Detectable dimension-6 proton decay in SUSY SO(10) GUT at Hyper-Kamiokande

In the minimal SUSY SU(5) GUT with $O(1)$ TeV SUSY particles and $O(1)$ or below self-coupling for the GUT-breaking Higgs field, the width of the dimension-6 proton decay is suppressed below the reach of Hyper-Kamiokande. In this paper, we point out that a SUSY SO(10) GUT which adopts only ${\bf 45}_{\rm H}+{\bf 16}_{\rm H}+{\bf \overline{16}}_{\rm H}$ GUT-breaking Higgs fields leads to an enhanced dimension-6 proton decay width detectable at Hyper-Kamiokande. The enhancement is because the SU(5)-breaking VEV of ${\bf 45}_{\rm H}$ arises due to Planck-suppressed terms, $W\propto ({\bf 45}_{\rm H}^2)^2/M_*+{\bf 45}_{\rm H}^4/M_*$, and is therefore substantially larger than the other VEVs that conserve SU(5). As a result, the $({\bf 3},{\bf 2},1/6)$ GUT gauge boson mass is about $1/5$ smaller than the $({\bf 3},{\bf 2},-5/6)$ GUT gauge boson mass and can induce a fast dimension-6 proton decay. Through a numerical analysis on threshold corrections of the GUT gauge bosons and the physical components of the GUT-breaking Higgs fields, we confirm that the dimension-6 proton decay can be within the reach of Hyper-Kamiokande.


Introduction
The dimension-6 proton decay is an important prediction of the grand unified theory (GUT) [1]. The Super-Kamiokande experiment currently gives the bound of the partial proton lifetime τ (p → π 0 e + ) > 1.6 × 10 34 years (90% confidence level) [2], and it will be searched up to 6.3 × 10 34 years at 3σ level by a 10 year exposure of one 187 kton fiducial volume detector at Hyper-Kamiokande (HK) [3]. Now that the HK experiment is scheduled to start in 2026, it is time to survey GUT models which predict the dimension-6 proton decay within the discovery reach.
In the minimal supersymmetric (SUSY) SU(5) GUT [4,5,6] with O(1) TeV SUSY particles, the partial lifetime for the dimension-6 proton decay via GUT gauge boson exchange is predicted to be more than a few times 10 35 years naively. The gauge coupling unification condition does not directly give the mass of the GUT gauge boson, because the mass of the physical components of the SU(5)-breaking Higgs field 24 H cannot be determined theoretically. The GUT gauge boson mass becomes heavier (the proton lifetime is longer) if the self-coupling of 24 H is smaller. 1 The SUSY GUTs also predict the dimension-5 proton decay via colored Higgs exchange [7], such as p → K +ν , whose current bound reads τ (p → K +ν ) > 5.9 × 10 33 years [8] and which often gives a severe constraint on the model construction. There are several ways to suppress the dimension-5 decay to a harmless level, e.g., by enhancing the colored Higgs mass with SUSY particle threshold with large wino/gluino mass ratio [9], with non-renormalizable superpotential of adjoint representations [10], or with GUT particle thresholds in non-minimal models for the gauge coupling evolutions [11]. Other ways include assuming heavy squarks, or utilizing a cancellation among multiple Higgs couplings. Compared to the dimension-5 decay, the dimension-6 proton decay involves less parameters and its naive prediction is above the current experimental bound. Therefore, it is worth pursuing the possibility that p → π 0 e + will be observed at HK. In fact, as the LHC results imply that the SUSY particles have mass above multi-TeV scale, some people revisit the unification conditions in the context of the high-scale SUSY scenario [12,13,14,15]. As the wino and gluino are heavier, the unification scale becomes lower, and it can reach the discovery range of HK for ∼10-100 TeV wino and gluino masses.
