, and three matter fields Ψ i in 16 representation (i = 1, 2, 3 denotes flavor index) [7]. The model also contains chiral superfields responsible for breaking SU(5) subgroup of SO(10), but we do not specify them in this paper. The most general renormalizable Yukawa couplings are given by where (Ỹ 10 ) ij and (Ỹ 126 ) ij are 3×3 complex symmetric matrices. The Higgs fields of the minimal SUSY Standard Model (MSSM), H u , H d , are linear combinations of (1, 2, ± 1 2 ) components of H, ∆ and other fields. Accordingly, the MSSM Yukawa coupling for up-type quarks, Y u , that for down-type quarks, Y d , and that for charged leptons, Y e , and the Dirac Yukawa coupling for neutrinos, Y D , are derived from W Yukawa as at a SO(10) breaking scale. Here Y 10 ∝Ỹ 10 , Y 126 ∝Ỹ 126 , and r 1 , r 2 are numbers. By a phase redefinition, we take r 1 to be real positive. In principle, r 1 , r 2 are determined from the mass matrix for (1, 2, ± 1 2 ) components [33]- [38], but in this paper we treat them as independent parameters.
Majorana mass for the right-handed neutrinos is proportional to (Y 126 Integrating out N c i yields an effective operator L i H u L j H u , which we call the Type-1 seesaw contribution. Additionally, if the (1, 3, 1) component of ∆ mixes with that of 54 representation field, after integrating out these components, we get an effective operator L i H u L j H u , which we call the Type-2 seesaw contribution.
H,∆ and other fields contain pairs of (3, C , we get dimension-5 operators contributing to proton decay, (in the first term, isospin indices are summed in each bracket) where and M H C denotes the mass matrix of H A C , H B C fields and M H C represents a typical value of the eigenvalues of M H C .
We concentrate on the contribution of the (Q k Q l )(Q i L j ) operators to the p → K +ν µ and p → K 0 µ + decays, and calculate the ratio of their partial widths in the minimal renormalizable SUSY SO(10) GUT. It should be noted that the (Q k Q l )(Q i L j ) and the E c k U c l U c i D c j operators contribute to the p → K +ν τ decay, which is experimentally indistinguishable from the p → K +ν µ decay. Hence, our prediction on Γ(p → K 0 µ + )/Γ(p → K +ν µ ) should be regarded as the maximum of the following measurable quantity: The maximum is attained if the (Q k Q l )(Q i L j ) operators' contribution and the E c k U c l U c i D c j operators' contribution to the p → K +ν τ decay cancel each other. This cancellation is always possible by adjusting the ratio of the Wino mass and the µ-term.
The contribution of the C ijkl 5L (Q k Q l )(Q i L j ) term to the p → K +ν µ and the p → K 0 µ + decays is given by [39] where , β H denotes a hadronic matrix element, D, F are parameters of the baryon chiral Lagrangian, and C LL , C LL are Wilson coefficients of the effective Lagrangian ) (ψ denotes a SM Weyl fermion and spinor index is summed in each bracket). We have neglected the mass splittings among nucleons and hyperons. The Wilson coefficients C LL , C LL satisfy 4 where F is a common loop function factor F = 1 x−y ( x 1−x log x− y 1−y log y)/16π 2 + 1 x−1 ( x 1−x log x+ 1)/16π 2 with x = |M W | 2 /m 2 q and y = m 2 ℓ /m 2 q , and mq denotes the 1st and 2nd generation lefthanded squark mass and ml denotes the mass of the left-handed smuon and muon sneutrino.
