Moderately suppressed dimension-five proton decay in a flipped SU(5) model

We study colored Higgsino-mediated proton decay (dimension-five proton decay) in a model based on the flipped SU(5) GUT. In the model, the GUT-breaking 10, 10¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \overline{\mathbf{10}} $$\end{document} fields have a GUT-scale mass term and gain VEVs through higher-dimensional operators, which induces an effective mass term between the color triplets in the 5, 5¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \overline{\mathbf{5}} $$\end{document} Higgs fields that is not much smaller than the GUT scale. This model structure gives rise to observable dimension-five proton decay, and at the same time achieves moderate suppression on dimension-five proton decay that softens the tension with the current bound on Γ(p → K+ν¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \overline{\nu} $$\end{document}). We investigate the flavor dependence of the Wilson coefficients of the operators relevant to dimension-five proton decay, by relating them with diagonalized Yukawa couplings and CKM matrix components in MSSM, utilizing the fact that the GUT Yukawa couplings are in one-to-one correspondence with the MSSM Yukawa couplings in flipped models. Then we numerically evaluate the Wilson coefficients, and predict the distributions of the ratios of the partial widths of various proton decay modes.


Introduction
Proton decay mediated by colored Higgsinos in supersymmetric grand unified theories (SUSY GUTs) [1,2], called dimension-five proton decay, is a primary target in the proton decay searches at HyperKamiokande [3], JUNO [4], and DUNE [5,6]. This is because the GUT gauge boson mass is predicted to be ∼ 2 · 10 16 GeV in usual SUSY GUTs and the corresponding partial width of GUT gauge boson-mediated proton decay is out of the sensitivity ranges of the above experiments (however, proton decay mediated by (3, 2, 1/6) gauge boson in SUSY SO(10) GUT can be accessible [7]). In non-SUSY GUTs, GUT gauge boson-mediated proton decay can be within the experimental reach, and one can even set upper bounds on the proton lifetime in some cases [8]. Nevertheless, dependence on the choice of split SU(5) multiplets that assist the gauge coupling unification, is inevitable. In contrast, if Nature favors as light SUSY particles as possible (e.g., the SUSY particle mass spectrum is such that the p → K +ν partial width is narrowly above the current experimental bound [9]) for the naturalness of the electroweak scale, there is a great chance that dimension-five proton decay is observed in forthcoming experiments. (For study on proton decay and the LHC bounds on SUSY particle masses and the Higgs particle mass, see refs. [10,11].) In this situation, it is important to study the flavor dependence of dimensionfive proton decay in various SUSY GUT models, and compare the partial widths of different decay modes, as in ref. [12], to bridge theory and proton decay search experiments.
In this paper, we focus on dimension-five proton decay in a model based on the SUSY flipped SU(5) GUT [13,14]. 1 Although the flipped SU(5) GUT by itself cannot address the origin of the U(1) hypercharge quantization, it has attractive features such as the realization of the doublet-triplet splitting, and the suppression of dimension-five proton decay that allows one to lower the sfermion mass without conflicting the current bound JHEP01(2022)061 on the p → K +ν decay. In pervasive models of the SUSY flipped SU(5) GUT, dimensionfive proton decay is in effect totally suppressed, because the mass term between the color triplets in the 5, 5 Higgs fields and that between the color triplets in the GUT-breaking 10, 10 fields, are at the soft SUSY breaking scale, not at the GUT scale. However, if the GUT-breaking 10, 10 fields are allowed to possess a GUT-scale mass term and gain vacuum expectation values (VEVs) through higher-dimensional operators, then the color triplets in the 5,5 fields gain an effective mass term not much smaller than the GUT scale, which gives rise to observable dimension-five proton decays. Meanwhile, the operators obtained after integrating out the colored Higgsinos can be O(100) times suppressed compared to those in non-flipped models, which mitigates the tension with the current experimental bound on the p → K +ν mode. We materialize the above possibility in our model of the SUSY flipped SU(5) GUT, and investigate the flavor dependence of dimension-five proton decay in the model. Interestingly, since the GUT Yukawa couplings for the 5,5 Higgs fields are in one-to-one correspondence with the Yukawa couplings of the minimal SUSY Standard Model (MSSM), we have a strong predictive power on the flavor structure of the Wilson coefficients of the operators relevant to dimension-five proton decay. We take advantage of the above feature and express the Wilson coefficients in terms of diagonalized Yukawa couplings and Cabibbo-Kobayashi-Maskawa (CKM) matrix components in MSSM, plus one unknown unitary matrix and several unknown phases. Then we vary the unknown unitary matrix and phases and predict the distributions of the partial width ratios of different proton decay modes.
Previously, colored Higgsino-mediated proton decay in the SUSY flipped SU(5) GUT has been studied in ref. [16]. However, since the mass term between the color triplets of the 5,5 Higgs fields and that between the color triplets of the GUT-breaking 10,10 fields are highly suppressed, only chirality non-flipping colored Higgsino exchange is considered, unlike the present paper where we focus on chirality flipping colored Higgsino exchange. For reference, GUT gauge boson-mediated proton decay in SUSY flipped SU(5) GUT models has been investigated in refs. [17]- [24]. This paper is organized as follows. In section 2, we describe our model of the SUSY flipped SU(5) GUT. In section 3, we present the expressions for the Wilson coefficients of dimension-five operators obtained after integrating out colored Higgs fields and dimensionsix operators obtained after integrating out the SUSY particles which contribute to proton decay. In section 4, we investigate the flavor dependence of the Wilson coefficients by relating them with diagonalized Yukawa couplings and CKM matrix in MSSM. In section 5, we numerically evaluate the Wilson coefficients using the values of diagonalized Yukawa couplings and CKM matrix based on experimental data, and randomly varying the remaining unknown parameters. The results are presented as a prediction for the distributions of proton decay partial width ratios. Section 6 summarizes the paper.