What about the dimension-6 proton decay in SUSY SO(10) GUTs? The breaking pattern of the SO(10) symmetry has room for the existence of intermediate scales, and thus the prediction of the dimension-6 proton decay varies in a wide range. Among various choices of the Higgs representations to break SO(10) to the SM gauge symmetry, the simplest choice is 45 H + 16 H + 16 H , which is also the most economical in view of the total contribution to the beta coefficient for gauge couplings. The above choice of the Higgs representations gives characteristic vacua where the GUT gauge boson with SM charge (3, 2, 1/6), which is absent in SU (5) GUT, is about 1/5 lighter than the GUT gauge boson with SM charge (3, 2, −5/6), which is also present in SU(5) GUT. In the vacua, therefore, the dimension-6 proton decay width is enlarged compared to the minimal SU(5) model due to the exchange of the light (3, 2, 1/6) gauge boson. So, it is worth scrutinizing the prediction of the dimension-6 proton decay in the above model, since the predicted proton lifetime can be in the range of HK. To our best knowledge, this simple SO(10) model has not been investigated in light of experimental accessibility of the dimension-6 proton decay. In this paper, we will show a numerical calculation of the dimension-6 proton decay p → π 0 e + in the SO(10) model with 45 H + 16 H + 16 H GUT-breaking Higgs fields.
We also find that in the characteristic vacua of the above model, the colored Higgs mass is enhanced by about 576 compared to the minimal SU(5) model due to threshold corrections of GUT gauge bosons and physical components of GUT-breaking Higgs fields. 2 So, this SO(10) model exhibits an interesting tendency that the dimension-6 decay width is enhanced and the dimension-5 decay width is suppressed.
This paper is organized as follows: In Section 2, we present the spectrum of the SO ( where X, Q, U and E denote SO(10) gauge bosons whose SM charges are X : (3, 2, −5/6), The dimension-6 proton decay operators are generated not only by the X gauge boson exchange but also by the Q gauge boson exchange. The partial width of the dimension-6 proton decay is given by where A L,R are the renormalization factors for qℓ(u c ) † (d c ) † and qq(e c ) † (u c ) † operators. One finds that the Q gauge boson exchange gives much larger contribution when M X : M Q ≃ 5 : 1. The ratio of the decay width in SU(5) GUT (M Q → ∞) and in the SO for A L ≃ A R , if the X gauge boson masses are the same. Since the naive prediction of p → π 0 e + partial lifetime in SU (5) GUT is τ p ∼ 10 36 years, the prediction in the SO(10) with a 24 ≫ a 1 , v R is 10 34 years, which is on the current experimental bound at SK.