A LL (µ had , µ SUSY ) accounts for renormalization group (RG) corrections in the evolution 5 from soft SUSY breaking scale µ SUSY to a hadronic scale where the value of β H is reported. C 5L are 4 When writing C uµ us 5L , we mean that Q i is in the flavor basis where the up-type quark Yukawa coupling Y u is diagonal and that the up-type quark component of Q i is exactly u quark (then the down-type quark component of Q i is a mixture of d, s, b). Likewise, Q k is in the flavor basis where the down-type quark Yukawa coupling Y d is diagonal and its down-type quark component is exactly s quark, and Q l is in the flavor basis where the up-type quark Yukawa coupling is diagonal and its up-type quark component is exactly u quark. The same rule applies to other Wilson coefficients. 5 RG corrections involving SM Yukawa couplings are negligible for C sµ du LL , C dµ su LL , C uµ us LL , and hence their RG corrections are approximately flavor-universal. related to the colored Higgs Yukawa couplings as where A L (µ SUSY , µ H C ) accounts for RG corrections in the evolution from colored Higgs mass We relate the flavor-dependent part of Eqs. (18)- (20) to Y 10 , Y 126 . Since Y A L , Y A L are proportional to either Y 10 or Y 126 , we can write without loss of generality where M H C is a typical value of the eigenvalues of M H C , and a, b, c, d are numbers common for Eqs. In principle, numbers a, b, c, d are determined from the colored Higgs mass matrix [33]- [38].
However, as we do not specify fields responsible for breaking SU(5) subgroup of SO(10), we treat a, b, c, d as independent O(1) parameters.
We observe that each term in Eq. (23) is given by (22), as advertised in Introduction. For example, the term (21), and also equals ( in the minimal SU(5) GUT and in the minimal renormalizable SUSY SO(10) GUT without numerically determining Y 10 , Y 126 .

Estimate in the Minimal SU (5) GUT
In the minimal SU(5) GUT, we have only one pair of colored Higgs fields, and Y L and Y L are proportional to the Yukawa couplings for 5 and 5 Higgs fields, respectively. Hence, Eqs. (21)- where Y 5 and Y 5 denote the Yukawa couplings for 5 and 5 Higgs fields, respectively, and M H C denotes the mass for the colored Higgs fields.
The key fact is that since Y 5 is identical to the up-type quark Yukawa coupling matrix, the components of Y 5 with flavor index u L are given by the up quark Yukawa coupling times a mixing angle. Hence, they are estimated to be where µ H C ∼ M H C , and λ denotes the Cabibbo angle λ ≃ |V us | ≃ |V cd | ≃ 0.22. On the other hand, (Y 5 ) d L s L is estimated to be the second generation Yukawa coupling times a mixing angle as Although the unification of down-type quark Yukawa coupling and charged lepton Yukawa coupling is unsuccessful, we can estimate components of Y 5 as From formulas Eqs. (13)- (20) and estimates Eqs. (24)-(31), we estimate the partial widths as 7 where C is a common constant, c 1 , c 2 , c 3 are O(1) numbers, and y u , y c , y µ , y s are the up, charm, muon and strange quark Yukawa couplings at scale µ = µ H C . We have discarded subleading terms. The partial width ratio is then estimated as where the variation is due to unknown relative phase between c 1 and c 2 . Numerically, the above estimate becomes 7 We neglect the small difference between hyperon masses and the nucleon mass.
We find that p → K 0 µ + partial width is quite suppressed compared to p → K +ν µ partial width because of the factor 0.002 coming from the ratio of y u and λ 2 y c , namely, the large hierarchy between the up and charm quark Yukawa couplings suppresses the partial width ratio. Also, baryon chiral Lagrangian parameters give ( 3, and they provide further suppression.