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where HHHH stands for the coupling where SU(5) indices are summed cyclically and (H10 i 1 )(H10 j 1 ) the couplings where SU(5) indices are summed separately in each bracket. The HHHH operator, obtained by integrating out Σ, triggers GUT breaking, while the (H10 i 1 )(H10 j 1 ) operators, obtained by integrating out S i 's, generate the Majorana mass of the singlet neutrinos. The isospin-doublet components of h, h gain mass only from the µ h hh term, while the color-triplet components additionally gain GUT-scale mass from the λ HHh + λ H H h terms after GUT-breaking, which achieves the doublet-triplet splitting. Note that the effective superpotential eq. (2.2) does not contain higher-dimensional operators giving rise to proton decay 10 i 1 10 j 1 10 k 15 −3 ; proton decay occurs only through the exchange of colored Higgsinos, colored Higgs bosons and GUT gauge bosons (the contributions of the latter two can be neglected in the present model).
Let in the basis where the lepton-doublet components of5 i −3 diagonalize the charged lepton mass matrix. Here U M N S denotes Maki-Nakagawa-Sakata matrix [30]. In this case, we get where m 1 , m 2 , m 3 are the active neutrino masses. For | N c H | 3 · 10 16 GeV, v u 246 GeV, and for the normal hierarchy with m 2 m 1 , the following numerical values reproduce the measured neutrino mass differences: We expect that in general cases suppression of some components of Y S at O(10 −3 ) suffices to reproduce the measured neutrino mass differences.