SO(10) breaking vacua in the model
We consider a superpotential for the GUT breaking Higgs fields 45 H (A), 16 H (χ) and 16 H (χ), where we define A 2 ≡ A ab A ab /2, and A 4 ≡ A ac A ad A bc A bd /2 so that the multiplication of contraction of 2-anti-symmetric indices is removed by dividing by 2. The superpotential in terms of canonically-normalized SM singlets a 1 , a 24 in A and v R in χ (Clebsch-Gordan coefficients for 16 representation can be found in [17]) is given by The F -flat conditions read where a is a solution to In Eq. (10), the condition ∂W/∂v R = (m χ + √ 5κa 1 )v R fixes the VEV of a 1 to be around m χ . In Eq. (13), on the other hand, the VEV of a 24 (proportional to a) is fixed by a balance between the quadratic mass term and the quartic non-renormalizable term, and |a| is large if Thus, vacua with |v R |, |a 1 | ≪ |a 24 | are obtained 3 with a feasible assumption m χ , m A ≪ M * .

GUT-scale threshold corrections for the gauge coupling unification
The gauge coupling unification conditions [24] in SUSY SO(10) GUT are written as 4 where M X,Q,U,E are the SO(10) gauge boson masses which we have already defined, M H C is the colored Higgs mass, and i stands for the degree of physical modes under the SM decompositions.
We define l A = 5 12 (2l 3 − 3l 2 + l 1 ) and l B = 1 6 (2l 3 + 3l 2 − 5l 1 ) where l i gives the beta coefficient contribution of the respective multiplet, l i = ∆b SUSY i . Because the would-be-Goldstone modes which are eaten by the gauge bosons lack from the multiplets, we obtain The RGEs give M H ∼ 10 15 − 10 16 GeV, where µ H , m H are higgsino and heavier Higgs masses, Mg and Mw are gluino and wino masses. From these equations, one finds that the colored Higgs mass is larger for a smaller ratio of Mg/Mw, and the unification scale M G becomes smaller for heavier wino and gluino masses.
(16 H ) is absorbed by the gauge bosons Q, U, E. For |v R | ≪ |a 24 |, the linear combination to be absorbed mainly comes from 45 H . The other linear combination is a physical mode and we denote its components by χ Q , χ U , χ E (which respectively have the same SM charge as Q, U, E).
For |m A |, |m χ | ≪ M * , their masses satisfy the ratio (see Appendix A for the derivation) The 24 representation in 45 H contains a SU(3) c adjoint (8, 1, 0) and a SU(2) L adjoint (1, 3, 0) as physical modes. Their masses can be calculated (using the minimization conditions) as and when |a 1 | ≪ |a 24 |, we find In the vacua with |v R |, |a 1 | ≪ |a 24 |, we obtain from Eqs. (5),(23),(26), Due to the factor 1/576, the colored Higgs mass can be much larger than in the minimal SU(5) model. As for the gauge boson mass, in the minimal SU(5) model, one has M 6 G = M 4 X M 8 M 3 and M 8 = M 3 = λM X where λ is proportional to the self-coupling of the SU(5) adjoint representation. λ is arbitrary unless it far exceeds O(1), and people often assume λ ∼ 1, which gives M X ∼ M G . In the current SO(10)-breaking vacua |v R |, |a 1 | ≪ |a 24 |, if we write 4M 8 ≃ M 3 = ρM X , ρ is always much smaller than 1 because the masses of the SU(3) c adjoint and SU(2) L adjoint particles are roughly m A , while the X gauge boson mass is roughly (m A M * ) 1/2 . To be specific, we get from Eqs. (28),(30), and from Eqs. (1),(25), Therefore we find which equals 0.1 for λ 2 = 1 and M * = 2 × 10 18 GeV. It follows that M X is a little larger than M G . Nevertheless, the Q gauge boson satisfying M X : M Q ≃ 5 : 1 enhances the dimension-6 proton decay width compared to the minimal SU(5) model.
To summarize, in the SO(10)-breaking vacua with |a 24 | ≫ |a 1 |, |v R |, the colored Higgs is made heavier by the GUT-scale threshold corrections, and the dimension-5 proton decay is suppressed compared to the minimal SU(5) model. On the other hand, the dimension-6 proton decay width is roughly 100 times enlarged and we have τ p ∼ 10 34 years, which is in the scope of HK.
Suppression of the dimension-5 proton decay is also achieved by making the ratio of gluino and wino masses Mg/Mw smaller, and enhancement of the dimension-6 proton decay is achieved by increasing their product MgMw, as seen from the SUSY particle threshold correction formulas. Hence, in the high scale SUSY scenario, the dimension-6 proton decay is detectable at HK even in the minimal SUSY SU(5) model. In contrast, in our SO(10)-breaking vacua |a 24 | ≫ |a 1 |, |v R |, the GUT-scale threshold corrections enhance the dimension-6 proton decay width to a detectable level, even if SUSY particle masses are a few TeV.
We comment on the case when the 16 H is replaced by 126 H representation. In this case, when a vacuum with |a 1 | ≪ |a 24 | is chosen, (6, 3, 1/3) multiplet is about 1/3 lighter than the other components in the representation. Since this multiplet has l A = −33/2, it gives a large threshold correction and renders the colored Higgs too light.