Estimate in the Minimal Renormalizable SUSY SO(10) GUT
In the minimal renormalizable SUSY SO(10) GUT, we can rewrite the right-hand side of where The components of Y u with flavor index u L are always given by the up Yukawa coupling y u times a mixing angle, and hence we get In contrast, the components of Y 126 do not follow the rule and are estimated as We have estimated (Y 126 ) s L µ L to be y s (µ H C )/r 1 , because we empirically have y µ /y s | µ=10 16 GeV ≃ 4 and this factor 4 is mostly explained by the factor 3 in Eq. (5). We have estimated (Y 126 ) u L u L to be y d (µ H C )/r 1 , not y u (µ H C ), based on the following argument: Recall that components of Y 10 and Y 126 reproduce the up and down Yukawa couplings as Since the unification of the top and bottom Yukawa couplings requires tan β/r 1 ≃ m t /m b ≃ 50, we get Then, the only way to realize Eq. (51) is to take and impose a fine-tuning between (Y 10 ) u R u L and r 2 (Y 126 ) u R u L to realize the small value 0.01 in Eq. (51). Here we cannot assume r 2 ≃ 0 because we need |r 2 | = O(1) to reproduce the charged lepton Yukawa coupling, as will be confirmed numerically in Fig. 1. From Eq. (52), we find Using From formulas Eqs. (13)- (20) and estimates Eqs. (42)-(48), we estimate the partial widths as where C is a common constant, y u , y s , y c are the up, strange and charm quark Yukawa couplings at scale µ = µ H C , and β 1 , β 2 , β 3 , β 4 , γ 1 , γ 2 , γ 3 , γ 4 , δ 1 , δ 2 , δ 3 , δ 4 are O(1) numbers. We have used empirical relation m s λ 2 ≃ m d and let y s λ 2 represent both y s λ 2 and y d . In Eqs. (55),(56), y s λ 2 /r 2 1 and y c y s λ 2 /r 1 are much larger than the other terms containing y u . Hence, in generic cases where d ′ = O(1) and/or c ′ = O(1), the partial width ratio is estimated as where the variation is due to unknown relative phases among β 2 , β 3 , β 4 , γ 2 , γ 3 , γ 4 . We find that the suppression factor of 0.002 in Eq. (37) is absent in Eq. (57). This means that in the minimal renormalizable SUSY SO(10) GUT with d ′ = O(1) and/or c ′ = O(1), Γ(p → K 0 µ + )/Γ(p → K +ν µ ) is highly enhanced compared to the minimal SU(5) GUT. In the non-generic case where c ′ and d ′ are both fine-tuned to 0, the partial width ratio is quite suppressed as which is the same as in the minimal SU(5) GUT. This is reasonable because when c ′ = d ′ = 0, the contribution of (3, 1, − 1 3 ) fields to dimension-5 proton decay is dictated by the up-type quark Yukawa matrix, just as in the minimal SU(5) GUT.
In the next section, we numerically confirm the estimates Eqs.

Overview
Our first task is to fit the MSSM Yukawa matrices with Y 10 , Y 126 , r 1 , r 2 through Eqs. (3)-(5), and fit the neutrino mass matrix with Y 10 , Y 126 , r 2 . When calculating the Type-1 seesaw contribution to the Weinberg operator L i H u L j H u , we have to integrate out each right-handed neutrino N c i at its respective mass scale. This requires information on the eigenvalues of Y 126 , but it is obtained only after the fitting is complete. Hence, it is technically difficult to integrate out each righthanded neutrino separately. In this paper, therefore, we make an approximation that the three right-handed neutrinos are integrated out at one scale. Accordingly, the neutrino mass matrix M ν is related to Y 126 and Y D in Eq. (6) as where r L is a complex number that parametrizes the ratio of the Type-1 and Type-2 seesaw contributions, and R ij denotes the flavor-dependent RG correction to the coefficient of the Weinberg operator L i H u L j H u when it evolves from a SO(10) breaking scale to electroweak scale. Since the flavor-dependent RG correction R ij is at most 3% (see Table 1) while the errors of the neutrino data we employ are much larger (see Table 2), we expect that the approximation of integrating out right-handed neutrinos at one scale does not affect the results. We repeat the above fitting analysis many times and obtain as many fitting results. We compute Γ(p → K +ν µ ) and Γ(p → K 0 µ + ) from each fitting result of Y 10 , Y 126 , r 1 , r 2 , r L using Eqs. (13)- (20) and Eqs. (38)-(40), with coefficients a, b ′ , c ′ , d ′ treated as independent O(1) parameters. The fitting results are plotted with respect to the ratio Γ(p → K 0 µ + )/Γ(p → K +ν µ ). From the plot, we read out the range of the ratio Γ(p → K 0 µ + )/Γ(p → K +ν µ ) predicted by the minimal renormalizable SO(10) GUT.