Wilson coefficients contributing to proton decay
We focus on proton decay mediated by colored Higgsinos. The mass matrix for the colored Higgs fields reads The colored Higgs fields couple to the matter fields as Integrating out the colored Higgs fields, we obtain the following dimension-five operators responsible for proton decay: where in the first term isospin indices are summed in each bracket, and the Wilson coefficients satisfy where µ denotes the renormalization scale and µ H C is about the colored Higgs mass eigenvalues. The effective inverse of the 11 , is obtained from eq. (3.1) as where we have used the fact that µ h is negligible compared to the GUT-scale, and used eq. (2.3) in the second equality. When |κ 2 /(2λλ )| = 1 and M Σ = M P , the Wilson coefficients C ijkl 5L , C ijkl 5R are about 100 times suppressed compared to non-flipped models, where (M −1 H C ) 11 is given by the inverse of the colored Higgs mass ∼ 2·10 16 GeV. The resulting 10 4 suppression on proton decay partial widths allows the model to evade the current stringent experimental bound on the p → K +ν decay without enormously raising sfermion masses. Still, the suppression is not strong, and leaves the possibility of observing proton decay in near-future experiments.
Integrating out the SUSY particles, we obtain the following operators responsible for proton decay: where ψ denotes a SM Weyl spinor and spinor index is summed in each bracket. Those Wilson coefficients which contribute to proton decay are C dα ud with α = e, µ, τ and β = e, µ. They satisfy, at the soft SUSY breaking scale µ = µ SUSY , 2 (3.14) Here F ,F are loop functions given by , we mean that Qi is in the flavor basis where the down-type quark Yukawa coupling is diagonal and that the down-type quark component of Qi is exactly s quark (the up-type quark component of Qi is a mixture of u, c, t). Likewise, Q k is in the flavor basis where the up-type quark Yukawa coupling is diagonal and its up-type component is exactly u quark, and Q l is in the flavor basis where the down-type quark Yukawa coupling is diagonal and its down-type quark component is exactly d quark. The same rule applies to C uα ds 5L and others.

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isospin-doublet squarks (assumed degenerate), mass of isospin-doublet slepton of flavor α, mass of isospin-singlet top squark, and mass of isospin-singlet tau slepton (mixings between isospin-doublet and singlet sfermions are neglected). y t , y τ , V ckm denote the top quark and tau lepton Yukawa couplings and CKM matrix, respectively. Finally (4.8) Here C 0 LLα (α = e, µ, τ ), C 0 RL are defined as where y d , y s , y b denote the diagonalized Yukawa couplings for the down, strange and bottom quarks, respectively, and φ 1 , φ 2 , φ 3 are unknown phases. Combining eq. (4.11) with the definition of the CKM matrix, we get where V ckm ij denote components of the CKM matrix. From the definition of Y u and the CKM matrix, we get, for α = e, µ, τ , where y u , y c , y t denote the diagonalized Yukawa couplings for the up, charm and top quarks, respectively, and U ij is a component of an unknown unitary matrix U that transforms the flavor basis of5 i −3 's as (4.14) Combining the definition of Y e with eq. (4.14), we get Eq. (4.11) gives that the flavor basis of 10 i 1 is transformed as Combining eq. (4.16) with the definition of CKM matrix, we get

Numerical analysis
We numerically evaluate the right-hand sides of eqs. (4.18)-(4.25) by randomly varying the unknown unitary matrix U and unknown phases φ 1 , φ 2 , φ 3 . The result is presented in the form of the ratios of the proton decay partial widths below, , µ), (5.5) which are suitable for the presentation because they are observable quantities.