Numerical result
In the previous section, we have used 1-loop relations to describe qualitative behaviors. In this section, we will show a numerical result using 2-loop RGE evolutions [18,19]. In the result, we use the central value of the 5-flavor strong coupling, α (5) s (M Z ) MS = 0.1181 ± 0.0011 [20]. The colored Higgs mass is sensitive to the value of the strong coupling, while the GUT gauge boson masses are less sensitive. The proton lifetime is about 50% larger if we use the value +3σ. We assume all the SUSY particle masses to be 2 TeV except for the wino mass, which is taken to be 500 GeV.
The decay width of p → π 0 e + is [21] the partial proton lifetime is found to be As discussed in the previous section, ρ ≪ 1 in the current SO(10)-breaking vacua because the VEV of a 24 is roughly the geometrical average of m A and M * while M 3 , M 8 are roughly m A , and we get ρ ≃ 0.1 for λ 2 = 1 and M * = 2 × 10 18 GeV.
It is interesting to compare the above estimate with the prediction of the minimal SU(5) model. In the minimal SU(5), we define M 8 = M 3 = λM X where λ is proportional to the self-coupling of the adjoint field that breaks SU (5). Then, the partial proton lifetime is found to be τ SU(5) p ≃ λ −4/3 × 5.5 × 10 35 years. (39) We observe that the partial lifetime decreases by 1/20 in our SO(10)-breaking vacua compared to the minimal SU(5) model, for natural values of ρ = 0.1 and λ = 1.
The estimate for our SO(10)-breaking vacua, Eq. (38), receives corrections from the small VEVs of a 1 , v R that perturb the mass ratios. In Table 1, we show precise numerical values.
Here, we fix M * = 2 × 10 18 GeV, and take benchmark values for λ 1 , λ 2 , κ and m χ . We solve the  The p → π 0 e + partial lifetime and the mass spectrum for various input values of λ 1 , λ 2 , κ and m χ . The masses and VEV are in units of 10 16 GeV. From (i-A) to (ii-B), m A < 0 and thus a 24 is real and the mass spectrum splits into two, for which (i-A) and (ii-A) correspond to the cases with a 24 > 0, and (i-B) and (ii-B) to the cases with a 24 < 0. In (iii), m A > 0 and thus a 24 is complex. We change one of  Table 1, we find that the mass spectrum is not sensitive to λ 1 . This is because the relation |a 1 |, |v R | ≪ |a 24 | gives M 3 /a 24 ∝ λ 2 a 24 /M * . Although a 24 depends on λ 1 , the ratio M 3 /M X does not depend on λ 1 for |a 1 |, |v R | ≪ |a 24 |. As a result, once M X is chosen to realize the gauge coupling unification, the mass spectrum is almost independent of λ 1 . On the other hand, when λ 2 is smaller, the SU(3) c adjoint and SU(2) L adjoint particles become lighter (ρ = M 3 /M X is smaller), and the proton lifetime becomes longer, as seen from (iii) and (iii ′ ) of Table 1. Consequently, the proton lifetime cannot be bounded from above theoretically. Still, it is interesting that for λ 2 ∼ 1, the dimension-6 proton decay is detectable at HK.
In the benchmarks of Table 1, the effective colored Higgs mass, , is 2 × 10 17 GeV. The relation M H T > M X is realized with a large coupling of AH 1 H 2 (see Appendix A). Since the dimension-5 proton decay amplitudes also depend on details of the Yukawa coupling unification, we do not discuss the dimension-5 decay in this paper.

Conclusion
We have studied the dimension-6 proton decay in a SUSY SO (10)  The mass matrix of each component of the 10 + 10's can be written as where where (C Q , C U , C E ) = (1, −4, 6) and (D Q , D U , D E ) = (19,4,9). We can verify that one eigenvalue is zero when the F -flat conditions are used. The mass of the physical mode is M AA + M χχ . In the limit with m A , m χ ≪ M * , M χχ dominates, but M AA can be nonnegligible for χ Q due to the large factor D Q /C Q . Using the minimization condition, we obtain The masses of isospin doublet and color triplet Higgses are obtained from the superpotential and the mass term is where and (c D , c T ) = (3, −2). The doublet-triplet splitting needs fine-tuning. Without loss of generality, λ 1 χ is set to zero by a rotation of (H 1 , H 2 ). In this basis, by the fine-tuning M 11 = M 12 + λ H A D = 0, we have one pair of doublets massless. H 1 in this basis should dominantly give the large top quark Yukawa coupling. The mass of the corresponding triplet is roughly ∼ 5/3λ H A D for |a 1 |, |v R | ≪ |a 24 |.

B Renormalizable model obtained by employing 54 H
In the main text, we have considered the model with 45 H +16 H +16 H and with non-renormalizable quartic terms of 45 H . In this appendix, for readers who prefer renormalizable models, we show that a renormalizable superpotential with 54 H (whose SM singlet component is denoted by E) can also provide the wanted vacua where |a 24 | ≫ |a 1 |, |v R | (and |a 24 | ≫ |E|).
The superpotential for the SM singlets is From the F -flat conditions, we obtain where a is a solution of the following equation: Vacua with |E|, |a 1 |, |v R | ≪ |a 24 | are obtained by assuming m χ , m A ≪ m E , which gives The 54 H is decomposed as 54 H = 24 + 15 + 15 under SU (5). The mass matrices of the adjoint representations after SU(5) breaking are for (8, 1, 0) and m A + 3κ 2 E 3κ 2 a 24 + √ 6κ 2 a 1 3κ 2 a 24 + √ 6κ 2 a 1 m E + 6κ 1 E