We assume a benchmark SUSY particle mass spectrum to evaluate the MSSM Yukawa couplings at a SO(10) breaking scale as well as R ij , and to compute the individual partial widths Γ(p → K +ν µ ) and Γ(p → K 0 µ + ). However, we emphasize that the purpose of this paper is to predict the ratio Γ(p → K 0 µ + )/Γ(p → K +ν µ ), which is not much dependent on the SUSY particle mass spectrum due to the cancellations of the RG corrections and the factors coming from Wino-dressing.

Procedures
First, we numerically calculate the MSSM Yukawa matrices Y u , Y d , Y e at scale µ = 2·10 16 GeV in DR scheme, and the flavor-dependent RG correction to the coefficient of the Weinberg operator R ij . Specifically, we calculate R ij for the evolution from µ = 2 · 10 16 GeV to µ = M Z . We assume a high-scale split SUSY particle mass spectrum below for concreteness, For the calculation of the quark Yukawa couplings, we adopt the following input values for quark masses and CKM matrix parameters: The isospin-averaged quark mass and strange quark mass in MS scheme are obtained from lattice calculations in Refs. [41,42,43,44,45,46] [50]. For the QCD and QED gauge couplings, we use α The results are given in terms of the singular values of Y u , Y d , Y e and the CKM mixing angles and CP phase at µ = 2 · 10 16 GeV, as well as R ij in the flavor basis where Y e is diagonal (R ij is also diagonal in this basis), tabulated in Table 1. For each singular value of Y u , Y d , we present 1σ error that has propagated from experimental error of the corresponding input quark mass.
For the CKM mixing angles and CP phase, we present 1σ errors that have propagated from experimental errors of the input Wolfenstein parameters.
where a 2 , a 3 , b 1 , b 2 , b 3 are unknown phases. In the same flavor basis, Y e is written from Eqs. (3)- with Y d given in Eq. (60). We can also write Finally, we perform the singular value decomposition of Y e as and calculate the active neutrino mass matrix (up to overall constant) as Now we perform the fitting with Y 10 , Y 126 , r 1 , r 2 , r L . It proceeds as follows. We fix y u , y c , y t and CKM matrix by the values in Table 1, while we vary y d /r 1 , y s /r 1 , y b /r 1 , unknown phases a 2 , a 3 , b 1 , b 2 , b 3 in Eq. (60) and complex number r 2 . Here we eliminate r 1 by requiring that the central value of the electron Yukawa coupling y e be reproduced. In this way, we try to reproduce the correct values of y d , y s , y µ , y τ , θ pmns 12 , θ pmns 13 , θ pmns 23 and neutrino mass difference ratio ∆m 2 21 /∆m 2 32 . Specifically, we require y d , y s to fit within their respective 3σ ranges, while we do not constrain y b because y b can receive sizable SUSY particle and GUT-scale threshold corrections. Since the experimental errors of y µ , y τ are tiny, we only require that their reproduced values fit within ±0.1% ranges of their central values. We require sin 2 θ pmns 12 , sin 2 θ pmns 13 , sin 2 θ pmns 23 , ∆m 2 21 /∆m 2 32 to fit within their respective 3σ ranges reported by NuFIT 4.1 [52,53]. However, we do not constrain the Dirac CP phase δ pmns , since its measurement is still at a primitive stage. We only consider the normal mass hierarchy case, because we cannot obtain a good fitting with the inverted mass hierarchy. We have confirmed that our fitting analysis always gives small values for m 1 that are not in tension with cosmological observations or searches for neutrinoless double-beta decay, and hence no constraint is imposed on α 2 , α 3 , m 1 . The constraints are summarized in Table 2.
We collect sets of values of Y 10 , Y 126 , r 1 , r 2 , r L that satisfy the constraints of Table 2. From these values, we reconstruct the MSSM Yukawa couplings Y u , Y d , Y e , perform flavor basis changes, and calculate the following components: From the values above, we calculate Γ(p → K +ν µ ) and Γ(p → K 0 µ + ) through Eqs.  (40). Here we take M H C = 2 · 10 16 GeV and assume the SUSY particle mass spectrum of Eq. (59). We employ the following data and formulas. For the hadronic matrix element β H , we adopt the value in Ref. [54], which reads β H = 0.0144 GeV 3 at µ = 2 GeV in MS scheme. The baryon chiral Lagrangian parameters are given by D = 0.804, F = 0.463, and we include the mass splittings among nucleon and hyperon masses found in Particle Data Group [51]. When computing RG corrections to the dimension-5 operators and the dimension-6 operators after Wino-dressing, we choose µ SUSY = 2000 TeV and µ H C = 2·10 16 GeV, and use one-loop formulas in Ref. [40].