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The partial width of each mode is given by , and α H , β H denote hadronic matrix elements, and D, F are parameters of the baryon chiral Lagrangian. The mass splittings among nucleons and hyperons are neglected.
In the evaluation of the proton decay partial widths, the baryon chiral Lagrangian parameters are given by D = 0.804, F = 0.463, and the hadronic matrix elements are taken from ref. [31] as α H (µ had ) = −β H (µ had ) = −0.0144 GeV 3 for µ had = 2 GeV.
In the calculation of C 0 LLα , C 0 RL , defined in eqs. (4.9), (4.10), we assume two benchmark SUSY particle mass spectra. In one spectrum, the pole masses and tan β satisfy and in the other spectrum, they satisfy where all the sfermions are mass-degenerate. The relative phase between M W and µ h , which determines the relative phase between C 0 LLα and C 0 RL , is varied randomly. The The values of the Wolfenstein parameters are taken from the CKM fitter result [42]. For the QCD and QED gauge couplings, we use α (5) s (M Z ) = 0.1181 and α (5) (M Z ) = 1/127.95. For the lepton and W, Z, Higgs pole masses, we use the values in Particle Data Group [44].
We comment on the impact of the choice of the benchmark spectrum in eq. (5.13). If the spectrum deviates from eq. (5.13) and the masses of isospin-doublet sleptons˜ α are split, this gives rise to a splitting in C 0 LLα 's, which affects proton decay partial width ratios. However, since the right-hand sides of eqs. (4.18)-(4.25) have a large hierarchy, the possible splitting in C 0 LLα 's has only a minor impact on proton decay partial width ratios. Likewise, a splitting in the 1st and 2nd generation isospin-doublet squark masses does not change the result significantly.
The unknown unitary matrix U is varied with the Haar measure given by [45,46] U = e i η e i ω 1 λ 3 +i ω 2 λ 8 We present the result of the numerical analysis. Since Γ(p → K +ν ) is the largest partial width in the entire parameter space, the phenomenologically most meaningful quantities are the ratios of Γ(p → K +ν ) and the other partial widths. Therefore, we show the distributions of (5.19) corresponding to randomly varied values of the unknown unitary matrix U , unknown phases φ 1 , φ 2 , φ 3 , and relative phase between M W and µ h . Figures 1 and 2 are the distributions for tan β = 5, 50, respectively. From figures 1, 2, we find that for low tan β such as tan β = 5, the ratio Γ(p→π +ν ) can be in the range 0.1-0.2 with O(0.1) probability. Therefore, future sensitivity study on the p → π +ν mode should be performed seriously, to examine the possibility of observing both decay modes. The other partial width ratios are mostly below 0.05 for both low and high tan β. However, for low tan β such as tan β = 5, there is a non-negligible probability that Γ(p→π 0 β + ) Γ(p→K +ν ) and Γ(p→ηβ + ) Γ(p→K +ν ) are in the range 0.05-0.1. In this case, the current bound on Γ(p → K +ν ) [9] and the future sensitivity reach for Γ(p → π 0 β + ) [3] imply that p → π 0 β + ,

Summary
We have studied dimension-five proton decay in a model based on the flipped SU (5) GUT.
In the model, the GUT-breaking 10, 10 fields have a GUT-scale mass term and gain VEVs through operators suppressed by the Planck scale. This structure induces an effective mass term not much smaller than the GUT scale between the color triplets in the 5,5 Higgs fields. This mass term gives rise to observable dimension-five proton decay, and at the same time achieves moderate suppression on dimension-five proton decay amplitudes, which is estimated to be 0.01 if the coefficients in the superpotential eq. (2.2) satisfies |γ 1 /(λλ )| = 1.
We have investigated the flavor structure of the Wilson coefficients of the operators contributing to dimension-five proton decay, and expressed them in terms of diagonalized Yukawa couplings and CKM matrix components in MSSM plus an unknown unitary matrix U and unknown phases. We have numerically evaluated the Wilson coefficients by randomly varying U and the unknown phases, and presented the result in the form of the distributions of the partial width ratios of various proton decay modes for a benchmark SUSY particle spectrum. We have found that the ratio Γ(p→π +ν ) Γ(p→K +ν ) can be in the range 0.1-0.2 with O(0.1) probability for low tan β such as tan β = 5. Also, for such low tan β, it is possible that Γ(p→π 0 β + ) Γ(p→K +ν ) and Γ(p→ηβ + ) Γ(p→K +ν ) are in the range 0.05-0.1, and there is a non-zero probability that Γ(p→K 0 β + ) Γ(p→K +ν ) = O(0.1).