Results
We have obtained 158 sets of values of Y 10 , Y 126 , r 1 , r 2 , r L that satisfy the constraints of Table 2.
Before presenting the main results, we show in Fig. 1 Table 2. Now we plot the sets of values of Y 10 , Y 126 , r 1 , r 2 , r L satisfying Table 2, on the plane of p → K +ν µ partial lifetime versus the ratio of the partial widths of p → K 0 µ + and p → K +ν µ . From the plots, we read out the range of the partial width ratio predicted by the model.
• When d ′ = O(1), the partial width Γ(p → K +ν µ ) is dominated by the contribution from the term with coefficient d ′ . Since the partial width ratio Γ(p → K 0 µ + )/Γ(p → K +ν µ ) with (a, b ′ , c ′ , d ′ ) = (0, 0, 0, 1) is comparable to or larger than in the other cases, we expect that Γ(p → K 0 µ + ) is also dominated by the contribution from the term with d ′ . Therefore, we conclude that when d ′ = O(1), irrespectively of the values of a, b ′ , c ′ , the prediction on the partial width ratio is given by the lower-right panel of Fig. 2, where the partial width ratio mostly varies in the range 0.05-0.6. This result is consistent with our estimate Eq. (57).
Hence, when c ′ = O(1) and b ′ = O(1), the partial width ratio Γ(p → K 0 µ + )/Γ(p → K +ν µ ) is suppressed if the contributions of the terms with c ′ and b ′ to Γ(p → K +ν µ ) interfere constructively, and the partial width ratio is enhanced if they interfere destructively. To examine these possibilities, we present plots for cases with (a, b ′ , c ′ , d ′ ) = (0, 1, 1, 0), Fig. 3. We observe that when d ′ = 0, c ′ = O(1) and b ′ = O(1), the prediction on the partial width ratio varies considerably with the relative phase of b ′ and c ′ and with different fitting results. Still, we can assert that the ratio is above 0.01. The absence of strong suppression factor 0.3 · 0.002 is consistent with our estimate Eq. (57).
• Only in the very special case with d ′ = c ′ = b ′ = 0 do we obtain the distribution of the upper-left panel of Fig. 2, where the ratio is above 0.05.

Summary
The ratio of the partial widths of some dimension-5 proton decay modes can be predicted without knowledge of SUSY particle masses, and thus serves as a probe for various SUSY GUT models even when SUSY particles are not discovered. We have focused on the partial width ratio Γ(p → K 0 µ + )/Γ(p → K +ν µ ) in the minimal renormalizable SUSY SO (10) GUT. In the model, the Wilson coefficients of dimension-5 operators responsible for the p → K 0 µ + and the p → K +ν µ decays are on the same order, and Γ(p → K 0 µ + )/Γ(p → K +ν µ ) is largely determined by the ratio of baryon chiral Lagrangian parameters and is estimated to be O(0.1). This is in striking contrast to the minimal SU(5) GUT, where this partial width ratio is further suppressed by factor y 2 u /(λ 2 y c ) 2 ≃ 0.002. To confirm that Γ(p → K 0 µ + )/Γ(p → K +ν µ ) = O(0.1) in the minimal renormalizable SUSY SO(10) GUT, we have numerically determined Y 10 , Y 126 through a fitting of the quark and charged lepton Yukawa couplings and neutrino mass matrix, and calculated the partial width ratio based on the fitting results. Our most important finding is that the partial width ratio generally varies in the range 0.6 Γ(p → K 0 µ + )/Γ(p → K +ν µ ) 0.05 in the most generic case where d ′ = O(1) in Eqs. (38)- (40).